bluuebunny/crossref_metadata_2025_split huggingface dataset:
The mod ℭ Suspension Theorem
Authors: Brown Benson Samuel 
Published: December 2010

Our aim in this paper is to prove the general mod ℭ suspension theorem: Suppose that X and Y are CW-complexes,ℭ is a class offinite abelian groups, and that(i) πi(Y) ∈ℭfor all i < n,(ii) H*(X; Z) is finitely generated,(iii) Hi(X;Z) ∈ℭfor all i > k.Then the suspension homomorphismis a(mod ℭ) monomorphism for 2 ≦ r ≦ 2n – k – 2 (when r= 1, ker E is a finite group of order d, where Zd∈ ℭ and is a (mod ℭ) epimorphism for 2 ≦ r ≦ 2n – k – 2The proof is basically the same as the proof of the regular suspension theorem. It depends essentially on (mod ℭ) versions of the Serre exact sequence and of the Whitehead theorem.
A Spectral Sequence for Cohomotopy
Authors: Brown Benson Samuel 
Published: December 2010

For a prime number p let be the class of finite abelian groups whose orders are prime to p. For a finitely generated abelian group G, let Gp be the sum of the free and p-primary components of G. Our aim in this paper is to prove the following theorem.Theorem. Suppose that(i) Hi(X;Z) = 0 for i > k,(ii) for i > k – dThen there exists a spectral sequence withand the differential is given by
A One-Parameter Subsemigroup which meets many Regular -Classes
Authors: Tymchatyn E. D. 
Published: December 2010

We give an example which answers affirmatively the following problem posed by Hofmann and Mostert (1, p. 200):P. 10. Let S be a compact semigroup with identity 1, zero 0, and totally ordered -class space . Suppose that there is a one-parameter semigroup containing 0 and 1. Can S have any regular -classes aside from D(0) and D(1)?I am indebted to Professor K. H. Hofmann and Professor P. S. Mostert for their many helpful suggestions.1. We first give a method of embedding a given semigroup S in a semigroup R so that in the commutative diagramthe bottom map is an isomorphism of partially ordered spaces and some non-regular -class of S is embedded into a regular -class of R.
A Tauberian Theorem for Borel-Type Methods of Summability
Authors: Borwein D. 
Published: December 2010

Suppose throughout that α >0, β is real, and Nis a non-negative integer such that αN+ β> 0. A series of complex terms is said to be summable (B, α,β) to l if, as x→ ∞,where sn= a0 + a1 + … + an.The Borel-type summability method (B, α, β) is regular, i.e., all convergent series are summable (B, α,β) to their natural sums; and (B,1, 1) is the standard Borel exponential method B.Our aim in this paper is to prove the following Tauberian theorem.THEOREM. Iƒ(i) p ≧ – ½, an = o(np), and(ii) is summable (B, α,β) to l, then the series is summable by the Cesaro method(C, 2p + 1) to l.
Remetrization in Strongly Countabledimensional Spaces
Authors: B. R. Wenner  
Published: December 2010

Although the Lebesgue dimension function is topologically invariant, the dimension-theoretic properties of a metric space can sometimes be made clearer by the introduction of a new, topologically equivalent metric. A considerable amount of effort has been devoted to the problem of constructing such metrics; one example of the fruits of this research is the following theorem by Nagata (2, Theorem 5).In order that dim R ≦ n for a metrizable space R it is necessary and sufficient to be able to define a metric p(x, y) agreeing with the topology of R such that for every ∊ > 0 and for every point x oƒ R,implyA metric ρ which satisfies the condition of this theorem is called Nagata's metric (this term was introduced, to the best of the author's knowledge, by Nagami (1, Definition 9.3)).
On a Transformation Group
Authors: Kaul S. K. 
Published: December 2010

0. Let Γ denote a group of real linear fractional transformations (the constants defining any element of Γ are real numbers); see (3, § 2, p. 10). Then it is known that Γ is discontinuous if and only if it is discrete (3, Theorem 2F, p. 13).Now Γ may also be regarded, equivalently, as a group of homeomorphisms of a disc D onto itself; and if Γ is discrete, then, except for elements of finite order, each element of Γ is either of type 1 or type 2 (see Definitions 0.1 and 0.2 below).We wish to generalize the result quoted above in purely topological terms. Thus, throughout this paper we denote by X a compact metric space with metric d, and by G a topological transformation group on X each element of which, except the identity e, is either of type 1 or type 2. Let L = ﹛a ∈ X: g(a) = a for some g in G — e﹜, and . We assume furthermore that 0 is non-empty.
Some Results Relating the Behaviour of Fourier Transforms Near the Origin and at Infinity
Authors: Nasim C. 
Published: December 2010

