415 results
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2. SOME NORMALITY CRITERIA AND A COUNTEREXAMPLE TO THE CONVERSE OF BLOCH’S PRINCIPLE
- Author
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Kuldeep Singh Charak and S.D. Sharma
- Subjects
Pure mathematics ,Distribution (number theory) ,Mathematics::Complex Variables ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Holomorphic function ,01 natural sciences ,010101 applied mathematics ,Converse ,0101 mathematics ,Differential (infinitesimal) ,Value (mathematics) ,Normality ,Mathematics ,Meromorphic function ,Counterexample ,media_common - Abstract
In this paper we continue our earlier investigations on normal families of meromorphic functions\cite{CS2}. Here, we prove some value distribution results which lead to some normality criteria for a family of meromorphic functions involving the sharing of a holomorphic function by more general differential polynomials generated by members of the family and get some recently known results extended and improved. In particular, the main result of this paper leads to a counterexample to the converse of Bloch's principle.
- Published
- 2016
3. DISCRETENESS CRITERIA FOR MÖBIUS GROUPS ACTING ON II
- Author
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Xian-Tao Wang and Liu-Lan Li
- Subjects
Pure mathematics ,Inequality ,Group (mathematics) ,General Mathematics ,media_common.quotation_subject ,Ambiguity ,Fixed point ,Relation (history of concept) ,Mathematics ,media_common - Abstract
Jørgensen’s famous inequality gives a necessary condition for a subgroup of PSL(2,ℂ) to be discrete. It is also true that if Jørgensen’s inequality holds for every nonelementary two-generator subgroup, the group is discrete. The sufficient condition has been generalized to many settings. In this paper, we continue the work of Wang, Li and Cao (‘Discreteness criteria for Möbius groups acting on $\overline {\mathbb {R}}^n$’, Israel J. Math.150 (2005), 357–368) and find three more (infinite) discreteness criteria for groups acting on $\overline {\mathbb {R}}^n$; we also correct a linguistic ambiguity of their Theorem 3.3 where one of the necessary conditions might be vacuously fulfilled. The results of this paper are obtained by using known results regarding two-generator subgroups and a careful analysis of the relation among the fixed point sets of various elements of the group.
- Published
- 2009
4. THE SET OF SOLUTIONS OF INTEGRODIFFERENTIAL EQUATIONS IN BANACH SPACES
- Author
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Aneta Sikorska-Nowak, Donal O'Regan, and Ravi P. Agarwal
- Subjects
Discrete mathematics ,Set (abstract data type) ,Pure mathematics ,General Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Banach space ,Existence theorem ,Integral equation ,Mathematics - Abstract
In this paper, we first prove an existence theorem for the integrodifferential equation(*)wheref,k,xare functions with values in a Banach spaceEand the integral is taken in the sense of Henstock–Kurzweil–Pettis. In the second part of the paper we show that the setSof all solutions of the problem (*) is compact and connected in (C(Id,E),ω), where$I_{d} \subset I_{a} $.
- Published
- 2008
5. A new system of variational inclusions with (H, η)-monotone operators
- Author
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Jianrong Huang and Jian-Wen Peng
- Subjects
symbols.namesake ,Pure mathematics ,Monotone polygon ,General Mathematics ,Resolvent operator ,Convergence (routing) ,Hilbert space ,symbols ,Uniqueness ,Operator theory ,Mathematics - Abstract
In this paper, We introduce and study a new system of variational inclusions involving(H, η)-monotone operators in Hilbert spaces. By using the resolvent operator method associated with (H, η)-monotone operators, we prove the existence and uniqueness of solutions and the convergence of some new three-step iterative algorithms for this system of variational inclusions and its special cases. The results in this paper extends and improves some results in the literature.
- Published
- 2006
6. Parallel metrics and reducibility of the holonomy group
- Author
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Richard Atkins
- Subjects
Combinatorics ,Pure mathematics ,Group (mathematics) ,Computer Science::Information Retrieval ,General Mathematics ,Holonomy ,Mathematics::Differential Geometry ,Mathematics - Abstract
In this paper we investigate the relationship between the existence of parallel semi-Riemannian metrics of a connection and the reducibility of the associated holonomy group. The question as to whether the holonomy group necessarily reduces in the presence of a specified number of independent parallel semi-Riemannian metrics is completely determined by the the signature of the metrics and the dimension d of the manifold, when d ≠ 4. In particular, the existence of two independent, parallel semi-Riemannian metrics, one of which having signature (p,q) with p ≠ q, implies the holonomy group is reducible. The (p,p) cases, however, may allow for more than one parallel metric and yet an irreducible holonomy group: for n = 2m, m ≥ 3, there exist connections on Rn with irreducible infinitesimal holonomy and which have two independent, parallel metrics of signature (m,m). The case of four-dimensional manifolds, however, depends on the topology of the manifold in question: the presence of three parallel metrics always implies reducibility but reducibility in the case of two metrics of signature (2,2) is guaranteed only for simply connected manifolds. The main theorem in the paper is the construction of a topologically non-trivial four-dimensional manifold with a connection that admits two independent metrics of signature (2,2) and yet has irreducible holonomy. We provide a complete solution to the general problem.
