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In this paper, we consider the generalized Motzkin paths whose step set consists of $$E = (1, 0), N= (0,1), U = (1,1)$$ and $$D = (1,-1)$$ . In the general case, for the number of such paths running from (0, 0) to $$(k,n-2k)$$ , we define a number triangle, which turns out to be a common extension of Pascal triangle and Delannoy triangle. Under the restriction of above or below the x-axis, these paths can be seen as an unified generalization of the well-known Dyck paths, Motzkin paths, and Schroder paths. We also consider the counting of such paths above the main diagonal. In every condition, we treat with two classes of paths, which are restricted and unrestricted paths. For each class of paths, the corresponding counting array is a Riordan array. Numerous Combinatorial matrices such as the Catalan matrix, Motzkin matrix, and Schroder matrix are special cases of these Riordan arrays.

The Lyapunov–Floquet (L–F) theory, a powerful result of Floquet theory, is one of the main tools for the stability analysis and study of nonlinear periodic systems. In this paper, Lyapunov–Floquet and modal transformations of the nonlinear approximation model of spacecraft relative motion, containing periodic state dependent coefficients (SDC), are developed. The transformations, advantageously, retain the stability characteristics of the original nonlinear relative motion dynamics. To provide a form to which averaging method can be applied, the nonlinear approximated equation of motion is scaled using the chief’s eccentricity. Through the application of L–F transformation, the linear part with periodic coefficients is reduced to time-invariant form and the transformed periodic nonlinear part has coefficients with true anomaly as the independent variable. The modal transformation reduced LF transformed equation into a form containing Jordan canonical form. Afterward, the averaging method is applied to the L–F transformed equation to describe the evolution of the motion in terms of the average values of the dynamical variables. Both the LF and modal transformed relative motion equations and the averaged equations, with chief in elliptical orbit, present forms that are useful and applicable for spacecraft guidance, navigation and control, formation flying, rendezvous and proximity operations.

This paper proposes a class of randomized Kaczmarz algorithms for obtaining isolated solutions of large-scale well-posed or overdetermined nonlinear systems of equations. This type of algorithm improves the classic Newton method. Each iteration only needs to calculate one row of the Jacobian instead of the entire matrix, which greatly reduces the amount of calculation and storage. Therefore, these algorithms are called matrix-free algorithms. According to the different probability selection patterns of choosing a row of the Jacobian matrix, the nonlinear Kaczmarz (NK) algorithm, the nonlinear randomized Kaczmarz (NRK) algorithm and the nonlinear uniformly randomized Kaczmarz (NURK) algorithm are proposed. In addition, the NURK algorithm is similar to the stochastic gradient descent (SGD) algorithm in nonlinear optimization problems. The only difference is the choice of step size. In the case of noise-free data, theoretical analysis and the results of numerical based on the classical tangential cone conditions show that the algorithms proposed in this paper are superior to the SGD algorithm in terms of iterations and calculation time.

In Treschev (Proc Steklov Math Inst 310:262–290, 2020) we introduce the concept of a $$\mu $$ -norm for a bounded operator in a Hilbert space. The main motivation is the extension of the measure entropy to the case of quantum systems. In this paper we recall the basic results from Treschev (2020) and present further results on the $$\mu $$ -norm. More precisely, we specify three classes of unitary operators for which the $$\mu $$ -norm generates a bistochastic operator. We plan to use the latter in the construction of quantum entropy.

In this paper, we propose a series expansion of the baroclinic torque in low-Mach-number flows, so that the accuracy and universality of any buoyancy term could be examined analytically, and new types of buoyancy terms could be constructed and validated. We first demonstrate that the purpose of introducing a buoyancy term is to approximate the baroclinic torque, and straightforwardly the error of any buoyancy term could be defined with the deviation of its curl from the corresponding baroclinic torque. Then a regular perturbation method is introduced for the elliptic equation of the hydrodynamic pressure in low-Mach-number flows, resulting in a sequence of Poisson equations, whose solutions lead to the series representation of the baroclinic torque and the new types of buoyancy terms. It is found that the frame invariance of the momentum equation is maintained with one of the new types of buoyancy terms. With the error definition of buoyancy terms and the series representation of the baroclinic torque, the validity and accuracy of previous and new buoyancy terms are examined. Finally, numerical simulations confirm that, with a decreasing density variation or an increasing order of our new buoyancy term, the simplified equations can converge to the original low-Mach-number equations.

