bluuebunny/crossref_metadata_2025_split huggingface dataset:
Calderón–Zygmund Operators Associated to Ultraspherical Expansions
Authors: Buraczewski Dariusz , Martinez Teresa , Torrea José L. 
Published: December 2010

AbstractWe define the higher order Riesz transforms and the Littlewood–Paley g-function associated to the differential operator Lλf(θ) = –f′′(θ)–2λ cot θ f′(θ) + λ2f(θ). We prove that these operators are Calderón–Zygmund operators in the homogeneous type space ((0, π), (sin t)2λdt). Consequently, Lp weighted, H1 – L1 and L∞ – BMO inequalities are obtained.
Generic Extensions and Canonical Bases for Cyclic Quivers
Authors: Deng Bangming , Du Jie , Xiao Jie 
Published: December 2010

AbstractWe use the monomial basis theory developed by Deng and Du to present an elementary algebraic construction of the canonical bases for both the Ringel–Hall algebra of a cyclic quiver and the positive part U+of the quantum affine. This construction relies on analysis of quiver representations and the introduction of a new integral PBW–like basis for the Lusztig ℤ[v,v–1]-form of U+.
On an Elementary Problem in Number Theory
Authors: Erdös Paul 
Published: December 2010

A question which Chalk and L. Moser asked me several years ago led me to the following problem: Let 0 < x ≤ y. Estimate the smallest f(x) so that there should exist integers u and v satisfying1I am going to prove that for every ∊ > 0 there exist arbitrarily large values of x satisfying2but that for a certain c > 0 and all x3A sharp estimation of f(x) seems to be a difficult problem. It is clear that f(p) = 2 for all primes p. I can prove that f(x)→ ∞ and f(x)/loglog x → 0 if we neglect a sequence of integers of density 0, but I will not give the proof here.
A Remark by Philip Hall
Authors: Robinson G. de B. 
Published: December 2010

The relationship between the representation theory of the full linear group GL(d) of all non-singular linear transformations of degree d over a field of characteristic zero and that of the symmetric group Sn goes back to Schur and has been expounded by Weyl in his classical groups, [4; cf also 2 and 3]. More and more, the significance of continuous groups for modern physics is being pressed on the attention of mathematicians, and it seems worth recording a remark made to the author by Philip Hall in Edmonton.As is well known, the irreducible representations of Sn are obtainable from the Young diagrams [λ]=[λ1, λ2 ,..., λr] consisting of λ1 nodes in the first row, λ2 in the second row, etc., where λ1≥λ2≥ ... ≥λr and Σ λi = n. If we denote the jth node in the ith row of [λ] by (i,j) then those nodes to the right of and below (i,j), constitute, along with the (i,j) node itself, the (i,j)-hook of length hij.
A Note on the Compositions of an Integer
Authors: Narayana T. V. , Fulton G. E. 
Published: December 2010

Given an integer n, we define an r-composition of n as follows:An r-composition of n, (t1, ... , tr), is a set of ti where ti ≥ 1 is an integer for i = 1, ... , r such thatIf r is an integer such that 1 ≤ r ≤ n, we have, obviously, distinct r - compositions of n.
Notes on Numerical Analysis I. Polynomial Iteration
Authors: Schwerdtfeger Hans 
Published: December 2010

Let f(x) be a real analytic function of the real variable x and α a simple root of the equation f(x) = 0. It is well known that a function ϕ(x) can be associated with the equation in many different ways such that(i)α is a root of the equation ϕ(x), i.e. α is a fixed point (invariant point) of the function ϕ(x);(ii)
On Continuous Linear Transformations of Integral Type
Authors: Ellis H.W. , Mott T.E. 
Published: December 2010

