bluuebunny/crossref_metadata_2025_split huggingface dataset:
A Maximum Principle for Bounded Harmonic Functions on Riemannian Spaces
Authors: Kwon Y. K. , Sario L. 
Published: December 2010

Harmonic functions with certain boundedness properties on a given open Riemann surfaceRattain their maxima and minima on the harmonic boundaryΔBofR.The significance of such maximum principles lies in the fact that the classification theory of Riemann surfaces related to harmonic functions reduces to a study of topological properties of Δ(cf. [11; 8; 3; 12].For the corresponding problem in higher dimensions we shall first show that the complement of ΔRwith respect to the Royden boundary ΓRof a Riemannian N-spaceRis harmonically negligible: given any non-empty compact subsetEof ΓR– ΔRthere exists an Evans superharmonic functionv,i.e., a positive continuous function onR* = R∪ΓR,superharmonic onR,withv= 0 on ΔR, v≡ ∞ onE,and with a finite Dirichlet integral overR.
On Positive Definite Functions over a Locally Compact Group
Authors: Price J. F. 
Published: December 2010

In this note we are concerned with several questions on positive definite functions over a Hausdorff locally compact group. The main result, Theorem A, gives some necessary and sufficient conditions for to be a positive definite function when μ is a (complex Radon) measure. In particular, is a positive definite function if and only if μ ∊ L2, and Theorem B then follows by giving a complete characterization of functions of the type , where f ∊ L2. Perhaps the most interesting aspect of these results is that they provide further examples of results over a non-abelian, non-compact group, which otherwise are simple consequences (with μ, a bounded measure in Theorem A) of the theorems of Plancherel and Bochner.Unless otherwise specified, all notation and definitions will follow [1;2]. The underlying group will always be G, a Hausdorff locally compact group with identity e, and with left Haar measure dx.
Perfect Mappings and Spaces of Countable Type
Authors: Vaughan J. E. 
Published: December 2010

In [1, p. 41, Theorem 3.10] Arhangel'skiï proved that the
perfect image of a completely regular space of countable type is of
countable type, and he asked [1, p. 60, problem 4] if a similar
result held for regular or Hausdorff spaces. In this paper, it is proved
that the perfect image of a space of countable type is of countable type,
provided that the image is Hausdorff or regular. An affirmative answer to
both of Arhangel'skiï's questions follows immediately from this.
Arhangel'skiï made use of the Stone-Čech compactification in the proof of
his result, but the proofs below are of a different nature.Let X be a topological space and let K ⊂
X. A collection of open sets
is called a base at K provided that for every open set
W ⊃ K there exists such that K ⊂ U ⊂
W. Clearly, we may assume that every member of
contains K.
Representation of certain Linear Operators in Hilbert
Space
Authors: Harvey Bernard Niel 
Published: December 2010

In this paper we represent certain linear operators in a space with
indefinite metric. Such a space may be a pair (H, B), where
H is a separable Hilbert space, B is a
bilinear functional on H given by
B(x, y) = [Jx, y], [,
] is the Hilbert inner product in H, and J
is a bounded linear operator such that J = J*
and J2 = I. If T is a linear operator
in H, then ‖T‖ is the usual operator norm.
The operator J above has two eigenspaces corresponding to
the eigenvalues + 1 and –1.In case the eigenspace in which J induces a positive
operator has finite dimension k, a general spectral theory
is known and has been developed principally by Pontrjagin [25],
Iohvidov and Kreĭn [13], Naĭmark [20], and
others.
A Remark on the Proof of a Theorem of Laufer and Tomber
Authors: Myung Hyo Chul 
Published: December 2010

In this note we give a correction to the proof of the following theorem [1, Theorem 2].THEOREM. Letbe a flexible, power-associative algebra, over an arbitrary algebraically closed field Ω of characteristic 0. Ifis a simple Lie algebra, thenis a simple Lie algebra isomorphic to.Step (i) of the proof, which proves that the Cartan subalgebra of is a nil subalgebra of , is incomplete. Assuming that is not a nil subalgebra of , there exists an idempotent e ≠ 0 in .
On Reductive Lie Admissible Algebras
Authors: Sagle Arthur A. 
Published: December 2010

