31,196 results
Search Results
2. New numerical simulations for some real world problems with Atangana–Baleanu fractional derivative
- Author
-
Behzad Ghanbari, Wei Gao, and Haci Mehmet Baskonus
- Subjects
Work (thermodynamics) ,General Mathematics ,Applied Mathematics ,Numerical analysis ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Lipschitz continuity ,Differential operator ,01 natural sciences ,010305 fluids & plasmas ,Fractional calculus ,Superposition principle ,Operator (computer programming) ,Kernel (statistics) ,0103 physical sciences ,Applied mathematics ,010301 acoustics ,Mathematics - Abstract
In this work, we introduce ABC-Caputo operator with ML kernel and its main characteristics are discussed. Viral diseases models for AIDS and Zika are considered, and finally, as third model, the macroeconomic model involving ABC fractional derivatives is investigated, respectively. It is presented that the AB Caputo derivatives satisfy the Lipschitz condition along with superposition property. The numerical methods for solving the fractional models are presented by means of ABC fractional derivative in a detailed manner. Finally the simulation results obtained in this paper according to the suitable values of parameters are also manifested.
- Published
- 2019
3. On high-frequency limits of $U$-statistics in Besov spaces over compact manifolds
- Author
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Claudio Durastanti and Solesne Bourguin
- Subjects
Connection (fibred manifold) ,Pure mathematics ,General Mathematics ,Mathematics - Statistics Theory ,Probability density function ,Statistics Theory (math.ST) ,Wavelets ,Poisson distribution ,01 natural sciences ,Point process ,010104 statistics & probability ,symbols.namesake ,60F05 ,Poisson point process ,FOS: Mathematics ,Compact Riemannian manifolds ,60B05 ,0101 mathematics ,Stein-Malliavin method ,Central limit theorem ,Mathematics ,62E20 ,U-Statistics, Poisson random measures, High-frequency limit theorems, Wavelets, Compact Riemannian manifolds, Besov spaces, Stein-Malliavin method ,60B05, 60F05, 60G57, 62E20 ,010102 general mathematics ,Manifold ,U-Statistics ,Besov spaces ,symbols ,60G57 ,Besov space ,Poisson random measures ,High-frequency limit theorems - Abstract
In this paper, quantitative bounds in high-frequency central limit theorems are derived for Poisson based $U$-statistics of arbitrary degree built by means of wavelet coefficients over compact Riemannian manifolds. The wavelets considered here are the so-called needlets, characterized by strong concentration properties and by an exact reconstruction formula. Furthermore, we consider Poisson point processes over the manifold such that the density function associated to its control measure lives in a Besov space. The main findings of this paper include new rates of convergence that depend strongly on the degree of regularity of the control measure of the underlying Poisson point process, providing a refined understanding of the connection between regularity and speed of convergence in this framework., Comment: 19 pages
- Published
- 2017
4. Powers of the szegö Kernel and Hankel operators on hardy spaces
- Author
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Marco M. Peloso, Frédéric Symesak, Aline Bonami, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Dipartimento di Matematica 'Giuseppe Peano' [Torino], Università degli studi di Torino (UNITO), Groupes de Lie et Géométrie, Laboratoire de Mathématiques, and Université de Poitiers
- Subjects
Discrete mathematics ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,Hilbert space ,Microlocal analysis ,32A25 ,Spectral theorem ,Hardy space ,Operator theory ,01 natural sciences ,Fourier integral operator ,Compact operator on Hilbert space ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,46E15 ,0101 mathematics ,47B35 ,Operator norm ,Mathematics - Abstract
In this paper we study the action of certain integral operators on spaces of holomorphic functions on some domains in Cn: These integral operators are defined by using powers of the Szego kernel as integral kernel. We show that they act like differential operators, or like pseudo-differential operators of not necessarily integral order. These operators may be used to give equivalent norms for the Besov spaces Bp of holomorphic functions. As a consequence we prove that, when 1 p < 1; the small Hankel operators hf on Hardy and weighted Bergman spaces are in the Schatten class Sp if and only if the symbol f belongs to Bp: The type of domains we deal with are the smoothly bounded strictly pseudoconvex domains in Cn and a class of complex ellipsoids in Cn: Our results for strictly pseudo-convex domains depend on Fefferman's expansion of the Szego kernel. In this case, its powers act like a power of the derivation in the normal direction. The ellipsoids we consider are the simplest examples of domains of finite type. In this case, the symmetries of the domains can be exploited to use methods of harmonic analysis and describe the pseudo-differential operators involved.