Let (XxY, SxT, μxν) denote the completion of the Cartesian product of the σ-finite and complete "measure spaces (X, S, μ.) and (Y, T, ν) [3]. Let λx and λy denote arbitrary length functions defined on (X, S, μ.) and (Y, T, ν) respectively, the conjugate length functions [2]. We suppose that1is defined for every f(x, y) measurable (SxT). The Fubini theorem implies that f(x, y) is measurable (T) for almost all x. Thus λ xy(f) will be defined when λy(f) Is measurable (S). If Lλ y = Lp, 1 ≤ p < ∞, this is implied by the Fubini theorem. General conditions ensuring that λ y (f) is measurable (S) are given in [l, Theorem 3.2] When λxy(f) is defined for every f(x, y) measurable (SxT), it is a length function and Lλ xy is a Banach space [l, Theorem 3.l].
Self-Injective Rings
Authors: Wong E.T. , Johnson R.E. 
Published: December 2010

Historically, the first example of a ring of quotients was the quotient field of an integral domain. Later on, conditions were found under which a noncommutative integral domain has a quotient division ring. More recently, R.E. Johnson [4], Y. Utumi [5], and G.D. Findlay and J. Lambek [3] have discussed the existence and structure of a maximal ring of quotients of any ring.The present paper uses the methods of Findlay and Lambek to recast the results of Johnson on the quotient ring of a ring with zero singular ideal. It is also shown that such a ring has a unique left-right maximal ring of quotients.
Some Properties of Beatty Sequences I*
Authors: Connell Ian G. 
Published: December 2010

Two sequences of natural numbers are said to be complementary if they contain all the positive integers without repetition or omission. S. Beatty [l] observed that the sequences(1)(2)(where square brackets denote the integral part function) are complementary if and only if α > 0 and α is irrational. We call the pair (1),(2) Beatty sequences of argument α.
A Non-Hamiltonian Graph
Authors: Tutte W. T. 
Published: December 2010

There is a remarkable cubic graph of 28 vertices, discovered by Prof. H.S.M. Coxeter, which has no circuit of fewer than 7 edges and in which all oriented arcs made up of three edges are equivalent under the automorphism group.The structure of the graph is as follows. There are three disjoint heptagons A(a1a2a3a4a5a6a7a1), B(b1b5b2b6b3b7b4b1) and C(c1c6c4c2c7c5c3c1). We denote the seven remaining vertices by d1, d2,…, d7. For each suffix i the three edges incident with dijoin it to ai, bi and ci. In drawing a diagram of the graph it seems best to show only the heptagons A, B and C,
leaving the rest of the figure to the imagination.The purpose of this note is to establish another property of the graph, that it has no Hamiltonian circuit.
A Note on Prime-Power Groups
Authors: Struik Ruth Rebekka 
Published: December 2010

In this paper an unsuccessful attempt to classify prime - power groups is described. The attempt consisted in combining some ideas of P.Hall and O. N. Golovin.Golovin [2] defined nilpotent products of groups and showed that every nilpotent group is a factor group of a nilpotent product of cyclic groups.
Elementary Proofs of the Extremal Properties of the Eigenvalues of the Sturm-Liouville Equation
Authors: Beesack Paul R. 
Published: December 2010

If the Sturm-Liouville eigenvalue problem1is first approached from the standpoint of differential equations theory - as opposed, say, to the calculus of variations, or the theory of integral equations - the extremal properties of the eigenvalues seem to be generally regarded as lying beyond the scope of the theory. Thus, neither in the standard work of Bocher [l], nor in the recent work of Coddington and Levinson [2] is any mention made of this topic. Collatz [3, 166-8] gives an elementary proof of the minimum property of the least positive eigenvalue of (1.1), and a brief indication of how this argument can be extended to the higher eigenvalues. The purpose of this paper is to consolidate this elementary approach, and to extend it to cover the singular cases where either the interval is infinite, or one or more of the coefficients are singular at the end-points.
Sequence and Series Transformations
Authors: Macphail M. S. 
Published: December 2010

The summability methodswhere bnk=ank+an,k+1+ …,are regarded as the sequenceto-sequence and series-to-sequence forms of the same method, and if sk=uo + … + uk, we speak of the series or the sequence {sk} indifferently, as summable A or B.
A Note an a Group Defined by a Quadratic Form
Authors: Marcus Marvin , Khan Nisar A. 
Published: December 2010