A Lie admissible algebra is a non-associative algebra A such that A− is a Lie algebra where A− denotes the anti-commutative algebra with vector space A and with commutation [X, Y] = XY – YX as multiplication; see [1; 2; 5]. Next let L−(X): A− → A−: Y → [X, Y] and H = {L−(X): X ∊ A−}; then, since A− is a Lie algebra, we see that H is contained in the derivation algebra of A− and consequently the direct sum g = A − ⊕ H can be naturally made into a Lie algebra with multiplication [PQ] given by: P = X + L−(U), Q = Y + L−(V) ∊ g, thenand note that for any P, [PP] = 0 so that [PQ] = −[QP] and the Jacobi identity for g follows from the fact that A− is Lie.
Maximal Abelian Subgroups of the Symmetric Groups
Authors: Dixon John D. 
Published: December 2010

Our aim is to present some global results about the set of maximal abelian subgroups of the symmetric group Sn. We shall show that certain properties are true for “almost all” subgroups of this set in the sense that the proportion of subgroups which have these properties tends to 1 as n → ∞. In this context we consider the order and the number of orbits of a maximal abelian subgroup and the number of generators which the group requires.Earlier results of this kind may be found in the papers [1; 2; 3; 4; 5]; the papers of Erdös and Turán deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turán who posed the general problem: given a specific class of subgroups (e.g., the abelian subgroups or the solvable subgroups) of Sn, what kind of properties hold for almost all subgroups of the class?
On Additive Operators
Authors: Friedman N. A. , Tong A. E. 
Published: December 2010

Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh} which take their values in X**. If X is weakly sequentially complete, then {μh} can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4]):
A Best Possible Tauberian Theorem for the Collective Continuous Hausdorff Summability Method
Authors: Peterson G. E. 
Published: December 2010

The purpose of this paper is to prove that o(l/x) is the best possible Tauberian condition for the collective continuous Hausdorff method of summation. The analogue of this result for the collective (discrete) Hausdorff method is known [1, pp. 229, ff.; 7, p. 318; 8, p. 254]. Our method involves generalizing a well-known Abelian theorem of Agnew [2] to locally compact spaces and then applying the analogue for integrals of a result Lorentz obtained for series [6, Theorem 1].Let T and X denote locally compact, non compact, σ-compact Hausdorff spaces. Let T′ = T ∪ (∞) and X′ = X ∪ (∞) denote the onepoint compactifications of T and X, respectively. Let B(T) denote the set of locally bounded, complex valued Borel functions on T and let B∞(T) denote the bounded functions in B(T).
Characterizations of Modules
Authors: Fieldhouse David J. 
Published: December 2010

In this paper we use the Bourbaki [2] conventions for rings and modules. All rings are associative but not necessarily commutative and have a 1; all modules are unital.Bass [1] calls a ring A left perfect if and only if every left A -module has a projective cover, which he shows is equivalent to every flat left A -module being projective. Bass calls a ring A semi-perfect if and only if every finitely generated module has a projective cover and shows that this concept is leftright symmetric.We will define a ring A to be quasi-perfect if and only if every finitely generated flat left A -module is projective.An exercise [6, Exercise 10, p. 136] is given by Lambek to show that every semi-perfect ring is quasi-perfect.
On Primitive Solvable Linear Groups
Authors: Wright C. R. B. 
Published: December 2010

Let V be a vector space over the field K. A group G of K-linear transformations of V onto itself is primitive in case no proper nontrivial subspace of V is G-invariant and V cannot be written as a direct sum of proper subspaces permuted among themselves by G. Equivalently, G is primitive on V in case G is irreducible and is not induced from a proper subgroup.Suprunenko showed [3, Theorem 12, p. 28] that the n-dimensional general linear group GL(n, K) has a solvable primitive subgroup only if(1) there is a divisor, m, of n such that K has an extension field of degree m containing a primitive p-th root of 1 for each prime p dividing n/m.The main result of this note is the converse fact.
Multilinear Functions of Row Stochastic Matrices
Authors: Pierce Stephen 
Published: December 2010

In the study of inequalities, the cases of equality are often the most difficult and interesting part. The case of equality is, in some sense, a measure of the tightness of the inequality. In this paper, we generalize two inequalities of Brualdi and Newman [1, Theorems 3, 4], but the instances of equality are probably more interesting because of the variety of cases which can occur.Let A = (aij) be an n × n matrix. Define the permanent of A byWe say that A is row stochastic if all entries are non-negative and all row sums are 1. In [1], several inequalities involving permanents of row stochastic matrices were proved. In two of these results, the case of equality was not determined. We will generalize both of these results to a class of functions which includes the permanent, and determine all cases of equality.
Obstructions to Liftings in Commutative Squares
Authors: Pressman Irwin S. 
Published: December 2010

