bluuebunny/crossref_metadata_2025 huggingface dataset:
The Maximum Term and the Rank of an Entire Function
Authors: Sreenivasulu V.
Published: December 2010

1. For an entire function , let M(r, f), μ(r, f), and v(r, f) denote the maximum modulus, the maximum term, and the rank for |z\ = r, respectively. Also, let

and λ(r) the lower proximate order relative to log M(r, f). For the properties of the lower proximate order, we refer the reader to the paper by Shah (1).

2. We prove the following theorems.

THEOREM 1. For an entire function

where μ(r, f1) and M(r, f1) correspond to fl(z), the derivative of f(z), provided (n + l)Rn > nRn+1, when L(f) > 1.
A Presentation of the Groups PSL(2, p) with Three Defining Relations
Authors: Zassenhaus Hans J.
Published: December 2010

H. Behr and J. Mennicke (1) have proven that the group PSL(2, p) can be presented by the following system of generators and relations:

1

From this presentation, it follows that the three relations

2

for the same generators S and T suffice if p > 3, p ≠ 17. If p = 3, it is well known that the relations S3 = 1, T2 = 1, and (ST)3 = 1 define PSL(2, 3). For p = 2, the relations S3 = 1, T2 = 1, and (ST)2 = 1 define PSL(2, 2). For p = 17, the three relations

3

will suffice.

Indeed, the group G, with generators S, T and defining relations (2), contains the subgroup 〈Sp〉 in its centre.
On Arithmetic Properties of the Taylor Series of Rational Functions
Authors: Cantor David G.
Published: December 2010

Pólya (3) has shown that if bnis a sequence of algebraic integers and is a rational function, then so is . This result was generalized by Uchiyama (5) who showed that one may replace the assumption that the bn are algebraic integers by the assumption that the bn lie in a finitely generated submodule of the complex numbers, and by the author (1) who showed that if p is a non-zero polynomial with complex coefficients and if bn is a sequence of algebraic integers such that is a rational function, then so is . Our aim in this note is to give a common generalization of all of these theorems.
Abelian Groups with a Vanishing Homology Group
Authors: Schafer James A.
Published: December 2010

In this paper, we wish to characterize those abelian groups whose integral homology groups vanish in some positive dimension. We obtain a complete characterization provided the dimension in which the homology vanishes is odd; in fact, we prove that the only abelian groups which possess a vanishing homology group in an odd dimension are, up to isomorphism, subgroups of Qn, where Q denotes the additive group of rational numbers. The case of vanishing in an even dimension is much more complicated. We exhibit a class of groups whose homology vanishes in even dimensions and is otherwise very nice, namely the subgroups of Q/Z, and then show that unless we impose further restrictions, there exist abelian groups which possess the homology of subgroups of Q/Z without being isomorphic to a subgroup of Q/Z.
On the Isomorphism of Integral Group Rings. I
Authors: Sehgal Sudarshan K.
Published: December 2010

In this note we study the question of automorphisms of the integral group ring Z(G) of a finite group G. We prove that if G is nilpotent of class two, any automorphism of Z(G) is composed of an automorphism of G and an inner automorphism by a suitable unit of Q(G), the group algebra of G with rational coefficients. In § 3, we prove that if two finitely generated abelian groups have isomorphic integral group rings, then the groups are isomorphic. This is an extension of the classical result of Higman (2) for the case of finite abelian groups. In the last section we give a new proof of the fact that an isomorphism of integral group rings of finite groups preserves the lattice of normal subgroups. Other proofs are given in (1;4).
Linear Transformations on Grassmann Spaces
Authors: Westwick Roy
Published: December 2010

1. Let U denote an n-dimensional vector space over a field F and let Gnr denote the set of non-zero decomposable r-vectors of the Grassmann product space ΛrU. Let T be a linear transformation of ΛrU into itself which maps Gnr into itself. If F is algebraically closed, or if T is non-singular, then the structure of T is known. In this paper we show that if T is singular, then the image of ΛrU has a very special form with dimension equal to the larger of the integers r + 1 and n – r + 1. We give an example to show that this can occur.
Lower Bounds for the Essential Spectrum of Fourth-Order Differential Operators
Authors: Kreith Kurt
Published: December 2010

In this paper, we seek to determine the greatest lower bound of the essential spectrum of self-adjoint singular differential operators of the form

1

where 0 ≦ x < ∞. In the event that this bound is + ∞, our results will yield criteria for the discreteness of the spectrum of (1).

Such bounds have been established by Friedrichs (3) for Sturm-Liouville operators of the form

and our techniques will be closely related to those of (3). However, instead of studying the solutions of

2

directly, we shall exploit the intimate connection between the infimum of the essential spectrum of (1) and the oscillation properties of (2).
A Double-Centralizer Theorem for Simple Associative Algebras
Authors: Werner W. L.
Published: December 2010

Consider the following result.

PROPOSITION. Let D be a finite-dimensional central division algebra over a field F, and let Dn be the algebra (over F) of all n × n matrices with entries in D. Let A and B be in Dn, and suppose that BX = XB for every X in Dn such that XA = AX. Then B is a polynomial in A with coefficients in F.