- Published
- 2006
7. Finite presentability of some metabelian Hopf algebras
- Author
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Dessislava H. Kochloukova
- Subjects
Pure mathematics ,Group action ,Quantum group ,General Mathematics ,Lie algebra ,Representation theory of Hopf algebras ,Lie theory ,Abelian group ,Quasitriangular Hopf algebra ,Hopf algebra ,Mathematics - Abstract
The purpose of this paper is to try to unite some existing methods used in the classification results of metabelian Lie algebras and metabelian discrete groups of homological type FP2 via the language of Hopf algebras. This sheds more light on the similarities between the Lie and group cases and explains partially the differences. Still some of the results in the group case have homotopical flavour, using methods from covering spaces to establish that having homological type FP2 imposes strong condition on the first Σinvariant of the group ([4]). These methods do not have a purely algebraic counterpart. The Lie case was treated in [5, 6] with algebraic methods, and a Lie invariant (with a valuation flavour) for metabelian Lie algebras was proposed. This plays the same role in the Lie theory as the Bieri-Strebel Σ-invariant for metabelian groups. In this paper we do not suggest a new invariant but establish that the main result of [5] holds for some metabelian Hopf algebras. It is interesting to note that in both the Lie and group cases calculations with the second homology group of Abelian objects (Lie algebras or Abelian groups) viewed as modules over a commutative ring via the corresponding diagonal action was always quite helpful. The definition of the diagonal Lie and group actions can be united via the comultiplication map of Hopf algebras, and this was the starting point of our considerations. We study Hopf algebras H = U(L)#kG over a field k, that is, smash products of universal enveloping algebras U(L) of Lie algebras L over k by group rings kG, where G acts via conjugation on L and write X for the category of such Hopf algebras. This category is quite important. If char(k) = 0 it coincides with the category of cocommutative
- Published
- 2005
8. A characterisation of Hilbert spaces via orthogonality and proximinality
- Author
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Fathi B. Saidi
- Subjects
Pure mathematics ,Hilbert manifold ,Computer Science::Information Retrieval ,General Mathematics ,Mathematical analysis ,Hilbert space ,Banach space ,Rigged Hilbert space ,symbols.namesake ,Orthogonality ,symbols ,Projective Hilbert space ,Subspace topology ,Mathematics ,Reproducing kernel Hilbert space - Abstract
In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.
- Published
- 2005
9. Hardy-type inequalities for means
- Author
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Lars-Erik Persson and Zsolt Páles
- Subjects
Algebra ,Pure mathematics ,General Mathematics ,Of the form ,Type (model theory) ,Mathematics - Abstract
In this paper we consider inequalities of the form, WhereMis a mean. The main results of the paper offer sufficient conditions onMso that the above inequality holds with a finite constantC. The results obtained extend Hardy's and Carleman's classical inequalities together with their various generalisations in a new dirction.
- Published
- 2004
10. From surfaces in the 5-sphere to 3-manifolds in complex projective 3-space
- Author
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Luc Vrancken, John Bolton, and Christine Scharlach
- Subjects
Pure mathematics ,symbols.namesake ,Minimal surface ,General Mathematics ,Complex projective space ,symbols ,Projective test ,Curvature ,Ellipse ,Submanifold ,Lagrangian ,Pencil (mathematics) ,Mathematics - Abstract
In a previous paper it was shown how to associate with a Lagrangian submanifold satisfying Chen's equality in 3-dimensional complex projective space, a minimal surface in the 5-sphere with ellipse of curvature a circle. In this paper we focus on the reverse construction.
- Published
- 2002
11. On totally paranormal operators
- Author
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Christoph Schmoeger
- Subjects
Pure mathematics ,General Mathematics ,Paranormal ,Mathematics education ,Mathematics - Abstract
A continuous linear operator on a complex Banach space is said to be paranormal if ‖Tx‖2 ≤ ‖T2x‖ ‖x‖ for all x ∈ X. T is called totally paranormal if T–λ is paranormal for every λ ∈ C. In this paper we investigate the class of totally paranormal operators. We shall see that Weyl's theorem holds for operators in this class. We also show that for totally paranormal operators the Weyl spectrum satisfies the spectral mapping theorem. In Section 5 of this paper we investigate the operator equations eT = eS and eTeS = eSeT for totally paranormal operators T and S.
- Published
- 2002
12. Covariance factorisation and abstract representation of generalised random fields
- Author
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José M. Angulo, Vo Anh, and María D. Ruiz-Medina
- Subjects
Algebra ,Pure mathematics ,Random field ,Covariance function ,Multivariate random variable ,Covariance matrix ,General Mathematics ,Duality (mathematics) ,Random element ,White noise ,Covariance ,Mathematics - Abstract
This paper introduces a new concept of duality of generalised random fields using the geometric properties of Sobolev spaces of integer order. Under this duality condition, the covariance operators of a generalised random field and its dual can be factorised. The paper also defines a concept of generalised white noise relative to the geometries of the Sobolev spaces, and via the covariance factorisation, obtains a representation of the generalised random field as a stochastic equation driven by a generalised white noise. This representation is unique except for isometric isomorphisms on the parameter space.
- Published
- 2000
13. ON CERTAIN PRODUCTS OF PERMUTABLE SUBGROUPS
- Author
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T. M. Mudziiri Shumba, Sesuai Yash Madanha, A. Ballester-Bolinches, and M. C. Pedraza-Aguilera
- Subjects
Pure mathematics ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,General Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Permutable prime ,Mathematics - Abstract
In this paper, we study the structure of finite groups $G=AB$ which are a weakly mutually $sn$ -permutable product of the subgroups A and B, that is, A permutes with every subnormal subgroup of B containing $A \cap B$ and B permutes with every subnormal subgroup of A containing $A \cap B$ . We obtain generalisations of known results on mutually $sn$ -permutable products.
- Published
- 2021
14. Generic Gateaux differentiability via smooth perturbations
- Author
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Pando Georgiev and Nadia Zlateva
- Subjects
Bump function ,Mathematics::Functional Analysis ,Pure mathematics ,Continuous function ,Variational principle ,General Mathematics ,Banach space ,Christian ministry ,Differentiable function ,Space (mathematics) ,Mathematics - Abstract
We prove that in a Banach space with an uniformly Gateaux smooth bump function, every continuous function which is directionally differentiable on a dense Gδ subset of the space, is Gateaux differentiable on a dense Gδ subset of the space. Applications of this result are given. The usual applications of the variational principles in Banach spaces refer to differentiability of real valued functions. For example the papers of [B-P] and [D-G-Z] contain results about Gateaux differentiability on dense sets. An application of Ekeland’s variational principle to generic Frechet differentiability is given in the proff of famous Ekeland-Lebourg’s theorem (see [E-L]). In [Ge] an application of the smooth variational principle to generic Gateaux differentiability is presented. In this paper we prove some results about generic Gateaux differentiability of directionally differentiable functions. The tool for proving the main ∗The research was partially supported by the Bulgarian Ministry of Education and Science under contract number MM 506/1995.