A commutative square (1) of morphisms is said to have a lifting if there is a morphism λ: B1 → A2 such that λϕ1 = α and ϕ2λ = β1Let us assume that we are working in a fixed abelian category . Therefore, ϕi will have a kernel “Ki” and a cokernel “Ci” for i = 1, 2. Let k : K1 → K2 and c: C1 → C2 denote the canonical morphisms induced by α and β.We shall construct a short exact sequence (s.e.s.)2using the data of (1). We shall prove that (1) has a lifting if and only if k = 0, c = 0, and (2) represents the zero class in Ext1(C1, K2). Furthermore, if (1) has one lifting, then the liftings will be in one-to-one correspondence with the elements of the set |Hom(G1, K2)|.
Isomorphic Subgroups of Finite p-Groups. II
Authors: Glauberman George 
Published: December 2010

Suppose that we are given an isomorphism ϕ between two subgroups of index p in a finite p-group P. Let N(ϕ) be the largest subgroup of P fixed by ϕ. By a result of Sims [2, Proposition 2.1], n(ϕ) is a normal subgroup of P. In [2], we showed that P/N(ϕ) has nilpotence class at most two if p = 2, and at most three if p is odd. We then applied this result to investigate certain cases of the following question. Suppose that P is contained in a finite group G and that some subgroup of index p in P is a normal subgroup of G. Let α be an automorphism of P. Then, does α fix some nonidentity normal subgroup of P that is normal in G?In this paper, we consider characteristic subgroups of P rather than normal subgroups.
On Screenability and Metrizability of Moore Spaces
Authors: Reed G. M. 
Published: December 2010

After showing that each screenable Moore space is pointwise paracompact and that the converse is not true, Heath in [4] asked for a necessary and sufficient condition for a pointwise paracompact Moore space to be screenable. In [12], Traylor asked for a necessary and sufficient condition for a pointwise paracompact Moore space to be metrizable. It is the purpose of this paper to provide such conditions, and to establish relationships between those conditions and metrization problems in Moore spaces.A Moore space S is a space (all spaces are T1) in which there exists a sequence G = (G1, G2, …) of open coverings of S, called a development, which satisfies the first three parts of Axiom I in [7]. The statement that a collection H of subsets of the space S is point finite (point countable) means that no point of S belongs to infinitely (uncountably) many elements of H.
Conjugate p-Subgroups of Finite Groups
Authors: Currano John J. 
Published: December 2010

Throughout this paper, let p be a prime, P be a p-group of order pt , and ϕ be an isomorphism of a subgroup R of P of index p onto a subgroup Q which fixes no non-identity subgroup of P, setwise. In [2, Lemma 2.2], Glauberman shows that P can be embedded in a finite group G such that ϕ is effected by conjugation by some element g of G. We assume that P is thus embedded. Then Q = P ∩ Pg. Let H = 〈P,Pg〉 and V = [H,Z(Q)], so Q ⊲ H and V ⊲ H.Let E(p) be the non-abelian group of order p3 which is generated by two elements of order p. Then E(p) is dihedral if p = 2 and has exponent p if p is odd. If p is odd, then E* (p) is defined in § 2 to be a particular group of order p6 and nilpotence class three.
Injectivity in Equational Classes of Algebras
Authors: Day Alan 
Published: December 2010

The concept of injectivity in classes of algebras can be traced back to Baer's initial results for Abelian groups and modules in [1]. The first results in non-module types of algebras appeared when Halmos [14] described the injective Boolean algebras using Sikorski's lemma on extensions of Boolean homomorphisms [19]. In recent years, there have been several results (see references) describing the injective algebras in other particular equational classes of algebras.In [10], Eckmann and Schopf introduced the fundamental notion of essential extension and gave the basic relations that this concept had with injectivity in the equational class of all modules over a given ring. They developed the notion of an injective hull (or envelope) which provided every module with a minimal injective extension or equivalently, a maximal essential extension. In [6] and [9], it was noted that these relationships hold in any equational class with enough injectives.
Limit Point Criteria for Differential Equations
Authors: Hinton Don 
Published: December 2010

For certain ordinary differential operators L of order 2n, this paper considers the problem of determining the number of linearly independent solutions of class L2[a, ∞) of the equation L(y) = λy. Of central importance is the operator0.1where the coefficients pi are real. For this L, classical results give that the number m of linearly independent L2[a, ∞) solutions of L(y) = λy is the same for all non-real λ, and is at least n [10, Chapter V]. When m = n, the operator L is said to be in the limit-point condition at infinity. We consider here conditions on the coefficients pi of L which imply m = n. These conditions are in the form of limitations on the growth of the coefficients.
S-Maximal Subgroups of πl(M3)
Authors: Feustel C. D. 
Published: December 2010

