174 Results Found

The purpose of this paper is to prove certain refinements of Ostrowski’s inequality in an inner product space. We study extensions of Ostrowski type inequalities in a 2-inner product space. Finally, some applications which are related to the Chebyshe...

The aim of this paper is to investigate when a linear normed space is an inner product space. Several conditions in a linear normed space are formulated with the help of inequalities. Some of them are from the literature and others are new. We prove...

The aim of this paper is to provide a suitable definition for the concept of fuzzy inner product space. In order to achieve this, we firstly focused on various approaches from the already-existent literature. Due to the emergence of various studies o...

First, we show that the non-trivial fuzzy inner product space under the linearity condition does not exist, which means a fuzzy inner product space with linearity produces only a crisp real number for each pair of vectors. If the positive-definitenes...

There is increasing focus on the difficult challenge of realizing coordinated development of production, living and ecological spaces within the regional development process. An ecological–production–living space evaluation index system w...

The paper is concerned with complex fuzzy numbers and complex fuzzy inner product spaces. In the classical complex number set, a complex number can be expressed using the Cartesian form or polar form. Both expressions are needed because one expressio...

Let $G_p(x,y,z)={\parallel x+y+z\parallel }^p+{\parallel z\parallel }^p{-\parallel x+z\parallel }^p-{\parallel y+z\parallel }^p$ be defined on a normed space $\mathcal{X}$. The special case $G_2(x,y,z)=0,\forall z\in \mathcal{X}$, where $\mathcal{X}$ is a real normed linear space, coincides with the trapezoid o...

In this article, we prove an orthogonal decomposition theorem for real inner product gyrogroups, which unify some well-known gyrogroups in the literature: Einstein, Möbius, Proper Velocity, and Chen’s gyrogroups. This leads to the study of...

Let (X,d) be a metric linear space and a∈X. The point a divides the space into three sets: H_a = {x ∈ X: d(0,x) < d(x,a)}, M_a = {x ∈ X: d(0,x) = d(x,a)} and L_a = {x ∈ X: d(0,x) > d(x,a)}. If the distance is generated by a nor...

The aim of this article is to establish several estimates of the triangle inequality in a normed space over the field of real numbers. We obtain some improvements of the Cauchy–Schwarz inequality, which is improved by using the Tapia semi-inner...

In this article, we established new results related to a 2-pre-Hilbert space. Among these results we will mention the Cauchy-Schwarz inequality. We show several applications related to some statistical indicators as average, variance and standard dev...

We consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a time-independent Hilbe...

We show how to obtain new results on the Ulam stability of the quadratic equation $q(a+b)+q(a-b)=2q\left(a\right)+2q\left(b\right)$ using the Banach limit and the fixed point theorem obtained quite recently for some function spaces. The equation is modeled on the para...

In this paper, we propose a definition for a semi-inner product in the space of p-summable sequences equipped with an n-norm. Using this definition, we introduce the concepts of $p_n$-orthogonality and the $p_n$-angle between two vectors in the s...

Rapid urbanization is causing ecological and environmental issues to worsen. The stability of the ecosystem function of the farming–pastoral ecotone (FPE) in Inner Mongolia is essential to ensuring the sustained growth of the nearby cities, act...

This work aims to develop a new class of accretive mappings and investigate its associated class of proximal mappings. This new class of accretive mappings is known as generalized $(H^k,\phi )$-$\eta$-accretive mappings. Further, the research work includ...

A unitary-evolution process leading to an ultimate collapse and to a complete loss of observability alias quantum phase transition is studied. A specific solvable $N\mathrm{-}$state model is considered, characterized by a non-stationary non-Hermitian Hami...

In this study, we defined a kind of Fourier expansion of set-valued square-integrable functions. In fact, we have seen that the classical Fourier basis also constitutes a basis for the Hilbert quasilinear space $L_2(\left[-\pi ,\pi \right],\Omega \left(\mathbb{C}\right))$ of $\&Ome$...

We introduce the generalized helical hypersurface having a space-like axis in five-dimensional Minkowski space. We compute the first and second fundamental form matrices, Gauss map, and shape operator matrix of the hypersurface. Additionally, we comp...

In this paper, the generalized helical hypersurfaces $\mathbf{x}=\mathbf{x}(u,v,w)$ with a time-like axis in Minkowski spacetime ${\mathbb{E}}_1^4$ are considered. The first and the second fundamental form matrices, the Gauss map, and the shape operator matrix of $\mathbf{x}$ are calculated. Mor...

This study takes a detailed look at various inequalities related to the Euclidean operator radius. It examines groups of n-tuple operators, studying how they add up and multiply together. It also uncovers a unique power inequality specific to the Euc...

We analyze the notion of reproducing pairs of weakly measurable functions, a generalization of continuous frames. The aim is to represent elements of an abstract space Y as superpositions of weakly measurable functions belonging to a space $Z:=$...

For a given operator $D\left(t\right)$ of an observable in theoretical parity-time symmetric quantum physics (or for its evolution-generator analogues in the experimental gain-loss classical optics, etc.) the instant $t_{critical}$ of a sp...

