In fact, 2 is the only prime that ramifies in Z[i], since every prime that ramifies must divide the discriminant of Z[i], which is −4\.
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In the example above, the discriminant of the number field Q(x) with x3 − x − 1 = 0 is −23, and as we have seen the 23-adic place ramifies. The Dedekind discriminant tells us it is the only ultrametric place which does. The other ramified place comes from the absolute value on the complex embedding of F.
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If fj = ej = 1 for every j (and thus g = [L : K]), we say that p splits completely in L. If g = 1 and f1 = 1 (and so e1 = [L : K]), we say that p ramifies completely in L. Finally, if g = 1 and e1 = 1 (and so f1 = [L : K]), we say that p is inert in L.
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The masseteric nerve passes laterally, above the pterygoideus externus, in front of the temporomandibular articulation, and behind the tendon of the temporalis; it crosses the mandibular notch with the masseteric artery, to the deep surface of the masseter, in which it ramifies nearly as far as its anterior border. It is a branch of the mandibular nerve (V3). It gives a filament to the temporomandibular joint.
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The anterior tibial recurrent artery is a small artery in the leg. It arises from the anterior tibial artery, as soon as that vessel has passed through the interosseous space. It ascends in the tibialis anterior muscle, ramifies on the front and sides of the knee-joint, and assists in the formation of the patellar plexus by anastomosing with the genicular branches of the popliteal, and with the highest genicular artery.
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Much of the significance of the discriminant lies in the fact that ramified ultrametric places are all places obtained from factorizations in Qp where p divides the discriminant. This is even true of the polynomial discriminant; however the converse is also true, that if a prime p divides the discriminant, then there is a p-place which ramifies. For this converse the field discriminant is needed. This is the Dedekind discriminant theorem.
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The auricular branch of posterior auricular artery is a small artery in the head. It branches off the posterior auricular artery and ascends behind the ear, beneath the posterior auricular muscle, and is distributed to the back of the auricula, upon which it ramifies minutely, some branches curving around the margin of the cartilage, others perforating it, to supply the anterior surface. It anastomoses with the parietal and anterior auricular branches of the superficial temporal artery.
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Unlike most other mosses, the endostome (inner row) does not divide into teeth, but rather is a continuous pleated membrane around the capsule opening. Only the genus Diphyscium has a similar peristome structure, although that genus has only 16 pleats in its endostome, in contrast to the 32 pleats in Buxbaumia. Diphyscium shares with Buxbaumia one other oddity of the sporophyte; the foot (stalk base) ramifies as a result of outgrowths, so much so that they may be mistaken for rhizoids.
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The anterior medial malleolar artery (medial anterior malleolar artery, internal malleolar artery) is an artery in the ankle. It arises about 5 cm. above the ankle-joint from the anterior tibial artery. The anterior medial malleolar artery passes behind the tendons of the extensor hallucis longus and tibialis anterior muscles, to the medial side of the ankle, upon which it ramifies, anastomosing with branches of the posterior tibial and medial plantar arteries and with the medial calcaneal from the posterior tibial.
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The anterior tympanic artery (glaserian artery) is a small artery in the head that supplies the middle ear. It usually arises as a branch of the first part of the maxillary artery. It passes upward behind the temporomandibular articulation, enters the tympanic cavity through the petrotympanic fissure, and ramifies upon the tympanic membrane, forming a vascular circle around the membrane with the stylomastoid branch of the posterior auricular, and anastomosing with the artery of the pterygoid canal and with the caroticotympanic branch from the internal carotid.
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It ramifies towards aesthetics, acoustics, hermeneutics, symbolism, metaphysics, sociology, music therapy, pedagogy, psycho - pedagogy, and rhetoric, and includes research on the interpretation (Aufführungspraxis) allowing to execute the old repertoires according to their authenticity. It revalues the orality as a living vehicle of traditional practices and knowledge, bearing precious values: vital, human, spiritual (cf. Marcel Jousse). This broad, open, spiritualist conception of musical science confers their originality to the works that Jacques Viret has published since 2004 at , about gregorian chant, Medieval music, Richard Wagner, musicothérapy, Baroque music, and opera.
