51 Sentences With "analyticity" | Random Sentence Generator

For a proof of this theorem, see analyticity of holomorphic functions.

And to explain this logical necessity we must appeal to analyticity once again.

The Cauchy-Kovalevskaya theorem is used in the proof, so the analyticity is necessary.

In Philosophical Analysis in the Twentieth Century, Volume 1: The Dawn of Analysis, Scott Soames has pointed out that Quine's circularity argument needs two of the logical positivists' central theses to be effective: :All necessary (and all a priori) truths are analytic :Analyticity is needed to explain and legitimate necessity. It is only when these two theses are accepted that Quine's argument holds. It is not a problem that the notion of necessity is presupposed by the notion of analyticity if necessity can be explained without analyticity. According to Soames, both theses were accepted by most philosophers when Quine published "Two Dogmas".

Most of Quine's argument against analyticity in the first four sections is focused on showing that different explanations of analyticity are circular. The main purpose is to show that no satisfactory explanation of analyticity has been given. Quine begins by making a distinction between two different classes of analytic statements. The first one is called logically true and has the form: :(1) No unmarried man is married A sentence with that form is true independent of the interpretation of "man" and "married", so long as the logical particles "no", "un-" and "is" have their ordinary English meaning.

Linsky, J. Analytical/Synthetic and Semantic TheoryQuine, W. v. O.: On a Suggestion of KatzKatz, J: Where Things Stand Now with the Analytical/Synthetic Distinction In his book Philosophical Analysis in the Twentieth Century, Volume 1 : The Dawn of Analysis Scott Soames (pp 360–361) has pointed out that Quine's circularity argument needs two of the logical positivists' central theses to be effective: :All necessary truths (and all a priori truths) are analytic :Analyticity is needed to explain and legitimate necessity. It is only when these two theses are accepted that Quine's argument holds. It is not a problem that the notion of necessity is presupposed by the notion of analyticity if necessity can be explained without analyticity.

The statements in the second class have the form: :(2) No bachelor is married. A statement with this form can be turned into a statement with form (1) by exchanging synonyms with synonyms, in this case "bachelor" with "unmarried man". It is the second class of statements that lack characterization according to Quine. The notion of the second form of analyticity leans on the notion of synonymy, which Quine believes is in as much need of clarification as analyticity.

The first four focus on analyticity, the last two on reductionism. There, Quine turns the focus to the logical positivists' theory of meaning. He also presents his own holistic theory of meaning.

1 (1995), pp. 44–74; [2014a], Essay 5, with postscript; [2014b]. and Willard Van Orman Quine."Quine and Gödel on analyticity", Parsons [2014a], Essay 6; also Essays 8 and 9, and [1983], Essay 7.

Her paper on partial differential equations contains what is now commonly known as the Cauchy–Kovalevskaya theorem, which proves the existence and analyticity of local solutions to such equations under suitably defined initial/boundary conditions.

Analyticity would be acceptable if we allowed for the verification theory of meaning: an analytic statement would be one synonymous with a logical truth, which would be an extreme case of meaning where empirical verification is not needed, because it is "confirmed no matter what". "So, if the verification theory can be accepted as an adequate account of statement synonymy, the notion of analyticity is saved after all." The problem that naturally follows is how statements are to be verified. An empiricist would say that it can only be done using empirical evidence.

In Speech Acts, John Searle argues that from the difficulties encountered in trying to explicate analyticity by appeal to specific criteria, it does not follow that the notion itself is void. Considering the way which we would test any proposed list of criteria, which is by comparing their extension to the set of analytic statements, it would follow that any explication of what analyticity means presupposes that we already have at our disposal a working notion of analyticity. In "'Two Dogmas' Revisited", Hilary Putnam argues that Quine is attacking two different notions: Analytic truth defined as a true statement derivable from a tautology by putting synonyms for synonyms is near Kant's account of analytic truth as a truth whose negation is a contradiction. Analytic truth defined as a truth confirmed no matter what, however, is closer to one of the traditional accounts of a priori.