Let M be a compact, connected, irreducible 3-manifold. Let S be a closed, connected, 2-manifold other than the 2-sphere or projective plane. Let f be a map of S into M such thatSuppose for every closed, connected surface S1 and every map g:S1 → M such that(1) is an injection,(1) Then we shall say that the subgroup is a surface maximal or S-maximal subgroup of π1(M). We may also say that the map f is S-maximal.Let M be an irreducible 3-manifold which does not admit any embedding of the projective plane. Then we shall say that M is p2-irreducible. Throughout this paper all spaces will be simplicial complexes and all maps will be piecewise linear.
On Multipliers with Unconditionally Converging Fourier Series
Authors: Bachelis Gregory F. , Pigno Louis 
Published: December 2010

Let G be a compact abelian group with dual group Γ. For 1 ≦ p < ∞, 1 ≦ q < ∞, let denote the Banach space of complex-valued functions on Γ which are multipliers of type (p, q) and the subspace of compact multipliers.Grothendieck [10; 11] has proven that a function in LP(G), 1 ≦ p < 2, has an unconditionally converging Fourier series in LP(G) if and only if it is in L2(G), and Helgason [12] has proven that the derived algebra of LP(G), 1 ≦ p < 2, is L2(G). Using these results we show in § 2 that a multiplier of type (p, g), 1 ≦ p ≦ 2, 1 ≦ q ≦ 2, has an unconditionally converging Fourier series in if and only if it is in (Theorem 2.1), and that, for 1 ≦ p ≦ q ≦ 2, the derived algebra of is (Theorem 2.2). Statements equivalent to the above are also given.
δ-Continuous Selections of Small
Multifunction
Authors: Schirmer Helga 
Published: December 2010

A multifunction φ : X → Y from a
topological space X into a topological space
Y is a correspondence such that φ(x) is
a non-empty subset of Y for every x ∊ X. A
single-valued function f : X → Y is called
a selection of φ if f(x)
∊ φ(x) for all x ∊ X; it is called a
continuous selection if f is continuous. It is well-known
that not every semi-continuous or even continuous multifunction has a
continuous selection (see e.g. [4] for a survey on selection
theory).We investigate here some connections between multifunctions which are
'almost single-valued” and selections which are ‘almost continuous”.
Absolute Continuity of Some Vector Functions and
Measures
Authors: Knight William J. 
Published: December 2010

In the theory of vector valued functions there is a theorem which states
that if a function from a compact interval I into a normed
linear space X is of weak bounded variation, then it is of
bounded variation. The proof uses in a straightforward way the Uniform
Boundedness Principle (see [2, p. 60]). The present paper grew
from the question of whether an analogous theorem holds for absolutely
continuous functions. The answer is in the negative, and an example will be
given (Theorem 7). But it will also be shown that if X is
weakly sequentially complete (e.g. an Lp space, 1 ≦ p < ∞ ), then a weakly absolutely
continuous point function from / into X is absolutely
continuous. The method of proof involves the construction of a countably
additive set function in the standard Lebesgue-Stieltjes fashion.The paper is divided into three parts. In Section 1 extensions of finitely
additive, absolutely continuous set functions are carried out in an abstract
setting. Section 2 applies this to vector valued (point) functions on the
real line.
Integral Means and Zero Distributions of Blaschke
Products
Authors: Linden C. N. 
Published: December 2010

A sequence {zn} in D = {z: |z|
< 1} is a Blaschke sequence if and only ifIf 0 appears m times in {zn}
thenis the Blaschke product defined by {zn}. The set
of all Blaschke products will be denoted by . If B ∊ it is well-known that B is regular
in D, and |B(z, {zn})| < 1 when z ∊ D.For a given pair of values p in (0, ∞) and
q in [0, ∞) we denote by ℐ(p,
g) the class of all Blaschke products
B(z, {zn}) such thatas r → 1 — 0. In the case q ≦ max(p — 1,0) the
classes of functions B and ℐ(p,
q) are identical: this is a particular case of an elementary
theorem for functions subharmonic in a disc, the analogous theorem for
functions subharmonic in a half-plane appearing in [1],
Generalisation de la Formule de Riemann-Hurwitz
Authors: Van Quê Ngô 
Published: December 2010

