bluuebunny/crossref_metadata_2025_split huggingface dataset:
On Explicit Bounds In Landau's Theorem
Authors: Jenkins J. A. 
Published: December 2010

1. The theorem of Landau in question may be stated in the form that if the function F(Z) is regular for |Z| < 1 and does not take the values 0 and 1, whileF(Z) = a0 + a1Z + …is its Taylor expansion about Z = 0, then |a1| has a bound depending only on a0. In fact |a1| has a bound depending only on |a0| and Hayman (1) gave the explicit bound
The Cauchy Problem For Linear Partial Differential Equations With Restricted Boundary Conditions
Authors: Miles E. P. , Williams Ernest 
Published: December 2010

We shall discuss solutions of linear partial differential equations of the form1where Ψ is an ordinary differential operator of order s with respect to t. Our first theorem gives a solution of (1) for the Cauchy data;2j = 1,2, ߪ,s − 1,whenever the function P is annihilated by a finite iteration of the operator Φ.
Extremal Properties Of Hermitian Matrices
Authors: Marcus M. , McGregor J. L. 
Published: December 2010

In (1) Fan showed that if A is a Hermitian matrix with eigenvalues λ1 ≤ … ≤ λn then, for k ≤ n,,,where X1, … , xk run over all sets of k orthonormal (o.n.) vectors in unitary n-space V.
On Ward's Perron-Stieltjes Integral
Authors: Henstock Ralph 
Published: December 2010

In the paper (5), Ward defines an integral of Perron type of a finite
function f with respect to another finite function
g, where g need not be of bounded
variation. There arise two problems, (a) and
(b) below, that have not been dealt with in (5).If f = j at a countable number of points
everywhere dense in (a, b), where
f and j are both integrable with
respect to g, then f — j
can be nonzero on a large set of points of (a,
b).
Geodesic Groups of Minimal Surfaces
Authors: Helfenstein H. G. 
Published: December 2010

In a previous paper (6) we have studied those minimal surfaces which admit
geodesic mappings without isometries or similarities on another, not necessarily
minimal, surface. Here we determine all pairs of minimal surfaces which can be
geodesically mapped on each other. We find that two such surfaces are
either:
(i) similar Bonnet associates of each other, or
(ii) both Poisson surfaces (that is, isometric to a plane), or
(iii) both Scherk surfaces (2).
On the Convergence of Mean Values Over Lattices
Authors: Schmidt Wolfgang 
Published: December 2010

Recently C. A.Rogers (2, Theorem 4) proved the following theorem which
applies to many problems in geometry of numbers:
Let
f(X1,X2, …
, Xk) be a non-negative B or el-measurable function in the
nk-dimensional space of points (X1,X2.
Further, let Λo be the fundamental lattice, Ω a linear
transformation of determinant 1, F a fundamental region in the space of linear
transformations of determinant 1, defined with respect to the subgroup of
unimodular transformations and μ(Ω) the invariant measure1 on the space of linear
transformations of determinant 1 in Rn.
On the Problème Des Ménages
Authors: Wyman Max , Moser Leo 
Published: December 2010

The classical problème des
ménages asks for the number of ways of seating at a
circular table n married couples, husbands and wives
alternating, so that no husband is next to his own wife.
An outline of the history of the problem to 1946 was given by Kaplansky and
Riordan (11). They also presented a bibliography, which is augmented and brought
up to date in the bibliography of the present paper.
Series of Products of Bessel Polynomials
Authors: Ragab F. M. 
Published: December 2010

The Bessel polynomials, which arise as solution of the classical wave equation in spherical co-ordinates, are defined by Krall and Frink (3) by the equation1The purpose of this paper is to present some series of products of these polynomials when the two arguments are different as in the case of Legendre and Hermite polynomials. Such an explanation was given by Brafman (2), namely :2These series will be stated and proved in § 2.
On the Representation of Functions as Fourier Transforms
Authors: Rooney P. G. 
Published: December 2010

If f ∈ Lp (— ∞, ∞), 1 < p ≤ 2, then f has a Fourier-Plancherel transform F ∈ Lq (— ∞, ∞) where p-1 + q-1 = 1. Also if ∣x∣1-2/qf(x) ∈ Lq (— ∞, ∞), q ≥ 2, then / has a Fourier-Plancherel transform F ∈ Lq (— ∞, ∞). These results can be found in (2, Theorems 74 and 79). In neither case, however, does the collection of transforms cover Lq, except when p = q = 2, and in neither case, with the same exception, has the collection of transforms been characterized.Further, if f ∈ Lp, (— ∞, ∞), 1 < p ≤ 2, then its transform F has the property |x|1-2/pF(x) ∈ Lp (— ∞, ∞) (see 2, Theorem 80) but, except when p = 2, the collection of transforms does not cover the set of functions with this property, and again, except when p = 2, the collection of transforms has not been characterized.
The Elliptic Integrals of the Third Kind
Authors: Neville E. H. 
Published: December 2010