Putnam, Hilary, "'Two dogmas' revisited." In Gilbert Ryle, Contemporary Aspects of Philosophy. Stocksfield: Oriel Press, 1976, 202–213. Jerrold Katz countered the arguments of "Two Dogmas" directly by trying to define analyticity non-circularly on the syntactical features of sentences.

The Kramers–Kronig relations are bidirectional mathematical relations, connecting the real and imaginary parts of any complex function that is analytic in the upper half-plane. The relations are often used to compute the real part from the imaginary part (or vice versa) of response functions in physical systems, because for stable systems, causality implies the condition of analyticity, and conversely, analyticity implies causality of the corresponding stable physical system. The relation is named in honor of Ralph Kronig and Hans Kramers. In mathematics, these relations are known by the names Sokhotski–Plemelj theorem and Hilbert transform.

In mathematics, the FBI transform or Fourier-Bros-Iagolnitzer transform is a generalization of the Fourier transform developed by the French mathematical physicists Jacques Bros and Daniel Iagolnitzer in order to characterise the local analyticity of functions (or distributions) on Rn. The transform provides an alternative approach to analytic wave front sets of distributions, developed independently by the Japanese mathematicians Mikio Sato, Masaki Kashiwara and Takahiro Kawai in their approach to microlocal analysis. It can also be used to prove the analyticity of solutions of analytic elliptic partial differential equations as well as a version of the classical uniqueness theorem, strengthening the Cauchy–Kowalevski theorem, due to the Swedish mathematician Erik Albert Holmgren (1872–1943).

So some form of reductionism - "the belief that each meaningful statement is equivalent to some logical construct upon terms which refer to immediate experience" - must be assumed in order for an empiricist to 'save' the notion of analyticity. Such reductionism, says Quine, presents just as intractable a problem as did analyticity. In order to prove that all meaningful statements can be translated into a sense-datum language, a reductionist would surely have to confront "the task of specifying a sense-datum language and showing how to translate the rest of significant discourse, statement by statement, into it." To illustrate the difficulty of doing so, Quine describes Rudolf Carnap's attempt in his book Der logische Aufbau der Welt.

While the first four sections of Quine's paper concern analyticity, the last two concern a priority. Putnam considers the argument in the two last sections as independent of the first four, and at the same time as Putnam criticizes Quine, he also emphasizes his historical importance as the first top rank philosopher to both reject the notion of a priority and sketch a methodology without it. Jerrold Katz, a one-time associate of Noam Chomsky, countered the arguments of "Two Dogmas" directly by trying to define analyticity non- circularly on the syntactical features of sentences. Chomsky himself critically discussed Quine's conclusion, arguing that it is possible to identify some analytic truths (truths of meaning, not truths of facts) which are determined by specific relations holding among some innate conceptual features of the mind/brain.

516 on the other hand a new method for studying partial waves using the Laplace transform.A.Martin, « Analyticity of partial waves obtained from the Schrödinger equation », Nuovo cimento, 14, (1959), p. 516 After the proof, due to Froissart, that the total effective cross section cannot grow faster than the logarithm squared of the energy, using the Mandelstam representation,M.Froissart, « Asymptotic Behavior and Subtractions in the Mandelstam Representation », Phys.

He founded the theory of Markov random fields, which represented a new direction in the theory of random fields. Later, the theory of Markov random fields was further developed in works related to problems of statistical physics and quantum field theory. Yadrenko studied the analytic properties of choice functions of random fields. In particular, he established conditions for the continuity, analyticity, and quasianalyticity of random fields.

Most of Quine's following arguments are focused on showing how explanations of synonymy end up being dependent on the notions of analyticity, necessity, or even synonymy itself. How do we reduce sentences from the second class to a sentence of the first class? Some might propose definitions. "No bachelor is married" can be turned into "No unmarried man is married" because "bachelor" is defined as "unmarried man".