In diverse branches of mathematics, several inequalities have been studied and applied. In this article, we improve Furuta’s inequality. Subsequently, we apply this improvement to obtain new radius inequalities that not been reported in the cur...

This paper introduces several refinements of the classical Selberg inequality, which is considered a significant result in the study of the spectral theory of symmetric spaces, a central topic in the field of symmetry studies. By utilizing the contra...

In this paper, we use soft linear operators to introduce the notion of discrete frames on soft Hilbert spaces, which extends the classical notion of frames on Hilbert spaces to the context of algebraic structures on soft sets. Among other results, we...

We study the Ulam-type stability of a generalization of the Fréchet functional equation. Our aim is to present a method that gives an estimate of the difference between approximate and exact solutions of this equation. The obtained estimate depends o...

Two orthonormal bases in the d-dimensional Hilbert space are said to be unbiased if the square modulus of the inner product of any vector of one basis with any vector of the other equals 1 d. The presence of a modulus in the problem of finding a set...

This paper presents new results related to Bombieri’s generalization of Bessel’s inequality in a semi-inner product space induced by a positive semidefinite operator A. Specifically, we establish new inequalities that generalize the class...

In this paper, we prove the Chebyshev-Steffensen inequality involving the inner product on the real m-space. Some upper bounds for the weighted Chebyshev-Steffensen functional, as well as the Jensen-Steffensen functional involving the inner product u...

A non-Hermitian operator H defined in a Hilbert space with inner product $⟨·|·⟩$ may serve as the Hamiltonian for a unitary quantum system if it is $\eta$ -pseudo-Hermitian for a metric operator (positive-definit...

We introduce a quantum geometric tensor in a curved space with a parameter-dependent metric, which contains the quantum metric tensor as the symmetric part and the Berry curvature corresponding to the antisymmetric part. This parameter-dependent metr...

As a key task in machine learning, data classification is essential to find a suitable coordinate system to represent the data features of different classes of samples. This paper proposes the mutual-energy inner product optimization method for const...

We investigate the Hyers–Ulam stability of the well-known Fréchet functional equation that comes from a characterization of inner product spaces. We also show its hyperstability on a restricted domain. We work in the framework of quasi-B...

It is shown that, for both compact and non-compact Lie groups, vector-coherent-state methods provide straightforward derivations of holomorphic representations on symmetric spaces. Complementary vector-coherent-state methods are introduced to derive...

Vectors are almost always introduced as objects having magnitude and direction. Following that idea, textbooks and courses introduce the concept of a vector norm and the angle between two vectors. While this is correct and useful for vectors in two-...

In this paper, we consider a discrete Sobolev inner product involving the Jacobi weight with a twofold objective. On the one hand, since the orthonormal polynomials with respect to this inner product are eigenfunctions of a certain differential opera...

In this paper, we study the learnability of the Boolean inner product by a systematic simulation study. The family of the Boolean inner product function is known to be representable by neural networks of threshold neurons of depth 3 with only $2n+1$ un...

For the inner product space, we have Appolonius’ identity. From this identity, Park and Th. M. Rassias induced and investigated the quadratic functional equation of the Apollonius type. And Park and Th. M. Rassias first introduced an Apollonius...

We address the reasons why the “Wick-rotated”, positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic fra...

We present a novel derivation of the multipole interaction (energies, forces and fields) in spherical harmonics, which results in an expression that is able to exactly reproduce the results of earlier Cartesian formulations. Our method follows the de...

This work investigates the fractional Dirac system that has transmission conditions, and its boundary condition contains an eigenparameter. Defining a convenient inner product space and a new operator that has the same eigenvalues as the considered p...

In this article, we investigate new findings on Boas–Bellman-type inequalities in semi-Hilbert spaces. These spaces are generated by semi-inner products induced by positive and positive semidefinite operators. Our objective is to reveal signifi...

Herpes simplex virus 1 (HSV-1) capsids are assembled in the nucleus bud at the inner nuclear membrane into the perinuclear space, acquiring envelope and tegument. In theory, these virions are de-enveloped by fusion of the envelope with the outer nuc...

In generalized inner product Sobolev spaces we investigate elliptic differential problems with additional unknown functions or distributions in boundary conditions. These spaces are parametrized with a function OR-varying at infinity. This characteri...

In this research, obtaining of approximate solution for fractional-order Burgers’ equation will be presented in reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces are identified according to inner products and norms...

We present a generalization of Bregman divergences in finite-dimensional symplectic vector spaces that we term symplectic Bregman divergences. Symplectic Bregman divergences are derived from a symplectic generalization of the Fenchel–Young ineq...

A quasi-least-squares (QLS) mixed finite element method (MFE) based on the $L^2$-inner product is utilized to solve an incompressible magnetohydrodynamic (MHD) model. These models are associated with the three unknown terms, i.e., fluid velocity, fluid...

In this paper, we will introduce a new geometric constant $L_{\mathrm{YJ}}(\lambda ,\mu ,X)$ based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constan...

The average helicity of a given electromagnetic field measures the difference between the number of left- and right-handed photons contained in the field. Here, the average helicity is derived using the conformally invariant inner product for Maxwell...