It is known that under special conditions, Fourier sine transforms and Fourier cosine transforms behave asymptotically like a power of x, either as x → 0 or as x → ∞ or both. For example (3),where f(x) = x–αϕ(x), 0 < α < 1, and ϕ(x) is of bounded variation in (0, ∞) and Fc(x) is the Fourier cosine transform of f(x). This suggests that other results connecting the behaviour of a function at infinity with the behaviour of its Fourier or Watson transform near the origin might exist. In this paper wre derive various such results. For example, a special case of these results iswhere f(x) is the Fourier sine transform of g(x). It should be noted that the Fourier inversion formula fails to give f(+0) directly in this case. Some applications of these results to show the relationships between various forms of known summation formulae are given.
A Summation Formula Involving σk(n), k > 1
Authors: Nasim C. 
Published: December 2010

The existence of certain formulae analogous to Poisson's summation formula (9, pp. 60-64),where αβ = 2π, α > 0, and Fc(x) is the Fourier cosine transform of f(x), but involving number-theoretic functions as coefficients, was first demonstrated by Voronoï (10) in 1904. He proved thatwhere r(n) is an arithmetic function,/(x) is continuous in (a, b) and a(x) and i?(x) are analytic functions dependent on τ(n). Later, numerous papers were published by various authors giving formulae of this type involving d(n), the number of divisors of n (3), and rp(n), the number of ways of expressing n as the sum of p squares of integers (8).In 1937, Ferrar (4) developed a general theory of summation formulae, using complex analysis. Around that time, Guinand (5) also published papers where he developed the general theory from a different point of view. He applied the theory of mean convergence for the transforms of class L2(0, ∞ ). Later in 1950, Bochner (1) gave a general summation formula.
Finite Groups whose Powers have no Countably Infinite Factor Groups
Authors: Billis M. J. 
Published: December 2010

Let P be the class of all finite groups G whose powers GI have no countably infinite factor groups. Neumann and Yamamuro (1) proved that if G is a finite non-Abelian simple group, then G ∈ P. We generalize this result by proving the following theorem.THEOREM. A finite group G ∈ P if and only if G is perfect2. Inheritance properties ofP.P1. If G ∈ P and N is normal in G, then G/N ∈ P.Proof. Since (G/N)I is isomorphic to GI/NI it is clear that factor groups of (G/N)I are isomorphic to factor groups of GI, and hence finite or uncountable.P2. If G = HK, where H ∈ P and K ∈ P, then G ∈ P.Proof. We show that homomorphic images of GI are either finite or uncountable. Let ϕ be a homomorphism of GI. Then GIϕ = (HK)Iϕ = (HI/KI)ϕ = (HIϕ) (KIϕ). Since HIϕ and KIϕ must be finite or uncountable, the conclusion follows.
On the Irreducible Lattices of Orders
Authors: Roggenkamp Klaus W. 
Published: December 2010

We shall use the following notation:R = Dedekind domain;K = quotient field of R;Rp = ring of p-adic integers in K, p being a prime ideal in R;A = finite-dimensional separable k-algebra;G – R-order in A (for the definition cf. (3)).All modules that occur are assumed to be finitely generated unitary left modules, unless otherwise specified. By a G-lattice we mean a G-module which is torsion-free as R-module. A G-lattice is called irreducible if it does not contain a proper G-submodule of smaller R-rank. If p is a prime ideal in R we shall write Gp = Rp ⊗RG; Mp = RP⊗RM for a G-lattice M, and KM = K ⊗R M. Two G-lattices M and N are said to lie in the same genus (notation M ∨ N) if Mp ≌ Np for every prime ideal p in R.For any A-module L, let S(L) be the collection of G-lattices M, for which KM ≌ L. Suppose that S(L) splits into rg(L) genera, and into ri(L) classes under G-isomorphism. Maranda (6) has shown: If L is an absolutely irreducible A-module, then
Mean Value and Limit Theorems for Generalized Matrix Functions
Authors: Nikolai Paul J. 
Published: December 2010

Let A = [aij] denote an n-square matrix with entries in the field of complex numbers. Denote by H a subgroup of Sn, the symmetric group on the integers 1, …, n, and by a character of degree 1 on H. Thenis the generalized matrix function of A associated with H and x; e.g., if H = Sn and χ = 1, then the permanent function. If the sequences ω = (ω1, …, ωm) and ϒ = (ϒ1, …, ϒm) are m-selections, m ≦ w, of integers 1, …, n, then A [ω| ϒ] denotes the m-square generalized submatrix [aωiϒj], i, j = 1, …, m, of the n-square matrix A. If ω is an increasing m-combination, then A [ω|ω] is an m-square principal submatrix of A.
Projective Ideals of Finite Type
Authors: Smith William W. 
Published: December 2010

