184 Sentences With "infinitesimally" | Random Sentence Generator

It is infectious because an infinitesimally small amount can cause illness.

The odds that this storm would intersect Earth's orbit were infinitesimally small.

The odds of striking a huge Facebook-level success are infinitesimally tiny.

These systems are designed to make the likelihood of collisions infinitesimally small.

Prices will become infinitesimally divisible, optimized in some cases for fractional cent values.

But an infinitesimally small contribution to an infinitely large climate catastrophe is still significant.

The line between what's considered beyond the pale and what's everyday targeting is infinitesimally thin.

As our planet's rotation infinitesimally slows, it puts our clocks ever so slightly out of whack.

Because an infinitesimally small amount of the virus can cause illness, it is considered extremely infectious.

It divided labor into infinitesimally smaller functions and thrived on creating inequality between forms of art.

But an infinitesimally different stressing of the second syllable will give you the character which means "heartbeat".

He concluded that, while two people might share one measurement, the odds of sharing eleven were infinitesimally small.

"The donor class, the elites, the pundits represent an infinitesimally small portion of the Democratic electorate," Murray said.

It's such a minute, infinitesimally small amount because often you don't know quite what happened to you, honestly.

At the beginning of time, all the matter in the universe was compressed into an infinitesimally small point.

You'll see your character absorb the yellow orbs that spill out, each making you infinitesimally more deadly with explosives.

Instead, it finds that the Thunder now have an infinitesimally better chance, though both clubs are right around 36%.

The odds of getting funded for an idea with no current revenue and no current growth are infinitesimally slim.

These extremely, infinitesimally isolated occurrences are almost invariably the result of cancer that has spread from elsewhere in the body.

The fact is that undocumented immigrants commit an infinitesimally small number of violent crimes compared to the native born population.

If the company can pull that off, it will make the odds of a sensing mistake "infinitesimally low," Shashua said.

Each studio I saw seemed to only infinitesimally reduce the number still left to see as listed in the printed guide.

However, when the risk is boiled down to a matter of immigrants, refugees, and American citizens, the threat becomes infinitesimally small.

Only Elliott Carter, the grand paterfamilias of American modernism, managed to squeeze through the infinitesimally small needle's eye of Boulezian approval.

That musty smell that old homes have, made infinitesimally worse by the piles of slowly decaying papers, photos, boxes, books, and whatnot.

At the end of an eight-week program, the group that did split training had increased their biceps and triceps size infinitesimally.

And I wonder if that clashes with this idea that we are wildly, infinitesimally insignificant compared to the size of the universe?

But at least there is now a small hope—even if it is an infinitesimally small one—for a Nixon-in-China moment.

Still, while the number of attacks has increased each year, the likelihood of being killed by a shark is "infinitesimally small," Burgess said.

But the Archbishop of Canterbury said the chance of this was "infinitesimally small" because political wrangling will prevent the detailed work that is needed.

This is because the behaviour is the result of the aggregate effort of hundreds or thousands of genes, whose individual effects are infinitesimally weak.

So Poke decided he'd ride the metro until he ran Rod down, because their city may have been big but their orbits were infinitesimally small.

The tragically powerful but infinitesimally nuanced resonances of "Trilogie" are as enthrallingly pleasurable to listen to as observing glistening icicles melting in the winter sun.

That is, even the infinitesimally small changes in time wrought by orbiting the Earth can make a difference in a GPS system determining your location.

GrimaultWhen the seismometer originally underwent this battery of tests, engineers spotted an infinitesimally slow leak in the vacuum sphere that provides thermal shielding for its sensitive pendulums.

"Once you get into something like 17.8 tons, your market is infinitesimally small," Mr. Twelker said in a telephone interview from his home in Port Townsend, Wash.

Soon a solo tenor (Pascal Charbonneau) portraying Gorlaeus, a pioneering atomic theorist who died in 1612, sings urgent, clarion phrases explaining that matter consists of infinitesimally small particles.

That said, the likelihood of being hit by a falling piece of space debris is infinitesimally small — just one in a trillion, according to the nonprofit Aerospace Corporation.

The resulting approximations were gemlike, faceted objects that yielded fantastic insights into the original shapes, especially when he imagined using infinitely many, infinitesimally small facets in the process.

Whenever you look at your Facebook newsfeed, you are granted access to some infinitesimally small section of that database, seeing only the information that is relevant to you.

Unless the governing bodies want to create an infinitesimally complicated handicap system, they should back off and allow the games to celebrate human diversity in all of its forms.

Those companies with users and page views can dominate, and accumulating those users is everything, something only an infinitesimally small number of companies can find the key to doing.

Considering the benefits of modern-day sanitation, vaccinations and health care, the likelihood of getting sick from our shoes is "infinitesimally small as to almost be unwarranted," one pediatrician said.

The ethos behind traditional block printing runs counter to the very nature of big business: Designs are stamped into textiles by hand, leaving each product infinitesimally different from the next.

Because there is only an infinitesimally small chance that any one vote can influence the result of an election, even most smart people usually have little motivation to follow politics closely.

Considering the number of people around the world who go to the beach and the billions of hours of humans in the water, though, those numbers are infinitesimally small, Collier said.

In their stead would be a single unified technology that fills all of the memory requirements of a computer—that is, it will be ultrafast, infinitesimally small, and non-volatile, e.g.

Basically, the infinitesimally small odds of winning the big jackpots became even worse — they are now one in roughly 292 million for Powerball and one in 259 million for Mega Millions.

" She can even make a pathetic fallacy work: "Outside her breath rises in a fine mist and the snow keeps falling, like a ceaseless repetition of the same infinitesimally small mistake.

You've certainly heard of the dangers of microbeads—the infinitesimally small plastic spheres that plague our environment, being found in fish lungs and various other annoyingly small corners of our planet.

And yet The Young Pope somehow gets into the nervous system and under the skin and maybe — by somehow penetrating those infinitesimally small crevices that reveal infinite space — even into the soul.

That said, know the likelihood of any of us being hit by a falling piece of space debris was infinitesimally small — just one in a trillion, according to the nonprofit Aerospace Corporation.

Where much of Semsroth's discography has emphasized precision and the cerebral effects of infinitesimally subtle variation, this record is pure body music, thrashing and bucking about in pure recklessness, at least by Semsroth's standards.

And the provision, part of the Senate Republicans' tax plan under consideration this week, is expected by one tally to cost federal coffers $500,000 over 10 years, an infinitesimally small amount in congressional bookkeeping.

And since US population and GDP growth are already extremely low in comparison to the rest of the world, marginally raising fertility will have an infinitesimally small impact on the growth path of carbon emissions.

"Basically, it's a risk worth taking if you are a factory owner...There is an incentive not to pay minimum wage because the chances of getting caught are infinitesimally small," she told the Thomson Reuters Foundation.

Here's data on the religious beliefs of Iranian refugees versus Iranians generally: As you can see, despite Christians representing an infinitesimally small share of Iranians, they make up almost 60 percent of the 36,000 refugees from Iran.

Hahnemann's pre-science idea is derived from the so-called "law of similars," the idea that the key to curing an illness lies in infinitesimally small doses of substances that are harmful to healthy people in large doses.

The fluctuation in the power supply is infinitesimally small — not nearly enough to make a meaningful difference for most powered devices — and if it were a brief disturbance, the effect on clocks might be too little to worry about.