This paper develops a case for adopting as the standard elliptic integrals of the third kind the function IIs(u, a) defined byand the three functions lIs (u, a + Kc), Πs(u, a + Kn), Πss(u, a+ Kn) where KC, Kn, Kn are the three quarter-periods of the Jacobian system. The function Πs(u, a) is the same function whether qs u is cs u, ns u, or ds u.The origin of the paper was a wish to understand how it has come about that the integrals commonly accepted as standard are not related symmetrically to the theta functions in terms of which they are expressed.
Subspaces of a Generalized Metric Space
Authors: Eliopoulos H. A. 
Published: December 2010

In a paper published in 1956, Rund (4) developed the differential geometry of a hypersurface of n— 1 dimensions imbedded in a Finsler space of n dimensions, considered as locally Minkowskian.The purpose of the present paper is to provide an extension of the results of (4) and thus develop a theory for the case of m-dimensional subspaces imbedded in a generalized (Finsler) metric space.We consider an n-dimensional differentiable manifold Xn and we restrict our attention to a suitably chosen co-ordinate neighbourhood of Xn in which a co-ordinate system xi (i= 1, 2, … , n), is defined. A system of equations of the type xi = xi(t) defines a curve C of Xn the tangent vector dxi/dt of which is denoted by xi.
The Term and Stochastic Ranks of a Matrix
Authors: Mendelsohn N. S. , Dulmage A. L. 
Published: December 2010

The term rank p of a matrix is the order of the largest minor which has a non-zero term in the expansion of its determinant. In a recent paper (1), the authors made the following conjecture. If S is the sum of all the entries in a square matrix of non-negative real numbers and if M is the maximum row or column sum, then the term rank p of the matrix is greater than or equal to the least integer which is greater than or equal to S/M. A generalization of this conjecture is proved in § 2.The term doubly stochastic has been used to describe a matrix of nonnegative entries in which the row and column sums are all equal to one. In this paper, by a doubly stochastic matrix, the, authors mean a matrix of non-negative entries in which the row and column sums are all equal to the same real number T.
Separation and Approximation in Topological Vector Lattices
Authors: Leader Solomon 
Published: December 2010

Spectral theory in its lattice-theoretic setting proves abstractly that the indicators of measurable sets generate the space L of Lebesgue-integrable functions on an interval. We are concerned here with abstractions suggested by the fact that indicators of intervals suffice to generate L. Our results show that the approximation of arbitrary elements of a topological vector lattice rests upon the ability to separate disjoint elements/ and g by an operation that behaves in the limit like a projection annihilating/ and leaving g invariant.The introduction of this concept of separation together with the notion of limit unit leads (via the Fundamental Lemma) to abstract generalizations of the Radon-Nikodym Theorem (Theorem 1) and the Stone-Weierstrass Theorem (Theorem 3).
Supersoluble Immersion
Authors: Baer Reinhold 
Published: December 2010

Supersoluble immersion of a normal subgroup K of a finite group G shall be defined by the following property:If σ is a homomorphism of G, and if the minimal normal subgroup J of Gσ is part of Kσ then J is cyclic (of order a prime).Our principal aim in the present investigation is the proof of the equivalence of the following three properties of the normal subgroup K of the finite group G:(i)K is supersolubly immersed in G.(ii)K/ϕK is supersolubly immersed in G/ϕK.(iii)If θ is the group of automorphisms induced in the p-subgroup U of K by elements in the normalizer of U in G, then θ' θp-1 is a p-subgroup of θ.Though most of our discussion is concerned with the proof of this theorem, some of our concepts and results are of independent interest.
Extremal Properties of Hermitian Matrices. II
Authors: Marcus M. , Moyls B. N. , Westwick R. 
Published: December 2010