The main results in this paper relate the concepts of flatness and projectiveness for finitely generated ideals in a commutative ring with unity. In this discussion the idea of a multiplicative ideal is used.Definition.An ideal Jis multiplicative if and only if whenever I is an ideal with I ⊂ J there exists an ideal Csuch that I = JC.Throughout this paper Rwill denote a commutative ring with unity. If I and Jare ideals of R,then I: J = {x| xJ ⊂ I}. By “prime ideal” we will mean “proper prime ideal” and Specie will denote this set of ideals. Ris called a local ring if it has a unique maximal ideal (the ring need not be Noetherian). If P is in Spec R then RPis the quotient ring formed using the complement of P.
Automorphism Groups of Unary Algebras on Groups
Authors: Wenzel G. H. 
Published: December 2010

This paper presents a systematic study of the automorphism groups of those unary (universal) algebras whose carrier set G is the carrier set of some group ) and whose automorphism set contains the right translations of the latter group. These algebras appear, apart from the known classical contexts, repeatedly in characterization theorems of endomorphism semigroups (End) and automorphism groups (Aut) of algebras due to Grätzer (3; 4; 5), Makkai (7), Armbrust and Schmidt (1), Birkhoff (2), and others.Our main result (Theorem 1) constitutes an essential strengthening of a theorem of Birkhoff and represents the automorphism group of a unary algebra (where F is contained in the set of left translations of the group as wreath product of two groups that are easily determined from F and G.
An Inequality Concerning Analytic Functions with a Positive Real Part
Authors: MacGregor Thomas H. 
Published: December 2010

This paper contains an inequality about functions which are analytic and have a positive real part in the unit disk. A first consequence of the inequality is the fact that if is analytic for |z| < 1 and has values lying in a strip of width δ. This result is known and was first proved by Tammi (1).Our second theorem is a generalization of this. Namely, ifis analytic for |z| < 1 and satisfies Re{zmf(m>(z)} ≧ A andthenconverges.Another application of our fundamental inequality is the following. Let be analytic for |z| < 1 and satisfy Re p(z) ≧ 0 and set and .
The Reflection Principle for Banach Space-Valued Analytic Functions
Authors: Finkelstein Mark 
Published: December 2010

We give sufficient conditions for the continuation of an analytic function with values in a Branch space. For analytic functions taking complex numbers as values, the principle is known as the Schwarz Reflection Principle.A function defined on a domain of the complex plane with values in a Banach space X is analytic if it possesses at each point Z0 of the domain a convergent power series in z, with coefficients in X.THEOREM. Let D be a domain in the upper half-plane, and E a regular subset of the boundary of D. Suppose that E is an interval of the real axis (a,b). Let f be an analytic function defined on D, continuous up to E, taking values in a Banach space X. Let the image of D under f be Ω, and let Γ be the part of the boundary of Ω which is the image of E under f.
On Plane Sections and Projections of Convex Sets
Authors: Groemer H. 
Published: December 2010

Let K be a three-dimensional convex body. It has been conjectured (cf. 3) that one can always find a plane H such that the intersection K ∩ H is, in a certain sense, fairly circular. Instead of the plane section K ∩ H one can also consider the orthogonal projection of K onto H. Our aim in this paper is to prove some results concerning this type of problems. It appears that John has found similar theorems (cf. the remarks of Behrend, 1, p. 717). His proof of the first inequality of our Theorem 1 has been published (6). It is based on a property of the ellipse of inertia which will not be used in the present paper.A non-empty compact convex set 5 which is contained in some plane of euclidean three-dimensional space E3 will be called a convex domain.
On a Theorem of Herstein
Authors: Chacron M. 
Published: December 2010

Throughout this paper, Z is the ring of integers, ƒ*(t) (ƒ(t)) is an integer monic (co-monic) polynomial in the indeterminate t (i.e., each coefficient of ƒ* (ƒ) is in Z and its highest (lowest) coefficient is 1 (5, p. 121, Definition) and M* (M) is the multiplicative semigroup of all integer monic (co-monic) polynomials ƒ* (ƒ) having no constant term. In (3, Theorem 2), Herstein proved that if R is a division ring with centre C such that1then R = C. In this paper we seek a generalization of Herstein's result to semi-simple rings. We also study the following condition:(1)*Our results are quite complete for a semi-simple ring R in which there exists a bound for the codegree ofƒ (ƒ*) (i.e., the degree of the lowest monomial of ƒ(ƒ*)) appearing in the left-hand side of (1) ((1)*).
Some Remarks Concerning (f, dn) and [F, dn] Summability Methods
Authors: Koch C. F. 
Published: December 2010