5. Global results for semilinear hyperbolic equations with damping term on manifolds with conical singularity
- Author
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Gholamreza Karamali, Morteza Koozehgar Kalleji, and Mohsen Alimohammady
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,General Engineering ,Hyperbolic manifold ,Conical surface ,01 natural sciences ,Manifold ,010101 applied mathematics ,Sobolev space ,Singularity ,Gravitational singularity ,Boundary value problem ,0101 mathematics ,Hyperbolic partial differential equation ,Mathematics - Abstract
In this paper, we apply the family of potential wells to the initial boundary value problem of semilinear hyperbolic equations on the cone Sobolev spaces. We not only give some results of global existence and nonexistence of solutions but also obtain the vacuum isolating of solutions. Finally, we show blow-up in finite time of solutions on a manifold with conical singularities. Copyright © 2017 John Wiley & Sons, Ltd.
- Published
- 2017
6. The Duty of Exposition with Special Reference to the Cauchy-Heaviside Expansion Theorem
- Author
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Francis D. Murnaghan
- Subjects
business.industry ,Mathematical society ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Mathematical association ,01 natural sciences ,Mathematical research ,law.invention ,Epistemology ,Publishing ,law ,Nothing ,0103 physical sciences ,CLARITY ,010307 mathematical physics ,0101 mathematics ,business ,Mathematical economics ,Duty ,Competence (human resources) ,media_common ,Mathematics - Abstract
As the speaker representing the Mathematical Association of America at this session I propose this morning to call your attention to one of the more important duties of a mathematician, namely, the duty of explaining as clearly as possible mathematical truths and discoveries both to his fellow mathematicians and to students of mathematics in general. The American Mathematical Society is especially devoted to the encouragement of research and the secretary of that society has called to your attention during this meeting the remarkable increase in the number of papers presented annually to the Society during the past five years. This increase has been such as to make the problem of publishing the results of mathematical research in America a very acute and difficult one. At the same time it has become more desirable than ever before that the papers published in our American journals should be as clear and as easily readible as possible. The Mathematical Association of America is especially concerned with the teaching and exposition of mathematical truth and it is pretty generally agreed that it is highly desirable that care should be taken to make this teaching and exposition to students as good as possible. I fear, however, that some of my friends who are particularly interested in research agree to this in a rather condescending manner; their tone implying that, whilst nothing should be done to discourage anyone beginning the study of mathematics, such care is not so necessary nor even so desirable when writing for fellow mathematicians. The underlying idea is that a competent mathematician usually prefers to glance at a paper, see the results arrived at, and then derive these results in his own individual manner. I believe that this opinion is not justified by the facts and in support of this belief I shall mention two instances which have recently come to my attention where mathematicians of great competence failed, through a lack of detail or of clarity in available expositions of known results, to arrive at immediate and important corollaries of these results. I shall be happy if my talk this morning tends to make the editors of our mathematical journals insist more definitely on clearness of exposition when considering papers submitted for publication. One can surmise that the writers of at least some of our papers set down their results with the referee of the paper more in mind than the prospective readers. They neglect, therefore, to state or. emphasize points which they think will be familiar to the referee. Of course this is unfortunate since there is us'ually a one-to-one
- Published
- 1927
7. The structured coalescent process with weak migration
- Author
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Morihiro Notohara
- Subjects
Statistics and Probability ,Coalescence (physics) ,education.field_of_study ,Recurrence relation ,General Mathematics ,Weak solution ,010102 general mathematics ,Population ,Linear system ,Perturbation (astronomy) ,Moment-generating function ,01 natural sciences ,010104 statistics & probability ,Statistics ,Quantitative Biology::Populations and Evolution ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Coalescent process ,education ,Mathematics - Abstract
The aim of this paper is to study genealogical processes in a geographically structured population with weak migration. The coalescence time for sampled genes from different colonies diverges to infinity as the migration rates among colonies are close to zero. We investigate the moment generating functions of the coalescence time, the number of segregating sites and the number of allele types in sampled genes when there is low migration. Employing a perturbation method, we obtain a system of recurrence relations for the approximate solutions of these moment generating functions and solve them in some cases.
- Published
- 2001
8. Groups with Representations of Bounded Degree
- Author
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Irving Kaplansky
- Subjects
Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Combinatorics ,Matrix (mathematics) ,Identity (mathematics) ,Compact group ,Bounded function ,Completeness (order theory) ,0103 physical sciences ,Converse ,010307 mathematical physics ,0101 mathematics ,Unit (ring theory) ,Commutative property ,Mathematics - Abstract
Let G be a compact group. According to the celebrated theorem of Peter-Weyl there exists a complete set of finite-dimensional irreducible unitary representations of G, the completeness meaning that for any group element other than the identity there is a representation sending it into a matrix other than the unit matrix. If G is commutative, the representations are necessarily one-dimensional. It is an immediate consequence of the Peter-Weyl theorem that the converse also holds: if every representation is one-dimensional, G is commutative. The main theorem in the present paper is a generalization of this result to the case where the representations have bounded degree.