- Published
- 1997
15. Metrisation of Moore spaces and abstract topological manifolds
- Author
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David L. Fearnley
- Subjects
Topological manifold ,Pure mathematics ,Topological algebra ,Computer Science::Information Retrieval ,General Mathematics ,Topological tensor product ,Hausdorff space ,Moore space (algebraic topology) ,Banach manifold ,Topological space ,Manifold ,Mathematics - Abstract
The problem of metrising abstract topological spaces constitutes one of the major themes of topology. Since, for each new significant class of topological spaces this question arises, the problem is always current. One of the famous metrisation problems is the Normal Moore Space Conjecture. It is known from relatively recent work that one must add special conditions in order to be able to get affirmative results for this problem. In this paper we establish such special conditions. Since these conditions are characterised by local simplicity and global coherence they are referred to in this paper generically as “abstract topological manifolds.” In particular we establish a generalisation of a classical development of Bing, giving a proof which is complete in itself, not depending on the result or arguments of Bing. In addition we show that the spaces recently developed by Collins designated as “W satisfying open G(N)” are metrisable if they are locally separable and locally connected and regular. Finally, we establish a new necessary and sufficient condition for spaces to be metrisable.
- Published
- 1997
16. Two characterisations of additive *-automorphisms of B(H)
- Author
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Lajos Molnár
- Subjects
symbols.namesake ,Pure mathematics ,General Mathematics ,Bounded function ,Linear operators ,Hilbert space ,symbols ,Bijection ,Algebra over a field ,Automorphism ,Mathematics - Abstract
Let H be a complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. In this paper we give two necessary and sufficient conditions for an additive bijection of B(H) to be a *-automorphism. Both of the results in the paper are related to the so-called preserver problems.
- Published
- 1996
17. Trace functions on inverse semigroup algebras
- Author
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W. D. Munn and David Easdown
- Subjects
Discrete mathematics ,Pure mathematics ,Inverse semigroup ,Trace (linear algebra) ,Semigroup ,Principal ideal ,General Mathematics ,Bicyclic semigroup ,Semilattice ,Group algebra ,Subring ,Mathematics - Abstract
Let S be an inverse semigroup and let F be a subring of the complex field containing 1 and closed under complex conjugation. This paper concerns the existence of trace functions on F[S], the semigroup algebra of S over F. Necessary and sufficient conditions on S are found for the existence of a trace function on F[S] that takes positive integral values on the idempotents of S. Although F[S] does not always admit a trace function, a weaker form of linear functional is shown to exist for all choices of S. This is used to show that the natural involution on F[S] is special. It also leads to the construction of a trace function on F[S] for the case in which F is the real or complex field and 5 is completely semisimple of a type that includes countable free inverse semigroups. The concept of a trace function on a real or complex algebra had its origin in matrix theory and is of central importance in many algebraic and analytical contexts. In the case of a group algebra, the trace of an element is defined simply to be the coefficient of the identity and is easily seen to possess all the standard properties. With the growth of interest in inverse semigroups (a class of involution semigroups with many group-like features), it is natural to ask whether the corresponding semigroup algebras also admit trace functions. In this paper we consider the semigroup algebra F[S] of an inverse semigroup 5 over a subring F of C that contains 1 and is closed under complex conjugation. In Section 1, where the basic definitions appear, two simple necessary conditions are obtained for the existence of a trace function on F[S] and attention is drawn to those trace functions (called 'strong') with the property that their values on the idempotents of S are positive integers. The main result of Section 2 provides a necessary and sufficient condition for F[S] to admit a strong trace function - namely that each principal ideal of the semilattice of S be finite. Section 3 comprises two examples. The notion of a pseudotrace function relative to a submodule is introduced in Section 4 and it is shown that, for any nonempty finite subset T of 5, F[S] admits a
- Published
- 1995
18. Haar measure and compact right topological groups
- Author
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Paul Milnes
- Subjects
Pure mathematics ,Compact group ,General Mathematics ,Calculus ,Euclidean group ,Lie group ,Peter–Weyl theorem ,Topological group ,Locally compact space ,Locally compact group ,Haar measure ,Mathematics - Abstract
The consideration of compact right topological groups goes back at least to a paper of Ellis in 1958, where it is shown that a flow is distal if and only if the enveloping semigroup of the flow is such a group (now called the Ellis group of the distal flow). Later Ellis, and also Namioka, proved that a compact right topological group admits a left invariant probability measure. As well, Namioka proved that there is a strong structure theorem for compact right topological groups. More recently, John Pym and the author strengthened this structure theorem enough to be able to establish the existence of Haar measure on a compact right topological group, a probability measure that is invariant under all continuous left and right translations, and is unique as such. Examples of compact right topological groups have been considered earlier. In the present paper, we give concrete representations of several Ellis groups coming from low dimensional nilpotent Lie groups. We study these compact right topological groups, and two others, in some detail, paying attention in particular to the structure theorem and Haar measure, and to the question: is Haar measure uniquely determined by left invariance alone? (It is uniquely determined by right invariance alone.) To assist in answering this question, we develop some sufficient conditions for a positive answer. We suspect that one of the examples, a compact right topological group coming from the Euclidean group of the plane, does not satisfy these conditions; we don't know if the question has a positive answer for this group.
- Published
- 1992
19. A factor theorem for locally convex differentiability spaces
- Author
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Roger Eyland and Bernice Sharp
- Subjects
Factor theorem ,Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,Mathematical analysis ,Regular polygon ,Differentiable function ,Mathematics - Abstract
The main result of this paper is that a continuous convex function with domain in a locally convex space factors through a normed space. In a recent paper by Sharp, topological linear spaces are categorised according to the differentiability properties of their continuous convex functions; we show that under suitable conditions the classification is preserved by linear maps. A technique for deducing results for locally convex spaces from Banach space theory is an immediate consequence. Examples are given and Asplund C(S) spaces are characterised.