In a recent series of papers [3,4,5], H. Zassenhaus considered the structure of those linear transformations T on real 4-space, R4, into itself that preserve the quadratic form . That is,1.1Define a function ϕ on R4 to the space M 2 of 2-square matrices over the complex numbers as follows:1.2Let G2 be the vector space of matrices generated by all real linear combinations of1.3
On Duo Rings
Authors: Thierrin G. 
Published: December 2010

Following E. H. Feller [l], a ring R is called a duo ring if every one-sided ideal of R is a two-sided ideal.In the first part of this paper, we give some properties of duo rings and we show that the set of the nilpotent elements of a duo ring R is an ideal, the intersection of the completely prime ideals of R.It is easy to see that every duo ring is a subdirect sum of subdirectly irreducible duo rings. We give in the second part of this paper a characterization of the subdirectly irreducible duo rings. This characterization is quite similar to the characterization of the subdirectly irreducible commutative rings, due to N. H. McCoy [2], whose methods we use.
On Schauder Bases for Spaces of Continuous Functions1)
Authors: Ellis H. W. , Kuehner D. G. 
Published: December 2010

In a finite dimensional vector space V a set xi, i = 1, 2, …, n of vectors of V is said to be a basis, base, or coordinate system for V if the vectors xi are linearly independent and if each vector in V is a linear combination of the elements x1 with real coefficients. If a topology for V is defined in terms of a norm ||.|| then {xi} is a basis for V if and only if to each x ϵ V corresponds a unique set of constants ai such thatIn infinite dimensional normed vector spaces the above concepts of basis have different generalizations. The first or algebraic definition gives a Hamel basis which is a maximal linearly independent set [l, p. 2]. We shall be interested in the other or topological definition.
Odd and Even Permutations
Authors: Halperin Israel 
Published: December 2010

This note gives a proof for the familiar elementary theorem that if a permutation of the integers 1, …, n (with n ≥ 2) is expressed as a product π1 of N1 transpositions and also as a product π2 of N2 transpositions, then N1 and N2 are both even or both odd (equivalently: N1 + N2 is even).Here, a transposition means an interchange of two of the integers. If the interchange is between adjacent integers it is called an adjacent-trans position.
Hesse's Theorem for a Quadrilateral Whose Sides Touch a Conic
Authors: Brown William G. 
Published: December 2010

Hesse's theorem states that “if two pairs of opposite vertices of a quadrilateral are respectively conjugate with respect to a given polarity, then the remaining pair of vertices are also conjugate ”.In the real projective plane there cannot exist such a quadrilateral, all four sides of which are self-conjugate [1, §5.54]. We shall show that such a quadrilateral exists in PG(2, 3), and that any geometry in which such a quadrilateral exists contains the configuration 134 of PG(2,3). We shall thus provide a synthetic proof of Hesse1 s theorem for a quadrilateral of this type, which, together with [1, § 5.55], constitutes a complete proof of the theorem valid in general Desarguesian projective geometry. We shall also show analytically that a finite Desarguesian geometry which admits a Hessian quadrilateral all of whose sides touch a conic must be of type PG(2, 3n).
On the Watson and Laplace Transformations
Authors: Rooney P. G. , Schubert C. 
Published: December 2010

Let k be the kernel of a Watson transformation; that is k(x)/x ϵ L2(0,∞), and if x and y are positive,1Then if g is the transform of F ϵ L2(0,∞), that is if2it i s known that g 2(0,∞), that3and that4For these results see [1; theorem 79].
A Generalization of Cox's Chain of Theorems
Authors: Al-Dhahir M. W. 
Published: December 2010

