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 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.
We investigate singularities of parallel curves and evolutes of frontal curvesin the Euclidean plane admitting certain singular points. Moreover, we studyassociated lifted surfaces to parallel curves to understand propagation ofplanar waves whose initial curve may have singularities.
We introduce the notion of theta-omega-balancedand delta theta-omega-balancedsemitopologicalgroups and prove that both classes, the class of theta-omega-balancedsemitopologi-cal groups and the class of delta theta-omega-balancedsemitopological groups, are closedunder taking subgroups and products. We show that a regular (Hausdorff,T1) semitopological groupGadmits a homeomorphic embedding as a sub-group into a product of regular (Hausdorff,T1) semitopological groups witha theta-base if and only ifGis theta-omega-balanced andIr(G)<=omega(Hs(G)<=omega,Sm(G)<=omega). Similarly, a regular (Hausdorff,T1) semitopological groupGadmits a homeomorphic embedding as a subgroup into a product of regu-lar (Hausdorff,T1) semitopological groups with a delta theta-base if and only ifGis delta theta-omega-balanced andIr(G)<=omega(Hs(G)<=omega,Sm(G)<=omega). Moreover, we givea characterization of a semitopological group that can be embedded as a sub-group into a product of regular (Hausdorff,T1) quasi-developable semitopo-logical groups. In the end of this article, we give the relationship betweenweakly omega-balancedness [14],M-omega-balancedness [15] and omega s-balancedness[13], and their examples
We study differentiable functionsffrom compact perfect subsetsCofRontoCwith vanishing derivative, that is, withf '(x) =0 for everyx is an element of C. We showthat the domain of such a function can have Hausdorff dimensiondfor anyd is an element of[0, 1)and that it can be extended to a differentiable functionF:R -> Rsuch thatFis alpha-H older for every alpha is an element of(0, 1). This last part is deduced from anovel generalization of Jarn & imath;k's differentiable extension theorem stating thatevery differentiable mapf:P -> R, whereP subset of Ris compact, admits adifferentiable extensionF:R -> Rwhich preserves H older continuity off
We propose an energy dissipative finite element scheme for the strongly coupled nonlinear space fractional wave equations with damping which generalize the coupled Klein-Gordon systems. We provide the discrete energy dissipation property of the proposed scheme, and existence and uniqueness of the numerical solution. We also show that the scheme is unconditionally convergent and stable. Finally, we give some numerical experiments which confirm the theoretical results.
Some new results on K-convex and (K1, K2)-convex functions are investigated. Also, a type of K-upper (lower) semicontinuity for the (K1, K2)-convex functions is introduced and examined. These concepts are compared with other generalized convex functions from the literature. Furthermore, the Hermite-Hadamard inequality is extended for (K1, K2)-convex functions.
The super-biderivations on the -super pound Galilean conformal algebra are completely determined. We prove that they are all inner.
We study two complementary themes on the derivative Hardy space S-2: the Hyers-Ulam stability of coefficient multipliers and the long-time dynamics of multiplication operators. For M phi acting on S-2, we give symbol-theoretic characterizations of power boundedness (PB), Cesa`ro boundedness (CB), uniform Kreiss boundedness (UKB), and mean ergodicity (ME) by reducing these properties to explicit S-2-norm bounds for the powers phi (n) and the Cesa`ro kernels (uniformly in unimodular scalings). In particular, we identify sharp conditions under which mean ergodicity is equivalent to Cesa`ro boundedness. On the coefficient side, we establish sharp boundedness and Hyers-Ulam stability criteria for diagonal multipliers on the model scale l(alpha)(2) (alpha > 0) and specialize them to S-2. Altogether, the results unify ergodic and stability phenomena on S-2 within a single intrinsic S-2-norm framework.
We obtain results related to (k, r)-integers over Piatetski-Shapiro sequences, which improve previous results by Srichan [16]. We also consider the largest prime factors in Piatetski-Shapiro sequences and other related problems.