Let H be an n-square Hermitian matrix with eigenvalues h1 ≥ h2 ≥ . . . ≥ hn. Fan (2) showed that1k — 1,2, … , n, where the max and min are taken over all sets of k orthonormal (o.n.) vectors in unitary w-space Vn. Marcus and McGregor (3) have generalized this result in the case that H is non-negative Hermitian. For vectors X1, … , xr r ≤ n, in Vn, let X1 Λ X2 Λ … Λ xT denote the Grassmann exterior product of the xt; it is a vector in Vm,, where
Sur un Theorème de G. Szekeres, Concernant les Fonctions Monotones et Convexes
Authors: Marcus S. 
Published: December 2010

Dans une note récente (1), Szekeres a donné le théorème suivant:Sif est une fonction réelle, définie, strictement croissante et ayant la deuxième dérivée continue sur (a, b) ( — ∞ ≤ a < b ≤ ≤) alors il existe une fonction réelle ϕ, définie et strictement croissante sur (a, b), et une fonction réelle ψ définie et strictement croissante sur (ϕ(a), ϕ(b)), telles que ϕ˝,(x) ≥ 0 pour a < x < b, ψ ˝(u) < 0 pour ϕ(a) < u < ϕ(b) et f(x) = ψ(ϕ(x)) pour a < x < b.Dans la démonstration de ce théorème, Szekeres utilise une expression où la dérivée f‘(x) apparaît au dénominateur, ce qui exige qu'on ait f'(x) ≠ 0 pour a < x < b, condition qui ne figure pas dans l'énoncé du théorème. Il est naturel alors de poser le problème si le théorème ci-dessus reste en vigueur si cette condition n'est pas remplie. Il y a lieu aussi de se demander si les autres conditions, par exemple l'existence et la continuité def“(x) sont nécessaires pour qu'une fonction strictement croissante puisse être représentée comme une superposition de deux fonctions croissantes, l'une convexe, l'autre concave.
The Asymptotic Behaviour of the Laurent Coefficients
Authors: Wyman Max 
Published: December 2010

LetG(z)be a function of a complex variable, regular in the annulus 0 ≤ a ≤ |z| <b≤ ∞. We shall assume there exists a curve within the annulus for whichprovidedzis restricted to be a point of this curve. Under these restrictionsG (z)has a Laurent expansion of the form1.1where the Laurent coefficientsanhave the integral representation1.2andCcan be any contour, within the domain of regularity, that enclosesz= 0. We shall also assume that theanare all real numbers. Using the usual complex conjugate notation, we can, therefore, write1.3The problem of determining the asymptotic behaviour ofanasn —> co is very old in mathematical literature and appears in many forms and disguises.
Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture
Authors: Bose R. C. , Shrikhande S. S. , Parker E. T. 
Published: December 2010

Ifis the prime power decomposition of an integer v, and we define the arithmetic function n(v) bythen it is known, MacNeish (10) and Mann (11), that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v. We shall denote by N(v) the maximum possible number of mutually orthogonal Latin squares of order v. Then the Mann-MacNeish theorem can be stated asMacNeish conjectured that the actual value of N(v) is n(v).
On the Maximal Number of Pairwise Orthogonal Latin Squares of a Given Order
Authors: Chowla S. , Erdös P. , Straus E. G. 
Published: December 2010

In the preceding paper Bose, Shrikhande, and Parker give their important discovery of the disproof of Euler's conjecture on Latin squares. In this paper we show that their results can be strengthened to imply that N(n), the maximal number of pairwise orthogonal Latin squares of order n, tends to infinity with n. In fact there exists a positive constant c, such that N(n) > nc for all sufficiently large n.Our proof involves no new combinatorial insights, but is based entirely on a number-theoretical investigation of the following inequality due to Bose and Shrikhande.
A Theorem on Pure Submodules
Authors: Kolettis George 
Published: December 2010

In (1) Baer studied the following problem: If a torsion-free abelian group G is a direct sum of groups of rank one, is every direct summand of G also a direct sum of groups of rank one? For groups satisfying a certain chain condition, Baer gave a solution. Kulikov, in (3), supplied an affirmative answer, assuming only that G is countable. In a recent paper (2), Kaplansky settles the issue by reducing the general case to the countable case where Kulikov's solution is applicable. As usual, the result extends to modules over a principal ideal ring R (commutative with unit, no divisors of zero, every ideal principal).The object of this paper is to carry out a similar investigation for pure submodules, a somewhat larger class of submodules than the class of direct summands. We ask: if the torsion-free i?-module M is a direct sum of modules of rank one, is every pure submodule N of M also a direct sum of modules of rank one? Unlike the situation for direct summands, here the answer depends heavily on the ring R.
Generalized Lie Elements
Authors: Ree Rimhak 
Published: December 2010