The case D = F is a well-known classical result. Recently, the particular case where D is the algebra of real quaternions was established by Cullen and Carlson (2). In this note, the general proposition is proved by reduction to the classical case by way of tensor products.
Inequalities for some Monotone Matrix Functions
Authors: Marcus Marvin , Nikolai Paul J.
Published: December 2010

Let V denote a unitary vector space with inner product (x, y). A self-adjoint linear map T: V → V is positive (positive definite) if (Tx, x) ≧ 0 ((Tx, x) ≧ 0) for all x ≠ 0 in V. We write S ≧ T(S > T) if S and T are self-adjoint and S – T ≧ 0 (S – T > 0). If U is a unitary vector space, a function f: Hom(V, V) → Hom(U, U) is monotone idf S ≧ T implies that f(S) ≧ f(T). If both U and V are taken to be the n-dimensional unitary space Cn of n-tuples of complex numbers with standard inner product, then f is a monotone matrix junction, a notion introduced for a more restrictive class of functions by Löwner (3) which has important applications in pure and applied mathematics. For orientation we refer the reader to (1), where several interesting examples of monotone and related functions are displayed in detail.
Adjoint Abelian Operators on Banach Space
Authors: Stampfli J. G.
Published: December 2010

I. In the first part of this paper we introduce a new class of operators, mentioned in the title. It is easy to say that these are a generalization of self-adjoint operators for Hilbert space. This is deceptive since it implies that the definition of self-adjointness is forced into the unnatural setting of a Banach space. We feel that the definition of adjoint abelian preserves the obvious distinction between a space and its dual. Certain attractive properties of self-adjoint operators have already been singled out and carried over to Banach space. Specifically, we mention the notion of hermitian (see 17; 11), and spectral type operators (see 4). There is some comparison of these concepts later.
Some Lemmas on Interpolating Blaschke Products and a Correction
Authors: Kerr-Lawson A.
Published: December 2010

A Blaschke product on the unit disc,

where |c|= 1 and kis a non-negative integer, is said to be interpolatingif the
condition

C

is satisfied for a constant δ independent of m.A Blaschke product always belongs to the set I of inner functions; it has norm 1 and radial limits of modulus 1 almost everywhere. The most general inner function can be uniquely factored into a product BS,where Bis a Blaschke product and

for some positive singular measure μ(θ) on the unit circle. The discussion will be carried out in terms of the hyperbolic geometry on the open unit disc D,its metric

and its neighbourhoods N(x, ∈) = ﹛z′ ∈ D: Ψ(z, z′) < ∈ ﹜
On the Absolute Cesaro Summability of Negative Order of a Series Associated with the Conjugate Series of a Fourier Series
Authors: Mohanty R. , Ray B. K.
Published: December 2010

1. Definition. Let λ ≡ λ(ω) be continuous, differentiable, and monotonie increasing in (0, ∞) and let it tend to infinity as ω → ∞. A series an is summable |R, λ, r|, where r > 0, if

where A is a fixed positive number (6, Definition B).

Let f(t) be a periodic function with period 2π and Lebesgue integrable over (–π, π) and let

1.1

The series conjugate to (1.1), at t = x, is

1.2
On Maximal Regular Ideals and Identities in the Tensor Product of Commutative Banach Algebras
Authors: Lardy L. J. , Lindberg J. A.
Published: December 2010

Let A1and A2be commutative Banach algebras and A1 ⊙ A2 their algebraic tensor product over the complex numbers C.There is always a t least one norm, namely the greatest cross-norm γ (2), on A1 ⊙ A2 that renders it a normed algebra. We shall write A1 ⊗αA2 for the α-completion of A1⊙ A2when αis an algebra norm on A1⊙ A2.Gelbaum (2; 3), Tomiyama (9), and Gil de Lamadrid (4) have shown that for certain algebra norms α on A1⊙ A2 every complex homomorphism on A1 ⊙ A2 is α-continuous. In § 3 of this paper, we present a condition on an algebra norm α which is equivalent to the α-continuity of every complex homomorphism on A1⊙ A2.
Conilpotency and Weak Category
Authors: Hoo C. S.
Published: December 2010

Let ƒ : X → Y be a map and let e′: Y → ΩΣY be the usual embedding. Then we prove the following results.

Theorem 1. cat ƒ = cat(e′ƒ), w cat ƒ = w cat(e′ƒ) if Y is an H-space.

Theorem 2. conil ƒ = w Σ cat(e′ƒ) ≦ Σ w cat(e′ƒ) ≦ w cat(e′ƒ), where Σ the suspension functor. If we take X = Y and ƒ= lx, this result yields conil X ≦ w cat e', a result due to Ganea, Hilton, and Peterson(4).

Theorem 3. Suppose that Y is (m– 1)-connected and

Then conil ƒ = w Σ cat (e′ƒ) = Σ w cat(e′ƒ) = w cat(e′ƒ).
On the Decomposition of Groups
Authors: Hill Paul
Published: December 2010

The problem in which we are interested is the following. Call an additively written group G finitely decomposable if G = Σ Gi is the weak sum of finite groups Gi, Consider the following property.

Property P. Each subgroup of G having cardinality less than G is contained in a finitely decomposable direct summand of G.

Does Property P imply that G is finitely decomposable? We shall demonstrate that the answer is negative even in the commutative case. Our question is closely related to (1, Problem 5). In (4), an abelian group is called a Fuchs 5-group if every infinite subgroup of the group can be embedded in a direct summand of the same cardinality. The question of whether or not a Fuchs 5-group is in fact a direct sum of countable groups has been open for several years.
The Distribution of Primitive Roots
Authors: Elliott P. D. T. A.
Published: December 2010

Notation,p and q are generic symbols for prime numbers. N(H, p) denotes the number of primes q, not exceeding Hy which are primitive roots (mod p).

g(p) denotes the least positive primitive root (mod p).

g*(p) is the least prime primitive root (mod p).

v(m) denotes the number of distinct prime divisors of the integer m.

τk(m) is the number of ways of representing the integer m as the product of k integers, order being important.