In [5, p. 105] attention has been called to a set of propositions, due to H. Cox [3, p. 67], which are related to another set, due to Clifford [2, p. 145; 4, p. 447], concerning points and circles in the plane or on the sphere. One may state Cox's chain of theorems as follows:In a projective 3-space, S3, let (1), (2), (3), (4) be four points lying in a plane α such that no three of them are collinear. Every two determine a line; let one plane such as [12], pass through each line. There are six such planes. The planes [12], [23], [13] determine a point (123); there are four such points. The first theorem of the chain states that they all lie in one plane [1234], It is not difficult to see that this is, in fact, a rewording of Möbius's theorem on mutually inscribed pairs of tetrahedra [4, p. 444].
A Determinantal Inequality for Positive Definite Matrices
Authors: Thompson R. C. 
Published: December 2010

Let H = (Hi, j) (1 ≦ i, j ≦ n) be an nk × nk matrix with complex coefficients, where each Hi, j is itself a k × k matrix (n, k ≧ 2). Let |H| denote the determinant of H and let ∥H∥ = |(|H i, j|)| (1 ≦ i, j ≦ n ). The purpose of this note is to prove the following theorem.Theorem. If H is positive definite Hermitian then |H| ≦∥H∥. Moreover, |H| = ∥H∥ if and only if Hi, j = 0 whenever i ≠ j.The case n = 2 of this theorem is contained in [1].
A Norm Inequality for Linear Transformations
Authors: Moyls B.N. , Khan N.A. 
Published: December 2010

In 1949 Ky Fan [1] proved the following result: Let λ1…λn be the eigenvalues of an Hermitian operator H on an n-dimensional vector space Vn. If x1, …, xq is an orthonormal set in V1, and q is a positive integer such n that 1 ≤ q ≤ n, then1
Extensions Radicales et Quasi-Radicales Dans Les
Anneaux
Authors: Thierrin G. 
Published: December 2010

Soient un anneau A et un sous-ensemble (non vide) B de A. L'anneau A est
dit:Extension radicale de B, si pour tout a ∊ A, il existe un entier n(a)>0,
dépendant de a, tel que an(a)∊B.Extension quasi-radicale de B, si pour tout a ∊ A, il existe un entier
n(a)>1, dépendant de a, tel que an(a)-a∊B.En imposant certaines conditions à A ou B, on peut alors obtenir certains
renseignements sur la structure de A. Dans le cas où B est un sous-anneau de
A, plusieurs travaux ont été consacrés récemment à ce sujet. Notamment, on a
cherché quelles conditions, imposées à A ou B, entraînent la commutativité
de A. (Voir en particulier [2], [3], [4], [5] et [6].)
On the Fundamental Theorem of Affine Geometry
Authors: Scherk P. 
Published: December 2010

The fundamental theorem of affine geometry is an easy corollary of the
corresponding projective theorem 2.26 in Artin's Geometric Algebra. However,
a simple direct proof based on Lipman's paper [this Bulletin, 4, 265−278]
and his axioms 1 and 2 may be of some interest.Lipman's [desarguian] affine geometry G determined a left linear vector
space L={a, b,…} over a skew field F. We wish to construct 1−1
transformations γ of G onto itself such that γ and γ-1 map
straight lines onto straight lines preserving parallelism. Designate any
point 0 as the origin of G. Multiplying γ with a suitable translation, we
may assume γ0=0. Thus γ will then be equivalent to a 1−1 transformation Γ of
L onto itself which preserves linear dependence. Since Γ-1 will
have the same properties, Γ must also preserve linear independence.
A New Kind of Free Extension for Projective
Planes
Authors: Ditor Seymour 
Published: December 2010

Marshall Hall [l] shows how projective planes of very general structure may
be constructed and at the same time exhibits an extensive class which are
non-Desarguesian. Here we shall indicate how his method of free extension
can be generalized to yield a class of planes which seem to be distinct from
those which he obtains.A partial plane is a system consisting of two distinct sets of elements, a
set of "points" P, Q,… and a set of "lines" l, m,…, and a
relation between these two sets, called "incidence", such that for any two
distinct points, there is at most one line incident with both (or,
equivalently, for any two distinct lines, there is at most one point
incident with both). A partial plane is complete if every two distinct
points are joined by a line and every two distinct lines intersect in a
point.
The Five and Six Dimensional Magic Hypercubes of Order
3
Authors: Hendricks John R. 
Published: December 2010