- Published
- 1949
9. Mean-Continuous Integrals
- Author
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H. W. Ellis
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Descriptive definitions of Cesàro-Denjoy integrals (CD-integrals) equivalent to the Cesàro-Perron integrals (CP-integrals) introduced by J. C. Burkill [1, 2] have been given by Miss Sargent [6] (see §2). The CD§integrals are generalizations of the special Denjoy integral [5, p. 201]. They are somewhat complicated in that modifications of the definitions of continuity, generalized absolute continuity in the restricted sense (ACG*) [5, p. 231], and of derivatives are required for each order. In the present paper a scale of integrals is obtained which is based on the descriptive definition of the general Denjoy integral [5, p. 241]. The approximate derivative and a slightly modified definition of generalized absolute continuity (ACG) are used for all orders so that the only concept generalized for increasing orders is that of continuity.
- Published
- 1949
10. Completely Indecomposable Modules
- Author
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Ernst Snapper
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Indecomposable module ,01 natural sciences ,Mathematics - Abstract
The purpose of this paper is to investigate completely indecomposable modules. A completely indecomposable module is an additive abelian group with a ring A as operator domain, where the following four conditions are satisfied.1-1. A is a commutative ring and has a unit element which is unit operator for .1-2. The submodules of satisfy the ascending chain condition. (Submodule will always mean invariant submodule.)
- Published
- 1949
11. On the Disjoint Product of Irreducible Representations of the Symmetric Group
- Author
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G. de B. Robinson
- Subjects
Discrete mathematics ,Disjoint union ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Specht module ,Irreducible element ,Disjoint sets ,01 natural sciences ,Combinatorics ,Representation theory of the symmetric group ,Representation theory of SU ,Symmetric group ,Product (mathematics) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The results of the present paper can be interpreted (a) in terms of the theory of the representations of the symmetric group, or (b) in terms of the corresponding theory of the full linear group. In the latter connection they give a solution to the problem of the expression of an invariant matrix of an invariant matrix as a sum of invariant matrices, in the sense of Schur's Dissertation. D. E. Littlewood has pointed out the significance of this problem for invariant theory and has attacked it via Schur functions, i.e. characters of the irreducible representations of the full linear group. We shall confine our attention here to the interpretation (a).
- Published
- 1949
12. Direct Theorems on Methods of Summability
- Author
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George G. Lorentz
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
1.1. A regular Toeplitz method of summability is given by a transformation m = 0, 1, 2 , … of the sequence sn into the sequence σm. According to the definition of regularity, every such method sums a convergent sequence sn to the value lim sn. The question naturally arises, whether there are more extensive classes of sequences summable by all regular methods 1.1(1) or at least by all such methods subject to some simple additional conditions. Questions of this kind have been treated by the author (Lorentz [2], [5]) and, from another point of view, by R. P. Agnew [2] [3] ; in this paper we wish to discuss the problem systematically.
- Published
- 1949
13. Factorization Ladders and Eigenfunctions
- Author
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G. F. D. Duff
- Subjects
Algebra ,Factorization ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Eigenfunction ,01 natural sciences ,Mathematics - Abstract
The eigenfunctions of a boundary value problem are characterized by two quite distinct properties. They are solutions of ordinary differential equations, and they satisfy prescribed boundary conditions. It is a definite advantage to combine these two requirements into a single problem expressed by a unified formula. The use of integral equations is an example in point. The subject of this paper, namely the Schrödinger-Infeld Factorization Method, which is applicable to certain restricted. Sturm-Liouville problems, is based upon another combination of the two properties. The Factorization Method prescribes a manufacturing process.
- Published
- 1949
14. Extremum Properties of the Regular Polyhedra
- Author
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László Fejes Tóth
- Subjects
Combinatorics ,Regular polyhedron ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
1. Historical remarks. In this paper we extend some well-known extremum properties of the regular polygons to the regular polyhedra. We start by mentioning some known results in this direction.First, let us briefly consider the problem which has received the greatest attention among all the extremum problems for polyhedra. It is the determination of the polyhedron of greatest volume F of a class of polyhedra of equal surface areas F, i.e., the isepiphan problem.
- Published
- 1950
15. Elementarfunktionen Auf Riemannschen Flächen Als Hilfsmittel Für Die Funktionentheorie MEHRERER Veränderlichen
- Author
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Karl Stein and H. C. Heinrich Behnke
- Subjects
Continuity theorem ,Pure mathematics ,General Mathematics ,Riemann surface ,010102 general mathematics ,Pole–zero plot ,01 natural sciences ,Domain (mathematical analysis) ,symbols.namesake ,0103 physical sciences ,Several complex variables ,symbols ,010307 mathematical physics ,0101 mathematics ,Direct product ,Mathematics ,Analytic function - Abstract
In the present paper, the authors extend the Cousin theorems and the continuity theorem, using some previous results on analytic functions connected with open Riemann surfaces.The Cousin theorems, concerning the existence of analytic functions of several complex variables with prescribed poles and zeros in a given domain, have been generalized in various manners, but only in the case where the domain is schlicht. The authors proceed to the case where the given domain is the direct product of n open Riemann surfaces. They prove the following two theorems.