π(x, k, r) is the number of primes p, not exceeding x, which satisfy p ≡ r(mod k); while π(x) denotes the total number of p ≦ x.

logm x denotes the mth iterated logarithmic function which is defined, when possible, by
Federer-Čech Couples
Authors: Dyer Micheal
Published: December 2010

In (5),I considered two-term conditions in π-exact couples, of which the exact couple of Federer (7) is an example. Let M(X, Y)be the space of all maps from X to Y with the compact-open topology. Our aim in this paper is to construct a π-exact couple , where Xis a finite-dimensional (in the sense of Lebesgue) metric space and , a certain (rather large) class of spaces. Specifically, is the class of all topological spaces Xwhich possess the following property (P).

(P) Let Y be a (possibly infinite) simplicial complex. There exists x0 ∈ X and y0 ∊ Y such that [X, x0]≃ [Y, y0].

In § 5 it will be seen that contains all CW complexes and all metric absolute neighbourhood retracts (ANR)s.
Axioms for Absolute Geometry. II
Authors: Rigby J. F.
Published: December 2010

In this paper I continue the process, begun in (2), of reducing and weakening the axioms of congruence needed for absolute geometry. The congruence axioms Cl*–C4*, C4**, and C5a–C5c (frequently referred to below) can all be found in (2) and will not be quoted again here. (This paper should be read in conjunction with (2); any attempt to make it self-contained would result in the repetition of large parts of (2).) The notation of (2) will be used throughout the paper.

The main result here is that axiom C5c is unnecessary. This is shown in § 1. In § 2 I discuss three other points arising from (2).

Note added in proof.Since writing this paper, I have constructed examples of (a) Archimedean planes satisfying Cl*-C4* in which not all points are isometric, (b) non-Archimedean planes satisfying Cl*-C4*but not C4**,and (c) one-dimensional geometries in which 2.1 (with “plane” replaced by “line“) is false.
Stone Lattices. I: Construction Theorems
Authors: Chen C. C. , Grätzer G.
Published: December 2010

Stone lattices were (named and) first studied in 1957 (5). Since then, a great number of papers have been written on Stone lattices and a very satisfactory theory evolved. Despite the fact that all chains with 0, 1 as well as all Boolean algebras are Stone lattices, it turns out that many of the nice theorems on Boolean algebras have analogues, in fact, generalizations for Stone lattices. To give just two examples: the characterization of Boolean algebras in terms of prime ideals (Nachbin (6)) is generalized in (5) (see also (9)); Stone's representation theory (8) is generalized in (4); see also (2).
Nil Subrings of Goldie Rings are Nilpotent
Authors: Lanski Charles
Published: December 2010

Herstein and Small have shown (1) that nil rings which satisfy certain chain conditions are nilpotent. In particular, this is true for nil (left) Goldie rings. The result obtained here is a generalization of their result to the case of any nil subring of a Goldie ring.

Definition. Lis a left annihilator in the ring R if there exists a subset S ⊂ R with L = {x∈ R|xS= 0}. In this case we write L= l(S). A right annihilator K = r(S) is defined similarly.

Definition. A ring R satisfies the ascending chain condition on left annihilators if any ascending chain of left annihilators terminates at some point. We recall the well-known fact that this condition is inherited by subrings.

Definition. R is a Goldie ring if R has no infinite direct sum of left ideals and has the ascending chain condition on left annihilators.
On the Automorphisms of Infinite Chevalley Groups
Authors: Humphreys J. E.
Published: December 2010

In (8, § 3.2) Steinberg proved the following result.

THEOREM. Let K be a finite field, G′ a simple Chevalley group (“normal type1”) over K. Then every automorphism of G’ is the composite of inner, graph, field, and diagonal automorphisms.

For the meaning of these notions, see (8). Our aim in this note is to indicate how the Theorem may be extended to arbitrary infinite fields K, provided we replace G′ by the group denoted G in (5) and Ĝ in (8). This amounts to proving the Theorem for automorphisms of G′ which are induced by automorphisms of G; when K is finite, Steinberg's results show that all automorphisms of G′ arise in this way. As Steinberg points out, the sole use made of the finiteness of K in his argument is in the proof of the following statement: Let U be the subgroup of G′ corresponding to the set of positive roots, and let σ be any automorphism of G′; then Uσ is conjugate to U in G′.
Graphs with Maximal Even Girth
Authors: Gewirtz A.
Published: December 2010

In this paper we examine the classGof simple undirected, connected graphs of diameterd> 1, girth2d,and for anygɛG, if a pair of nodes are at distancedfrom each other, then that pair of nodes is connected bytdistinct paths of lengthd, t> 1. (The girth ofgis the length of the smallest circuit ing.)

We establish, in § 2, that for allgɛG, gis regular.

We establish necessary conditions for the existence of elements ofG.IfgɛG,we adopt the notationg=g(d, t, v, n),wherevis the valence ofgandnis the number of nodes. It is of course possible forg, hɛG, g ≠ h,and for givend, t, v, nto have bothg(d, t, v, n)andh(d, t, v, n).

In particular, we show that ifd= 2,t≠ 2, 4 or 6, then there is at most a finite number of graphs with a particular giventvalue.
A Note about Locally Spherical Spheres
Authors: Eaton W. T.
Published: December 2010

A 2-sphere S in E3 is said to be locally spherical if for each point p in S and each ∈ > 0 there is a 2-sphere S' such that p Ç Int S”, S' ᴖ S is a continuum, and Diam S' < ∈. It is not known whether locally spherical spheres are tame; however, there are several partial results. Burgess (2) showed that S is tame if S' C\ S is a simple closed curve and Loveland (3) proved that S is tame if S can be side approximated missing the continuum S ᴖ S'. In this paper we demonstrate that S is tame if the continuum S P\ S' is irreducible with respect to separating S. This result is stated more precisely in Theorem 3. Theorem 2, which is used in the proof of Theorem 3, is a generalization of a theorem recently proved by Loveland (4).
Finite Linear Groups of Prime Degree
Authors: Wales David B.
Published: December 2010

If G is a finite group which has a faithful complex representation of degree nit is said to be a linear group of degree n. It is convenient to consider only unimodular irreducible representations. For n ≦ 4 these groups have been known for a long time. An account may be found in Blichfeldt's book (1). For n= 5 they were determined by Brauer in (4). In (4), many properties of linear groups of prime degree pwere determined for pa prime greater than or equal to 5.