To put that in perspective, infinitesimally small neutrinos interact with matter at an energy level of about 10 raised to the minus 5 GeV squared, which is still incredibly weak but is 10 nonillion times bigger than this new limit.

The line between what's considered beyond the pale and what's everyday targeting is infinitesimally thin "It humanizes the consumer and it brings to the forefront behaviors and attitudes," says Alanna Gombert, general manager of trade group Interactive Advertising Bureau's tech lab.

LONDON (Reuters) - The chance of Britain's exit from the European Union taking place by March 2019 is "infinitesimally small" because domestic political wrangling will prevent the detailed work that is needed for Brexit, the Archbishop of Canterbury said on Monday.

"If each one of those has to be argued as a point of confidence on the floor of the (parliament's) House of Commons, the chance of getting this done in what's now roughly 18 months is infinitesimally small," he said.

What would be, if the world was infinitesimally better in six months" — I use the six-month timeframe because that seems to be reasonable — "and the team was just knocking it out of the park and everybody was standing back like, 'Yes!

LONDON, July 31 (Reuters) - The chance of Britain's exit from the European Union taking place by March 2019 is "infinitesimally small" because domestic political wrangling will prevent the detailed work that is needed for Brexit, the Archbishop of Canterbury said on Monday.

In a co-authored 1998 paper, Hawking gave some ground, replacing the mechanism by which chronology would be protected with a mathematical improbability, concluding that the chance that appropriately curved space might be turned into a time machine was real, although infinitesimally small.

If Liverpool is to overhaul Manchester City, if it is to end its 29-year wait for an English championship, it needs the player it had last season, rather than the one who is almost the same, but somehow, infinitesimally, not quite.

They argue that individual footprint reductions are such an infinitesimally small part of the problem as to be meaningless, or that it is not fair to ask struggling working Americans to buy expensive hybrid or electric cars or pay to install solar panels on their houses.

Affluent parents devote extraordinary resources — money, time, string-pulling, to getting their kids into College A, which is infinitesimally better on some measures than College B. The first 18 years in the lives of the children of these parents have become an expensive, extended college preparation course.

Mainly, it's because Although neutrinos are incredibly abundant, they're also incredibly difficult to detect: they're just a fraction of the size of an electron, they have no electric charge, and they have a mass so infinitesimally small that physicists won the Nobel Prize in 2015 for proving they had any mass at all.

French argued that Shelby's killing of Crutcher was reasonable because of the infinitesimally small possibility Crutcher could have managed to reach into his SUV, retrieve a weapon and immediately turn around and shoot the small contingent of officers on the scene, even though they had their weapons drawn and Crutcher boxed in.

This kind of populist thinking furthermore begs the question: it simply assumes that questions about the size and scale of our economic institutions ought to be decided by a small (indeed, infinitesimally small as a percentage of the population) group of politicians and bureaucrats, whom we ought to consider interested parties, and therefore conflicted, given their close ties to the rich and powerful.

Sure, that isn't very much, but it's hard to imagine a worse use of that speck of carbon than this missive from the official BP account: If you followed that link, in doing so consuming ever so slightly more carbon and therefore infinitesimally hastening the heating of our planet, you'd find yourself on a website decorated with drawings of planes and bikes and whatever.

The amplitwist associated with a given function is its derivative in the complex plane. More formally, it is a complex number z such that in an infinitesimally small neighborhood of a point a in the complex plane, f(\xi) = z \xi for an infinitesimally small vector \xi. The complex number z is defined to be the derivative of f at a.

When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum gauge theory is recovered.

Differential formulations of the conservation laws apply Stokes' theorem to yield an expression which may be interpreted as the integral form of the law applied to an infinitesimally small volume (at a point) within the flow.

A quasistatic process is an idealized or fictive model of a thermodynamic "process" considered in theoretical studies. It does not occur in physical reality. It may be imagined as happening infinitely slowly so that the system passes through a continuum of states that are infinitesimally close to equilibrium.

This condition may be expressed in linear algebra as a constraint that the gradient vector of the motion of the vertices must have zero inner product with the row of the rigidity matrix that represents the given edge. Thus, the family of gradients of (infinitesimally) rigid motions is given by the nullspace of the rigidity matrix. For frameworks that are not in generic position, it is possible that some infinitesimally rigid motions (vectors in the nullspace of the rigidity matrix) are not the gradients of any continuous motion, but this cannot happen for generic frameworks. A rigid motion of the framework is a motion such that, at each point in time, the framework is congruent to its original configuration.

The nonlinear graviton construction encodes only anti-self-dual, i.e., left-handed fields. A first step towards the problem of modifying twistor space so as to encode a general gravitational field is the encoding of right-handed fields. Infinitesimally, these are encoded in twistor functions or cohomology classes of homogeneity −6.

Surface integrals have applications in physics, particularly with the theories of classical electromagnetism. The definition of surface integral relies on splitting the surface into small surface elements. An illustration of a single surface element. These elements are made infinitesimally small, by the limiting process, so as to approximate the surface.

2001 Oxford, p 12. For compositions within this curve, infinitesimally small fluctuations in composition and density will lead to phase separation via spinodal decomposition. Outside of the curve, the solution will be at least metastable with respect to fluctuations. In other words, outside the spinodal curve some careful process may obtain a single phase system.

However, the explanation is essentially the same, because in any GUT that breaks down into a U(1) gauge group at long distances, there are magnetic monopoles. The argument is topological: # The holonomy of a gauge field maps loops to elements of the gauge group. Infinitesimal loops are mapped to group elements infinitesimally close to the identity.

Orbital debris expert Darren McKnight stated that the car poses no risk because it is far from Earth orbit. He added: "The enthusiasm and interest that [Musk] generates more than offsets the infinitesimally small 'littering' of the cosmos."Is the Tesla Roadster Flying on the Falcon Heavy's Maiden Flight Just Space Junk?. Leonard David, Space, 5 February 2018.

In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space of all geodesics. They are named after Carl Jacobi.

It was the more notable because it used the method of adequality, which may be understood in retrospect as finding the point where the slope of an infinitesimally short chord is zero,Sabra, 1981, pp.144–5. without the intermediate step of finding a general expression for the slope (the derivative). It was also immediately controversial.

Continuous congestion games are the limiting case as n \rightarrow \infty. In this setup, we consider players as "infinitesimally small." We keep E a finite set of congestible elements. Instead of recognizing n players, as in the discrete case, we have n types of players, where each type i is associated with a number r_i, representing the rate of traffic for that type.

This formalises the classical theory of the "moving frame", favoured by French authors. Lifts of loops about a point give rise to the holonomy group at that point. The Gaussian curvature at a point can be recovered from parallel transport around increasingly small loops at the point. Equivalently curvature can be calculated directly infinitesimally in terms of Lie brackets of lifted vector fields.

Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric. This is a concept of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, although they still resemble the Euclidean space at each point infinitesimally, i.e. in the first order of approximation.

At the 1910 meeting which originally defined the curie, it was proposed to make it equivalent to 10 nanograms of radium (a practical amount). But Marie Curie, after initially accepting this, changed her mind and insisted on one gram of radium. According to Bertram Boltwood, Marie Curie thought that "the use of the name 'curie' for so infinitesimally small [a] quantity of anything was altogether inappropriate".

Each plug of differential volume is considered as a separate entity, effectively an infinitesimally small continuous stirred tank reactor, limiting to zero volume. As it flows down the tubular PFR, the residence time (\tau) of the plug is a function of its position in the reactor. In the ideal PFR, the residence time distribution is therefore a Dirac delta function with a value equal to \tau.