This paper studies the propagation of the short pulse optics model governed by the higher-order nonlinear Schrödinger equation (NLSE) with non-Kerr nonlinearity. Exact one-soliton solutions are derived for a generalized case of the NLSE with the aid of software symbolic computations. The modified Kudryashov simple equation method (MSEM) is employed for this purpose under some parametric constraints. The computational work shows the difference, effectiveness, reliability, and power of the considered scheme. This method can treat several complex higher-order NLSEs that arise in mathematical physics. Graphical illustrations of some obtained solitons are presented.

In this paper, we investigate $$L^{p}$$ estimates ( $$1\leq p\leq+\infty$$ ) and $$f$$ -Holder estimates for the Cauchy–Riemann equation in a class of convex domains of finite or infinite type with piecewise smooth boundary in $$\mathbb{C}^{2}$$ .

In this paper, we investigate a class of Schrödinger–Poisson systems with critical growth. By the principle of concentration compactness and variational methods, we prove that the system has radially symmetric solutions, which improve the related results on this topic.

In this paper, we consider a microorganism flocculation model with stochastic perturbation and time delay. By constructing some suitable Lyapunov functions, the asymptotic behaviors of the solution of the stochastic model around the equilibria of the corresponding deterministic model are investigated. We find that the average in time of the second moment of the solutions of the stochastic system is bounded to a relatively small noise.

In this paper, we investigate the Dirichlet and Neumann eigenvalue problems in bounded domains on a complete self-shrinker, then we get some lower bound estimates for the first non-zero eigenvalue of $${\mathfrak {L}}$$ .

This paper is concerned with the event-triggered positive l 1 -gain non-fragile filter design of positive Markov jump systems. By using the upper and lower bounds of the error vector of system measurement outputs, the positive l 1 -gain filtering system to be designed is transformed into interval uncertain systems. Combining linear Lyapunov functions and linear programming, the positivity and stability of such systems can be obtained. Then, the proposed approach is developed to investigate positive l 1 -gain additive and multiplicative non-fragile filters, respectively. Furthermore, two classes of non-fragile positive filters are proposed for positive Markov jump systems with incomplete measurements and sensors saturation, respectively. Finally, two examples are provided to verify the effectiveness of the proposed design.

This paper proposed a chaotic system of N-level logistic cascaded homogenization mapping (N-LLCHM). The cascade structure of multiple one-dimensional logistic mappings results in an exponential increase in the number of fixed points in the mapping, which greatly increases the initial error divergence of the system during the iteration process, and ultimately increases the complexity of the system to generate time series. In the experiment, the homogenization adjustment function of the mapping is derived according to the maximum entropy theorem. The homogenization adjustment function and the N-level logistic cascade structure are cascaded again to construct a complete dynamics system. The analyses of indicators such as information entropy, Lyapunov exponent, spectral entropy and NIST SP800-22 randomness test reveal that the mapping generates a chaotic time series with good aperiodic and uniform distribution characteristics under the condition that N is greater than or equal to 4. Then, N-LLCHM is used as the core of pseudo-random number generator, and the color image encryption algorithm is constructed by using three steps of pixel bidirectional cross-coding, position scrambling and pixel diffusion as the main system. According to the evaluation of histogram, correlation, information entropy and anti-differential attack test, it is verified that this algorithm has better encryption effect and higher security than the present image encryption algorithms.

We develop a unified convergence analysis of three-step iterative schemes for solving nonlinear Banach space valued equations. The local convergence order has been shown before to be five on the finite dimensional Euclidean space assuming Taylor expansions and the existence of the sixth derivative not on these schemes. So, the usage of them is restricted six or higher differentiable mappings. But in our paper only the first Frèchet derivative is utilized to show convergence. Consequently, the scheme is expanded. Numerical applications are also given to test convergence.

In this paper, a vector-borne disease model with two delays and reinfection is established and considered. First of all, the existence of the equilibrium of the system, under different cases of two delays, is discussed through analyzing the corresponding characteristic equation of the linear system. Some conditions that the system undergoes Hopf bifurcation at the endemic equilibrium are obtained. Furthermore, by employing the normal form method and the center manifold theorem for delay differential equations, some explicit formulas used to describe the properties of bifurcating periodic solutions are derived. Finally, the numerical examples and simulations are presented to verify our theoretical conclusions. Meanwhile, the influences of the degree of partial protection for recovered people acquired by a primary infection on the endemic equilibrium and the critical values of the two delays are analyzed.