In a forthcoming series of papers these results will be extended and the linear groups of degree 7 determined. In the first paper, some general results on linear groups of degree p, p≧ 7, will be given. These results will later be
applied to the prime p = 7.
A Note on the Construction of Projective Planes from Groups
Authors: Kirkpatrick P. B.
Published: December 2010

André (1) gave a construction for translation planes from abelian groups possessing “congruences” of subgroups. Schwerdtfeger (3) constructed the plane over a field F from the group of substitutions x → ax + b (a, b ∊ F; a ≠ 0). In this note we describe a construction (inspired by Schwerdtfeger's work), from groups, of planes which are duals of near-field planes.

If a plane is (l, m)-transitive (cf. 2, p. 67) for some pair of distinct lines l, m, then the central collineations ϕ with axis m and centre on I may be identified with the “proper” points (that is, points not on I or m) of the plane once an origin O is chosen (not on l or m):
On the Semisimplicity of Modular Group Algebras. II
Authors: Passman D. S.
Published: December 2010

Let G be a discrete group, let Kbe an algebraically closed field of characteristic p > 0 and let KGdenote the group algebra of Gover K.In a previous paper (2) I studied the Jacobson radical JKGof KGfor groups Gwith big abelian subgroups or quotient groups. It is therefore natural to next consider metabelian groups, and I do this here. The main result is as follows.

THEOREM 1. Let K be an algebraically closed field of characteristic p and let a group G have a normal abelian subgroup A with G/A abelian. Then JKG ≠ {0} if and only if G has an element g of order p such that the A-conjugacy class gA is finite and such that the group is periodic.

Note that since and G/Ais abelian, we do in fact have .
On the Isomorphism of Integral Group Rings. II
Authors: Sehgal Sudarshan K.
Published: December 2010

Let Z(G) denote the integral group ring of a group G. Let be the class of groups G with the property that for any isomorphism θ: Z(G) → Z(H), we have θ(g) = ±h, h ∈ H, for all g ∈ G. We study this class in § 2 and establish that it contains classes of torsion-free abelian groups, torsion abelian groups, and ordered groups.

In § 4, we prove the following result.

THEOREM. Let G be a group which contains a normal abelian subgroup A such that. Suppose that θ: Z(G) → Z(H) is an isomorphism such that θ(Δ(G, A)) = Δ(H, B) for a suitable normal subgroup B of H. Then G ≃ H. (Here Δ(G, A) is the kernel of the natural map Z(G) → Z(G/A).)

Jackson (3) and Whitcomb (6) proved the special case of this theorem when G is supposed to be finite metabelian. The lemmas needed are given in §3.
On the Cones Associated with Biorthogonal Systems and Bases in Banach Spaces
Authors: Mcarthur C. W. , Singer Ivan , Levin Mark
Published: December 2010

1. Let E be a Banach space (by this we shall mean, for simplicity, a real Banach space) and (xn,fn) ({xn} ⊂ E, {fn} ⊂ E*) a biorthogonal system, such that {fn} is total on E (i.e. the relations x ∈ E,fn(x) = 0, n = 1, 2, …, imply x = 0). Then it is natural to consider the cone

1

which we shall call “the cone associated with the biorthogonal system (xn,fn)”. In particular, if {xn} is a basis of E and {fn} the sequence of coefficient functional associated with the basis {xn}, this cone is nothing else but

2

and we shall call it “the cone associated with the basis {xn}”.
Reconstruction of Cacti
Authors: Geller Dennis , Manvel Bennet
Published: December 2010

Following the work of Kelly (8), Harary and Palmer (5), and Bondy (1) on the reconstruction of trees, and of Manvel (10) on the reconstruction of connected graphs with a single cycle, it was a natural step to attempt to solve the reconstruction problem for cacti. The solution of this problem, presented here, assumes both Kelly's Theorem and the result of Manvel in (10). Any definitions not given here can be found in (2).

Let graph G have point set V = {v1 v2, …, vp} and line set X = {x1, x2, …, xq}. For each vi ∈ V, Gi = G – vi is the maximal subgraph of G which does not contain vi and is formed by deleting vi and all lines incident with it from G.
Generalized Discrete Valuation Rings
Authors: Brungs H.-H.
Published: December 2010

Jategaonkar (5) has constructed a class of rings which can be used to provide counterexamples to problems concerning unique factorization in non-commutative domains, the left-right symmetry of the global dimension for a right- Noetherian ring and the transhnite powers of the Jacobson radical of a right- Noetherian ring. These rings have the following property:

(W) Every non-empty family of right ideals of the ring R contains exactly one maximal element.

In the present paper we wish to consider rings, with unit element, which satisfy property (W). This property means that the right ideals are inverse well-ordered by inclusion, and it is our aim to describe these rings by their order type. Rings of this kind appear as a generalization of discrete valuation rings in R; see (1; 2).