- Published
- 1950
16. Squaring the Square
- Author
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William T. Tutte
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Squaring the square ,Arithmetic ,Object (computer science) ,01 natural sciences ,Square (algebra) ,Mathematics - Abstract
1. Introduction. It is the object of this paper to describe in more detail than has hitherto been done the general methods by which a square may be dissected into smaller unequal non-overlapping squares. Examples of such dissections are given
- Published
- 1950
17. A Generalized Integral II
- Author
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R. D. James
- Subjects
Generalized inverse ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Line integral ,Singular integral ,01 natural sciences ,Integral equation ,Volume integral ,Dirichlet integral ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The definition and some of the properties of what may be called a Perron second integral (P2-integral) were given in a previous paper [4]. This integral starts with a function f(x) defined in an interval (a, c) and goes directly to a second primitive F(x) with the property that the generalized second derivative D2F is equal to f(x) for almost all x in (a, c). In the present paper the definition is changed slightly and further properties are deduced.
- Published
- 1950
18. Note on a Paper by Robinson
- Author
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J. A. Todd
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematical economics ,Mathematics - Abstract
In a recent paper Robinson has obtained an explicit formula for the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices. The object of the present note is to show that this formula may be deduced from known properties of Schur functions, with the aid of a result which the author has proved elsewhere.
- Published
- 1950
19. Induced Representations and Invariants
- Author
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G. de B. Robinson
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
1. Introduction. The problem of the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices is intimately associated with the representation theory of the full linear group on the one hand and with the representation theory of the symmetric group on the other. In a previous paper the author gave an explicit formula for this reduction in terms of characters of the symmetric group. Later J. A. Todd derived the same formula using Schur functions, i.e. characters of representations of the full linear group.
- Published
- 1950
20. Iterates of Fractional Order
- Author
-
Rufus Isaacs
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,Function (mathematics) ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,Iterated function ,0103 physical sciences ,Order (group theory) ,Algebraic topology (object) ,Function composition ,010307 mathematical physics ,0101 mathematics ,Mathematics ,Function mapping - Abstract
The body of this paper is a complete answer to the following question:Let E be any space whatever. g(x) is a function mapping E into E. When does there exist a function f(x), of the same type, such that(1)
- Published
- 1950
21. Incidence Relations in Multicoherent Spaces II
- Author
-
A. H. Stone
- Subjects
General Mathematics ,Incidence (epidemiology) ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics ,Demography - Abstract
Introduction. One standard method of studying the incidences of a system of sets A1, A2, .… , An is to consider the nerveof the system. However, this gives no direct information as to the numbers of components of the various intersections of the sets—information which would be desirable in several geometrical problems. The object of the present paper is to modify the definition of the nerve so that these numbers of components can be taken into account, and to study this modified nervefor systems of sets in a connected, locally connected, normal T1space S of a given degree of multicoherence r(S). The principal result (Theorem 6, 6.4) is a refinement of a theorem of Eilenberg [4, p. 107], and asserts that, ifthen under suitable hypotheses we have(1)
- Published
- 1950
22. Stochastic Differential Equations in a Differentiable Manifold
- Author
-
Kiyosi Itô
- Subjects
Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Malliavin calculus ,01 natural sciences ,60.0X ,Integrating factor ,Stochastic partial differential equation ,symbols.namesake ,Stochastic differential equation ,0103 physical sciences ,Runge–Kutta method ,symbols ,Finsler manifold ,0101 mathematics ,Differential algebraic equation ,Mathematics ,Algebraic differential equation - Abstract
The theory of stochastic differential equations in a differentiate manifold has been established by many authors from different view-points, especially by R Lévy [2], F. Perrin [1], A. Kolmogoroff [1] [2] and K. Yosida [1] [2]. It is the purpose of the present paper to discuss it by making use of stochastic integrals.
- Published
- 1950
23. Faithful Representations of Lie Groups II
- Author
-
Morikuni Gotô
- Subjects
010308 nuclear & particles physics ,General Mathematics ,Simple Lie group ,010102 general mathematics ,Lie group ,20.0X ,(g,K)-module ,Mathematical proof ,01 natural sciences ,Algebra ,Continuation ,Representation of a Lie group ,0103 physical sciences ,Fundamental representation ,0101 mathematics ,Supergroup ,Mathematics - Abstract
The present paper is a continuation of Part I. In the introduction of Part I we have explained our main problems and sketched their results. In the present part we shall proceed to give complete proofs of them.