In this paper we note a simple connection between the (ƒ, dn) method of summability (defined by Smith (8)) and a composition of [F, dn] (defined by Jakimovski (4)) and Sonnenschein methods (9; 10). This connection is then used to supply some sufficient conditions for the regularity of (ƒ, dn) methods by using known regularity conditions for various [F, dn] and Sonnenschein methods. Finally, the connection is further exploited to obtain information about the Lebesgue constants for a certain class of [F, dn] methods by investigating related (ƒ, dn) methods.2. Definitions and the regularity theorem. The (ƒ, dn) method of summability is defined by Smith (8) essentially as follows. Letƒ(z) be a nonconstant entire function satisfying ƒ (l) = 1 and let {dn} be a sequence of complex numbers satisfying dt ≠ – 1, di ≠ – ƒ(0) (i ≧ 1).
Right Bol Quasi-Fields
Authors: Kallaher Michael J. 
Published: December 2010

We shall consider quasi-fields which satisfy the multiplicative Identity1.1(1.1) will be called the right Bol law and a quasi-field satisfying it will be called a right Bol quasi-held. Moufang quasi-fields, i.e., those satisfying the Moufang identity1.2were studied in (5). Quasi-fields satisfying the left Bol identity1.3were studied by Burn (3) and the author (6). Such quasi-fields are called Bol quasi-fields.Our investigation will parallel the investigations in (5; 6). In § 2 we derive necessary and sufficient conditions for a right Bol quasi-field to be an alternative division ring and also criteria for it to be a near-field. With this information we derive in §§ 3 and 4 new characterizations of Moufang planes similar to those in (5; 6).Loops satisfying (1.1) have been studied by Robinson (10). He calls such loops Bol loops.
The Width of a Module
Authors: Wichman Michael 
Published: December 2010

An R-module N is said to have finite width n if n is the smallest integer such that for any set of n + 1 elements of N, at least one of the elements is in the submodule generated by the remaining n. The width of N over R will be denoted by W(R, N).The notion of width was introduced by Brameret [2, p. 3605]. However, Cohen [3] investigated rings of finite rank, which, in the case that R is a local Noetherian domain, is equivalent to width (Proposition 1.6). He showed that finite width of R was both equivalent to R having Krull dimension one, and to R having the restricted minimum condition (Theorem 1.12).
Homotopy of Natural Transformations
Authors: Hardie K. A. 
Published: December 2010

Let C be a full subcategory of T, the category of based topological spaces and based maps, and let Cn be the corresponding category of n-tuples. Let S, T: Tn → T be covariant functors which respect homotopy classes and let u, v: S → T be natural transformations, u and v are homotopic inC, denoted u ≃ v(C), if uX ≃ vX: SX → TX (X ∈ Cn), that is to say, for every X ∈ C, uX and vX are homotopic (all homotopies are required to respect base points), u and v are naturally homotopic inC, denoted u ≃n v; (C), if there exist morphismssuch that, for every X ∈ C, utX is a homotopy from uX to vX and such that, for every t ∈ I, ut:S → T is a natural transformation.
A Generalization of Commutative and Alternative Rings
Authors: Kleinfeld Erwin , Kleinfeld Margaret Humm , Kosier Frank 
Published: December 2010

In [3] Schafer has defined generalized standard rings as rings satisfying the identities(1)(2)(3)and observed that these identities imply (y,y, (x, z)) = 0 and if the characteristic is not three, (x, y, x2) = 0. Schafer determined the structure of simple, finite-dimensional generalized standard algebras of characteristic not two or three by showing that they must be either commutative, Jordan, or alternative.Previously one of us [2] had studied accessible rings, which are defined by the identities (x,y,z) + (z,x,y) – (x,z,y) = 0 and ((w,x), y,z) = 0.
Cercles De Remplissage and Asymptotic Behaviour along Circuitous Paths
Authors: Gauthier P. M. 
Published: December 2010