In the following, R will always denote a ring with unit element satisfying (W).
Euclidean Lie Algebras
Authors: Moody Robert V.
Published: December 2010

Our aim in this paper is to study a certain class of Lie algebras which arose naturally in (4). In (4), we showed that beginning with an indecomposable symmetrizable generalized Cartan matrix (A ij) and a field Φ of characteristic zero, we could construct a Lie algebra E((A ij)) over Φ patterned on the finite-dimensional split simple Lie algebras. We were able to show that E((A ij)) is simple providing that (A ij) does not fall in the list given in (4, Table). We did not prove the converse, however.

The diagrams of the table of (4) appear in Table 2. Call the matrices that they represent Euclidean matrices and their corresponding algebras Euclidean Lie algebras. Our first objective is to show that Euclidean Lie algebras are not simple.
Distinct Values of a Polynomial in Subsets of a Finite Field
Authors: Williams Kenneth S.
Published: December 2010

If A is a set with only a finite number of elements, we write |A| for the number of elements in A. Let p be a large prime and let m be a positive integer fixed independently of p. We write [pm] for the finite field with pm elements and [pm]′ for [pm] – {0}. We consider in this paper only subsets H of [pm] for which |H| = h satisfies

1.1

If f(x) ∈ [pm, x] we let N(f; H) denote the number of distinct values of y in H for which at least one of the roots of f(x) = y is in [pm]. We write d(d ≥ 1) for the degree of f and suppose throughout that d is fixed and that p ≧ p0(d), for some prime p0, depending only on d, which is greater than d.
Differentiable Montgomery-Samelson Fiberings with Finite Singular Sets
Authors: Antonelli Peter L.
Published: December 2010

In 1946 Montgomery and Samelson (11) introduced a generalization of the notion of a differentiable group action with one type of orbit besides fixed points. Such an object is essentially a locally trivial fibering except on a certain singular set over which fibres are pinched to points. In recent years there has been a fair amount of research on these MS-fiberings and similar singular fiberings. This paper is another effort in this direction. For a fairly complete bibliography of the literature, the reader should consult the references, and in particular, (5).

Let f: Mn → Sp, with Mn a closed connected n-manifold and Sp the unit p-sphere with standard differentiable structure, be the projection map of a smooth MS-fibering with finite non-empty singular set.
Notes on Locally Compact Connected Topological Lattices
Authors: Choe Tae Ho
Published: December 2010

I t was shown in (2) that if

(1) L is a locally compact connected topological lattice and if

(2) L is topologically contained in R2, the Euclidean plane, then each compact subset of L has an upper bound and a lower bound in L. I t was also asked whether this result could be proved without assuming condition (2). In this note, we show that this result continues to hold if condition (2) is weakened to: L is finite-dimensional.

In (11), it was shown that the centre of a compact topological lattice is totally disconnected. We shall prove t h a t this result is also true even in a locally compact, locally convex topological lattice with 0 and 1. This yields that any locally compact topological Boolean algebra is totally disconnected.
Regular Polytopes and Harmonic Polynomials
Authors: Flatto Leopold , Wiener Sister Margaret M.
Published: December 2010

In this paper we study the following problem originally proposed by Walsh (8). To determine the class of functions f(x) continuous in a given n-dimensional region R and having the property that the value of f(x) be equal to the average of f(x) over the vertices of all sufficiently small regular polytopes similar to a given one, which are centred at x. This problem has been studied by several mathematicians (1; 6; 8) and has been completely solved except for the four-dimensional regular polytopes {3, 4, 3}, {3, 3, 5}, {5, 3, 3} (see 3, p. 129, for the meaning of these symbols) and the n-dimensional cube. In each case, the class of functions is identical with a class of harmonic polynomials which can be specified. In § 2, we solve the problem for the four-dimensional figures, thus leaving the problem open only for the n-dimensional cube.
Semigroups of Operators in C(S)
Authors: Sentilles F. Dennis
Published: December 2010

Our study in this paper is two-fold: One is that of a semigroup of linear operators on the space C(S) of bounded continuous functions on a locally compact Hausdorff space S, while the other is that of a transition function of measures in the Banach space M(S) of bounded regular Borel measures on S. It will be seen that an informative and essentially non-restrictive theory of the former can be obtained when C(S) is given the strict topology rather than the usual supremum norm topology and that, in this setting, the natural relationship between semigroups and transition functions obtained when S is compact is maintained, essentially because the dual of C(S) with the strict topology is M(S).
On the Nørlund Summability of a Class of Fourier Series
Authors: Badri N. Sahney
Published: December 2010

1. Our aim in this paper is to determine a necessary and sufficient condition for N∅rlund summability of Fourier series and to include a wider class of classical results. A Fourier series, of a Lebesgue-integrable function, is said to be summable at a point by N∅rlund method (N, pn), as defined by Hardy [1], if pn → Σpn → ∞, and the point is in a certain subset of the Lebesgue set. The following main results are known.
Residually Finite Rings
Authors: Chew Kim Lin , Lawn Sherry
Published: December 2010

Throughout this paper a ring will always be an associative, not necessarily commutative ring with an identity. It is tacitly assumed that the identity of a subring coincides with that of the whole ring. A ring R is said to be residually finite if it satisfies one of the following equivalent conditions:

(1) Every non-zero ideal of R is of finite index in R;

(2) For each non-zero ideal A of R, the residue class ring R/A is finite;

(3) Every proper homomorphic image of R is finite.

The class of residually finite rings is large enough to merit our investigation. All finite rings and all simple rings are trivially residually finite. Other residually finite rings are said to be proper.
On Absolute Summability by Riesz and Generalized Cesàro Means. II
Authors: Körle H.-H.
Published: December 2010

1. We will use the terminology of part I [9], including the general assumptions of [9, § 1]. In that paper we had proved that |R, λ, κ| = |C, λ, κ| in case that κ is an integer. Now, we turn to non-integral orders κ.

