691 Sentences With "differentiable" | Random Sentence Generator

DeepMind tested its differentiable neural computer on the London Underground and was successful at generating routes from the structured data.

While identity politics tends to trade in pathological assumptions of behavior, again, Black voters or women voters are far from a non-differentiable monolith.

In a new paper published in Nature, the Google subsidiary DeepMind explained a new approach to machine learning that uses something called a differentiable neural computer.

This stunt is also barely differentiable from Eminem's 2009 Relapse promotion, which involved sending out candy pills and launching a website for a fake rehab center called Popsomp Hills (a gag that felt a little corny, even at the time).

It's hardly differentiable from the bright-white home in Canary's smart security camera promotional images, which is displayed in a foyer decorated with little other than an animal bone and a copy of a hardcover catalog of specialty paint finishes.

It is loaded with readily differentiable components piled up the way you might stack a bunch of small tables, pedestals, and step-stools — if you had to get them out of the way quickly or clumsily to barricade a door, or to create a veritable Arcadian playground for a cat.

Four-dimensional manifolds are the most unusual: they are not geometrizable (as in lower dimensions), and surgery works topologically, but not differentiably. Since topologically, 4-manifolds are classified by surgery, the differentiable classification question is phrased in terms of "differentiable structures": "which (topological) 4-manifolds admit a differentiable structure, and on those that do, how many differentiable structures are there?" Four-manifolds often admit many unusual differentiable structures, most strikingly the uncountably infinitely many exotic differentiable structures on R4. Similarly, differentiable 4-manifolds is the only remaining open case of the generalized Poincaré conjecture.

At zero, the function is continuous but not differentiable. If is differentiable at a point , then must also be continuous at . In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable.

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

A classic example of a pathological structure is the Weierstrass function, which is continuous everywhere but differentiable nowhere. The sum of a differentiable function and the Weierstrass function is again continuous but nowhere differentiable; so there are at least as many such functions as differentiable functions. In fact, by the Baire category theorem, one can show that continuous functions are generically nowhere differentiable. In layman's terms, the majority of functions are nowhere differentiable, and relatively few can ever be described or studied.

If M is a differentiable manifold, a real or complex-valued function f on M is said to be differentiable at a point p if it is differentiable with respect to some (or any) coordinate chart defined around p. More generally, if M and N are differentiable manifolds, a function f: M → N is said to be differentiable at a point p if it is differentiable with respect to some (or any) coordinate charts defined around p and f(p).

This formula is true whenever is differentiable and its inverse ' is also differentiable. This formula can fail when one of these conditions is not true. For example, consider . Its inverse is , which is not differentiable at zero.

Manifold. A topological manifold is a locally Euclidean Hausdorff space. (In Wikipedia, a manifold need not be paracompact or second-countable.) A Ck manifold is a differentiable manifold whose chart overlap functions are k times continuously differentiable. A C∞ or smooth manifold is a differentiable manifold whose chart overlap functions are infinitely continuously differentiable.

If a complex function is differentiable just once on an open set, it is both infinitely differentiable and analytic on that set.

Differentiable functions can be locally approximated by linear functions. The function f:\R\to\R with f(x)=x^2\sin\left(\tfrac 1x\right) for x eq 0 and f(0)=0 is differentiable. However, this function is not continuously differentiable. A function f is said to be continuously differentiable if the derivative '(x) exists and is itself a continuous function.

The maps that relate the coordinates defined by the various charts to one another are called transition maps. Differentiability means different things in different contexts including: continuously differentiable, k times differentiable, smooth, and holomorphic. Furthermore, the ability to induce such a differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields.

Differentiable programming has been applied in areas such as combining deep learning with physics engines in robotics, differentiable ray tracing, image processing, and probabilistic programming.

In the following we assume all manifolds are differentiable manifolds of class Cr for a fixed r ≥ 1, and all morphisms are differentiable of class Cr.

Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology.

Differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics, general relativity, and Yang–Mills theory. It is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the exterior calculus.

In differential topology, a critical value of a differentiable function between differentiable manifolds is the image (value of) ƒ(x) in N of a critical point x in M.

A function differentiable at a point is continuous at that point. Differentiation is a linear operation in the following sense: if f and g are two maps V → W which are differentiable at x, and r and s are scalars (two real or complex numbers), then rf + sg is differentiable at x with D(rf + sg)(x) = rDf(x) + sDg(x). The chain rule is also valid in this context: if f : U → Y is differentiable at x in U, and g : Y → W is differentiable at y = f(x), then the composition g o f is differentiable in x and the derivative is the composition of the derivatives: :D(g \circ f)(x) = Dg(f(x))\circ Df(x).

Continuity of u need not be assumed, but it follows instead from the definition of the quasi- derivative. If f is Fréchet differentiable at x0, then by the chain rule, f is also quasi-differentiable and its quasi-derivative is equal to its Fréchet derivative at x0. The converse is true provided E is finite-dimensional. Finally, if f is quasi-differentiable, then it is Gateaux differentiable and its Gateaux derivative is equal to its quasi-derivative.

However, the early controllers of such memories were not differentiable.

The same argument works for a piecewise differentiable Jordan curve.

A differentiable function In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. The differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. More generally, for x0 as an interior point in the domain of a function f, then f is said to be differentiable at x0 if and only if the derivative f ′(x0) exists.

If and are real-valued functions differentiable at a point , then the product rule asserts that the product is differentiable at , and : abla (fg)(a) = f(a) abla g(a) + g(a) abla f(a).

For instance, even though all manifolds look locally the same (as Rn for some n) in the topological sense, it is natural to ask whether their differentiable structures behave in the same manner locally. For example, one can impose two different differentiable structures on R that make R into a differentiable manifold, but both structures are not locally diffeomorphic (see below). Although local diffeomorphisms preserve differentiable structure locally, one must be able to "patch up" these (local) diffeomorphisms to ensure that the domain is the entire (smooth) manifold. For example, there can be no local diffeomorphism from the 2-sphere to Euclidean 2-space although they do indeed have the same local differentiable structure.

The Klein bottle, immersed in 3-space. :For a closed immersion in algebraic geometry, see closed immersion. In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective.This definition is given by , , , , , , , .

In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion.

The study of calculus on differentiable manifolds is known as differential geometry.

If a real-valued, differentiable function f, defined on an interval I of the real line, has zero derivative everywhere, then it is constant, as an application of the mean value theorem shows. The assumption of differentiability can be weakened to continuity and one-sided differentiability of f. The version for right differentiable functions is given below, the version for left differentiable functions is analogous.

The existence of smooth but non-analytic functions represents one of the main differences between differential geometry and analytic geometry. In terms of sheaf theory, this difference can be stated as follows: the sheaf of differentiable functions on a differentiable manifold is fine, in contrast with the analytic case. The functions below are generally used to build up partitions of unity on differentiable manifolds.

Holomorphic functions are analytic and vice versa. This means that, in complex analysis, a function that is complex- differentiable in a whole domain (holomorphic) is the same as an analytic function. This is not true for real differentiable functions.

A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the center and right charts, the Tropic of Cancer is a smooth curve, whereas in the left chart it has a sharp corner. The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable.

Since continuity of partial derivatives implies differentiability of the function, F is indeed differentiable.

Here we typically make the assumption that the function f is continuous and differentiable.

The traditional definition of velocity makes no sense in the non-differentiable fractal spacetime.

The theorem (and its proof below) is more general than the intuition in that it doesn't require the function to be differentiable over a neighbourhood around \displaystyle x_0. It is sufficient for the function to be differentiable only in the extreme point.

The points and do not have disjoint neighborhoods in X. Any covering space of a differentiable manifold may be equipped with a (natural) differentiable structure that turns p (the covering map in question) into a local diffeomorphism – a map with constant rank n.

Harmonic functions are infinitely differentiable in open sets. In fact, harmonic functions are real analytic.

Banach manifolds and Fréchet manifolds, in particular manifolds of mappings are infinite dimensional differentiable manifolds.

In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If is an open subset of and is Lipschitz continuous, then is differentiable almost everywhere in ; that is, the points in at which is not differentiable form a set of Lebesgue measure zero.

The gradient is linear in the sense that if and are two real-valued functions differentiable at the point , and and are two constants, then is differentiable at , and moreover : abla\left(\alpha f+\beta g\right)(a) = \alpha abla f(a) + \beta abla g (a).

Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds. More specifically, differential topology considers the properties and structures that require only a smooth structure on a manifold to be defined. Smooth manifolds are "softer" than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology.

An example of this is given by the derivative g of the (differentiable but not absolutely continuous) function f(x)=x²·sin(1/x²) (the function g is not Lebesgue-integrable around 0). The Denjoy integral corrects this lack by ensuring that the derivative of any function f that is everywhere differentiable (or even differentiable everywhere except for at most countably many points) is integrable, and its integral reconstructs f up to a constant; the Khinchin integral is even more general in that it can integrate the approximate derivative of an approximately differentiable function (see below for definitions). To do this, one first finds a condition that is weaker than absolute continuity but is satisfied by any approximately differentiable function. This is the concept of generalized absolute continuity; generalized absolutely continuous functions will be exactly those functions which are indefinite Khinchin integrals.

If X is a C^k manifold (i.e., a manifold whose charts are k times continuously differentiable), then a C^k curve in X is such a curve which is only assumed to be C^k (i.e. k times continuously differentiable). If X is an analytic manifold (i.e.

If (E, M) is a vector bundle over a differentiable manifold M then (TE, E, TM, M) is a double vector bundle when considering its secondary vector bundle structure. If M is a differentiable manifold, then its double tangent bundle (TTM, TM, TM , M) is a double vector bundle.

Classical fields as above, such as the electromagnetic field, are usually infinitely differentiable functions, but they are in any case almost always twice differentiable. In contrast, generalized functions are not continuous. When dealing carefully with classical fields at finite temperature, the mathematical methods of continuous random fields are used, because thermally fluctuating classical fields are nowhere differentiable. Random fields are indexed sets of random variables; a continuous random field is a random field that has a set of functions as its index set.

Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance, the function given by is not differentiable at . In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative. Most functions that occur in practice have derivatives at all points or at almost every point. Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points.

Furthermore, if F is (complex) Gateaux differentiable at each u \in U with derivative :DF(u) : \psi\mapsto dF(u;\psi) then F is Fréchet differentiable on U with Fréchet derivative DF . This is analogous to the result from basic complex analysis that a function is analytic if it is complex differentiable in an open set, and is a fundamental result in the study of infinite dimensional holomorphy. ;Continuous differentiability Continuous Gateaux differentiability may be defined in two inequivalent ways. Suppose that F : U \to Y is Gateaux differentiable at each point of the open set U. One notion of continuous differentiability in U requires that the mapping on the product space :dF:U\times X \rightarrow Y \, be continuous.

Differentiable programming is a programming paradigm in which a numeric computer program can be differentiated throughout via automatic differentiation. This allows for gradient based optimization of parameters in the program, often via gradient descent. Differentiable programming has found use in a wide variety of areas, particularly scientific computing and artificial intelligence.

Probabilistic programming in Julia has also been combined with differentiable programming by combining the Julia package Zygote.jl with Turing.jl.

Suppose that the function has a zero at , i.e., , and is differentiable in a neighborhood of . If is continuously differentiable and its derivative is nonzero at , then there exists a neighborhood of such that for all starting values in that neighborhood, the sequence will converge to .. If the function is continuously differentiable and its derivative is not 0 at and it has a second derivative at then the convergence is quadratic or faster. If the second derivative is not 0 at then the convergence is merely quadratic.

Dirichlet has shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points). Riemann gave an example of a Fourier series representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet. He also proved the Riemann–Lebesgue lemma: if a function is representable by a Fourier series, then the Fourier coefficients go to zero for large n. Riemann's essay was also the starting point for Georg Cantor's work with Fourier series, which was the impetus for set theory.

Let us write for . In 1995, Luis Rodríguez-Piazza proved that the isometry can be chosen so that every non-zero function in the image is nowhere differentiable. Put another way, if consists of functions that are differentiable at at least one point of , then can be chosen so that This conclusion applies to the space itself, hence there exists a linear map that is an isometry onto its image, such that image under of (the subspace consisting of functions that are everywhere differentiable with continuous derivative) intersects only at : thus the space of smooth functions (with respect to the uniform distance) is isometrically isomorphic to a space of nowhere-differentiable functions. Note that the (metrically incomplete) space of smooth functions is dense in .

For all Hermitian × matrices and and all differentiable convex functions : ℝ → ℝ with derivative , or for all positive-definite Hermitian × matrices and , and all differentiable convex functions :(0,∞) → ℝ, the following inequality holds, In either case, if is strictly convex, equality holds if and only if = . A popular choice in applications is , see below.

Miller's website explains how CGP works. He edited a book entitled Cartesian Genetic Programming, published in 2011 by Springer. The open source project dCGP implements a differentiable version of CGP developed at the European Space Agency by Dario Izzo, Francesco Biscani and Alessio Mereta Izzo, D. and Biscani, F. and Mereta, A.: Differentiable Genetic Programming.

It also has a standard differentiable structure on it, making it a differentiable manifold. (Up to diffeomorphism, there is only one differentiable structure that the topological space supports.) The real line is a locally compact space and a paracompact space, as well as second-countable and normal. It is also path-connected, and is therefore connected as well, though it can be disconnected by removing any one point. The real line is also contractible, and as such all of its homotopy groups and reduced homology groups are zero.

Furthermore, ' is differentiable at by assumption, so is continuous at , by definition of the derivative. The function ' is continuous at ' because it is differentiable at ', and therefore is continuous at '. So its limit as ' goes to ' exists and equals , which is . This shows that the limits of both factors exist and that they equal and , respectively.

Like the heat flow, Ricci flow tends towards uniform behavior. Unlike the heat flow, the Ricci flow could run into singularities and stop functioning. A singularity in a manifold is a place where it is not differentiable: like a corner or a cusp or a pinching. The Ricci flow was only defined for smooth differentiable manifolds.

By definition, differentiable manifolds of a fixed dimension are all locally diffeomorphic to Euclidean space, so aside from dimension, there are no local invariants. Thus, differentiable structures on a manifold are topological in nature. By contrast, the curvature of a Riemannian manifold is a local (indeed, infinitesimal) invariant (and is the only local invariant under isometry).

In differential topology, a branch of mathematics, a stratifold is a generalization of a differentiable manifold where certain kinds of singularities are allowed. More specifically a stratifold is stratified into differentiable manifolds of (possibly) different dimensions. Stratifolds can be used to construct new homology theories. For example, they provide a new geometric model for ordinary homology.

In this article, several kinds of surfaces are considered and compared. An unambiguous terminology is thus necessary to distinguish them. Therefore, we call topological surfaces the surfaces that are manifolds of dimension two (the surfaces considered in Surface (topology)). We call differentiable surfaces the surfaces that are differentiable manifolds (the surfaces considered in Surface (differential geometry)).

The class C∞ of infinitely differentiable functions, is the intersection of the classes Ck as k varies over the non-negative integers.

The limit of this tower yields a topological but not differentiable map, hence surgery works topologically but not differentiably in dimension 4.

When p(x; θ) is non-differentiable, the Fisher information is not defined, and hence the Cramér–Rao bound does not exist.

Intuitively, a differentiable function is approximated by its derivative – a differentiable function behaves infinitesimally like a linear function a+bx, or more precisely, f(x_0) + f'(x_0)(x-x_0). Thus, from the perspective that "if f is differentiable and has non-vanishing derivative at x_0, then it does not attain an extremum at x_0," the intuition is that if the derivative at x_0 is positive, the function is increasing near x_0, while if the derivative is negative, the function is decreasing near x_0. In both cases, it cannot attain a maximum or minimum, because its value is changing. It can only attain a maximum or minimum if it "stops" – if the derivative vanishes (or if it is not differentiable, or if one runs into the boundary and cannot continue).

The class C1 consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a C1 function is exactly a function whose derivative exists and is of class C0. In general, the classes Ck can be defined recursively by declaring C0 to be the set of all continuous functions, and declaring Ck for any positive integer k to be the set of all differentiable functions whose derivative is in Ck−1. In particular, Ck is contained in Ck−1 for every k > 0, and there are examples to show that this containment is strict (Ck ⊊ Ck−1).

The collection of K-derivations of A into an A-module M is denoted by . Derivations occur in many different contexts in diverse areas of mathematics. The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on Rn. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold. It follows that the adjoint representation of a Lie algebra is a derivation on that algebra.

For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly. Most functions that occur in practice have derivatives at all points or at almost every point. However, a result of Stefan Banach states that the set of functions that have a derivative at some point is a meager set in the space of all continuous functions.. Cited by Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function.

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.

In particular, if f is infinitely-often differentiable, then so is u. Any differential operator exhibiting this property is called a hypoelliptic operator; thus, every elliptic operator is hypoelliptic. The property also means that every fundamental solution of an elliptic operator is infinitely differentiable in any neighborhood not containing 0. As an application, suppose a function f satisfies the Cauchy–Riemann equations.

The term compact basically means that it is finite in extent ("bounded") and complete. The term "without boundary" means that the space has no edges. Moreover, so that calculus can be applied, the universe is typically assumed to be a differentiable manifold. A mathematical object that possesses all these properties, compact without boundary and differentiable, is termed a closed manifold.

When the limit exists, is said to be differentiable at . Here is one of several common notations for the derivative (see below). From this definition it is obvious that a differentiable function is increasing if and only if its derivative is positive, and is decreasing iff its derivative is negative. This fact is used extensively when analyzing function behavior, e.g.

In mathematics, the principal orbit type theorem states that compact Lie group acting smoothly on a connected differentiable manifold has a principal orbit type.

All definitions above can be made in the topological category instead of the category of differentiable manifolds, and this does not change the objects.

In geometric measure theory an approximate tangent space is a measure theoretic generalization of the concept of a tangent space for a differentiable manifold.

Throughout this section, M is assumed to be a differentiable manifold, and ∇ a covariant derivative on the tangent bundle of M unless otherwise noted.

This occurs in two main cases. When is a complex manifold . In this case, one considers the algebra of holomorphic functions, i.e., complex differentiable functions.

Another important example is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations, which is an important idea in general relativity.

Then F is uniformly continuous on [a, b] and differentiable on the open interval and :F'(x) = f(x)\, for all x in (a, b).

Under convexity, these conditions are also sufficient. If some of the functions are non-differentiable, subdifferential versions of Karush–Kuhn–Tucker (KKT) conditions are available.

Köthe showed in 1983 that a normed space is smooth at a point if and only if the norm is Gateaux differentiable at that point.

Similarly, if is a real differentiable function over , then defines a map from to . If both maps happen to be inverses of each other, we say we have a Legendre transform. The notion of the tautological one-form is commonly used in this setting. When the function is not differentiable, the Legendre transform can still be extended, and is known as the Legendre-Fenchel transformation.

In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth. These are the defining symmetry transformations of General Relativity since the theory is formulated only in terms of a differentiable manifold. In general relativity, general covariance is intimately related to "diffeomorphism invariance".

In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their composition on chart intersections in the atlas must be differentiable functions on the corresponding linear space. In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas.

Differentiable neural computers (DNCs) are an extension of Neural Turing machines, allowing for usage of fuzzy amounts of each memory address and a record of chronology.

Then is complex-differentiable at that point if and only if the partial derivatives of u and v satisfy the Cauchy–Riemann equations (1a) and (1b) at that point. The sole existence of partial derivatives satisfying the Cauchy–Riemann equations is not enough to ensure complex differentiability at that point. It is necessary that u and v be real differentiable, which is a stronger condition than the existence of the partial derivatives, but in general, weaker than continuous differentiability. Holomorphy is the property of a complex function of being differentiable at every point of an open and connected subset of ℂ (this is called a domain in ℂ). Consequently, we can assert that a complex function f, whose real and imaginary parts u and v are real-differentiable functions, is holomorphic if and only if, equations (1a) and (1b) are satisfied throughout the domain we are dealing with.

It is continuous everywhere, but differentiable nowhere. It is not rectifiable. It has a Lebesgue measure of 0. The type 1 curve has a dimension of ≈ 1.46.

A turning point is a point at which the derivative changes sign. A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). If the function is differentiable, then a turning point is a stationary point; however not all stationary points are turning points. If the function is twice differentiable, the stationary points that are not turning points are horizontal inflection points.

Letting z2 → z1 shows the resolvent map is (complex-) differentiable at each z1 ∈ ρ(T); so the integral in the expression of functional calculus converges in L(X).

Let V be the vector space of all differentiable functions of a real variable t. Then the functions e^t and e^{2t} in V are linearly independent.

A triple where and are differentiable manifolds and is a surjective submersion, is called a fibered manifold. E is called the total space, B is called the base.

Like many categories, the category Manp is a concrete category, meaning its objects are sets with additional structure (i.e. a topology and an equivalence class of atlases of charts defining a Cp-differentiable structure) and its morphisms are functions preserving this structure. There is a natural forgetful functor :U : Manp -> Top to the category of topological spaces which assigns to each manifold the underlying topological space and to each p-times continuously differentiable function the underlying continuous function of topological spaces. Similarly, there is a natural forgetful functor :U′ : Manp -> Set to the category of sets which assigns to each manifold the underlying set and to each p-times continuously differentiable function the underlying function.

Constantin Carathéodory's alternative definition of the differentiability of a function can be used to give an elegant proof of the chain rule. Under this definition, a function is differentiable at a point if and only if there is a function , continuous at and such that . There is at most one such function, and if is differentiable at then . Given the assumptions of the chain rule and the fact that differentiable functions and compositions of continuous functions are continuous, we have that there exist functions , continuous at and , continuous at and such that, :f(g(x))-f(g(a))=q(g(x))(g(x)-g(a)) and :g(x)-g(a)=r(x)(x-a).

A continuous function fails to be absolutely continuous if it fails to be uniformly continuous, which can happen if the domain of the function is not compact – examples are tan(x) over [0, /2), x2 over the entire real line, and sin(1/x) over (0, 1]. But a continuous function f can fail to be absolutely continuous even on a compact interval. It may not be "differentiable almost everywhere" (like the Weierstrass function, which is not differentiable anywhere). Or it may be differentiable almost everywhere and its derivative f ′ may be Lebesgue integrable, but the integral of f ′ differs from the increment of f (how much f changes over an interval).

In metric geometry, the space of directions at a point describes the directions of curves that start at the point. It generalizes the tangent space in a differentiable manifold.

Tangent bundle, the vector bundle of tangent spaces on a differentiable manifold. Tangent field, a section of the tangent bundle. Also called a vector field. Tangent space Torus Transversality.

Adiaphoron (, plural: adiaphora from the Greek ἀδιάφορα (pl. of ἀδιάφορον), the negation of διάφορα, meaning "not different or differentiable")., . In Cynicism "adiaphora" represents indifference to the vicissitudes of life.

More general questions of this kind on an arbitrary differentiable manifold are the subject of de Rham cohomology, which allows one to obtain purely topological information using differential methods.

The Clairaut-Schwarz theorem is the key fact needed to prove that for every C^\infty (or at least twice differentiable) differential form \omega\in\Omega^k(M), the second exterior derivative vanishes: d^2\omega := d(d\omega) = 0. This implies that every differentiable exact form (i.e., a form \alpha such that \alpha = d\omega for some form \omega) is closed (i.e., d\alpha = 0), since d\alpha = d(d\omega) = 0.

Neural Turing machines couple LSTM networks to external memory resources, with which they can interact by attentional processes. The combined system is analogous to a Turing machine but is differentiable end-to-end, allowing it to be efficiently trained by gradient descent. Preliminary results demonstrate that neural Turing machines can infer simple algorithms such as copying, sorting and associative recall from input and output examples. Differentiable neural computers (DNC) are an NTM extension.

The proofs were first obtained in the early 1960s by Stephen Smale, for differentiable manifolds. The development of handlebody theory allowed much the same proofs in the differentiable and PL categories. The proofs are much harder in the topological category, requiring the theory of Robion Kirby and Laurent C. Siebenmann. The restriction to manifolds of dimension greater than four are due to the application of the Whitney trick for removing double points.

The term "manifold" comes from German Mannigfaltigkeit, by Riemann. In English, "manifold" refers to spaces with a differentiable or topological structure, while "variety" refers to spaces with an algebraic structure, as in algebraic varieties. In Romance languages, manifold is translated as "variety" – such spaces with a differentiable structure are literally translated as "analytic varieties", while spaces with an algebraic structure are called "algebraic varieties". Thus for example, the French word "variété topologique" means topological manifold.

Let be a continuously differentiable function. Write for the solid of revolution of the graph about the -axis. If the surface area of is finite, then so is the volume.

In multivariate calculus, a differential is said to be exact or perfect, as contrasted with an inexact differential, if it is of the form dQ, for some differentiable function Q.

One infinite-dimensional generalization is as follows. Let and be Banach spaces, and a pair of open sets. Let :F:A\times B \to L(X,Y) be a continuously differentiable function of the Cartesian product (which inherits a differentiable structure from its inclusion into X × Y) into the space of continuous linear transformations of into Y. A differentiable mapping u : A → B is a solution of the differential equation :(1) \quad y' = F(x,y) if :\forall x \in A: \quad u'(x) = F(x, u(x)). The equation (1) is completely integrable if for each (x_0, y_0)\in A\times B, there is a neighborhood U of x0 such that (1) has a unique solution defined on U such that u(x0)=y0.

In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold provides a possibility of extending the theory of manifolds to infinite-dimensional setting. Analogously to the finite-dimensional situation, one can define a differentiable Hilbert manifold by considering a maximal atlas in which the transition maps are differentiable.

Similar examples show that a function can have a th derivative for each non-negative integer but not a th derivative. A function that has successive derivatives is called times differentiable. If in addition the th derivative is continuous, then the function is said to be of differentiability class . (This is a stronger condition than having derivatives, as shown by the second example of .) A function that has infinitely many derivatives is called infinitely differentiable or smooth.

Differentiable manifolds also generalize smoothness. They are normally defined as topological manifolds with an atlas, whose transition maps are smooth, which is used to pull back the differential structure. Every smooth manifold defined in this way has a natural diffeology, for which the plots correspond to the smooth maps from open subsets of Rn to the manifold. With this diffeology, a map between two smooth manifolds is smooth if and only if it is differentiable in the diffeological sense.

It can be proved that any complex function differentiable (in the complex sense) in an open set is analytic. Consequently, in complex analysis, the term analytic function is synonymous with holomorphic function.

In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below.

Algebraic varieties have continuous moduli spaces, hence their study is algebraic geometry. These are finite-dimensional moduli spaces. The space of Riemannian metrics on a given differentiable manifold is an infinite-dimensional space.

The function has its local and global minimum at x=0, but on no neighborhood of 0 is it decreasing down to or increasing up from 0 – it oscillates wildly near 0. This pathology can be understood because, while the function is everywhere differentiable, it is not continuously differentiable: the limit of g'(x) as x \to 0 does not exist, so the derivative is not continuous at 0. This reflects the oscillation between increasing and decreasing values as it approaches 0.

This process is repeated until an optimal integer solution is found. Cutting-plane methods for general convex continuous optimization and variants are known under various names: Kelley's method, Kelley–Cheney–Goldstein method, and bundle methods. They are popularly used for non-differentiable convex minimization, where a convex objective function and its subgradient can be evaluated efficiently but usual gradient methods for differentiable optimization can not be used. This situation is most typical for the concave maximization of Lagrangian dual functions.

The conditions of the Frobenius theorem depend on whether the underlying field is or . If it is R, then assume F is continuously differentiable. If it is , then assume F is twice continuously differentiable. Then (1) is completely integrable at each point of if and only if :D_1F(x,y)\cdot(s_1,s_2) + D_2F(x,y)\cdot(F(x,y)\cdot s_1,s_2) = D_1F(x,y) \cdot (s_2,s_1) + D_2F(x,y)\cdot(F(x,y)\cdot s_2,s_1) for all .

In calculus, a branch of mathematics, the notions of one-sided differentiability and semi-differentiability of a real-valued function f of a real variable are weaker than differentiability. Specifically, the function f is said to be right differentiable at a point a if, roughly speaking, a derivative can be defined as the function's argument x moves to a from the right, and left differentiable at a if the derivative can be defined as x moves to a from the left.

Rolle's theorem is a property of differentiable functions over the real numbers, which are an ordered field. As such, it does not generalize to other fields, but the following corollary does: if a real polynomial factors (has all of its roots) over the real numbers, then its derivative does as well. One may call this property of a field Rolle's property. More general fields do not always have differentiable functions, but they do always have polynomials, which can be symbolically differentiated.

However, even if a function is continuous at a point, it may not be differentiable there. For example, the absolute value function given by is continuous at , but it is not differentiable there. If is positive, then the slope of the secant line from 0 to is one, whereas if is negative, then the slope of the secant line from 0 to is negative one. This can be seen graphically as a "kink" or a "cusp" in the graph at .

A differentiable manifold is a Hausdorff and second countable topological space , together with a maximal differentiable atlas on . Much of the basic theory can be developed without the need for the Hausdorff and second countability conditions, although they are vital for much of the advanced theory. They are essentially equivalent to the general existence of bump functions and partitions of unity, both of which are used ubiquitously. The notion of a manifold is identical to that of a topological manifold.

2-dimensional section of Reeb foliation 3-dimensional model of Reeb foliation In mathematics (differential geometry), a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space Rn into the cosets x + Rp of the standardly embedded subspace Rp. The equivalence classes are called the leaves of the foliation.Candel and Conlon 2000, Foliations I, p. 5 If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable (of class Cr), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1 (otherwise, C0 is a topological foliation).

Usually one would want G(Z) to be continuously differentiable with nonsingular Jacobian matrix. Quasiconvex functions and quasiconcave functions extend the concept of unimodality to functions whose arguments belong to higher-dimensional Euclidean spaces.

If the function is not continuously differentiable in a neighborhood of the root then it is possible that Newton's method will always diverge and fail, unless the solution is guessed on the first try.

There is no specific way to segment market. However, businesses can follow generalized rules like geographic, demographic, psychographic, and behavioral. A good market segmentation should be sustainable, accessible, actionable, measurable, and differentiable (Karlsson,2012).

There are various versions of the surgery exact sequence. One can work in either of the three categories of manifolds: differentiable (smooth), PL, topological. Another possibility is to work with the decorations s or h.

The Standard Model unifies the description of electromagnetism, weak interactions and strong interactions in the language of gauge theory. In the 1970s, Michael Atiyah began studying the mathematics of solutions to the classical Yang–Mills equations. In 1983, Atiyah's student Simon Donaldson built on this work to show that the differentiable classification of smooth 4-manifolds is very different from their classification up to homeomorphism. Michael Freedman used Donaldson's work to exhibit exotic R4s, that is, exotic differentiable structures on Euclidean 4-dimensional space.

It is often not possible to provide one consistent coordinate system for an entire space. In this case, a collection of coordinate maps are put together to form an atlas covering the space. A space equipped with such an atlas is called a manifold and additional structure can be defined on a manifold if the structure is consistent where the coordinate maps overlap. For example, a differentiable manifold is a manifold where the change of coordinates from one coordinate map to another is always a differentiable function.

For ensuring more regularity, the function that defines a curve is often supposed to be differentiable, and the curve is then said to be a differentiable curve. A plane algebraic curve is the zero set of a polynomial in two indeterminates. More generally, an algebraic curve is the zero set of a finite set of polynomials, which satisfies the further condition of being an algebraic variety of dimension one. If the coefficients of the polynomials belong to a field , the curve is said to be defined over .

In differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one (hence the name "exotic"). The first exotic spheres were constructed by in dimension n = 7 as S^3-bundles over S^4. He showed that there are at least 7 differentiable structures on the 7-sphere.

Thus, it may take an unreasonable length of time for it to ascend the ridge (or descend the alley). By contrast, gradient descent methods can move in any direction that the ridge or alley may ascend or descend. Hence, gradient descent or the conjugate gradient method is generally preferred over hill climbing when the target function is differentiable. Hill climbers, however, have the advantage of not requiring the target function to be differentiable, so hill climbers may be preferred when the target function is complex.

In conjunction with computational geometry, a computational synthetic geometry has been founded, having close connection, for example, with matroid theory. Synthetic differential geometry is an application of topos theory to the foundations of differentiable manifold theory.

In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite.

When studying geometrical objects, the arising groupoids often carry some differentiable structure, turning them into Lie groupoids. These can be studied in terms of Lie algebroids, in analogy to the relation between Lie groups and Lie algebras.

In mathematics, the Malliavin derivative is a notion of derivative in the Malliavin calculus. Intuitively, it is the notion of derivative appropriate to paths in classical Wiener space, which are "usually" not differentiable in the usual sense.

The curvature of a differentiable curve was originally defined through osculating circles. In this setting, Augustin-Louis Cauchy showed that the center of curvature is the intersection point of two infinitely close normal lines to the curve.

Thus, the model is fully differentiable and trains end-to-end. The key characteristic of these models is that their depth, the size of their short-term memory, and the number of parameters can be altered independently.

The polymorphism of C++ makes it possible to envisage a programming system in which all mathematical operators and functions can be overloaded to automatically compute the derivative contributions of every differentiable numerical operation in any computer program.

Somewhat surprisingly, a differential equation may have solutions which are not differentiable; and the weak formulation allows one to find such solutions. Weak solutions are important because a great many differential equations encountered in modelling real-world phenomena do not admit of sufficiently smooth solutions, and the only way of solving such equations is using the weak formulation. Even in situations where an equation does have differentiable solutions, it is often convenient to first prove the existence of weak solutions and only later show that those solutions are in fact smooth enough.

In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology. Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography.

These results generalize and formalize the 'rule' stated in the previous section. Using the two previous results and the fact that there exists a Morse function on any differentiable manifold, one can prove that any differentiable manifold is a CW complex with an n-cell for each critical point of index n. To do this, one needs the technical fact that one can arrange to have a single critical point on each critical level, which is usually proven by using gradient-like vector fields to rearrange the critical points.

In many mathematical branches, several structures defined on a topological space X (e.g., a differentiable manifold) can be naturally localised or restricted to open subsets U \subset X: typical examples include continuous real-valued or complex-valued functions, n times differentiable (real-valued or complex- valued) functions, bounded real-valued functions, vector fields, and sections of any vector bundle on the space. The ability to restrict data to smaller open subsets gives rise to the concept of presheaves. Roughly speaking, sheaves are then those presheaves, where local data can be glued to global data.

There is also a quick proof, by Morris Hirsch, based on the impossibility of a differentiable retraction. The indirect proof starts by noting that the map f can be approximated by a smooth map retaining the property of not fixing a point; this can be done by using the Weierstrass approximation theorem, for example. One then defines a retraction as above which must now be differentiable. Such a retraction must have a non-singular value, by Sard's theorem, which is also non-singular for the restriction to the boundary (which is just the identity).

A rectangular grid (top) and its image under a Conformal map f (bottom). In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is, at every point of its domain, complex differentiable in a neighborhood of the point. The existence of a complex derivative in a neighbourhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal, locally, to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis.

In 1956, John Milnor constructed an exotic sphere in 7 dimensions and showed that there are at least 7 differentiable structures on the 7-sphere. In 1963 he showed that the exact number of such structures is 28.

Neural network pushdown automata (NNPDA) are similar to NTMs, but tapes are replaced by analogue stacks that are differentiable and that are trained. In this way, they are similar in complexity to recognizers of context free grammars (CFGs).

The Jacobian matrix is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. A further generalization for a function between Banach spaces is the Fréchet derivative.

In optimization, Newton's method is applied to the derivative of a twice- differentiable function to find the roots of the derivative (solutions to ), also known as the stationary points of . These solutions may be minima, maxima, or saddle points.

The absolute value function is continuous (i.e. it has no gaps). It is differentiable everywhere except at the point = 0, where it makes a sharp turn as it crosses the -axis. cusp on the graph of a continuous function.

The exterior algebra of differential forms, equipped with the exterior derivative, is a cochain complex whose cohomology is called the de Rham cohomology of the underlying manifold and plays a vital role in the algebraic topology of differentiable manifolds.

Even though the hinge-loss is not differentiable, it can also give rise to a tractable variant of the 0/1- loss based learning problem, since the hinge-loss allows it to recast to the equivalent constrained optimization problem.

Arrow motivated his paper by reference to the need to extend proofs to cover equilibria at the edge of the space, and Debreu by the possibility of indifference curves being non-differentiable. Modern texts follow their style of proof.

In computational engineering, Luus–Jaakola (LJ) denotes a heuristic for global optimization of a real-valued function. In engineering use, LJ is not an algorithm that terminates with an optimal solution; nor is it an iterative method that generates a sequence of points that converges to an optimal solution (when one exists). However, when applied to a twice continuously differentiable function, the LJ heuristic is a proper iterative method, that generates a sequence that has a convergent subsequence; for this class of problems, Newton's method is recommended and enjoys a quadratic rate of convergence, while no convergence rate analysis has been given for the LJ heuristic. In practice, the LJ heuristic has been recommended for functions that need be neither convex nor differentiable nor locally Lipschitz: The LJ heuristic does not use a gradient or subgradient when one be available, which allows its application to non-differentiable and non-convex problems.

The measure-theoretic version of differentiation under the integral sign also applies to summation (finite or infinite) by interpreting summation as counting measure. An example of an application is the fact that power series are differentiable in their radius of convergence.

Mathematically, a pseudoscalar is an element of the top exterior power of a vector space, or the top power of a Clifford algebra; see pseudoscalar (Clifford algebra). More generally, it is an element of the canonical bundle of a differentiable manifold.

Svensson proved another sufficient condition for the existence of PEEF allocations. Again all preferences are represented by continuous utility functions. Moreover, all utility functions are continuously differentiable in the interior of the consumption space. The main concept is sigma-optimality.

A superlative index is defined technically as "an index that is exact for a flexible functional form that can provide a second-order approximation to other twice-differentiable functions around the same point."Export and Import manual, Chapter 18, p. 23.

A real-valued, continuously differentiable function f is positive-definite on a neighborhood of the origin, D, if f(0) = 0 and f(x) > 0 for every non-zero x \in D. This definition is in conflict with the one above.

Surprisingly, a mapping between open subset of Fréchet spaces is smooth (infinitely often differentiable) if it maps smooth curves to smooth curves; see Convenient analysis. Moreover, smooth curves in spaces of smooth functions are just smooth functions of one variable more.

In a so- called concrete category, the objects are associated with mathematical structures like sets, magmas, groups, rings, topological spaces, vector spaces, metric spaces, partial orders, differentiable manifolds, uniform spaces, etc., and morphisms between two objects are associated with structure- preserving functions between them. In the examples above, these would be functions, magma homomorphisms, group homomorphisms, ring homomorphisms, continuous functions, linear transformations (or matrices), metric maps, monotonic functions, differentiable functions, and uniformly continuous functions, respectively. As an algebraic theory, one of the advantages of category theory is to enable one to prove many general results with a minimum of assumptions.

We don't need to assume continuity of f on the whole interval. Part I of the theorem then says: if f is any Lebesgue integrable function on and x0 is a number in such that f is continuous at x0, then :F(x) = \int_a^x f(t)\, dt is differentiable for with We can relax the conditions on f still further and suppose that it is merely locally integrable. In that case, we can conclude that the function F is differentiable almost everywhere and almost everywhere. On the real line this statement is equivalent to Lebesgue's differentiation theorem.

The derivative operator P : C∞([0,1]) → C∞([0,1]) defined by P(ƒ) = ƒ′ is itself infinitely differentiable. The first derivative is given by :D(P)(f)(h) = h' for any two elements ƒ and h in C∞([0,1]). This is a major advantage of the Fréchet space C∞([0,1]) over the Banach space Ck([0,1]) for finite k. If P : U → Y is a continuously differentiable function, then the differential equation :x'(t) = P(x(t)),\quad x(0) = x_0\in U need not have any solutions, and even if does, the solutions need not be unique.

The function on an open disk around 0 is not a local homeomorphism at 0 when n is at least 2. In that case 0 is a point of "ramification" (intuitively, n sheets come together there). Using the inverse function theorem one can show that a continuously differentiable function (where U is an open subset of ) is a local homeomorphism if the derivative Dxf is an invertible linear map (invertible square matrix) for every . (The converse is false, as shown by the local homeomorphism with .) An analogous condition can be formulated for maps between differentiable manifolds.

The whorls are rounded, shouldered by the strong posterior primary spiral thread . The siphonal canal is nearly straight, very wide, hardly differentiable from the aperture. The columella is nearly straight with little callus. The outer lip is thin and crenulated by the sculpture.

It is a natural question to ask if a Sobolev function is continuous or even continuously differentiable. Roughly speaking, sufficiently many weak derivatives (i.e. large p) result in a classical derivative. This idea is generalized and made precise in the Sobolev embedding theorem.

A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a non-negative derivative at each point and exactly one inflection point. A sigmoid "function" and a sigmoid "curve" refer to the same object.

The coefficient 2 before P(\xi>a) is in fact the Euler characteristic of the sphere (for the torus it vanishes). It is assumed that X is twice continuously differentiable (almost surely), and reaches its maximum at a single point (almost surely).

In mathematical optimization, Wolfe duality, named after Philip Wolfe, is type of dual problem in which the objective function and constraints are all differentiable functions. Using this concept a lower bound for a minimization problem can be found because of the weak duality principle.

In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the Lp space L^1([a,b]). See distributions for a more general definition.

D.26 in . Systems derived from the Franklin system give bases in the space C1([0, 1]2) of differentiable functions on the unit square. and The existence of a Schauder basis in C1([0, 1]2) was a question from Banach's book.see p.

More in general, the concept can be applied to representing positions on the boundary of a strictly convex bounded subset of k-dimensional Euclidean space, provided that that boundary is a differentiable manifold. In this general case, the n-vector consists of k parameters.

In mathematics, Moreau's theorem is a result in convex analysis. It shows that sufficiently well-behaved convex functionals on Hilbert spaces are differentiable and the derivative is well-approximated by the so-called Yosida approximation, which is defined in terms of the resolvent operator.

Keep the same field and vector space as before, but now consider the set Diff(R) of all differentiable functions. The same sort of argument as before shows that this is a subspace too. Examples that extend these themes are common in functional analysis.

Since , Rolle's theorem applies, and indeed, there is a point where the derivative of is zero. Note that the theorem applies even when the function cannot be differentiated at the endpoints because it only requires the function to be differentiable in the open interval.

On the real line, every polynomial function is infinitely differentiable. By standard differentiation rules, if a polynomial of degree is differentiated times, then it becomes a constant function. All of its subsequent derivatives are identically zero. In particular, they exist, so polynomials are smooth functions.

Let M and N be two real, smooth manifolds. Furthermore, let C∞(M,N) denote the space of smooth mappings between M and N. The notation C∞ means that the mappings are infinitely differentiable, i.e. partial derivatives of all orders exist and are continuous.

If, moreover, f is analytic or continuously differentiable k times in a neighborhood of (a, b), then one may choose U in order that the same holds true for g inside U. In the analytic case, this is called the analytic implicit function theorem.

In case of a double knot, the length of the knot span becomes zero and the peak reaches one exactly. The basis function is no longer differentiable at that point. The curve will have a sharp corner if the neighbour control points are not collinear.

By Fermat's theorem, all local maxima and minima of a differentiable function occur at critical points. Therefore, to find the local maxima and minima, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the Hessian matrix at these zeros.

In 3 dimensions, self-avoiding approximation curves can even contain knots. Approximation curves remain within a bounded portion of n-dimensional space, but their lengths increase without bound. Space-filling curves are special cases of fractal curves. No differentiable space-filling curve can exist.

Furthermore, the interpolant is a polynomial and thus infinitely differentiable. So, we see that polynomial interpolation overcomes most of the problems of linear interpolation. However, polynomial interpolation also has some disadvantages. Calculating the interpolating polynomial is computationally expensive (see computational complexity) compared to linear interpolation.

Let be a continuous real-valued function defined on a closed interval . Let be the function defined, for all in , by :F(x) = \int_a^x f(t)\, dt. Then, is continuous on , differentiable on the open interval , and :F'(x) = f(x) for all in .

Despite never being differentiable, the function is continuous: Since the terms of the infinite series which defines it are bounded by ±an and this has finite sum for 0 < a < 1, convergence of the sum of the terms is uniform by the Weierstrass M-test with Mn = an. Since each partial sum is continuous, by the uniform limit theorem, it follows that f is continuous. Additionally, since each partial sum is uniformly continuous, it follows that f is also uniformly continuous. It might be expected that a continuous function must have a derivative, or that the set of points where it is not differentiable should be "small" in some sense.

While the first examples of curves that are met are mostly plane curves (that is, in everyday words, curved lines in two-dimensional space), there are obvious examples such as the helix which exist naturally in three dimensions. The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. In general relativity, a world line is a curve in spacetime. If X is a differentiable manifold, then we can define the notion of differentiable curve in X. This general idea is enough to cover many of the applications of curves in mathematics.

To calculate derivatives, one needs to have local coordinate systems defined consistently in X. Mathematicians were surprised in 1956 when Milnor showed that consistent coordinate systems could be set up on the 7-sphere in two different ways that were equivalent in the continuous sense, but not in the differentiable sense. Milnor and others set about trying to discover how many such exotic spheres could exist in each dimension and to understand how they relate to each other. No exotic structures are possible on the 1-, 2-, 3-, 5-, 6-, 12-, 56- or 61-spheres. Some higher-dimensional spheres have only two possible differentiable structures, others have thousands.

DNC system diagram DNC networks were introduced as an extension of the Neural Turing Machine (NTM), with the addition of memory attention mechanisms that control where the memory is stored, and temporal attention that records the order of events. This structure allows DNCs to be more robust and abstract than a NTM, and still perform tasks that have longer-term dependencies than some predecessors such as Long Short Term Memory (LSTM). The memory, which is simply a matrix, can be allocated dynamically and accessed indefinitely. The DNC is differentiable end-to-end (each subcomponent of the model is differentiable, therefore so is the whole model).

If a structure has a discrete moduli (if it has no deformations, or if a deformation of a structure is isomorphic to the original structure), the structure is said to be rigid, and its study (if it is a geometric or topological structure) is topology. If it has non- trivial deformations, the structure is said to be flexible, and its study is geometry. The space of homotopy classes of maps is discrete, so studying maps up to homotopy is topology. Similarly, differentiable structures on a manifold is usually a discrete space, and hence an example of topology, but exotic R4s have continuous moduli of differentiable structures.

For a continuously differentiable function of several real variables, a point P (that is, a set of values for the input variables, which is viewed as a point in Rn) is critical if all of the partial derivatives of the function are zero at P, or, equivalently, if its gradient is zero. The critical values are the values of the function at the critical points. If the function is smooth, or, at least twice continuously differentiable, a critical point may be either a local maximum, a local minimum or a saddle point. The different cases may be distinguished by considering the eigenvalues of the Hessian matrix of second derivatives.

The original derivation of the equations by Kolmogorov starts with the Chapman-Kolmogorov equation (Kolmogorov called it Fundamental equation) for time-continuous and differentiable Markov processes on a finite, discrete state space. In this formulation, it is assumed that the probabilities P(i,s;j,t) are continuous and differentiable functions of t > s . Also adequate limit properties for the derivatives are assumed. Feller Feller, Willy (1940) "On the Integro-Differential Equations of Purely Discontinuous Markoff Processes", Transactions of the American Mathematical Society, 48 (3), 488-515 derives the equations under slightly different conditions, starting with the concept of purely discontinuous Markov process and formulating them for more general state spaces.

No solution therefore offers a "short cut". This is under the assumption that the search space is a probability density function. It does not apply to the case where the search space has underlying structure (f.e. is a differentiable function) that can be exploited more efficiently (f.e.

If we attempt to use the above formula to compute the derivative of ' at zero, then we must evaluate . Since and , we must evaluate 1/0, which is undefined. Therefore, the formula fails in this case. This is not surprising because ' is not differentiable at zero.

Although the definition of a manifold does not require that its model space should be , this choice is the most common, and almost exclusive one in differential geometry. On the other hand, Whitney embedding theorems state that any real differentiable -dimensional manifold can be embedded into .

A circular sector is shaded in green. Its curved boundary of length L is a circular arc. In Euclidean geometry, an arc (symbol: ⌒) is a connected subset of a differentiable curve. Arcs of lines are called segments or rays, depending whether they are bounded or not.

Who Invented Backpropagation? when Linnainmaa introduced the reverse mode of automatic differentiation (AD), in order to efficiently compute the derivative of a differentiable composite function that can be represented as a graph, by recursively applying the chain rule to the building blocks of the function.Griewank, Andreas (2012).

"Implementation and Evaluation of a Background Music Reactive Game." IE Conference 2007 - Tempere University of Technology and Nokia Research Center. 2007. music video games such as Vib- Ribbon, Audiosurf, or Dance Factory lack a differentiable underlying genre and as such cannot be considered hybrid music games.

Risken (1984) However, d W_t / dt does not exist because the Wiener process is nowhere differentiable, and so the Langevin equation is, strictly speaking, only heuristic. In physics and engineering disciplines, it is nevertheless a common representation for the Ornstein–Uhlenbeck process and similar stochastic differential equations.

They occur also in configuration spaces of physical systems. Beside Euclidean geometry, Euclidean spaces are also widely used in other areas of mathematics. Tangent spaces of differentiable manifolds are Euclidean vector spaces. More generally, a manifold is a space that is locally approximated by Euclidean spaces.

Similarly, the composition of onto functions is always onto. It follows that the composition of two bijections is also a bijection. The inverse function of a composition (assumed invertible) has the property that . Derivatives of compositions involving differentiable functions can be found using the chain rule.

Any differentiable function may be used for . The examples that follow use a variety of elementary functions; special functions may also be used. Note that multi-valued functions such as the natural logarithm may be used, but attention must be confined to a single Riemann surface.

The idea of a local homeomorphism can be formulated in geometric settings different from that of topological spaces. For differentiable manifolds, we obtain the local diffeomorphisms; for schemes, we have the formally étale morphisms and the étale morphisms; and for toposes, we get the étale geometric morphisms.

SageManifolds (following styling of SageMath) is an extension fully integrated into SageMath, to be used as a package for differential geometry and tensor calculus. The official page for the project is sagemanifolds.obspm.fr. It can be used on CoCalc. SageManifolds deals with differentiable manifolds of arbitrary dimension.

Vertical tangent on the function ƒ(x) at x = c. In mathematics, particularly calculus, a vertical tangent is a tangent line that is vertical. Because a vertical line has infinite slope, a function whose graph has a vertical tangent is not differentiable at the point of tangency.

It has an instruction page, which explains the use of the plotter and the function syntax. About 180 functions are predefined. These belong to the categories basic functions, trigonometric and hyperbolic functions, non-differentiable functions, probability functions, special functions, programmable functions, iterations and fractals, differential and integral equations.

Another major contribution of Neyman was the introduction of the Neyman value,Neyman, A., 2001, "Values of non-atomic vector measure games," Israel Journal of Mathematics, 124, pp 1–27 a far-reaching generalization of the Aumann–Shapley value to the case of non-differentiable vector measure games.

A regular homotopy between two immersions f and g from a manifold M to a manifold N is defined to be a differentiable function such that for all t in the function defined by for all is an immersion, with , . A regular homotopy is thus a homotopy through immersions.

As well as developing Jarník's algorithm, he found tight bounds on the number of lattice points on convex curves, studied the relationship between the Hausdorff dimension of sets of real numbers and how well they can be approximated by rational numbers, and investigated the properties of nowhere-differentiable functions.

The Weierstrass function is continuous everywhere but differentiable nowhere. In mathematics, a pathological object is one which possesses deviant, irregular or counterintuitive property, in such a way that distinguishes it from what is conceived as a typical object in the same category. The opposite of pathological is well-behaved.

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth. image of a rectangular grid on a square under a diffeomorphism from the square onto itself.

Every differentiable surface is a topological surface, but the converse is false. For simplicity, unless otherwise stated, "surface" will mean a surface in the Euclidean space of dimension 3 or in . A surface that is not supposed to be included in another space is called an abstract surface.

Genomovar is a term commonly used within the genera Burkholderia and Agrobacterium to denote strains which are phylogenetically differentiable, but are phenotypically indistinguishable. A genomovar cannot be identified by standard biochemical tests, but it is classified as a species when a biochemical test allows it to be identified.

The function \sin(1/x) – it oscillates increasingly rapidly between -1 and 1 as x approaches 0. Consequently, the function f(x) = (1 + \sin(1/x))x^2 oscillates increasingly rapidly between 0 and 2x^2 as x approaches 0. If one extends this function by defining f(0) = 0 then the extended function is continuous and everywhere differentiable (it is differentiable at 0 with derivative 0), but has rather unexpected behavior near 0: in any neighborhood of 0 it attains 0 infinitely many times, but also equals 2x^2 (a positive number) infinitely often. Continuing in this vein, one may define g(x) = (2 + \sin(1/x))x^2, which oscillates between x^2 and 3x^2.

Let A, B be smooth manifolds and let \Phi: A \rightarrow B be a C^r-diffeomorphism between them, that is: \Phi is a r times continuously differentiable, bijective map from A to B with r times continuously differentiable inverse from B to A. Here r may be any natural number (or zero), \infty (smooth) or \omega (analytic). The map \Phi is called a regular coordinate transformation or regular variable substitution, where regular refers to the C^r-ness of \Phi. Usually one will write x = \Phi(y) to indicate the replacement of the variable x by the variable y by substituting the value of \Phi in y for every occurrence of x.

These phrases are often pleonasms and form irreversible binomials. In other cases, the two components have differences which are subtle, appreciable only to lawyers, or obsolete. For example, ways and means, referring to methods and resources respectively, are differentiable, in the same way that tools and materials, or equipment and funds, are differentiable—but the difference between them is often practically irrelevant to the contexts in which the irreversible binomial ways and means is used today in non-legal contexts as a mere cliché. Doublets may also have arisen or persisted because the solicitors and clerks who drew up conveyances and other documents were paid by the word, which tended to encourage verbosity.

Jarník's work in real analysis was sparked by finding, in the unpublished works of Bernard Bolzano, a definition of a continuous function that was nowhere differentiable. Bolzano's 1830 discovery predated the 1872 publication of the Weierstrass function, previously considered to be the first example of such a function. Based on his study of Bolzano's function, Jarník was led to a more general theorem: If a real-valued function of a closed interval does not have bounded variation in any subinterval, then there is a dense subset of its domain on which at least one of its Dini derivatives is infinite. This applies in particular to the nowhere-differentiable functions, as they must have unbounded variation in all intervals.

Based on the inverse function theorem in Banach spaces, it is possible to extend the implicit function theorem to Banach space valued mappings. Let X, Y, Z be Banach spaces. Let the mapping f : X × Y → Z be continuously Fréchet differentiable. If (x_0,y_0)\in X\times Y, f(x_0,y_0)=0, and y\mapsto Df(x_0,y_0)(0,y) is a Banach space isomorphism from Y onto Z, then there exist neighbourhoods U of x0 and V of y0 and a Fréchet differentiable function g : U → V such that f(x, g(x)) = 0 and f(x, y) = 0 if and only if y = g(x), for all (x,y)\in U\times V.

An implicit surface in a Euclidean space (or, more generally, in an affine space) of dimension 3 is the set of the common zeros of a differentiable function of three variables :f(x, y, z)=0. Implicit means that the equation defines implicitly one of the variables as a function of the other variables. This is made more exact by the implicit function theorem: if , and the partial derivative in of is not zero at , then there exists a differentiable function such that :f(x,y,\varphi(x,y))=0 in a neighbourhood of . In other words, the implicit surface is the graph of a function near a point of the surface where the partial derivative in is nonzero.

Suppose M and N are two differentiable manifolds with dimensions m and n, respectively, and f is a function from M to N. Since differentiable manifolds are topological spaces we know what it means for f to be continuous. But what does "f is " mean for ? We know what that means when f is a function between Euclidean spaces, so if we compose f with a chart of M and a chart of N such that we get a map that goes from Euclidean space to M to N to Euclidean space we know what it means for that map to be . We define "f is " to mean that all such compositions of f with charts are .

In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry, the same definition is used, but the defining function is required to be differentiable Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one.

C. McGovern, ″Commoditization of nanomaterials″. Nanotechnology Perceptions 6 (2010) 155–178. There is a spectrum of commoditization, rather than a binary distinction of "commodity versus differentiable product". Few products have complete undifferentiability and hence fungibility; even electricity can be differentiated in the market based on its method of generation (e.g.

In practice, these results are local, and the neighborhood of convergence is not known in advance. But there are also some results on global convergence: for instance, given a right neighborhood of , if is twice differentiable in and if , in , then, for each in the sequence is monotonically decreasing to .

A theorem of Hassler Whitney Th. Bröcker, Differentiable Germs and Catastrophes, London Mathematical Society. Lecture Notes 17. Cambridge, (1975)Bruce and Giblin, Curves and singularities, (1984, 1992) , (paperback) states :Theorem. Any closed set in Rn occurs as the solution set of f −1(0) for some smooth function f:Rn→R.

Cambridge University Press, , pp. 117–119. In classical mechanics for instance, in the action formulation, extremal solutions to the variational principle are on shell and the Euler–Lagrange equations give the on-shell equations. Noether's theorem regarding differentiable symmetries of physical action and conservation laws is another on-shell theorem.

One immediate advantage is that the UT can be applied with any given function whereas linearization may not be possible for functions that are not differentiable. A practical advantage is that the UT can be easier to implement because it avoids the need to derive and implement a linearizing Jacobian matrix.

In addition, there tends to be a high degree of concordance (agreement, or cross-reference between different parts of the sentence). Therefore, morphology in synthetic languages is more important than syntax. Most Indo-European languages are moderately synthetic. There are two subtypes of synthesis, according to whether morphemes are clearly differentiable or not.

When the fixed points are paired such that :d_0(p_1) = d_1(p_0) then it may be shown that the resulting curve p_x is a continuous function of x. When the curve is continuous, it is not in general differentiable. In the remaining of this page, we will assume the curves are continuous.

Giles' work on neural networks showed that fundamental computational structures such as regular grammars and finite state machines could be theoretically represented in recurrent neural networks. Another contribution was the Neural Network Pushdown Automata and the first analog differentiable stack. Some of these publications are cited as early work in "deep" learning.

In differential geometry, an almost symplectic structure on a differentiable manifold M is a two-form ω on M that is everywhere non-singular.. If, in addition, ω is closed, then it is a symplectic form. An almost symplectic manifold is an Sp-structure; requiring ω to be closed is an integrability condition.

In mathematics, a branched manifold is a generalization of a differentiable manifold which may have singularities of very restricted type and admits a well-defined tangent space at each point. A branched n-manifold is covered by n-dimensional "coordinate charts", each of which involves one or several "branches" homeomorphically projecting into the same differentiable n-disk in Rn. Branched manifolds first appeared in the dynamical systems theory, in connection with one-dimensional hyperbolic attractors constructed by Smale and were formalized by R. F. Williams in a series of papers on expanding attractors. Special cases of low dimensions are known as train tracks (n = 1) and branched surfaces (n = 2) and play prominent role in the geometry of three-manifolds after Thurston.

Though long used informally, this term has found a formal definition in category theory. ; pathological:An object behaves pathologically (or, somewhat more broadly used, in a degenerated way) if it either fails to conform to the generic behavior of such objects, fails to satisfy certain context-dependent regularity properties, or simply disobeys mathematical intuition. In many occasions, these can be and often are contradictory requirements, while in other occasions, the term is more deliberately used to refer to an object artificially constructed as a counterexample to these properties. :Note for that latter quote that as the differentiable functions are meagre in the space of continuous functions, as Banach found out in 1931, differentiable functions are colloquially speaking a rare exception among the continuous ones.

Note that, for each , f^{(j)}(a)=P^{(j)}(a). Hence each of the first k−1 derivatives of the numerator in h_k(x) vanishes at x=a, and the same is true of the denominator. Also, since the condition that the function f be k times differentiable at a point requires differentiability up to order k−1 in a neighborhood of said point (this is true, because differentiability requires a function to be defined in a whole neighborhood of a point), the numerator and its k − 2 derivatives are differentiable in a neighborhood of a. Clearly, the denominator also satisfies said condition, and additionally, doesn't vanish unless x=a, therefore all conditions necessary for L'Hopital's rule are fulfilled, and its use is justified.

In real analysis, this example shows that there are infinitely differentiable functions whose Taylor series are not equal to even if they converge. By contrast, the holomorphic functions studied in complex analysis always possess a convergent Taylor series, and even the Taylor series of meromorphic functions, which might have singularities, never converge to a value different from the function itself. The complex function , however, does not approach 0 when approaches 0 along the imaginary axis, so it is not continuous in the complex plane and its Taylor series is undefined at 0. More generally, every sequence of real or complex numbers can appear as coefficients in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of Borel's lemma.

Earlier in his career he taught at the University of Michigan and at Princeton University. Among Munkres' contributions to mathematics is the development of what is sometimes called the Munkres assignment algorithm. A significant contribution in topology is his obstruction theory for the smoothing of homeomorphisms.Obstructions to the smoothing of piecewise- differentiable homeomorphisms, Ann.

By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions).

The same holds true for several variables. If V is some topological space, for example a subset of some Rn, real- or complex-valued continuous functions on V form a commutative ring. The same is true for differentiable or holomorphic functions, when the two concepts are defined, such as for V a complex manifold.

In mathematics, a statistical manifold is a Riemannian manifold, each of whose points is a probability distribution. Statistical manifolds provide a setting for the field of information geometry. The Fisher information metric provides a metric on these manifolds. Following this definition, the log-likelihood function is a differentiable map and the score is an inclusion.

Teiji Takagi (高木貞治 Takagi Teiji, April 21, 1875 – February 28, 1960) was a Japanese mathematician, best known for proving the Takagi existence theorem in class field theory. The Blancmange curve, the graph of a nowhere- differentiable but uniformly continuous function, is also called the Takagi curve after his work on it.

A comparison of gradient descent (green) and Newton's method (red) for minimizing a function (with small step sizes). Newton's method uses curvature information (i.e. the second derivative) to take a more direct route. In calculus, Newton's method is an iterative method for finding the roots of a differentiable function , which are solutions to the equation .

The subset of the space of all functions from R to R consisting of (sufficiently differentiable) functions that satisfy a certain differential equation is a subspace of RR if the equation is linear. This is because differentiation is a linear operation, i.e., (a f + b g)′ = a f′ + b g′, where ′ is the differentiation operator.

For nuclear experiments, high carbon contamination would result in extremely high background and the experimental results would be skewed or less differentiable with the background. With ERDA and heavy ion projectiles, valuable information can be obtained on the light element content of thin foils even if only the energy of the recoils is measured.