A recent publication entitled "Magic Squares and Cubes", by W. S. Andrews,
published by the Dover Publications, New York, is an excellent authoritative
book on magic squares, and magic cubes. In fact, chapter XIV entitled "Magic
Octrahedroids", has shown examples of the extension into four-dimensional
space. Andrews' method seems to be that of extending symmetrical
considerations, which he calls reversions, in forming higher dimensional
forms of magic squares. Neither he, nor any other author, to the best of my
knowledge, has extended magic squares to higher than four dimensions.La Hire must be given full credit for his method of breaking down magic
squares into component squares, and conversely constructing magic squares
from component squares. The approach of this author is to extend La Hire's
method for n-dimensional space by means of a relation:
Polynomials with Real Roots
Authors: Dixon J. D. 
Published: December 2010

In a recent issue of this Bulletin a problem equivalent to the following is proposed by Moser and Pounder [1]:If ax2+bx+c is a polynomial with real coefficients and real roots then a+b+c ≤9/4 max (a, b, c).The object of this note is to prove the following theorems which generalise this result.Theorem 1. Let αn be the smallest constant such that n for all polynomials1
A Generalization of Certain Rings of A. L.
Foster
Authors: Yaqub Adil 
Published: December 2010

The concept of a Boolean ring, as a ring A in which every element is
idempotent (i. e., a2 = a for all a in A), was first introduced
by Stone [4]. Boolean algebras and Boolean rings, though historically and
conceptually different, were shown by Stone to be equationally
interdefinable. Indeed, let (A, +, x) be a Boolean ring with unit 1, and let
(A, ∪, ∩, ') be a Boolean algebra, where ∩, ∪, ', denote "union", "
intersection", and "complement". The equations which convert the Boolean
ring into a Boolean algebra are:IConversely, the equations which convert the Boolean algebra into a
Boolean ring are:II
Triangular Dissections of N-Gons
Authors: Moon J. W. , Moser L. 
Published: December 2010

Let f(n) denote the number of dissections of a regular n-gon into n-2 triangles by n-3 non-intersecting diagonals. It is known thatand that1for n = 3, 4, . . . , where f(2) = 1 by definition. (For pertinent references on this and related problems see, e. g. , Motzkin [2].) The object of this note is to obtain a simple expression for g(n) , the number of such dissections remaining when those which differ only by a rotation, reflection, or both are not considered as being different. For convenience we shall let g(2) = 1 and f(k) = 0 when k is not an integer.
Generating Reflections For U(2, p2n). II, p =
2
Authors: Crowe D. W. 
Published: December 2010

It is known [4] that the finite two- dimensional unitary group U(2,
p2n) is generated by two reflections if p ≠ 2. The present
note completes that result by giving two generating reflections for U(2,
22n), n > 1. As in [4] this implies that the points of the
"unit circle " in the
unitary plane over GF(22n), n > 1, are the vertices of a
"regular unitary polygon" whose group of auto- 2n morphisms is U(2,
22n).The final section gives abstract definitions for the particular groups
U(2,24) and U(2,52) in terms of their generating
reflections.
Permanents of (0, 1)-Circulants
Authors: Minc Henryk 
Published: December 2010

The permanent of an n-square matrix A = (aij) is defined bywhere the summation extends over all permutations σ of the symmetric group
Sn. A matrix is said to be a (0, 1)-matrix if each of its
entries is either 0 or 1. A (0, 1)-matrix of n-1 the form , where θj = 0 or 1, j = 1,…, n, and
Pn is the n-square permutation matrix with ones in the (1, 2),
(2, 3),…, (n-1, n), (n, 1) positions, is called a (0, 1)-circulant. Denote
the (0, 1)-circulant . It has
been conjectured that1
New Proofs for two Theorems of Capelli
Authors: Rowlinson Elizabeth 
Published: December 2010