- Published
- 1950
24. Finite Nets, I. Numerical Invariants
- Author
-
R. H. Bruck
- Subjects
Degree (graph theory) ,General Mathematics ,010102 general mathematics ,Graeco-Latin square ,Net (mathematics) ,01 natural sciences ,Combinatorics ,Loop (topology) ,symbols.namesake ,0103 physical sciences ,Line (geometry) ,Euclidean geometry ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,symbols ,Order (group theory) ,010307 mathematical physics ,Affine transformation ,0101 mathematics ,Mathematics - Abstract
A finite net N of degree k, order n, is a geometrical object of which the precise definition will be given in §1. The geometrical language of the paper proves convenient, but other terminologies are perhaps more familiar. A finite affine (or Euclidean) plane with n points on each line is simply a net of degree n+ 1, order n (Marshall Hall [1]). A loop of order n is essentially a net of degree 3, order n (Baer [1], Bates [1]). More generally, for , a set of k —2 mutually orthogonal n ⨯ n latin squares may be used to define a net of degree k, order n (and conversely) by paralleling Bose's correspondence (Bose [1]) between affine planes and complete sets of orthogonal latin squares.
- Published
- 1951
25. The Mathieu Groups
- Author
-
R. G. Stanton
- Subjects
Combinatorics ,Transitive relation ,Position (vector) ,General Mathematics ,Simple group ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Permutation group ,01 natural sciences ,Mathematics - Abstract
An enumeration of known simple groups has been given by Dickson [17]; to this list, he made certain additions in later papers [15], [16]. However, with but five exceptions, all known simple groups fall into infinite families; these five unusual simple groups were discovered by Mathieu [21], [22] and, after occasioning some discussion [20], [23], [27], were relegated to the position, which they still hold, of freakish groups without known relatives. Further interest is attached to these Mathieu groups in virtue of their providing the only known examples (other than the trivial examples of the symmetric and alternating groups) of quadruply and quintuply transitive permutation groups.
- Published
- 1951
26. The Representations of GL(3,q), GL(4,q), PGL(3,q), and PGL(4,q)
- Author
-
Robert Steinberg
- Subjects
Combinatorics ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,PSL ,01 natural sciences ,Mathematics - Abstract
This paper is a result of an investigation into general methods of determining the irreducible characters of GL(n, q), the group of all non-singular linear substitutions with marks in GF(q), and of the related groups, SL(n, q), PGL(n, q), PSL(n, q), the corresponding group of determinant unity, projective group, projective group of determinant unity, respectively.
- Published
- 1951
27. On Euclid's Algorithm in Cyclic Fields
- Author
-
H. Heilbronn
- Subjects
Pure mathematics ,Quadratic equation ,General Mathematics ,Quartic function ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Finite set ,Algorithm ,Mathematics - Abstract
In two papers I have proved that there are only a finite number of quadratic fields [6] and of cyclic cubic fields [7] in which Euclid's algorithm (E.A.) holds. Davenport has shown by a different method that there are only a finite number of quadratic fields [1, 2], of non-totally real cubic fields [3, 4] and of totally complex quartic fields in which E.A. holds.
- Published
- 1951
28. On the Topological Theory of Functions
- Author
-
James A. Jenkins
- Subjects
Topological quantum field theory ,Topological algebra ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Symmetry protected topological order ,Topological entropy in physics ,Topological vector space ,Homeomorphism ,Theoretical physics ,0103 physical sciences ,Topological ring ,010307 mathematical physics ,0101 mathematics ,Topological quantum number ,Mathematics - Abstract
The present paper constitutes a continuation of the ideas and methods of M. Morse and M. Heins [1]. As in that work the subject treated is the theory of deformation classes of meromorphic functions and interior transformations. There the functions considered were defined over the open disc < 1 and had only a finite number of zeros, poles and branch point antecedents. It is possible to transfer the results obtained to the situation where the domain of definition is any simply-connected domain of hyperbolic type or, alternatively, of parabolic type.
- Published
- 1951
29. Modular Representations of the Symmetric Group
- Author
-
J. H. Chung
- Subjects
Modular representation theory ,Pure mathematics ,business.industry ,General Mathematics ,010102 general mathematics ,Modular invariance ,Modular design ,01 natural sciences ,Covering groups of the alternating and symmetric groups ,Representation theory of the symmetric group ,Symmetric group ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,business ,SL2(R) ,Hecke operator ,Mathematics - Abstract
The theory of modular representations of the symmetric group was studied first by Nakayama (5, 6), and later by Thrall and Nesbitt (11) and Robinson (7, 8, 9). Nakayama built up his elaborate theory of hooks for the express purpose of studying this problem, while Robinson's extensive work on the various phases of the relationship between Young diagrams, skew diagrams and star diagrams on the one hand, and representations of the symmetric group on the other, culminating in a set of relations among the degrees of the representations, serves as a starting point for this paper.
- Published
- 1951
30. Darboux Properties and Applications to Non-Absolutely Convergent Integrals
- Author
-
H. W. Ellis
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Absolute convergence ,01 natural sciences ,Mathematics - Abstract
This paper consists of two parts. The first contains an outline of the theorems and principal results and the second (§§2-6) gives proofs of the theorems and additional details. The theorems concern properties of Darboux continuous functions and functions having generalized Darboux properties. The corresponding results are shown to have interesting applications to the theory of non-absolutely convergent integrals.
