Class Of Generalized Functions Research Articles

On some inequalities for uniformly convex mapping with estimations to normal distributions

In this paper, we introduce notable Jensen–Mercer inequality for a general class of convex functions, namely uniformly convex functions. We explore some interesting properties of such a class of functions along with some examples. As a result, we establish Hermite–Jensen–Mercer inequalities pertaining uniformly convex functions by considering the class of fractional integral operators. Moreover, we establish Mercer–Ostrowski inequalities for conformable integral operator via differentiable uniformly convex functions. Finally, we apply our inequalities to get estimations for normal probability distributions (Gaussian distributions).

Monge solutions and uniqueness in multi-marginal optimal transport via graph theory

We study a multi-marginal optimal transport problem with surplus b(x1,…,xm)=∑{i,j}∈Pxi⋅xj, where P⊆Q:={{i,j}:i,j∈{1,2,...m},i≠j}. We reformulate this problem by associating each surplus of this type with a graph with m vertices whose set of edges is indexed by P. We then establish uniqueness and Monge solution results for two general classes of surplus functions. Among many other examples, these classes encapsulate the Gangbo and Świȩch surplus [15] and the surplus ∑i=1m−1xi⋅xi+1+xm⋅x1 studied in an earlier work by the present authors [26].

Two types of the second Hankel determinant for the class U and the general class S

In this paper we determine the upper bounds of the Hankel determinants of special type H2(3)(f) and H2(4)(f) for the general class of univalent functions and for the class U.

Parameterized transformations and truncation: When is the result a copula?

Our starting point are several general classes of real functions defined on the unit square satisfying some basic properties such as a boundary condition or several types of monotonicity and continuity. Applying to these functions some parameterized transformations and other constructions such as the transpose and flipping (which describe different aspects of symmetry) and truncation, we ask for conditions yielding (again) a bivariate copula. Some of these transformations are involutive (on one or more classes of functions), others are not even injective, and occasionally they induce additional properties, yielding, e.g., a (quasi-)copula. For several typical scenarios we identify the (not necessarily convex) sets of parameters leading to a copula and conditions imposing a minimal set of parameters.

High-order asymptotic expansions of Gaussian quadrature rules with classical and generalized weight functions

Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and explicitly list asymptotic expressions for the points and weights of Gaussian quadrature rules for three general classes of positive weight functions: analytic functions on a bounded interval with algebraic singularities at the endpoints, analytic weight functions on the halfline with exponential decay at infinity and an algebraic singularity at the finite endpoint, and analytic functions on the real line with exponential decay in both directions at infinity. The results include the Gaussian rules of classical orthogonal polynomials (Legendre, Jacobi, Laguerre and Hermite) as special cases. Explicit expressions for these cases are included in the appendix. We present experiments indicating the range of the number of points at which these expressions achieve high precision. We provide an algorithm that can compute arbitrarily many terms in these expansions for the classical cases, and many though not all terms for the generalized cases.

A Generalized Version of Polynomial Convex Functions and Some Interesting Inequalities with Applications

In this article, we introduce a general class of convex functions and proved some of its basic properties. We establish Hermite-Hadamard type inequalities as well as fractional version of Hermite-Hadamard type inequalities by using Riemann-Liouville integral operator. At the end, some application to special means of real numbers are also given. It can be observed from the remarks given in this paper that several exiting results of ligature can be obtained immediacy from our results by taking suitable involved parameters.

Preparing valence-bond-solid states on noisy intermediate-scale quantum computers

Quantum state preparation is a key step in all digital quantum simulation algorithms. Here we propose methods to initialize on a gate-based quantum computer a general class of quantum spin wave functions, the so-called valence-bond-solid (VBS) states, that are important for two reasons. First, VBS states are the exact ground states of a class of interacting quantum spin models introduced by Affleck, Kennedy, Lieb, and Tasaki (AKLT). Second, the two-dimensional VBS states are universal resource states for measurement-based quantum computing. We find that schemes to prepare VBS states based on their tensor-network representations yield quantum circuits that are too deep to be within reach of noisy intermediate-scale quantum (NISQ) computers. We then apply the general nondeterministic method herein proposed to the preparation of the spin-1 and spin-$3/2$ VBS states, the ground states of the AKLT models defined in one dimension and in the honeycomb lattice, respectively. Shallow quantum circuits of depth independent of the lattice size are explicitly derived for both cases, making use of optimization schemes that outperform standard basis gate decomposition methods. The probabilistic nature of the proposed routine translates into an average number of repetitions to successfully prepare the VBS state that scales exponentially with the number of lattice sites $N$. However, two strategies to quadratically reduce this repetition overhead for any bipartite lattice are devised. Our approach should permit to use NISQ processors to explore the AKLT model and variants thereof, outperforming conventional numerical methods in the near future.