- Published
- 1991
20. NOTE ON THE CONVOLUTION OF HARMONIC MAPPINGS
- Author
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Saminathan Ponnusamy and Liulan Li
- Subjects
Pure mathematics ,Mathematics - Complex Variables ,General Mathematics ,010102 general mathematics ,Regular polygon ,Harmonic (mathematics) ,01 natural sciences ,Convolution ,010101 applied mathematics ,FOS: Mathematics ,Primary: 31A05, Secondary: 30C45, 30C20 ,Complex Variables (math.CV) ,0101 mathematics ,Convex mapping ,Mathematics - Abstract
Dorff et al. \cite{DN} formulated a question concerning the convolution of two right half-plane mappings, where the normalization of the functions was considered incorrectly. In this paper, we have reformulated the open problem in correct form and provided a solution to it in a more general form. In addition, we also obtain two new theorems which correct and improve some other results., 11 pages; An extended version of this article was in a couple of conferences, and also in later workshops in Chennai during 2017 in India. This version will appear in Bulletin of the Australian Mathematical Society
- Published
- 2019
21. A COMPACT QUALITATIVE UNCERTAINTY PRINCIPLE FOR SOME NONUNIMODULAR GROUPS
- Author
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Wassim Nasserddine
- Subjects
Pure mathematics ,Uncertainty principle ,Fourier algebra ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Connection (mathematics) ,Harmonic analysis ,symbols.namesake ,Identity (mathematics) ,Fourier transform ,010201 computation theory & mathematics ,symbols ,Locally compact space ,0101 mathematics ,Abelian group ,Mathematics - Abstract
Let $G$ be a separable locally compact group with type $I$ left regular representation, $\widehat{G}$ its dual, $A(G)$ its Fourier algebra and $f\in A(G)$ with compact support. If $G=\mathbb{R}$ and the Fourier transform of $f$ is compactly supported, then, by a classical Paley–Wiener theorem, $f=0$. There are extensions of this theorem for abelian and some unimodular groups. In this paper, we prove that if $G$ has no (nonempty) open compact subsets, $\hat{f}$, the regularised Fourier cotransform of $f$, is compactly supported and $\text{Im}\,\hat{f}$ is finite dimensional, then $f=0$. In connection with this result, we characterise locally compact abelian groups whose identity components are noncompact.
- Published
- 2018
22. ON VANISHING CRITERIA THAT CONTROL FINITE GROUP STRUCTURE II
- Author
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Qingjun Kong and Julian Brough
- Subjects
Pure mathematics ,Finite group ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Structure (category theory) ,010307 mathematical physics ,0101 mathematics ,Control (linguistics) ,01 natural sciences ,Mathematics - Abstract
The first author [J. Brough, ‘On vanishing criteria that control finite group structure’, J. Algebra458 (2016), 207–215] has shown that for certain arithmetical results on conjugacy class sizes it is enough to consider only the vanishing conjugacy class sizes. In this paper we further weaken the conditions to consider only vanishing elements of prime power order.
- Published
- 2018
23. MAPPING PROPERTIES OF A SCALE INVARIANT CASSINIAN METRIC AND A GROMOV HYPERBOLIC METRIC
- Author
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Manas Ranjan Mohapatra and Swadesh Kumar Sahoo
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,51M10, 26A15, 30C20, 30C65, 30F45 ,A domain ,Metric Geometry (math.MG) ,Scale invariance ,Mathematics::Geometric Topology ,01 natural sciences ,Modulus of continuity ,010101 applied mathematics ,Euclidean distance ,Uniform continuity ,Mathematics - Metric Geometry ,FOS: Mathematics ,Identity function ,Ball (mathematics) ,0101 mathematics ,Mathematics - Abstract
In this paper, we consider a scale invariant Cassinian metric and a Gromov hyperbolic metric. We discuss a distortion property of the scale invariant Cassinian metric under M\"obius maps of a punctured ball onto another punctured ball. We obtain a modulus of continuity of the identity map from a domain equipped with the scale invariant Cassinian metric (or the Gromov hyperbolic metric) onto the same domain equipped with the Euclidean metric. The quasi-invariance properties of both the metrics under quasiconformal maps are also established., Comment: 13 pages (to appear in Bull. Aust. Math. Soc.)
- Published
- 2017
24. SHARP INEQUALITIES FOR THE VARIATION OF THE DISCRETE MAXIMAL FUNCTION
- Author
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José Madrid
- Subjects
010101 applied mathematics ,Sobolev space ,Pure mathematics ,Variation (linguistics) ,General Mathematics ,010102 general mathematics ,Bounded variation ,Maximal function ,Derivative ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper we establish new optimal bounds for the derivative of some discrete maximal functions, in both the centred and uncentred versions. In particular, we solve a question originally posed by Bober et al. [‘On a discrete version of Tanaka’s theorem for maximal functions’, Proc. Amer. Math. Soc.140 (2012), 1669–1680].
- Published
- 2016
25. The Cauchy problem for a second-order nonlinear hyperbolic equation with initial data on a line of parabolicity
- Author
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John Michael Rassias
- Subjects
Pure mathematics ,Nonlinear system ,Fourth order ,General Mathematics ,Mathematical analysis ,Line (geometry) ,Interval (graph theory) ,Order (ring theory) ,Initial value problem ,Continuous derivative ,Hyperbolic partial differential equation ,Mathematics - Abstract
In this paper we study the Cauchy problem for the second order nonlinear hyperbolic partial differential equationwith initial conditionswhereand |u|, |ux|, |uy| < ∞, y ≥ 0, r = r(x) ∈ C4(·), ν = ν(x) ∈ C4(·).These conditions on k, H, f, r, and ν are assumed to be satisfied in some sufficiently small neighborhood of the segment I, y = 0, in the upper half-plane y > 0This paper generalizes the results obtained by N.A. Lar'kin (Differencial'nye Uravnenija 8 (1972), 76–84), who has treated the special case H = H(x, y, u); that is, the quasi-linear hyperbolic equation (*).