- Published
- 1951
31. On a type of subgroups of a compact Lie group
- Author
-
Yozô Matsushima
- Subjects
Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,Simple Lie group ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,010102 general mathematics ,0103 physical sciences ,Lie group ,Maximal torus ,20.0X ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Let G be a connected compact Lie group and H a connected closed subgroup. Then H is an orientable submanifold of G and we may consider H as a cycle in G. In his interesting paper on the topology of group manifolds H. Samelson has proved that, if H is not homologous to 0, then the homology ring of the coset space G/H is isomorphic to the homology ring of a product space of odd dimensional spheres and the homology ring of G is isomorphic to that of the product of the spaces H and G/H. On the other hand, in a recent investigation of fibre bundles’ T. Kudo has shown that, if the homology ring of the coset space G/H is isomorphic to that of an odd dimensional sphere, then H is not homologous to 0.
- Published
- 1951
32. On the canonical form of turbulence
- Author
-
Seizô Itô
- Subjects
010308 nuclear & particles physics ,Turbulence ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Canonical coordinates ,Canonical transformation ,Canonical form ,0101 mathematics ,01 natural sciences ,60.0X ,Mathematical physics ,Mathematics - Abstract
In K, Itô’s paper [1] on the theory of turbulence, the problem to determine the canonical form of the moment tensor of temporally homogeneous and isotropic turbulence, has not been solved. In the present paper, the author will solve the problem by making use of the result of his preceding paper [2]. We shall treat the turbulence in R3 but the similar argument is possible in Rn.
- Published
- 1951
33. An Application of Cornu’s Spiral to the Mathematical Theory of the Motion of an Unrotated Rocket
- Author
-
V. Mauranen and E. T. J. Davies
- Subjects
Mathematical theory ,business.product_category ,Rocket ,General Mathematics ,010102 general mathematics ,Motion (geometry) ,Mechanics ,0101 mathematics ,business ,01 natural sciences ,Spiral ,Mathematics - Abstract
The mathematical theory of rocket motion has been given by Rosser, Newton and Gross (1947) for the case of unrotated, and slowly rotated, rockets. A rigorous treatment of the subject has been given also by Rankin (1949) in a recent paper which is applicable to both unrotated and rotated rockets.One of the main objects of such work is the derivation of formulae which may be used to predict the behaviour of the rocket under the action of various disturbing influences. Thus, an angular deviation of the rocket from its normal trajectory may be caused by the thrust not passing through the centre of gravity, by the action of a crosswind (usually assumed constant in the theory), or by the rocket being launched with an initial yaw or initial angular velocity about an axis at right angles to the rocket axis. Malaligned fins may also cause such a deviation.
- Published
- 1951
34. Integration in Locally Compact Spaces II
- Author
-
Herbert S. Zuckerman and Edwin Hewitt
- Subjects
Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Locally compact space ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
The general problem of producing concrete representations for continuous linear functionals on normed linear spaces, ie., of identifying conjugate spaces, has of course attracted the attention of many mathematicians during the last five decades and has been solved in many cases [1, pp. 59-72]. Likewise, the problem of extending a linear functional defined on a linear subspace of a normed linear space may be regarded as solved by the Hahn-Banach theorem [1, p. 28], although problems involving “natural” extensions, like that yielding the Lebesgue integral from the Riemann integral, remain. In the present paper, we shall consider two “natural’ methods of extending a certain linear functional and show that they are in fact identical. As a by-product, we obtain a concrete representation both for the original functional and for its “natural” extension. In subsequent communications, the writers will consider topologies in certain families of linear functionals, canonical resolutions of linear functionals, and other extension problems.
- Published
- 1951
35. Some Studies on Semi-local Rings
- Author
-
Masayoshi Nagata
- Subjects
Noetherian ,Pure mathematics ,Ring (mathematics) ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Local ring ,Zero element ,Divisor (algebraic geometry) ,01 natural sciences ,09.1X ,Chain (algebraic topology) ,0103 physical sciences ,0101 mathematics ,Minimal prime ,Mathematics - Abstract
The concept of semi-local rings was introduced by C. Chevalley [1], which the writer has generalized in a recent paper [7] by removing the chain condition. The present paper aims mainly at the study of completions of semi-local rings. First in § 1 we investigate semi-local rings which are subdirect sums of semi-local rings, and we see in § 2 that a Noetherian semi-local ringRis complete if (and only if)R/pis complete for every minimal prime divisorpof zero ideal, together with some other properties. Further we consider in § 3 subrings of the completion of a semi-local ring. § 4 gives some supplementary remarks to [7], Chapter II, Proposition 8.