A closed system may lie in thermodynamic equilibrium in a static gravitational field, so that its pressure varies continuously with altitude, while, because of the equilibrium requirement, its temperature is invariant with altitude. (Correspondingly, the system's gravitational potential energy density also varies with altitude.) Then the enthalpy summation becomes an integral: : H = \int (\rho h) \, dV, where : ("rho") is density (mass per unit volume), : is the specific enthalpy (enthalpy per unit mass), : represents the enthalpy density (enthalpy per unit volume), : denotes an infinitesimally small element of volume within the system, for example, the volume of an infinitesimally thin horizontal layer, : the integral therefore represents the sum of the enthalpies of all the elements of the volume. The enthalpy of a closed homogeneous system is its cardinal energy function , with natural state variables its entropy and its pressure . A differential relation for it can be derived as follows.

Fluids are composed of molecules that collide with one another and solid objects. However, the continuum assumption assumes that fluids are continuous, rather than discrete. Consequently, it is assumed that properties such as density, pressure, temperature, and flow velocity are well-defined at infinitesimally small points in space and vary continuously from one point to another. The fact that the fluid is made up of discrete molecules is ignored.

Acousticians, in studying the radiation of sound sources have developed some concepts important to understanding how loudspeakers are perceived. The simplest possible radiating source is a point source, sometimes called a simple source. An ideal point source is an infinitesimally small point radiating sound. It may be easier to imagine a tiny pulsating sphere, uniformly increasing and decreasing in diameter, sending out sound waves in all directions equally, independent of frequency.

These definitions must be applied for infinitesimally small "elements" of area and solid angle, which must then be summed over both the source and the diaphragm as shown below. Etendue may be considered to be a volume in phase space. Etendue never decreases in any optical system where optical power is conserved.Lecture notes on Radiance A perfect optical system produces an image with the same etendue as the source.

Theoretically, an electric dipole is defined by the first-order term of the multipole expansion; it consists of two equal and opposite charges that are infinitesimally close together, although real dipoles have separated charge.Many theorists predict elementary particles can have very tiny electric dipole moments, possibly without separated charge. Such large dipoles make no difference to everyday physics, and have not yet been observed. (See electron electric dipole moment).

Figure 5: Saturn's rings are inside the orbits of its principal moons. Tidal forces oppose gravitational coalescence of the material in the rings to form moons. In the case of an infinitesimally small elastic sphere, the effect of a tidal force is to distort the shape of the body without any change in volume. The sphere becomes an ellipsoid with two bulges, pointing towards and away from the other body.

Oliver Heaviside The origin of the loading coil can be found in the work of Oliver Heaviside on the theory of transmission lines. Heaviside (1881) represented the line as a network of infinitesimally small circuit elements. By applying his operational calculus to the analysis of this network he discovered (1887) what has become known as the Heaviside condition.Heaviside, O, "Electromagnetic Induction and its propagation", The Electrician, 3 June 1887.

Milton Stanley Livingston, the lab's associate director, said in 1968, "The frontier of high energy and the infinitesimally small is a challenge to the mind of man. If we can reach and cross this frontier, our generations will have furnished a significant milestone in human history."Adrienne Kolb and Lillian Hoddeson, "A New Frontier in the Chicago Suburbs: Settling Fermilab, 1963–1972," Illinois Historical Journal (1995) 88#1 pp. 2–18, quotes on p.

The ZND detonation model is a one-dimensional model for the process of detonation of an explosive. It was proposed during World War II independently by Y. B. Zel'dovich, John von Neumann, and Werner Döring, hence the name. This model admits finite-rate chemical reactions and thus the process of detonation consists of the following stages. First, an infinitesimally thin shock wave compresses the explosive to a high pressure called the von Neumann spike.

A solid, spherically symmetric body can be modeled as an infinite number of concentric, infinitesimally thin spherical shells. If one of these shells can be treated as a point mass, then a system of shells (i.e. the sphere) can also be treated as a point mass. Consider one such shell (the diagram shows a cross-section): 500px (Note: the d\theta in the diagram refers to the small angle, not the arc length.

For example, an infinitesimal compression of a gas in a cylinder where there exists friction between the piston and the cylinder is a quasistatic, but not reversible process.Giancoli, D.C. (2000), Physics for Scientists and Engineers (with Modern Physics), 3rd edition (Prentice-Hall.) Although the system has been driven from its equilibrium state by only an infinitesimal amount, heat has been irreversibly lost due to friction, and cannot be recovered by simply moving the piston infinitesimally in the opposite direction.

The magnetic field of permanent magnets can be quite complicated, especially near the magnet. The magnetic field of a smallHere, "small" means that the observer is sufficiently far away from the magnet, so that the magnet can be considered as infinitesimally small. "Larger" magnets need to include more complicated terms in the and depend on the entire geometry of the magnet not just . straight magnet is proportional to the magnet's strength (called its magnetic dipole moment ).

Due to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a shear stress. However, fluids can exert pressure normal to any contacting surface. If a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal. If this were not the case, the fluid would move in the direction of the resulting force.

At one extreme, changes with infinitesimally small effect have a 50% chance of improving fitness. This argument led to the widely held position that evolution proceeds by small mutations. Furthermore, Orr discovered that both the fixation probability of a beneficial mutation and the fitness gain that is conferred by the fixation of the beneficial mutation decrease with organismal complexity. Thus, the predicted rate of adaptation decreases quickly with the rise in organismal complexity, a theoretical finding known as the ‘cost of complexity’.

The presence of an interface influences generally all thermodynamic parameters of a system. There are two models that are commonly used to demonstrate interfacial phenomena: the Gibbs ideal interface model and the Guggenheim model. In order to demonstrate the thermodynamics of an interfacial system using the Gibbs model, the system can be divided into three parts: two immiscible liquids with volumes and and an infinitesimally thin boundary layer known as the Gibbs dividing plane () separating these two volumes. Guggenheim Model.

The Jains envisioned the world as consisting wholly of atoms, except for souls. Atoms were considered as the basic building blocks of all matter. Each atom had "one kind of taste, one smell, one color, and two kinds of touch", though it is unclear what was meant by "kind of touch". Atoms can exist in one of two states: subtle, in which case they can fit in infinitesimally small spaces, and gross, in which case they have extension and occupy a finite space.

If the perturbation to the oscillatory cycle is infinitesimally small, it is possible to derive a response function of the neural oscillator. This response function can be classified into different classes (Type 1 and Type 2) based upon its response. #Type I Phase Response Curves are non-negative and strictly positive thus perturbations are only able to enhance a spike in phase, but never delay it. This occurs through a slight depolarization, such as postsynaptic potentials that increase the excitation of an axon.

Paul Tannery, early theoriser of a Pythagorean "unit-point atomism" According to some twentieth-century philosophers,Paul Tannery (1887), Pour l'histoire de la science Hellène (Paris), and J. E. Raven (1948), Pythagoreans and Eleatics (Cambridge: Cambridge University Press), are the major purveyors of this view. unit-point atomism was the philosophy of the Pythagoreans, a conscious repudiation of Parmenides and the Eleatics. It stated that atoms were infinitesimally small ("point") yet possessed corporeality. It was a predecessor of Democritean atomism.