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After invading a host larva, the fungus ramifies throughout the host and eventually kills it. Gradually the host larvae become rigid because of the production of fungal sclerotia. Fungal sclerotia are multihyphal structures that can remain dormant and then germinate to produce spores. After over-wintering, the fungus ruptures the host body, forming the fruiting body, a sexual sporulating structure (a perithecial stroma) from the larval head that is connected to the sclerotia (dead larva) below ground and grows upward to emerge from the soil to complete the cycle.
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The relative discriminant regulates the ramification data of the field extension K/L. A prime ideal p of L ramifies in K if, and only if, it divides the relative discriminant ΔK/L. An extension is unramified if, and only if, the discriminant is the unit ideal. The Minkowski bound above shows that there are no non-trivial unramified extensions of Q. Fields larger than Q may have unramified extensions: for example, for any field with class number greater than one, its Hilbert class field is a non- trivial unramified extension.
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The prime 2 of Z ramifies in Z[i]: :(2)=(1+i)^2 The ramification index here is therefore e = 2. The residue field is :O_L / (1+i)O_L which is the finite field with two elements. The decomposition group must be equal to all of G, since there is only one prime of Z[i] above 2. The inertia group is also all of G, since :a+bi\equiv a-bi\bmod1+i for any integers a and b, as a+bi = 2bi + a-bi =(1+i) \cdot (1-i)bi + a-bi \equiv a-bi \bmod 1+i .
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It is suggested that the system of levels proposed by Chomsky in Aspects of the Theory of Syntax has its antecedents in the works of Descartes, Kant, Carnap, Quine, and others. Certainly the criterion of adequacy found in rationalism, specifically, rational empiricism, bear some resemblance to Chomsky's formulation. Since one of the key issues which Chomsky treats in Aspects is a supposition of a congenital endowment of the language faculty in humans, the topic ramifies into questions of innateness and a priori knowledge, since it is by reference to those questions that the third level of adequacy is to be sought.
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The superior labial artery (superior labial branch of facial artery) is larger and more egregious than the inferior labial artery. It follows a similar course along the edge of the upper lip, lying between the mucous membrane and the orbicularis oris, and anastomoses with the artery of the opposite side. It supplies the upper lip, and gives off in its course two or three vessels which ascend to the nose; a septal branch ramifies on the nasal septum as far as the point of the nose, and an alar branch supplies the ala of the nose.
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The medial superior genicular, a branch of the popliteal artery, runs in front of the Semimembranosus and Semitendinosus, above the medial head of the Gastrocnemius, and passes beneath the tendon of the Adductor magnus. It divides into two branches, one of which supplies the vastus medialis, anastomosing with the highest genicular and medial inferior genicular arteries; the other ramifies close to the surface of the femur, supplying it and the knee-joint, and anastomosing with the lateral superior genicular artery. The medial superior genicular artery is frequently of small size, a condition, which is associated with an increase in the size of the highest genicular.
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These ultimately leave the trabecular sheaths, and terminate in the proper substance of the spleen in small tufts or pencils of minute arterioles, which open into the interstices of the reticulum formed by the branched sustentacular cells. Each of the larger branches of the artery supplies chiefly that region of the organ in which the branch ramifies, having no anastomosis with the majority of the other branches. The arterioles, supported by the minute trabeculae, traverse the pulp in all directions in bundles (penicilli) of straight vessels. Their trabecular sheaths gradually undergo a transformation, become much thickened, and converted into adenoid tissue; the bundles of connective tissue becoming looser and their fibrils more delicate, and containing in their interstices an abundance of lymph corpuscles.