- Published
- 1951
36. Open Riemann Surface with Null Boundary
- Author
-
Kiyoshi Noshiro
- Subjects
30.0X ,Geometric function theory ,010308 nuclear & particles physics ,General Mathematics ,Riemann surface ,010102 general mathematics ,Null (mathematics) ,Mathematical analysis ,Riemann sphere ,Riemann's differential equation ,01 natural sciences ,Riemann Xi function ,Riemann–Hurwitz formula ,symbols.namesake ,0103 physical sciences ,Uniformization theorem ,symbols ,0101 mathematics ,Mathematics - Abstract
Recently the writer has obtained some results concerning meromorphic or algebroidal functions with the set of essential singularities of capacity zero, with an aid of a theorem of Evans. In the present paper, suggested from recent interesting papers of Sario and Pfluger, the writer will extend his results to single-valued analytic functions defined on open abstract Riemann surfaces with null boundary in the sense of Nevanlinna, using a lemma instead of Evans’ theorem.
- Published
- 1951
37. On The Modular Representations of the Symmetric Group
- Author
-
G. de B. Robinson
- Subjects
Multidisciplinary ,business.industry ,General Mathematics ,010102 general mathematics ,Modular design ,01 natural sciences ,Algebra ,Symmetric group ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,business ,Mathematics - Abstract
The study of the modular representation theory of the symmetric group has been greatly facilitated lately by the introduction of the graph (9, III ), the q-graph (5) and the hook-graph (4) of a Young diagram [λ]. In the present paper we seek to coordinate these ideas and relate them to the r-inducing and restricting processes (9, II ).
- Published
- 1951
38. Zeta Functions on the Unitary Sphere
- Author
-
S. Minakshisundaram
- Subjects
Arithmetic zeta function ,Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Unitary state ,Mathematics - Abstract
In an earlier paper [5], the author defined a zeta function on the real sphere , whereas in the present paper it is proposed to define one on the unitary sphere where xi's are complex numbers and their complex conjugates. Following E. Cartan, harmonics on the unitary sphere are defined and then a zeta function formed just as in the case of a real sphere. The unitary sphere is seen to behave like an even-dimensional closed manifold, since results similar to the ones proved by the author and A. Pleijel [6] for closed manifolds (of even dimensions) are observed here also.
- Published
- 1952
39. Contributions to Noncommutative Ideal Theory
- Author
-
D. C. Murdoch
- Subjects
Theoretical physics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Noncommutative algebraic geometry ,010307 mathematical physics ,0101 mathematics ,Noncommutative quantum field theory ,01 natural sciences ,Noncommutative geometry ,Ideal theory ,Mathematics - Abstract
The well-known results of Krull concerning the minimal prime divisors and the radical of an ideal in a commutative ring have been extended to the noncommutative case in a recent paper [5] by N. H. McCoy. In that paper systematic use was made of the concept of an m-system, a set M of elements of the ring such that if a ∈ M and b ∈ M then axb ∈ M for some element x of the ring.
- Published
- 1952
40. A Theory of Normal Chains
- Author
-
Christine Williams Ayoub
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper we deal with a group-theoretic configuration of the following type: G is an additive group (not necessarily abelian) for which an operator system M and a complete lattice ø of M admissible subgroups are defined; we call G and M-ø group. In §1 we make various definitions and note that analogues of some of the classical theorems of group theory hold.
- Published
- 1952
41. A One-Regular Graph of Degree Three
- Author
-
Robert Frucht
- Subjects
Loop (graph theory) ,Degree (graph theory) ,General Mathematics ,010102 general mathematics ,Quartic graph ,01 natural sciences ,Distance-regular graph ,law.invention ,Combinatorics ,Vertex-transitive graph ,law ,0103 physical sciences ,Line graph ,Cubic graph ,Regular graph ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Soon after the publication of Tutte's paper [5] on m-cages, H. S. M. Coxeter asked in a letter to the author whether one-regular graphs of degree 3 exist. The purpose of the following paper is to show by an example that the answer is in the affirmative.
- Published
- 1952
42. Finite Projective Geometries
- Author
-
Gerald Berman
- Subjects
Pure mathematics ,Transitive relation ,Collineation ,General Mathematics ,Existential quantification ,010102 general mathematics ,01 natural sciences ,Linear subspace ,Set (abstract data type) ,Finite field ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Projective test ,Mathematics - Abstract
James Singer [12] has shown that there exists a collineation which is transitive on the (t - 1)-spaces, that is, (t - 1)-dimensional linear subspaces, of PG(t, pn). In this paper we shall generalize this result showing that there exist t - r collineations which together are transitive on the s-spaces of PG(t, pn). An explicit construction will be given for such a set of collineations with the aid of primitive elements of Galois fields. This leads to a calculus for the linear subspaces of finite projective geometries.