- Published
- 1979
26. Nonuniform dichotomy of evolutionary processes in Banach spaces
- Author
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Petre Preda and Mihail Megan
- Subjects
Pure mathematics ,Class (set theory) ,Semigroup ,General Mathematics ,Mathematical analysis ,Eberlein–Šmulian theorem ,Banach space ,Mathematics ,Exponential function - Abstract
In this paper we study nonuniform dichotomy concepts of linear evolutionary processes which are defined in a general Banach space and whose norms can increase no faster than an exponential. Connections between the dichotomy concepts and (B, D) admissibility properties are established. These connections have been partially accomplished in an earlier paper by the authors for the case when the process was a semigroup of class C0 and (B, D) = [(Lp, Lq).
- Published
- 1983
27. Spaces with homogeneous norms
- Author
-
A.J. Pryde
- Subjects
Sobolev space ,Elliptic operator ,Pure mathematics ,Constant coefficients ,Elliptic partial differential equation ,General Mathematics ,Bounded function ,Boundary value problem ,Extension (predicate logic) ,Domain (mathematical analysis) ,Mathematics - Abstract
Spaces with homogeneous norms are closely related to the Beppo Levi spaces of Deny and Lions, to spaces of Riesz potentials, and to Sobolev spaces. In this paper we survey the literature on them, give a broad extension of their definition, and present their basic theory. Many of the properties of Sobolev spaces have their analogues. In fact, the two families are locally equivalent. Spaces with homogeneous norms are especially suited to the study of boundary value problems on for homogeneous elliptic operators with constant coefficients. We will use them extensively in a forthcoming paper to study elliptic partial differential equations with mixed boundary conditions on a smoothly bounded domain.
- Published
- 1980
28. Continuity of Cima and Rung's extension and non normal meromorphic functions
- Author
-
Douglas M. Campbell
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,Non normality ,Extension (predicate logic) ,Meromorphic function ,Mathematics - Abstract
A function meromorphic in |z| < 1 is constructed such that on every curve in |z| < 1 which goes to |z| = 1 the set of limit points of the function is the entire complex plane. This example is used to prove the existence of non-normal meromorphic functions in |z| < 1 which have continuous set valued extensions. Cima and Rung had introduced a set valued extension for meromorphic functions and proved that all normal meromorphic functions have a continuous extension while all functions with a continuous extension have the Lindelöf property. For a long time it was thought that this might characterize normal meromorphic functions. This paper proves that it is not possible to determine the normality of a meromorphic function by the continuity of Cima and Rung's set valued extension. The paper closes with the open problem: do there exist non-normal analytic functions for which Cima and Rung's set valued extension is continuous?
- Published
- 1979
29. Autoclinisms and automorphisms of finite groups
- Author
-
Jürgen Tappe and Ratje Reimers
- Subjects
Group isomorphism ,Pure mathematics ,Profinite group ,Group of Lie type ,Automorphisms of the symmetric and alternating groups ,General Mathematics ,Extra special group ,Simple group ,Hurwitz's automorphisms theorem ,Cycle graph (algebra) ,Mathematics - Abstract
Let Γ be a family of isoclinic finite groups, and let acl(Γ) be the group of autoclinisms of Γ. In this paper we prove the following formula:where the sum is taken over a complete system of stem groups S in Γ. This result is due to P. Hall, who outlined a proof in his paper, “On groups of automorphisms” (J. reine angew. Math). 182 (1940), 194–204, using presentations, whereas in this paper we consider stem groups in terms of central extensions.
- Published
- 1975
30. Approximative compactness and continuity of metric projections
- Author
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O.P. Kapoor and B.B. Panda
- Subjects
Set (abstract data type) ,Metric space ,Pure mathematics ,Compact space ,Computer Science::Information Retrieval ,General Mathematics ,Metric (mathematics) ,Banach space ,Uniformly convex space ,Space (mathematics) ,Topology ,Chebyshev filter ,Mathematics - Abstract
In the paper “Some remarks on approximative compactness”, Rev. Roumaine Math. Pures Appl. 9 (1964), Ivan Singer proved that if K is an approximatively compact Chebyshev set in a metric space, then the metric projection onto K is continuous. The object of this paper is to show that though, in general, the continuity of the metric projection supported by a Chebyshev set does not imply that the set is approximatively compact, it is indeed so in a large class of Banach spaces, including the locally uniformly convex spaces. It is also proved that in such a space X the metric projection onto a Chebyshev set is continuous on a set dense in X.
- Published
- 1974
31. A classification of groups with a centralizer condition II: Corrigendum and addendum
- Author
-
Zvi Arad and Marcel Herzog
- Subjects
Pure mathematics ,General Mathematics ,Addendum ,Centralizer and normalizer ,Mathematics - Abstract
The aim of this note is to prove a theorem, which extends the results of the authors' earlier paper, Bull. Austral. Math. Soc. 16 (1977), 55–60. As one of the corollaries we prove Theorem 2 of that paper, the proof of which was incomplete.
- Published
- 1977
32. On some Gelfand-Mazur like theorems in p-normed algebras
- Author
-
V.K. Srinivasan and Hu Shaing
- Subjects
Discrete mathematics ,Normed algebra ,Pure mathematics ,General Mathematics ,Unique factorization domain ,Domain (ring theory) ,Field (mathematics) ,Complex number ,Mathematics - Abstract
The main theorem of this paper shows that a complex p-normed algebra which is a pre-Bezout domain is isomorphic to the field of complex numbers, if it is a generalized unique factorization domain. This theorem generalizes the previous result of the authors proved by them in their paper Bull. Austral. Math. Soc. 20 (1979), 247–252. Some applications are then given.