As to ordinary summation, the following inclusion relations (in the customary sense; see [9, end of § 1]) for non-integral κ have been established so far. (Since we are comparing Riesz methods of the same type λ and order κ only, (R, λ, κ) is written (R), etc., for the moment.) (R) ⊆ (C) is a result by Borwein and Russell [2]. (C) ⊆ (R) was proved by Jurkat [3] in the case 0 < κ < 1, and, after Borwein [1], it holds in the case 1 < κ < 2 if

(1)

(2)

(i.e. decreases in the wide sense).
Extensions of Certain Maps to Automorphisms of m
Authors: Berg I. D.
Published: December 2010

In this paper we consider Banach space automorphisms of m, the space of bounded sequences, which map c, the space of convergent sequences, into itself. In particular, we consider the problem of determining which maps from C0, the space of sequences converging to 0, to c can be extended to such automorphisms.

The origin of this note lies in an incorrect conjecture of mine. If the automorphism T: m → m is given by a matrix, that is, a sequence of elements of ll, and if T is conservative, that is, T(c) ⊂ c, then T(c) = c. That is, T restricted to c is an automorphism of c. We had hoped this would hold even if T were not a matrix. We can see, for example, that if the conservative automorphism T is bounded on the unit cube of m by 1 and ρ, where , then T(c) = c. However, in general it is possible for a conservative automorphism of m to map c properly into c.
On the Radius of Curvature for Convex Analytic Functions
Authors: Eenigenburg Paul
Published: December 2010

Definition 1.1. Let be analytic for |z| < 1. If ƒ is univalent, we say that ƒ belongs to the class S.

Definition 1.2. Let ƒ ∈ S, 0 ≦ α < 1. Then ƒ belongs to the class of convex functions of order α, denoted by Kα, provided

(1)

and if > 0 is given, there exists Z0, |Z0| < 1, such that

Let ƒ ∈ Kα and consider the Jordan curve ϒτ = ƒ(|z| = r), 0 < r < 1. Let s(r, θ) measure the arc length along ϒτ; and let ϕ(r, θ) measure the angle (in the anti-clockwise sense) that the tangent line to ϒτ at ƒ(reiθ) makes with the positive real axis.
A Problem of Gelfand on Rings of Operators and Dynamical Systems
Authors: Kallman Robert R.
Published: December 2010

Let G be a separable locally compact group (separable in the sense that the topology of G has a countable base). Let S be a standard Borel space on which G acts on the right such that:

(1) s · g1g2 = (s · g1) · g2;

(2) s · e = s;

(3) (s, g) → s · g is a Borel function from S × G to S.

If μ is a Borel measure on S, let μg be the Borel measure on S defined by μg(E) = μ(E · g).

Let μ be a Borel measure on S which is quasi-invariant under the action of G; i.e., μg and μ are absolutely continuous (g ∈ G). The triple (G, S, μ) is called a dynamical system [11; 8].

Consider the following general problem. Let (G, S, μ) be a dynamical system.
On Direct Products of Abelian Groups
Authors: Irwin John M. , O'Neill John D.
Published: December 2010

In this paper we investigate the properties of the product (or complete direct sum) of torsion Abelian groups. The chief results concern products of Abelian primary groups (p-groups). Given a set of p-groups, [Gλ], over an index set Λ, the product of these groups is written λλ∈ΔGλ, the torsion subgroup of the product of these p-groups is written T[λGλ], and the discrete direct sum of the p-groups is written Σ Gλ.

Definition. Σ Gλ is said to be an essentially bounded decomposition if and only if there exists an integer M > 0 such that MGλ = 0 for all but a finite number of Gλs; otherwise the decomposition is essentially unbounded.
On a Conjecture Concerning Semigroup Homomorphisms
Authors: Plemmons R. J.
Published: December 2010

In this paper we settle (with a counterexample) the question raised by Clifford and Preston in [2, p. 275], concerning maximal group homomorphic images of semigroups. We also consider the question in a more general context and characterize all such examples. The notation and definitions follow [1; 2].

By a type of semigroups we mean a class of semigroups, closed under isomorphisms and containing the one-element semigroup. If S is any semigroup and is a type, then a semigroup S* is defined, in [1, p. 18], to be a maximal homomorphic image of S having type if

(i) ,

(ii) S* is a homomorphic image of S, and

(iii) whenever and T is a homomorphic image of S, then there exists a homomorphism from S* onto T.
Screening Properties of the Subbase of all Closed Connected Subsets of a Connectedly Generated Space
Authors: Hursch J. L.
Published: December 2010

In [1] de Groot has introduced the notation “connectedly generated” (or eg) for those spaces in which the closed connected sets form a subbase for the topology. He pointed out that these are the semi-locally connected spaces of Whyburn. See [5; 6].

If X is cg, then, since X is closed, X is the union of a finite number of closed connected sets and, thus, has only a finite number of components. If p is any point in a eg space, and Nv is any neighbourhood of p, then the complement of Nv may be covered by a finite number of closed connected sets, none of which contain p.