There is a clear connection with complex analysis. Let us write a complex number z in terms of its real and imaginary parts, say x and y respectively, i.e. . Since , from the point of view of complex analysis, the quotient tends to a limit as dz tends to 0. In other words, the definition of ω∗ was chosen for its connection with the concept of a derivative (analyticity).

Sequent calculi and systems of natural deduction have been developed for several modal logics, but it has proven hard to combine generality with other features expected of good structural proof theories, such as purity (the proof theory does not introduce extra-logical notions such as labels) and analyticity (the logical rules support a clean notion of analytic proof). More complex calculi have been applied to modal logic to achieve generality.

Lars Hörmander (pictured) proved a variant of Rådström's embedding theorem using support functions. Per Enflo (pictured) wrote his doctoral thesis under the supervision of Hans Rådström. Rådström was interested in Hilbert's fifth problem on the analyticity of the continuous operation of topological groups. The solution of this problem by Andrew Gleason used constructions of subsets of topological vector spaces, (rather than simply points), and inspired Rådström's research on set-valued analysis.

"Mathematical Methods and Theory in Games, Programming, and Economics." Dover Publications, 1992. . Another theorem named after Pringsheim gives an analyticity criterion for a C∞ function on a bounded interval, based on the behaviour of the radius of convergence of the Taylor expansion around a point of the interval. However, Pringsheim's original proof had a flaw (related to uniform convergence), and a correct proof was provided by Ralph P. Boas.

In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (1894–1964). The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent Schwartz.

With French physicists Jacques Bros and Henri Epstein he worked on setting up analyticity properties required for the use of dispersion relations in high energy collisions. Epstein, Glaser and Arthur Jaffe proved that (Wightman) quantum fields can necessarily have negative energy density values. Together with Henri Epstein, he found a new approach to renormalization theory called causal perturbation theory, where ultraviolet divergences are avoided in the calculation of Feynman diagrams by using mathematically well-defined quantities only.

Investigations was influential in the development of "ordinary language philosophy," which was mainly promoted by Gilbert Ryle and J. L. Austin. In the United States, meanwhile, the philosophy of Willard Van Orman Quine was having a major influence, with the paper "Two Dogmas of Empiricism". In that paper Quine criticizes the distinction between analytic and synthetic statements, arguing that a clear conception of analyticity is unattainable. Patricia Churchland, 2005 Notable students of Quine include Donald Davidson and Daniel Dennett.

In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called. As with complex and real analysis, it is possible to study the concepts of analyticity, holomorphy, harmonicity and conformality in the context of quaternions. Unlike the complex numbers and like the reals, the four notions do not coincide.

The logical positivists' initial stance was that a statement is "cognitively meaningful" only if some finite procedure conclusively determines its truth.For a classic survey of other versions of verificationism, see Carl G Hempel, "Problems and changes in the empiricist criterion of meaning", Revue Internationale de Philosophie, 1950;41:41–63. By this verifiability principle, only statements verifiable either by their analyticity or by empiricism were cognitively meaningful. Metaphysics, ontology, as well as much of ethics failed this criterion, and so were found cognitively meaningless.

Schröder's equation was solved analytically if is an attracting (but not superattracting) fixed point, that is by Gabriel Koenigs (1884). In the case of a superattracting fixed point, , Schröder's equation is unwieldy, and had best be transformed to Böttcher's equation. There are a good number of particular solutions dating back to Schröder's original 1870 paper. The series expansion around a fixed point and the relevant convergence properties of the solution for the resulting orbit and its analyticity properties are cogently summarized by Szekeres.

Thus, interpreting the Bible through the typological system means emphasizing literary analysis and the flow of the overarching Story through each of the smaller, individual stories. This method of interpretation has been around since the Church Fathers, and writers such as Geerhardus Vos and other 19th century Presbyterian theologians have contributed to the present Presbyterian understanding. In 19th century German Protestantism, typological interpretation was distinguished from rectilinear interpretation of prophecy. The former was associated with Hegelian theologians and the latter with Kantian analyticity.