- Published
- 1980
33. Convex sets, fixed points, variational and minimax inequalities
- Author
-
Tzu-Chu Lin
- Subjects
Convex analysis ,Lemma (mathematics) ,Pure mathematics ,Quasiconvex function ,General Mathematics ,Variational inequality ,Mathematical analysis ,Convex set ,Fixed-point theorem ,Fixed point ,Minimax ,Mathematics - Abstract
Recently, Ky Fan extended his will known lemma (which is an extension of the classical theorem of Knaster, Kuratowski and Mazurkiewica) to the noncompact case. Using this result, another interesting lemma of Fan is generalized in this paper. As applications of our theorem, we obtain a generalizationof Browder's variational inequality and derive Fan's other recent results directly from our theorem. Also, in this paper, we give a slight extension recent results of K. K. Tan, which themselves are generalizations of many well-known results on minimax and variational inequalities.
- Published
- 1986
34. Dedekind-finite fields
- Author
-
J.L. Hickman
- Subjects
Pure mathematics ,Finite field ,Field (physics) ,General Mathematics ,Dedekind cut ,Mathematics - Abstract
Let p be a prime and let (mk)k be a strictly increasing sequence of positive integers such that m0 = 1 and mk divides mk+1. A field F is said to be of type (p, (mk)k) if it is the union of an increasing sequence (Fk)k of fields such that Fk has pmk elements. A set X is called “finite” if it has n elements for some nonnegative integer n, and “Dedekind-finite” if every injection f: X → X is a bijection. If the Axiom of Choice is rejected, then it is relatively consistent to assume the existence of medial (that is, infinite, Dedekind-finite) sets. In this paper it is shown that given any type (p, (mk)k) as above, it is relatively consistent with the usual axioms of set theory (minus Choice) to assume the existence of a medial field of type (p, (mk)k). Conversely, it is shown that any medial field must be of type (p, (mk)k) for some (p, (mk)k) as above. The paper concludes with a few observations on Dedekind-finite rings. In the first part of the paper, a general knowledge of Fraenkel-Mostowski set theory and of the Jech-Sochor Embedding Theorems is assumed.
- Published
- 1978
35. On the structure of a real crossed group algebra
- Author
-
K. Buzási
- Subjects
Discrete mathematics ,Ring (mathematics) ,Pure mathematics ,Incidence algebra ,General Mathematics ,Subalgebra ,Algebra representation ,Division algebra ,Infinite dihedral group ,Central simple algebra ,Group ring ,Mathematics - Abstract
The main result of this paper is that there exist non-principal left ideals in a certain twisted group algebraAof the infinite dihedral group ≤a, b|b−1ab=a−1,b2= 1 ≥ over the field R of real numbers: namely in theAdefined byb−lab = a−1, b2= −1, and λa=aλ, λb=bλ for all real λ.The motivation comes from the study (in a series of papers by Berman and the author) of finitely generated torsion-free RG-modules for groupsGwhich have an infinite cyclic subgroup of finite index. In a sense, this amounts to studying modules over (full matrix algebras over) a finite set of R-algebras [namely, for the groups in question, these algebras take on the role played by R, C and H (the real quaternions) in the theory of real representations of finite groups]. For all but two algebras in that finite set, satisfying results have been obtained by exploiting the fact that each of them is either a ring with zero divisors or a principal left ideal ring. The other two are known to have no zero divisors. One of them is the presentA. The point of the main result is that new ideas will be needed for understandingA-modules.A number of subsidiary results are concerned with convenient generating sets for left ideals inA.
- Published
- 1988
36. A uniqueness theorem for the Chaplygin-Frankl problem
- Author
-
John Michael Rassias
- Subjects
Pure mathematics ,Picard–Lindelöf theorem ,Fundamental theorem ,General Mathematics ,Compactness theorem ,Fixed-point theorem ,Danskin's theorem ,Brouwer fixed-point theorem ,Squeeze theorem ,Mathematics ,Carlson's theorem - Abstract
In a paper dealing with trans-sonic jet flows Frankl (Bull. Acad. Sci. URSS Sér. Math. [Izv. Akad. Nauk SSSE] 9 (1945), 121–143) considered the following problem (T) by applying the conditionwhere k = k(y) is a monotone increasing function with a continuous second derivative, k(0) = 0, F(0) > 0, k′(y) ≠ 0 for y < 0. Consider an equation of the formwhich is elliptic for y > 0, hyperbolic for y < 0, and parabolic for y = 0. Consider equation (2) in a bounded simply connected region D ⊂ R2 which is bounded by the following three curves: a piecewise smooth curve Γ0 lying in the half-plane y > 0 which intersects the line y = 0 at the points A(0, 0) and B(l, 0); for y < 0 by a smooth curve Γ2 through B which meets the characteristic of (2) issuing from A(0, 0) at the point P; and the curve Γ1 which consists of the portion PA of the characteristic through A. The problem (T) (or problem of Tricomi-Frankl) consists of finding a solution u = u(x, y) ∈ C2(D) assuming prescribed values on Γ0 ∪ Γ2. In the present paper we generalize Frankl's uniqueness theorem; our uniqueness theorem includes cases where F(y) may be negative.
- Published
- 1979
37. On analytic functions with reference to the Bernardi integral operator
- Author
-
K. S. Padmanabhan and G. Lakshma Reddy
- Subjects
Pure mathematics ,Positive-definite kernel ,General Mathematics ,Hypoelliptic operator ,Global analytic function ,Daniell integral ,Shift operator ,Compact operator ,Fourier integral operator ,Mathematics ,Analytic function - Abstract
Bernardi has proved that if f if starlike univalent in the uni disc E, then so is the function g given byIn the first part of the paper, we extend Bernardi's theorem to a certain class of p-valent starlike functions in E. We prove that if then g, defined byalso belongs to . In the second part of the paper we examine the converse problem for functions with negative coefficients, satisfying certain conditions.
- Published
- 1982
38. On a new finite non–abelian simple group of Janko
- Author
-
S. K. Wong
- Subjects
Pure mathematics ,General Mathematics ,Simple group ,Abelian group ,Mathematics - Abstract
Two new simple groups have recently been discovered by Z. Janko. One of these groups has order 50,232,960. As a first step in showing that there is precisely one (up to isomorphism) simple group of order 50,232,960, the author proves in this paper the following result: If G is a non-abelian simple group of order 50,232,960, then the structure of the centralizer of an element of order two in G is uniquely determined.In a note added on 21 April 1969 to this paper, the author announces that he has proved the uniqueness of the simple group of order 50,232,960.