In [1] and in [2] the concept of “screening” is introduced and shown to be usefully related to local connectedness and construction of compactifications for completely regular spaces. We review this concept in § 2.
A Mapping Problem and Jp-Index. I
Authors: Wakae Masami , Hamara Oma
Published: December 2010

Indices of normal spaces with countable basis for equivariant mappings have been investigated by Bourgin [4; 6] and by Wu [11; 12] in the case where the transformation groups are of prime order p. One of us has extended the concept to the case where the transformation group is a cyclic group of order pt and discussed its applications to the Kakutani Theorem (see [10]). In this paper we will define the Jp-index of a normal space with countable basis in the case where the transformation group is a cyclic group of order n, where n is divisible by p. We will decide, by means of the spectral sequence technique of Borel [1; 2], the Jp-index of SO(n) where n is an odd integer divisible by p. The method used in this paper can be applied to find the Jp-index of a classical group G whose cohomology ring over Jp has a system of universally transgressive generators of odd degrees.
Central Idempotent Measures on Unitary Groups
Authors: Rider Daniel
Published: December 2010

Let G be a locally compact group and M(G) the space of finite regular Borel measures on G. If μ and v are in M(G), their convolution is defined by

Thus, if f is a continuous bounded function on G,

μ is central if μ(Ex) = μ(xE) for all x ∈ G and all measurable sets E. μ is idempotent if μ * μ = μ.

The idempotent measures for abelian groups have been classified by Cohen [1]. In this paper we will show that for a certain class of compact groups, containing the unitary groups, the central idempotents can be characterized. The method consists of showing that, in these cases, the central idempotents arise from idempotents on abelian groups and applying Cohen's result.
A Set of Plane Measure Zero Containing all Finite Polygonal Arcs
Authors: Ward D. J.
Published: December 2010

We say a (plane) set A contains all sets of some type if, for each B of type , there is a subset of A that is congruent to B. Recently, Besicovitch and Rado [3] and independently, Kinney [5] have constructed sets of plane measure zero containing all circles. In these papers it is pointed out that the set of all similar rectangles, some sets of confocal conies and other such classes of sets can be contained in sets of plane measure zero, but all these generalizations rely in some way on the symmetry, or similarity of the sets within the given type.

In this paper we construct a set of plane measure zero containing all finite polygonal arcs (i.e., the one-dimensional boundaries of all polygons with a finite number of sides) with slightly stronger results if we restrict our attention to k-gons for some fixed k.
Embedding Theorems for Countable Groups
Authors: McCool James
Published: December 2010

A group P is said to be a CEF-group if, for every countable group G, there is a factor group of P which contains a subgroup isomorphic to G. It was shown by Higman, Neumann, and Neumann [5] that the free group of rank two is a CEF-group. More recently, Levin [6] proved that if P is the free product of two cyclic groups, not both of order two, then P is a CEF-group. Later, Hall [3] gave an alternative proof of Levin's result.

In this paper we give a new proof of Levin's result (Theorem 2). The proof given yields information about the factor group H of P in which a given countable group G is embedded; for example, if G is given by a recursive presentation (this concept is denned in [4]), then a recursive presentation is obtained for H, and certain decision problems (in particular, the word problem) are solvable for the recursive presentation obtained for H if and only if they are solvable for the given recursive presentation of G.
On Metanilpotent Varieties of Groups
Authors: Gupta Narain
Published: December 2010

Let denote the variety of all groups which are extensions of a nilpotent-of-class-c group by a nilpotent-of-class-d group, and let denote the variety of all metabelian groups. The main result of this paper is the following theorem.

THEOREM. Let be a subvariety of which does not contain . Then every -group is an extension of a group of finite exponent by a nilpotent group by a group of finite exponent. In particular, a finitely generated torsion-free -group is a nilpotent-by-finite group.

This generalizes the main theorem of Ŝmel′kin [4], where the same result is proved for subvarieties of , where is the variety of abelian groups. See also Lewin and Lewin [2] for a related discussion.
On Baer Involutions of Finite Projective Planes
Authors: Cofman Judita
Published: December 2010

1. An involution of a projective plane π is a collineation X of π such that λ2 = 1. Involutions play an important röle in the theory of finite projective planes. According to Baer [2], an involution λ of a finite projective plane of order n is either a perspectivity, or it fixes a subplane of π of order in the last case, λ is called a Baer involution.

While there are many facts known about collineation groups of finite projective planes containing perspectivities (see for instance [4; 5]), the investigation of Baer involutions seems rather difficult. The few results obtained about planes admitting Baer involutions are restricted only to special cases. Our aim in the present paper is to investigate finite projective planes admitting a large number of Baer involutions. It is known (see for instance [3, p. 401]) that in a finite Desarguesian projective plane of square order, the vertices of every quadrangle are fixed by exactly one Baer involution.
Equational Classes of Distributive Pseudo-Complemented Lattices
Authors: Lee K. B.
Published: December 2010

A pseudo-complemented lattice is a lattice L with zero such that for every a ∊ L there exists a* ∊ L such that, for all x ∊ L, a ∧ x = 0 if and only if x ≦ a*. a* is called a pseudo-complement of a. It is clear that for each element a of a pseudo-complemented lattice L, a* is uniquely determined by a. Thus * can be regarded as a unary operation on L. Moreover, each pseudo-complemented lattice contains the unit, namely 0*. It follows that every pseudo-complemented lattice L can be regarded as an algebra (L; (∧, ∨, *, 0, 1)) of type (2, 2, 1, 0, 0). In this paper, we consider only distributive pseudo-complemented lattices. For simplicity, we call such a lattice a p-algebra.
Fixed Points of Automorphisms of Compact Riemann Surfaces
Authors: Moore M. J.
Published: December 2010

In his fundamental paper [3], Hurwitz showed that the order of a group of biholomorphic transformations of a compact Riemann surface S into itself is bounded above by 84(g – 1) when S has genus g ≧ 2. This bound on the group of automorphisms (as we shall call the biholomorphic self-transformations) is attained for Klein's quartic curve of genus 3 [4] and, from this, Macbeath [7] deduced that the Hurwitz bound is attained for infinitely many values of g.