Among the exponential functions of the form αx, setting α = e2/e = 2.0870652... results in a sharp upper bound; the slightly smaller choice α = 2 fails to produce an upper bound, since then α3 = 8 < 32. In applied fields the word "tight" is often used with the same meaning. ; smooth: Smoothness is a concept which mathematics has endowed with many meanings, from simple differentiability to infinite differentiability to analyticity, and still others which are more complicated. Each such usage attempts to invoke the physically intuitive notion of smoothness.

But for salva veritate to hold as a definition of something more than extensional agreement, i.e., cognitive synonymy, we need a notion of necessity and thus of analyticity. So, from the above example, it can be seen that in order for us to distinguish between analytic and synthetic we must appeal to synonymy; at the same time, we should also understand synonymy with interchangeability salva veritate. However, such a condition to understand synonymy is not enough so we not only argue that the terms should be interchangeable, but necessarily so.

Quine maintains that there is no distinction between universally known collateral information and conceptual or analytic truths. Another approach to Quine's objection to analyticity and synonymy emerges from the modal notion of logical possibility. A traditional Wittgensteinian view of meaning held that each meaningful sentence was associated with a region in the "logical space" Tractatus Logico-Philosophicus 1.13. Quine finds the notion of such a space problematic, arguing that there is no distinction between those truths which are universally and confidently believed and those which are necessarily true.

The former was associated with Hegelian theologians and the latter with Kantian analyticity. Several groups favoring typology today include the Christian Brethren beginning in the 19th century, where typology was much favoured and the subject of numerous books and the Wisconsin Evangelical Lutheran Synod. Notably, in the Eastern Orthodox Church, typology is still a common and frequent exegetical tool, mainly due to that church's great emphasis on continuity in doctrinal presentation through all historical periods. Typology was frequently used in early Christian art, where type and antitype would be depicted in contrasting positions.

Letters, 10, (1963), p. 460 Finally, in 1966 he succeeded in demonstrating the validity of the Froissart bound using local field theory, without postulating the Mandelstam representation.A.Martin, « Extension of the axiomatic analyticity domain of scattering amplitudes by unitarity: », Nuovo Cimento, 42, (1966), p. 930 In the meantime, in 1964, he obtains an absolute bound on the pion-pion scattering amplitude,An absolute bound on the pion pion scattering amplitude, Stanford Preprint ITP-1 (1964) non publié this bound was considerably improved later.B. Bonnier, C Lopez et G.Mennessier, « Improved absolute bounds on the π0π0 amplitude », Physics Letters B, 60, (1), 22 December 1975, p.

Using different degrees of freedom, we have to assure that observables calculated in the EFT are related to those of the underlying theory. This is achieved by using the most general Lagrangian that is consistent with the symmetries of the underlying theory, as this yields the ‘‘most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition and the assumed symmetry. In general there is an infinite number of terms which meet this requirement. Therefore in order to make any physical predictions, one assigns to the theory a power-ordering scheme which organizes terms by some pre-determined degree of importance.

Hilbert's definition of a regular variational problem is stronger than the currently used one, found, for example, in . property means that such kind of variational problems are minimum problems, property is the ellipticity condition on the Euler–Lagrange equations associated to the given functional, while property is a simple regularity assumption the function .Since Hilbert considers all derivatives in the "classical", i.e. not in the weak but in the strong, sense, even before the statement of its analyticity in , the function is assumed to be at least , as the use of the Hessian determinant in implies.

The principle behind the Regge theory hypothesis (also called analyticity of the second kind or the bootstrap principle) is that all strongly interacting particles lie on Regge trajectories. This was considered the definitive sign that all the hadrons are composite particles, but within S-matrix theory, they are not thought of as being made up of elementary constituents. The Regge theory hypothesis allowed for the construction of string theories, based on bootstrap principles. The additional assumption was the narrow resonance approximation, which started with stable particles on Regge trajectories, and added interaction loop by loop in a perturbation series.