- Published
- 1969
39. Extensions of finite nilpotent groups
- Author
-
John Poland
- Subjects
Pure mathematics ,Nilpotent ,General Mathematics ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Nilpotent group ,Unipotent ,Central series ,Mathematics - Abstract
If G is a finite group and P is a group-theoretic property, G will be called P-max-core if for every maximal subgroup M of G, M/MG has property P where MG = ∩ is the core of M in G. In a joint paper with John D. Dixon and A.H. Rhemtulla, we showed that if p is an odd prime and G is (p-nilpotent)-max-core, then G is p-solvable, and then using the techniques of the theory of solvable groups, we characterized nilpotent-max-core groups as finite nilpotent-by-nilpotent groups. The proof of the first result used John G. Thompson's p-nilpotency criterion and hence required p > 2. In this paper I show that supersolvable-max-core groups (and hence (2-nilpotent)-max-core groups) need not be 2-solvable (that is, solvable). Also I generalize the second result, among others, and characterize (p-nilpotent)-max-core groups (for p an odd prime) as finite nilpotent-by-(p-nilpotent) groups.
- Published
- 1970
40. Conditions for a plane projective metric to be a norm
- Author
-
B.B. Phadke
- Subjects
Pure mathematics ,Blocking set ,Plane curve ,Real projective plane ,General Mathematics ,Complex projective space ,Mathematical analysis ,Projective space ,Line at infinity ,Projective plane ,Pencil (mathematics) ,Mathematics - Abstract
Let R be a metrization with distance xy of an open convex set D in the 2-dimensional real affine plane such that xy + yz = xz whenever x, y, z lie on an affine line with y between x and z and such that all the balls px ≤ ρ are compact. The study of such metrics, called open plane projective metrics falls under the topic of Hilbert's Problem IV of his famous mathematical problems. In this paper it is proved that if in R the sets of points equidistant from lines lie again on lines then D must be the entire affine plane and the distance must in fact be a norm. The paper contributes to and gives extensions of similar results proved earlier. The novel features of the present result are that in the space collinearity of points x, y, z is taken only as a sufficient condition for the equality xy + yz = xz. Consequently the solution encompasses all normed linear planes, that is, norms whose unit circles are not necessarily strictly convex are also admitted.
- Published
- 1973
41. On a cardinal equation in set theory
- Author
-
J.L. Hickman
- Subjects
Pure mathematics ,Infinite set ,Class (set theory) ,Rank-into-rank ,General Mathematics ,Zermelo–Fraenkel set theory ,Equinumerosity ,Uncountable set ,Universal set ,Cardinality of the continuum ,Mathematics - Abstract
We work in a Zermelo-Fraenkel set theory without the Axiom of Choice. In the appendix to his paper “Sur les ensembles finis”, Tarski proposed a finiteness criterion that we have called “C-finiteness”: a nonempty set is called “C-finite” if it cannot be partitioned into two blocks, each block being equivalent to the whole set. Despite the fact that this criterion can be shown to possess several features that are undesirable in a finiteness criterion, it has a fair amount of intrinsic interest. In Section 1 of this paper we look at a certain class of C-finite sets; in Section 2 we derive a few consequences from the negation of C-finiteness; and in Section 3 we show that not every C-infinite set necessarily possesses a linear ordering. Any unexplained notation is given in my paper, “Some definitions of finiteness”, Bull. Austral. Math. Soc. 5 (1971).
- Published
- 1972
42. GAPS IN TAYLOR SERIES OF ALGEBRAIC FUNCTIONS
- Author
-
Seth Dutter
- Subjects
Pure mathematics ,Mathematics - Complex Variables ,General Mathematics ,Local parameter ,Rational function ,Upper and lower bounds ,Mathematics - Algebraic Geometry ,symbols.namesake ,FOS: Mathematics ,Taylor series ,symbols ,Algebraic function ,Algebraic curve ,Complex Variables (math.CV) ,Algebraic Geometry (math.AG) ,Complex number ,Variable (mathematics) ,Mathematics - Abstract
Let $f$ be a rational function on an algebraic curve over the complex numbers. For a point $p$ and local parameter $x$ we can consider the Taylor series for $f$ in the variable $x$. In this paper we give an upper bound on the frequency with which the terms in the Taylor series have $0$ as their coefficient.
- Published
- 2015
43. WARPED PRODUCTS IN RIEMANNIAN MANIFOLDS
- Author
-
Kwang-Soon Park
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,53C40, 53C42, 53C50 ,General Mathematics ,Second fundamental form ,Clifford torus ,Space (mathematics) ,Curvature ,Manifold ,Warping function ,Differential Geometry (math.DG) ,FOS: Mathematics ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,ComputingMethodologies_COMPUTERGRAPHICS ,Mathematics - Abstract
In this paper we prove two inequalities relating the warping function to various curvature terms, for warped products isometrically immersed in Riemannian manifolds. This extends work of B. Y. Chen for the case of immersions into space forms. Finally we give an application where the target manifold is the Clifford torus., Comment: 10 pages, to appear in the Bulletin of the Australian Mathematical Society
- Published
- 2014
44. NONTRIVIAL SOLUTIONS FOR STURM–LIOUVILLE SYSTEMS VIA A LOCAL MINIMUM THEOREM FOR FUNCTIONALS
- Author
-
Gabriele Bonanno, Donal O'Regan, and Shapour Heidarkhani
- Subjects
Class (set theory) ,Pure mathematics ,General Mathematics ,Mathematical analysis ,Sturm–Liouville theory ,Differentiable function ,Sturm separation theorem ,Mathematics - Abstract
In this paper, employing a very recent local minimum theorem for differentiable functionals, the existence of at least one nontrivial solution for a class of systems of $n$ second-order Sturm–Liouville equations is established.