The simplest theory to predict the behaviour of detonations in gases is known as Chapman-Jouguet (CJ) theory, developed around the turn of the 20th century. This theory, described by a relatively simple set of algebraic equations, models the detonation as a propagating shock wave accompanied by exothermic heat release. Such a theory confines the chemistry and diffusive transport processes to an infinitesimally thin zone. A more complex theory was advanced during World War II independently by Zel'dovich, von Neumann, and W. Doering.

An impulse response is a recording of the reverberation that is caused by an acoustic space when an ideal impulse is played. However, an ideal impulse is a mathematical construct, and cannot exist in reality, as it would have to be infinitesimally narrow in time. Therefore, approximations have to be used: the sound of an electric spark, starter pistol shot or the bursting of a balloon, for instance. A recording of this approximated ideal impulse may be used directly as an impulse response.

She was one of only a few female artists to be lauded as the equal of male artists. One 1909 critic went so far as to say that gender was irrelevant in her case: 'Sex is an accident — the capacity for expressing the infinitely large or the infinitesimally little cannot be gauged by outward measurements. The soul frequently bears little relation to its case. Else, why does Florence Rodway, tall, slight and blonde, revel in peopling large spaces with the Titanic creatures of her imagination.

The materials are merely an extension of the > gesture. I start with the gesture and have spent decades distilling these > concepts down to the essential. So whether I’m using vanguard, illustrious > materials and high technology or centuries old metal techniques — my > interest is in communicating the most direct way I can. There is a > minimalistic simplicity to the work, yet it expresses something infinite — > it’s a pluralistic paradox. I’ve tried to visually demonstrate the > colossally vast and the infinitesimally small — the cosmos and the divinity > inside oneself.

Strain is the relative internal change in shape of an infinitesimally small cube of material and can be expressed as a non- dimensional change in length or angle of distortion of the cube. Strains are related to the forces acting on the cube, which are known as stress, by a stress-strain curve. The relationship between stress and strain is generally linear and reversible up until the yield point and the deformation is elastic. The linear relationship for a material is known as Young's modulus.

Hele-Shaw flow is defined as Stokes flow between two parallel flat plates separated by an infinitesimally small gap, named after Henry Selby Hele-Shaw, who studied the problem in 1898. Various problems in fluid mechanics can be approximated to Hele-Shaw flows and thus the research of these flows is of importance. Approximation to Hele-Shaw flow is specifically important to micro-flows. This is due to manufacturing techniques, which creates shallow planar configurations, and the typically low Reynolds numbers of micro-flows.

This unseen quarter of the world has no geographic and/or cultural access (or infinitesimally limited access) to churches and Christians. In spite of the great advances made by missionaries over the years, 1/4 of the world’s entire population still has not encountered a Christian, let alone the Gospel itself. These people will likely never have a chance to hear the Gospel before they die, unless intentional efforts are made to reach them. Yet only 3% of the Church’s missionaries and 0.1% of its resources are going to evangelize these unreached people groups.

The modes that had more energy than the thermal energy of the substance itself were not considered, and because of quantization modes having infinitesimally little energy were excluded. Thus for shorter wavelengths very few modes (having energy more than h u) were allowed, supporting the data that the energy emitted is reduced for wavelengths less than the wavelength of the observed peak of emission. Notice that there are two factors responsible for the shape of the graph. Firstly, longer wavelengths have a larger number of modes associated with them.

More refined consideration distinguishes classical and quantum mechanics on the basis that classical mechanics fails to recognize that matter and energy cannot be divided into infinitesimally small parcels, so that ultimately fine division reveals irreducibly granular features. The criterion of fineness is whether or not the interactions are described in terms of Planck's constant. Roughly speaking, classical mechanics considers particles in mathematically idealized terms even as fine as geometrical points with no magnitude, still having their finite masses. Classical mechanics also considers mathematically idealized extended materials as geometrically continuously substantial.

Some theorems of standard and nonstandard analysis are false in smooth infinitesimal analysis, including the intermediate value theorem and the Banach–Tarski paradox. Statements in nonstandard analysis can be translated into statements about limits, but the same is not always true in smooth infinitesimal analysis. Intuitively, smooth infinitesimal analysis can be interpreted as describing a world in which lines are made out of infinitesimally small segments, not out of points. These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved.

Corpuscular theories, or corpuscularianism, are similar to the theories of atomism, except that in atomism the atoms were supposed to be indivisible, whereas corpuscles could in principle be divided. Corpuscles are single, infinitesimally small, particles which have shape, size, color, and other physical properties which alter their functions and effects in phenomena in the mechanical and biological sciences. This later led to the modern idea that compounds have secondary properties different from the elements of those compounds. Gassendi asserts that corpuscles are particles that carry other substance or substances and are of different types.

Numerous research has suggested two primary assumptions that allow the use of PRCs to be used to predict the occurrence of synchrony within neural oscillation. These assumptions work to show synchrony within coupled neurons that are linked to other neurons. The first assumption claims that coupling between neurons must be weak and requires an infinitesimally small phase change in response to a perturbation. The second assumption assumes coupling between neurons to be pulsatile where the perturbation to calculate PRC should only include those inputs that are received within the circuit.

Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum QCD is recovered. Analytic or perturbative solutions in low-energy QCD are hard or impossible to obtain due to the highly nonlinear nature of the strong force and the large coupling constant at low energies.

If the SU(2) has infinitesimally weak couplings, so that it only confines at enormous distances, then the representation of the SU(2) group and the U(1) charge both are superselection rules. But if the SU(2) has a nonzero coupling then the superselection sectors are separated by infinite mass because the mass of any state in a nontrivial representation is infinite. By changing the temperature, the Higgs fluctuations can zero out the expectation value at a finite temperature. Above this temperature, the SU(2) and U(1) quantum numbers describe the superselection sectors.

Jessen's icosahedron is vertex-transitive (or isogonal), meaning that it has symmetries taking any vertex to any other vertex. Its dihedral angles are all right angles. One can use it as the basis for the construction of a large family of polyhedra with right dihedral angles, formed by gluing copies of Jessen's icosahedron together on their equilateral-triangle faces. net for Jessen's icosahedron, suitable for making a (shaky) physical model Although it is not a flexible polyhedron, Jessen's icosahedron is also not infinitesimally rigid; that is, it is a "shaky polyhedron".

This theory, now known as ZND theory, admits finite-rate chemical reactions and thus describes a detonation as an infinitesimally thin shock wave followed by a zone of exothermic chemical reaction. With a reference frame of a stationary shock, the following flow is subsonic, so that an acoustic reaction zone follows immediately behind the lead front, the Chapman-Jouguet condition. Continued in Continued in There is also some evidence that the reaction zone is semi- metallic in some explosives. Both theories describe one-dimensional and steady wave fronts.

Construction of the envelope of a family of curves. In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two "infinitesimally adjacent" curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.

Because computer-generated imagery reflects only the outside, or skin, of the object being rendered, it fails to capture the infinitesimally small interactions between interlocking muscle groups used in fine motor control, like speaking. The constant motion of the face as it makes sounds with shaped lips and tongue movement, along with the facial expressions that go along with speaking are difficult to replicate by hand. Motion capture can catch the underlying movement of facial muscles and better replicate the visual that goes along with the audio, like Josh Brolin's Thanos.

