bluuebunny/crossref_metadata_2025_split huggingface dataset:
On Quadruple Systems
Authors: Hanani Haim 
Published: December 2010

Given a set E of n elements we denote by S(l, m, n), (l ≤ m ≤ n) a system of subsets of E, having m elements each, such that every subset of E having l elements is contained in exactly one set of the system S (l, m, n).It is clear (3), that a necessary condition for the existence of S (l, m, n) is that1is the number of elements of S(l, m, n) andis the number of those elements of S (l, m, n) which contain h fixed elements of E.It is known that condition (1) is not sufficient for S(l, m, n) to exist. It has been proved that no finite projective geometry exists with 7 points on every line. This implies non-existence of S(2, 7, 43).
On the Composition of Balanced Incomplete Block Designs
Authors: Bose R. C. , Shrikhande S. S. 
Published: December 2010

The object of this paper is to develop a method of constructing balanced incomplete block designs. It consists in utilizing the existence of two balanced incomplete block designs to obtain another such design by what may be called the method of composition.1. Preliminary results on orthogonal arrays and balanced incomplete block designs. Consider a matrixA =(aij) of k rows andNcolumns, where eachaijrepresents one of the integers 1, 2, … ,s. Consider all t-rowed submatrices ofNcolumns, which can be formed from this array,t≤k.Each column of any Crowed submatrix can be regarded as an orderedt-plet.
On Null-Recurrent Markov Chains
Authors: Lamperti John 
Published: December 2010

Throughout this paper, the symbol P = [Pij] will represent the transition probability matrix of an irreducible, null-recurrent Markov process in discrete time. Explanation of this terminology and basic facts about such chains may be found in (6, ch. 15). It is known (3) that for each such matrix P there is a unique (except for a positive scalar multiple) positive vector Q = {qi} such that QP = Q, or1this vector is often called the "invariant measure" of the Markov chain.The first problem to be considered in this paper is that of determining for which vectors U(0) = {μi(0)} the vectors U(n) converge, or are summable, to the invariant measure Q, where U(n) = U(0)Pn has components2In § 2, this problem is attacked for general P. The main result is a negative one, and shows how to form U(0) for which U(n) will not be (termwise) Abel summable.
Lattice Octahedra
Authors: Mordell L. J. 
Published: December 2010

Let Ai, A2, … , An be n linearly independent points in n-dimensional Euclidean space of a lattice Λ. The points ± A1, ±A2, . . , ±An define a closed n-dimensional octahedron (or “cross poly tope“) K with centre at the origin O. Our problem is to find a basis for the lattices Λ which have no points in K except ±A1, ±A2, … , ±An.Let the position of a point P in space be defined vectorially by1where the p are real numbers. We have the following results.When n = 2, it is well known that a basis is2When n = 3, Minkowski (1) proved that there are two types of lattices, with respective bases3When n = 4, there are six essentially different bases typified by A1, A2, A3 and one of4In all expressions of this kind, the signs are independent of each other and of any other signs. This result is a restatement of a result by Brunngraber (2) and a proof is given by Wolff (3).
A Note on Continued Fractions
Authors: Oppenheim A. 
Published: December 2010

Any real number y leads to a continued fraction of the type(1)where ai, bi are integers which satisfy the inequalities(2)by means of the algorithm(3)the a's being assigned positive integers. The process terminates for rational y; the last denominator bk satisfying bk ≥ ak + 1. For irrational y, the process does not terminate. For a preassigned set of numerators ai ≥ 1, this C.F. development of y is unique; its value being y.Bankier and Leighton (1) call such fractions (1), which satisfy (2), proper continued fractions. Among other questions, they studied the problem of expanding quadratic surds in periodic continued fractions. They state that “it is well-known that not only does every periodic regular continued fraction represent a quadratic irrational, but the regular continued fraction expansion of a quadratic irrational is periodic.
The Equivalence of Two Extremum Problems
Authors: Kiefer J. , Wolfowitz J. 
Published: December 2010

Let f1 , …, fk be linearly independent real functions on a space X, such that the range R of (f1, …, fk) is a compact set in k dimensional Euclidean space. (This will happen, for example, if the fi are continuous and X is a compact topological space.) Let S be any Borel field of subsets of X which includes X and all sets which consist of a finite number of points, and let C = {ε} be any class of probability measures on S which includes all probability measures with finite support (that is, which assign probability one to a set consisting of a finite number of points), and which are such thatis defined. In all that follows we consider only probability measures ε which are in C.
Traces of Matrices of Zeros and Ones
Authors: Ryser H. J. 
Published: December 2010