Let λ(ij),i,j =1, 2, … ,m, bem2elements in a fieldKof characteristic zero such thatλ(ij)λ(ji) = 1 for alliandj, and X1,x2, … , xmnon-commutative associative indeterminates overK. Define the elements [xi1Xi2…xin] inductively by [xi] =xiandAny linear combination of the elementswith coefficients inKwill be called ageneralized Lie elememt. Generalized Lie elements reduce to ordinary Lie elements ifλ(ij) = 1 for alliandj.The purpose of this paper is to generalize to the generalized Lie elements the following: a theorem of Friedrichs, a theorem of Dynkin-Specht-Wever (2), and the Witt formula on the dimension of the space spanned by homogeneous Lie elements of a fixed degree. The set of all generalized Lie elements will be made into an algebra which generalizes the ordinary free Lie algebra. This algebra turns out to be free in a certain sense. We shall also generalize the algebra associated with shuffles in (2).
On Immersion of Manifolds
Authors: Samelson Hans 
Published: December 2010

In (3) R. Lashof and S. Smale proved among other things the following theorem. If the compact oriented manifold M is immersed into the oriented manifold M', with dim M' ≥ dim M + 2, then the normal degree of the immersion is equal to the Euler-Poincaré characteristic x of M reduced module the characteristic x’ of M'. If M’ is not compact, x' is replaced by 0. “Manifold” always means C∞-manifold. An immersion is a differentiable (that is, C∞) map f whose differential df is non-singular throughout. The normal degree is defined in a certain fashion using the normal bundle of M in M', derived from f, and injecting it into the tangent bundle of M'It is our purpose to give an elementary proof, using vector fields, of this theorem, and at the same time to identify the homology class that represents the normal degree (Theorem I), and to give a proof, using the theory of Morse, for the special case M’ = Euclidean space (Theorem II).
On Orientations, Connectivity and Odd-Vertex-Pairings in Finite Graphs
Authors: Nash-Williams C. ST. J. A. 
Published: December 2010

The integer part of a non-negative real number p will be denoted by [p]. For any integer n, n* will denote the greatest even integer less than or equal to n, that is, n* = n or n — 1 according as n is even or odd respectively.The order of a set A, denoted by |A|, is the number of elements in A. The set whose elements are a1, a2, … , an will be denoted by {a1, a2 … , an. The empty set will be denoted by Λ. A set will be said to include each of its elements. A set separates two elements if it includes one but not both of them.An unoriented graph U consists of two disjoint sets V(U), E(U), the elements of V(U) being called vertices of U and the elements of V(U) being called edges of U, together with a relationship whereby with each edge is associated an unordered pair of distinct vertices which the edge is said to join.
On the Projective Centres of Convex Curves
Authors: Kelly Paul , Straus E. G. 
Published: December 2010

We consider a closed curve C in the projective plane and the projective involutions which map C into itself. Any such mapping γ, other than the identity, is a harmonic homology whose axis η we call a projective axis of C and whose centre p we call an interior or exterior projective centre according as it is inside or outside C. The involutions are the generators of a group Γ, and the set of centres and the set of axes are invariant under Γ. The present paper is concerned with the type of centre sets which can exist and with the relationship between the nature of C and its centre set.If C is a conic, then every point which is not on C is a projective centre. Conversely, it was shown by Kojima (4) that if C has a chord of interior centres, or a full line of exterior centres, then C is a conic.
Homotopy and Isotopy Properties of Topological Spaces
Authors: Hu Sze-Tsen 
Published: December 2010

The most important notion in topology is that of ahomeomorphism f: X→Yfrom a topological spaceXonto a topological spaceY. If a homeomorphism f:X→Yexists, then the topological spaces X andFare said to behomeomorphic(ortopologically equivalent), in symbols,X ≡ Y.The relation ≡ among topological spaces is obviously reflexive, symmetric, and transitive; hence it is an equivalence relation. For an arbitrary familyFof topological spaces, this equivalence relation ≡ divides /Mnto disjoint equivalence classes called thetopology typesof the familyF. Then, the main problem in topology is the topological classification problem formulated as follows.The topological classification problem:Given a familyF oftopological spaces, find an effective enumeration of the topology types of the familyFand exhibit a representative space in each of these topology types.
Properties of Solutions of Parabolic Equations and Inequalities
Authors: Protter M. H. 
Published: December 2010