Etant un polynôme Φ ∊ R[X1 … ,Xn] à n
indéterminés, Φ est dit de poids homogène m siavec m =Si E est un fibre vectoriel complexe de rang
n (i.e., n est la dimension sur
C de la fibre Ex) sur une
variété differentiate orientée compacte M de dimension
m, à tout polynôme Φ à n indéterminés
de poids homogène m, on associe un nombre de Chern:où [M] est le cycle fondamental défini par l'orientation de
M et Ci(E) la ième classe de
Chern de E. Dans le cas où le poids homogène de Φ est
différent de la dimension de M, on pose: Φ(E, M)
= 0.
Cyclic Maps From Suspensions to Suspensions
Authors: Hoo C. S. 
Published: December 2010

In [7] Varadarajan denned the notion of a cyclic map f
: A → X. The collection of all homotopy classes of such cyclic
maps forms the Gottlieb subset G(A, X) of
[A, X]. If A = S1 this reduces to the group G(X,
X0) of Gottlieb [5]. We show that a cyclic map
f maps ΩA into the centre of
ΩX in the sense of Ganea [4]. If
A and X are both suspensions, we then
show that if f : A → X
maps ΩA into the centre of ΩX, then
f is cyclic. Thus for maps from suspensions to
suspensions, Varadarajan's cyclic maps are just those maps considered by
Ganea. We also define G (Σ4, ΣX) in terms
of the generalized Whitehead product [1], This gives the
computations for G(Sn+k, Sn) in terms of Whitehead products in
π2n+k(Sn).We work in the category of spaces with base points and having the homotopy
type of countable CW-complexes. All maps and homotopies are
with respect to base points. For simplicity, we shall frequently use the
same symbol for a map and its homotopy class.
The Equality (A∩B)n=An∩Bnfor Ideals
Authors: Gilmer Robert , Grams Anne 
Published: December 2010

LetDbe an integral domain with identity, and letRbe a commutative ring. Ifnis a positive integer,Rwill be said tohave property (n), (n)′,or(n)″according asproperty(n):For anyx, y∊R,(x,y)n= (xn, yn).property(n)′: For anyx ∊ Rand any idealAofRsuch that xn∊ An, it follows thatx ∊ A.property(n)′: For any idealsA, BofR, (A∩ B)n= An∩ Bn.J. Ohm introduced property (n) in [7] in connection with the question: Ifn≦ 2 and ifDhas property (n), mustDbe a Prüfer domain? (The integral domainDwith identity is a Prüfer domain if each nonzero finitely generated ideal ofDis invertible; equivalently,DPis a valuation ring for each proper prime idealPofD.)Prior to Ohm's paper, it was known that ifDhas property (2) and ifDis integrally closed, thenDis Prüfer.
Maximal Quotient Rings and
S-Rings
Authors: Armendariz E. P. , McDonald Gary R. 
Published: December 2010

Throughout, we assume all rings are associative with identity and all
modules are unitary. See [7] for undefined terms and [3] for
all homological concepts.Let R be a ring, E(R) the
injective envelope of RR, and H =HomR(E(R),E(R)).
Then we obtain a bimodule RE(R)H. Let Q = HomH(E(R), E(R)). Q
is called the maximal left quotient ring of R. Q has the
property that if p, q ∈ Q, p ≠ 0, then
there exists r ∈ R such that
rp ≠ 0, rq ∈ R, i.e.,
Q is a ring of left quotients of R.A left ideal I of R is
dense if for every x,y ∈ R,x ≠ 0, there
exists r ∈ R such that rx
≠ 0, ry ∈ I. An alternate description of Q is Q
= {x ∈ E(RR) : (R :
x) is a dense left ideal of R{, where (R : x) =
{r ∈ R : rx ∈ R}.The left singular ideal of R is Zl(R) =
{r ∈ R : lR(r) is an essential left ideal of R},
where lR(r) = {x ∈ R : xr = 0}. If
Zl(R) = (0), then Q is a
left self-injective von Neumann regular ring [7, § 4.5]. Most of the
previous work on maximal left quotient rings has been done in this case.
Polynomial Rings with the Outer Product Property
Authors: Geramita A. V. 
Published: December 2010

In [3] Lissner defined a class of rings called outer
product rings, (OP-rings). These are commutative rings
R with identity for which every
exterior vector is
decomposable, i.e., with
.If we look only at those vectors whose
co-ordinates with respect to any basis of generate
the unit ideal in R and consider those rings
R for which all vectors of this type are decomposable,
we obtain the class of rings which have been referred to as Her
mite-rings (H-rings, see also Lissner
[3]). This class of H-rings evidently
contains the class of OP-rings.