This paper continues the study appearing in (9) and (10) of the combinatorial properties of a matrix A of m rows and n columns, all of whose entries are 0's and l's. Let the sum of row i of A be denoted by ri and let the sum of column j of A be denoted by Sj. We call R = (r1, … , rm) the row sum vector and S = (s1 . . , sn) the column sum vector of A. The vectors R and S determine a class1.1consisting of all (0, 1)-matrices of m rows and n columns, with row sum vector R and column sum vector S. The majorization concept yields simple necessary and sufficient conditions on R and S in order that the class 21 be non-empty (4; 9). Generalizations of this result and a critical survey of a wide variety of related problems are available in (6).
A Metrical Theorem in Diophantine Approximation
Authors: Schmidt Wolfgang 
Published: December 2010

In this paper we prove a sharpening and generalization of the following Theorem of Khintchine (4):Let ψ1(q), …, ψnq) be n non-negative junctions of the positive integer q and assumeis monotonically decreasing. Then the set of inequalities1has an infinity of integer solutions q > 0 and p1, … , pn for almost all or no sets of numbers θ1, … , θ2, according as Σψ(q) diverges or converges.
A Local Property of Measurable Sets
Authors: Eames W. 
Published: December 2010

Let Ω be a metric space with metric ρ, let C be a class of closed sets from Ω and let τ be a non-negative real-valued set function on C. We assume that the empty set ϕ is in C and that τ(I)= 0 if and only if I = ϕ. For each set A in Ω, we define φ(A), 0 ≤ φ(A) ≤ ∞ by:where the infimum is taken for all possible countable collections of sets I(n) from C such that:and the diameter of I(n), d(I(n)), is less than ∈ for every n.
On a Discriminant Inequality
Authors: Mordell L. J. 
Published: December 2010

The following result has been conjectured by Dr. Birch. Let z1 z2, . . . , zn be any n complex numbers such that(1)Then(2)attains its greatest value when the z are at the vertices of a regular n-sided polygon inscribed in the circle |z| =1.It seems to be difficult to prove this but Dr. Birch informs me that some work by Mullholland shows that the result is false for large n. I can, however, prove that the result is true for n = 3, and then Δ ≤ 27. The suggested general result would be Δ ≤ nn.
Construction of Primitives of Generalized Derivatives with Applications to Trigonometric Series
Authors: Bullen P. S. 
Published: December 2010

This paper is an extension of the ideas discussed in (3, §§ 14-16); the extension consisting of the use of the third and fourth symmetric Riemann derivative instead of the Schwarz or second symmetric Riemann derivative.The J2integral, due to James (1), is defined in (3) as follows. Let f(x) be measurable on [a, b] and finite at each point; if there exists a continuous function F(x) such that D2F = f everywhere on (a, b),then1.1The definition is unique since if F(x) and G(x) are continuous and D2F =D2G everywhere then
Simple Algebras of Type (1,1) are Associative
Authors: Kleinfeld Erwin 
Published: December 2010

In the classification of almost alternative algebras relative quasiequivalence by Albert two new classes of algebras of type (γ, δ) were introduced, namely those of type (1, 1) and ( —1, 0) (1, equations (34), (35), and Theorem 6). Since rings of type (1, 1) and ( — 1, 0) are anti-isomorphic it suffices to consider those of type (1, 1). They may be defined as rings satisfying1and2for all elements x, y, and z of the ring, where the associator (a, b, c) is given by (a, b, c) = (ab)c — a (bc).Actually the identity2'together with (1) implies (2) whenever the characteristic of the ring is different from 2. This may readily be verified by linearizing (2’). Consequently we may use (1) and (2’) as the defining relations for a ring of type (1, 1).
On the Derivation Algebras of Lie Algebras
Authors: Tôgô Shigeaki 
Published: December 2010

LetLbe a Lie algebra over a field of characteristic 0 and letD(L)be the derivation algebra ofL, that is, the Lie algebra of all derivations ofL. Then it is natural to ask the following questions: What is the structure ofD(L)?What are the relations of the structures ofD(L)andL? It is the main purpose of this paper to present some results onD(L)as the answers to these questions in simple cases.Concerning the questions above, we give an example showing that there exist non-isomorphic Lie algebras whose derivation algebras are isomorphic (Example 3 in § 5). Therefore the structure of a Lie algebraLis not completely determined by the structure ofD(L). However, there is still some intimate connection between the structure ofD(L)and that ofL.
The Expansion Problem with Boundary Conditions at a Finite Set of Points
Authors: Cole Randal H. 
Published: December 2010