Neural Turing machines (NTMs) are a method of extending recurrent neural networks by coupling them to external memory resources which they can interact with by attentional processes. The combined system is analogous to a Turing machine or Von Neumann architecture but is differentiable end-to-end, allowing it to be efficiently trained with gradient descent.

The stationary points are the red circles. In this graph, they are all relative maxima or relative minima. The blue squares are inflection points. In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero.

If a sequence of analytic functions converges uniformly in a region S of the complex plane, then the limit is analytic in S. This example demonstrates that complex functions are more well-behaved than real functions, since the uniform limit of analytic functions on a real interval need not even be differentiable (see Weierstrass function).

Georg Landsberg (January 30, 1865 – September 14, 1912) was a German-Jewish mathematician, known for his work in the theory of algebraic functions and on the Riemann–Roch theorem.. The Takagi–Landsberg curve, a fractal that is the graph of a nowhere-differentiable but uniformly continuous function, is named after Teiji Takagi and him.

A. koschevnikovi hosts a unique species of the honey bee parasitic mite genus Varroa, named Varroa rindereri. Although this parasite species is quite similar to Varroa jacobsoni it is perfectly differentiable. V. rindereri is larger (1 180 x 1 698 micrometers). V.rindereri also has a fewer number of setae and pores on the sternal shield.

However, despite the total absence of examples, certain interesting results were proved in the mid-1960s in the pioneering work of Edmond Bonan, Alfred Gray, and Vivian Kraines. Simultaneously and independently, Edmond Bonan Bonan Edmond. (1965) Structure presque quaternale sur une variété differentiable, Comptes Rendus de l'Académie des Sciences, 261, 1965, 5445–5448. and Vivian Yoh KrainesKraines,Vivian Yoh .

In many cases, it is impossible to define a single frame of reference that valid globally. To overcome this, frames are commonly pieced together to form an atlas, thus arriving at the notion of a local frame. In addition, it is often desirable to endow these atlases with a smooth structure, so that the resulting frame fields are differentiable.

As the first order derivatives are arbitrary, the wave function can be a continuously differentiable function of space, since at any boundary the gradient of the wave function can be matched. On the contrary, wave equations in physics are usually second order in time, notable are the family of classical wave equations and the quantum Klein–Gordon equation.

When this happens, the limit of the product of these two factors will equal the product of the limits of the factors. The two factors are and . The latter is the difference quotient for at , and because ' is differentiable at ' by assumption, its limit as ' tends to ' exists and equals . As for , notice that is defined wherever ' is.

In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold. If M is already a topological manifold, it is required that the new topology be identical to the existing one.

Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3. Rotations are linear transformations of \R^3 and can therefore be represented by matrices once a basis of \R^3 has been chosen.

This paper calculates the structure of the group of smooth structures on an n-sphere for n > 4. It is suspected that certain differentiable structures on the 4-sphere, called Gluck twists, are not isomorphic to the standard one, but at the moment there are no known invariants capable of distinguishing different smooth structures on a 4-sphere.

The orientation of a volume may be determined by the orientation on its boundary, indicated by the circulating arrows. Each point p on an n-dimensional differentiable manifold has a tangent space TpM which is an n-dimensional real vector space. Each of these vector spaces can be assigned an orientation. Some orientations "vary smoothly" from point to point.

Perhaps the simplest partial differential relation is for the derivative to not vanish: f'(x) eq 0. Properly, this is an ordinary differential relation, as this is a function in one variable. A holonomic solution to this relation is a function whose derivative is nowhere vanishing. I.e., a strictly monotone differentiable functions, either increasing or decreasing.

In mathematics, noncommutative residue, defined independently by M. and , is a certain trace on the algebra of pseudodifferential operators on a compact differentiable manifold that is expressed via a local density. In the case of the circle, the noncommutative residue had been studied earlier by M. and Y. in the context of one-dimensional integrable systems.

The concept of semimartingales, and the associated theory of stochastic calculus, extends to processes taking values in a differentiable manifold. A process X on the manifold M is a semimartingale if f(X) is a semimartingale for every smooth function f from M to R. Stochastic calculus for semimartingales on general manifolds requires the use of the Stratonovich integral.

Gemination, the doubling of a consonantal sound, is meaningful in Tigrinya, i.e. it affects the meaning of words. While gemination plays an important role in the morphology of the Tigrinya verb, it is normally accompanied by other marks. But there is a small number of pairs of words which are only differentiable from each other by gemination, e.g.

Logarithmic differentiation relies on the chain rule as well as properties of logarithms (in particular, the natural logarithm, or the logarithm to the base e) to transform products into sums and divisions into subtractions. The principle can be implemented, at least in part, in the differentiation of almost all differentiable functions, providing that these functions are non-zero.

Suppose that is a complex-valued function which is differentiable as a function . Then Goursat's theorem asserts that f is analytic in an open complex domain Ω if and only if it satisfies the Cauchy–Riemann equation in the domain . In particular, continuous differentiability of f need not be assumed . The hypotheses of Goursat's theorem can be weakened significantly.

Symplectic geometry is the study of symplectic manifolds. An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate skew-symmetric bilinear form on each tangent space, i.e., a nondegenerate 2-form ω, called the symplectic form. A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed: .

Historically, body schema and body image were generally lumped together, used interchangeably, or ill-defined. In science and elsewhere, the two terms are still commonly misattributed or confused. Efforts have been made to distinguish the two and define them in clear and differentiable ways. A body image consists of perceptions, attitudes, and beliefs concerning one's body.

In convex analysis and the calculus of variations, branches of mathematics, a pseudoconvex function is a function that behaves like a convex function with respect to finding its local minima, but need not actually be convex. Informally, a differentiable function is pseudoconvex if it is increasing in any direction where it has a positive directional derivative.

Let M be a differentiable manifold that is second-countable and Hausdorff. The diffeomorphism group of M is the group of all Cr diffeomorphisms of M to itself, denoted by Diffr(M) or, when r is understood, Diff(M). This is a "large" group, in the sense that—provided M is not zero-dimensional—it is not locally compact.

In differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold. The structure of a projective connection is modeled on the geometry of projective space, rather than the affine space corresponding to an affine connection. Much like affine connections, projective connections also define geodesics. However, these geodesics are not affinely parametrized.

See Zariski tangent space. Once the tangent spaces of a manifold have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving in space. A vector field attaches to every point of the manifold a vector from the tangent space at that point, in a smooth manner. Such a vector field serves to define a generalized ordinary differential equation on a manifold: A solution to such a differential equation is a differentiable curve on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field. All the tangent spaces of a manifold may be ‘glued together’ to form a new differentiable manifold with twice the dimension of the original manifold, called the tangent bundle of the manifold.

A manifold is a generalization of the concepts of curve and surface. In topology, a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. In differential geometry, a differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory.

There is one requirement for this to be a functor, namely that the derivative of a composite must be the composite of the derivatives. This is exactly the formula . There are also chain rules in stochastic calculus. One of these, Itō's lemma, expresses the composite of an Itō process (or more generally a semimartingale) dXt with a twice-differentiable function f.

The main difficulty lies in showing that a distance- preserving map, which is a priori only continuous, is actually differentiable. The second theorem, which is much more difficult to prove, states that the isometry group of a Riemannian manifold is a Lie group. For instance, the group of isometries of the two-dimensional unit sphere is the orthogonal group O(3).

It can also be shown that the eigenfunctions are infinitely differentiable functions. More generally, these results hold for the Laplace–Beltrami operator on any compact Riemannian manifold with boundary, or indeed for the Dirichlet eigenvalue problem of any elliptic operator with smooth coefficients on a bounded domain. When is the -sphere, the eigenfunctions of the Laplacian are the spherical harmonics.

Spike based activation of SNNs is not differentiable thus making it hard to develop gradient descent based training methods to perform error backpropagation, though a few recent algorithms such as NormAD and multilayer NormAD have demonstrated good training performance through suitable approximation of the gradient of spike based activation. SNNs have much larger computational costs for simulating realistic neural models than traditional ANNs.

Schlömilch's series is a Fourier series type expansion of twice continuously differentiable function in the interval (0,\pi) in terms of the Bessel function of the first kind, named after the German mathematician Oskar Schlömilch, who derived the series in 1857Schlomilch, G. (1857). On Bessel's function. Zeitschrift fur Math, and Pkys., 2, 155-158.Whittaker, E. T., & Watson, G. N. (1996).

Mishra specialised in differential geometry, relativity and fluid mechanics and his contributions to these fields have been documented. He was known to have elucidated the complete solutions to the unified field theory of Albert Einstein. He also added to the index-free notations and developed his own notations in differential geometry. He also wrote structures for Differentiable manifolds and Almost Contact Metric Manifolds.

Like all deterministic global optimization software, ANTIGONE requires the user to provide the explicit mathematical expressions for all the functions used in the problem, as well as initial bounds for all variables. If initial bounds are not supplied, ANTIGONE will attempt to infer bounds, but global optimality is not guaranteed. ANTIGONE can only solve differentiable functions, and can not solve trigonometric problems.

Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds and Donaldson–Thomas theory. He is currently a permanent member of the Simons Center for Geometry and Physics at Stony Brook University in New York, and a Professor in Pure Mathematics at Imperial College London.

Later, after learning of a result by Stefan Banach and Stefan Mazurkiewicz that generic functions (that is, the members of a residual set of functions) are nowhere differentiable, Jarník proved that at almost all points, all four Dini derivatives of such a function are infinite. Much of his later work in this area concerned extensions of these results to approximate derivatives..

In mathematics, an exotic \R^4 is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space \R^4. The first examples were found in 1982 by Michael Freedman and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds.Kirby (1989), p. 95Freedman and Quinn (1990), p.

The one-form dθ (defined on the complement of the origin) is closed but not exact, and it generates the first de Rham cohomology group of the punctured plane. In particular, if ω is any closed differentiable one-form defined on the complement of the origin, then the integral of ω along closed loops gives a multiple of the winding number.

Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions R → R, or the space of all distributions on R, are complete but are not normed vector spaces and hence not Banach spaces. In Fréchet spaces one still has a complete metric, while LF-spaces are complete uniform vector spaces arising as limits of Fréchet spaces.

Root-finding algorithms are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is differentiable and the derivative is known, then Newton's method is a popular choice.Ezquerro Fernández, J. A., & Hernández Verón, M. Á. (2017). Newton’s method: An updated approach of Kantorovich’s theory. Birkhäuser.

Various forms of the implicit function theorem exist for the case when the function f is not differentiable. It is standard that local strict monotonicity suffices in one dimension. The following more general form was proven by Kumagai based on an observation by Jittorntrum. Consider a continuous function f : R^n \times R^m \to R^n such that f(x_0, y_0) = 0.

In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the form of physical laws under arbitrary differentiable coordinate transformations. The essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws.

Inflection points sufficient conditions: 1) A sufficient existence condition for a point of inflection is: :If is times continuously differentiable in a certain neighborhood of a point with odd and , while for and then has a point of inflection at . 2) Another sufficient existence condition requires and to have opposite signs in the neighborhood of x (Bronshtein and Semendyayev 2004, p. 231).

Under the viscosity solution concept, u does not need to be everywhere differentiable. There may be points where either Du or D^2 u does not exist and yet u satisfies the equation in an appropriate generalized sense. The definition allows only for certain kind of singularities, so that existence, uniqueness, and stability under uniform limits, hold for a large class of equations.

Topological spaces are of analytic nature. Open sets, given in a topological space by definition, lead to such notions as continuous functions, paths, maps; convergent sequences, limits; interior, boundary, exterior. However, uniform continuity, bounded sets, Cauchy sequences, differentiable functions (paths, maps) remain undefined. Isomorphisms between topological spaces are traditionally called homeomorphisms; these are one-to-one correspondences continuous in both directions.

A vector manifold is a special set of vectors in the UGA.Chapter 1 of: [D. Hestenes & G. Sobczyk] From Clifford Algebra to Geometric Calculus These vectors generate a set of linear spaces tangent to the vector manifold. Vector manifolds were introduced to do calculus on manifolds so one can define (differentiable) manifolds as a set isomorphic to a vector manifold.

There are various ways to define the derivative of a function on a differentiable manifold, the most fundamental of which is the directional derivative. The definition of the directional derivative is complicated by the fact that a manifold will lack a suitable affine structure with which to define vectors. Therefore, the directional derivative looks at curves in the manifold instead of vectors.

Sylvester's formula applies for any diagonalizable matrix with distinct eigenvalues, 1, …, λk, and any function defined on some subset of the complex numbers such that is well defined. The last condition means that every eigenvalue is in the domain of , and that every eigenvalue with multiplicity i > 1 is in the interior of the domain, with being () times differentiable at .

LSE is convex but not strictly convex. We can define a strictly convex log-sum-exp type function by adding an extra argument set to zero: :LSE_0^+(x_1,...,x_n) = LSE(0,x_1,...,x_n) This function is a proper Bregman generator (strictly convex and differentiable). It is encountered in machine learning, for example, as the cumulant of the multinomial/binomial family.

More precisely, suppose that the point is moving on the curve at a constant speed of one unit, that is, the position of the point is a function of the parameter , which may be thought as the time or as the arc length from a given origin. Let be a unit tangent vector of the curve at , which is also the derivative of with respect to . Then, the derivative of with respect to is a vector that is normal to the curve and whose length is the curvature. For being meaningful, the definition of the curvature and its different characterizations require that the curve is continuously differentiable near , for having a tangent that varies continuously; it requires also that the curve is twice differentiable at , for insuring the existence of the involved limits, and of the derivative of .

For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero.

Shub obtained his Ph.D. degree at the University of California, Berkeley with a thesis entitled Endomorphisms of Compact Differentiable Manifolds on 1967. His advisor was Stephen Smale. From 1967 to 1985 he worked at Brandeis University, the University of California, Santa Cruz and the Queens College at the City University of New York. From 1985 to 2004 he joined IBM's Thomas J. Watson Research Center.

Cesàro's main contributions are in the field of differential geometry. Lessons of intrinsic geometry, written in 1894, explains in particular the construction of a fractal curve. After that, Cesàro also studied the "snowflake curve" of Koch, continuous but not differentiable in all its points. Among his other works are Introduction to the mathematical theory of infinitesimal calculus (1893), Algebraic analysis (1894), Elements of infinitesimal calculus (1897).

The game was closed the following July. The regular game took place in a dark fantasy world called Aerynth (the world will sometimes depend on the servers, many of which have unique world maps). Gameplay features many aspects typical of role-playing video games, such as experience points, character classes, and fantasy races. Character creation was fairly extensive, allowing for detailed, differentiable characters to be created.

More generally, for any sufficiently differentiable functions f and g : S(f \circ g) = \left( S(f)\circ g\right ) \cdot(g')^2+S(g). This makes the Schwarzian derivative an important tool in one-dimensional dynamics Weisstein, Eric W. "Schwarzian Derivative." From MathWorld—A Wolfram Web Resource. since it implies that all iterates of a function with negative Schwarzian will also have negative Schwarzian.

MPS is used for multidimensional real-valued functions but does not use the gradient of the problem being optimized, which means MPS does not require for the optimization problem to be differentiable as is required by classic optimization methods such as gradient descent and quasi-newton methods. MPS can therefore also be used on optimization problems that are not even continuous, are noisy, change over time, etc.

If a real-valued function is continuous on a proper closed interval , differentiable on the open interval , and , then there exists at least one in the open interval such that :f'(c) = 0. This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of Taylor's theorem.

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the metric induced by the norm). All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.

The principle expresses the idea that natural things and properties change gradually, rather than suddenly. In a mathematical context, this allows one to assume that the solutions of the governing equations are continuous, and also does not preclude their being differentiable (differentiability implies continuity). Modern day quantum mechanics is sometimes seen as violating the principle, with its idea of a quantum leap.Marxists.org and Arizona.

Let be a differentiable function, and let be its derivative. The derivative of (if it has one) is written and is called the second derivative of . Similarly, the derivative of the second derivative, if it exists, is written and is called the third derivative of . Continuing this process, one can define, if it exists, the th derivative as the derivative of the th derivative.

In non- differentiable terms, the law of supply can be expressed as: :(p - p')(y-y') \geq 0 where y is the amount that would be supplied at some price p, and y' is the amount that would be supplied at some other price p' . Thus for example if p > p' then y > y' .Mas-Colell, d., lucrezi, M. Green, J.: Principles of Microeconomics.

The equations are one way of looking at the condition on a function to be differentiable in the sense of complex analysis: in other words they encapsulate the notion of function of a complex variable by means of conventional differential calculus. In the theory there are several other major ways of looking at this notion, and the translation of the condition into other language is often needed.

The problem of finding the local maxima and minima subject to constraints can be generalized to finding local maxima and minima on a differentiable manifold M. In what follows, it is not necessary that M be a Euclidean space, or even a Riemannian manifold. All appearances of the gradient abla (which depends on a choice of Riemannian metric) can be replaced with the exterior derivative d.

Differentiable inverse rendering caustic design At first, the target pattern is designed and compute the forward pass to get the synthetic pattern. It's compared to the target pattern and get the loss. The objection is to let the synthetic pattern is similar to the target pattern as much as possible. And then do the back propagation to get the optimized properties need to use in caustic manufacturing.

One can also understand the differentiability properties of the original function f in terms of the asymptotics of Sff(ℓ). In particular, if Sff(ℓ) decays faster than any rational function of ℓ as ℓ → ∞, then f is infinitely differentiable. If, furthermore, Sff(ℓ) decays exponentially, then f is actually real analytic on the sphere. The general technique is to use the theory of Sobolev spaces.

In practice the equations for electromagnetic and strong interactions are invariant, while the weak interaction is not invariant under the parity transformation. For example, the Maxwell equation is invariant, while the corresponding equation for the weak field explicitly contains left currents and thus is not invariant under the parity transformation. In general relativity the covariance group consists of all arbitrary (invertible and differentiable) coordinate transformations.

A Gaussian quadrature rule is typically more accurate than a Newton–Cotes rule, which requires the same number of function evaluations, if the integrand is smooth (i.e., if it is sufficiently differentiable). Other quadrature methods with varying intervals include Clenshaw–Curtis quadrature (also called Fejér quadrature) methods, which do nest. Gaussian quadrature rules do not nest, but the related Gauss–Kronrod quadrature formulas do.

Sigma algebras are a special case of a topology, and so thereby allow notions such as continuous and differentiable functions to be defined. These are the basic ingredients to a dynamical system: a phase space, a topology (sigma algebra) on that space, a measure, and an invertible function providing the time evolution. Conservative systems are those systems that do not shrink their phase space over time.

In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real division algebras: R, C, the quaternions H and the octonions O. The cotangent bundle of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the cotangent space. Sections of that bundle are known as differential one-forms.

For physical reasons, it is usually assumed that reactant concentrations cannot be negative, and that each reaction only takes place if all its reactants are present, i.e. all have non-zero concentration. For mathematical reasons, it is usually assumed that V(x) is continuously differentiable. It is also commonly assumed that no reaction features the same chemical as both a reactant and a product (i.e.

In Riemannian geometry and relativity theory, an orthonormal frame is a tool for studying the structure of a differentiable manifold equipped with a metric. If M is a manifold equipped with a metric g, then an orthonormal frame at a point P of M is an ordered basis of the tangent space at P consisting of vectors which are orthonormal with respect to the bilinear form gP..

In mathematics, the Markus–Yamabe conjecture is a conjecture on global asymptotic stability. The conjecture states that if a continuously differentiable map on an n-dimensional real vector space has a fixed point, and its Jacobian matrix is everywhere Hurwitz, then the fixed point is globally stable. The conjecture is true for the two-dimensional case. However, counterexamples have been constructed in higher dimensions.

For a sample set, the maximum function is non-smooth and thus non-differentiable. For optimization problems that occur in statistics it often needs to be approximated by a smooth function that is close to the maximum of the set. A smooth maximum, for example, : g(x1, x2, …, xn) = log( exp(x1) + exp(x2) + … + exp(xn) ) is a good approximation of the sample maximum.

Libgober's early work studies the diffeomorphism type of complete intersections in complex projective space. This later led to the discovery of relations between Hodge and Chern numbers.A.Libgober, J.Wood, Differentiable structures on complete intersections I, Topology, 21 (1982),469-482 He introduced the technique of Alexander polynomialA.Libgober,Development of the theory of Alexander invariants in algebraic geometry, Topology of algebraic varieties and singularities, 3–17, Contemp. Math.

In stochastic calculus, the Doléans-Dade exponential, Doléans exponential, or stochastic exponential, of a semimartingale X is defined to be the solution to the stochastic differential equation with initial condition . The concept is named after Catherine Doléans-Dade. It is sometimes denoted by Ɛ(X). In the case where X is differentiable, then Y is given by the differential equation to which the solution is .

The dual space of a vector space is the set of real valued linear functions on the vector space. The cotangent space at a point is the dual of the tangent space at that point, and the cotangent bundle is the collection of all cotangent spaces. Like the tangent bundle, the cotangent bundle is again a differentiable manifold. The Hamiltonian is a scalar on the cotangent bundle.

This implicit equation defines f as a function of x only if -1 \leq x \leq 1 and one considers only non- negative (or non-positive) values for the values of the function. The implicit function theorem provides conditions under which some kinds of relations define an implicit function, namely relations defined as the indicator function of the zero set of some continuously differentiable multivariate function.

Plot of the data with spline interpolation applied Remember that linear interpolation uses a linear function for each of intervals [xk,xk+1]. Spline interpolation uses low-degree polynomials in each of the intervals, and chooses the polynomial pieces such that they fit smoothly together. The resulting function is called a spline. For instance, the natural cubic spline is piecewise cubic and twice continuously differentiable.

This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension. Analytically, fractals are usually nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is still 1-dimensional, its fractal dimension indicates that it also resembles a surface. Sierpinski carpet (to level 6), a fractal with a topological dimension of 1 and a Hausdorff dimension of 1.893 Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass, and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century.

An exotic R4 is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space R4. The first examples were found in the early 1980s by Michael Freedman, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds.. There is a continuum of non-diffeomorphic differentiable structures of R4, as was shown first by Clifford Taubes.Theorem 1.1 of Prior to this construction, non-diffeomorphic smooth structures on spheres—exotic spheres—were already known to exist, although the question of the existence of such structures for the particular case of the 4-sphere remained open (and still remains open as of 2018). For any positive integer n other than 4, there are no exotic smooth structures on Rn; in other words, if n ≠ 4 then any smooth manifold homeomorphic to Rn is diffeomorphic to Rn.Corollary 5.2 of .

If g : I → R is a Lebesgue- integrable function on some interval I = [a,b], and if :f(x) = \int_a^x g(t)\,dt is its Lebesgue indefinite integral, then the following assertions are true: #f is absolutely continuous (see below) #f is differentiable almost everywhere #Its derivative coincides almost everywhere with g(x). (In fact, all absolutely continuous functions are obtained in this manner.) The Lebesgue integral could be defined as follows: g is Lebesgue-integrable on I iff there exists a function f that is absolutely continuous whose derivative coincides with g almost everywhere. However, even if f : I → R is differentiable everywhere, and g is its derivative, it does not follow that f is (up to a constant) the Lebesgue indefinite integral of g, simply because g can fail to be Lebesgue-integrable, i.e., f can fail to be absolutely continuous.

Every differentiable manifold has a cotangent bundle. That bundle can always be endowed with a certain differential form, called the canonical one-form. This form gives the cotangent bundle the structure of a symplectic manifold, and allows vector fields on the manifold to be integrated by means of the Euler-Lagrange equations, or by means of Hamiltonian mechanics. Such systems of integrable differential equations are called integrable systems.

Because the above expression is equal to the difference , by the definition of the derivative is differentiable at a and its derivative is The role of Q in the first proof is played by η in this proof. They are related by the equation: :Q(y) = f'(g(a)) + \eta(y - g(a)). The need to define Q at g(a) is analogous to the need to define η at zero.

As noted above, any analytic function (real or complex) is infinitely differentiable (also known as smooth, or C∞). (Note that this differentiability is in the sense of real variables; compare complex derivatives below.) There exist smooth real functions that are not analytic: see non-analytic smooth function. In fact there are many such functions. The situation is quite different when one considers complex analytic functions and complex derivatives.

If M is a supermanifold of dimension (p,q), then the underlying space M inherits the structure of a differentiable manifold whose sheaf of smooth functions is OM/I, where I is the ideal generated by all odd functions. Thus M is called the underlying space, or the body, of M. The quotient map OM → OM/I corresponds to an injective map M → M; thus M is a submanifold of M.

GeNMR also makes use of genetic algorithms to allow configurational sampling and structural refinement using non-differentiable scores, such as ShiftX chemical shift scores. GeNMR's genetic algorithm creates a population of initial structures and then uses combinations of mutations, cross-overs, segment swaps and writhe movements to comprehensively sample conformation space. The 25 lowest energy structures are then selected, duplicated and carried to the next round of conformational sampling.

However, no such allocation is PE, since it is Pareto-dominated by the allocation [(4,0);(0,2)] whose utility vector is (4,2). Non-existence remains even if we weaken envy-freeness to no domination -- no agent gets more of each good than another agent. Proposition 8 (Maniquet): There exist 2-good 3-agent division economies with strictly monotonic, continuous and even differentiable preferences, where there is domination at every Pareto efficient allocation.

A recursive neural network is created by applying the same set of weights recursively over a differentiable graph-like structure by traversing the structure in topological order. Such networks are typically also trained by the reverse mode of automatic differentiation. They can process distributed representations of structure, such as logical terms. A special case of recursive neural networks is the RNN whose structure corresponds to a linear chain.

In geometric topology, obstruction theory is concerned with when a topological manifold has a piecewise linear structure, and when a piecewise linear manifold has a differential structure. In dimension at most 2 (Rado), and 3 (Morse), the notions of topological manifolds and piecewise linear manifolds coincide. In dimension 4 they are not the same. In dimensions at most 6 the notions of piecewise linear manifolds and differentiable manifolds coincide.

Pompeiu's contributions were mainly in the field of mathematical analysis, complex functions theory, and rational mechanics. In an article published in 1929, he posed a challenging conjecture in integral geometry, now widely known as the Pompeiu problem. Among his contributions to real analysis there is the construction, dated 1906, of non-constant, everywhere differentiable functions, with derivative vanishing on a dense set. Such derivatives are now called Pompeiu derivatives.

As of 2011 Remy Denis, president of the All India Catholic Union, was a professor in the Department of Mathematics. Research is being done in the fields of Theory of Relativity, Differential Geometry, Fluid Dynamics, Special Functions, Summability Theory, Functional Analysis, Differentiable Manifolds, Number Theory, and Graph Theory and in Statistics, Bayesian inference, Life Testing, Reliability Theory Demography, etc. The Department of Mathematics and Statistics has made great strides.

To have an envelope, it is necessary that the individual members of the family of curves are differentiable curves as the concept of tangency does otherwise not apply, and there has to be a smooth transition proceeding through the members. But these conditions are not sufficient – a given family may fail to have an envelope. A simple example of this is given by a family of concentric circles of expanding radius.

Finsler geometry has Finsler manifolds as the main object of study. This is a differential manifold with a Finsler metric, that is, a Banach norm defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold is a function such that: # for all in and all , # is infinitely differentiable in }, # The vertical Hessian of is positive definite.

"Exceptional" object is reserved for objects that are unusual, meaning rare, the exception, not for unexpected or non-standard objects. These unexpected-but-typical (or common) phenomena are generally referred to as pathological, such as nowhere differentiable functions, or "exotic", as in exotic spheres — there are exotic spheres in arbitrarily high dimension (not only a finite set of exceptions), and in many dimensions most (differential structures on) spheres are exotic.

Given a manifold in three dimensions that is smooth and differentiable over a patch containing the point p, where k and m are defined as the principal curvatures and K(x) is the Gaussian curvature at a point x, if k has a max at p, m has a min at p, and k is strictly greater than m at p, then K(p) is a non-positive real number..

The smoothness requirements on the transition functions can be weakened, so that we only require the transition maps to be k-times continuously differentiable; or strengthened, so that we require the transition maps to be real-analytic. Accordingly, this gives a C^k or (real-)analytic structure on the manifold rather than a smooth one. Similarly, we can define a complex structure by requiring the transition maps to be holomorphic.

By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions fn with desirable properties that approximates a given limit function, is equally crucial. Early analysis, in the guise of the Taylor approximation, established an approximation of differentiable functions f by polynomials. By the Stone–Weierstrass theorem, every continuous function on can be approximated as closely as desired by a polynomial.

It is, however, different from the cylinder , because the latter is orientable whereas the former is not. Properties of certain vector bundles provide information about the underlying topological space. For example, the tangent bundle consists of the collection of tangent spaces parametrized by the points of a differentiable manifold. The tangent bundle of the circle S1 is globally isomorphic to , since there is a global nonzero vector field on S1.

The tensor bundle is the direct sum of all tensor products of the tangent bundle and the cotangent bundle. Each element of the bundle is a tensor field, which can act as a multilinear operator on vector fields, or on other tensor fields. The tensor bundle is not a differentiable manifold in the traditional sense, since it is infinite dimensional. It is however an algebra over the ring of scalar functions.

The curve must be continuous (no jump) between the two ends. The sinuosity value is really significant when the line is continuously differentiable (no angular point). The distance between both ends can also be evaluated by a plurality of segments according to a broken line passing through the successive inflection points (sinuosity of order 2). The calculation of the sinuosity is valid in a 3-dimensional space (e.g.

A manifold is a topological space that near each point resembles Euclidean space. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot be realized in three dimensions, but can be realized in four dimensions. Typically, results in algebraic topology focus on global, non- differentiable aspects of manifolds; for example Poincaré duality.

Calculus on Manifolds is a brief monograph on the theory of vector-valued functions of several real variables (f : Rn→Rm) and differentiable manifolds in Euclidean space. In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and Riemann integration (including Fubini's theorem) to functions of several variables, the book treats the classical theorems of vector calculus, including those of Cauchy–Green, Ostrogradsky–Gauss (divergence theorem), and Kelvin–Stokes, in the language of differential forms on differentiable manifolds embedded in Euclidean space, and as corollaries of the generalized Stokes' theorem on manifolds-with-boundary. The book culminates with the statement and proof of this vast and abstract modern generalization of several classical results: The cover of Calculus on Manifolds features snippets of a July 2, 1850 letter from Lord Kelvin to Sir George Stokes containing the first disclosure of the classical Stokes' theorem (i.e., the Kelvin–Stokes theorem).

This symmetry is one of the defining features of the theory. However, it is a common misunderstanding that "diffeomorphism invariance" refers to the invariance of the physical predictions of a theory under arbitrary coordinate transformations; this is untrue and in fact every physical theory is invariant under coordinate transformations this way. Diffeomorphisms, as mathematicians define them, correspond to something much more radical; intuitively a way they can be envisaged is as simultaneously dragging all the physical fields (including the gravitational field) over the bare differentiable manifold while staying in the same coordinate system. Diffeomorphisms are the true symmetry transformations of general relativity, and come about from the assertion that the formulation of the theory is based on a bare differentiable manifold, but not on any prior geometry — the theory is background-independent (this is a profound shift, as all physical theories before general relativity had as part of their formulation a prior geometry).

This would prove to be a major area of research in the first half of the 20th century. The 19th century saw great advances in the theory of real analysis, including theories of convergence of functions and Fourier series. Mathematicians such as Karl Weierstrass began to construct functions that stretched intuition, such as nowhere-differentiable continuous functions. Previous conceptions of a function as a rule for computation, or a smooth graph, were no longer adequate.

