Finiteness Conditions Research Articles

Conditions for finiteness and bounds on moments of record values from iid continuous life distributions

We consider the standard and kth record values arising in sequences of independent identically distributed continuous and positive random variables with finite expectations. We determine necessary and sufficient conditions on the type of record k, its number n and moment order r so that the rth moment of the n value of kth record is finite for every parent distribution. Under the conditions we present the optimal upper bounds on these moments expressed in the scale units being the respective powers of the first population moment. The theoretical results are illustrated by some numerical evaluations.

Inhomogeneous chiral condensation under rotation in the holographic QCD

We investigate inhomogeneous chiral condensation under rotation considering finite size effects and boundary conditions in the holographic QCD model. The rotational suppression effect determined by $\Omega r$ is confirmed in the holographic model which is not influenced by the boundary conditions. For chiral condensation at the center, it is found that under Neumann boundary condition the finite size exhibits two opposite effects, i.e., catalysis at high temperatures and inverse catalysis at low temperatures. In contrast, under Dirichlet boundary condition, the effect of finite size on condensation is inverse catalysis, and small size induces a phase transition from inhomogeneous to homogeneous phase. The temperature-angular velocity phase diagrams of QCD are obtained for different boundary conditions and sizes, and it is found that the critical temperature decreases with angular velocity.

Impulsive Boundedness for Nonautonomous Dynamic Complex Networks with Constraint Nonlinearity

This paper discusses time-scale type global boundedness of nonautonomous complex networks (NCNs) under impulsive control. By using constraint nonlinearity of NCNs, properties of exponential function and impulsive inequalities, we establish new sufficient conditions for global boundedness and asymptotic stability of NCNs on time scales, and the upper bound of the system can be calculated easily. It is shown that the constraint nonlinearity and impulse interval length will affect the estimation of state bound of NCNs under different network size. It provides a unified criteria for global boundedness of NCNs on continuous, discrete and mixed-time domains and hence improves corresponding existing ones. The nonlinear constraints studied in this paper are widely applied to Lurie systems, ball motion models and so on. Numerical examples are given to illustrate the correctness of the results. Finally, the theoretical results are applied to Lurie systems and bounded driving of unmanned vehicles.

The impact of conformational sampling on first-principles calculations of vicinal COCH J-couplings in carbohydrates.

Dihedral angles in organic molecules and biomolecules are vital structural parameters that can be indirectly probed by nuclear magnetic resonance (NMR) measurements of vicinal J-couplings. The empirical relations that map the measured couplings to dihedral angles are typically determined by fitting using static structural models, but this neglects the effects of thermal fluctuations at the finite temperature conditions under which NMR measurements are often taken. In this study, we calculate ensemble-averaged J-couplings for several structurally rigid carbohydrate derivatives using first-principles molecular dynamics simulations to sample the thermally accessible conformations around the minimum energy structure. Our results show that including thermal fluctuation effects significantly shifts the predicted couplings relative to single-point calculations at the energy minima, leading to improved agreement with experiments. This provides evidence that accounting for conformational sampling in first-principles calculations can improve the accuracy of NMR-based structure determination for structurally complex carbohydrates.

On the Convergence of Overlapping Schwarz Decomposition for Nonlinear Optimal Control

We study the convergence properties of an overlapping Schwarz decomposition algorithm for solving nonlinear optimal control problems (OCPs). The algorithm decomposes the time domain into a set of overlapping subdomains, and solves all subproblems defined over subdomains in parallel. The convergence is attained by updating primal-dual information at the boundaries of overlapping subdomains. We show that the algorithm exhibits local linear convergence, and that the convergence rate improves exponentially with the overlap size. We also establish global convergence results for a general quadratic programming, which enables the application of the Schwarz scheme inside second-order optimization algorithms (e.g., sequential quadratic programming). The theoretical foundation of our convergence analysis is a sensitivity result of nonlinear OCPs, which we call “exponential decay of sensitivity” (EDS). Intuitively, EDS states that the impact of perturbations at domain boundaries (i.e., initial and terminal time) on the solution decays exponentially as one moves into the domain. Here, we expand a previous analysis available in the literature by showing that EDS holds for both primal and dual solutions of nonlinear OCPs, under uniform second-order sufficient condition, controllability condition, and boundedness condition. We conduct experiments with a quadrotor motion planning problem and a partial differential equations (PDE) control problem to validate our theory, and show that the approach is significantly more efficient than alternating direction method of multipliers and as efficient as the centralized interior-point solver.

