We prove that the function space $C_p(X)$ of a Tychonoff space $X$ is cofinally $\sigma$-compact if and only if it is $\sigma$-pseudocompact. It is also established that cofinal $\sigma$-compactness of $C_p(X, [0,1])$ is equivalent to its pseudocompactness. If $X$ is a locally Lindelöf domain representable GO space of countable extent, then it is cofinally Polish. In particular, any locally Lindelöof Cech-complete GO space of countable extent is cofinally Polish. Our results provide answers to two published open questions.
The main objective of this paper is to estimate asymptotic property of the logarithmic differences of a transcendental meromorphic function of order less than 1 in several complex variables. The results extend the fundamental lemmas of Bergweiler and Langley [1] in several complex variables. As an application of our main results, we estimate the growth of entire solutions in Cm of a certain type of complex linear difference equation.
Recently, employing Gasper and Rahman's quadratic summation and applying the method of creative microscoping in a novel way, Gu and Wang proved a q-supercongruence modulo the fourth power of a cyclotomic polynomial conjectured by Tang. Making use of Gasper and Rahman's quadratic summation and the creative microscoping method, we present a further generalization of Gu and Wang's result.
Euler's pioneering work on linear and non-linear harmonic sums has continued to inspire extensive research, yielding many new results. This paper studies representations of linear harmonic Euler sums of both even and odd weight. This work establishes new identites linking sums with positive terms to alternating series, introduces two symmetry properties at half integer value, and extends known recurrence relations for linear harmonic Euler sums of even and odd weight.
We prove that the function space Cp(X) of a Tychonoff space X is cofinally cr-compact if and only if it is cr-pseudocompact. It is also established that cofinal cr-compactness of Cp(X, [0, 1]) is equivalent to its pseudocompactness. If X is a locally Lindelo & uml;f domain representable GO space of countable extent, GO space of countable extent is cofinally Polish. Our results provide answers to two published open questions.
This paper is the first of a series of two papers where we develop a point-free Lebesgue integration theory for functions on o-locales. This series describes the integral of localic general functions with respect to a measure defined on the coframe of all o-sublocales, moving beyond the constraints of Boolean algebras. It also extends the notion of integrable function, usually reserved for measurable functions, to localic general functions. In this first part, we present the theory for the case of simple functions, laying the groundwork to develop the theory for more general functions.
We establish the existence of an infinite-dimensional linear subspace consisting of functions (taking values in certain Banach spaces) that are bounded and Riemann integrable but non-measurable. We also show existence of an infinite-dimensional linear subspace consisting of bounded functions which are Riemann integrable and measurable but Darboux non-integrable, with values in certain Banach spaces.
We investigate the Lp - Lq estimates of the solution operator for higher-order Schro & uml;dinger equations of the form ut(t, x) = iP(D)u(t, x), where P(D) is a real degenerate elliptic partial differential operator. We use the global pointwise time-space estimates of the fundamental solutions for the above equations to obtain improved Lp - Lq estimates of the solution operator. As an application, we obtain local Strichartz estimates for the above equations and the unique-existence of solutions for higher-order Schro & uml;dinger equations with a time-dependent potential.
The two operations of Minkowski addition and subtraction, defined both for sets and for functions, have found many applications in integral geometry, morphological image processing and spatial logic, generally within the digital framework of subsets of Zn and functions Zn Z. However, in the Euclidean framework of subsets of Rn and functions Rn R U {-infinity, +infinity}, they suffer from a defect: applying them to two sets or functions that differ only in a set of zero Lebesgue measure can lead to fundamentally different results. In this paper, we remedy to this problem. Our framework is the complete lattice of equivalence classes of Borel functions under equality almost everywhere. We replace the usual operations of numerical supremum and infimum by their essential forms with respect to a sigma-finite Borel measure applied to the second argument of the addition or subtraction, called the "structuring element"; we examine two particular cases, the Lebesgue measure and the discrete measure for a countable structuring element. The algebraic and lattice-theoretical properties of the Minkowski operations are preserved in this framework. For the sake of completeness, we recall in detail the lattice-theoretical background of our theory and give the precise formalism for Minkowski operations on functions Rn R U {-infinity, +infinity}.
We prove that there exists a subspace of C( [0, 1]) isometric to c (the space of convergent sequences) such that every nonzero function in the subspace is Besicovitch, i.e. the function does not have a one-sided derivative (not even infinite) at any point.
It is shown that PU(1, n), for n >= 2, does not admit non-elementary representations into the group of isometries of an infinite-dimensional real hyperbolic space.
We introduce and study new transformations between two functions satisfying some basic growth properties and generalize the known lower and upper Legendre conjugate (or envelope). We also investigate how these transformations modify recently defined growth indices for weight functions. A special but important and useful situation, to which the knowledge is then applied, is when considering associated weight functions which are expressed in terms of an underlying weight sequence. In this case these transformations precisely correspond to the point-wise product resp. point-wise division of the given sequences. Therefore, the new approach studied in this work illustrates the genuineness and importance and suggests applications for weighted spaces in different directions.
We exhibit an obstruction for groups with Relative Property (T) to act on the real line by bi-Lipschitz homeomorphisms. This condition is expressed in terms of the Lipschitz and Kazhdan constants associated to finite generating subsets. As an application, we obtain an explicit lower bound for the Lipschitz constants associated to actions of the semidirect product F2 x Z2. We also obtain an upper bound for the Kazhdan constants of pairs of orderable groups, depending only on the cardinal of the generating subset.
This paper is a comprehensive survey of characterizations of the tracial func-tionals among all positive linear functionals on the full matrix algebras,C & lowast;-algebras, and von Neumann algebras. It also includes new character-izations of the tracial property. We explore Thompson's triangle inequalityand demonstrate its connection to the characterization of the tracial property.Moreover, we provide rather simple proofs for various characterizations ofthe standard trace on the full matrix algebraMn. We establish a new charac-terization of the tracial functionals in the framework ofC & lowast;-algebras by show-ing that the following conditions are equivalent for a state phi on aC & lowast;-algebraA: (i)phi is tracial; (ii)phi(A1/2B A1/2+lambda(AB+B A))>= 0 for some number lambda >=-1/2,lambda 6=0 and allA,B is an element of A+. As a consequence, we introduce a newcriterion for the commutativity ofC & lowast;-algebras.
