Inclusion Theorems Research Articles

Spectrum of inner product type integral transformers on Hilbert spaces

In this paper, we have determined the spectrum of inner product type integral transformers on the Hilbert spaces. This has been achieved using the abelian properties of sets and the concept of commutators as in the von Neumann algebras. We have also shown that the spectrum of the inner product type integral transformers is contained in the closure of its numerical range using the classical spectral inclusion theorem.

On Statistical Riemann-Stieltjes Integrablility and Deferred Cesàro Summabilit

In this paper, we first introduce and study the notion of statistical Riemann-Stieltjes sum for the sequence of functions and establish some elementary results based on this notion. Subsequently, we extend this notion to the probability space and demonstrate some new results for sequence of distribution functions. Furthermore, we suggest the deferred Cesàro summability method for Riemann-Stieltjes sum. We then establish various inclusion theorems based on our proposed methods in association with the Riemann-Stieltjes sum for the sequence of usual functions as well as distribution functions in Rn.

Lacunary Statistically Convergence via Modulus Function Sequences

Modifying the definition of density functions is one method used to generalise statistical convergence. In the present study, we use sequences of modulus functions and order $\alpha \in \left( 0,1\right] $ to introduce a new density. Based on this density framework, we define strong $(f_k)$-lacunary summability of order $\alpha $ and $(f_k)$-lacunary statistical convergence of order $\alpha $ for a sequence of modulus functions $(f_k)$. This concept holds an intermediate position between the usual convergence and the statistical convergence for lacunary sequences. We also establish inclusion theorems and relations between these two concepts in the study.

A Novel Clustering Algorithm Integrating Gershgorin Circle Theorem and Nonmaximum Suppression for Neural Spike Data Analysis

(1) Problem Statement: The development of clustering algorithms for neural recordings has significantly evolved, reaching a mature stage with predominant approaches including partitional, hierarchical, probabilistic, fuzzy logic, density-based, and learning-based clustering. Despite this evolution, there remains a need for innovative clustering algorithms that can efficiently analyze neural spike data, particularly in handling diverse and noise-contaminated neural recordings. (2) Methodology: This paper introduces a novel clustering algorithm named Gershgorin—nonmaximum suppression (G–NMS), which incorporates the principles of the Gershgorin circle theorem, and a deep learning post-processing method known as nonmaximum suppression. The performance of G–NMS was thoroughly evaluated through extensive testing on two publicly available, synthetic neural datasets. The evaluation involved five distinct groups of experiments, totaling eleven individual experiments, to compare G–NMS against six established clustering algorithms. (3) Results: The results highlight the superior performance of G–NMS in three out of five group experiments, achieving high average accuracy with minimal standard deviation (SD). Specifically, in Dataset 1, experiment S1 (various SNRs) recorded an accuracy of 99.94 ± 0.01, while Dataset 2 showed accuracies of 99.68 ± 0.15 in experiment E1 (Easy 1) and 99.27 ± 0.35 in experiment E2 (Easy 2). Despite a slight decrease in average accuracy in the remaining two experiments, D1 (Difficult 1) and D2 (Difficult 2) from Dataset 2, compared to the top-performing clustering algorithms in these categories, G–NMS maintained lower SD, indicating consistent performance. Additionally, G–NMS demonstrated robustness and efficiency across various noise-contaminated neural recordings, ranging from low to high signal-to-noise ratios. (4) Conclusions: G–NMS’s integration of deep learning techniques and eigenvalue inclusion theorems has proven highly effective, marking a significant advancement in the clustering domain. Its superior performance, characterized by high accuracy and low variability, opens new avenues for the development of high-performing clustering algorithms, contributing significantly to the body of research in this field.

On ρ-Statistically Convergence Defined by a Modulus Function in Fuzzy Difference Sequences

In this study, we first introduced the definition ∆_ρ^m-statistical convergence for sequences of fuzzy numbers using the generalized difference operator ∆^m. Furthermore, we defined the strong N_F^ρ (∆^m,q)-summable sequence set and the strong N_F^ρ (∆^m,f,q)-summable sequence set for fuzzy difference sequences aided by a modulus function f. Subsequently, we provided certain inclusion theorems between these sets and the S_F^ρ (∆^m ) set.

A new approach to Korovkin-type theorems based on deferred Nörlund summability mean

This paper aims to introduce the notions of deferred Nörlund statistical Riemann integrability and statistical deferred Nörlund Riemann summability for sequence of real-valued functions and to apply them in Korovkin-type new approximations. First, we present an inclusion theorem to understand the connection between these new notions. Then, based on these potential notions we establish new versions of Korovkin-type theorems with three algebraic test functions. Finally, we compute an example, under the consideration of a positive linear operator in association with the Bernstein polynomials to exhibit the effectiveness of our findings.

Bounds for the spectral radius of Schur product of matrices

We establish a new upper bound on the spectral radius for the Schur product of two matrices with nonnegative elements by utilizing the eigenvalue inclusion theorem. The new bound is more useful in practical applications because it is simply based on the elements of two matrices. Additionally, a numerical example is taken into account to show that, in some circumstances, the result achieved is superior to some previously reported results.

On a multivalued prescribed mean curvature problem and inclusions defined on dual spaces

This article addresses two main objectives. First, it establishes a functional analytic framework and presents existence results for a quasilinear inclusion describing a prescribed mean curvature problem with homogeneous Dirichlet boundary conditions, involving a multivalued lower order term. The formulation of the problem is done in the space of functions with bounded variation. The second objective is to introduce a general existence theory for inclusions defined on nonreflexive Banach spaces, which is specifically applicable to the aforementioned prescribed mean curvature problem. This problem can be formulated as a multivalued variational inequality in the space of functions with bounded variation, which, under suitable conditions, is equivalent to an inclusion involving a maximal monotone mapping of type (D) and a generalized pseudomonotone mapping. We prove an abstract existence theorem for inclusions of this form, under some coercivity conditions involving both the maximal monotone and the generalized pseudomonotone mappings.