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.

This work presents some extensions and improvements of former results that allow proving asymptotic stability, uniform stability and global uniform asymptotic stability of zero solution to a class of non-linear Volterra integro-differential equations (VIDEs). Via the Lyapunov–Krasovskiĭ and the Lyapunov–Razumikhin methods, three new results are proved on the mentioned concepts. These results are proved using Lyapunov functional and quadratic Lyapunov function. The results of this paper improve and extend the known ones in the literature. Some examples are given to validate these results and the concepts introduced.

In this paper, a stochastic eco-epidemiological system with patchy structure and transport-related infection is proposed and the stochastic dynamical behaviors are investigated. Firstly, by constructing suitable Lyapunov functions, it is revealed that there is a unique globally positive solution starting from the positive initial value. Secondly, it is proved that the presented system is stochastically ultimately bounded and the average in time of the second moment of solution is bounded. Thirdly, we prove that the large enough stochastic perturbations may lead the predator population and the diseases in the predator to be extinct while it is persistent in the deterministic system. Finally, some numerical simulations are given to test our theoretical results.

This paper studies an interaction between one prey and one predator following Lotka-Volterra model with additive Allee effect in predator. The Atangana-Baleanu fractional-order derivative is used for the operator. Since the theoretical ways to investigate the model using this operator are limited, the dynamical behaviors are identified numerically. By simulations, the influence of the order of the derivative on the dynamical behaviors is given. The numerical results show that the order of the derivative may impact the convergence rate, the occurrence of Hopf bifurcation, and the evolution of the diameter of the limit-cycle.

Let be a connected graph. Let be a subset of V with an order imposed on W. The k-vector is called the resolving vector of v with respect to W. The set W is called a resolving set if for any two distinct vertices In this paper we investigate the existence of independent resolving sets in Cartesian product and corona of graphs.

It is generally admitted that many financial time series have heavy tailed marginal distributions. When time series models are fitted on such data, the non-existence of appropriate moments may invalidate standard statistical tools used for inference. Moreover, the existence of moments can be crucial for risk management, for instance when risk is measured through the expected shortfall. This paper considers testing the existence of moments in the framework of GARCH processes. While the second-order stationarity condition does not depend on the distribution of the innovation, higher-order moment conditions involve moments of the independent innovation process. We propose tests for the existence of high moments of the returns process which are based on the joint asymptotic distribution of the Quasi-Maximum Likelihood (QML) estimator of the volatility parameters and empirical moments of the residuals. A bootstrap procedure is proposed to improve the finite-sample performance of our test. To achieve efficiency gains we consider non Gaussian QML estimators founded on reparameterizations of the GARCH model, and we discuss optimality issues. Monte Carlo experiments and an empirical study illustrate the asymptotic results.

In this paper, we consider symmetric $$\alpha $$ -stable processes on (unbounded) horn-shaped regions which are non-uniformly $$C^{1,1}$$ near infinity. By using probabilistic approaches extensively, we establish two-sided Dirichlet heat kernel estimates of such processes for all time. The estimates are very sensitive with respect to the reference function corresponding to each horn-shaped region. Our results also cover the case that the associated Dirichlet semigroup is not intrinsically ultracontractive. A striking observation from our estimates is that, even when the associated Dirichlet semigroup is intrinsically ultracontractive, the so-called Varopoulos-type estimates do not hold in general for symmetric stable processes on horn-shaped regions.

Poisson shot noise processes are natural generalizations of compound Poisson processes that have been widely applied in insurance, neuroscience, seismology, computer science and epidemiology. In this paper we study sharp deviations, fluctuations and the stable probability approximation of Poisson shot noise processes. Our achievements extend, improve and complement existing results in the literature. We apply the theoretical results to Poisson cluster point processes, including generalized linear Hawkes processes, and risk processes with delayed claims. Many examples are discussed in detail.

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic edge coloring conjecture by Fiamcik (1978) and Alon, Sudakov and Zaks (2001) states that every simple graph with maximum degree Δ is acyclically edge ( Δ + 2 ) -colorable. Despite many milestones, the conjecture remains open even for planar graphs. In this paper, we confirm affirmatively the conjecture on planar graphs without intersecting triangles. We do so by first showing, by discharging methods, that every planar graph without intersecting triangles must have at least one of the six specified groups of local structures, and then proving the conjecture by recoloring certain edges in each such local structure and by induction on the number of edges in the graph.