The following two theorems are due to Capelli.THEOREM 1. Let g(x) and h(x) be polynomials over a field R of characteristic 0; let f(x) = g(h(x)). Then f(x) is irreducible over R if and only if(i)g(x) is irreducible over Rand(ii)h(x) - β is irreducible over R(β), where β is a root of g(x).
On the Number of Positive Entries in the Powers of a Non-Negative Matrix
Authors: Pullman N. 
Published: December 2010

A real matrix A is said to be non-negative if and only if none of its entries is negative. Suppose A is an r by r non-negative matrix. We want to examine:(A)The first power of A to maximize the number of positive entries in An,(B)For each 1 ≤ i ≤ r the first power of A to maximize the number of positive entries in the i-th row of An.We shall call the former first power the index of A and the latter the i-th row index of A (index (i, A)).
On the Exactness of the Homotopy Sequence
Authors: Scherk P. 
Published: December 2010

This note is based on Hu's Homotopy Theory (New York, 1959), pp. 115 f.The theorem that the homotopy sequence is exact splits into six statements. The fifth in Hu's exposition can be stated as follows: Let f:In, In-1, Jn-1→X, A, x0,1Then there exists a map2such that f~j g: In, In-1, Jn-1→X, A, x0.
On Certain Simultaneous Functional Equations
Authors: Howroyd T. D. 
Published: December 2010

G. de Rham [1] has shown that certain remarkable curves, defined by limits of sequences of inscribed polygons, are solutions of functional equations of the formIf the functions FP (z) (p = 1,2) are contraction mappings of the complex plane into itself, i.e. for fixed r < 1
Some General Instability Criteria for Ordinary Differential Equations
Authors: Burton T. A. 
Published: December 2010

We consider a system of differential equations1where 0 = (o, o) is an isolated singular point. Thus, there exists B > o such that S(0, B) contains only one singular point. Here, S(0, B) denotes a sphere centered at 0 with radius B. We shall denote the boundary of S(0, B) by ∂ S(0, B).
An Application of Ramsay's Theorem to a Problem of Erdos and Hajnal
Authors: Abbott H. L. 
Published: December 2010

A family of sets is said to possess property if there exists a set such that and F ⊄ B for each In [1], P. Erdos and A. Hajnal ask the following question: Does there exist for every positive integer k a finite family of finite sets satisfying(i)|F|=k for each (ii)| F∩ G| ≤ 1 for each F, , F ≠ G(iii) does not possess property ?
On Semigroups of Transformations Acting Transitively on a Set
Authors: Tully E.J. 
Published: December 2010

We call a semigroup S transitive if S is isomorphic to a semigroup T of transformations of some set M into itself, where T acts on M transitively, that is in such a manner that for all x, y ∊ M we have Xπ = y for some transformation π∊T. In [4] the author showed that S is transitive if and only if there exists a right congruence σ (i.e., an equivalence relation for which a σ b always implies ac σ bc for all c ∊ S) on S, satisfying:(1)There exists a left identity modulo σ, that is an element e such that ea σ a for all a ∊ S .(2)Each σ-class meets each right ideal, or, equivalently, for all a, b ∊ S we have ac σ b for some c ∊ S .(3)The relation σ contains ( i. e. , is less fine than) no left congruence except the identity relation (in which each class consists of a single element).
On Closed, Totally Bounded Sets
Authors: Murdeshwar M. G. 
Published: December 2010

C. Goffman asserts that "… in a metric space X a set S is compact if and only if it is closed and totally bounded." [1] and "… every totally bounded sequence in a metric space has convergent subsequence." [2].The statements (incidentally, equivalent to each other) are both wrong, as the following counter-example shows. Take the set of all reals in the open interval (0, 1) with the usual metric. This space is closed and totally bounded, but not compact.