On ρ-statistical convergence in neutrosophic normed spaces

Abstract In this study, the concept of ρ-statistical convergence with respect to the neutrosophic norm in the neutrosophic normed spaces is introduced. Some properties and some inclusion theorems related to this concept are investigated.

Double lacunary statistical convergence of Δ-measurable functions on product time scales

Abstract We first present a notion of a double lacunary sequence on product time scales. Using this notion, we define the notions of the double lacunary statistical convergence and double lacunary strongly p-Cesàro summability of 2-multiple functions on product time scales and we study some fundamental properties of both notions. We also present a theorem that connects the above-mentioned two concepts. Furthermore, we define a refinement of a double lacunary sequence on product time scales and provide some fundamental properties as well as inclusion theorems for a refined and a non-refined double lacunary sequence on product time scales.

Filippov’s Theorem for mutational inclusions in a metric space

Filippov’s Theorem for mutational inclusions in a metric space

New summability methods via $\widetilde{\phi}$ functions

In 1971, the definition of Orlicz $\widetilde{\phi}$ functions was introduced by Lindenstrauss and Tzafriri and moreover in 2006, the notion of double lacunary sequences was presented by Savaş and Patterson. The primary focus of this article is to introduce the double statistically $\widetilde{\phi }$-convergence and double lacunary statistically $\widetilde{\phi }$-convergence which are generalizations of the double statistically convergence [19] and double lacunary statistically convergence [24]. Additionally, some essential inclusion theorems are examined.

Geometric Properties of a Linear Complex Operator on a Subclass of Meromorphic Functions: An Analysis of Hurwitz-Lerch-Zeta Functions

Abstract Geometric function theory (GFT) is one of the richest research disciplines in complex analysis. This discipline also deals with the extended differential inequality theory, known as the differential subordination theory. Based on these theories, this study focuses on analyzing intriguing aspects of the geometric subclass of meromorphic functions in terms of a linear complex operator and a special class of Hurwitz-Lerch-Zeta functions. Hence, several of its geometric attributes are deduced. Furthermore, the paper highlights the different fascinating advantages and applications of various new geometric subclasses in relation to the subordination and inclusion theorems.

Fixed-time synchronization for quaternion-valued memristor-based neural networks with mixed delays

In this paper, the fixed-time synchronization (FXTSYN) of unilateral coefficients quaternion-valued memristor-based neural networks (UCQVMNNs) with mixed delays is investigated. A direct analytical approach is suggested to obtain FXTSYN of UCQVMNNs utilizing one-norm smoothness in place of decomposition. When dealing with drive–response system discontinuity issues, use the set-valued map and the differential inclusion theorem. To accomplish the control objective, innovative nonlinear controllers and the Lyapunov functions are designed. Furthermore, some criteria of FXTSYN for UCQVMNNs are given using inequality techniques and the novel FXTSYN theory. And the accurate settling time is obtained explicitly. Finally, in order to show that the obtained theoretical results are accurate, useful, and applicable, numerical simulations are presented at the conclusion.

New Z-eigenvalue inclusion theorem of tensors with application to the geometric measure of entanglement

New Z-eigenvalue inclusion theorem of tensors with application to the geometric measure of entanglement

Comparison theorems for evolution inclusions with maximal monotone operators. $L^2$-theory

An evolution inclusion with time-dependent family of maximal monotone operators in considered in a separable Hilbert space. If the elements with minimum norm of the family of maximal monotone operators satisfy certain growth conditions, then the domains of definition of this family are closed convex sets. Hence the sweeping process is well defined, whose values are the normal cones of the domains of definition of maximal monotone operators. It is shown that if the sweeping process has a solution for each single-valued perturbation from the space of integrable functions, then the evolution inclusion with the maximal monotone operators and single-valued perturbations from the space of integrable functions is also solvable. Quite general conditions in terms of the properties of the family of maximal monotone operators that ensure the existence of solutions for the sweeping process are presented. All results obtained and the approach presented are new. They are used to prove an existence theorem for evolution inclusions with multivalued perturbations, whose values are closed nonconvex sets. Bibliography: 19 titles.

Some new inequalities for nonnegative matrices involving Schur product

<abstract> <p>In this study, we focused on the spectral radius of the Schur product. Two new types of the upper bound of $ \rho \left({M \circ N} \right) $, which is the spectral radius of the Schur product of two matrices $ M, N $ with nonnegative elements, were established using the Hölder inequality and eigenvalue inclusion theorem. In addition, the obtained new type upper bounds were compared with the classical conclusions. Numerical examples demonstrated that the new type of upper formulas improved the result of Johnson and Horn effectively in some cases, and were sharper than other existing results.</p> </abstract>

Deferred statistical boundedness on time scales

In this paper, we first define deferred statistical boundedness of $ \Delta $-measurable real valued functions on an arbitrary time scale. Some theorems relating deferred statistical convergence and deferred statistical boundedness on time scales have been included. Furthermore, we established some inclusion theorems relating statistical boundedness and deferred statistical boundedness on time scales. Finally, we have provided a general description of inclusion between two deferred statistical methods of boundedness on time scales.

Domain of Jordan Totient Matrix in the Space of Almost Convergent Sequences

In this paper, the notion of almost convergence is used to obtain a space as the domain of a regular matrix. After defining a new type of core for complex-valued sequences, certain inclusion theorems are proved.

Joint spectrum and a spectral inclusion theorem for tensor product of semigroups on locally convex spaces

Joint spectrum and a spectral inclusion theorem for tensor product of semigroups on locally convex spaces