Classical Goncarov polynomials arose in numerical analysis as a basis for the solutions of the Goncarov interpolation problem. These polynomials provide a natural algebraic tool in the enumerative theory of parking functions. By replacing the differentiation operator with a delta operator and using the theory of finite operator calculus, Lorentz, Tringali and Yan introduced the sequence of generalized Goncarov polynomials associated to a pair ( Δ , Z ) of a delta operator Δ and an interpolation grid Z . Generalized Goncarov polynomials share many nice algebraic properties and have a connection with the theories of binomial enumeration and order statistics. In this paper we give a complete combinatorial interpretation for any sequence of generalized Goncarov polynomials. First we show that they can be realized as weight enumerators in partition lattices. Then we give a more concrete realization in exponential families and show that these polynomials enumerate various enriched structures of vector parking functions.

In this paper, we extend existing population growth models and propose a model based on a nonlinear cubic differential equation that reveals itself as a special subclass of Abel differential equations of first kind. We first summarize properties of the time-continuous problem formulation. We state the boundedness, global existence, and uniqueness of solutions for all times. Proofs of these properties are thoroughly given in the Appendix to this paper. Subsequently, we develop an explicit–implicit time-discrete numerical solution algorithm for our time-continuous population growth model and show that many properties of the time-continuous case transfer to our numerical explicit–implicit time-discrete solution scheme. We provide numerical examples to illustrate different behaviors of our proposed model. Furthermore, we compare our explicit–implicit discretization scheme to the classical Eulerian discretization. The latter violates the nonnegativity constraints on population sizes, whereas we prove and illustrate that our explicit–implicit discretization algorithm preserves this constraint. Finally, we describe a parameter estimation approach to apply our algorithm to two different real-world data sets.

In this paper, we focus on the synchronization of fractional-order coupled neural networks (FCNNs). First, by taking information on activation functions into account, we construct a convex Lur’e–Postnikov Lyapunov function. Based on the convex Lyapunov function and a general convex quadratic function, we derive a novel Mittag-Leffler synchronization criterion for the FCNNs with symmetrical coupled matrix in the form of linear matrix inequalities (LMIs). Then we present a robust Mittag-Leffler synchronization criterion for the FCNNs with uncertain parameters. These two Mittag-Leffler synchronization criteria can be solved easily by LMI tools in Matlab. Moreover, we present a novel Lyapunov synchronization criterion for the FCNNs with unsymmetrical coupled matrix in the form of LMIs, which can be easily solved by YALMIP tools in Matlab. The feasibilities of the criteria obtained in this paper are shown by four numerical examples.

The constant elasticity of substitution (CES for short) is a basic property widely used in some areas of economics that involves a system of second-order nonlinear partial differential equations. One of the most remarkable results in mathematical economics states that under homogeneity condition i.e. the production function is a homogeneous function of a certain degree, there are no other production models with the CES property apart from the famous Cobb–Douglas and Arrow–Chenery–Minhas–Solow production functions. In this paper we generalize this classification result to a much wider framework of production functions under quasi-homogeneity conditions, showing in particular the existence of three new classes of production models with the CES property.

Our aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed with a Caputo–Fabrizio fractional derivative. By means of such an approach, we utilize the Gram–Schmidt orthogonalization process to create an orthonormal set of bases that leads to an appropriate solution in the Hilbert space $\mathcal{H}^{2}[a,b]$ H 2 [ a , b ] . We investigate and discuss stability and convergence of the proposed method. The n-term series solution converges uniformly to the analytic solution. We present several numerical examples of potential interests to illustrate the reliability, efficacy, and performance of the method under the influence of the Caputo–Fabrizio derivative. The gained results have shown superiority of the reproducing kernel algorithm and its infinite accuracy with a least time and efforts in solving the fractional Abel-type model. Therefore, in this direction, the proposed algorithm is an alternative and systematic tool for analyzing the behavior of many nonlinear temporal fractional differential equations emerging in the fields of engineering, physics, and sciences.

In this paper, we consider the superlinear Schrödinger equation with bounded potential well. The potential here is allowed to be sign-changing. Without assuming the Ambrosetti–Rabinowitz-type condition, we prove the existence of a nontrivial solution and multiplicity results.