In his essay Two Dogmas of Empiricism, the philosopher W. V. O. Quine called into question the distinction between analytic and synthetic statements. It was this second class of analytic statements that caused him to note that the concept of analyticity itself stands in need of clarification, because it seems to depend on the concept of synonymy, which stands in need of clarification. In his conclusion, Quine rejects that logical truths are necessary truths. Instead he posits that the truth-value of any statement can be changed, including logical truths, given a re-evaluation of the truth-values of every other statement in one's complete theory.

A simple consequence of the Bros and Iagolnitzer characterisation of local analyticity is the following regularity result of Lars Hörmander and Mikio Sato (). Theorem. Let P be an elliptic partial differential operator with analytic coefficients defined on an open subset X of Rn. If Pf is analytic in X, then so too is f. When "analytic" is replaced by "smooth" in this theorem, the result is just Hermann Weyl's classical lemma on elliptic regularity, usually proved using Sobolev spaces (Warner 1983). It is a special case of more general results involving the analytic wave front set (see below), which imply Holmgren's classical strengthening of the Cauchy–Kowalevski theorem on linear partial differential equations with real analytic coefficients.

In operator theory, the Gelfand–Mazur theorem is a theorem named after Israel Gelfand and Stanisław Mazur which states that a Banach algebra with unit over the complex numbers in which every nonzero element is invertible is isometrically isomorphic to the complex numbers, i. e., the only complex Banach algebra that is a division algebra is the complex numbers C. The theorem follows from the fact that the spectrum of any element of a complex Banach algebra is nonempty: for every element a of a complex Banach algebra A there is some complex number λ such that λ1 − a is not invertible. This is a consequence of the complex-analyticity of the resolvent function. By assumption, λ1 − a = 0.

Note that the foregoing proof of analyticity derived an expression for a system of n different function elements fi(x), provided that x is not a critical point of p(x, y). A critical point is a point where the number of distinct zeros is smaller than the degree of p, and this occurs only where the highest degree term of p vanishes, and where the discriminant vanishes. Hence there are only finitely many such points c1, ..., cm. A close analysis of the properties of the function elements fi near the critical points can be used to show that the monodromy cover is ramified over the critical points (and possibly the point at infinity).

In Quine's view, the indeterminacy of translation leads to the inability to separate analytic statements whose validity lies in the usage of language from synthetic statements, those that assert facts about the world. The argument hinges on the role of synonymy in analytic statements, "A natural suggestion, deserving close examination, is that the synonymy of two linguistic forms consists simply in their interchangeability in all contexts without change of truth value". However, Quine argues, because of the indeterminacy of translation, any attempt to define 'analyticity' on a substitutional basis invariably introduces assumptions of the synthetic variety, resulting in a circular argument. Thus, this kind of substitutability does not provide an adequate explanation of synonyms.

Between 1933 and 1938 he applied his results to elliptic equations, establishing the majorizing limits for their solutions, generalizing the two- dimensional case of Felix Bernstein. At the same time he studied analytic functions of several complex variables, that is, analytic functions whose domain belongs to the vector space , proving in 1933 the fundamental theorem on normal families of such functions: if a family is normal with respect to every complex variable, it is also normal with respect to the set of the variables. He also proved a logarithmic residue formula for functions of two complex variables in 1949. In 1935 Caccioppoli proved the analyticity of class solutions of elliptic equations with analytic coefficients.

Analytic truth defined as a true statement derivable from a tautology by putting synonyms for synonyms is near Kant's account of analytic truth as a truth whose negation is a contradiction. Analytic truth defined as a truth confirmed no matter what however, is closer to one of the traditional accounts of a priori. While the first four sections of Quine's paper concern analyticity, the last two concern apriority. Putnam considers the argument in the two last sections as independent of the first four, and at the same time as Putnam criticizes Quine, he also emphasizes his historical importance as the first top rank philosopher to both reject the notion of apriority and sketch a methodology without it.

