We investigate a new class of function spaces called variable bounded variation spaces of higher order by generalizing the classical notion of bounded variation. Our approach incorporates concepts from higher-order variations, bounded variation in the sense of generalized norms, and variations defined through modular function spaces. We focus on analysing the approximation properties of convolution-type nonlinear integral operators within these newly defined function spaces. In particular, we study these operators on variable bounded variation spaces of a fixed order in the sense of higher variations. Several approximation results are obtained using regular summability methods, highlighting the effectiveness of the proposed approach in improving convergence behaviour.
Using the J_P^2 -power series method, we redefine the sets of statistical limit points and statistical cluster points for double sequences of real numbers. We compare these sets with the set of usual limit points of double sequences and provide examples to demonstrate that the inclusions between these sets can be strict.
Tonks’ projection from the permutohedron to the associahedron and the Loday-Ronco map both send permutations to planar binary trees. We give a syntactic account of these maps in the equational calculus of the free non-symmetric, non-unital operad on one binary generator. The vertex restriction of Tonks’ projection is obtained by evaluating the head-insertion encoding on the reversed permutation, while the Loday-Ronco map is obtained by evaluating the decreasing encoding. We also give a local operadic proof that Tonks’ vertex map is order-preserving from the weak Bruhat order to the Tamari order.
We prove a Steinhaus-type theorem for four-dimensional matrix transformations of double sequences, establishing that (L_1, L_1, P ) \cap (L_s, L_1) = \emptyset , s > 1 . This extends Fridy?s classical result for single sequences. Our results hold for sequences of bounded variation, bounded, Pringsheim convergent, bounded Pringsheim convergent, and regularly convergent sequence spaces. We show this theorem fails in non-archimedean fields through a counterexample.
If p(z) is a polynomial of degree n having all its zeros in |z| <= k, k >= 1, then for r >= 1, Aziz [J. Approx. Theory, 55 (1988), 232-239] proved { integral(2 pi)(0) |1 + k(n) e(i theta)|(R) d theta}(1/r) max(|z|=1) |p ' (z)| >= n { integral(2 pi)(0) |p(e(i theta))|(R) d theta}(1/r) , whereas Devi et al. [Note Mat., 41 (2021), 19-29] proved that if p(z) is a polynomial of degree n having no zero in |z| < k, k <= 1, then for r > 0, k(n)n{(2 pi) (0) |p(e(i theta))|(R) d theta}(1/r) <= {(2 pi) (0) |e(i theta) + k(n)|(R) d theta} (1/r) {n max (|z|=1) |p(z)| - max (|z|=1) |p ' (z)|} , provided |p ' (z)| and |q ' (z)| attain their maxima at the same point on |z| = 1, where q(z) = z (n)p (1/z) (SIC). We do not only obtain improved extensions of the above inequalities into polar derivative by involving the leading coefficient and the constant term of the polynomial, but also give integral analogues of inequalities on polar derivative recently proved by Mir and Dar [Filomat, 36(16) (2022), 5631- 5640].
We deal with the approximation operators which are generalization of the operators of exponential type. We construct some composition operators and discuss the convergence behaviour. Also we estimate some direct results with the aid of the characteristic functions of such operators. Also, difference estimates in terms of modulus of continuity are established.
Here we introduce topologically multiply 3-recurrent collections {T-t}(t is an element of S) of bounded linear operators on a Banach space X, where 3 is a Furstenberg family for a semigroup S. Then, we give some equivalent conditions for a collection {T-at,T-wt }(t is an element of S) of weighted translation operators on an Orlicz space in the context of a locally compact hypergroup to be topologically multiply 3-recurrent.
. The Angel problem, introduced by Conway in 1982, is a two-player game played on an infinite board where an Angel of power k competes against the Devil. We examine variations of this game on triangular and hexagonal boards. Using M & aacute;th & eacute;'s proof technique originally developed for the square board, we prove that the King (Angel of power 1) can win on a triangular board. Through a mapping between hexagonal and square boards, we then establish two results for the hexagonal board: first, that the Devil can defeat the King, and second, that an Angel of power 2 can win. Our proof for the triangular board involves analyzing the perimeter bounds of connected sets and developing a wall-following strategy for the King, while our hexagonal board results utilize transformations that map to known results from the square board case.
We introduce and explore two novel types of contractions, namely the (psi, a, k)-SM-Bianchini and the generalized (Psi, a, k)-SM-Bianchini type contractions. These contractions represent an extension and generalization of existing contraction principles, allowing for a broader and more flexible framework within the study of fixed-point theory. By incorporating the functions , a, and k, we develop a more comprehensive approach that encompasses and extends various classical results. Moreover, to emphasize the practical significance of our theoretical contributions, we present an application of the generalized (Psi, a, k)-SM-Bianchini type contractions to the split feasibility problem.