In the 19th century, mathematicians became aware of logical gaps and inconsistencies in their field. It was shown that Euclid's axioms for geometry, which had been taught for centuries as an example of the axiomatic method, were incomplete. The use of infinitesimals, and the very definition of function, came into question in analysis, as pathological examples such as Weierstrass' nowhere- differentiable continuous function were discovered. Cantor's study of arbitrary infinite sets also drew criticism.

One can also think of adapting this parametrization during the optimization. Should the objective function be based on a norm other than the Euclidean norm, we have to leave the area of quadratic optimization. As a result, the optimization problem becomes more difficult. In particular, when the L^1 norm is used for quantifying the data misfit the objective function is no longer differentiable: its gradient does not make sense any longer.

In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let f : D → D′ be an orientation-preserving homeomorphism between open sets in the plane. If f is continuously differentiable, then it is K-quasiconformal if the derivative of f at every point maps circles to ellipses with eccentricity bounded by K.

Microeconomic theory typically begins with the study of a single rational and utility maximizing individual. To economists, rationality means an individual possesses stable preferences that are both complete and transitive. The technical assumption that preference relations are continuous is needed to ensure the existence of a utility function. Although microeconomic theory can continue without this assumption, it would make comparative statics impossible since there is no guarantee that the resulting utility function would be differentiable.

Let I be a closed interval, f\colon I\to \R a real-valued differentiable function. Then f' has the intermediate value property: If a and b are points in I with a, then for every y between f'(a) and f'(b), there exists an x in (a,b) such that f'(x)=y.Apostol, Tom M.: Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd edition, Addison-Wesley Longman, Inc. (1974), page 112.

The term "structural stability" is due to Solomon Lefschetz, who oversaw translation of their monograph into English. Ideas of structural stability were taken up by Stephen Smale and his school in the 1960s in the context of hyperbolic dynamics. Earlier, Marston Morse and Hassler Whitney initiated and René Thom developed a parallel theory of stability for differentiable maps, which forms a key part of singularity theory. Thom envisaged applications of this theory to biological systems.

Ehresmann first investigated the topology and homology of manifolds associated with classical Lie groups, such as Grassmann manifolds and other homogeneous spaces. He developed the concept of fiber bundle, building on work by Herbert Seifert and Hassler Whitney. Norman Steenrod was working in the same direction in the USA, but Ehresmann was particularly interested in differentiable (smooth) fiber bundles, and in differential-geometric aspects of these. He was a pioneer of differential topology.

In any given dimension, there are only two possibilities: either all manifolds have Arf–Kervaire invariant equal to 0, or half have Arf–Kervaire invariant 0 and the other half have Arf–Kervaire invariant 1. The Kervaire invariant problem is the problem of determining in which dimensions the Kervaire invariant can be nonzero. For differentiable manifolds, this can happen in dimensions 2, 6, 14, 30, 62, and possibly 126, and in no other dimensions.

The term "cytokine storm" is often loosely used interchangeably with cytokine release syndrome (CRS) but is more precisely a differentiable syndrome that may represent a severe episode of cytokine release syndrome or a component of another disease entity, such as macrophage activation syndrome. When occurring as a result of a therapy, CRS symptoms may be delayed until days or weeks after treatment. Immediate-onset (fulminant) CRS appears to be a cytokine storm.

The cutting-plane method is an umbrella term for optimization methods which iteratively refine a feasible set or objective function by means of linear inequalities, termed cuts. Such procedures are popularly used to find integer solutions to mixed integer linear programming (MILP) problems, as well as to solve general, not necessarily differentiable convex optimization problems. The use of cutting planes to solve MILP was introduced by Ralph E. Gomory and Václav Chvátal.

TPS has been widely used as the non-rigid transformation model in image alignment and shape matching. An additional application is the analysis and comparisons of archaeological findings in 3D and was implemented for triangular meshes in the GigaMesh Software Framework. The thin plate spline has a number of properties which have contributed to its popularity: #It produces smooth surfaces, which are infinitely differentiable. #There are no free parameters that need manual tuning.

The notion of a continuously differentiable function on a family of level sets can be made rigorous by means of the implicit function theorem. The level sets corresponding to the maximal independent solution sets of (1) are called the integral manifolds because functions on the collection of all integral manifolds correspond in some sense to constants of integration. Once one of these constants of integration is known, then the corresponding solution is also known.

The Greek philosopher Pyrrho traveled to India as part of Alexander the Great's entourage where he was influenced by the Indian gymnosophists, which inspired him to create the philosophy of Pyrrhonism. Philologist Christopher Beckwith has demonstrated that Pyrrho based his philosophy on his translation of the three marks of existence into Greek, and that adiaphora (not logically differentiable, not clearly definable, negating Aristotle's use of "diaphora") reflects Pyrrho's understanding of the Buddhist concept of anatta.

This function does not have a derivative at the marked point, as the function is not continuous there (specifically, it has a jump discontinuity). If is differentiable at , then must also be continuous at . As an example, choose a point and let be the step function that returns the value 1 for all less than , and returns a different value 10 for all greater than or equal to . cannot have a derivative at .

At about the same time as Hironaka's work, the catastrophe theory of René Thom was receiving a great deal of attention. This is another branch of singularity theory, based on earlier work of Hassler Whitney on critical points. Roughly speaking, a critical point of a smooth function is where the level set develops a singular point in the geometric sense. This theory deals with differentiable functions in general, rather than just polynomials.

The Newton–Fourier method is Joseph Fourier's extension of Newton's method to provide bounds on the absolute error of the root approximation, while still providing quadratic convergence. Assume that is twice continuously differentiable on and that contains a root in this interval. Assume that on this interval (this is the case for instance if , , and , and on this interval). This guarantees that there is a unique root on this interval, call it .

Michel André Kervaire (26 April 1927 – 19 November 2007) was a French mathematician who made significant contributions to topology and algebra. He introduced the Kervaire semi-characteristic. He was the first to show the existence of topological n-manifolds with no differentiable structure (using the Kervaire invariant), and (with John Milnor) computed the number of exotic spheres in dimensions greater than four. He is also well known for fundamental contributions to high-dimensional knot theory.

It gives a rigorous foundation of infinitesimal calculus based on the set of real numbers, arguably resolving the Zeno paradoxes and Berkeley's arguments. Mathematicians such as Karl Weierstrass (1815–1897) discovered pathological functions such as continuous, nowhere-differentiable functions. Previous conceptions of a function as a rule for computation, or a smooth graph, were no longer adequate. Weierstrass began to advocate the arithmetization of analysis, to axiomatize analysis using properties of the natural numbers.

Due to , : :For any abstract elliptic operator on a closed, oriented, topological manifold, the analytical index equals the topological index. The proof of this result goes through specific considerations, including the extension of Hodge theory on combinatorial and Lipschitz manifolds , , the extension of Atiyah–Singer's signature operator to Lipschitz manifolds , Kasparov's K-homology and topological cobordism . This result shows that the index theorem is not merely a differentiable statement, but rather a topological statement.

Inflection points in differential geometry are the points of the curve where the curvature changes its sign. For example, the graph of the differentiable function has an inflection point at if and only if its first derivative, , has an isolated extremum at . (This is not the same as saying that has an extremum). That is, in some neighborhood, is the one and only point at which has a (local) minimum or maximum.

Consequently, :\sin^2 x + \cos^2 x = 1 \ , which is the Pythagorean trigonometric identity. When the trigonometric functions are defined in this way, the identity in combination with the Pythagorean theorem shows that these power series parameterize the unit circle, which we used in the previous section. This definition constructs the sine and cosine functions in a rigorous fashion and proves that they are differentiable, so that in fact it subsumes the previous two.

The transfer functions usually have a sigmoid shape, but they may also take the form of other non-linear functions, piecewise linear functions, or step functions. They are also often monotonically increasing, continuous, differentiable and bounded. The thresholding function has inspired building logic gates referred to as threshold logic; applicable to building logic circuits resembling brain processing. For example, new devices such as memristors have been extensively used to develop such logic in recent times.

A Van Hove singularity is a singularity (non-smooth point) in the density of states (DOS) of a crystalline solid. The wavevectors at which Van Hove singularities occur are often referred to as critical points of the Brillouin zone. For three-dimensional crystals, they take the form of kinks (where the density of states is not differentiable). The most common application of the Van Hove singularity concept comes in the analysis of optical absorption spectra.

Brownian motion or the Wiener process was discovered to be exceptionally complex mathematically. The Wiener process is almost surely nowhere differentiable; thus, it requires its own rules of calculus. There are two dominating versions of stochastic calculus, the Itô stochastic calculus and the Stratonovich stochastic calculus. Each of the two has advantages and disadvantages, and newcomers are often confused whether the one is more appropriate than the other in a given situation.

A vector manifold is always a subset of Universal Geometric Algebra by definition and the elements can be manipulated algebraically. In contrast, a manifold is not a subset of any set other than itself, but the elements have no algebraic relation among them. The differential geometry of a manifold can be carried out in a vector manifold. All quantities relevant to differential geometry can be calculated from if it is a differentiable function.

In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. All manifolds are topological manifolds by definition, but many manifolds may be equipped with additional structure (e.g. differentiable manifolds are topological manifolds equipped with a differential structure).

An NTM has a neural network controller coupled to external memory resources, which it interacts with through attentional mechanisms. The memory interactions are differentiable end-to-end, making it possible to optimize them using gradient descent. An NTM with a long short- term memory (LSTM) network controller can infer simple algorithms such as copying, sorting, and associative recall from examples alone. The authors of the original NTM paper did not publish their source code.

The drug was coded as CP-690,550 during development. Its original recommended INN (rINN) was tasocitinib, but that was overruled during the INN approval process as being not optimally differentiable from other existing INNs, so the name "tofacitinib" was proposed and became the INN. In November 2012, the FDA approved tofacitinib for treatment of rheumatoid arthritis. Two rheumatologists interviewed by the magazine Nature Biotechnology complained that they were "shocked" and "disappointed" at the $2,055 a month wholesale price.

In mathematics, Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the differentiation of another function has the intermediate value property: the image of an interval is also an interval. When ƒ is continuously differentiable (ƒ in C1([a,b])), this is a consequence of the intermediate value theorem. But even when ƒ′ is not continuous, Darboux's theorem places a severe restriction on what it can be.

A level surface, or isosurface, is the set of all points where some function has a given value. If is differentiable, then the dot product of the gradient at a point with a vector gives the directional derivative of at in the direction . It follows that in this case the gradient of is orthogonal to the level sets of . For example, a level surface in three-dimensional space is defined by an equation of the form .

In complex analysis (a branch of mathematics), zeros of holomorphic functions—which are points where —play an important role. For meromorphic functions, particularly, there is a duality between zeros and poles. A function of a complex variable is meromorphic in the neighbourhood of a point if either or its reciprocal function is holomorphic in some neighbourhood of (that is, if or is differentiable in a neighbourhood of ). If is a zero of , then it is a pole of .

In the situation of the chain rule, such a function ε exists because g is assumed to be differentiable at a. Again by assumption, a similar function also exists for f at g(a). Calling this function η, we have :f(g(a) + k) - f(g(a)) = f'(g(a)) k + \eta(k) k. The above definition imposes no constraints on η(0), even though it is assumed that η(k) tends to zero as k tends to zero.

In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant. The directional derivative is a special case of the Gateaux derivative.

A function f defined on some subset of the real line is said to be real analytic at a point x if there is a neighborhood D of x on which f is real analytic. The definition of a complex analytic function is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complex plane". A function is complex analytic if and only if it is holomorphic i.e. it is complex differentiable.

This is a natural inverse of the linear approximation to tetration. Authors like Holmes recognize that the super- logarithm would be a great use to the next evolution of computer floating- point arithmetic, but for this purpose, the function need not be infinitely differentiable. Thus, for the purpose of representing large numbers, the linear approximation approach provides enough continuity (C^0 continuity) to ensure that all real numbers can be represented on a super-logarithmic scale.

In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold which is a homotopy sphere a sphere. More precisely, one fixes a category of manifolds: topological (Top), piecewise linear (PL), or differentiable (Diff). Then the statement is :Every homotopy sphere (a closed n-manifold which is homotopy equivalent to the n-sphere) in the chosen category (i.e. topological manifolds, PL manifolds, or smooth manifolds) is isomorphic in the chosen category (i.e.

A metric induces a topology on a set, but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable. One important source of metrics in differential geometry are metric tensors, bilinear forms that may be defined from the tangent vectors of a differentiable manifold onto a scalar. A metric tensor allows distances along curves to be determined through integration, and thus determines a metric.

Normal maps also apply to the study of the uniqueness of manifold structures within a homotopy type, which was pioneered by Sergei Novikov. The cobordism classes of normal maps on X are called normal invariants. Depending on the category of manifolds (differentiable, piecewise-linear, or topological), there are similarly defined, but inequivalent, concepts of normal maps and normal invariants. It is possible to perform surgery on normal maps, meaning surgery on the domain manifold, and preserving the map.

Sequential quadratic programming (SQP) is an iterative method for constrained nonlinear optimization. SQP methods are used on mathematical problems for which the objective function and the constraints are twice continuously differentiable. SQP methods solve a sequence of optimization subproblems, each of which optimizes a quadratic model of the objective subject to a linearization of the constraints. If the problem is unconstrained, then the method reduces to Newton's method for finding a point where the gradient of the objective vanishes.

If two functions are weak derivatives of the same function, they are equal except on a set with Lebesgue measure zero, i.e., they are equal almost everywhere. If we consider equivalence classes of functions such that two functions are equivalent if they are equal almost everywhere, then the weak derivative is unique. Also, if u is differentiable in the conventional sense then its weak derivative is identical (in the sense given above) to its conventional (strong) derivative.

The term Weierstrass function is often used in real analysis to refer to any function with similar properties and construction to Weierstrass's original example. For example, the cosine function can be replaced in the infinite series by a piecewise linear "zigzag" function. G. H. Hardy showed that the function of the above construction is nowhere differentiable with the assumptions 0 < a < 1, ab ≥ 1.Hardy G. H. (1916) "Weierstrass's nondifferentiable function," Transactions of the American Mathematical Society, vol.

Genetic data has suggested that B. aquilonas is simply an allopatric population of B. tryoni. Additionally, B. tryoni mate at night, while B. neohumeralis mate during the day. More pertinently, B. neohumeralis are not pests; they do not destroy crops. Despite this behavioral difference, B. neohumeralis and B. tyroni are nearly genetically identical: the two species are only differentiable based on newly-developed microsatellite technology.. The evolutionary relationship between the species within the B. tryoni complex is unknown.

It states that if is continuously differentiable, then around most points, the zero set of looks like graphs of functions pasted together. The points where this is not true are determined by a condition on the derivative of . The circle, for instance, can be pasted together from the graphs of the two functions . In a neighborhood of every point on the circle except and , one of these two functions has a graph that looks like the circle.

A manifold is a space that in the neighborhood of each point resembles a Euclidean space. In technical terms, a manifold is a topological space, such that each point has a neighborhood that is homeomorphic to an open subset of a Euclidean space. Manifold can be classified by increasing degree of this "resemblance" into topological manifolds, differentiable manifolds, smooth manifolds, and analytic manifolds. However, none of these types of "resemblance" respect distances and angles, even approximately.

The derivative at different points of a differentiable function. In this case, the derivative is equal to:\sin \left(x^2\right) + 2x^2 \cos\left(x^2\right) Let be a function that has a derivative at every point in its domain. We can then define a function that maps every point x to the value of the derivative of f at x. This function is written and is called the derivative function or the derivative of .

More generally, if the objective function is not a quadratic function, then many optimization methods use other methods to ensure that some subsequence of iterations converges to an optimal solution. The first and still popular method for ensuring convergence relies on line searches, which optimize a function along one dimension. A second and increasingly popular method for ensuring convergence uses trust regions. Both line searches and trust regions are used in modern methods of non-differentiable optimization.

Continuously differentiable functions are sometimes said to be of class C1. A function is of class C2 if the first and second derivative of the function both exist and are continuous. More generally, a function is said to be of class Ck if the first k derivatives '(x), '(x), ..., f (k)(x) all exist and are continuous. If derivatives f (n) exist for all positive integers n, the function is smooth or equivalently, of class C∞.

Loper and Black popularized "differentiable rendering" which has become an important component of self- supervised training of neural networks for problems like facial analysis. Classical methods for analysis by synthesis formulate an objective function and then differentiate it. The OpenDR method was more generic in that it (approximately) differentiated a graphics rendering engine using Automatic differentiation. This provided a framework for posing a forward synthesis problem and automatically obtaining an optimization method to solve the inverse problem.

In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below. One of the most important applications of smooth functions with compact support is the construction of so-called mollifiers, which are important in theories of generalized functions, such as Laurent Schwartz's theory of distributions.

Suppose X, Y, and Z are Banach spaces (which includes Euclidean space) and B : X × Y → Z is a continuous bilinear operator. Then B is differentiable, and its derivative at the point (x,y) in X × Y is the linear map D(x,y)B : X × Y → Z given by : (D_\left( x,y \right)\,B)\left( u,v \right) = B\left( u,y \right) + B\left( x,v \right)\qquad\forall (u,v)\in X \times Y.

Simon Brendle has solved major open problems regarding the Yamabe equation in conformal geometry. This includes his counterexamples to the compactness conjecture for the Yamabe problem, and the proof of the convergence of the Yamabe flow in all dimensions (conjectured by Richard Hamilton). In 2007, he proved the differentiable sphere theorem (in collaboration with Richard Schoen), a fundamental problem in global differential geometry. In 2012, he proved the Hsiang–Lawson's conjecture, a longstanding problem in minimal surface theory.

For any function that is continuous on [a,b] and differentiable on (a,b) there exists some c in the interval (a,b) such that the secant joining the endpoints of the interval [a,b] is parallel to the tangent at c . In mathematics, the mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval. More precisely, if f is a continuous function on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in (a,b) such that the tangent at c is parallel to the secant line through the endpoints (a,f(a)) and (b,f(b)), that is, It is one of the most important results in real analysis.

The HJB equation is usually solved backwards in time, starting from t = T and ending at t = 0. When solved over the whole of state space and V(x) is continuously differentiable, the HJB equation is a necessary and sufficient condition for an optimum when the terminal state is unconstrained. If we can solve for V then we can find from it a control u that achieves the minimum cost. In general case, the HJB equation does not have a classical (smooth) solution.

The proof of L'Hôpital's rule is simple in the case where and are continuously differentiable at the point and where a finite limit is found after the first round of differentiation. It is not a proof of the general L'Hôpital's rule because it is stricter in its definition, requiring both differentiability and that c be a real number. Since many common functions have continuous derivatives (e.g. polynomials, sine and cosine, exponential functions), it is a special case worthy of attention.

Interface conditions describe the behaviour of electromagnetic fields; electric field, electric displacement field, and the magnetic field at the interface of two materials. The differential forms of these equations require that there is always an open neighbourhood around the point to which they are applied, otherwise the vector fields and H are not differentiable. In other words, the medium must be continuous. On the interface of two different media with different values for electrical permittivity and magnetic permeability, that condition does not apply.

Symptoms also must be present for the majority of the length of a day and present for a majority of the days in the two-week period. Diagnosis can only occur if the symptoms cause "clinically significant distress or impairment". Dysthymia consists of the same depressive symptoms, but its main differentiable feature is its longer-lasting nature as compared to minor depressive disorder. Dysthymia was replaced in the DSM-5 by persistent depressive disorder, which combined dysthymia with chronic major depressive disorder.

In numerical optimization, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm is an iterative method for solving unconstrained nonlinear optimization problems. The BFGS method belongs to quasi-Newton methods, a class of hill-climbing optimization techniques that seek a stationary point of a (preferably twice continuously differentiable) function. For such problems, a necessary condition for optimality is that the gradient be zero. Newton's method and the BFGS methods are not guaranteed to converge unless the function has a quadratic Taylor expansion near an optimum.

Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can be regarded as a stochastic approximation of gradient descent optimization, since it replaces the actual gradient (calculated from the entire data set) by an estimate thereof (calculated from a randomly selected subset of the data). Especially in high- dimensional optimization problems this reduces the computational burden, achieving faster iterations in trade for a lower convergence rate.

All cotangent spaces at points on a connected manifold have the same dimension, equal to the dimension of the manifold. All the cotangent spaces of a manifold can be "glued together" (i.e. unioned and endowed with a topology) to form a new differentiable manifold of twice the dimension, the cotangent bundle of the manifold. The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore isomorphic to each other via many possible isomorphisms.

So, by the previous result, will be in harmonic coordinate charts. As a further application of Lanczos' formula, it follows that an Einstein metric is analytic in harmonic coordinates. In particular, this shows that any Einstein metric on a smooth manifold automatically determines an analytic structure on the manifold, given by the collection of harmonic coordinate charts. Due to the above analysis, in discussing harmonic coordinates it is standard to consider Riemannian metrics which are at least twice-continuously differentiable.

This follows from the triangle inequality and homogeneity of the norm. Thus all Banach spaces and Hilbert spaces are examples of topological vector spaces. ;Non-normed spaces There are topological vector spaces whose topology is not induced by a norm, but are still of interest in analysis. Examples of such spaces are spaces of holomorphic functions on an open domain, spaces of infinitely differentiable functions, the Schwartz spaces, and spaces of test functions and the spaces of distributions on them.

Then it became a book. The public interest in this work was such that The New York Times ran a front-page story. In this book, von Neumann declared that economic theory needed to use functional analysis, especially convex sets and the topological fixed-point theorem, rather than the traditional differential calculus, because the maximum-operator did not preserve differentiable functions. Independently, Leonid Kantorovich's functional analytic work on mathematical economics also focused attention on optimization theory, non-differentiability, and vector lattices.

In 2000 he developed the halo model for dark matter and galaxy clustering statistics. Much of Seljak's recent work has been focused on how to extract fundamental properties of our universe from cosmological observations using analytical methods and numerical simulations. He has developed cosmological generative models of dark matter, stars and cosmic gas distributions, including differentiable FastPM code and its extensions. Seljak is actively developing methods for accelerated approximate Bayesian methodologies, and applying them to cosmology, astronomy, and other sciences.

This strategy forms the rudiment of the Galerkin method (a finite element method) for numerical solution of partial differential equations.More detail on finite element methods from this point of view can be found in . A typical example is the Poisson equation with Dirichlet boundary conditions in a bounded domain in . The weak formulation consists of finding a function such that, for all continuously differentiable functions in vanishing on the boundary: : \int_\Omega abla u\cdot abla v = \int_\Omega gv\,.

In our analogy, the above theorem says that the two hikers will depart in directions perpendicular to each other. A consequence of this theorem (and its proof) is that if is differentiable, a level set is a hypersurface and a manifold outside the critical points of . At a critical point, a level set may be reduced to a point (for example at a local extremum of ) or may have a singularity such as a self-intersection point or a cusp.

The Conjecture claims that any convex, closed and sufficiently smooth surface in three dimensional Euclidean space needs to admit at least two umbilic points. In the sense of the Conjecture, the spheroid with only two umbilic points and the sphere, all points of which are umbilic, are examples of surfaces with minimal and maximal numbers of the umbilicus. For the conjecture to be well posed, or the umbilic points to be well-defined, the surface needs to be at least twice differentiable.

Work by Michael Mandler (1999) has challenged this claim. The Arrow–Debreu–McKenzie model is neutral between models of production functions as continuously differentiable and as formed from (linear combinations of) fixed coefficient processes. Mandler accepts that, under either model of production, the initial endowments will not be consistent with a continuum of equilibria, except for a set of Lebesgue measure zero. However, endowments change with time in the model and this evolution of endowments is determined by the decisions of agents (e.g.

Klein surfaces were introduced by Felix Klein in 1882. A Klein surface is a surface (i.e., a differentiable manifold of real dimension 2) on which the notion of angle between two tangent vectors at a given point is well-defined, and so is the angle between two intersecting curves on the surface. These angles are in the range [0,π]; since the surface carries no notion of orientation, it is not possible to distinguish between the angles α and −α.

Alexander also proved that the theorem does hold in three dimensions for piecewise linear/smooth embeddings. This is one of the earliest examples where the need for distinction between the categories of topological manifolds, differentiable manifolds, and piecewise linear manifolds became apparent. Now consider Alexander's horned sphere as an embedding into the 3-sphere, considered as the one-point compactification of the 3-dimensional Euclidean space R3. The closure of the non-simply connected domain is called the solid Alexander horned sphere.

Rye flour is a constituent of bread in central and northern Europe. Cereal flour consists either of the endosperm, germ, and bran together (whole-grain flour) or of the endosperm alone (refined flour). Meal is either differentiable from flour as having slightly coarser particle size (degree of comminution) or is synonymous with flour; the word is used both ways. For example, the word cornmeal often connotes a grittier texture whereas corn flour connotes fine powder, although there is no codified dividing line.

In functional data analysis, time series are regarded as discretizations of smooth (differentiable) functions of time. By viewing the observed samples at smooth functions, one can utilize continuous mathematics for analyzing data. Smoothness and monotonicity of time warp functions may be obtained for instance by integrating a time-varying radial basis function, thus being a one-dimensional diffeomorphism. Optimal nonlinear time warping functions are computed by minimizing a measure of distance of the set of functions to their warped average.

Informally, it is a point where the function "stops" increasing or decreasing (hence the name). For a differentiable function of several real variables, a stationary point is a point on the surface of the graph where all its partial derivatives are zero (equivalently, the gradient is zero). Stationary points are easy to visualize on the graph of a function of one variable: they correspond to the points on the graph where the tangent is horizontal (i.e., parallel to the -axis).

A physical law expressed in a generally covariant fashion takes the same mathematical form in all coordinate systems,More precisely, only coordinate systems related through sufficiently differentiable transformations are considered. and is usually expressed in terms of tensor fields. The classical (non-quantum) theory of electrodynamics is one theory that has such a formulation. Albert Einstein proposed this principle for his special theory of relativity; however, that theory was limited to space-time coordinate systems related to each other by uniform inertial motion.

As a result, the radius of convergence of a Taylor series can be zero. There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere. A function cannot be written as a Taylor series centred at a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable ; see Laurent series. For example, can be written as a Laurent series.

In his 1881 work Mathematical Psychics,Mathematical Psychics Francis Ysidro Edgeworth presented the indifference curve, deriving its properties from marginalist theory which assumed utility to be a differentiable function of quantified goods and services. But it came to be seen that indifference curves could be considered as somehow given, without bothering with notions of utility. In 1915, Eugen Slutsky derived a theory of consumer choice solely from properties of indifference curves.Eugen Slutsky; "Sulla teoria del bilancio del consumatore", Giornale degli Economisti 51 (1915).

Similarly, the covariance group for classical mechanics will be any coordinate systems that are obtained from one another by shifts in position as well as other translations allowed by a Galilean transformation. In the classical case, the invariance, or symmetry, group and the covariance group coincide, but they part ways in relativistic physics. The symmetry group of the general theory of relativity includes all differentiable transformations, i.e., all properties of an object are dynamical, in other words there are no absolute objects.

In 1946 he defended his Ph.D. thesis "Integral images of continuously differentiable functions of a complex variable". In 1949, he began working at the University of Kyiv, first as an associate professor of the Department of Mathematical Physics, and from 1951 to 1958 – its head. In 1953 he defended his doctoral dissertation on the topic "On some methods of the theory of functions in the mechanics of a continuous medium". In 1958 Georgiy Polozhia was elected head of the Department of Computational Mathematics.

In some cases, the necessary conditions are also sufficient for optimality. In general, the necessary conditions are not sufficient for optimality and additional information is required, such as the Second Order Sufficient Conditions (SOSC). For smooth functions, SOSC involve the second derivatives, which explains its name. The necessary conditions are sufficient for optimality if the objective function f of a maximization problem is a concave function, the inequality constraints g_j are continuously differentiable convex functions and the equality constraints h_i are affine functions.

Johnsonbaugh earned a bachelor's degree in mathematics from Yale University, and then moved to the University of Oregon for graduate study.Author biography from Discrete Mathematics (8th ed.) He completed his Ph.D. at Oregon in 1969. His dissertation, I. Classical Fundamental Groups and Covering Space Theory in the Setting of Cartan and Chevalley; II. Spaces and Algebras of Vector-Valued Differentiable Functions, was supervised by Bertram Yood. He also has a second master's degree in computer science from the University of Illinois at Chicago.

In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let f : D → D′ be an orientation-preserving homeomorphism between open sets in the plane. If f is continuously differentiable, then it is K-quasiconformal if the derivative of f at every point maps circles to ellipses with eccentricity bounded by K. If K is 0, then the function is conformal.

Rössler has authored hundreds of scientific papers in fields as wide-ranging as biogenesis, the origin of language, differentiable automata, chaotic attractors, endophysics, micro relativity, artificial universes, the hypertext encyclopedia, and world-changing technology. His most heavily cited publication is the 1976 paper. in which he studied what is now known as the Rössler attractor, a system of three linked differential equations that exhibit chaotic dynamics... Rössler discovered his system after a series of exchanges with Arthur Winfree as detailed by .

The curve generated by Brownian motion in the plane, at any fixed time, has probability 1 of having a convex hull whose boundary forms a continuously differentiable curve. However, for any angle \theta in the range \pi/2<\theta<\pi, there will be times during the Brownian motion where the moving particle touches the boundary of the convex hull at a point of angle \theta. The Hausdorff dimension of this set of exceptional times is (with high probability) 1-\pi/2\theta.

Fermat's theorem is central to the calculus method of determining maxima and minima: in one dimension, one can find extrema by simply computing the stationary points (by computing the zeros of the derivative), the non-differentiable points, and the boundary points, and then investigating this set to determine the extrema. One can do this either by evaluating the function at each point and taking the maximum, or by analyzing the derivatives further, using the first derivative test, the second derivative test, or the higher-order derivative test.

A subtle misconception that is often held in the context of Fermat's theorem is to assume that it makes a stronger statement about local behavior than it does. Notably, Fermat's theorem does not say that functions (monotonically) "increase up to" or "decrease down from" a local maximum. This is very similar to the misconception that a limit means "monotonically getting closer to a point". For "well-behaved functions" (which here means continuously differentiable), some intuitions hold, but in general functions may be ill-behaved, as illustrated below.

Under changes of coordinates, the volume element changes by the absolute value of the Jacobian determinant of the coordinate transformation (by the change of variables formula). This fact allows volume elements to be defined as a kind of measure on a manifold. On an orientable differentiable manifold, a volume element typically arises from a volume form: a top degree differential form. On a non- orientable manifold, the volume element is typically the absolute value of a (locally defined) volume form: it defines a 1-density.

From this viewpoint there are no distances, but collinearity and ratios of distances on any line are preserved. Differential geometry views a plane as a 2-dimensional real manifold, a topological plane which is provided with a differential structure. Again in this case, there is no notion of distance, but there is now a concept of smoothness of maps, for example a differentiable or smooth path (depending on the type of differential structure applied). The isomorphisms in this case are bijections with the chosen degree of differentiability.

Below is an example of a learning algorithm for a single-layer perceptron. For multilayer perceptrons, where a hidden layer exists, more sophisticated algorithms such as backpropagation must be used. If the activation function or the underlying process being modeled by the perceptron is nonlinear, alternative learning algorithms such as the delta rule can be used as long as the activation function is differentiable. Nonetheless, the learning algorithm described in the steps below will often work, even for multilayer perceptrons with nonlinear activation functions.