Fixed-Domain Posterior Contraction Rates for Spatial Gaussian Process Model with Nugget

Spatial Gaussian process regression models typically contain finite dimensional covariance parameters that need to be estimated from the data. We study the Bayesian estimation of covariance parameters including the nugget parameter in a general class of stationary covariance functions under fixed-domain asymptotics, which is theoretically challenging due to the increasingly strong dependence among spatial observations. We propose a novel adaptation of the Schwartz’s consistency theorem for showing posterior contraction rates of the covariance parameters including the nugget. We derive a new polynomial evidence lower bound, and propose consistent higher-order quadratic variation estimators that satisfy concentration inequalities with exponentially small tails. Our Bayesian fixed-domain asymptotics theory leads to explicit posterior contraction rates for the microergodic and nugget parameters in the isotropic Matérn covariance function under a general stratified sampling design. We verify our theory and the Bayesian predictive performance in simulation studies and an application to sea surface temperature data. Supplementary materials for this article are available online.

Estimates for singular numbers of Hausdorff–Zhu operators and applications

We continue the study of the so‐called Hausdorff–Zhu operators. In this paper, we study the behavior of the singular values of such operators. We prove the general fact that the sequence of singular numbers tends to zero, as well as we prove power‐type estimates for such behavior under additional conditions on the kernel of the operator. We give application of these results to the boundedness of Hausdorff–Zhu operators in general classes of analytic functions in the unit disc and also in some special classes of analytic functions defined in terms of conditions on Taylor or Fourier coefficients.

Advanced Information Securing by Combining Fortified Binary Image Steganography and Asymmetric Encryption Standard

In information security, image steganography is one of the information securing process. Image steganography is mostly used in medical fields for hiding medical prescription data behind the medical images. Designing steganographic algorithms for empirical cover sources, such as digital images, is very perplexing due to the lack of accurate models. The most successful approach today avoids estimating the cover source distribution because this task is infeasible for complex and non-stationary sources. Instead, the steganography problem is formulated as source coding with fidelity constraint the sender entrenches the message with minimizing the distortion. Practical algorithms that entrench near the theoretical payload–distortion bound are available for a very general class of distortion functions. With the current framework, the only task left to the sender is essential for the design of the distortion function. Here information is which wants to be confidential that information's are stored behind the image pixels. However, due to increasing crypt analysis attacks and binary image de stenographic process image steganography has very less security. So improving the security in image steganography process, the current system provide the asymmetric encryption standard for improving more security in steganography.

Multipleμ‐stability analysis of time‐varying delayed quaternion‐valued neural networks

This article addresses the multiple ‐stability analysis of ‐dimensional quaternion‐valued neural networks (QVNNs) with unbounded time‐varying delays (UTVD) and two general classes of activation functions (AFs). Firstly, the QVNNs are decomposed into four equivalent real‐valued systems, and based on the geometrical configuration of the AFs, the state space is divided into disjoint regions. Considering the properties of AFs, several sufficient conditions are derived to ensure the coexistence of equilibria, out of which are locally ‐stable. Moreover, some sufficient conditions for multiple exponential stability, multiple power stability, and multiple log stability are also derived in this article. Finally, two numerical examples are presented. The first example validates the effectiveness of the proposed theoretical results while the second example illustrates the application of QVNNs on the associative memory, which shows that QVNNs have the ability to reliably retrieve true‐color image patterns.

Multistability of Fuzzy Neural Networks With a General Class of Activation Functions and State-Dependent Switching Rules

This paper addresses the multistability of switched fuzzy neural networks with a general class of activation functions under state-dependent switching. The existence, stability, and attraction basins of equilibria are analyzed via state-space decomposition based on Brouwer fixed point theorem and M-matrix properties. It is shown that there exist <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$5^{k_{1}}3^{k_{2}}$</tex-math></inline-formula> equilibria, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$3^{k_{1}}2^{k_{2}}$</tex-math></inline-formula> of them are locally exponentially stable under four sets of sufficient conditions for an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -neuron switched network, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k_{1}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k_{2}$</tex-math></inline-formula> are nonnegative integers such that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$0&lt; k_{1}+k_{2}\leq n$</tex-math></inline-formula> . The results reveal that the switched fuzzy neural networks have much more equilibria than conventional fuzzy neural networks. Four numerical examples with simulation results are discussed to substantiate the theoretical results.

Fusion of Distance Measurements Between Agents With Unknown Correlations

Cooperative localization is a promising solution to improve the accuracy and overcome the shortcomings of GNSS. Cooperation is often achieved by measuring the distance between users. To optimally integrate a distance measurement between two users into a navigation filter, the correlation between the errors of their estimates must be known. Unfortunately, in large scale networks the agents cannot compute these correlations and must use consistent filters. A consistent filter provides an upper bound on the covariance of the error of the estimator taking into account all the possible correlations. In this paper, a consistent linear filter for integrating a distance measurement is derived using Split Covariance Intersection. Its analysis shows that a distance measurement between two agents can only benefit one of them, i.e., only one of the two can use the distance measurement to improve its estimator. Furthermore, in some cases, none can. A necessary condition for an agent to benefit from the measurement is given for a general class of objective functions. When the objective function is the trace or the determinant, necessary and sufficient conditions are given.