We study various types of generating functions for higher-order derangement numbers. We present a generating function for a new class of binomial-type polynomials with coefficients given by higher-order derangement numbers. Using this generating function, we derive explicit formulas for these polynomials, as well as derivative and integral formulas involving Stirling numbers and Cauchy numbers of the first kind. By employing this generating function along with other special formulas, we obtain several new results involving higher-order derangement numbers, Bernoulli numbers, Stirling numbers, combinatorial numbers associated with Daehee numbers, Peters-type Simsek numbers, and special finite sums.
We prove a Steinhaus-type theorem for four-dimensional matrix transformations of double sequences, establishing that (L-1, L-1, P) boolean AND (L-s, L-1) = empty set, s > 1. This extends Fridy's classical result for single sequences. Our results hold for sequences of bounded variation, bounded, Pringsheim convergent, bounded Pringsheim convergent, and regularly convergent sequence spaces. We show this theorem fails in non-archimedean fields through a counterexample.
Let K be a complete, non-trivially valued, ultrametric (or non-archimedean) field. Entries of sequences, infinite series and infinite matrices are in K. In this paper are defined the notions of speed-Maddox spaces over K, where the speed is defined by a sequence μ = {μn} in K with the property 0 < |μn| ↗ ∞, n → ∞. Let λ be another speed in K. The necessary and sufficient conditions for a matrix A would transform all sequences that are λ-convergent to zero over K into the speed-Maddox spaces over K, where the speed is defined by μ.
The rate of convergence of negative order generalized de la Vallee Poussin means of the Fourier series of a 2?-periodic function of p-bounded variation, is estimated. The result can be viewed as generalization of a result of well-known Weiner?s theorem and quantitative version of it.
We generalize a theorem dealing with absolute summability factors of an infinite series to absolute matrix summability under weaker conditions by using an almost increasing sequence.
We first review and analyze the golden integral and its definitions and some properties. Then we introduce a new generalization of the Hermite polynomials via the golden exponential function (called Fibonacci-Hermite polynomials) and investigate several properties and relations. We derive some explicit and implicit summation formulas for mentioned polynomials. Then, we analyze derivative properties and provide a higher-order difference equation of the Fibonacci-Hermite polynomials. Moreover, we examine a recurrence relation and integral representation. In addition, we obtain some properties of Fibonacci-Bernstein polynomials. Lastly, we obtain a correlation between the Fibonacci-Hermite polynomials and the Fibonacci-Bernstein polynomials
We give some conditions for the self-adjoint operators associated with the q-Sturm-Liouville expression ?y := ? 1/q Dq ?1 (p(x)Dqy(x)) + r(x), ? ? < x < ?. to have a discrete spectrum, and investigate the continuous spectra of these operators. We also prove that the regular symmetric q-Sturm-Liouville operator is semi-bounded from below which is not studied in literature yet.
We use the equivariant version of factorization homology constructed using the parametrized higher category theory and show that it can be used to describe the results used in the series of papers.
The bicategorical point of view provides a natural setting for many concepts in the representation theory of monoidal categories. We show that centers of twisted bimodule categories correspond to categories of 2dimensional natural transformations and modifications between the deloopings of the twisting functors. This explains conceptually the lifting of (rigid) dualities to centers of twisted bimodule categories. Inspired by the notion of (pre)bimonoidal functors due to McCurdy and Street and by bilax functors of Aguiar and Mahajan, we study 2-dimensional functors which are simultaneously lax and colax with a compatibility condition. Our approach is build upon a 2-categorical Yang-Baxter operator. We show how this concept, which we call a bilax functor, generalizes many known notions from the theory of Hopf algebras. We propose a 2-category of bilax functors whose 1-cells generalize Yetter-Drinfel'd modules in ordinary categories. We prove that the 2-category of bilax functors from the trivial 2-category is isomorphic to the 2-category of bimonads, and construct a faithful 2-functor from the latter to the 2-category of mixed distributive laws of Power and Watanabe.
Under the condition that the Prym map is injective in characteristic p, we prove that the special subvarieties in the moduli space of abelian varieties of dimension l and polarization type D, Al,D , arising from families of abelian covers of P 1 are of a very restrictive nature. In other words, if the family is one-dimensional or if it contains an eigenspace of certain type for the group action on the cohomology of fibers, then the Shimura varieties arising from such families can only be constructed by the group action of the family.
We consider fixed-circle problem in C*-algebra valued metric spaces and prove some fixed-circle theorems for self-mappings by defining the notion of fixed-circle on such spaces with geometric interpretation. Furthermore, we give some illustrative examples to substantiate the importance of our newly obtained results.