The problem of finding an analytical solution for the circuit formed by a diode with series resistance (D-R circuit) has been investigated for a long time and several authors have proposed approximate solutions. Most solutions use the Lambert function and numerical methods. However, an analytical solution independent of experimental parameters is still missing to replace the D-R circuit with an analytical expression, in which only the diode and circuit parameters appear. In this paper, using a Taylor series expansion of the equation VD = f (VS, R, ID), we propose a new analytical solution that fits the exact solution with more accuracy than all known approximations. The average of the absolute error vector is the best, with an order of magnitude of difference compared to the other 3 approximations studied. This new function can be expressed by voltage analysis, which allows analytical solutions. Our solution may have many applications, such as a simple and accurate way, in circuit simulation programs or numerical calculation programs. In particular, it can be used as well in circuits with D-R branches, exponential amplifiers, logarithmic amplifiers and waveform shapers. On another level, a comparison with other state of the art approximations is presented.

This paper proposes a simple, accurate and effective empirical formula to determine the number of supporting nodes in a newly-developed method, the localized method of fundamental solutions (LMFS). The LMFS has the merits of meshless, high-accuracy and easy-to-simulation in large-scale problems, but the number of supporting nodes has a certain impact on the accuracy and stability of the scheme. By using the curve fitting technique, this study established a simple formula between the number of supporting nodes and the node spacing. Based on the developed formula, the reasonable number of supporting nodes can be determined according to the node spacing. Numerical experiments confirmed the validity of the proposed methodology. This paper perfected the theory of the LMFS, and provided a quantitative selection strategy of method parameters.

Consider two digital spaces $ (X_i, k_i), i \in \{1, 2\} $, (in the sense of Rosenfeld model) satisfying the almost fixed point property(AFPP for brevity). Then, the problem of whether the AFPP for the digital spaces is, or is not necessarily invariant under Cartesian products plays an important role in digital topology, which remains open. Given a Cartesian product $ (X_1 \times X_2, k) $ with a certain $ k $-adjacency, after proving that the AFPP for digital spaces is not necessarily invariant under Cartesian products, the present paper proposes a certain condition of which the AFPP for digital spaces holds under Cartesian products. Indeed, we find that the product property of the AFPP is strongly related to both the sets $ X_i $ and the $ k_i $-adjacency, $ i \in \{1, 2\} $. Eventually, assume two $ k_i $-connected digital spaces $ (X_i, k_i), i \in \{1, 2\} $, and a digital product $ X_1 \times X_2 $ with a normal $ k $-adjacency such that $ N_k^\star(p, 1) = N_k(p, 1) $ for each point $ p \in X_1 \times X_2 $ (see Remark 4.2(1)). Then we obtain that each of $ (X_i, k_i), i \in \{1, 2\} $, has the AFPP if and only if $ (X_1 \times X_2, k) $ has the AFPP

A class of Dirichlet-Morrey spaces $ D_{\beta, \lambda} $ is introduced in this paper. For any positive Borel measure $ \mu $, the boundedness and compactness of the identity operator from $ D_{\beta, \lambda} $ into the tent space $ \mathcal{T}_s^1(\mu) $ are characterized. As an application, the boundedness of the Volterra integral operator $ T_g: D_{\beta, \lambda} \to F(1, \beta-s, s) $ is studied. Moreover, the essential norm and the compactness of the operator $ T_g $ are also investigated.

In this paper, we study the following kind of Schrodinger-Poisson system in $ { \mathbb{R}}^{2} $ $ \begin{equation*} \left\{\begin{array}{ll} -\Delta u+V(x)u+\phi u = K(x)f(u), \ \ \ x\in{ \mathbb{R}}^{2}, \\ \Delta \phi = u^{2}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in{ \mathbb{R}}^{2}, \end{array}\right. \end{equation*} $ where $ f\in C({ \mathbb{R}}, { \mathbb{R}}) $, $ V(x) $ and $ K(x) $ are both axially symmetric functions. By constructing a new variational framework and using some new analytic techniques, we obtain an axially symmetric solution for the above planar system. Our result improves and extends the existing works.