By 1718, he came to regard as a function "any expression made up of a variable and some constants." Alexis Claude Clairaut (in approximately 1734) and Leonhard Euler introduced the familiar notation {f(x)} for the value of a function. The functions considered in those times are called today differentiable functions. For this type of function, one can talk about limits and derivatives; both are measurements of the output or the change in the output as it depends on the input or the change in the input.

Some conservation laws are partial, in that they hold for some processes but not for others. One particularly important result concerning conservation laws is Noether's theorem, which states that there is a one-to-one correspondence between each one of them and a differentiable symmetry of nature. For example, the conservation of energy follows from the time-invariance of physical systems, and the conservation of angular momentum arises from the fact that physical systems behave the same regardless of how they are oriented in space.

Coordinate descent is an optimization algorithm that successively minimizes along coordinate directions to find the minimum of a function. At each iteration, the algorithm determines a coordinate or coordinate block via a coordinate selection rule, then exactly or inexactly minimizes over the corresponding coordinate hyperplane while fixing all other coordinates or coordinate blocks. A line search along the coordinate direction can be performed at the current iterate to determine the appropriate step size. Coordinate descent is applicable in both differentiable and derivative-free contexts.

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions. A function is analytic if and only if its Taylor series about x0 converges to the function in some neighborhood for every x0 in its domain.

Delay Coordinate Embedding Attractor Reconstruction of the strange nonchaotic dynamics of the pulsating star KIC 5520878 In mathematics, a strange nonchaotic attractor (SNA) is a form of attractor which, while converging to a limit, is strange, because it is not piecewise differentiable, and also non- chaotic, in that its Lyapunov exponents are non-positive. SNAs were introduced as a topic of study by Grebogi et al. in 1984. SNAs can be distinguished from periodic, quasiperiodic and chaotic attractors using the 0-1 test for chaos.

Actually, if the division in this definition is carried out in the class of functions of Y continuous at X, it will recapture the classical definition of the derivative. If it is carried out in the class of functions continuous in both X and Y, we get uniform differentiability, and our function f will be continuously differentiable. Likewise, by choosing different classes of functions (say, the Lipschitz class), we get different flavors of differentiability. In this way, differentiation becomes a part of algebra of functions.

These algorithms are differentiable, allowing the use of gradient descent to optimize them. Supernetwork-based search has been shown to produce competitive results using a fraction of the search-time required by RL-based search methods. For example, FBNet (which is short for Facebook Berkeley Network) demonstrated that supernetwork-based search produces networks that outperform the speed-accuracy tradeoff curve of mNASNet and MobileNetV2 on the ImageNet image-classification dataset. FBNet accomplishes this using over 400x less search time than was used for mNASNet.

Thus it can hardly be defended any- more to call non-differentiable continuous functions pathological. ; rigor (rigour):The act of establishing a mathematical result using indisputable logic — rather than informal descriptive argument. Rigor is a cornerstone quality of mathematics, and can play an important role in preventing mathematics from degenerating into fallacies. ; well-behaved:An object is well-behaved (in contrast with being pathological) if it satisfies certain prevailing regularity properties, or if it conforms to mathematical intuition (even though intuition can often suggest opposite behaviors as well).

Each branch L(z) of log z on an open set U is an inverse of a restriction of the exponential function, namely the restriction to the image of U under L. Since the exponential function is holomorphic (that is, complex differentiable) with nonvanishing derivative, the complex analogue of the inverse function theorem applies. It shows that L(z) is holomorphic at each z in U, and L′(z) = 1/z. Another way to prove this is to check the Cauchy–Riemann equations in polar coordinates.

Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of y' = xy. Given a family of curves, assumed to be differentiable, an isocline for that family is formed by the set of points at which some member of the family attains a given slope. The word comes from the Greek words ἴσος (isos), meaning "same", and the κλίνειν, meaning "make to slope". Generally, an isocline will itself have the shape of a curve or the union of a small number of curves.

Since the intensity function of a digital image is only known at discrete points, derivatives of this function cannot be defined unless we assume that there is an underlying differentiable intensity function that has been sampled at the image points. With some additional assumptions, the derivative of the continuous intensity function can be computed as a function on the sampled intensity function, i.e. the digital image. It turns out that the derivatives at any particular point are functions of the intensity values at virtually all image points.

In probability theory, a real valued stochastic process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process. Semimartingales are "good integrators", forming the largest class of processes with respect to which the Itô integral and the Stratonovich integral can be defined. The class of semimartingales is quite large (including, for example, all continuously differentiable processes, Brownian motion and Poisson processes). Submartingales and supermartingales together represent a subset of the semimartingales.

Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered as a continuous version of the simple random walk. The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, which is the subject of Donsker's theorem or invariance principle, also known as the functional central limit theorem. The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes.

In the words of Atiyah, the paper "stunned the mathematical world." Whereas Michael Freedman classified topological four-manifolds, Donaldson's work focused on four- manifolds admitting a differentiable structure, using instantons, a particular solution to the equations of Yang–Mills gauge theory which has its origin in quantum field theory. One of Donaldson's first results gave severe restrictions on the intersection form of a smooth four-manifold. As a consequence, a large class of the topological four-manifolds do not admit any smooth structure at all.

The matrix derivative is a convenient notation for keeping track of partial derivatives for doing calculations. The Fréchet derivative is the standard way in the setting of functional analysis to take derivatives with respect to vectors. In the case that a matrix function of a matrix is Fréchet differentiable, the two derivatives will agree up to translation of notations. As is the case in general for partial derivatives, some formulae may extend under weaker analytic conditions than the existence of the derivative as approximating linear mapping.

Cross-bedded sediments are recognized in the field by the many layers of "foresets", which are the series of layers that form on the downstream or lee side of the bedform (ripple or dune). These foresets are individually differentiable because of small-scale separation between layers of material of different sizes and densities. Cross-bedding can also be recognized by truncations in sets of ripple foresets, where previously- existing stream deposits are eroded by a later flood, and new bedforms are deposited in the scoured area.

Michael Freedman had previously shown that any unimodular symmetric bilinear form is realized as the intersection form of some closed, oriented four-manifold. Combining this result with the Serre classification theorem and Donaldson's theorem, several interesting results can be seen: 1) Any non-diagonalizable intersection form gives rise to a four-dimensional topological manifold with no differentiable structure (so cannot be smoothed). 2) Two smooth simply-connected 4-manifolds are homeomorphic, if and only if, their intersection forms have the same rank, signature, and parity.

In his 1881 work Mathematical Psychics, Francis Ysidro Edgeworth presented the indifference curve, deriving its properties from marginalist theory which assumed utility to be a differentiable function of quantified goods and services. Later work attempted to generalize to the indifference curve formulations of utility and marginal utility in avoiding unobservable measures of utility. In 1915, Eugen Slutsky derived a theory of consumer choice solely from properties of indifference curves.Слуцкий, Евгений Евгениевич (Slutsky, Yevgyeniy Ye.); "Sulla teoria del bilancio del consumatore", Giornale degli Economisti 51 (1915).

If is a differentiable function, a critical point of is a point where the rank of the Jacobian matrix is not maximal. This means that the rank at the critical point is lower than the rank at some neighbour point. In other words, let be the maximal dimension of the open balls contained in the image of ; then a point is critical if all minors of rank of are zero. In the case where , a point is critical if the Jacobian determinant is zero.

Differentiable stacks and topological stacks are defined in a way similar to algebraic stacks, except that the underlying category of affine schemes is replaced by the category of smooth manifolds or topological spaces. More generally one can define the notion of an n-sheaf or n–1 stack, which is roughly a sort of sheaf taking values in n–1 categories. There are several inequivalent ways of doing this. 1-sheaves are the same as sheaves, and 2-sheaves are the same as stacks.

A strongly continuous semigroup is called eventually norm continuous if there exists a t0 ≥ 0 such that the map t → T(t) is continuous from (t0, ∞) to L(X). The semigroup is called immediately norm continuous if t0 can be chosen to be zero. Note that for an immediately norm continuous semigroup the map t → T(t) may not be continuous in t = 0 (that would make the semigroup uniformly continuous). Analytic semigroups, (eventually) differentiable semigroups and (eventually) compact semigroups are all eventually norm continuous.

Larval shape The larva of L. trifolii are unique from those of many other flies in their shape, as the body of L. trifolii larva does not taper at the head end. The larva are uniform in thickness at both their anterior and posterior ends but additionally have a pair of spiracles at the posterior end. They do not have legs and are initially clear in color, but gradually become yellow as they mature. The larval instars are differentiable by the lengths of the body and mouthparts.

The first functional networks with many layers were published by Ivakhnenko and Lapa in 1965, as the Group Method of Data Handling. The basics of continuous backpropagation were derived in the context of control theory by Kelley in 1960 and by Bryson in 1961, using principles of dynamic programming. In 1970, Seppo Linnainmaa published the general method for automatic differentiation (AD) of discrete connected networks of nested differentiable functions. In 1973, Dreyfus used backpropagation to adapt parameters of controllers in proportion to error gradients.

The simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations. An important example in calculus is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial function, and the Stone–Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial function. Practical methods of approximation include polynomial interpolation and the use of splines.

In practice, showing the equicontinuity is often not so difficult. For example, if the sequence consists of differentiable functions or functions with some regularity (e.g., the functions are solutions of a differential equation), then the mean value theorem or some other kinds of estimates can be used to show the sequence is equicontinuous. It then follows that the limit of the sequence is continuous on every compact subset of G; thus, continuous on G. A similar argument can be made when the functions are holomorphic.

The Navier–Stokes equations are based on the assumption that the fluid, at the scale of interest, is a continuum – a continuous substance rather than discrete particles. Another necessary assumption is that all the fields of interest including pressure, flow velocity, density, and temperature are differentiable, at least weakly. The equations are derived from the basic principles of continuity of mass, momentum, and energy. Sometimes it is necessary to consider a finite arbitrary volume, called a control volume, over which these principles can be applied.

While the original proof of this result due to Richard Schoen and Shing-Tung Yau used variational methods, Witten's proof used ideas from supergravity theory to simplify the argument. A third area mentioned in Atiyah's address is Witten's work relating supersymmetry and Morse theory, a branch of mathematics that studies the topology of manifolds using the concept of a differentiable function. Witten's work gave a physical proof of a classical result, the Morse inequalities, by interpreting the theory in terms of supersymmetric quantum mechanics.

Zimmer's work centers on group actions on manifolds and more general spaces, with applications to topology and geometry. Much of his work is in the area now known as the "Zimmer Program" which aims to understand the actions of semisimple Lie groups and their discrete subgroups on differentiable manifolds. Crucial to this program is "Zimmer's cocycle superrigidity theorem", a generalization of Grigory Margulis's superrigidity theorem. Like Margulis's work, which greatly influenced Zimmer, it uses ergodic theory as a central technique in the case of invariant measures.

To endow the Grassmannian with the structure of a differentiable manifold, choose a basis for . This is equivalent to identifying it with with the standard basis, denoted (e_1, \dots, e_n) , viewed as column vectors. Then for any -dimensional subspace , viewed as an element of , we may choose a basis consisting of linearly independent column vectors (W_1, \dots, W_k) . The homogeneous coordinates of the element consist of the components of the rectangular matrix of maximal rank whose th column vector is W_i, \quad i=1, \dots, k .

Nair proved a convergence analysis. For twice continuously differentiable functions, the LJ heuristic generates a sequence of iterates having a convergent subsequence. For this class of problems, Newton's method is the usual optimization method, and it has quadratic convergence (regardless of the dimension of the space, which can be a Banach space, according to Kantorovich's analysis). The worst-case complexity of minimization on the class of unimodal functions grows exponentially in the dimension of the problem, according to the analysis of Yudin and Nemirovsky, however.

The Palais–Smale compactness condition, named after Richard Palais and Stephen Smale, is a hypothesis for some theorems of the calculus of variations. It is useful for guaranteeing the existence of certain kinds of critical points, in particular saddle points. The Palais-Smale condition is a condition on the functional that one is trying to extremize. In finite-dimensional spaces, the Palais–Smale condition for a continuously differentiable real-valued function is satisfied automatically for proper maps: functions which do not take unbounded sets into bounded sets.

In 1943 Alphonse Chapanis, a lieutenant in the U.S. Army, showed that this so-called "pilot error" could be greatly reduced when more logical and differentiable controls replaced confusing designs in airplane cockpits. After the war, the Army Air Force published 19 volumes summarizing what had been established from research during the war. In the decades since World War II, human factors has continued to flourish and diversify. Work by Elias Porter and others within the RAND Corporation after WWII extended the conception of human factors.

Flanders is known for advancing an approach to multivariate calculus that is independent of coordinates through treatment of differential forms. According to Shiing- Shen Chern, "an affine connection on a differentiable manifold gives rise to covariant differentiations of tensor fields. The classical approach makes use of the natural frames relative to local coordinates and works with components of tensor fields, thus giving the impression that this branch of differential geometry is a venture through a maze of indices. The author [Flanders] gives a mechanism which shows that this is not necessarily so."H.

We can loosely think of H_0^1(0,1) to be the absolutely continuous functions of (0,1) that are 0 at x=0 and x=1 (see Sobolev spaces). Such functions are (weakly) once differentiable and it turns out that the symmetric bilinear map \\!\,\phi then defines an inner product which turns H_0^1(0,1) into a Hilbert space (a detailed proof is nontrivial). On the other hand, the left-hand-side \int_0^1 f(x)v(x)dx is also an inner product, this time on the Lp space L^2(0,1).

Since Viacom became the market leader in music television in Germany in 2004 through the acquisition of VIVA Germany and VIVA Plus, all special formats on MTV2 Pop were cancelled. The reason given for this step was that the four programs were now to be made more widely available with differentiable formats. As a family-friendly program, MTV2 Pop occasionally recorded older videos in the program until it was discontinued. On September 11, 2005, MTV2 Pop was replaced by the children's channel Nickelodeon Deutschland, which started broadcasting after a 24-hour countdown.

In physics, one-parameter groups describe dynamical systems.Zeidler, E. (1995) Applied Functional Analysis: Main Principles and Their Applications Springer-Verlag Furthermore, whenever a system of physical laws admits a one-parameter group of differentiable symmetries, then there is a conserved quantity, by Noether's theorem. In the study of spacetime the use of the unit hyperbola to calibrate spatio-temporal measurements has become common since Hermann Minkowski discussed it in 1908. The principle of relativity was reduced to arbitrariness of which diameter of the unit hyperbola was used to determine a world-line.

Using ideas from Whitney's embedding theorem, A can be embedded in k-dimensional Euclidean space with :k > 2 d_A. That is, there is a diffeomorphism \phi that maps A into \R^k such that the derivative of \phi has full rank. A delay embedding theorem uses an observation function to construct the embedding function. An observation function \alpha must be twice-differentiable and associate a real number to any point of the attractor A. It must also be typical, so its derivative is of full rank and has no special symmetries in its components.

In mathematical analysis, a Pompeiu derivative is a real-valued function of one real variable that is the derivative of an everywhere differentiable function and that vanishes in a dense set. In particular, a Pompeiu derivative is discontinuous at any point where it is not 0. Whether non-identically zero such functions may exist was a problem that arose in the context of early-1900s research on functional differentiability and integrability. The question was affirmatively answered by Dimitrie Pompeiu by constructing an explicit example; these functions are therefore named after him.

A related but distinct use of second derivatives is to determine whether a function is concave up or concave down at a point. It does not, however, provide information about inflection points. Specifically, a twice-differentiable function f is concave up if f(x) > 0 and concave down if f(x) < 0. Note that if f(x) = x^4, then x = 0 has zero second derivative, yet is not an inflection point, so the second derivative alone does not give enough information to determine whether a given point is an inflection point.

Figure 4: Coordinate surfaces, coordinate lines, and coordinate axes of general curvilinear coordinates. To quote Bullo and Lewis: "Only in exceptional circumstances can the configuration of Lagrangian system be described by a vector in a vector space. In the natural mathematical setting, the system's configuration space is described loosely as a curved space, or more accurately as a differentiable manifold." Instead of Cartesian coordinates, when equations of motion are expressed in a curvilinear coordinate system, Christoffel symbols appear in the acceleration of a particle expressed in this coordinate system, as described below in more detail.

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a probability distribution by maximizing a likelihood function, so that under the assumed statistical model the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference. If the likelihood function is differentiable, the derivative test for determining maxima can be applied.

When dealing with large populations, as in the case of tuberculosis, deterministic or compartmental mathematical models are often used. In a deterministic model, individuals in the population are assigned to different subgroups or compartments, each representing a specific stage of the epidemic. The transition rates from one class to another are mathematically expressed as derivatives, hence the model is formulated using differential equations. While building such models, it must be assumed that the population size in a compartment is differentiable with respect to time and that the epidemic process is deterministic.

250x250px Historically, the curvature of a differentiable curve was defined through the osculating circle, which is the circle that best approximates the curve at a point. More precisely, given a point on a curve, every other point of the curve defines a circle (or sometimes a line) passing through and tangent to the curve at . The osculating circle is the limit, if it exists, of this circle when tends to . Then the center and the radius of curvature of the curve at are the center and the radius of the osculating circle.

Kreck, "Is the universe exotic?", Universitas 1988 4-manifolds with exotic differentiable structure and the interaction of differential geometry and topology. In his habilitation in 1977 he managed the complete classification of closed smooth manifolds with diffeomorphisms up to bordism: a problem that had already been worked on by René Thom, William Browder and Dennis Sullivan. Building on this work he developed a modified theory of surgery which is applicable under weaker conditions than classical surgery and he applied this theory to solve outstanding questions in differential geometry.

Chechen, an agglutinative language. Agglutinative languages have words containing several morphemes that are always clearly differentiable from one another in that each morpheme represents only one grammatical meaning and the boundaries between those morphemes are easily demarcated; that is, the bound morphemes are affixes, and they may be individually identified. Agglutinative languages tend to have a high number of morphemes per word, and their morphology is usually highly regular, with a notable exception being Georgian, among others. Agglutinative languages include Finnish, Hungarian, Turkish, Mongolian, Korean, Japanese, and Indonesian.

The space of such functions consists of two disjoint convex sets: the increasing ones and the decreasing ones, and has the homotopy type of two points. A non-holonomic solution to this relation would consist in the data of two functions, a differentiable function f(x), and a continuous function g(x), with g(x) nowhere vanishing. A holonomic solution gives rise to a non-holonomic solution by taking g(x) = f'(x). The space of non-holonomic solutions again consists of two disjoint convex sets, according as g(x) is positive or negative.

The cursive script ((), cǎoshū, literally "grass script") is used informally. The basic character shapes are suggested, rather than explicitly realized, and the abbreviations are sometimes extreme. Despite being cursive to the point where individual strokes are no longer differentiable and the characters often illegible to the untrained eye, this script (also known as draft) is highly revered for the beauty and freedom that it embodies. Some of the simplified Chinese characters adopted by the People's Republic of China, and some simplified characters used in Japan, are derived from the cursive script.

In terms of composition of the differential operator Di which takes the partial derivative with respect to xi: :D_i \circ D_j = D_j \circ D_i. From this relation it follows that the ring of differential operators with constant coefficients, generated by the Di, is commutative; but this is only true as operators over a domain of sufficiently differentiable functions. It is easy to check the symmetry as applied to monomials, so that one can take polynomials in the xi as a domain. In fact smooth functions are another valid domain.

A function f(x) is "S-unimodal" (often referred to as "S-unimodal map") if its Schwarzian derivative is negative for all \ x e c, where c is the critical point.See e.g. In computational geometry if a function is unimodal it permits the design of efficient algorithms for finding the extrema of the function. A more general definition, applicable to a function f(X) of a vector variable X is that f is unimodal if there is a one-to-one differentiable mapping X = G(Z) such that f(G(Z)) is convex.

In the context of differentiable dynamical systems, the notion of integrability refers to the existence of invariant, regular foliations; i.e., ones whose leaves are embedded submanifolds of the smallest possible dimension that are invariant under the flow. There is thus a variable notion of the degree of integrability, depending on the dimension of the leaves of the invariant foliation. This concept has a refinement in the case of Hamiltonian systems, known as complete integrability in the sense of Liouville (see below), which is what is most frequently referred to in this context.

The Bhaban consists of nine individual blocks: the eight peripheral blocks rise to a height of 110' while the central octagonal block rises to a height of 155'. All nine blocks include different groups of functional spaces and have different levels, inter-linked horizontally and vertically by corridors, lifts, stairs, light courts, and circular areas. The entire structure is designed to blend into one single, non-differentiable unit, that appears from the exterior to be a single story. The main committee rooms are located at level two in one of the peripheral blocks.

The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and root data. Lie theory has been particularly useful in mathematical physics since it describes the standard transformation groups: the Galilean group, the Lorentz group, the Poincaré group and the conformal group of spacetime.

His later studies centred mainly on the theory of differentiable economies, where he showed that, in general, aggregate excess demand functions vanish at a finite number of points – basically, he showed that economies have a finite number of price equilibria. In 1976, he received the French Legion of Honour. He was awarded the 1983 Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, for having incorporated new analytical methods into economic theory and for his rigorous reformulation of general equilibrium theory. He was a member of the International Academy of Science.

Because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to f(x) at x = a. While the concept of local linearity applies the most to points arbitrarily close to x = a, those relatively close work relatively well for linear approximations. The slope M should be, most accurately, the slope of the tangent line at x = a. An approximation of f(x)=x^2 at (x, f(x)) Visually, the accompanying diagram shows the tangent line of f(x) at x.

In mathematics, a function f is weakly harmonic in a domain D if :\int_D f\, \Delta g = 0 for all g with compact support in D and continuous second derivatives, where Δ is the Laplacian. This is the same notion as a weak derivative, however, a function can have a weak derivative and not be differentiable. In this case, we have the somewhat surprising result that a function is weakly harmonic if and only if it is harmonic. Thus weakly harmonic is actually equivalent to the seemingly stronger harmonic condition.

Differential forms are mathematical objects constructed via tangent spaces and multilinear forms that behave, in many ways, like differentials in the classical sense. Though conceptually and computationally useful, differentials are founded on ill-defined notions of infinitesimal quantities developed early in the history of calculus. Differential forms provide a mathematically rigorous and precise framework to modernize this long-standing idea. Differential forms are especially useful in multivariable calculus (analysis) and differential geometry because they possess transformation properties that allow them be integrated on curves, surfaces, and their higher-dimensional analogues (differentiable manifolds).

In mathematics, the Frölicher–Nijenhuis bracket is an extension of the Lie bracket of vector fields to vector-valued differential forms on a differentiable manifold. It is useful in the study of connections, notably the Ehresmann connection, as well as in the more general study of projections in the tangent bundle. It was introduced by Alfred Frölicher and Albert Nijenhuis (1956) and is related to the work of Schouten (1940). It is related to but not the same as the Nijenhuis–Richardson bracket and the Schouten–Nijenhuis bracket.

The graph of an arbitrary function y=f(x). The orange line is tangent to x=a, meaning at that exact point, the slope of the curve and the straight line are the same. The derivative at different points of a differentiable function The derivative of f(x) at the point x=a is defined as the slope of the tangent to (a,f(a)). In order to gain an intuition for this definition, one must first be familiar with finding the slope of a linear equation, written in the form y=mx+b.

Fig. 12. Price lines for a box with boundary equilibriaKenneth Arrow and Gérard Debreu published papers independently in 1951 drawing attention to limitations in the calculus proofs of equilibrium theorems.K. Arrow, ‘An Extension of the Basic Theorems of Classical Welfare Economics’ (1951); G. Debreu, ‘The Coefficient of Resource Utilization’ (1951). Arrow specifically mentioned the difficulty caused by equilibria on the boundary, and Debreu the problem of non- differentiable indifference curves. Without aiming for exhaustive coverage it is easy to see in intuitive terms how to extend our methods to apply to these cases.

In topology (a mathematical discipline) a prime manifold is an n-manifold that cannot be expressed as a non-trivial connected sum of two n-manifolds. Non- trivial means that neither of the two is an n-sphere. A similar notion is that of an irreducible n-manifold, which is one in which any embedded (n − 1)-sphere bounds an embedded n-ball. Implicit in this definition is the use of a suitable category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds.

Together with the other members of the genus Babyrousa, the North Sulawesi babirusa has usually been considered a subspecies of a widespread Babyrousa babyrussa, but recent work suggests that there may be several species, differentiable on the basis of geography, body size, amount of body hair, and the shape of the upper canine tooth of the male. Following the split, the "true" Babyrousa babyrussa is restricted to Buru and the Sula Islands.Meijaard, E. and Groves, C. P. (2002). Upgrading three subspecies of Babirusa (Babyrousa sp.) to full species level.

A 0-dimensional manifold (or differentiable or analytic manifold) is nothing but a discrete topological space. We can therefore view any discrete group as a 0-dimensional Lie group. A product of countably infinite copies of the discrete space of natural numbers is homeomorphic to the space of irrational numbers, with the homeomorphism given by the continued fraction expansion. A product of countably infinite copies of the discrete space {0,1} is homeomorphic to the Cantor set; and in fact uniformly homeomorphic to the Cantor set if we use the product uniformity on the product.

A differentiable map that is a submersion at each point p\in M is called a submersion. Equivalently, is a submersion if its differential Df_p has constant rank equal to the dimension of . A word of warning: some authors use the term critical point to describe a point where the rank of the Jacobian matrix of at is not maximal.. Indeed, this is the more useful notion in singularity theory. If the dimension of is greater than or equal to the dimension of then these two notions of critical point coincide.

Random search (RS) is a family of numerical optimization methods that do not require the gradient of the problem to be optimized, and RS can hence be used on functions that are not continuous or differentiable. Such optimization methods are also known as direct-search, derivative-free, or black-box methods. The name "random search" is attributed to Rastrigin who made an early presentation of RS along with basic mathematical analysis. RS works by iteratively moving to better positions in the search-space, which are sampled from a hypersphere surrounding the current position.

The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending without stretching or tearing a page of paper gives an isometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent. The first theorem is for continuously differentiable (C1) embeddings and the second for analytic embeddings or embeddings that are smooth of class Ck, 3 ≤ k ≤ ∞.

Pyrrho claimed that all pragmata (matters, affairs, questions, topics) are adiaphora (not differentiable, not clearly definable, negating Aristotle's use of "diaphora"), astathmēta (unstable, unbalanced, unmeasurable), and anepikrita (unjudgeable, undecidable). Therefore, neither our senses nor our beliefs and theories are able to identify truth or falsehood. Philologist Christopher Beckwith has demonstrated that Pyrrho's use of adiaphora reflects his effort to translate the Buddhist three marks of existence into Greek, and that adiaphora reflects Pyrrho's understanding of the Buddhist concept of anatta. Likewise he suggests that astathmēta and anepikrita may be compared to dukkha and anicca respectively.

If the limit exists, meaning that there is a way of choosing a value for that makes a continuous function, then the function is differentiable at , and its derivative at equals . In practice, the existence of a continuous extension of the difference quotient to is shown by modifying the numerator to cancel in the denominator. Such manipulations can make the limit value of for small clear even though is still not defined at . This process can be long and tedious for complicated functions, and many shortcuts are commonly used to simplify the process.

Under mild conditions, for example if the function is a monotone function or a Lipschitz function, this is true. However, in 1872 Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. This example is now known as the Weierstrass function. In 1931, Stefan Banach proved that the set of functions that have a derivative at some point is a meager set in the space of all continuous functions.. Cited by Informally, this means that hardly any random continuous functions have a derivative at even one point.

The uniquely defined dimension of every connected topological manifold can be calculated. A connected topological manifold is locally homeomorphic to Euclidean -space, in which the number is the manifold's dimension. For connected differentiable manifolds, the dimension is also the dimension of the tangent vector space at any point. In geometric topology, the theory of manifolds is characterized by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases are simplified by having extra space in which to "work"; and the cases and are in some senses the most difficult.

In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations. The essential idea is that coordinates are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws. A more significant requirement is the principle of general relativity that states that the laws of physics take the same form in all reference systems. This is a generalization of the principle of special relativity which states that the laws of physics take the same form in all inertial frames.

One may also use Newton's method to solve systems of (nonlinear) equations, which amounts to finding the zeroes of continuously differentiable functions . In the formulation given above, one then has to left multiply with the inverse of the Jacobian matrix instead of dividing by : :x_{n+1} = x_{n} - J_F(x_n)^{-1}F(x_n) Rather than actually computing the inverse of the Jacobian matrix, one may save time and increase numerical stability by solving the system of linear equations :J_F(x_n) (x_{n+1} - x_n) = -F(x_n) for the unknown .

The primary auditory cortex is the first region of cerebral cortex to receive auditory input. Perception of sound is associated with the left posterior superior temporal gyrus (STG). The superior temporal gyrus contains several important structures of the brain, including Brodmann areas 41 and 42, marking the location of the primary auditory cortex, the cortical region responsible for the sensation of basic characteristics of sound such as pitch and rhythm. We know from research in nonhuman primates that the primary auditory cortex can probably be divided further into functionally differentiable subregions.

In dynamical systems, if the dynamic is given by a differentiable map f then a point is hyperbolic if and only if the differential of ƒ n (where n is the period of the point) has no eigenvalue on the (complex) unit circle when computed at the point. Then a saddle point is a hyperbolic periodic point whose stable and unstable manifolds have a dimension that is not zero. A saddle point of a matrix is an element which is both the largest element in its column and the smallest element in its row.

From a mathematical point of view, the phases are merely regions in which the solutions of the underlying PDE are continuous and differentiable up to the order of the PDE. In physical problems such solutions represent properties of the medium for each phase. The moving boundaries (or interfaces) are infinitesimally thin surfaces that separate adjacent phases; therefore, the solutions of the underlying PDE and its derivatives may suffer discontinuities across interfaces. The underlying PDEs are not valid at the phase change interfaces; therefore, an additional condition—the Stefan condition—is needed to obtain closure.

In such coordinates the covariant derivative reduces to a partial derivative (at p only), and the geodesics through p are locally linear functions of t (the affine parameter). This idea was implemented in a fundamental way by Albert Einstein in the general theory of relativity: the equivalence principle uses normal coordinates via inertial frames. Normal coordinates always exist for the Levi-Civita connection of a Riemannian or Pseudo-Riemannian manifold. By contrast, in general there is no way to define normal coordinates for Finsler manifolds in a way that the exponential map are twice-differentiable .

For differentiable manifolds, this can happen in dimensions 2, 6, 14, 30, 62, and possibly 126, and in no other dimensions. The final case of dimension 126 remains open. These five or six framed cobordism classes of manifolds having Kervaire invariant 1 are exceptional objects related to exotic spheres. The first three cases are related to the complex numbers, quaternions and octonions respectively: a manifold of Kervaire invariant 1 can be constructed as the product of two spheres, with its exotic framing determined by the normed division algebra.

In mathematics, the Gibbs phenomenon, discovered by Available on-line at: National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. and rediscovered by , is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. The nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself. The overshoot does not die out as n increases, but approaches a finite limit.

In mathematics, the jet is an operation that takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain. Although this is the definition of a jet, the theory of jets regards these polynomials as being abstract polynomials rather than polynomial functions. This article first explores the notion of a jet of a real valued function in one real variable, followed by a discussion of generalizations to several real variables. It then gives a rigorous construction of jets and jet spaces between Euclidean spaces.