The problem of expanding an arbitrary function in a series of characteristic solutions of the ordinary differential equation(1.1)and the boundary relations(1.2)is well known. The various discussions are distinguished by the manner in which a parameter λ appears in the differential system and by the number of points at which the boundary conditions apply. The case in which the boundary conditions apply at intermediate as well as at the end points of a fundamental interval has been considered by Wilder (3).
Anticommuting Linear Transformations
Authors: Kestelman H. 
Published: December 2010

It is well known that any set of four anticommuting involutions (see §2) in a four-dimensional vector space can be represented by the Dirac matrices(1)where the B1,r are the Pauli matrices(2)(See (1) for a general exposition with applications to Quantum Mechanics.) One formulation, which we shall call the Dirac-Pauli theorem (2; 3; 1), isTheorem 1. If M1,M2, M3, M4 are 4 X 4 matrices satisfyingthen there is a matrix T such thatand T is unique apart from an arbitrary numerical multiplier.
On Finite Groups with an Abelian Sylow Group
Authors: Brauer Richard , Leonard Henry S. 
Published: December 2010

We shall consider finite groups of order of g which satisfy the following condition:(*) There exists a prime p dividing g such that if P ≠ 1 is an element of p-Sylow group ofthen the centralizer(P) of P incoincides with the centralizer() of in.This assumption is satisfied for a number of important classes of groups. It also plays a role in discussing finite collineation groups in a given number of dimensions.Of course (*) implies that is abelian. It is possible to obtain rather detailed information about the irreducible characters of groups in this class (§ 4).
On Certain Onto Maps
Authors: Namioka Isaac 
Published: December 2010

Let Δn (n > 0) denote the subset of the Euclidean (n + 1)-dimensional space defined byA subset σ of Δn is called a face if there exists a sequence 0 ≤ i1 ≤ i2 ≤ … < im ≤ n such thatand the dimension of σ is defined to be (n — m). Let denote the union of all faces of Δn of dimensions less than n. A topological space Y is called solid if any continuous map on a closed subspace A of a normal space X into Y can be extended to a map on X into Y. By Tietz's extension theorem, each face of Δn is solid. The present paper is concerned with a generalization of the following theorem which seems well known.
Finite Sublattices of a Free Lattice
Authors: Jónsson Bjarni , Kiefer James E. 
Published: December 2010

It is known that every sublattice A of a free lattice satisfies the following conditions:(W) For all a, b, c, d ∈ A, if ab ≤ c + d, then ab ≤ c or ab ≤ d or a ≤ c+dor b ≤ c + d.(SD) For all u, a, b, c ∈ A, if u = a + b = a + c, then u = a + bc.(SD′) For all u, a, b, c ∈ A, if u = ab = ac, then u = a(b + c).In fact, (W) is one of the four conditions used in Whitman (4) to characterize free lattices, and in Jónsson (3) it was shown that (SD) and (SD′) follow from Whitman's canonical representations of elements of a free lattice.This note is concerned with lattices that satisfy one or more of the above conditions, and especially with finite lattices that satisfy all three conditions.
The Embedding Theorems of Malcev and Lambek
Authors: Bush George C. 
Published: December 2010

A given semigroup is said to be embeddable in a group if there exists a group which contains a subsemigroup isomorphic to .It can easily be proved that cancellation is a necessary condition for embeddability. It can also be shown that we can adjoin an identity to a semigroup without identity in such a way that the new semigroup is embeddable if and only if the original was embeddable. Therefore, we can, without loss of generality, restrict our attention to cancellation semigroups with identity, whenever this is convenient.
Rational Points on Linear Subspaces. Representation of an Integer as a Sum of Squares with Accessory Conditions
Authors: Motzkin T. S. 
Published: December 2010

The present study was motivated by an investigation of algebraic conjugates in the complex plane (cf. 4 for one of the results) where some of its concepts are extended and applied.Let F be a flat (linear subspace) in real affine n-space. The points z = (ζ1, . . . , ζn) on F for which the least common denominator of the coordinates ζv is minimum form a grid G, the main grid of F, studied in § 3.
On Weakly Positive Matrices
Authors: Wigner Eugene P. 
Published: December 2010