After genus 3, the next smallest genus for which the bound is attained is the case g = 7. The equations of such a curve of genus 7 were determined by Macbeath [8] who also gave the equations of the transformations. The equations of these transformations were found by using the Lefschetz fixed point formula. If the number of fixed points of each element of a group of automorphisms is known, then the Lefschetz fixed point formula may be applied to deduce the character of the representation given by the group acting on the first homology group of the surface.
On a Generalization of Alternative Rings
Authors: Morgan Raymond V.
Published: December 2010

Bruck and Kleinfeld [3] proved that any alternative ring with characteristic prime to 2 must satisfy the identity

where the associator (x,y,z) is defined by (x, y, z) = (xy)z – x(yz) and . Linearization of the identity (x2, y, z) = 2x · (x, y, z) yields for characteristic prime to 2 an equivalent identity

(1)

Using the right alternative law (x, y, z) = –(y, x, z) and the flexible law (x, y, z) = –(z, y, x) which is satisfied in any alternative ring we obtain

(2)

and

(3)
Weak Normality and Related Properties
Authors: Ball Eugene S.
Published: December 2010

In [5], Zenor stated the definition of weakly normal. In the main, since weak normality does not imply either normality or regularity, various properties related to either normality or regularity will be considered in the context of weak normality.

Throughout this paper the word “space” will mean topological space. The closure of a point set M will be denoted by cl(M). The closure of a point set M with respect to the subspace K will be denoted by cl(M, K).

Definition 1. A space S is weakly normal provided that if is a monotonically decreasing sequence of closed sets in S with no common part and H is a closed set in S not intersecting H1, then there is a positive integer N and an open set D such that HN ⊂ D and cl(D) does not intersect H.
A Note on a Generic Hyperplane Section of an Algebraic Variety
Authors: Kuan Wei-Eihn
Published: December 2010

Let V be an irreducible algebraic variety of dimension > 1 defined over a field k in an affine n-space over k, and let H be the generic hyperplane defined by u0 + u1X1 + … + unXn = 0, where u0, u1, …, un are indeterminates over k. It is well known that:

(1) if V is normal over k, then V ∩ H is normal over k(u0, …, un) (see [6]), and

(2) if P is in the intersection V ∩ H, then P is absolutely simple on V ∩ H over k(u0, …, un) if and only if P is absolutely simple on V over k (see [2; 5]).

In this paper we prove:

(1′) if V is factorial over k, then V ∩ H is also factorial over k(u0, …, un) (Theorem 3), and

(2′) if P is in V ∩ H, then P is normal on V ∩ H over k(u0, …, un) if and only if P is normal on V over k (Theorem 2).
Gentle Perturbations of with Application to
Authors: Derzko N. A.
Published: December 2010

The theory of gentle perturbations was introduced by Friedrichs [3] as a tool to study the perturbation theory of the absolutely continuous spectrum of a self-adjoint operator H0 and developed in an abstract form by Rejto [7; 8]. Two examples of gentle structures are well knowTn. In the first of these, the gentle operators have Hölder continuous complex or operator-valued kernels, and in the second, the kernels are Fourier transforms of L1 functions [4].

The gentle structure has traditionally been verified in the case when H0 is in its spectral representation, that is, when H0 is the simple differentiation operator. This is not the natural setting for the second example mentioned above where one should consider the simple differentiation operator in a suitable L2-space and perturbations with L1 kernels.
Finite Subsghemes of Group Schemes
Authors: Shatz Stephen S.
Published: December 2010

If G is an ordinary group and H is a non-empty subset of G, then there are two elementary criteria for H to be a subgroup of G. The first and more general is that the mapping H × H → G × G → G, via 〈x, y〉 ⟼ xy–1 factor through H. The second is that H be finite and closed under multiplication.

In the category of group schemes, if one writes down the hypotheses for the first criterion in diagram form, one can supply the proof by a suitable translation of the classical arguments. The only point that causes any difficulty whatsoever is that one must assume that the structure morphism πH: H → S (S is the base scheme) is an epimorphism in order to factor the identity section through H. The second criterion is also true for group schemes under a mild finite presentation hypothesis. It is our aim to provide a simple proof for the following theorem.
Commutative Self-Injective Rings
Authors: Singh Surjeet , Wasan Kamlesh
Published: December 2010

All rings considered here are commutative containing at least two elements,
but may not have identity. A ring R is said to be
selfinjective if R as an R-module is
injective. A ring R is said to be pre-selfinjective if
every proper homomorphic image of R is self-injective
[9]. Study of pre-self-injective rings was initiated by Levy
[10], who established a characterization of Noetherian
pre-self-injective rings with identity in terms of other well-known types of
rings. Recently Klatt and Levy [9] have characterized all
pre-self-injective rings with identity. In this paper we are mainly
interested in Noetherian rings. For the sake of convenience we shall call a
pre-self-injective ring an (I)-ring. A ring R
will be said to be a (PMI)-ring if for each proper prime ideal
P with P2 ≠ 0, the ring R/P2 is self-injective. Clearly, an (I)-ring is a (PMI)-ring.
A Generalization of Final Rank of Primary Abelian
Groups
Authors: Cutler Doyle O. , Dubois Paul F.
Published: December 2010

Let G be a p-primary Abelian group. Recall
that the final rank of G is infn∈ω{r(pnG)}, where
r(pnG) is the rank of
pnG and ω is the first
limit ordinal. Alternately, if Γ is the set of all basic subgroups of
G, we may define the final rank of G by supB∈Γ
{r(G/B)}. In fact, it
is known that there exists a basic subgroup B of
G such that
r(G/B) is equal to the
final rank of G. Since the final rank of G
is equal to the final rank of a high subgroup of G plus the
rank of pωG, one could obtain the same
information if the definition of final rank were restricted to the class of
p-primary Abelian groups of length
ω.