The special theory of relativity describes the behavior of objects traveling relative to other objects at speeds approaching that of light in a vacuum. Newtonian mechanics is exactly revealed to be an approximation to reality, valid to great accuracy at lower speeds. As the relevant speeds increase toward the speed of light, acceleration no longer follows classical equations. As speeds approach that of light, the acceleration produced by a given force decreases, becoming infinitesimally small as light speed is approached; an object with mass can approach this speed asymptotically, but never reach it.

The magnetic field due to natural magnetic dipoles (upper left), magnetic monopoles (upper right), an electric current in a circular loop (lower left) or in a solenoid (lower right). All generate the same field profile when the arrangement is infinitesimally small. A magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric dipole, but the analogy is not perfect.

At locations inside the charge array, to connect an array of paired charges to an approximation involving only a dipole moment density p(r) requires additional considerations. The simplest approximation is to replace the charge array with a model of ideal (infinitesimally spaced) dipoles. In particular, as in the example above that uses a constant dipole moment density confined to a finite region, a surface charge and depolarization field results. A more general version of this model (which allows the polarization to vary with position) is the customary approach using electric susceptibility or electrical permittivity.

Some modern techniques in analysing spacetimes rely heavily on using spacetime symmetries, which are infinitesimally generated by vector fields (usually defined locally) on a spacetime that preserve some feature of the spacetime. The most common type of such symmetry vector fields include Killing vector fields (which preserve the metric structure) and their generalisations called generalised Killing vector fields. Symmetry vector fields find extensive application in the study of exact solutions in general relativity and the set of all such vector fields usually forms a finite-dimensional Lie algebra.

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.

Combinatorial game theory provides alternative reals as well, with infinite Blue-Red Hackenbush as one particularly relevant example. In 1974, Elwyn Berlekamp described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of data compression. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = . However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the surreal number , where ω is the first infinite ordinal; the relevant game is LRRRR... or 0.000...2.

An impulse (Dirac delta function) is defined as a signal that has an infinite magnitude and an infinitesimally narrow width with an area under it of one, centered at zero. An impulse can be represented as an infinite sum of sinusoids that includes all possible frequencies. It is not, in reality, possible to generate such a signal, but it can be sufficiently approximated with a large amplitude, narrow pulse, to produce the theoretical impulse response in a network to a high degree of accuracy. The symbol for an impulse is δ(t).

The Gibbs–Helmholtz equation is a thermodynamic equation used for calculating changes in the Gibbs energy of a system as a function of temperature. It is named after Josiah Willard Gibbs and Hermann von Helmholtz. The equation is:Physical chemistry, P. W. Atkins, Oxford University Press, 1978, where H is the enthalpy, T the absolute temperature and G the Gibbs free energy of the system, all at constant pressure p. The equation states that the change in the G/T ratio at constant pressure as a result of an infinitesimally small change in temperature is a factor H/T2.

Smaller compressors can approximate isothermal compression even without intercooling, due to the relatively high ratio of surface area to volume of the compression chamber and the resulting improvement in heat dissipation from the compressor body itself. When one obtains perfect isothermal storage (and discharge), the process is said to be "reversible". This requires that the heat transfer between the surroundings and the gas occur over an infinitesimally small temperature difference. In that case, there is no exergy loss in the heat transfer process, and so the compression work can be completely recovered as expansion work: 100% storage efficiency.

Each atom, according to Jain philosophy, has one kind of taste, one smell, one color, and two kinds of touch, though it is unclear what was meant by "kind of touch". Atoms can exist in one of two states: subtle, in which case they can fit in infinitesimally small spaces, and gross, in which case they have extension and occupy a finite space. Certain characteristics of Paramāņu correspond with that of sub-atomic particles. For example, Paramāņu is characterized by continuous motion either in a straight line or in case of attractions from other Paramāņus, it follows a curved path.

Laugwitz (1989) and Benis-Sinaceur (1973) point out that Cauchy continued to use infinitesimals in his own research as late as 1853. Cauchy gave an explicit definition of an infinitesimal in terms of a sequence tending to zero. There has been a vast body of literature written about Cauchy's notion of "infinitesimally small quantities", arguing they lead from everything from the usual "epsilontic" definitions or to the notions of non-standard analysis. The consensus is that Cauchy omitted or left implicit the important ideas to make clear the precise meaning of the infinitely small quantities he used.

Anaxagoras, depicted as a medieval scholar in the Nuremberg Chronicle According to Anaxagoras all things have existed in some way from the beginning, but originally they existed in infinitesimally small fragments of themselves, endless in number and inextricably combined throughout the universe. All things existed in this mass, but in a confused and indistinguishable form. There was an infinite number of homogeneous parts () as well as heterogeneous ones. The work of arrangement, the segregation of like from unlike and the summation of the whole into totals of the same name, was the work of Mind or Reason ().

This ensemble is therefore sometimes called the ensemble, as each of these three quantities is a constant of the ensemble. In simple terms, the micro-canonical ensemble is defined by assigning an equal probability to every micro-state whose energy falls within a range centered at . All other micro-states are given a probability of zero. Since the probabilities must add up to 1, the probability is the inverse of the number of micro-states within the range of energy, :P = 1/W, The range of energy is then reduced in width until it is infinitesimally narrow, still centered at .

This is in contrast to the more common lumped-element model, which assumes that these values are lumped into electrical components that are joined by perfectly conducting wires. In the distributed-element model, each circuit element is infinitesimally small, and the wires connecting elements are not assumed to be perfect conductors; that is, they have impedance. Unlike the lumped-element model, it assumes nonuniform current along each branch and nonuniform voltage along each wire. The distributed model is used where the wavelength becomes comparable to the physical dimensions of the circuit, making the lumped model inaccurate.

The electric field due to a point dipole (upper left), a physical dipole of electric charges (upper right), a thin polarized sheet (lower left) or a plate capacitor (lower right). All generate the same field profile when the arrangement is infinitesimally small. The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI units for electric dipole moment are coulomb-meter (C⋅m); however, a commonly used unit in atomic physics and chemistry is the debye (D).

Born rigidity is satisfied if the orthogonal spacetime distance between infinitesimally separated curves or worldlines is constant, or equivalently, if the length of the rigid body in momentary co-moving inertial frames measured by standard measuring rods (i.e. the proper length) is constant and is therefore subjected to Lorentz contraction in relatively moving frames.Gron (1981) Born rigidity is a constraint on the motion of an extended body, achieved by careful application of forces to different parts of the body. A body rigid in itself would violate special relativity, as its speed of sound would be infinite.

Propositions 70 and 71 consider the force acting on a particle from a hollow sphere with an infinitesimally thin surface, whose mass density is constant over the surface. The force on the particle from a small area of the surface of the sphere is proportional to the mass of the area and inversely as the square of its distance from the particle. The first proposition considers the case when the particle is inside the sphere, the second when it is outside. The use of infinitesimals and limiting processes in geometrical constructions are simple and elegant and avoid the need for any integrations.

An osculating circle Osculating circles of the Archimedean spiral, nested by the Tait–Kneser theorem. "The spiral itself is not drawn: we see it as the locus of points where the circles are especially close to each other." In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p. Its center lies on the inner normal line, and its curvature defines the curvature of the given curve at that point.

Instead, he suggested that his geometric systems were representations of reality but in a more fundamental way that transcends what one can perceive about reality. Jevons claimed that there was a flaw in Helmholtz's argument relating to the concept of infinitesimally small. This concept involves how these creatures reason about geometry and space at a very small scale, which is not necessarily the same as the reasoning that Helmholtz assumed on a more global scale. Jevons claimed that the Euclidean relations could be reduced locally in the different scenarios that Helmholtz created and hence the creatures should have been able to experience the Euclidean properties, just in a different representation.