A matrix is said to be positive definite if it is hermitian and if all of its characteristic values are positive. It is well known, and easy to prove, that the necessary and sufficient condition for a matrix P to be positive definite is that its hermitian quadratic formwith any vector v ≠ 0 be positive. (This will imply, in the present article, that it is real.) It is easy to see from (1) that if P1 and P2 are positive definite, the same holds of a1P1 + a2P2 if a1 and a2 are positive numbers.
On the Group Ring
Authors: Connell Ian G. 
Published: December 2010

LetRbe the discrete group ring of the groupGover the ringA. In this paper we attempt to find necessary and sufficient conditions onGandAso thatRwill have some standard ring-theoretic property ; among the properties considered are those of being artinian, regular, self-injective, and semi-prime.The contents of this paper form essentially the author's doctoral thesis. The author would like to thank his supervisor Dr. J. Lambek for his generous encouragement and continued interest.
Variation-Diminishing Transformations and General Orthogonal Polynomials
Authors: Hirschman I. I. 
Published: December 2010

Letα(dx) be a finite measure defined on the Borel subsets of [—1, 1], the spectrum of which is infinite. Letbe the family of orthonormal polynomials associated with a, so thatTheare uniquely determined by this and by the condition
Compounds of Skew-Symmetric Matrices
Authors: Marcus Marvin , Yaqub Adil 
Published: December 2010

In a recent interesting paper (3) H. Schwerdtfeger answered a question of W. R. Utz (4) on the structure of the real solutions A of A* = B, where A is skew-symmetric. (Utz and Schwerdtfeger call A* the "adjugate" of A ; A* is the n-square matrix whose (i, j) entry is (—1)i+j times the determinant of the (n — 1)-square matrix obtained by deleting row i and column j of A. The word "adjugate," however, is more usually applied to the matrix (AT)*, where AT denotes the transposed matrix of A ; cf. (1, 2).)The object of the present paper is to find all real n-square skew-symmetric solutions A to the equation
A Formal Solution of Σ∞i=1AieBix = X
Authors: Raney George N. 
Published: December 2010

We consider a combinatorial enumeration problem involving certain collections of labelled rooted trees having coloured nodes and edges. Notations and definitions are introduced in §§2 and 3, and the problem is described in §4. We give recursion formulas for its solution in §5. Then, by using a modification of a method of Priifer, we obtain a direct solution in terms of multinomial coefficients and power products in §§6 and 7. These results are combined in §8. Working in a formal power series algebra we find a formal solution of the equationwhich expresses the unknown X as a multiple power series in the Ai and the Bi.
Infinite Euler Graphs
Authors: Sabidussi Gert 
Published: December 2010

It is well known that for finite connected graphs the following are equivalent:(i)X is Euler (i.e., every vertex of X has positive even degree);(ii)X is traceable (i.e., the edges of X can be arranged in a sequence e1, . . . ,en such that ei ≠ ej if i ≠ j, and ei, ei+1 are adjacent, i = 1, . . . , n, subscripts considered mod n) ;(iii)X is cyclically coverable (i.e., X contains a family of non-overlapping circuits whose union is X).
A Census of Finite Automata
Authors: Harrison Michael A. 
Published: December 2010

A finite automaton may be thought of as a possible abstraction of a digital computer. Imagine a tape or sequence of letters from some alphabet being fed into a device with a finite number of internal states. When the device is in a particular state and receives an input letter, the system passes to another internal state and prints a letter on an output tape. This operation is continued until the entire input sequence has been processed.Both Harary (6) and Ginsburg (4) have focused attention on the previously unsolved problem of counting the number of equivalence classes of finite automata. In the present paper, this problem is solved completely by proving a variety of theorems about the enumeration of functions.
The Distribution of Irreducible Polynomials in Several Indeterminates II
Authors: Carlitz L. 
Published: December 2010

It is well known that the number of normalized irreducible polynomials of degree m in a single indeterminate, with coefficients in GF(q), is given by(1.1)where μ(r) is the Möbius function. It follows from (1) that(12)
Kurosh Radicals of Rings with Operators
Authors: Divinsky N. , Suliński A. 
Published: December 2010