In other words, Quine accepted that analytic statements are those that are true by definition, then argued that the notion of truth by definition was unsatisfactory. Quine's chief objection to analyticity is with the notion of synonymy (sameness of meaning), a sentence being analytic, just in case it substitutes a synonym for one "black" in a proposition like "All black things are black" (or any other logical truth). The objection to synonymy hinges upon the problem of collateral information. We intuitively feel that there is a distinction between "All unmarried men are bachelors" and "There have been black dogs", but a competent English speaker will assent to both sentences under all conditions since such speakers also have access to collateral information bearing on the historical existence of black dogs.

Quine emphasizes his naturalism, the doctrine that philosophy should be pursued as part of natural science. He argues in favor of naturalizing epistemology, supports physicalism over phenomenalism and mind-body dualism, and extensionality over intensionality, develops a behavioristic conception of sentence-meaning, theorizes about language learning, speculates on the ontogenesis of reference, explains various forms of ambiguity and vagueness, recommends measures for regimenting language to eliminate ambiguity and vagueness as well as to make perspicuous the logic and ontic commitments of theories, argues against quantified modal logic and the essentialism it presupposes, argues for Platonic realism in mathematics, rejects instrumentalism in favor of scientific realism, develops a view of philosophical analysis as explication, argues against analyticity and for holism, against countenancing propositions, and tries to show that the meanings of theoretical sentences are indeterminate and that the reference of terms is inscrutable.

Taking inputs from Vedic sciences, Hankey tries to resolve problems in theoretical physics like the nature of the Hawking-Penrose singularity, "holistic" processes in physics and biology, the interpretation of quantum theory, the origins of thermodynamics, the implications of dispersion relations and analyticity. He has developed a new complexity-based theory of cognition, and a Vedic approach to understanding quantum theory with new extensions of that theory. Due to his diverse research interests, he has been a guest speaker at many international and national conferences, and has been featured in the news several times for promoting traditional knowledge to serve mankind and restoring a substantial ethical basis for modern life, introducing proper preventative health programss in developed societies, etc., Hankey lives in Bangalore, India, where he guides PhD research on yoga, meditation, the mind-body connection, and electronic measurement of holistic aspects of organism function.

Once the spectrum of particles is known, the force law is known, and this means that the spectrum is constrained to bound states which form through the action of these forces. The simplest way to solve the consistency condition is to postulate a few elementary particles of spin less than or equal to one, and construct the scattering perturbatively through field theory, but this method does not allow for composite particles of spin greater than 1 and without the then undiscovered phenomenon of confinement, it is naively inconsistent with the observed Regge behavior of hadrons. Chew and followers believed that it would be possible to use crossing symmetry and Regge behavior to formulate a consistent S-matrix for infinitely many particle types. The Regge hypothesis would determine the spectrum, crossing and analyticity would determine the scattering amplitude (the forces), while unitarity would determine the self-consistent quantum corrections in a way analogous to including loops.

Biswas worked in several diverse areas of theoretical high energy physics and particle physics,, that includes his early work in collaboration with Herbert S. Green on the Bethe-Salpeter equation and its solution, several investigations in particle physics phenomenology, two- dimensional quantum electrodynamics, analysis of anharmonic oscillator in quantum mechanics, scattering theory, study of dispersion relations in collision processes of elementary particles based on unitarity and analyticity, geometric phases of wave function in in quantum mechanics and quantum optics, equation of state of neutron stars, quark stars, weak interaction processes, weak decays involving neutral currents, processes involving stellar energy loss, supersymmetry in weak currents, chiral anomalies, super-propagator for a non-polynomial field, phase transitions in gauge theories, development of supersymmetric classical mechanics, supersymmetric quantum mechanics, stochastic quantization, quark stars, continued fraction theory, role of of parastatistics in statistical mechanics,, Biswas has written over 90 scientific articles, which have received a large number of citations.