This, however, is not a complete resolution to Zeno's dichotomy paradox. Strictly speaking, unless we allow for time to move in reverse, where the step size begins with r = 1/2 and approaches zero as a limit, this infinite series would otherwise have to begin with an infinitesimally small step. Treating infinitesimals in this way is typically not something which is rigorously defined mathematically, outside of Nonstandard Calculus. So, while it is true that the entire infinite summation yields a finite number, we can not create a simple ordering of the terms when starting from an infinitesimal, and therefore we can not adequately describe the first step of any given action.

As n, the number of compounding periods per year, increases without limit, the case is known as continuous compounding, in which case the effective annual rate approaches an upper limit of , where is a mathematical constant that is the base of the natural logarithm. Continuous compounding can be thought of as making the compounding period infinitesimally small, achieved by taking the limit as n goes to infinity. See definitions of the exponential function for the mathematical proof of this limit. The amount after t periods of continuous compounding can be expressed in terms of the initial amount P0 as :P(t)=P_0 e ^ {rt}.

To be able to effectively address this you must consider the probability density for a set of points in a small region of phase space, or a set of trajectories. The fluctuation theorem considers the probability density for all of the trajectories that are initially in an infinitesimally small region of phase space. This leads directly to the probability of finding a trajectory, in either the forward or the reverse trajectory sets, depending upon the initial probability distribution as well as the dissipation which is done as the system evolves. It is this crucial difference in approach that allows the fluctuation theorem to correctly solve the paradox.

In the point of view of Cartan connections, however, the affine subspaces of Euclidean space are model surfaces -- they are the simplest surfaces in Euclidean 3-space, and are homogeneous under the affine group of the plane -- and every smooth surface has a unique model surface tangent to it at each point. These model surfaces are Klein geometries in the sense of Felix Klein's Erlangen programme. More generally, an -dimensional affine space is a Klein geometry for the affine group , the stabilizer of a point being the general linear group . An affine -manifold is then a manifold which looks infinitesimally like -dimensional affine space.

The interaction of a component to other components is defined by physical ports, called connectors, e.g., an electrical pin is defined as connector Pin "Electrical pin" Voltage v "Potential at the pin"; flow Current i "Current flowing into the component"; end Pin; When drawing connection lines between ports, the meaning is that corresponding connector variables without the "flow" prefix are identical (here: "v") and that corresponding connector variables with the "flow" prefix (here: "i") are defined by a zero-sum equation (the sum of all corresponding "flow" variables is zero). The motivation is to automatically fulfill the relevant balance equations at the infinitesimally small connection point.

The question becomes how often we would see 40 instead of 1 due to pure chance. According to the hypergeometric distribution, one would expect to try about 10^57 times (10 followed by 56 zeroes) before picking 39 or more of the cholesterol genes from a pool of 10,000 by drawing 200 genes at random. Whether one pays much attention to how infinitesimally small the probability of observing this by chance is, one would conclude that the regulated gene list is enriched in genes with a known cholesterol association. One might further hypothesize that the experimental treatment regulates cholesterol, because the treatment seems to selectively regulate genes associated with cholesterol.

In the second dialogue the same guest announces that pain is an evil. Cicero argues that its sufferings may be overcome, not by the use of Epicurean maxims,—"Short if severe, and light if long," but by fortitude and patience; and he censures those philosophers who have represented pain in too formidable colours, and reproaches those poets who have described their heroes as yielding to its influence. Pain can be neutralized only when moral evil is regarded as the sole evil, or as the greatest of evils that the ills of body and of fortune are held to be infinitesimally small in comparison with it.

A consequence of this apparent paradox is that the electric field of a point- charge can only be described in a limiting sense by a carefully constructed Dirac delta function. This mathematically inelegant but physically useful concept allows for the efficient calculation of the associated physical conditions while conveniently sidestepping the philosophical issue of what actually occurs at the infinitesimally-defined point: a question that physics is as yet unable to answer. Fortunately, a consistent theory of quantum electrodynamics removes the need for infinitesimal point charges altogether. A similar situation occurs in general relativity with the gravitational singularity associated with the Schwarzschild solution that describes the geometry of a black hole.

The tangent space to the 2-sphere at some point is the infinite plane touching the sphere in this point. The tangent plane to a surface at a point is naturally a vector space whose origin is identified with the point of contact. The tangent plane is the best linear approximation, or linearization, of a surface at a point.That is to say , the plane passing through the point of contact P such that the distance from a point P1 on the surface to the plane is infinitesimally small compared to the distance from P1 to P in the limit as P1 approaches P along the surface.

These are some of the many ways in which different quantizations of the same classical theory can result in inequivalent quantum theories, or even in the impossibility to carry quantization through. One can't distinguish between SO(3) and SU(2) or between SO(3,1) and SL(2,C) at this level: the respective Lie algebras are the same. In fact, all four groups have the same complexified Lie algebra, which makes matters even more confusing (these subtleties are usually ignored in elementary particle physics). The physical interpretation of the Lie algebra is that of infinitesimally small group transformations, and gauge bosons (such as the graviton) are Lie algebra representations, not Lie group representations.

For example, Advaita Vedanta holds that after attaining moksha a person knows their "soul, self" and identifies it as one with Brahman and everyone in all respects. The followers of Dvaita (dualistic) schools, in moksha state, identify individual "soul, self" as distinct from Brahman but infinitesimally close, and after attaining moksha expect to spend eternity in a loka (heaven). To theistic schools of Hinduism, moksha is liberation from samsara, while for other schools such as the monistic school, moksha is possible in current life and is a psychological concept. According to Deutsch, moksha is transcendental consciousness to the latter, the perfect state of being, of self-realization, of freedom and of "realizing the whole universe as the Self".

He was sponsored to come to Canada as a farm worker, but later, as a husband and a father of two sons, he supported the family by running a small grocery store in Montreal. Sidney Altman was later to look back on his parents' lives as an illustration of the value of the work ethic: "It was from them I learned that hard work in stable surroundings could yield rewards, even if only in infinitesimally small increments." As Altman reached adulthood, the family's financial situation had become secure enough that he was able to pursue a college education. He went to the United States to study physics at the Massachusetts Institute of Technology.

In this expression, the second integral is calculated first with respect to y and x is held constant—a strip of width dx is integrated first over the y-direction (a strip of width dx in the x direction is integrated with respect to the y variable across the y direction), adding up an infinite amount of rectangles of width dy along the y-axis. This forms a three dimensional slice dx wide along the x-axis, from y=a to y=x along the y axis, and in the z direction z=f(x,y). Notice that if the thickness dx is infinitesimal, x varies only infinitesimally on the slice. We can assume that x is constant.

For the 1918 flu pandemic, they hypothesized that cometary dust brought the virus to Earth simultaneously at multiple locations—a view almost universally dismissed by experts on this pandemic. In 1982 Hoyle presented Evolution from Space for the Royal Institution's Omni Lecture. After considering what he thought of as a very remote possibility of Earth-based abiogenesis he concluded: Published in his 1982/1984 books Evolution from Space (co-authored with Chandra Wickramasinghe), Hoyle calculated that the chance of obtaining the required set of enzymes for even the simplest living cell without panspermia was one in 1040,000. Since the number of atoms in the known universe is infinitesimally tiny by comparison (1080), he argued that Earth as life's place of origin could be ruled out.