Finiteness conditions for performance indices in generalized fractional-order systems defined based on the regularized Prabhakar derivative

In the present paper, we introduce the generalized fractional-order control systems with the regularized Prabhakar fractional derivative and investigate the BIBO stability for this kind of systems. For such kind of systems with step functions at their inputs, we establish some conditions imposed on the parameters of the corresponding fractional-order transfer functions under which the steady state error is zero and the performance indices are finite. Further, in order to examine the analytical obtained results, several generalized fractional-order transfer functions are considered. Then, we show that the parameters of the regularized Prabhakar fractional derivative play significant roles in finiteness of the performance indices in the systems.

Dynamical Analysis of Discrete-Time Two-Predators One-Prey Lotka–Volterra Model

This research manifesto has a comprehensive discussion of the global dynamics of an achievable discrete-time two predators and one prey Lotka–Volterra model in three dimensions, i.e., in the space R3. In some assertive parametric circumstances, the discrete-time model has eight equilibrium points among which one is a special or unique positive equilibrium point. We have also investigated the local and global behavior of equilibrium points of an achievable three-dimensional discrete-time two predators and one prey Lotka–Volterra model. The conversion of a continuous-type model into its discrete counterpart model has been completed by adopting a dynamically consistent nonstandard difference scheme with the end goal that the equilibrium points are conserved in twin cases. The difficulty lies in how to find all fixed points O,P,Q,R,S,T,U,V and the Jacobian matrix and its characteristic polynomial at the unique positive fixed point. For that purpose, we use Mathematica software to find the equilibrium points and all of the Jacobian matrices at those equilibrium points. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of the obtained system about all of its equilibrium points. The discrete Lotka–Volterra model in three dimensions is given by system (3), where parameters α,β,γ,δ,ζ,η,μ,ε,υ,ρ,σ,ω∈R+ and initial conditions x0,y0,z0 are positive real numbers. Additionally, the rate of convergence of a solution that converges to a unique positive equilibrium point is discussed. To represent theoretical perceptions, some numerical debates are introduced, including phase portraits.

Stability and Synchronization of Nonautonomous Reaction-Diffusion Neural Networks With General Time-Varying Delays.

This article investigates the stability and synchronization of nonautonomous reaction-diffusion neural networks with general time-varying delays. Compared with the existing works concerning reaction-diffusion neural networks, the main innovation of this article is that the network coefficients are time-varying, and the delays are general (which means that fewer constraints are posed on delays; for example, the commonly used conditions of differentiability and boundedness are no longer needed). By Green's formula and some analytical techniques, some easily checkable criteria on stability and synchronization for the underlying neural networks are established. These obtained results not only improve some existing ones but also contain some novel results that have not yet been reported. The effectiveness and superiorities of the established criteria are verified by three numerical examples.

ОБ ОГРАНИЧЕННОСТИ И КОМПАКТНОСТИ ОДНОГО КЛАССА ОПЕРАТОРОВ ТИПА СВЕРТКИ В ПРОСТРАНСТВАХ МОРРИ

In a Morrey space, we study the integral operator 𝐻𝑏 with the kernel 𝑏(𝑥, 𝑦)ℎ(𝑥 − 𝑦), where ℎ is the summable function. The function 𝑏 will be referred to as a characteristic. For such operator, a sufficient condition of boundedness is obtained. This condition is obviously fulfilled in the case of the essentially bounded function 𝑏(𝑥, 𝑦). It is shown that if the characteristic 𝑏(𝑥, 𝑦) is essentially bounded and has a given behavior at infinity, then the operator 𝐻𝑏 is compact in a Morrey space. As a consequence, we have obtained the criterion for the Fredholm property of the operator, which is the sum of the identity operator and the operator 𝐻𝑏 . Namely, it is shown that such operator is Fredholm if and only if its symbol does not vanish, and the index of the operator is equal to zero.