The principal result is that the set of the inflection points of an algebraic curve coincides with the intersection set of the curve with the Hessian curve. For a smooth curve given by parametric equations, a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e., changes sign. For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the second derivative has an isolated zero and changes sign.

Two main classes of solutions are known, namely plane waves and spherical waves. The plane waves may be viewed as the limiting case of spherical waves at a very large (ideally infinite) distance from the source. Both types of waves can have a waveform which is an arbitrary time function (so long as it is sufficiently differentiable to conform to the wave equation). As with any time function, this can be decomposed by means of Fourier analysis into its frequency spectrum, or individual sinusoidal components, each of which contains a single frequency, amplitude and phase.

If the functions are linearly dependent, then so are the columns of the Wronskian as differentiation is a linear operation, so the Wronskian vanishes. Thus, the Wronskian can be used to show that a set of differentiable functions is linearly independent on an interval by showing that it does not vanish identically. It may, however, vanish at isolated points. A common misconception is that everywhere implies linear dependence, but pointed out that the functions and have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of .

KZ filtration resolved the problem and enabled proof of Kolmogorov's law in that domain. Filter construction relied on the main concepts of the continuous Fourier transform and their discrete analogues. The algorithm of the KZ filter came from the definition of higher-order derivatives for discrete functions as higher-order differences. Believing that infinite smoothness in the Gaussian window was a beautiful but unrealistic approximation of a truly discrete world, Kolmogorov chose a finitely differentiable tapering window with finite support, and created this mathematical construction for the discrete case.

The resulting theorem asserts that any Riemannian metric on a closed manifold may be conformally rescaled (that is, multiplied by a suitable positive function) so as to produce a metric of constant scalar curvature. In 2007, Simon Brendle and Richard Schoen proved the differentiable sphere theorem, a fundamental result in the study of manifolds of positive sectional curvature. He has also made fundamental contributions to the regularity theory of minimal surfaces and harmonic maps. His students include Hubert Bray, José F. Escobar, Ailana Fraser, Chikako Mese, William Minicozzi, and André Neves.

Consider predicting the class label of a single data point by consensus of its k-nearest neighbours with a given distance metric. This is known as leave-one-out classification. However, the set of nearest-neighbours C_i can be quite different after passing all the points through a linear transformation. Specifically, the set of neighbours for a point can undergo discrete changes in response to smooth changes in the elements of A, implying that any objective function f(\cdot) based on the neighbours of a point will be piecewise-constant, and hence not differentiable.

In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is diffeomorphic to the quotient space N/H, the quotient of a nilpotent Lie group N modulo a closed subgroup H. This notion was introduced by Anatoly Mal'cev in 1951. In the Riemannian category, there is also a good notion of a nilmanifold. A Riemannian manifold is called a homogeneous nilmanifold if there exist a nilpotent group of isometries acting transitively on it.

A pictorial representation of the tangent space of a single point x on a sphere. A vector in this tangent space represents a possible velocity at x . After moving in that direction to a nearby point, the velocity would then be given by a vector in the tangent space of that point—a different tangent space that is not shown. In differential geometry, one can attach to every point x of a differentiable manifold a tangent space—a real vector space that intuitively contains the possible directions in which one can tangentially pass through x .

The companion notion of curvature measures how moving frames "roll" along a curve "without twisting". More generally, on a differentiable manifold equipped with an affine connection (that is, a connection in the tangent bundle), torsion and curvature form the two fundamental invariants of the connection. In this context, torsion gives an intrinsic characterization of how tangent spaces twist about a curve when they are parallel transported; whereas curvature describes how the tangent spaces roll along the curve. Torsion may be described concretely as a tensor, or as a vector-valued 2-form on the manifold.

TensorFlow is a free and open-source software library for dataflow and differentiable programming across a range of tasks. It is a symbolic math library, and is also used for machine learning applications such as neural networks."TensorFlow: Open source machine learning" "It is machine learning software being used for various kinds of perceptual and language understanding tasks" — Jeffrey Dean, minute 0:47 / 2:17 from YouTube clip It is used for both research and production at Google. TensorFlow was developed by the Google Brain team for internal Google use.

The metaphor of the floodtide that transforms the earth's > surface to a muddy mass is frequently employed by Shakespeare to designate > the undifferentiated state of the world that is also portrayed in Genesis. Girard goes on to say that "equilibrium invariably leads to violence," while justice is really the imbalance that shows the difference between "good" and "evil" or "pure" and "impure." When the past is not differentiable from the present, a violence called sacrificial crises occurs. There will always be differences despite the seeming similarities; otherwise, violence will ensue to "correct" the situation.

For a fixed configuration of noise, SDE has a unique solution differentiable with respect to the initial condition. Nontriviality of stochastic case shows up when one tries to average various objects of interest over noise configurations. In this sense, an SDE is not a uniquely defined entity when noise is multiplicative and when the SDE is understood as a continuous time limit of a stochastic difference equation. In this case, SDE must be complemented by what is known as "interpretations of SDE" such as Itô or a Stratonovich interpretations of SDEs.

In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it is a differential form defined on the differentiable manifold, rather than on an abstract principal bundle over it. Hence, despite their lack of tensoriality, connection forms continue to be used because of the relative ease of performing calculations with them., , In physics, connection forms are also used broadly in the context of gauge theory, through the gauge covariant derivative.

Due to their general nature and versatility, sheaves have several applications in topology and especially in algebraic and differential geometry. First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general cohomology theory, which encompasses also the "usual" topological cohomology theories such as singular cohomology.

One can show that if g is infinitely differentiable, then the numerical algorithm using Fast Fourier Transforms will converge faster than any polynomial in the grid size h. That is, for any n>0, there is a C_n<\infty such that the error is less than C_nh^n for all sufficiently small values of h. We say that the spectral method is of order n, for every n>0. Because a spectral element method is a finite element method of very high order, there is a similarity in the convergence properties.

All the constructions in classical differential calculus have an analog in secondary calculus. For instance, higher symmetries of a system of partial differential equations are the analog of vector fields on differentiable manifolds. The Euler operator, which associates to each variational problem the corresponding Euler–Lagrange equation, is the analog of the classical differential associating to a function on a variety its differential. The Euler operator is a secondary differential operator of first order, even if, according to its expression in local coordinates, it looks like one of infinite order.

Calculation of sinuosity for an oscillating curve. Laces on mountain road with high sinuosity at Luz Ardiden Rio Cauto at Guamo Embarcadero, Cuba, is not taking the shortest path downslope. Therefore, its sinuosity index is > 1. Two ski tracks with different degrees of sinuosity on the same slope Sinuosity, sinuosity index, or sinuosity coefficient of a continuously differentiable curve having at least one inflection point is the ratio of the curvilinear length (along the curve) and the Euclidean distance (straight line) between the end points of the curve.

Manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low- dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such as exotic differentiable structures on R4. Thus the topological classification of 4-manifolds is in principle easy, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the generalized Poincaré conjecture; see Gluck twists. The distinction is because surgery theory works in dimension 5 and above (in fact, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above is controlled algebraically by surgery theory. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work, and other phenomena occur.

Dedicated methods (see for instance Lemaréchal) from non differentiable optimization come in. Once the optimal model is computed we have to address the question: "Can we trust this model?" The question can be formulated as follows: How large is the set of models that match the data "nearly as well" as this model? In the case of quadratic objective functions, this set is contained in a hyper-ellipsoid, a subset of {R}^M (M is the number of unknowns), whose size depends on what we mean with "nearly as well", that is on the noise level.

If we integrate by parts using a form of Green's identities, we see that if u solves P2, then we may define \phi(u,v) for any v by :\int_\Omega fv\,ds = -\int_\Omega abla u \cdot abla v \, ds \equiv -\phi(u,v), where abla denotes the gradient and \cdot denotes the dot product in the two-dimensional plane. Once more \,\\!\phi can be turned into an inner product on a suitable space H_0^1(\Omega) of once differentiable functions of \Omega that are zero on \partial \Omega. We have also assumed that v \in H_0^1(\Omega) (see Sobolev spaces).

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers , or a subset of that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers. Nevertheless, the codomain of a function of a real variable may be any set.

Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector). More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field.

Therefore, :f(g(x))-f(g(a))=q(g(x))r(x)(x-a), but the function given by is continuous at , and we get, for this :(f(g(a)))'=q(g(a))r(a)=f'(g(a))g'(a). A similar approach works for continuously differentiable (vector-)functions of many variables. This method of factoring also allows a unified approach to stronger forms of differentiability, when the derivative is required to be Lipschitz continuous, Hölder continuous, etc. Differentiation itself can be viewed as the polynomial remainder theorem (the little Bézout theorem, or factor theorem), generalized to an appropriate class of functions.

During the last twenty years of his life Thom's published work was mainly in philosophy and epistemology, and he undertook a reevaluation of Aristotle's writings on science. In 1992, he was one of eighteen academics who sent a letter to Cambridge University protesting against plans to award Jacques Derrida an honorary doctorate. Beyond Thom's contributions to algebraic topology, he studied differentiable mappings, through the study of generic properties. In his final years, he turned his attention to an effort to apply his ideas about structural topography to the questions of thought, language, and meaning in the form of a "semiophysics".

For any integer k > 0 and any n−dimensional Ck−manifold, the maximal atlas contains a C∞−atlas on the same underlying set by a theorem due to Hassler Whitney. It has also been shown that any maximal Ck−atlas contains some number of distinct maximal C∞−atlases whenever n > 0, although for any pair of these distinct C∞−atlases there exists a C∞−diffeomorphism identifying the two. It follows that there is only one class of smooth structures (modulo pairwise smooth diffeomorphism) over any topological manifold which admits a differentiable structure, i.e. The C∞−, structures in a Ck−manifold.

SNNs avoid problems of batch normalization since the activations across samples automatically converge to mean zero and variance one. SNNs an enabling technology to (1) train very deep networks, that is, networks with many layers, (2) use novel regularization strategies, and (3) learn very robustly across many layers. In unsupervised deep learning, Generative Adversarial Networks (GANs) are very popular since they create new images which are more realistic than those of obtained from other generative approaches. Sepp Hochreiter proposed a two time-scale update rule (TTUR) for learning GANs with stochastic gradient descent on any differentiable loss function.

Furthermore, for every open subset A of the real line, there exist smooth functions that are analytic on A and nowhere else. It is useful to compare the situation to that of the ubiquity of transcendental numbers on the real line. Both on the real line and the set of smooth functions, the examples we come up with at first thought (algebraic/rational numbers and analytic functions) are far better behaved than the majority of cases: the transcendental numbers and nowhere analytic functions have full measure (their complements are meagre). The situation thus described is in marked contrast to complex differentiable functions.

In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain trivial line bundle, called the density bundle. An element of the density bundle at x is a function that assigns a volume for the parallelotope spanned by the n given tangent vectors at x. From the operational point of view, a density is a collection of functions on coordinate charts which become multiplied by the absolute value of the Jacobian determinant in the change of coordinates.

The intersection of the unit cube with the cutting plane x_1 + x_2 + x_3 \geq 2. In the context of the Traveling salesman problem on three nodes, this (rather weak) inequality states that every tour must have at least two edges. In mathematical optimization, the cutting-plane method is any of a variety of optimization methods that iteratively refine a feasible set or objective function by means of linear inequalities, termed cuts. Such procedures are commonly used to find integer solutions to mixed integer linear programming (MILP) problems, as well as to solve general, not necessarily differentiable convex optimization problems.

Given a twice continuously differentiable function f of one real variable, Taylor's theorem for the case n = 1 states that : f(x) = f(a) + f'(a)(x - a) + R_2\ where R_2 is the remainder term. The linear approximation is obtained by dropping the remainder: : f(x) \approx f(a) + f'(a)(x - a). This is a good approximation when x is close enough to a; since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of f at (a,f(a)).

This approach departs from the classical logic used in conventional mathematics by denying the law of the excluded middle, e.g., NOT (a ≠ b) does not imply a = b. In particular, in a theory of smooth infinitesimal analysis one can prove for all infinitesimals ε, NOT (ε ≠ 0); yet it is provably false that all infinitesimals are equal to zero. One can see that the law of excluded middle cannot hold from the following basic theorem (again, understood in the context of a theory of smooth infinitesimal analysis): :Every function whose domain is R, the real numbers, is continuous and infinitely differentiable.

The definitions of eigenvalue and eigenvectors of a linear transformation T remains valid even if the underlying vector space is an infinite-dimensional Hilbert or Banach space. A widely used class of linear transformations acting on infinite-dimensional spaces are the differential operators on function spaces. Let D be a linear differential operator on the space C∞ of infinitely differentiable real functions of a real argument t. The eigenvalue equation for D is the differential equation :D f(t) = \lambda f(t) The functions that satisfy this equation are eigenvectors of D and are commonly called eigenfunctions.

Gradient descent is a first-order iterative optimization algorithm for finding the minimum of a function. In neural networks, it can be used to minimize the error term by changing each weight in proportion to the derivative of the error with respect to that weight, provided the non-linear activation functions are differentiable. Various methods for doing so were developed in the 1980s and early 1990s by Werbos, Williams, Robinson, Schmidhuber, Hochreiter, Pearlmutter and others. The standard method is called “backpropagation through time” or BPTT, and is a generalization of back- propagation for feed-forward networks.

To adjust weights properly, one applies a general method for non-linear optimization that is called gradient descent. For this, the network calculates the derivative of the error function with respect to the network weights, and changes the weights such that the error decreases (thus going downhill on the surface of the error function). For this reason, back-propagation can only be applied on networks with differentiable activation functions. In general, the problem of teaching a network to perform well, even on samples that were not used as training samples, is a quite subtle issue that requires additional techniques.

In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for piecewise linear and topological manifolds. A cobordism between manifolds M and N is a compact manifold W whose boundary is the disjoint union of M and N, \partial W=M \sqcup N. Cobordisms are studied both for the equivalence relation that they generate, and as objects in their own right.

Shortly after its inception, TransLink's board of directors approved replacement of the old BC Transit colour with TransLink's new blue and yellow colour scheme, or livery. It also created brands for the agency's different services, each with a different logo based on these colours, with the exception of the West Coast Express. The board decided against changing West Coast Express's purple colour to blue, since purple and yellow create a premium brand differentiable from TransLink's blue and yellow livery. Repainting of vehicles did not incur any additional costs, as it was completed during regular maintenance repaints or new vehicle purchases.

Example of convergence of a direct search method on the Broyden function. At each iteration, the pattern either moves to the point which best minimizes its objective function, or shrinks in size if no point is better than the current point, until the desired accuracy has been achieved, or the algorithm reaches a predetermined number of iterations. Pattern search (also known as direct search, derivative-free search, or black-box search) is a family of numerical optimization methods that does not require a gradient. As a result, it can be used on functions that are not continuous or differentiable.

The proof of the classical result that for the vanishing of the Cotton tensor is equivalent to the metric being conformally flat is given by Eisenhart using a standard integrability argument. This tensor density is uniquely characterized by its conformal properties coupled with the demand that it be differentiable for arbitrary metrics, as shown by . Recently, the study of three-dimensional spaces is becoming of great interest, because the Cotton tensor restricts the relation between the Ricci tensor and the energy–momentum tensor of matter in the Einstein equations and plays an important role in the Hamiltonian formalism of general relativity.

The exponential function y = ex (red) and the corresponding Taylor polynomial of degree four (dashed green) around the origin. In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the kth-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order k of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. (2013).

If is negative, then is on the low part of the step, so the secant line from to is very steep, and as tends to zero the slope tends to infinity. If is positive, then is on the high part of the step, so the secant line from to has slope zero. Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist. The absolute value function is continuous, but fails to be differentiable at since the tangent slopes do not approach the same value from the left as they do from the right.

In mathematics, a diffeology on a set declares what the smooth parametrizations in the set are. In some sense a diffeology generalizes the concept of smooth charts in a differentiable manifold. The concept was first introduced by Jean-Marie Souriau in the 1980s and developed first by his students Paul Donato (homogeneous spaces and coverings) and Patrick Iglesias (diffeological fiber bundles, higher homotopy, etc.), later by other people. A related idea was introduced by Kuo-Tsaï Chen (陳國才, Chen Guocai) in the 1970s, using convex sets instead of open sets for the domains of the plots.

A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines. Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three- dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds.

Also spelled as Pei La Cain, this villain is possibly named for the Earth, Terra/Gaia. He is Mamoru's counterpart in the Sol Masters, made in the image of Cain, "the Green Planet's Protector" and Mamoru's father. He has telekinetic abilities similar in strength to Latio/Mamoru, and displays the ability to perform the same Hell and Heaven technique that Mamoru used in the series. Failure Cain is only differentiable from the true Cain by the inversion of the G-Stone symbol on his forehead, which also appears in the background when he performs Hell and Heaven.

Yang–Mills theories met with general acceptance in the physics community after Gerard 't Hooft, in 1972, worked out their renormalization, relying on a formulation of the problem worked out by his advisor Martinus Veltman. Renormalizability is obtained even if the gauge bosons described by this theory are massive, as in the electroweak theory, provided the mass is only an "acquired" one, generated by the Higgs mechanism. The mathematics of the Yang–Mills theory is a very active field of research, yielding e.g. invariants of differentiable structures on four-dimensional manifolds via work of Simon Donaldson.

In that way, the curve interpolates between the two points, and the resulting curve is a polygon, which is continuous, but not differentiable at the interval boundaries, or knots. Higher degree polynomials have correspondingly more continuous derivatives. Note that within the interval the polynomial nature of the basis functions and the linearity of the construction make the curve perfectly smooth, so it is only at the knots that discontinuity can arise. In many applications the fact that a single control point only influences those intervals where it is active is a highly desirable property, known as local support.

In complex analysis, the open mapping theorem states that if U is a domain of the complex plane C and f : U → C is a non-constant holomorphic function, then f is an open map (i.e. it sends open subsets of U to open subsets of C, and we have invariance of domain.). The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the real line, for example, the differentiable function f(x) = x2 is not an open map, as the image of the open interval (−1, 1) is the half-open interval [0, 1).

Insilico is known for hiring mainly through hackathons such as their own molhack online hackathon. The company has multiple collaborations in the applications of next-generation artificial intelligence technologies such as the generative adversarial networks and reinforcement learning to the generation of novel molecular structures with desired properties. In conjunction with Alan Aspuru-Guzik's group at Harvard, they have published a journal article about an improved GAN architecture for molecular generation which combines GANs, reinforcement learning, and a differentiable neural computer. Insilico's recent paper in Nature Communications describing the iPANDA dimensionality reduction algorithm included collaborators from 11 institutions.

A spin network is a one-dimensional graph, together with labels on its vertices and edges which encode aspects of a spatial geometry. A spin network is defined as a diagram like the Feynman diagram which makes a basis of connections between the elements of a differentiable manifold for the Hilbert spaces defined over them, and for computations of amplitudes between two different hypersurfaces of the manifold. Any evolution of the spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network. A spin foam is analogous to quantum history.

The obstruction theories due to Milnor, Kervaire, Kirby, Siebenmann, Sullivan, Donaldson show that only a minority of topological manifolds possess differentiable structures and these are not necessarily unique. Sullivan's result on Lipschitz and quasiconformal structures shows that any topological manifold in dimension different from 4 possesses such a structure which is unique (up to isotopy close to identity). The quasiconformal structures and more generally the Lp-structures, p > n(n+1)/2, introduced by M. Hilsum , are the weakest analytical structures on topological manifolds of dimension n for which the index theorem is known to hold.

On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. It allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus. If a differential -form is thought of as measuring the flux through an infinitesimal -parallelotope at each point of the manifold, then its exterior derivative can be thought of as measuring the net flux through the boundary of a -parallelotope at each point.

In a deeper way, this theorem relates the topology of the domain of integration to the structure of the differential forms themselves; the precise connection is known as de Rham's theorem. The general setting for the study of differential forms is on a differentiable manifold. Differential -forms are naturally dual to vector fields on a manifold, and the pairing between vector fields and -forms is extended to arbitrary differential forms by the interior product. The algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback under smooth functions between two manifolds.

Hermann Weyl gave an intrinsic definition for differentiable manifolds in 1912. During the 1930s Hassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through differential geometry and Lie group theory. The Whitney embedding theorem showed that manifolds intrinsically defined by charts could always be embedded in Euclidean space, as in the extrinsic definition, showing that the two concepts of manifold were equivalent. Due to this unification, it is said to be the first complete exposition of the modern concept of manifold.

The total space of a cotangent bundle has the structure of a symplectic manifold. Cotangent vectors are sometimes called covectors. One can also define the cotangent bundle as the bundle of 1-jets of functions from M to R. Elements of the cotangent space can be thought of as infinitesimal displacements: if f is a differentiable function we can define at each point p a cotangent vector dfp, which sends a tangent vector Xp to the derivative of f associated with Xp. However, not every covector field can be expressed this way. Those that can are referred to as exact differentials.

Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include the squaring function x^2 and the exponential function e^x. In simple terms, a convex function refers to a function that is in the shape of a cup \cup, and a concave function is in the shape of a cap \cap.

The first stable open-source implementation was published in 2018 at the 27th International Conference on Artificial Neural Networks, receiving a best-paper award. Other open source implementations of NTMs exist but are not sufficiently stable for production use. The developers either report that the gradients of their implementation sometimes become NaN during training for unknown reasons and cause training to fail; report slow convergence; or do not report the speed of learning of their implementation. Differentiable neural computers are an outgrowth of Neural Turing machines, with attention mechanisms that control where the memory is active, and improve performance.

A Lie group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse, page 3. Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics.

In mathematics, Fermat's theorem (also known as interior extremum theorem) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point (the function's derivative is zero at that point). Fermat's theorem is a theorem in real analysis, named after Pierre de Fermat. By using Fermat's theorem, the potential extrema of a function \displaystyle f, with derivative \displaystyle f', are found by solving an equation in \displaystyle f'. Fermat's theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points (not a maximum or minimum).

Values scales were first developed by an international group of psychologists whose goal was to create a unique self-report instrument that measured intrinsic and extrinsic values for use in the lab and in the clinic. The psychologists called their project the Work Importance Study (WIS). The original values scale measured the following values, listed in alphabetical order: ability utilization, achievement, advancement, aesthetics, altruism, authority, autonomy, creativity, cultural identity, economic rewards, economic security, life style, personal development, physical activity, physical prowess, prestige, risk, social interaction, social relations, variety, and working conditions. Some of the listed values were intended to be inter-related, but conceptually differentiable.

In mathematics, the dual bundle of a vector bundle is a vector bundle whose fibers are the dual spaces to the fibers of E. The dual bundle can be constructed using the associated bundle construction by taking the dual representation of the structure group. Specifically, given a local trivialization of E with transition functions t, a local trivialization of E is given by the same open cover of X with transition functions t = (t)−1 (the inverse of the transpose). The dual bundle E is then constructed using the fiber bundle construction theorem. For example, the dual to the tangent bundle of a differentiable manifold is the cotangent bundle.

The German Gemmological Association supports basic as well as applied research in all fields of gemology, especially through funding the German Gemstone Research Foundation (DSEF).Homepage der DSEF Major aspects of the research include: \- Identifying properties of new gemstone-materials \- Understanding locality-specific and genetical characteristics of gemstones \- Differentiable marks of natural and synthetic gemstones, their imitations and artificial products. \- Analysing of characteristic traits of artificially treated and enhanced gemstone-materials. All the results are worked into the current technological transfer of the market. This can be through publishing the results in the own magazine “GEMMOLOGIE”, lectures and meetings in Germany (own annual symposia) and abroad.

When on the green, the ball may be picked up to be cleaned or if it is in the way of an opponent's putting line; there are certain other circumstances in which a ball may be lifted. In these cases, the ball's position must first be marked using a ball marker; this is typically a round, flat piece of metal or plastic that is differentiable from others in use. Ball markers are often integrated into other accessories, such as divot tools, scorekeeping tools or tee holders, and in the absence of a purpose-made marker, a small coin such as a penny is acceptable.

In mathematics, the category of manifolds, often denoted Manp, is the category whose objects are manifolds of smoothness class Cp and whose morphisms are p-times continuously differentiable maps. This is a category because the composition of two Cp maps is again continuous and of class Cp. One is often interested only in Cp-manifolds modeled on spaces in a fixed category A, and the category of such manifolds is denoted Manp(A). Similarly, the category of Cp-manifolds modeled on a fixed space E is denoted Manp(E). One may also speak of the category of smooth manifolds, Man∞, or the category of analytic manifolds, Manω.

A differential variety defined by implicit equations in the n-dimensional space Rn is the set of the common zeros of a finite set of differentiable functions in n variables : f_1(x_1,\ldots,x_n), \ldots, f_k(x_1,\ldots,x_n). The Jacobian matrix of the variety is the k×n matrix whose i-th row is the gradient of fi. By the implicit function theorem, the variety is a manifold in the neighborhood of a point where the Jacobian matrix has rank k. At such a point P, the normal vector space is the vector space generated by the values at P of the gradient vectors of the fi.

For example, in calculus if f is a differentiable function defined on some interval, then it is sufficient to show that the derivative is always positive or always negative on that interval. In linear algebra, if f is a linear transformation it is sufficient to show that the kernel of f contains only the zero vector. If f is a function with finite domain it is sufficient to look through the list of images of each domain element and check that no image occurs twice on the list. A graphical approach for a real-valued function f of a real variable x is the horizontal line test.

Splines are piecewise-smooth, hence in PDIFF, but not globally smooth or piecewise-linear, hence not in DIFF or PL. In geometric topology, PDIFF, for piecewise differentiable, is the category of piecewise-smooth manifolds and piecewise-smooth maps between them. It properly contains DIFF (the category of smooth manifolds and smooth functions between them) and PL (the category of piecewise linear manifolds and piecewise linear maps between them), and the reason it is defined is to allow one to relate these two categories. Further, piecewise functions such as splines and polygonal chains are common in mathematics, and PDIFF provides a category for discussing them.

The black family is differentiable from the white family through this report's conceptions that black families are impoverished due to the manner in which they dissolve the typical white family structure. The role reversal within black families—that the mother is the primary and present authority in the household and the fathers are absent, according to the report—deserves culpability for black familial "deficiencies". Spillers' work is a critique of sexism and racism in psychoanalysis of black feminism. Through naming typical stereotypes ascribed to black women, Spillers begins to refute the negative perceptions ascribed to the black family and black familial matriarchal structure asserted throughout the Moynihan Report.

The classification of complex manifolds is much more subtle than that of differentiable manifolds. For example, while in dimensions other than four, a given topological manifold has at most finitely many smooth structures, a topological manifold supporting a complex structure can and often does support uncountably many complex structures. Riemann surfaces, two dimensional manifolds equipped with a complex structure, which are topologically classified by the genus, are an important example of this phenomenon. The set of complex structures on a given orientable surface, modulo biholomorphic equivalence, itself forms a complex algebraic variety called a moduli space, the structure of which remains an area of active research.

For example, if the input sequence is a speech signal corresponding to a spoken digit, the final target output at the end of the sequence may be a label classifying the digit. For each sequence, its error is the sum of the deviations of all activations computed by the network from the corresponding target signals. For a training set of numerous sequences, the total error is the sum of the errors of all individual sequences. To minimize total error, gradient descent can be used to change each weight in proportion to its derivative with respect to the error, provided the non-linear activation functions are differentiable.

An extremely special case of this is the following: if a differentiable function from reals to the reals has nonzero derivative at a zero of the function, then the zero is simple, i.e. it the graph is transverse to the x-axis at that zero; a zero derivative would mean a horizontal tangent to the curve, which would agree with the tangent space to the x-axis. For an infinite-dimensional example, the d-bar operator is a section of a certain Banach space bundle over the space of maps from a Riemann surface into an almost-complex manifold. The zero set of this section consists of holomorphic maps.

The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas, of which the midpoint rule is similar to the trapezoid rule. Simpson's rule is another member of the same family, and in general has faster convergence than the trapezoidal rule for functions which are twice continuously differentiable, though not in all specific cases. However, for various classes of rougher functions (ones with weaker smoothness conditions), the trapezoidal rule has faster convergence in general than Simpson's rule. Moreover, the trapezoidal rule tends to become extremely accurate when periodic functions are integrated over their periods, which can be analyzed in various ways.

This is not differentiable at t = 0, showing that the Cauchy distribution has no expectation. Also, the characteristic function of the sample mean of n independent observations has characteristic function , using the result from the previous section. This is the characteristic function of the standard Cauchy distribution: thus, the sample mean has the same distribution as the population itself. The logarithm of a characteristic function is a cumulant generating function, which is useful for finding cumulants; some instead define the cumulant generating function as the logarithm of the moment- generating function, and call the logarithm of the characteristic function the second cumulant generating function.

Gradient descent is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. To find a local minimum of a function using gradient descent, we take steps proportional to the negative of the gradient (or approximate gradient) of the function at the current point. But if we instead take steps proportional to the positive of the gradient, we approach a local maximum of that function; the procedure is then known as gradient ascent. Gradient descent is generally attributed to Cauchy, who first suggested it in 1847, but its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944.

The highest order of derivation that appears in a differentiable equation is the order of the equation. The term , which does not depend on the unknown function and its derivatives, is sometimes called the constant term of the equation (by analogy with algebraic equations), even when this term is a non-constant function. If the constant term is the zero function, then the differential equation is said to be homogeneous, as it is a homogeneous polynomial in the unknown function and its derivatives. The equation obtained by replacing, in a linear differential equation, the constant term by the zero function is the associated homogeneous equation.

SGLD can be applied to the optimization of non-convex objective functions, shown here to be a sum of Gaussians. Stochastic gradient Langevin dynamics (SGLD), is an optimization technique composed of characteristics from Stochastic gradient descent, a Robbins–Monro optimization algorithm, and Langevin dynamics, a mathematical extension of molecular dynamics models. Like stochastic gradient descent, SGLD is an iterative optimization algorithm which introduces additional noise to the stochastic gradient estimator used in SGD to optimize a differentiable objective function. Unlike traditional SGD, SGLD can be used for Bayesian learning, since the method produces samples from a posterior distribution of parameters based on available data.

Even in a three-dimensional Euclidean space, there is typically no natural way to prescribe a basis of the tangent plane, and so it is conceived of as an abstract vector space rather than a real coordinate space. The tangent space is the generalization to higher- dimensional differentiable manifolds. Riemannian manifolds are manifolds whose tangent spaces are endowed with a suitable inner product.. See also Lorentzian manifold. Derived therefrom, the Riemann curvature tensor encodes all curvatures of a manifold in one object, which finds applications in general relativity, for example, where the Einstein curvature tensor describes the matter and energy content of space-time.

As defined in the preceding section, the Lambert azimuthal projection of the unit sphere is undefined at (0, 0, 1). It sends the rest of the sphere to the open disk of radius 2 centered at the origin (0, 0) in the plane. It sends the point (0, 0, −1) to (0, 0), the equator z = 0 to the circle of radius centered at (0, 0), and the lower hemisphere z < 0 to the open disk contained in that circle. The projection is a diffeomorphism (a bijection that is infinitely differentiable in both directions) between the sphere (minus (0, 0, 1)) and the open disk of radius 2.

The exterior algebra has notable applications in differential geometry, where it is used to define differential forms. Differential forms are mathematical objects that evaluate the length of vectors, areas of parallelograms, and volumes of higher-dimensional bodies, so they can be integrated over curves, surfaces and higher dimensional manifolds in a way that generalizes the line integrals and surface integrals from calculus. A differential form at a point of a differentiable manifold is an alternating multilinear form on the tangent space at the point. Equivalently, a differential form of degree k is a linear functional on the k-th exterior power of the tangent space.