Miscellaneous results related to the Gaussian product inequality conjecture for the joint distribution of traces of Wishart matrices

This note reports partial results related to the Gaussian product inequality (GPI) conjecture for the joint distribution of traces of Wishart matrices. In particular, several GPI-related results from Wei [32] and Liu et al. [15] are extended in two ways: by replacing the power functions with more general classes of functions, and by replacing the usual Gaussian and multivariate gamma distributional assumptions by the more general trace-Wishart distribution assumption. These findings suggest that a Kronecker product form of the GPI holds for diagonal blocks of any Wishart distribution.

Optimal scaling of MCMC beyond Metropolis

AbstractThe problem of optimally scaling the proposal distribution in a Markov chain Monte Carlo algorithm is critical to the quality of the generated samples. Much work has gone into obtaining such results for various Metropolis–Hastings (MH) algorithms. Recently, acceptance probabilities other than MH are being employed in problems with intractable target distributions. There are few resources available on tuning the Gaussian proposal distributions for this situation. We obtain optimal scaling results for a general class of acceptance functions, which includes Barker’s and lazy MH. In particular, optimal values for Barker’s algorithm are derived and found to be significantly different from that obtained for the MH algorithm. Our theoretical conclusions are supported by numerical simulations indicating that when the optimal proposal variance is unknown, tuning to the optimal acceptance probability remains an effective strategy.

Michaelis–Menten networks are structurally stable

We consider a class of biological networks where the nodes are associated with first-order linear dynamics and their interactions, which can be either activating or inhibitory, are modelled by nonlinear Michaelis–Menten functions (i.e., Hill functions with unitary Hill coefficient), possibly in the presence of external constant inputs. We show that all the systems belonging to this class admit at most one strictly positive equilibrium, which is stable; this property is structural, i.e., it holds for any possible choice of the parameter values, and topology-independent, i.e., it holds for any possible topology of the interaction network. When the network is strongly connected, the strictly positive equilibrium is the only equilibrium of the system if and only if the network includes either at least one inhibiting function, or a strictly positive external input (otherwise, the zero vector is an equilibrium). The proposed stability results hold also for more general classes of interaction functions, and even in the presence of arbitrary delays in the interactions.

Certain Identities of a General Class of Hurwitz-Lerch Zeta Function of Two Variables

In this paper, we introduce a generalized double Hurwitz-Lerch Zeta function and then systematically investigate its properties and various integral representations. Further, we show that these results provide certain known as well as new extensions of earlier stated results of generalized Hurwitz-Lerch Zeta functions.

Characteristics of collective modes in anisotropic plasmas

We use a quasi-particle picture and the hard-loop approximation to study collective modes in anisotropic plasmas. We study a general class of anisotropic distribution functions and derive and solve the dispersion equations. We focus on imaginary modes because of their potential influence on plasma dynamics, and we study their dependence on the parameters that characterize the anisotropy of the system. We show that the strength of the imaginary modes is related to the oblateness of the distribution, and the size of the azimuthal Fourier coefficients.

Overdetermined elliptic problems in onduloid-type domains with general nonlinearities

In this paper, we prove the existence of nontrivial unbounded domains Ω⊂Rn+1,n≥1, bifurcating from the straight cylinder B×R (where B is the unit ball of Rn), such that the overdetermined elliptic problem{Δu+f(u)=0in Ω, u=0on ∂Ω, ∂νu=constanton ∂Ω, has a positive bounded solution. We will prove such result for a very general class of functions f:[0,+∞)→R. Roughly speaking, we only ask that the Dirichlet problem in B admits a nondegenerate solution. The proof uses a local bifurcation argument.

The Computational Complexity of ReLU Network Training Parameterized by Data Dimensionality

Understanding the computational complexity of training simple neural networks with rectified linear units (ReLUs) has recently been a subject of intensive research. Closing gaps and complementing results from the literature, we present several results on the parameterized complexity of training two-layer ReLU networks with respect to various loss functions. After a brief discussion of other parameters, we focus on analyzing the influence of the dimension d of the training data on the computational complexity. We provide running time lower bounds in terms of W[1]-hardness for parameter d and prove that known brute-force strategies are essentially optimal (assuming the Exponential Time Hypothesis). In comparison with previous work, our results hold for a broad(er) range of loss functions, including lp-loss for all p ∈ [0, ∞]. In particular, we improve a known polynomial-time algorithm for constant d and convex loss functions to a more general class of loss functions, matching our running time lower bounds also in these cases.