Equilibrium set-valued variational principles and the lower boundedness condition with application to psychology

We first give a pre-order principle whose form is very general. Combining the pre-order principle and generalized Gerstewitz functions, we establish a general equilibrium version of set-valued Ekeland variational principle (denoted by EVP), where the objective function is a set-valued bimap defined on the product of quasi-metric spaces and taking values in a quasi-ordered linear space, and the perturbation consists of a subset of the ordering cone multiplied by the quasi-metric. From this, we obtain a number of new results which essentially improve the related results. Particularly, the earlier lower boundedness condition has been weakened. Finally, we apply the new EVPs to Psychology.

Admissible edge-dependent finite-time ℋ∞ filtering for discrete-time switched systems with average dwell time

Considering that the property of finite-time convergence is of great significance for practical filters, this article is focused on the finite-time ℋ∞ filtering for discrete-time switched systems with average dwell time. Different from the case of the general mode-dependent switching, the information of the admissible switching edge is fully exploited to derive more preferable filtering scheme. In order to further reduce the design conservatism, the multiple discontinuous Lyapunov function approach, which incorporates the concept of admissible edge-dependent average dwell time, is used to analyze the finite-time stability and ℋ∞ performance of the filtering error system. This approach assigns each subsystem of the error system with a discontinuous function rather than a continuous one, so it is less conservative compared to the conventional multiple Lyapunov function approach. First, the finite-time boundedness and ℋ∞ performance is analyzed for general switched nonlinear systems with admissible edge-dependent average dwell time. Then, a sufficient condition of the finite-time boundedness and ℋ∞ performance is derived for the filtering error system based on the obtained criterion. Subsequently, an admissible edge-dependent finite-time filtering scheme is proposed for the underlying system in terms of linear matrix inequalities. Finally, the proposed filter design scheme is applied to a DC–DC power converter circuit, and its effectiveness and superiority are verified by the simulation results.

Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems

This paper is devoted to the solvability of an initial-boundary value problem for second-order parabolic equations in divergence form with variable order of nonlinearity. The characteristic feature of the considered class of Cauchy-Neumann parabolic problem is the fact that the variable exponent p(t, x) and the anisotropic diffusion tensor D(t, x) are not well predefined a priori, but instead these characteristic depend on a solution of this problem, i.e., p = p(t, x, u) and D = D(t, x, u). Recently, it has been shown that the similar models appear in a natural way as the optimality conditions for some variational problems related to the image restoration technique. However, from practical point of view, in this case some principle difficulties can appear because of the absence of the corresponding rigorous mathematically substantiation. Thus, in this paper, we study the solvability issues of the Cauchy-Neumann parabolic boundary value problem for which the corresponding principle operator is strongly non-linear, non-monotone, has a variable order of nonlinearity, and satisfies a nonstandard coercivity and boundedness conditions that do not fall within the scope of the classical method of monotone operators. To construct a weak solution, we apply the technique of passing to the limit in a special approximation scheme and the Schauder fixed-point theorem.

An adaptive local algorithm for solving the phase-field evolution equation in the phase-field model for fracture

In the phase-field model for fracture, material damage can be characterized by a variable called the crack phase-field. Usually-two sub-problems controlled by the equilibrium equation and the evolution equation, respectively, are needed to be solved in the whole domain. In this paper, an adaptive local algorithm is proposed for the unified phase-field model to solve the evolution equation in the local domain based on the fact that the value of phase-field is non-zero only around the crack. In the initial unified phase-field model, the boundedness and irreversibility conditions are not easy to be satisfied. In this paper, a history field is proposed to convert the inequality into equality in the evolution equation, and the boundedness and irreversibility conditions can be treated easily as in the classical phase-field model. To further obtain the local algorithm, some failure criteria are introduced to identify the damaged area, and the elements in the local domain are updated adaptively in each iteration step. The local algorithm can be implemented easily since the boundary conditions of the local domain are the same as in the non-local algorithm. Numerical examples have shown that the proposed local algorithm can obtain the same results as the non-local algorithm.