Pseudoathetosis is a movement disorder, very similar to athetosis, in which the symptoms are not differentiable from those of actual athetosis, however the underlying cause is different. While actual athetosis is caused by damage to the brain, specifically in the basal ganglia, pseudoathetosis is caused by the loss of proprioception. The loss in proprioception is caused by damage to the area between the primary somatosensory cortex and the muscle spindles and joint receptors. Additionally, when observing an MRI, it can be seen that in the brain of a pseudoathetoid patient, lesions on the brain are not seen in the basal ganglia, the area that is oftentimes the cause of athetosis.

Neighbourhood components analysis aims at "learning" a distance metric by finding a linear transformation of input data such that the average leave-one-out (LOO) classification performance is maximized in the transformed space. The key insight to the algorithm is that a matrix A corresponding to the transformation can be found by defining a differentiable objective function for A, followed by use of an iterative solver such as conjugate gradient descent. One of the benefits of this algorithm is that the number of classes k can be determined as a function of A, up to a scalar constant. This use of the algorithm therefore addresses the issue of model selection.

He died in Kyoto in September 2012. Nishida's approach, dubbed "philological linguistics" by Shōgaito Masahiro, involved the linguistic study of textual works and the integration of fieldwork on contemporary languages with philological study. In the context of this overall approach, one of his major theoretical contributions was the notion of "sonus grammae", the phonology that is implied by a script system as analytically as differentiable from the phonology of a language that uses the particular script system at a certain time and place.Yabu, Shirō 藪司郎 (2014). “Professor Nishida, Tatsuo and the study of Tibeto-Burman languages.” Memoirs of the research department of the Toyo Bunko 72: 181, 187.

In addition, there are two sharp bending points (non-differentiable discontinuities) within the probability distribution, which we will call b and c, which occur between a and d, such that a \leq b \leq c \leq d. The image to the right shows a perfectly linear trapezoidal distribution. However, not all trapezoidal distributions are so precisely shaped. In the standard case, where the middle part of the trapezoid is completely flat, and the side ramps are perfectly linear, all of the values between c and d will occur with equal frequency, and therefore all such points will be modes (local frequency maxima) of the distribution.

Bregman divergences correspond to convex functions on convex sets. Given a strictly convex, continuously-differentiable function on a convex set, known as the Bregman generator, the Bregman divergence measures the convexity of: the error of the linear approximation of from as an approximation of the value at : :D_F(p, q) = F(p)-F(q)-\langle abla F(q), p-q\rangle. The dual divergence to a Bregman divergence is the divergence generated by the convex conjugate of the Bregman generator of the original divergence. For example, for the squared Euclidean distance, the generator is , while for the relative entropy the generator is the negative entropy .

Gradient boosting is a machine learning technique for regression and classification problems, which produces a prediction model in the form of an ensemble of weak prediction models, typically decision trees. It builds the model in a stage-wise fashion like other boosting methods do, and it generalizes them by allowing optimization of an arbitrary differentiable loss function. The idea of gradient boosting originated in the observation by Leo Breiman that boosting can be interpreted as an optimization algorithm on a suitable cost function. Explicit regression gradient boosting algorithms were subsequently developed by Jerome H. Friedman, simultaneously with the more general functional gradient boosting perspective of Llew Mason, Jonathan Baxter, Peter Bartlett and Marcus Frean.

In complex analysis, a branch of mathematics, the identity theorem for holomorphic functions states: given functions f and g holomorphic on a domain D (open and connected subset), if f = g on some S \subseteq D, where S has an accumulation point, then f = g on D. Thus a holomorphic function is completely determined by its values on a single open neighborhood in D, or even a countable subset of D (provided this contains a converging sequence). This is not true for real-differentiable functions. In comparison, holomorphy, or complex-differentiability, is a much more rigid notion. Informally, one sometimes summarizes the theorem by saying holomorphic functions are "hard" (as opposed to, say, continuous functions which are "soft").

This has strikingly simple immediate corollaries, such as the fact that any closed smooth 3-manifold which admits a Riemannian metric of positive curvature also admits a Riemannian metric of constant positive sectional curvature. Such results are notable in highly restricting the topology of such manifolds; the space forms of positive curvature are largely understood. There are other corollaries, such as the fact that the topological space of Riemannian metrics of positive Ricci curvature on a closed smooth 3-manifold is path-connected. These "convergence theorems" of Hamilton have been extended by later authors, in the 2000s, to give a proof of the differentiable sphere theorem, which had been a major conjecture in Riemannian geometry since the 1960s.

If H forms a field under the usual addition and multiplication of functions then so will H modulo this equivalence relation under the induced addition and multiplication operations. Moreover, if every function in H is eventually differentiable and the derivative of any function in H is also in H then H modulo the above equivalence relation is called a Hardy field. Elements of a Hardy field are thus equivalence classes and should be denoted, say, [f]∞ to denote the class of functions that are eventually equal to the representative function f. However, in practice the elements are normally just denoted by the representatives themselves, so instead of [f]∞ one would just write f.

John von Neumann, working with Oskar Morgenstern on the theory of games, broke new mathematical ground in 1944 by extending functional analytic methods related to convex sets and topological fixed-point theory to economic analysis. Their work thereby avoided the traditional differential calculus, for which the maximum–operator did not apply to non-differentiable functions. Continuing von Neumann's work in cooperative game theory, game theorists Lloyd S. Shapley, Martin Shubik, Hervé Moulin, Nimrod Megiddo, Bezalel Peleg influenced economic research in politics and economics. For example, research on the fair prices in cooperative games and fair values for voting games led to changed rules for voting in legislatures and for accounting for the costs in public–works projects.

Informally, the tangent bundle of a manifold (in this case a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle S1, see Examples section: all tangents to a circle lie in the plane of the circle.

Another way of proving the chain rule is to measure the error in the linear approximation determined by the derivative. This proof has the advantage that it generalizes to several variables. It relies on the following equivalent definition of differentiability at a point: A function g is differentiable at a if there exists a real number g′(a) and a function ε(h) that tends to zero as h tends to zero, and furthermore :g(a + h) - g(a) = g'(a) h + \varepsilon(h) h. Here the left-hand side represents the true difference between the value of g at a and at , whereas the right-hand side represents the approximation determined by the derivative plus an error term.

The set of all complex numbers with absolute value 1 (corresponding to points on the circle of center 0 and radius 1 in the complex plane) is a Lie group under complex multiplication: the circle group. Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. One of the key ideas in the theory of Lie groups is to replace the global object, the group, with its local or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its Lie algebra. Lie groups play an enormous role in modern geometry, on several different levels.

LaTeX is typically distributed along with plain TeX under a free software license: the LaTeX Project Public License (LPPL). The LPPL is not compatible with the GNU General Public License, as it requires that modified files must be clearly differentiable from their originals (usually by changing the filename); this was done to ensure that files that depend on other files will produce the expected behavior and avoid dependency hell. The LPPL is DFSG compliant as of version 1.3. As free software, LaTeX is available on most operating systems, which include UNIX (Solaris, HP-UX, AIX), BSD (FreeBSD, macOS, NetBSD, OpenBSD), Linux (Red Hat, Debian, Arch, Gentoo), Windows, DOS, RISC OS, AmigaOS and Plan9.

Although the term well-behaved statistic often seems to be used in the scientific literature in somewhat the same way as is well-behaved in mathematics (that is, to mean "non-pathological") it can also be assigned precise mathematical meaning, and in more than one way. In the former case, the meaning of this term will vary from context to context. In the latter case, the mathematical conditions can be used to derive classes of combinations of distributions with statistics which are well-behaved in each sense. First Definition: The variance of a well-behaved statistical estimator is finite and one condition on its mean is that it is differentiable in the parameter being estimated.

The number is sometimes called the order or the multiplicity of the cusp, and is equal to the degree of the nonzero part of lowest degree of . These definitions have been generalized to curves defined by differentiable functions by René Thom and Vladimir Arnold, in the following way. A curve has a cusp at a point if there is a diffeomorphism of a neighborhood of the point in the ambient space, which maps the curve onto one of the above-defined cusps. In some contexts, and in the remainder of this article, the definition of a cusp is restricted to the case of cusps of order two—that is, the case where .

Alberti has studied at Scuola Normale Superiore under the guide of Giuseppe Buttazzo and Ennio De Giorgi; he is professor of mathematics at the University of Pisa. Alberti is mostly known for two remarkable theorems he proved at the beginning of his career, that eventually found applications in various branches of modern mathematical analysis. The first is a very general Lusin type theorem for gradients asserting that every Borel vector field can be realized as the gradient of a continuously differentiable function outside a closed subset of a priori prescribed (small) measure. The second asserts the rank-one property of the distributional derivatives of functions with bounded variation, thereby verifying a conjecture of De Giorgi.

Luis Caffarelli, Robert Kohn, and Nirenberg studied the three-dimensional incompressible Navier-Stokes equations, showing that the set of spacetime points at which weak solutions fail to be differentiable must, roughly speaking, fill less space than a curve. This is known as a "partial regularity" result. In his description of the conjectural regularity of the Navier-Stokes equations as a Millennium prize problem, Charles Fefferman refers to Caffarelli-Kohn-Nirenberg's result as the "best partial regularity theorem known so far" on the problem. As a by- product of their work on the Navier-Stokes equations, Caffarelli, Kohn, and Nirenberg (in a separate paper) extended Nirenberg's earlier work on the Gagliardo-Nirenberg interpolation inequality to certain weighted norms.

The fixed points of τ correspond to the boundary points of Σ/τ. The surface Σ is called an "analytic double" of Σ/τ. The Klein surfaces form a category; a morphism from the Klein surface X to the Klein surface Y is a differentiable map f:X→Y which on each coordinate patch is either holomorphic or the complex conjugate of a holomorphic map and furthermore maps the boundary of X to the boundary of Y. There is a one-to-one correspondence between smooth projective algebraic curves over the reals (up to isomorphism) and compact connected Klein surfaces (up to equivalence). The real points of the curve correspond to the boundary points of the Klein surface.

For a strictly convex function, the Legendre transformation can be interpreted as a mapping between the graph of the function and the family of tangents of the graph. (For a function of one variable, the tangents are well-defined at all but at most countably many points, since a convex function is differentiable at all but at most countably many points.) The equation of a line with slope and -intercept is given by . For this line to be tangent to the graph of a function at the point requires :f\left(x_0\right) = p x_0 + b and :p = f'(x_0). The function f'is strictly monotone as the derivative of a strictly convex function.

More precisely, the extrinsic geometry is controlled by the extrinsic geometry of the isometric embedding uniquely determined by the intrinsic geometry. Shi and Tam's proof adopts a method, due to Robert Bartnik, of using parabolic partial differential equations to construct noncompact Riemannian manifolds-with-boundary of nonnegative scalar curvature and prescribed boundary behavior. By combining Bartnik's construction with the given compact manifold-with-boundary, one obtains a complete Riemannian manifold which is non-differentiable along a closed and smooth hypersurface. By using Bartnik's method to relate the geometry near infinity to the geometry of the hypersurface, and by proving a positive energy theorem in which certain singularities are allowed, Shi and Tam's result follows.

As Gleason told the story, the key insight of his proof was to apply the fact that monotonic functions are differentiable almost everywhere. On finding the solution, he took a week of leave to write it up, and it was printed in the Annals of Mathematics alongside the paper of Montgomery and Zippin; another paper a year later by Hidehiko Yamabe removed some technical side conditions from Gleason's proof. The "unrestricted" version of Hilbert's fifth problem, closer to Hilbert's original formulation, considers both a locally Euclidean group G and another manifold M on which G has a continuous action. Hilbert asked whether, in this case, M and the action of G could be given a real analytic structure.

Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled the formulation of Einstein's general theory of relativity, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.

He has held postdoctoral positions and fellowships: at Harvard University, Cambridge MA, 1980 and 1981; Freie Universität Berlin, 1979; Università degli Studi di Milano, 1971–1978. His past research activities have explored: topological Quantum Field Theories in any dimension, BF theories, Cohomology of imbedded loops, Higher dimensional knots; Links, knots and quantum groups; Differential geometrical aspects of string and field theories, Virasoro and Krichever-Novikov algebras; Anomalies in Quantum field theory and their differential geometrical interpretation; and Foundational aspects of Quantum Mechanics. His past and present teaching activities include: Mathematical Methods of Physics: complex analysis and functional analysis; Differential Geometry in Mathematical Physics: differentiable manifolds, Lie groups, principal bundles and their connections, Dirac operators; and Calculus.

Illustration of critique of De fluxionibus libri duo published in Acta Eruditorum, 1747 Maclaurin used Taylor series to characterize maxima, minima, and points of inflection for infinitely differentiable functions in his Treatise of Fluxions. Maclaurin attributed the series to Brook Taylor, though the series was known before to Newton and Gregory, and in special cases to Madhava of Sangamagrama in fourteenth century India. Nevertheless, Maclaurin received credit for his use of the series, and the Taylor series expanded around 0 is sometimes known as the Maclaurin series. Colin Maclaurin (1698–1746) Maclaurin also made significant contributions to the gravitation attraction of ellipsoids, a subject that furthermore attracted the attention of d'Alembert, A.-C.

There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function. More precisely, given a system of equations (often abbreviated into ), the theorem states that, under a mild condition on the partial derivatives (with respect to the s) at a point, the variables are differentiable functions of the in some neighborhood of the point. As these functions can generally not be expressed in closed form, they are implicitly defined by the equations, and this motivated the name of the theorem.

The method was originally known as the method of multipliers, and was studied much in the 1970 and 1980s as a good alternative to penalty methods. It was first discussed by Magnus Hestenes, and by Michael Powell in 1969. The method was studied by R. Tyrrell Rockafellar in relation to Fenchel duality, particularly in relation to proximal-point methods, Moreau–Yosida regularization, and maximal monotone operators: These methods were used in structural optimization. The method was also studied by Dimitri Bertsekas, notably in his 1982 book, together with extensions involving nonquadratic regularization functions, such as entropic regularization, which gives rise to the "exponential method of multipliers," a method that handles inequality constraints with a twice differentiable augmented Lagrangian function.

While pure mathematics has existed as an activity since at least Ancient Greece, the concept was elaborated upon around the year 1900, after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need to renew the concept of mathematical rigor and rewrite all mathematics accordingly, with a systematic use of axiomatic methods. This led many mathematicians to focus on mathematics for its own sake, that is, pure mathematics. Nevertheless, almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories.

The Yudin-Nemirovsky analysis implies that no method can be fast on high- dimensional problems that lack convexity: > "The catastrophic growth [in the number of iterations needed to reach an > approximate solution of a given accuracy] as [the number of dimensions > increases to infinity] shows that it is meaningless to pose the question of > constructing universal methods of solving ... problems of any appreciable > dimensionality 'generally'. It is interesting to note that the same > [conclusion] holds for ... problems generated by uni-extremal [that is, > unimodal] (but not convex) functions." > > Page 7 summarizes the later discussion of : When applied to twice continuously differentiable problems, the LJ heuristic's rate of convergence decreases as the number of dimensions increases.

In the 70's, Jürgen Moser sketched a proof, based on K.A.M. theory, that outer billiards relative to a 6-times-differentiable shape of positive curvature has all orbits bounded. In 1982, Raphael Douady gave the full proof of this result. A big advance in the polygonal case came over a period of several years when three teams of authors, Vivaldi-Shaidenko (1987), Kolodziej (1989), and Gutkin-Simanyi (1991), each using different methods, showed that outer billiards relative to a quasirational polygon has all orbits bounded. The notion of quasirational is technical (see references) but it includes the class of regular polygons and convex rational polygons, namely those convex polygons whose vertices have rational coordinates.

Biogeography-based optimization (BBO) is an evolutionary algorithm (EA) that optimizes a function by stochastically and iteratively improving candidate solutions with regard to a given measure of quality, or fitness function. BBO belongs to the class of metaheuristics since it includes many variations, and since it does not make any assumptions about the problem and can therefore be applied to a wide class of problems. BBO is typically used to optimize multidimensional real-valued functions, but it does not use the gradient of the function, which means that it does not require the function to be differentiable as required by classic optimization methods such as gradient descent and quasi-newton methods. BBO can therefore be used on discontinuous functions.

Just as there are various types of manifolds, there are various types of maps of manifolds. PDIFF serves to relate DIFF and PL, and it is equivalent to PL. In geometric topology, the basic types of maps correspond to various categories of manifolds: DIFF for smooth functions between differentiable manifolds, PL for piecewise linear functions between piecewise linear manifolds, and TOP for continuous functions between topological manifolds. These are progressively weaker structures, properly connected via PDIFF, the category of piecewise-smooth maps between piecewise-smooth manifolds. In addition to these general categories of maps, there are maps with special properties; these may or may not form categories, and may or may not be generally discussed categorically.

Delta rule (DR) is similar to the Perceptron Learning Rule (PLR), with some differences: # Error (δ) in DR is not restricted to having values of 0, 1, or -1 (as in PLR), but may have any value # DR can be derived for any differentiable output/activation function f, whereas in PLR only works for threshold output function Sometimes only when the Widrow-Hoff is applied to binary targets specifically, it is referred to as Delta Rule, but the terms seem to be used often interchangeably. The delta rule is considered to a special case of the back-propagation algorithm. Delta rule also closely resembles the Rescorla-Wagner model under which Pavlovian conditioning occurs.

For smooth or polygonal curves, the Jordan curve theorem can be proved in a straightforward way. Indeed the curve has a tubular neighbourhood, defined in the smooth case by the field of unit normal vectors to the curve or in the polygonal case by points at a distance of less than ε from the curve. In a neighbourhood of a differentiable point on the curve, there is a coordinate change in which the curve becomes the diameter of an open disk. Taking a point not on the curve, a straight line aimed at the curve starting at the point will eventually meet the tubular neighborhood; the path can be continued next to the curve until it meets the disk.

Concretely, the integral from 0 to any particular t is a random variable, defined as a limit of a certain sequence of random variables. The paths of Brownian motion fail to satisfy the requirements to be able to apply the standard techniques of calculus. So with the integrand a stochastic process, the Itô stochastic integral amounts to an integral with respect to a function which is not differentiable at any point and has infinite variation over every time interval. The main insight is that the integral can be defined as long as the integrand H is adapted, which loosely speaking means that its value at time t can only depend on information available up until this time.

SSCP used to be a way to discover new DNA polymorphisms apart from DNA sequencing but is now being supplanted by sequencing techniques on account of efficiency and accuracy. These days, SSCP is most applicable as a diagnostic tool in molecular biology. It can be used in genotyping to detect homozygous individuals of different allelic states, as well as heterozygous individuals that should each demonstrate distinct patterns in an electrophoresis experiment. SSCP is also widely used in virology to detect variations in different strains of a virus, the idea being that a particular virus particle present in both strains will have undergone changes due to mutation, and that these changes will cause the two particles to assume different conformations and, thus, be differentiable on an SSCP gel.

The intuition is based on the behavior of polynomial functions. Assume that function f has a maximum at x0, the reasoning being similar for a function minimum. If \displaystyle x_0 \in (a,b) is a local maximum then, roughly, there is a (possibly small) neighborhood of \displaystyle x_0 such as the function "is increasing before" and "decreasing after"This intuition is only correct for continuously differentiable \left(C^1\right) functions, while in general it is not literally correct--a function need not be increasing up to a local maximum: it may instead be oscillating, so neither increasing nor decreasing, but simply the local maximum is greater than any values in a small neighborhood to the left or right of it. See details in the pathologies.

Provided that the speed c is a constant, not dependent on frequency (the dispersionless case), then the most general solution is :p = f(c t - x) + g(c t + x) where f and g are any two twice-differentiable functions. This may be pictured as the superposition of two waveforms of arbitrary profile, one (f) travelling up the x-axis and the other (g) down the x-axis at the speed c. The particular case of a sinusoidal wave travelling in one direction is obtained by choosing either f or g to be a sinusoid, and the other to be zero, giving :p=p_0 \sin(\omega t \mp kx). where \omega is the angular frequency of the wave and k is its wave number.

Suppose that the steepest slope on a hill is 40%. A road going directly uphill has slope 40%, but a road going around the hill at an angle will have a shallower slope. For example, if the road is at a 60° angle from the uphill direction (when both directions are projected onto the horizontal plane), then the slope along the road will be the dot product between the gradient vector and a unit vector along the road, namely 40% times the cosine of 60°, or 20%. More generally, if the hill height function is differentiable, then the gradient of dotted with a unit vector gives the slope of the hill in the direction of the vector, the directional derivative of along the unit vector.

Titchener is also remembered for coining the English word "empathy" in 1909 as a translation of the German word "Einfühlungsvermögen", a new phenomenon explored at the end of 19th century mainly by Theodor Lipps. "Einfühlungsvermögen" was later re-translated as "Empathie", and is still in use that way in German. It should be stressed that Titchener used the term "empathy" in a personal way, strictly intertwined with his methodological use of introspection, and to refer to at least three differentiable phenomena.Titchener E.B. (1909/2014) Introspection and empathy Dialogues in Philosophy, Mental and Neuro Sciences 2014; 7: 25–30 Titchener's effect on the history of psychology, as it is taught in classrooms, was partially the work of his student Edwin Boring.

In three dimensions, a surface normal, or simply normal, to a surface at point P is a vector perpendicular to the tangent plane of the surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality (right angles). The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at point P is the set of vectors which are orthogonal to the tangent space at P. Normal vectors are of special interest in the case of smooth curves and smooth surfaces.

After World War I finished, Vauck undertook study of mathematics and physics at the Technical University of Dresden. In 1922, he passed the exam for a higher education office, and two years later, under supervision of Gerhard Kowalewski, Vauck was promoted to Dr. phil with a thesis titled A generalisation of Bolzano's continuous but non-differentiable function () which is within Mathematics Subject Classification 26 Real functions. Vauck's first career position was as a teacher at the secondary school in Thum and then from 1926 as a teacher at the secondary school in Bautzen. After the war from 1948, he became a teacher in Bautzen and later became a lecturer in physics and electrical engineering at the engineering school in Bautzen.

The function f is said to be of (differentiability) class Ck if the derivatives f′, f″, ..., f(k) exist and are continuous (continuity is implied by differentiability for all the derivatives except for f(k)). The function f is said to be infinitely differentiable, smooth, or of class C∞, if it has derivatives of all orders. The function f is said to be of class Cω, or analytic, if f is smooth and if its Taylor series expansion around any point in its domain converges to the function in some neighborhood of the point. Cω is thus strictly contained in C∞. Bump functions are examples of functions in C∞ but not in Cω. To put it differently, the class C0 consists of all continuous functions.

In point set topology, a set A is closed if it contains all its boundary points. The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such as metric spaces, differentiable manifolds, uniform spaces, and gauge spaces. An alternative characterization of closed sets is available via sequences and nets. A subset A of a topological space X is closed in X if and only if every limit of every net of elements of A also belongs to A. In a first-countable space (such as a metric space), it is enough to consider only convergent sequences, instead of all nets.

Precisely, an inflection point of a doubly continuously differentiable (C^2) curve on the surface of a sphere is a point p with the following property: let I be the connected component containing p of the intersection of the curve with its tangent great circle at p. (For most curves I will just be p itself, but it could also be an arc of the great circle.) Then, for p to be an inflection point, every neighborhood of I must contain points of the curve that belong to both of the hemispheres separated by this great circle. The theorem states that every C^2 curve that partitions the sphere into two equal-area components has at least four inflection points in this sense.

Let U be an open set in a manifold , be the space of smooth, differentiable 1-forms on U, and F be a submodule of of rank r, the rank being constant in value over U. The Frobenius theorem states that F is integrable if and only if for every in the stalk Fp is generated by r exact differential forms. Geometrically, the theorem states that an integrable module of -forms of rank r is the same thing as a codimension-r foliation. The correspondence to the definition in terms of vector fields given in the introduction follows from the close relationship between differential forms and Lie derivatives. Frobenius' theorem is one of the basic tools for the study of vector fields and foliations.

This linearized approach, originally devised to solve medium-sized CGE models on early computers, has since proved capable (on modern computers) of solving very large models. Additionally it has lent itself to some interesting extensions, such as: a Gaussian quadrature methodDeVuyst, E. A. and P. V. Preckel (1997), "Sensitivity Analysis Revisited: A Quadrature-Based Approach", Journal of Policy Modeling, 19(2) pp. 175–185. of estimating confidence intervals for model results from known distributions of shock or parameter values; a way to formulate inequality constraints or non-differentiable equations as complementarities;Harrison, W. J., J. M. Horridge, K. R. Pearson and G. Wittwer (2004), "A Practical Method for Explicitly Modeling Quotas and Other Complementarities", Computational Economics, June 2004, Vol. 23(4), pp. 325–341.

In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the problem can be stated as follows: :Given a function f that has values everywhere on the boundary of a region in Rn, is there a unique continuous function u twice continuously differentiable in the interior and continuous on the boundary, such that u is harmonic in the interior and u = f on the boundary? This requirement is called the Dirichlet boundary condition.

A manifold is orientable if it has a consistent choice of orientation, and a connected orientable manifold has exactly two different possible orientations. In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing a formulation in terms of differential forms. An important generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a fiber bundle) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.

Stanisław Ruziewicz (29 August 1889 – 12 July 1941) was a Polish mathematician and one of the founders of the Lwów School of Mathematics. He was a former student of Wacław Sierpiński, earning his doctorate in 1913 from the University of Lwów; his thesis concerned continuous functions that are not differentiable. He became a professor at the same university (then named Jan Kazimierz University) and rector of the Academy of Foreign Trade in Lwów. During the Second World War, Ruziewicz's home city of Lwów was annexed by the Ukrainian Soviet Socialist Republic, but then taken over by the General Government of German-occupied Poland in July 1941; Ruziewicz was arrested and murdered by the Gestapo on 12 July 1941 in Lviv, during the Massacre of Lviv professors.

PAX6 is essential is the formation of the retina, lens and cornea due to its role in early cell determination when forming precursors of these structures such as the optic vesicle and overlying surface ectoderm. PAX10 mutations also hinder nasal cavity development due to the similar precursor structures that in small eye mice do not express PAX10 mRNA. Mice lacking any functional pax6 begin to be phenotypically differentiable from normal mouse embryos at about day 9 to 10 of gestation. The full elucidation of the precise mechanisms and molecular components by which the PAX6 gene influences eye, nasal and central nervous system development are still researched however, the study of PAX6 has brought more understanding to the development and genetic complexities of these mammalian body systems.

In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold. A metric tensor is called positive-definite if it assigns a positive value to every nonzero vector .

Emission spectrum of an ultraviolet deuterium arc lamp Deuterium is most commonly used in hydrogen nuclear magnetic resonance spectroscopy (proton NMR) in the following way. NMR ordinarily requires compounds of interest to be analyzed as dissolved in solution. Because of deuterium's nuclear spin properties which differ from the light hydrogen usually present in organic molecules, NMR spectra of hydrogen/protium are highly differentiable from that of deuterium, and in practice deuterium is not "seen" by an NMR instrument tuned for light- hydrogen. Deuterated solvents (including heavy water, but also compounds like deuterated chloroform, CDCl3) are therefore routinely used in NMR spectroscopy, in order to allow only the light-hydrogen spectra of the compound of interest to be measured, without solvent-signal interference.

For unconstrained problems with twice- differentiable functions, some critical points can be found by finding the points where the gradient of the objective function is zero (that is, the stationary points). More generally, a zero subgradient certifies that a local minimum has been found for minimization problems with convex functions and other locally Lipschitz functions. Further, critical points can be classified using the definiteness of the Hessian matrix: If the Hessian is positive definite at a critical point, then the point is a local minimum; if the Hessian matrix is negative definite, then the point is a local maximum; finally, if indefinite, then the point is some kind of saddle point. Constrained problems can often be transformed into unconstrained problems with the help of Lagrange multipliers.

In differential geometry, normal coordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of p obtained by applying the exponential map to the tangent space at p. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations. In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point p, and that the first partial derivatives of the metric at p vanish. A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection.

For real analytic functions, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function :f: (-1, 1) \to (-1, 1) \, given by ƒ(x) = x3. This function is clearly injective, but its derivative is 0 at x = 0, and its inverse is not analytic, or even differentiable, on the whole interval (−1, 1). Consequently, if we enlarge the domain to an open subset G of the complex plane, it must fail to be injective; and this is the case, since (for example) f(εω) = f(ε) (where ω is a primitive cube root of unity and ε is a positive real number smaller than the radius of G as a neighbourhood of 0).

For a real-valued smooth function f : M → R on a differentiable manifold M, the points where the differential of f vanishes are called critical points of f and their images under f are called critical values. If at a critical point b, the matrix of second partial derivatives (the Hessian matrix) is non-singular, then b is called a non-degenerate critical point; if the Hessian is singular then b is a degenerate critical point. For the functions :f(x)=a + b x+ c x^2+d x^3+\cdots from R to R, f has a critical point at the origin if b = 0, which is non-degenerate if c ≠ 0 (i.e. f is of the form a + cx2 + ...) and degenerate if c = 0 (i.e.

Relative utilitarianism can serve to rationalize using 2% as an intergenerationally fair social discount rate for cost-benefit analysis. Mertens and Rubinchik show that a shift- invariant welfare function defined on a rich space of (temporary) policies, if differentiable, has as a derivative a discounted sum of the policy (change), with a fixed discount rate, i.e., the induced social discount rate. (Shift- invariance requires a function evaluated on a shifted policy to return an affine transformation of the value of the original policy, while the coefficients depend on the time-shift only.) In an overlapping generations model with exogenous growth (with time being the whole real line), relative utilitarian function is shift-invariant when evaluated on (small temporary) policies around a balanced growth equilibrium (with capital stock growing exponentially).

In differential geometry and algebraic geometry it is often convenient to have a coordinate-independent description of differential operators between two vector bundles. Let E and F be two vector bundles over a differentiable manifold M. An R-linear mapping of sections is said to be a kth-order linear differential operator if it factors through the jet bundle Jk(E). In other words, there exists a linear mapping of vector bundles :i_P: J^k(E) \rightarrow F\, such that :P = i_P\circ j^k where is the prolongation that associates to any section of E its k-jet. This just means that for a given section s of E, the value of P(s) at a point x ∈ M is fully determined by the kth-order infinitesimal behavior of s in x.

Diffeomorphisms are by their Latin root structure preserving transformations, which are in turn differentiable and therefore smooth, allowing for the calculation of metric based quantities such as arc length and surface areas. Spatial location and extents in human anatomical coordinate systems can be recorded via a variety of Medical imaging modalities, generally termed multi-modal medical imagery, providing either scalar and or vector quantities at each spatial location. Examples are scalar T1 or T2 magnetic resonance imagery, or as 3x3 diffusion tensor matrices diffusion MRI and diffusion-weighted imaging, to scalar densities associated to computed tomography (CT), or functional imagery such as temporal data of functional magnetic resonance imaging and scalar densities such as Positron emission tomography (PET). Computational anatomy is a subdiscipline within the broader field of neuroinformatics within bioinformatics and medical imaging.

Much of his work on stratified sets was developed so as to understand the notion of topologically stable maps, and to eventually prove the result that the set of topologically stable mappings between two smooth manifolds is a dense set. Thom's lectures on the stability of differentiable mappings, given at the University of Bonn in 1960, were written up by Harold Levine and published in the proceedings of a year long symposium on singularities at Liverpool University during 1969-70, edited by C. T. C. Wall. The proof of the density of topologically stable mappings was completed by John Mather in 1970, based on the ideas developed by Thom in the previous ten years. A coherent detailed account was published in 1976 by Christopher Gibson, Klaus Wirthmüller, Andrew du Plessis, and Eduard Looijenga.