Many problems in engineering and social sciences can be transformed into system of nonlinear equations. As a result, a lot of methods have been proposed for solving the system. Some of the classical methods include Newton and Quasi Newton methods which have rapid convergence from good initial points but unable to deal with large scale problems due to the computation of Jacobian matrix or its approximation. Spectral and conjugate gradient methods proposed for unconstrained optimization, and later on extended to solve nonlinear equations do not require any computation of Jacobian matrix or its approximation, thus, are suitable to handle large scale problems. In this paper, we proposed a spectral conjugate gradient algorithm for solving system of nonlinear equations where the operator under consideration is monotone. The search direction of the proposed algorithm is constructed by taking the convex combination of the Dai-Yuan (DY) parameter and a modified conjugate descent (CD) parameter. The proposed search direction is sufficiently descent and under some suitable assumptions, the global convergence of the proposed algorithm is proved. Numerical experiments on some test problems are presented to show the efficiency of the proposed algorithm in comparison with an existing one. Finally, the algorithm is successfully applied in signal recovery problem arising from compressive sensing.

The properties of solutions of the following differential equation $ f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f = F(z) $ are studied, where $ A_{j}(z) $ and $ F(z) $ are analytic in the unit disc $ \mathbb{D} = \{z:|z| < 1\} $, $ j = 0, 1, \ldots, k-1 $. First, the growth of solutions of the equation is estimated. Second, some coefficient's conditions such that the solution of the equation belong to Hardy type spaces are showed. Finally, some related question are studied in this paper.

The subject of mutually orthogonal Latin squares (MOLSs) has fascinated researchers for many years. Although there is a number of intriguing results in this area, many open problems remain to which the answers seem as elusive as ever. Mutually orthogonal graph squares (MOGSs) are considered a generalization to MOLS. MOLS are considered an area of combinatorial design theory which has many applications in optical communications, cryptography, storage system design, wireless communications, communication protocols, and algorithm design and analysis, to mention just a few areas. In this paper, we introduce a technique for constructing the mutually orthogonal disjoint union of graphs squares and the generalization of the Kronecker product of MOGS as a generalization to the MacNeish’s Kronecker product of MOLS. These are useful for constructing many new results concerned with the MOGS.

We investigate the nonlinear boundary value problems by reproducing kernel Hilbert space technique in this paper. We construct some reproducing kernel Hilbert spaces. We define a bounded linear operator to obtain the solutions of the problems. We demonstrate our numerical results by some tables. We compare our numerical results with some results exist in the literature to present the efficiency of the proposed method.

It is common practice to treat small jumps of L\'evy processes as Wiener noise and thus to approximate its marginals by a Gaussian distribution. However, results that allow to quantify the goodness of this approximation according to a given metric are rare. In this paper, we clarify what happens when the chosen metric is the total variation distance. Such a choice is motivated by its statistical interpretation. If the total variation distance between two statistical models converges to zero, then no tests can be constructed to distinguish the two models which are therefore equivalent, statistically speaking. We elaborate a fine analysis of a Gaussian approximation for the small jumps of L\'evy processes with infinite L\'evy measure in total variation distance. Non asymptotic bounds for the total variation distance between $n$ discrete observations of small jumps of a L\'evy process and the corresponding Gaussian distribution is presented and extensively discussed. As a byproduct, new upper bounds for the total variation distance between discrete observations of L\'evy processes are provided. The theory is illustrated by concrete examples., Comment: Important and necessary changes have been made in this new version, this version supersedes version 1

In complex statistical models, in which exact computation of the likelihood is intractable, Monte Carlo methods can be applied to approximate maximum likelihood estimates. In this paper we consider approximation obtained via Markov chain Monte Carlo. We prove consistency and asymptotic normality of the resulting estimator, when both sample sizes (the initial and Monte Carlo one) tend to infinity. Our results can be applied to models with intractable normalizing constants and missing data models. We also investigate properties of estimators in numerical experiments.

This paper investigates the problem of signal estimation from undersampled noisy sub-Gaussian measurements under the assumption of a cosparse model. Based on generalized notions of sparsity, we derive novel recovery guarantees for the $\ell^1$-analysis basis pursuit, enabling accurate predictions of its sample complexity. The corresponding bounds on the number of required measurements do explicitly depend on the Gram matrix of the analysis operator and therefore particularly account for its mutual coherence structure. Our findings defy conventional wisdom which promotes the sparsity of analysis coefficients as the crucial quantity to study. In fact, this common paradigm breaks down completely in many situations of practical interest, for instance, when applying a redundant (multilevel) frame as analysis prior. By extensive numerical experiments, we demonstrate that, in contrast, our theoretical sampling-rate bounds reliably capture the recovery capability of various examples, such as redundant wavelets systems, total variation, or random frames. The proofs of our main results build upon recent achievements in the convex geometry of data mining problems. More precisely, we establish a sophisticated upper bound on the conic Gaussian mean width that is associated with the underlying $\ell^1$-analysis polytope. Due to a novel localization argument, it turns out that the presented framework naturally extends to stable recovery, allowing us to incorporate compressible coefficient sequences as well.