Finite and infinite vertex-transitive cubic graphs and their distinguishing cost and density

A set S of vertices in a graph G with nontrivial automorphism group is distinguishing if the identity mapping is the only automorphism that preserves S as a set. If such sets exist, then their minimum cardinality is the distinguishing cost ρ(G) of G. A closely related concept is the distinguishing density δ(G). For finite G it is the quotient of ρ(G) by the order of G. We consider connected, vertex-transitive, cubic graphs G and show that either ρ(G) ≤ 5 or ρ(G) = ∞ and δ(G) = 0 if G has one or three arc-orbits, or two arc-orbits and vertex-stabilisers of order at most 2. For the case of two arc-orbits and vertex stabilizers of order > 2 we show the existence of finite graphs with ρ(G) > 5 and infinite graphs with δ(G) > 0. We also prove that two well known results about finite, vertex-transitive, cubic graphs hold without the finiteness condition and construct infinitely many cubic GRRs.

Necessary and sufficient condition for boundedness of translation operator in de Branges spaces

Necessary and sufficient condition for boundedness of translation operator in de Branges spaces

Hilbert evolution algebras and its connection with discrete-time Markov chains

Evolution algebras are non-associative algebras. In this work we provide an extension of this class of algebras, in a framework of Hilbert spaces, and illustrate the applicability of our approach by discussing a connection with discrete-time Markov chains with infinite countable state space. Specifically, if we associate to each possible state of such a Markov process a generator of the Hilbert evolution algebra structure, then the whole dynamics of the process can be described through consecutive applications of the evolution operator, provided certain boundedness conditions on the transition probabilities hold.

Local dimensions of self-similar measures satisfying the finite neighbour condition* *KEH was supported by NSERC Grant 2016-03719. AR was supported by this grant as well as EPSRC Grant EP/V520123/1.

We study sets of local dimensions for self-similar measures in satisfying the finite neighbour condition, which is formally stronger than the weak separation condition (WSC) but satisfied in all known examples. Under a mild technical assumption, we establish that the set of attainable local dimensions is a finite union of (possibly singleton) compact intervals. The number of intervals is bounded above by the number of non-trivial maximal strongly connected components of a finite directed graph construction depending only on the governing iterated function system. We also explain how our results allow computations of the sets of local dimensions in many explicit cases. This contextualises and generalises a vast amount of prior work on sets of local dimensions for self-similar measures satisfying the WSC.

On finite‐time stability analysis of homogeneous vector fields with multiplicative perturbations

AbstractThe problem of finite‐time and fixed‐time stability analysis is considered for a class of nonlinear systems where is homogeneous of a degree and is bounded. It is shown that under certain conditions on the asymptotic stability of the system implies its finite‐time stability for or nearly fixed‐time stability for . The so‐called homogeneous extensions are utilized for the analysis. The results are generalized to a class of systems where for every are homogeneous of some (possibly different) degrees and satisfy certain boundedness conditions. An example of a mechanical system is presented in the last section to illustrate the obtained results.

Necessary and sufficient conditions for boundedness of commutators associated with Calderón–Zygmund operators on slice spaces

Necessary and sufficient conditions for boundedness of commutators associated with Calderón–Zygmund operators on slice spaces

Feynman-Kac formula under a finite entropy condition

Motivated by entropic optimal transport, we investigate an extended notion of solution to the parabolic equation $( \partial_t + b\cdot \nabla + \Delta _{ a}/2 +V)g =0$ with a final boundary condition. It is well-known that the viscosity solution $g$ of this PDE is represented by the Feynman-Kac formula when the drift $b$, the diffusion matrix $a$ and the scalar potential $V$ are regular enough and not growing too fast. In this article, $b$ and $V$ are not assumed to be regular and their growth is controlled by a finite entropy condition, allowing for instance $V$ to belong to some Kato class. We show that the Feynman-Kac formula represents a solution, in an extended sense, to the parabolic equation. This notion of solution is trajectorial and expressed with the semimartingale extension of the Markov generator $ b\cdot \nabla + \Delta _{ a}/2.$ Our probabilistic approach relies on stochastic derivatives, semimartingales, Girsanov's theorem and the Hamilton-Jacobi-Bellman equation satisfied by $\log g$.