On 8 March, Anaesthesia published a report by John Carlisle, a consultant anesthetist in the United Kingdom, on his statistical analysis of the data reported in 168 papers by Fujii. Carlisle used statistical methods to evaluate whether the published distributions of various variables were consistent with the distributions that could be expected to result from random chance. He found that many of the data sets were "extremely unlikely to have arisen by chance", noting that many of the distributions had "likelihoods that are infinitesimally small", citing a calculated probability of 6.78 × 10−9, or about 1 in 150 million. Accordingly, he recommended that data published by Fujii should be "excluded from meta-analyses or reviews" until such time that the unlikely results could be satisfactorily explained.

In Europe, the foundational work was a treatise written by Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to Archimedes' in The Method, but this treatise is believed to have been lost in the 13th century, and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Cavalieri's work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time.

"The Clash Boomkat stated: "An engrossing and entirely out-of-its-time album from an always intriguing fixture of the German leftfield."Boomkat American journalist Philip Sherburne wrote on eMusic "both rhythmic slippage and rubbery room tone point to "real" instruments in "real" space and time, a matrix whose shorthand is "rock 'n' roll." But synthesizers, electronic effects and the very fabric of the music itself always pulls the live-band fantasy back inside a different kind of logic: an ideal form mapped to an infinitesimally pixellated grid. It's hard to put your finger on, but you can feel it, and that's precisely what gives the record its charge: humans imitating machines imitating humans, or maybe the other way round.

The lines PM and KQ are arcs of parallel circles of length (a\cos\varphi)\delta\lambda with\lambda in radian measure. In deriving a point property of the projection at P it suffices to take an infinitesimal element PMQK of the surface: in the limit of Q approaching P such an element tends to an infinitesimally small planar rectangle. Infinitesimal elements on the sphere and a normal cylindrical projection Normal cylindrical projections of the sphere have x=a\lambda and y equal to a function of latitude only. Therefore, the infinitesimal element PMQK on the sphere projects to an infinitesimal element P'M'Q'K' which is an exact rectangle with a base \delta x=a\,\delta\lambda and height \delta y.

A Klein geometry consists of a Lie group G together with a Lie subgroup H of G. Together G and H determine a homogeneous space G/H, on which the group G acts by left-translation. Klein's aim was then to study objects living on the homogeneous space which were congruent by the action of G. A Cartan geometry extends the notion of a Klein geometry by attaching to each point of a manifold a copy of a Klein geometry, and to regard this copy as tangent to the manifold. Thus the geometry of the manifold is infinitesimally identical to that of the Klein geometry, but globally can be quite different. In particular, Cartan geometries no longer have a well-defined action of G on them.

The Standard Model of cosmology is based on a model of spacetime called the Friedmann–Lemaître–Robertson–Walker (FLRW) metric. A metric provides a measure of distance between objects, and the FLRW metric is the exact solution of Einstein field equations (EFE) if some key properties of space such as homogeneity and isotropy are assumed to be true. The FLRW metric very closely matches overwhelming other evidence, showing that the universe has expanded since the Big Bang. If the FLRW metric equations are assumed to be valid all the way back to the beginning of the universe, they can be followed back in time, to a point where the equations suggest all distances between objects in the universe were zero or infinitesimally small.

The length of basic curves is more complicated but can also be calculated. Measuring with rulers, one can approximate the length of a curve by adding the sum of the straight lines which connect the points: center Using a few straight lines to approximate the length of a curve will produce an estimate lower than the true length; when increasingly short (and thus more numerous) lines are used, the sum approaches the curve's true length. A precise value for this length can be found using calculus, the branch of mathematics enabling the calculation of infinitesimally small distances. The following animation illustrates how a smooth curve can be meaningfully assigned a precise length: center However, not all curves can be measured in this way.

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).

Curious to find out what was held within the Eye, but having lost its signal, the Nomai built an orbital cannon to launch probes to visually find the Eye. The chance of visually finding the object with a random shot into space was infinitesimally small, so they also developed a device, the Ash Twin Project, to send the results of the probe's scan 22 minutes back in time, so that the cannon could be "reused" an infinite number of times. The amount of power required to go back in time could only be obtained from a supernova, so they attempted to artificially induce the sun to explode, but were unsuccessful. The Nomai were wiped out by an extinction-level event after completing construction of these projects but before setting the time-loop process into motion.

With this type of game playing with contradictory visual codes (positive and negative space), Metzinger discerns the past and the present infinitesimally separated in time; while the perception of the future is left to the intellectual discretion of the observer to contemplate. Cubist and Futurist devices appear superimposed in Metzinger's At the Cycle-Race Track, creating an image that is readable yet essentially anti-naturalistic. Cubist elements include the reduction of the geometric schema to simplified shapes, and the juxtaposition of rotating planes to define spatial qualities, printed-paper collage, the incorporation of a granular surface and multiple perspective. Parallels with Futurism include the choice of a subject in motion (the bicyclist), the suggestion of velocity (motion blur on the wheel spokes), and the fusing of forms in a static picture plane.

"Structures métriques pour les variétés riemanniennes", edited by Lafontaine and Pierre Pansu, 1981.M. Gromov, Groups of Polynomial growth and Expanding Maps, Publications mathematiques I.H.É.S., 53, 1981 This distance measures how far two compact metric spaces are from being isometric. If X and Y are two compact metric spaces, then dGH (X, Y) is defined to be the infimum of all numbers dH(f(X), g(Y)) for all metric spaces M and all isometric embeddings f : X → M and g : Y → M. Here dH denotes Hausdorff distance between subsets in M and the isometric embedding is understood in the global sense, i.e. it must preserve all distances, not only infinitesimally small ones; for example no compact Riemannian manifold admits such an embedding into Euclidean space of the same dimension.

The period depends on the length of the pendulum, and also to a slight degree on its weight distribution (the moment of inertia about its own center of mass) and the amplitude (width) of the pendulum's swing. For a point mass on a weightless string of length L swinging with an infinitesimally small amplitude, without resistance, the length of the string of a seconds pendulum is equal to L = g/π2 where g is the acceleration due to gravity, with units of length per second squared, and L is the length of the string in the same units. Using the SI recommended acceleration due to gravity of g0 = 9.80665 m/s2, the length of the string will be approximately 993.6 millimeters, i.e. less than a centimeter short of one meter everywhere on Earth.

Buksh was instrumental in making [each] scheme a success on the practical side. By 1966, his leadership had produced 600 organization officers and 11,000 agents, about 50 percent of whom belonged to the Eastern Zone. His value being measured by the amount of business he produced, Buksh secured so many clients that the company was able to “advertise that every second person having life insurance policy in the country was insured with EFU.” Buksh visited London in 1963, 1964 and 1967 and was the main driving force of the EFU's business in London. “His hard work and tremendous expertise had mobilized the nation’s economy, even if infinitesimally.” As business grew, Buksh became deputy general manager in December, 1963 and general manager in January, 1966 expanding the EFU's sales and influence throughout the region.

In chemistry an activated complex is defined by the International Union of Pure and Applied Chemistry (IUPAC) as "that assembly of atoms which corresponds to an arbitrary infinitesimally small region at or near the col (saddle point) of a potential energy surface". In other words, it refers to a collection of intermediate structures in a chemical reaction that persist while bonds are breaking and new bonds are forming. It therefore represents not one defined state, but rather a range of transient configurations that a collection of atoms passes through in between clearly defined products and reactants. It is the subject of transition state theory - also known as activated complex theory - which studies the kinetics of reactions that pass through a defined intermediate state with standard Gibbs energy of formation .