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For example, ramification is a phenomenon of codimension 1 (in the geometry of complex manifolds, reflecting as for Riemann surfaces that ramify at single points that it happens in real codimension two). A classical result, Zariski-Nagata purity of Masayoshi Nagata and Oscar Zariski, called also purity of the branch locus, proves that on a non-singular algebraic variety a branch locus, namely the set of points at which a morphism ramifies, must be made up purely of codimension 1 subvarieties (a Weil divisor). There have been numerous extensions of this result into theorems of commutative algebra and scheme theory, establishing purity of the branch locus in the sense of description of the restrictions on the possible "open subsets of failure" to be an étale morphism.
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It gives twigs to the muscles attached to the ischial tuberosity and anastomoses with the inferior gluteal artery. It also supplies an articular branch which enters the hip-joint through the acetabular notch, ramifies in the fat at the bottom of the acetabulum and sends a twig along the ligament of head of femur (ligamentum teres) to the head of the femur. The blood supply to the femoral head and neck is enhanced by the artery of the ligamentum teres derived from the obturator artery. In adults, this is small and doesn't have much importance, but in children whose epiphyseal line is still made of cartilage (which doesn't allow blood supply through it), it helps to supply the head and neck of the femur on its own.
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Branching veins on underside of taro leaf The venation within the bract of a lime tree Micrograph of a leaf skeleton Veins (sometimes referred to as nerves) constitute one of the more visible leaf traits or characteristics. The veins in a leaf represent the vascular structure of the organ, extending into the leaf via the petiole and providing transportation of water and nutrients between leaf and stem, and play a crucial role in the maintenance of leaf water status and photosynthetic capacity.They also play a role in the mechanical support of the leaf. Within the lamina of the leaf, while some vascular plants possess only a single vein, in most this vasculature generally divides (ramifies) according to a variety of patterns (venation) and form cylindrical bundles, usually lying in the median plane of the mesophyll, between the two layers of epidermis.
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Inline-Hockey Puck Although inline hockey appears to simply be ice hockey on inline skates, this single change ramifies through the rest of the game, resulting in important differences between the two sports. Inline hockey is typically played at room temperature on a surface that, rather than being made from (frozen) water, is kept dry to protect the bearings in the skate wheels. Several surface materials are used, including plastic tiles (sometimes known as sport-court flooring), wood, and sealed concrete; in general, surfaces try to balance the ability of wheels to grip against the ability of the puck to slide freely. None of these surfaces, however, is as smooth as ice; as a result, the puck is made of a much lighter plastic material, and rests on small nylon or poly-plastic nubs to reduce friction with the rink surface.
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Besides distributing branches to the sternocleidomastoid (which, however, mainly is supplied by the occipital artery and the superior thyroid artery), subclavius (which mainly is supplied by the thoracoacromial artery), and neighboring muscles, it gives off a suprasternal branch, which crosses over the sternal end of the clavicle to the skin of the upper part of the chest; and an acromial branch, which pierces the trapezius and supplies the skin over the acromion, anastomosing with the thoracoacromial artery. Just as with supplying the subclavius muscle, it anastomoses with the thoracoacromial artery in supplying skin areas. As the artery passes over the superior transverse scapular ligament, it sends a branch into the subscapular fossa, where it ramifies beneath the subscapularis, and anastomoses with the subscapular artery and with the dorsal scapular artery. It also sends articular branches to the acromioclavicular joint and the shoulder joint, and a nutrient artery to the clavicle.
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According to Carsetti, the result is a surfacing "countenance" and an extended arborization revealing itself at surface level (in that life traverses it) as a design-web of connections unified by a living intentionality finally expressing itself as true harmony. "This harmony - he writes - is that of signs created by a “hand” which ramifies (as instantiated, for example, by Auguste Rodin in a famous sculpture) to become countenance, a hand that individuates (and perceives) itself, through the operated design, as the combined and productive features of meaning in action". At the level of the membrane (or at the level of the visual cortex, for example), a (neural) geometry of the interacting connections is created by means of a specific self-organization process: a neurogeometry that will be to weave itself on the basis of the nourishment offered by the original web of the programs. Hence the birth of an intersection between the research carried out by Carsetti and the in-depth analysis work carried out by Jean Petitot and his school, an intersection that has given rise over the years to a variety of distinct publications.
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