In this paper we consider the value distribution of a meromorphic function whose behaviour is prescribed along a spiral. The existence of extremely wild holomorphic functions is established. Indeed a very weak form of one of our results would be that there are holomorphic functions (in the unit disc or the plane) for which every curve “tending to the boundary” is a Julia curve.The theorems in this paper generalize results of Gavrilov [7], Lange [9], and Seidel [11].I wish to express my thanks to Professor W. Seidel for his guidance and encouragement.2. Preliminaries. For the most part we will be dealing with the metric space (D, ρ) where D is the unit disc, |z| < 1, and ρ is the non-Euclidean hyperbolic metric on D. The chordal metric on the Riemann sphere will be denoted by x.
Normal and Canonical Representations in Free Products of Lattices†
Authors: Lakser H. 
Published: December 2010

In solving the word problem for free lattices, Whitman [4] showed that free lattices admit canonical representations, that is, of all polynomials over the generating set representing an element of the lattice, the polynomial of shortest length is unique up to commutativity and associativity. These well-defined shortest polynomials have proved very important in analyzing the internal structure of free lattices in detail; see, e.g., [5].Sorkin [3] proved that the free product of chains also admits canonical representations; these were exploited by Rolf [2]. In the above-mentioned paper, Sorkin also suggested that the free product of two copies of 22 does not admit canonical representations.
The Dimensions of Irreducible Representations of Linear Groups
Authors: King R. C. 
Published: December 2010

The theory of the relationship between the symmetric group on a symbols, Σa, and the general linear group in n-dimensions, GL(n), was greatly developed by Weyl [4] who, in this connection, made use of tensor representations of GL(n). The set of mixed tensorsforms the basis of a representation of GL(n) if all the indices may take the values 1, 2, …, n, and if the linear transformationis associated with every non-singular n × n matrix A. The representation is irreducible if the tensors are traceless and if the sets of covariant indices (α)a and contra variant indices (β)b themselves form the bases of irreducible representations (IRs) of Σa and Σb, respectively. These IRs of Σa and Σb may be specified by Young tableaux [μ]a and [v]b in the usual way [4].
Surfaces Embedded in M2 × S1
Authors: Jaco William 
Published: December 2010

In this paper we study incompressible and injective (see § 2 for definitions) surfaces embedded in M2 × S1, where M2 is a surface and S1 is the 1-sphere. We are able to characterize embeddings which are incompressible in M2 × S1 when M2 is closed and orientable. Namely, a necessary and sufficient condition for the closed surface F to be incompressible in M2 × S1, where M2is closed and orientable, is that there exists an ambient isotopy ht, 0 ≦ t ≦ 1, of M2 × S1onto itself so that either(i) there is a non-trivial simple closed curve J ⊂ M2 and h1(F) = J × S1, or(ii) p\h1(F) is a covering projection of h1(F) onto M2, where p is the natural projection of M2 × S1onto M2.
Sublattices and Initial Segments of the Degrees of Unsolvability
Authors: Thomason S. K. 
Published: December 2010

In this paper we shall prove that every finite lattice is isomorphic to a sublattice of the degrees of unsolvability, and that every one of a certain class of finite lattices is isomorphic to an initial segment of degrees.Acknowledgment. I am grateful to Ralph McKenzie for his assistance in matters of lattice theory.1. Representation of lattices. The equivalence lattice of the set S consists of all equivalence relations on S, ordered by setting θ ≦ θ’ if for all a and b in S, a θ b ⇒ a θ’ b. The least upper bound and greatest lower bound in are given by the ⋃ and ⋂ operations:
Subgroups of Conjugate Classes in Extensions
Authors: Burroughs John E. , Schafer James A. 
Published: December 2010

Often in various mathematical problems one encounters an extension B of the group G by the group π in which one wishes to extract certain information about B from information given in terms of G, π, the action of π on G, and the class of the extension in H2(π, centre G). An example of this type of problem is to determine some intrinsically defined subgroup of B, for instance the centre of B, given knowledge of the corresponding subgroup for G and π, and, of course, the usual information concerning the extension.In this paper we shall use the fact that any extension is congruent to a crossed product extension [2] to investigate a class of subgroups which naturally generalizes the notion of the centre. The definition of this class appears in § 3.1. Letby an extension of G by π.
On a Theorem of Heilbronn Concerning the Fractional Part of θn2
Authors: Liu Ming-Chit 
Published: December 2010

1. In 1948 Heilbronn [4] proved the following theorem.THEOREM H. For every real θ and every positive integer N, there is an integer n satisfying(1.1)whereis an arbitrarily small number,depends only on, and ‖t‖ means the distance from t to the nearest integer.The interest of the result (1.1) is that the inequality is uniform in θ, and is therefore analogous to the classical inequality of Dirichlet for the fractional part of θn. In this paper we shall prove the following theorem.THEOREM. For every real θ and every positive integer N, there is an integer n satisfying(1.2)where A is an absolute constant and.