In order to find the volume for this same shape, an integral with bounds a and b such that a and b are intersections of the line y = -x^2 + 4 and y = -1 would be used as follows:\pi \int_a^b (-x^2 + 5)^2 \, dxThe components of the above integral represent the variables in the equation for the volume of a cylinder, \pi r^2 h . The constant pi is factored out, while the radius, -x^2 + 5, is squared within the integral. The height, represented in the volume formula by h, is given in this integral by the infinitesimally small (in order to approximate the volume with the greatest possible accuracy) term dx. Integrals are also used in physics, in areas like kinematics to find quantities like displacement, time, and velocity.

Bollack wrote three books on this language: La Langue Bleue Bolak: langue internationale pratique (1899), Abridged Grammar of the Blue Language (1900) and Premier vocabulaire de la langue bleue Bolak (1902). Bollack caught the attention of H.G. Wells, who wrote in A Modern Utopia: > The language of Utopia will no doubt be one and indivisible; all mankind > will, in the measure of their individual differences in quality, be brought > into the same phase, into a common resonance of thought, but the language > they will speak will still be a living tongue, an animated system of > imperfections, which every individual man will infinitesimally modify. > Through the universal freedom of exchange and movement, the developing > change in its general spirit will be a world-wide change; that is the > quality of its universality. I fancy it will be a coalesced language, a > synthesis of many.

In a resonant antenna, the current and voltage form standing waves along the length of the antenna element, so the magnitude of the current in the antenna varies sinusoidally along its length. The feedpoint, the place where the feed line from the transmitter is attached, can be located at different points along the antenna element. Since radiation resistance depends on the input current, it varies with the feedpoint. It is lowest for feedpoints located at a point of maximum current (an antinode), and highest for feedpoints located at a point of minimum current, a node, such as at the end of the element (theoretically, in an infinitesimally thin antenna element, radiation resistance is infinite at a node, but the finite thickness of actual antenna elements gives it a high but finite value, on the order of thousands of ohms).

For any surface embedded in Euclidean space of dimension 3 or higher, it is possible to measure the length of a curve on the surface, the angle between two curves and the area of a region on the surface. This structure is encoded infinitesimally in a Riemannian metric on the surface through line elements and area elements. Classically in the nineteenth and early twentieth centuries only surfaces embedded in were considered and the metric was given as a 2×2 positive definite matrix varying smoothly from point to point in a local parametrization of the surface. The idea of local parametrization and change of coordinate was later formalized through the current abstract notion of a manifold, a topological space where the smooth structure is given by local charts on the manifold, exactly as the planet Earth is mapped by atlases today.

Indeed, authors may recommend "further research" when, given the existing evidence, further research would be extremely unlikely to be approved by an ethics committee. Studies finding that a treatment has no noticeable effects are sometimes greeted with statements that "more research is needed" by those convinced that the treatment is effective, but the effect has not yet been found. Since even the largest study can never rule out an infinitesimally small effect, an effect can only ever be shown to be insignificant, not non- existent. Similarly, Trish Greenhalgh, Professor of Primary Care Health Sciences at the University of Oxford, argues that FRIN is often used as a way in which a "[l]ack of hard evidence to support the original hypothesis gets reframed as evidence that investment efforts need to be redoubled", and a way to avoid upsetting hopes and vested interests.

Historically, the symbol dx was taken to represent an infinitesimally "small piece" of the independent variable x to be multiplied by the integrand and summed up in an infinite sense. While this notion is still heuristically useful, later mathematicians have deemed infinitesimal quantities to be untenable from the standpoint of the real number system.In the 20th century, nonstandard analysis was developed as a new approach to calculus that incorporates a rigorous concept of infinitesimals by using an expanded number system called the hyperreal numbers. Though placed on a sound axiomatic footing and of interest in its own right as a new area of investigation, nonstandard analysis remains somewhat controversial from a pedagogical standpoint, with proponents pointing out the intuitive nature of infinitesimals for beginning students of calculus and opponents criticizing the logical complexity of the system as a whole.

The entropy of a system depends on its internal energy and its external parameters, such as its volume. In the thermodynamic limit, this fact leads to an equation relating the change in the internal energy U to changes in the entropy and the external parameters. This relation is known as the fundamental thermodynamic relation. If external pressure p bears on the volume V as the only external parameter, this relation is: : dU = T \, dS - p \, dV Since both internal energy and entropy are monotonic functions of temperature T, implying that the internal energy is fixed when one specifies the entropy and the volume, this relation is valid even if the change from one state of thermal equilibrium to another with infinitesimally larger entropy and volume happens in a non-quasistatic way (so during this change the system may be very far out of thermal equilibrium and then the whole-system entropy, pressure, and temperature may not exist).

The convexity can be used to interpret derivative pricing: mathematically, convexity is optionality – the price of an option (the value of optionality) corresponds to the convexity of the underlying payout. In Black–Scholes pricing of options, omitting interest rates and the first derivative, the Black–Scholes equation reduces to \Theta = -\Gamma, "(infinitesimally) the time value is the convexity". That is, the value of an option is due to the convexity of the ultimate payout: one has the option to buy an asset or not (in a call; for a put it is an option to sell), and the ultimate payout function (a hockey stick shape) is convex – "optionality" corresponds to convexity in the payout. Thus, if one purchases a call option, the expected value of the option is higher than simply taking the expected future value of the underlying and inputting it into the option payout function: the expected value of a convex function is higher than the function of the expected value (Jensen inequality).

The problem with the latter option is that every time a player commits such a play, it will infinitesimally increase the percentage of spins a player must win to complete the system. The reason this is so is because the player is adding two numbers (which both will be crossed out in the event of wins) where only one loss was sustained. To prove this, if a player were to play the Labouchère system the same way with the exception being that the player always added half of the wager lost to the bottom of the list twice for every wager lost where: x = Number of wins y = Number of losses z = Numbers originally on the list When: y + (z / 2) ≤ x The result is the list being completed. The player would actually have to win in excess of 50% of the time (the actual percentage of wins necessary, given and , being dependent on ) in order to complete the list, or more than the player could actually be expected to win.

Critics largely reviewed The First Moderns favorably, appreciating Everdell's interdisciplinary approach, in publications including the New York Review of Books, the New York Times, the Los Angeles Times, and the Washington Post.Jim Holt, "Infinitesimally Yours", review of William R. Everdell, The First Moderns: Profiles in the Origins of Twentieth Century Thought, New York Review of Books, May 20, 1999. "Drawing together such disparate manifestations as Seurat's pointillism, Muybridge's stop-motion photography, the poetry of Whitman, Rimbaud, and Laforgue, the tone rows of Schoenberg, and the novels of Joyce, the author [Everdell] makes an engrossing and persuasive case for his claim that 'the heart of Modernism is the postulate of ontological discontinuity'"Hugh Kenner, "A Change of Mind, review of William R. Everdell, The First Moderns: Profiles in the Origins of Twentieth Century Thought, The New York Times, June 29, 1997. "The change started to happen in the 1870s, and not, as William R. Everdell arrestingly demonstrates in The First Moderns, in painting or in literature but in number theory.