- Published
- 2013
45. THE CATEGORIFICATION OF THE KAUFFMAN BRACKET SKEIN MODULE OF
- Author
-
Boštjan Gabrovšek
- Subjects
Khovanov homology ,symbols.namesake ,Pure mathematics ,Skein ,General Mathematics ,Euler characteristic ,Categorification ,symbols ,Bracket polynomial ,Invariant (mathematics) ,Homology (mathematics) ,Mathematics - Abstract
Khovanov homology, an invariant of links in ${ \mathbb{R} }^{3} $, is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda et al. [‘Categorification of the Kauffman bracket skein module of $I$-bundles over surfaces’, Algebr. Geom. Topol. 4 (2004), 1177–1210] generalised this construction by defining a double graded homology theory that categorifies the Kauffman bracket skein module of links in $I$-bundles over surfaces, except for the surface $ \mathbb{R} {\mathrm{P} }^{2} $, where the construction fails due to strange behaviour of links when projected to the nonorientable surface $ \mathbb{R} {\mathrm{P} }^{2} $. This paper categorifies the missing case of the twisted $I$-bundle over $ \mathbb{R} {\mathrm{P} }^{2} $, $ \mathbb{R} {\mathrm{P} }^{2} \widetilde {\times } I\approx \mathbb{R} {\mathrm{P} }^{3} \setminus \{ \ast \} $, by redefining the differential in the Khovanov chain complex in a suitable manner.
- Published
- 2013
46. ON THE -LENGTH AND THE WIELANDT LENGTH OF A FINITE -SOLUBLE GROUP
- Author
-
Ning Su and Yanming Wang
- Subjects
Discrete mathematics ,Pure mathematics ,Nilpotent ,Locally finite group ,General Mathematics ,Sylow theorems ,Permutable prime ,Invariant (mathematics) ,Mathematics - Abstract
The $p$-length of a finite $p$-soluble group is an important invariant parameter. The well-known Hall–Higman $p$-length theorem states that the $p$-length of a $p$-soluble group is bounded above by the nilpotent class of its Sylow $p$-subgroups. In this paper, we improve this result by giving a better estimation on the $p$-length of a $p$-soluble group in terms of other invariant parameters of its Sylow $p$-subgroups.
- Published
- 2013
47. JEŚMANOWICZ’ CONJECTURE REVISITED
- Author
-
Zhi-Juan Yang and Min Tang
- Subjects
Pure mathematics ,Conjecture ,General Mathematics ,Calculus ,Mathematics - Abstract
Let $a, b, c$ be relatively prime positive integers such that ${a}^{2} + {b}^{2} = {c}^{2} $. In 1956, Jeśmanowicz conjectured that for any positive integer $n$, the only solution of $\mathop{(an)}\nolimits ^{x} + \mathop{(bn)}\nolimits ^{y} = \mathop{(cn)}\nolimits ^{z} $ in positive integers is $(x, y, z)= (2, 2, 2)$. In this paper, we consider Jeśmanowicz’ conjecture for Pythagorean triples $(a, b, c)$ if $a= c- 2$ and $c$ is a Fermat prime. For example, we show that Jeśmanowicz’ conjecture is true for $(a, b, c)= (3, 4, 5)$, $(15, 8, 17)$, $(255, 32, 257)$, $(65535, 512, 65537)$.
- Published
- 2013
48. THE MULTIPLIER ALGEBRA AND BSE PROPERTY OF THE DIRECT SUM OF BANACH ALGEBRAS
- Author
-
Mahmood Lashkarizadeh Bami and Zeinab Kamali
- Subjects
Symmetric algebra ,Algebra ,Filtered algebra ,Pure mathematics ,General Mathematics ,Subalgebra ,Division algebra ,Algebra representation ,Cellular algebra ,Universal enveloping algebra ,Banach *-algebra ,Mathematics - Abstract
The notion of BSE algebras was introduced and first studied by Takahasi and Hatori and later studied by Kaniuth and Ülger. This notion depends strongly on the multiplier algebra $M( \mathcal{A} )$ of a commutative Banach algebra $ \mathcal{A} $. In this paper we first present a characterisation of the multiplier algebra of the direct sum of two commutative semisimple Banach algebras. Then as an application we show that $ \mathcal{A} \oplus \mathcal{B} $ is a BSE algebra if and only if $ \mathcal{A} $ and $ \mathcal{B} $ are BSE. We also prove that if the algebra $ \mathcal{A} \hspace{0.167em} {\mathop{\times }\nolimits}_{\theta } \hspace{0.167em} \mathcal{B} $ with $\theta $-Lau product is a BSE algebra and $ \mathcal{B} $ is unital then $ \mathcal{B} $ is a BSE algebra. We present some examples which show that the BSE property of $ \mathcal{A} \hspace{0.167em} {\mathop{\times }\nolimits}_{\theta } \hspace{0.167em} \mathcal{B} $ does not imply the BSE property of $ \mathcal{A} $, even in the case where $ \mathcal{B} $ is unital.
- Published
- 2013
49. HEISENBERG–PAULI–WEYL UNCERTAINTY INEQUALITY FOR THE DUNKL TRANSFORM ON ℝd
- Author
-
Fethi Soltani
- Subjects
symbols.namesake ,Pure mathematics ,Pauli exclusion principle ,Inequality ,General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,symbols ,Conjugate variables ,Mathematics ,media_common - Abstract
In this paper, we give analogues of the local uncertainty inequality for the Dunkl transform on ℝd, and indicate how the local uncertainty inequality implies a global uncertainty inequality.
- Published
- 2012
50. CANCELLING COMPLEX POINTS IN CODIMENSION TWO
- Author
-
Marko Slapar
- Subjects
Pure mathematics ,General Mathematics ,Mathematical analysis ,Isotopy ,Point (geometry) ,Codimension ,Complex manifold ,Submanifold ,Sign (mathematics) ,Mathematics - Abstract
A generically embedded real submanifold of codimension two in a complex manifold has isolated complex points that can be classified as either elliptic or hyperbolic. In this paper we show that a pair consisting of one elliptic and one hyperbolic complex point of the same sign can be cancelled by a $\mathcal {C}^{0}$small isotopy of embeddings.
- Published
- 2012
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