- Published
- 1952
43. Groups with a Certain Condition on Conjugates
- Author
-
Franklin Haimo
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,010307 mathematical physics ,0101 mathematics ,GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries) ,01 natural sciences ,Combinatorial chemistry ,Conjugate ,Mathematics - Abstract
In this paper, we shall show that if is a nilpotent [5] group and if M, a positive integer, is a uniform bound on the number of conjugates that any element of may have, then there exist “large” integers n for which x → xn is a central endomorphism of . If is not necessarily nilpotent, if the above condition on the conjugates is retained, and if we can find a member of the lower central series [1], every element of which lies in some member of the ascending central series, then we shall show that every non-unity element of the “high” derivatives has finite order.
- Published
- 1952
44. On a Conjecture by J. H. Chung
- Author
-
G. de B. Robinson
- Subjects
Combinatorics ,Conjecture ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,Beal's conjecture ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
The present paper is a sequel to that of J. H. Chung (2) and contains a proof of a conjecture made by him, namely, that the number of ordinary (modular) irreducible representations contained in a given p-block of Sn is independent of the p-core. A summary of the results contained herein appeared in the Proceedings of the National Academy of Sciences (9).
- Published
- 1952
45. A Coefficient Problem for Functions Regular in an Annulus
- Author
-
M. S. Robertson
- Subjects
Annulus (mycology) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,Mechanics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
1. Introduction A solution will be given in this paper for the following problem.Let(1.1)
- Published
- 1952
46. Multiply Subadditive Functions
- Author
-
George G. Lorentz
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Representation (systemics) ,Boolean ring ,Of the form ,Zero element ,01 natural sciences ,Complemented lattice ,Distributive property ,0103 physical sciences ,Subadditivity ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Let S denote a Boolean ring with elements e, that is, a distributive, relatively complemented lattice with zero element 0 [2, p. 153]. In this paper we study real-valued functions which have a representation of the form1.1
- Published
- 1952
47. On D. E. Littlewood's Algebra of S-Functions
- Author
-
D. G. Duncan
- Subjects
Filtered algebra ,Algebra ,Jordan algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Division algebra ,Cellular algebra ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,01 natural sciences ,Mathematics - Abstract
Several papers have been written on the “new” multiplication of S-functions since Littlewood [3, p. 206] first suggested the problem. M. Zia-ud-Din [13] calculated the case {m} ⊗ {n} for mn ≤ 12, making use of the tables of the characters of the symmetric group of degree mn. Later Thrall [10,pp. 378-382] developed explicit formulae for the cases {m} ⊗ {2},{m} ⊗ {3}, {2} ⊗ {m} (where m is any integer).
- Published
- 1952
48. Classification of Mappings of an (n + 2)-Complex into an (n − 1)-Connected Space with Vanishing (n + 1)-st Homotopy Group
- Author
-
Hiroshi Uehara and Nobuo Shimada
- Subjects
Combinatorics ,Homotopy group ,Connected space ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Extension (predicate logic) ,0101 mathematics ,Space (mathematics) ,01 natural sciences ,Finite set ,Mathematics - Abstract
The present paper is concerned with the classification and corresponding extension theorem of mappings of the (n+-2)-complex Kn+2 (n>2) into the space Y whose homotopy groups πi(Y) vanish for i < n and i = n+1, and the n-th homotopy group πn(Y) of which has a finite number of generators. Our methods followed here are essentially analogous to those of Steenrod [2].
- Published
- 1952
49. The Coordinate Conditions and the Equations of Motion
- Author
-
Leopold Infeld
- Subjects
Curvilinear coordinates ,Constant of motion ,General Mathematics ,010102 general mathematics ,Equations of motion ,Ellipsoidal coordinates ,Coordinate conditions ,01 natural sciences ,Classical mechanics ,0103 physical sciences ,Stress–energy tensor ,010307 mathematical physics ,0101 mathematics ,Mechanics of planar particle motion ,Mathematics ,Elliptic coordinate system - Abstract
The problem of the field equations and the equations of motion in general relativity theory is now sufficiently clarified. The equations of motion can be deduced from pure field equations by treating matter as singularities, [2; 3], or from field equations with the energy momentum tensor [4]. Recently two papers appeared in which the problem of the coordinate system was considered [5; 8]. The two papers are in general agreement as far as the role of the coordinate system is concerned. Yet there are some differences which require clarification.
- Published
- 1953
50. Harmonic p-Tensors on Normal Hyperbolic Riemannian Spaces
- Author
-
G. F. D. Duff
- Subjects
Harmonic coordinates ,Pure mathematics ,Riemannian submersion ,General Mathematics ,010102 general mathematics ,Hyperbolic 3-manifold ,Hyperbolic manifold ,Fundamental theorem of Riemannian geometry ,01 natural sciences ,Relatively hyperbolic group ,symbols.namesake ,Hyperbolic set ,0103 physical sciences ,Metric (mathematics) ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The subject of this paper is the study of boundary value theorems for harmonic p-tensors on a Riemannian space with an indefinite metric of the normal hyperbolic signature.
- Published
- 1953
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