If A is an algebra over a commutative ring with unity, Φ, then the Jacobson radical of the algebra A is equal to the Jacobson radical of A, thought of as a ring (1, p. 18, Theorem 1). The present note extends this result to all radical properties (in the sense of Kurosh 2) and allows ϕ to be any set of operators on A.If A is a ring and Φ is an arbitrary set, we say that Φ is a set of operators for A if for any α in Φ and any x in A, the composition ax is defined and is an element of A, and if this composition satisfies the following two conditions:
Enumeration of Quadrangular Dissections of the Disk
Authors: Brown William G. 
Published: December 2010

A dissection of the disk will be a cell complex (1, p. 39) K with polyhedron the closed disk B2. It will further be required that: (a) every edge of K be incident with two distinct vertices (called its ends) ;(b) no two edges have the same ends ; and(b) no two edges have the same ends ; and
On the Structure of Group Algebras, I
Authors: Cohn James A. , Livingstone Donald 
Published: December 2010

With this paper we begin a study of the structure of the group algebra RG of a finite group G over the ring of algebraic integers R in an algebraic number field k. The basic question is whether non-isomorphic groups can have isomorphic algebras over R. We shall show that this is impossible if G is (a) abelian,(b) Hamiltonian,(c) one of a special class of p-groups.
On Some Diophantine Inequalities Involving the Exponential Function
Authors: Baker A. 
Published: December 2010

It is well known that for any real number θ there are infinitely many positive integersnsuch thatHere ||a|| denotes the distance of a from the nearest integer, taken positively. Indeed, since ||a|| < 1, this implies more generally that ifθ1,θ2, . . . ,θkare any real numbers, then there are infinitely many positive integersnsuch that
Characterizations of Developable Topological Spaces
Authors: Worrell J. M. , Wicke H. H. 
Published: December 2010

The class of developable topological spaces, which includes the metrizable spaces, has been fundamentally involved in investigations in point set topology. One example is the remarkable edifice of theorems relating to these spaces constructed by R. L. Moore (13). Another is the role played by the developable property in several metrization theorems, including Alexandroff and Urysohn's original solution of the general metrization problem (1).This paper presents an anslysis of the concept of developable space in terms of certain more extensive classes of spaces satisfying the first axiom of countability : spaces with a base of countable order and those having what is here called a θ-base. The analysis is given in the characterizations of Theorems 3 and 4 below.
The K-Product of Arithmetic Functions
Authors: Gioia A. A. 
Published: December 2010

In this note we introduce a natural generalization of the ordinary convolution of arithmetic functions: If f and g are arithmetic functions,defines the K-product of f and g. If the kernel K(n) ≡ E(n) = 1, the K-product is the ordinary convolution Σd|nf(d)g(n/d);, if K(n) ≡ ∊(n) = [1/n], then the K-product is the unitary product Σf(d)g(n/d), summed over d|n, (d, n/d) = 1 (1, 2). We give in Theorem 1 a characterization of all associative kernels, i.e., kernels for which the corresponding K-product is associative.
On Integer Matrices and Incidence Matrices of Certain Combinatorial Configurations, I: Square Matrices
Authors: Majindar Kulendra N. 
Published: December 2010

A few years ago, in a short paper (4) Ryser introduced an interesting topic in number theory, viz. the connection between integer matrices (i.e., matrices having only integers as their elements) satisfying certain conditions and 0-1 matrices (i.e., matrices that have no element different from 0 and 1). In this series of papers we shall pursue this topic further.To make the statements of our theorems short we introduce some terminology. We need the definitions of certain 0-1 matrices related to a few well-known combinatorial configurations. By an incidence matrix of a balanced incomplete block (b.i.b. for conciseness) design we mean a 0-1 matrix with v rows and b columns, such that the sum of the elements in each column of A is k, k < v, and the scalar product of any two row vectors of A is λ, λ ≠ 0.
Groups Associated with Certain Loci In [5]
Authors: Horadam A. F. 
Published: December 2010

If x- (i = 1, 2, … , 6) are homogeneous co-ordinates in [5] (the complex projective space of five dimensions), then the equation1.1represents the well-known (11) Perazzo cubic primalP43 of order 3 and dimension 4. With it is associated the Segre cubic threefoldS33 (12); specifically, S33 is the section of P43 by a tangent [4].
Classes of Functions on Algebras
Authors: Cullen C. G. , Hall C. A. 
Published: December 2010

Let be a finite-dimensional linear associative algebra over the real field R or the complex field C and let F be a function with domain and range in .Several classes of functions on have been discussed in the literature, and it is the purpose of this paper to discuss the relationships between these classes and to present some interesting examples. First we shall list the definitions of the classes we wish to consider here.