In this paper we shall be concerned with two problems: (i) the asymptotic behavior of solutions of parabolic inequalities and (ii) the uniqueness of the Cauchy problem for such inequalities when the data are prescribed on a portion of a time-like surface. The unifying feature of these rather separate problems is the employment of integral estimates of the same type in both cases.We consider parabolic operators in self-adjoint form(1)as well as the non-self-adjoint operator(2)where the coefficients aij(x, t) = aij (x1, x2... , xn, t) are C1 functions of x and t and the bij= bij(x, t) are C2 functions of x and t.
Tetraspheres. I
Authors: De Majo A. 
Published: December 2010

Les propriétés anallagmatiques de groupes de sphères ont été étudiées dans des contextes divers ces dernières années (voir (4), (5)). Dans la note qui suit nous étudions les propriétés anallagmatiques de certains groupes de 4 sphères, nous plaçant à un point de vue élémentaire.A fin de ne pas alourdir la rédaction nous omettrons de spécifier chaque fois que les points, droites ou sphères considérés sont toujours supposés être dans la position relative la plus générale possible compatible avec les définitions données.L'index i pourra toujours prendre les valeurs i = 1, 2, 3, 4.
On Stability in the Large for Systems of Ordinary Differential Equations
Authors: Hartman Philip 
Published: December 2010

Autonomous systems. This note concerns the stability of systems of (real) differential equations in the large on Euclidean space En and on certain Riemannian manifolds Mn. The results will be refinements of those of Krasovski (3), (4), (5) and of Markus and Yamabe (8) and will make clear the role of the various assumptions in the type of theorems under consideration.In this section, the main theorems are stated for autonomous systems(1)Their proofs are given in § 2, 3, 4. In § 5, 6, 7, generalizations to non-autonomous systems are made.
General Radicals that Coincide with the Classical Radical on Rings with D.C.C.
Authors: Divinsky N. 
Published: December 2010

General radical theories were obtained by Amitsur (1; 2; 3) and Kurosh (6). Following Kurosh we say that a property of rings is a radical property if:(a) Every homomorphic image of an -ring is an -ring;(b) Every ring R contains an -ideal S which contains every other -ideal of R;(c) The factor ring R/S is-semi-simple (that is, has no non-zero -ideals).
A Normal form for a Matrix under the Unitary Congruence Group
Authors: Youla D. C. 
Published: December 2010

Let C be a square matrix with complex elements. If C = C' (C' denotes the transpose of C) there exists a unitary matrix U such that(1)where the μ's are the non-negative square roots of the eigenvalues μ12, μ22, … , μn2 of C*C (C* is the adjoint of C) (2). If C is skew-symmetric, that is, C= — C”, there exists a unitary matrix U such that(2)where(3)and the α's are the positive square roots of the non-zero eigenvalues α12, α22, … , αk2 of C*C (1). Clearly rank C = 2k and the number of zeros appearing in (2) is n — 2k. Both (1) and (2) are classical. In a recent paper (3) Stander and Wiegman, apparently unaware of (1), give an alternative derivation of (2) with its appropriate generalization to quaternions.
Permanent Preservers on the Space of Doubly Stochastic Matrices
Authors: Moyls B. N. , Marcus Marvin , Minc Henryk 
Published: December 2010

Let Mn be the linear space of n-square matrices with real elements. For a matrix X = (xij) ∈ Mn the permanent is defined bywhere σ runs over all permutations of 1, 2, …, n. In (2) Marcus and May determine the nature of all linear transformations T of Mn into itself such that per T(X) = per X for all X ∈ Mn. For such a permanent preserver T, and for n < 3, there exist permutation matrices P, Q, and diagonal matrices D, L in Mn, such that per DL = 1 and eitherorHere X′ denotes the transpose of X. In the case n = 2, a different type of transformation is also possible.
On Finite Groups of the Form ABA
Authors: Gorenstein Daniel 
Published: December 2010

The class of finite groups G of the form ABA, where A and B are subgroups of G, is of interest since it includes the finite doubly transitive groups, which admit such a representation with A the subgroup fixing a letter and B of order 2. It is natural to ask for conditions on A and B which will imply the solvability of G. It is known that a group of the form AB is solvable if A and B are nilpotent. However, no such general result can be expected for ABA -groups, since the simple groups PSL(2,2n) admit such a representation with A cyclic of order 2n + 1 and B elementary abelian of order 2n. Thus G need not be solvable even if A and B are abelian.In (3) Herstein and the author have shown that G is solvable if A and B are cyclic of relatively prime orders; and in (2) we have shown that G is solvable if A and B are cyclic and A possesses a normal complement in G.