The generalized Poincaré conjecture is true topologically, but false smoothly in some dimensions. This results in constructions of manifolds that are homeomorphic, but not diffeomorphic, to the standard sphere, which are known as the exotic spheres: you can interpret these as non-standard smooth structures on the standard (topological) sphere. Thus the homotopy spheres that John Milnor produced are homeomorphic (Top-isomorphic, and indeed piecewise linear homeomorphic) to the standard sphere S^n, but are not diffeomorphic (Diff-isomorphic) to it, and thus are exotic spheres: they can be interpreted as non-standard differentiable structures on the standard sphere. Michel Kervaire and Milnor showed that the oriented 7-sphere has 28 different smooth structures (or 15 ignoring orientations), and in higher dimensions there are usually many different smooth structures on a sphere.

The picture is describing the visual perception in the human's field of view. It shows the differentiable Areas and Angels for Perception of Motion, Color, Shape & Text. While most of these factors mentioned above become problematic when both eyes are covered with displays, a single display resting in the Peripheral Vision can be considered to be unproblematic, since it does not permanently influence the perceived picture of the real world. As mentioned earlier, there are two types of information being perceivable with a peripheral head mounted display: (1) detailed information: when consciously focusing on the display and (2) peripheral information: through the human's visual perception, when focusing at the 'real world'. (see also picture above) Most obvious changes are “motion”, which can be perceived over the whole spectrum of the FOV.

The classical Fermat's theorem says that if a differentiable function attains its minimum at a point, and that point is an interior point of its domain, then its derivative must be zero at that point. For problems where a smooth function must be minimized subject to constraints which can be expressed in the form of other smooth functions being equal to zero, the method of Lagrange multipliers, another classical result, gives necessary conditions in terms of the derivatives of the function. The ideas of these classical results can be extended to nondifferentiable convex functions by generalizing the notion of derivative to that of subderivative. Further generalization of the notion of the derivative such as the Clarke generalized gradient allow the results to be extended to nonsmooth locally Lipschitz functions.

In general, one cannot rule out "ergodic" flows (which basically means that an orbit is dense in some open set), or "subergodic" flows (which an orbit dense in some submanifold of dimension greater than the orbit's dimension). We can't have self-intersecting orbits. For most "practical" applications of first-class constraints, we do not see such complications: the quotient space of the restricted subspace by the f-flows (in other words, the orbit space) is well behaved enough to act as a differentiable manifold, which can be turned into a symplectic manifold by projecting the symplectic form of M onto it (this can be shown to be well defined). In light of the observation about physical observables mentioned earlier, we can work with this more "physical" smaller symplectic manifold, but with 2n fewer dimensions.

A visual depiction of a vector X in a domain being multiplied by a complex number z, then mapped by f, versus being mapped by f then being multiplied by z afterwards. If both of these result in the point ending up in the same place for all X and z, then f satisfies the Cauchy-Riemann condition In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic. This system of equations first appeared in the work of Jean le Rond d'Alembert . Later, Leonhard Euler connected this system to the analytic functions .

While the first derivative test identifies points that might be extrema, this test does not distinguish a point that is a minimum from one that is a maximum or one that is neither. When the objective function is twice differentiable, these cases can be distinguished by checking the second derivative or the matrix of second derivatives (called the Hessian matrix) in unconstrained problems, or the matrix of second derivatives of the objective function and the constraints called the bordered Hessian in constrained problems. The conditions that distinguish maxima, or minima, from other stationary points are called 'second-order conditions' (see 'Second derivative test'). If a candidate solution satisfies the first-order conditions, then the satisfaction of the second-order conditions as well is sufficient to establish at least local optimality.

A Carathéodory- solution addresses the fundamental problem of defining a solution to a differential equation, : \dot x = g(x,t) when g(x,t) is not differentiable with respect to x. Such problems arise quite naturally Clarke, F. H., Ledyaev, Y. S., Stern, R. J., and Wolenski, P. R., Nonsmooth Analysis and Control Theory, Springer–Verlag, New York, 1998. in defining the meaning of a solution to a controlled differential equation, : \dot x = f(x,u) when the control, u, is given by a feedback law, : u = k(x,t) where the function k(x,t) may be non- smooth with respect to x. Non-smooth feedback controls arise quite often in the study of optimal feedback controls and have been the subject of extensive study going back to the 1960s.

He constructed a smooth 7-dimensional manifold (called now Milnor's sphere) that is homeomorphic to the standard 7-sphere but not diffeomorphic to it. There are, in fact, 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere (each of them is the total space of a fiber bundle over the 4-sphere with the 3-sphere as the fiber). More unusual phenomena occur for 4-manifolds. In the early 1980s, a combination of results due to Simon Donaldson and Michael Freedman led to the discovery of exotic R4s: there are uncountably many pairwise non-diffeomorphic open subsets of R4 each of which is homeomorphic to R4, and also there are uncountably many pairwise non- diffeomorphic differentiable manifolds homeomorphic to R4 that do not embed smoothly in R4.

Mathematically, scalar fields on a region U is a real or complex-valued function or distribution on U. The region U may be a set in some Euclidean space, Minkowski space, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order. A scalar field is a tensor field of order zero, and the term "scalar field" may be used to distinguish a function of this kind with a more general tensor field, density, or differential form. The scalar field of \sin (2\pi(xy+\sigma)) oscillating as \sigma increases. Red represents positive values, purple represents negative values, and sky blue represents values close to zero.

The key word in the applications of germs is locality: all local properties of a function at a point can be studied by analyzing its germ. They are a generalization of Taylor series, and indeed the Taylor series of a germ (of a differentiable function) is defined: you only need local information to compute derivatives. Germs are useful in determining the properties of dynamical systems near chosen points of their phase space: they are one of the main tools in singularity theory and catastrophe theory. When the topological spaces considered are Riemann surfaces or more generally complex-analytic varieties, germs of holomorphic functions on them can be viewed as power series, and thus the set of germs can be considered to be the analytic continuation of an analytic function.

Suvorov was the founder of a new branch of function theory concerned with the study of classes of plane and spatial mappings with bounded Dirichlet integral and of a new trend in mathematics at the border of the theory of functions and general topology that deals with the topological aspects of the boundary correspondence in a conformal mapping. Suvorov made major contributions on the theory of topological and metric mappings on 2-dimensional space. Later at Donetsk he extended Lavrent'ev's work on stability and differentiable function theorems, to more general classes of transformations. Suvorov introduced new methods to help in the understanding of metric properties of mappings with bounded Dirichlet integral. Suvorov and Oleg Ivanov collaborated on a number of papers culminating in a joint monograph “Complete lattices of conformally invariant compactifications of a domain”.

If f is continuously differentiable \left(C^1\right) on an open neighborhood of the point x_0, then f'(x_0) > 0 does mean that f is increasing on a neighborhood of x_0, as follows. If f'(x_0) = K > 0 and f \in C^1, then by continuity of the derivative, there is some \varepsilon_0 > 0 such that f'(x) > K/2 for all x \in (x_0 - \varepsilon_0, x_0 + \varepsilon_0). Then f is increasing on this interval, by the mean value theorem: the slope of any secant line is at least K/2, as it equals the slope of some tangent line. However, in the general statement of Fermat's theorem, where one is only given that the derivative at x_0 is positive, one can only conclude that secant lines through x_0 will have positive slope, for secant lines between x_0 and near enough points.

In calculus, Rolle's theorem says that if a real-valued function ƒ is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and ƒ(a) = ƒ(b), then there exists a c in the open interval (a, b) such that f(c) is a maximum or a minimum and the gradient at x=c is zero, meaning f'(c)=0 . Laffer explains the model in terms of two interacting effects of taxation: an "arithmetic effect" and an "economic effect". The "arithmetic effect" assumes that tax revenue raised is the tax rate multiplied by the revenue available for taxation (or tax base). Thus revenue R is equal to t×B where t is the tax rate and B is the taxable base (R=t×B). At a 0% tax rate, the model states that no tax revenue is raised.

A Langevin equation is a coarse- grained version of a more microscopic model; depending on the problem in consideration, Stratonovich or Itô interpretation or even more exotic interpretations such as the isothermal interpretation, are appropriate. The Stratonovich interpretation is the most frequently used interpretation within the physical sciences. The Wong–Zakai theorem states that physical systems with non-white noise spectrum characterized by a finite noise correlation time τ can be approximated by a Langevin equations with white noise in Stratonovich interpretation in the limit where τ tends to zero. Because the Stratonovich calculus satisfies the ordinary chain rule, stochastic differential equations (SDEs) in the Stratonovich sense are more straightforward to define on differentiable manifolds, rather than just on Rn. The tricky chain rule of the Itô calculus makes it a more awkward choice for manifolds.

The Thompson group F is generated by operations like this on binary trees. Here L and T are nodes, but A B and R can be replaced by more general trees. The group F also has realizations in terms of operations on ordered rooted binary trees, and as a subgroup of the piecewise linear homeomorphisms of the unit interval that preserve orientation and whose non-differentiable points are dyadic rationals and whose slopes are all powers of 2. The group F can also be considered as acting on the unit circle by identifying the two endpoints of the unit interval, and the group T is then the group of automorphisms of the unit circle obtained by adding the homeomorphism x→x+1/2 mod 1 to F. On binary trees this corresponds to exchanging the two trees below the root.

Differentiable manifolds (Stewart Cairns, J. H. C. Whitehead, L. E. J. Brouwer, Hans Freudenthal, James Munkres), and subanalytic sets (Heisuke Hironaka and Robert Hardt) admit a piecewise-linear triangulation, technically by passing via the PDIFF category. Topological manifolds of dimensions 2 and 3 are always triangulable by an essentially unique triangulation (up to piecewise-linear equivalence); this was proved for surfaces by Tibor Radó in the 1920s and for three-manifolds by Edwin E. Moise and R. H. Bing in the 1950s, with later simplifications by Peter Shalen. As shown independently by James Munkres, Steve Smale and J. H. C. Whitehead, each of these manifolds admits a smooth structure, unique up to diffeomorphism. In dimension 4, however, the E8 manifold does not admit a triangulation, and some compact 4-manifolds have an infinite number of triangulations, all piecewise-linear inequivalent.

DE is used for multidimensional real-valued functions but does not use the gradient of the problem being optimized, which means DE does not require the optimization problem to be differentiable, as is required by classic optimization methods such as gradient descent and quasi-newton methods. DE can therefore also be used on optimization problems that are not even continuous, are noisy, change over time, etc. DE optimizes a problem by maintaining a population of candidate solutions and creating new candidate solutions by combining existing ones according to its simple formulae, and then keeping whichever candidate solution has the best score or fitness on the optimization problem at hand. In this way the optimization problem is treated as a black box that merely provides a measure of quality given a candidate solution and the gradient is therefore not needed.

In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operatorsSee references and .), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives with respect to one real variable, when applied to holomorphic functions, antiholomorphic functions or simply differentiable functions on complex domains. These operators permit the construction of a differential calculus for such functions that is entirely analogous to the ordinary differential calculus for functions of real variables.Some of the basic properties of Wirtinger derivatives are the same ones as the properties characterizing the ordinary (or partial) derivatives and used for the construction of the usual differential calculus.

In mathematics, the Bony–Brezis theorem, due to the French mathematicians Jean-Michel Bony and Haïm Brezis, gives necessary and sufficient conditions for a closed subset of a manifold to be invariant under the flow defined by a vector field, namely at each point of the closed set the vector field must have non-positive inner product with any exterior normal vector to the set. A vector is an exterior normal at a point of the closed set if there is a real- valued continuously differentiable function maximized locally at the point with that vector as its derivative at the point. If the closed subset is a smooth submanifold with boundary, the condition states that the vector field should not point outside the subset at boundary points. The generalization to non-smooth subsets is important in the theory of partial differential equations.

The dual curve construction works even if the curve is piecewise linear (or piecewise differentiable, but the resulting map is degenerate (if there are linear components) or ill-defined (if there are singular points). In the case of a polygon, all points on each edge share the same tangent line, and thus map to the same vertex of the dual, while the tangent line of a vertex is ill- defined, and can be interpreted as all the lines passing through it with angle between the two edges. This accords both with projective duality (lines map to points, and points to lines), and with the limit of smooth curves with no linear component: as a curve flattens to an edge, its tangent lines map to closer and closer points; as a curve sharpens to a vertex, its tangent lines spread further apart.

If an equation can be put into the form f(x) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x1 in the basin of attraction of x, and let xn+1 = f(xn) for n ≥ 1, and the sequence {xn}n ≥ 1 will converge to the solution x. Here xn is the nth approximation or iteration of x and xn+1 is the next or n + 1 iteration of x. Alternately, superscripts in parentheses are often used in numerical methods, so as not to interfere with subscripts with other meanings. (For example, x(n+1) = f(x(n)).) If the function f is continuously differentiable, a sufficient condition for convergence is that the spectral radius of the derivative is strictly bounded by one in a neighborhood of the fixed point.

Given a subset S in Rn, a vector field is represented by a vector-valued function V: S → Rn in standard Cartesian coordinates (x1, ..., xn). If each component of V is continuous, then V is a continuous vector field, and more generally V is a Ck vector field if each component of V is k times continuously differentiable. A vector field can be visualized as assigning a vector to individual points within an n-dimensional space. Given two Ck-vector fields V, W defined on S and a real valued Ck- function f defined on S, the two operations scalar multiplication and vector addition : (fV)(p) := f(p)V(p)\, : (V+W)(p) := V(p) + W(p)\, define the module of Ck-vector fields over the ring of Ck-functions where the multiplication of the functions is defined pointwise (therefore, it is commutative with the multiplicative identity being fid(p) := 1).

Loomis and Sternberg's textbook Advanced Calculus, an abstract treatment of calculus in the setting of normed vector spaces and on differentiable manifolds, was tailored to the authors' Math 55 syllabus and served for many years as an assigned text. Instructors for Math 55 and Math 25 have also selected Rudin's Principles of Mathematical Analysis, Spivak's Calculus on Manifolds, Axler's Linear Algebra Done Right, and Halmos's Finite-Dimensional Vector Spaces as textbooks or references. From 2007 onwards, the scope of the course (along with that of Math 25) was changed to more strictly cover the contents of four semester-long courses in two semesters: Math 25a (linear algebra) and Math 122 (group theory) in Math 55a; and Math 25b (calculus, real analysis) and Math 113 (complex analysis) in Math 55b. The name was also changed to "Honors Abstract Algebra" (Math 55a) and "Honors Real and Complex Analysis" (Math 55b).

Let K be a closed convex subset of a real vector space V and ∂K be the boundary of K. The solid tangent cone to K at a point x ∈ ∂K is the closure of the cone formed by all half-lines (or rays) emanating from x and intersecting K in at least one point y distinct from x. It is a convex cone in V and can also be defined as the intersection of the closed half-spaces of V containing K and bounded by the supporting hyperplanes of K at x. The boundary TK of the solid tangent cone is the tangent cone to K and ∂K at x. If this is an affine subspace of V then the point x is called a smooth point of ∂K and ∂K is said to be differentiable at x and TK is the ordinary tangent space to ∂K at x.

It is by no means true that a finite-dimensional manifold of dimension n is globally homeomorphic to Rn, or even an open subset of Rn. However, in an infinite- dimensional setting, it is possible to classify “well-behaved” Fréchet manifolds up to homeomorphism quite nicely. A 1969 theorem of David Henderson states that every infinite-dimensional, separable, metric Fréchet manifold X can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, H (up to linear isomorphism, there is only one such space). The embedding homeomorphism can be used as a global chart for X. Thus, in the infinite-dimensional, separable, metric case, up to homeomorphism, the “only” topological Fréchet manifolds are the open subsets of the separable infinite- dimensional Hilbert space. But in the case of differentiable or smooth Fréchet manifolds (up to the appropriate notion of diffeomorphism) this fails.

Suppose X and Y are locally convex topological vector spaces (for example, Banach spaces), U \subset X is open, and F : X \to Y. The Gateaux differential dF(u; \psi) of F at u \in U in the direction \psi \in X is defined as If the limit exists for all \psi \in X, then one says that F is Gateaux differentiable at u. The limit appearing in () is taken relative to the topology of Y. If X and Y are real topological vector spaces, then the limit is taken for real \tau. On the other hand, if X and Y are complex topological vector spaces, then the limit above is usually taken as \tau \to 0 in the complex plane as in the definition of complex differentiability. In some cases, a weak limit is taken instead of a strong limit, which leads to the notion of a weak Gateaux derivative.

Sabir Gusein-Zade (2010), El Escorial Sabir Medgidovich Gusein-Zade (; born 29 July 1950 in MoscowHome page of Sabir Gusein-Zade) is a Russian mathematician and a specialist in singularity theory and its applications.. He studied at Moscow State University, where he earned his Ph.D. in 1975 under the joint supervision of Sergei Novikov and Vladimir Arnold. Before entering the university, he had earned a gold medal at the International Mathematical Olympiad. Gusein-Zade co-authored with V. I. Arnold and A. N. Varchenko the textbook Singularities of Differentiable Maps (published in English by Birkhäuser). A professor in both the Moscow State University and the Independent University of Moscow, Gusein-Zade also serves as co-editor-in- chief for the Moscow Mathematical Journal.. He shares credit with Norbert A'Campo for results on the singularities of plane curves... Translated from the German original by John Stillwell, 2012 reprint of the 1986 edition.

He shows that the two arguments can be presented in the same terms, since the PPF plays the same role as the mirror-image indifference curve in an Edgeworth box. He also mentions that there’s no need for the curves to be differentiable, since the same result obtains if they touch at pointed corners. His definition of optimality was equivalent to Pareto’s: > If... it is possible to move one individual into a preferred position > without moving another individual into a worse position... we may say that > the relative optimum is not reached... The optimality condition for production is equivalent to the pair of requirements that (i) price should equal marginal cost and (ii) output should be maximised subject to (i). Lerner thus reduces optimality to tangency for both production and exchange, but does not say why the implied point on the PPF should be the equilibrium condition for a free market.

Gérard Debreu The Theory of Value: An axiomatic analysis of economic equilibrium, 1959 In the 1960s and 1970s, however, Gérard Debreu and Stephen Smale led a revival of the use of differential calculus in mathematical economics. In particular, they were able to prove the existence of a general equilibrium, where earlier writers had failed, through the use of their novel mathematics: Baire category from general topology and Sard's lemma from differential topology and differential geometry. Their publications initiated a period of research "characterized by the use of elementary differential topology": "almost every area in economic theory where the differential approach has been pursued, including general equilibrium" was covered by Mas-Colell's monograph on differentiable analysis and economics. Mas-Colell's book "offers a synthetic and thorough account of a major recent development in general equilibrium analysis, namely, the largely successful reconstruction of the theory using modern ideas of differential topology", according to its back cover.

3D computer generated fractal This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere differentiable". In a concrete sense, this means fractals cannot be measured in traditional ways. To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of measuring with a tape measure. But in measuring an infinitely "wiggly" fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve.

In the Travaux, Cartan breaks down his work into 15 areas. Using modern terminology, they are: # Lie theory # Representations of Lie groups # Hypercomplex numbers, division algebras # Systems of PDEs, Cartan–Kähler theorem # Theory of equivalence # Integrable systems, theory of prolongation and systems in involution # Infinite- dimensional groups and pseudogroups # Differential geometry and moving frames # Generalised spaces with structure groups and connections, Cartan connection, holonomy, Weyl tensor # Geometry and topology of Lie groups # Riemannian geometry # Symmetric spaces # Topology of compact groups and their homogeneous spaces # Integral invariants and classical mechanics # Relativity, spinors Cartan's mathematical work can be described as the development of analysis on differentiable manifolds, which many now consider the central and most vital part of modern mathematics and which he was foremost in shaping and advancing. This field centers on Lie groups, partial differential systems, and differential geometry; these, chiefly through Cartan's contributions, are now closely interwoven and constitute a unified and powerful tool.

Since holomorphic functions are much more rigid than smooth functions, the theories of smooth and complex manifolds have very different flavors: compact complex manifolds are much closer to algebraic varieties than to differentiable manifolds. For example, the Whitney embedding theorem tells us that every smooth n-dimensional manifold can be embedded as a smooth submanifold of R2n, whereas it is "rare" for a complex manifold to have a holomorphic embedding into Cn. Consider for example any compact connected complex manifold M: any holomorphic function on it is constant by Liouville's theorem. Now if we had a holomorphic embedding of M into Cn, then the coordinate functions of Cn would restrict to nonconstant holomorphic functions on M, contradicting compactness, except in the case that M is just a point. Complex manifolds that can be embedded in Cn are called Stein manifolds and form a very special class of manifolds including, for example, smooth complex affine algebraic varieties.

In numerical analysis, the Lax equivalence theorem is a fundamental theorem in the analysis of finite difference methods for the numerical solution of partial differential equations. It states that for a consistent finite difference method for a well-posed linear initial value problem, the method is convergent if and only if it is stable. The importance of the theorem is that while the convergence of the solution of the finite difference method to the solution of the partial differential equation is what is desired, it is ordinarily difficult to establish because the numerical method is defined by a recurrence relation while the differential equation involves a differentiable function. However, consistency—the requirement that the finite difference method approximates the correct partial differential equation—is straightforward to verify, and stability is typically much easier to show than convergence (and would be needed in any event to show that round-off error will not destroy the computation).

An (n+1)-dimensional cobordism is a quintuple (W; M, N, i, j) consisting of an (n+1)-dimensional compact differentiable manifold with boundary, W; closed n-manifolds M, N; and embeddings i\colon M \hookrightarrow \partial W, j\colon N \hookrightarrow\partial W with disjoint images such that :\partial W = i(M) \sqcup j(N)~. The terminology is usually abbreviated to (W; M, N).The notation "(n+1)-dimensional" is to clarify the dimension of all manifolds in question, otherwise it is unclear whether a "5-dimensional cobordism" refers to a 5-dimensional cobordism between 4-dimensional manifolds or a 6-dimensional cobordism between 5-dimensional manifolds. M and N are called cobordant if such a cobordism exists. All manifolds cobordant to a fixed given manifold M form the cobordism class of M. Every closed manifold M is the boundary of the non-compact manifold M × [0, 1); for this reason we require W to be compact in the definition of cobordism.

In general, the error in approximating a function by a polynomial of degree k will go to zero much faster than (x{-}a)^k as x tends to a. However, there are functions, even infinitely differentiable ones, for which increasing the degree of the approximating polynomial does not increase the accuracy of approximation: we say such a function fails to be analytic at x = a: it is not (locally) determined by its derivatives at this point. Taylor's theorem is of asymptotic nature: it only tells us that the error Rk in an approximation by a k-th order Taylor polynomial Pk tends to zero faster than any nonzero k-th degree polynomial as x → a. It does not tell us how large the error is in any concrete neighborhood of the center of expansion, but for this purpose there are explicit formulae for the remainder term (given below) which are valid under some additional regularity assumptions on f.

Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manifolds, there is a natural class of functions on an algebraic set, called regular functions or polynomial functions. A regular function on an algebraic set V contained in An is the restriction to V of a regular function on An. For an algebraic set defined on the field of the complex numbers, the regular functions are smooth and even analytic. It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal topological space, where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space. Just as with the regular functions on affine space, the regular functions on V form a ring, which we denote by k[V].

In topological applications, this vector space is usually real or complex. The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the positive and negative reals each to a point; whereas removing the origin from the complex plane yields the 1×1 invertible complex matrices, which have the homotopy type of a circle. From the perspective of homotopy theory, a real line bundle therefore behaves much the same as a fiber bundle with a two-point fiber, that is, like a double cover. A special case of this is the orientable double cover of a differentiable manifold, where the corresponding line bundle is the determinant bundle of the tangent bundle (see below).

After an initial section of the book, introducing computable analysis and leading up to an example of John Myhill of a computable continuously differentiable function whose derivative is not computable, the remaining two parts of the book concerns the authors' results. These include the results that, for a computable self-adjoint operator, the eigenvalues are individually computable, but their sequence is (in general) not; the existence of a computable self- adjoint operator for which 0 is an eigenvalue of multiplicity one with no computable eigenvectors; and the equivalence of computability and boundedness for operators. The authors' main tools include the notions of a computability structure, a pair of a Banach space and an axiomatically-characterized set of its sequences, and of an effective generating set, a member of the set of sequences whose linear span is dense in the space. The authors are motivated in part by the computability of solutions to differential equations.

If f and g are germ equivalent at x, then they share all local properties, such as continuity, differentiability etc., so it makes sense to talk about a differentiable or analytic germ, etc. Similarly for subsets: if one representative of a germ is an analytic set then so are all representatives, at least on some neighbourhood of x. Algebraic structures on the target Y are inherited by the set of germs with values in Y. For instance, if the target Y is a group, then it makes sense to multiply germs: to define [f]x[g]x, first take representatives f and g, defined on neighbourhoods U and V respectively, and define [f]x[g]x to be the germ at x of the pointwise product map fg (which is defined on U\cap V). In the same way, if Y is an abelian group, vector space, or ring, then so is the set of germs.

Scheinkman is perhaps most closely associated with his classic six page paper from 1979 with L. M. Benveniste "On the Differentiability of the Value Function in Dynamic Models of Economics", which provides conditions on model primitives allowing for the standard differentiable treatment of infinite-horizon dynamic models. At least as influential, however, is his very different work with David Kreps in 1983 showing that "Quantity precommitment and Bertrand competition yield Cournot outcomes" and thus providing the canonical modern foundation of Cournot equilibrium as the result of capacity pre-commitments. Building on his interest in the intersection between economics and physics, he helped draw out and test some of the most salient implication of the theory of social interactions, in a series of papers with his star student Edward Glaeser (among others), for "Growth in Cities" (1992), crime (1996) and "Measuring Trust" (2000). Perhaps one of the best loved of Scheinkman's papers is his work with Kevin Murphy and Sherwin Rosen on "Cattle Cycles" (1994), which provides one of the sharpest applications of natural economic theory to explain cyclical variations.

Together with Pradeep Dubey and Robert James Weber he studied the theory of semivalues, and separately demonstrated its importance in political economy.Dubey, P., Neyman, A., and Weber, R.J. , 1981, "Value theory without efficiency," Mathematics of Operations Research, 6, pp 122–128Neyman, A., 1985, "Semi-values of political economic games," Mathematics of Operations Research, 10, pp 390–402 Together with Pradeep Dubey Dubey. P. and Neyman, A., 1984, "Payoffs in nonatomic economies: An axiomatic approach," Econometrica, 52, pp 1129–1150Dubey, P. and Neyman, A., 1997, "An equivalence principle for perfectly competitive economies," Journal of Economic Theory, 75, pp 314–344 he characterized the well-known phenomenon of value correspondence, a fundamental notion in economics, originating already in Edgeworth's work and Adam Smith before him. In loose terms, it essentially states that in a large economy consisting of many economically insignificant agents, the core of the economy coincides with the perfectly competitive outcomes, which in the case of differentiable preferences is a unique element that is the Aumann–Shapley value.

In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in dynamic programming (the Hamilton–Jacobi–Bellman equation), differential games (the Hamilton–Jacobi–Isaacs equation) or front evolution problems, as well as second-order equations such as the ones arising in stochastic optimal control or stochastic differential games. The classical concept was that a PDE : F(x,u,Du,D^2 u) = 0 over a domain x\in\Omega has a solution if we can find a function u(x) continuous and differentiable over the entire domain such that x, u, Du, D^2 u satisfy the above equation at every point. If a scalar equation is degenerate elliptic (defined below), one can define a type of weak solution called viscosity solution.

Packages of butter thus wrapped were called prints (for example, pound prints weighed one pound each). The conceptual distinction of print butter, as opposed to any other type of butter, merited a separate name up until the mid-20th century, as before that time many people got their dairy products (milk, butter, cheese), eggs, and produce in ways that did not involve much branding or packaging—for example, either produced at home (in the case of family farms, which were formerly widespread), directly from a farmer that produced them (via either a regular delivery route or at the town market), or from any of various resellers who bought from farmers and resold (for example, grocer, huckster, or sutler). Unlike today when even bulk foods at a supermarket are usually labeled to show who produced them, in the past, the pickles or peanuts bought from a barrel at the general store, or the produce bought from a huckster's cart, were usually not labeled, let alone branded. Thus the idea of prepackaged units with branded labels was worthy of a differentiating name, somewhat similar to how "name-brand merchandise" is still differentiable from "generic merchandise" or "bulk commodities" today.

An arbitrary topological space X can be considered a locally ringed space by taking OX to be the sheaf of real-valued (or complex-valued) continuous functions on open subsets of X (there may exist continuous functions over open subsets of X that are not the restriction of any continuous function over X). The stalk at a point x can be thought of as the set of all germs of continuous functions at x; this is a local ring with maximal ideal consisting of those germs whose value at x is 0. If X is a manifold with some extra structure, we can also take the sheaf of differentiable, or complex-analytic functions. Both of these give rise to locally ringed spaces. If X is an algebraic variety carrying the Zariski topology, we can define a locally ringed space by taking OX(U) to be the ring of rational mappings defined on the Zariski-open set U that do not blow up (become infinite) within U. The important generalization of this example is that of the spectrum of any commutative ring; these spectra are also locally ringed spaces.

Let K be a metrizable space, together with: # a collection {Ui} of closed subsets of K; # for each Ui, a finite collection {Dij} of closed subsets of Ui; # for each i, a map πi: Ui → Din to a closed n-disk of class Ck in Rn. These data must satisfy the following requirements: # ∪j Dij = Ui and ∪i Int Ui = K; # the restriction of πi to Dij is a homeomorphism onto its image πi(Dij) which is a closed class Ck n-disk relative to the boundary of Din; # there is a cocycle of diffeomorphisms {αlm} of class Ck (k ≥ 1) such that πl = αlm · πm when defined. The domain of αlm is πm(Ul ∩ Um). Then the space K is a branched n-manifold of class Ck. The standard machinery of differential topology can be adapted to the case of branched manifolds. This leads to the definition of the tangent space TpK to a branched n-manifold K at a given point p, which is an n-dimensional real vector space; a natural notion of a Ck differentiable map f: K → L between branched manifolds, its differential df: TpK → Tf(p)L, the germ of f at p, jet spaces, and other related notions.

The tangent at a point (a, b) of the curve is the line of equation (x-a)p'_x(a,b)+(y-b)p'_y(a,b)=0, like for every differentiable curve defined by an implicit equation. In the case of polynomials, another formula for the tangent has a simpler constant term and is more symmetric: :xp'_x(a,b)+yp'_y(a,b)+p'_\infty(a,b)=0, where p'_\infty(x,y)=P'_z(x,y,1) is the derivative at infinity. The equivalence of the two equations results from Euler's homogeneous function theorem applied to P. If p'_x(a,b)=p'_y(a,b)=0, the tangent is not defined and the point is a singular point. This extends immediately to the projective case: The equation of the tangent of at the point of projective coordinates (a:b:c) of the projective curve of equation P(x, y, z) = 0 is :xP'_x(a,b,c)+yP'_y(a,b,c)+zP'_z(a,b,c)=0, and the points of the curves that are singular are the points such that :P'_x(a,b,c)=P'_y(a,b,c)=P'_z(a,b,c)=0.