The consecutive-interpolation procedure (CIP) has been recently proposed as an enhanced technique for traditional finite element method (FEM) with various desirable properties such as continuous nodal gradients and higher accuracy without increasing the total number of degrees of freedom (DOFs). It is common knowledge that linear finite elements, e.g., four-node quadrilateral (Q4) or eight-node hexahedral (HH8) elements, are not highly suitable for geometrically nonlinear analysis. The elements with quadratic interpolation functions have to be used instead. In this paper, the CIP-enhanced four-node quadrilateral element (CQ4), and the CIP-enhanced eight-node hexahedral element (CHH8), are for the first time extended to investigate geometrically nonlinear problems of two- (2D) and three-dimensional (3D) structures. To further enhance the efficiency of the present approaches, novel numerical integration schemes based on the concept of mid-point rules, namely element mid-points (EM) and element mid-edges (EE) are integrated into the present CQ4 element. For CHH8, the 3D-version of EM (namely 3D-EM) and the element mid-faces (EF) scheme are investigated. The accuracy and computational efficiency of the two novel quadrature schemes in both regular and irregular (distorted) meshes are analyzed. Numerical results indicate that the new integration approaches perform more efficiently than the well-known Gaussian quadrature while gaining equivalent accuracy. The performance of the CIP-enhanced elements, which is examined through numerical experiments, is found to be equivalent to that of quadratic Lagrangian finite element counterparts, while having the same discretization with that by the linear finite elements. In addition, we also apply the present CQ4 and CHH8 elements associated with different numerical integration techniques to nearly incompressible materials.

In this paper, we study the dynamics of a delay differential model of predator-prey system involving teams of two-prey and one-predator, with Monod-Haldane and Holling type II functional responses, and a cooperation between the two-teams of preys against predation. We assume that the preys grow logistically and the rate of change of the predator relies on the growth, death and intra-species competition for the predators. Two discrete time-delays are incorporated to justify the reaction time of predator with each prey. The permanence of such system is proved. Local and global stabilities of interior steady states are discussed. Hopf bifurcation analysis in terms of time-delay parameters is investigated, and threshold parameters τ 1 * and τ 2 * are obtained. Sensitivity analysis that measures the impact of small changes in the model parameters into the model predictions is also investigated. Some numerical simulations are provided to show the effectiveness of the theoretical results.

In this paper we studied the fractional Euler-Bernoulli beam equation including a composition of the left and right fractional Caputo derivatives. We analyzed the equation with two types of boundary conditions (for the fixed-supported and fixed-free ends). The differential equation is converted into an integral one, taking into account the assumed boundary conditions. The obtained exact solutions contain a composition of the left and right Riemann-Liouville integrals. Finally, we presented three particular solutions for a constant, power and trigonometric function.

This paper deals with phase lag (or time-lagged) heat conduction models: the Cattaneo-Vernotte (or thermal wave) model and the dual phase lag model. The main aim is to show that modal analysis of these second order partial differential equations provides a valid and effective approach for analysing and comparing the models. It is known that reliable values for the phase lags of the heat flux and the temperature gradient are not readily available. The modal solutions are used to determine a range of realistic values for these lag times. Furthermore, it is shown that using partial sums of the series solutions for calculating approximate solutions is an efficient procedure. These approximate solutions converge in terms of a so-called energy norm which is stronger than the maximum norm. A model problem where a single heat pulse is applied to a specimen, is used for comparing these models with the Fourier (or parabolic) heat conduction model.

In this article, we combine the local projection stabilization (LPS) technique in space and the discontinuous Galerkin (DG) method in time to investigate the time-dependent convection-diffusion-reaction problems. This kind of stabilized space-time finite element (STFE) scheme, based on approximation space enriched by bubble functions that can increase stability, is constructed. The existence, uniqueness and stability are proved with the properties of Lagrange interpolation polynomials established on Radau points in time direction. An error estimate in L ∞ ( L 2 ) -norm is given by introducing the elliptic projection operators in space direction. This estimate approach is different from the previous ones that construct a special interpolant into approximation space showing an extra orthogonality property on the projection space. Since the techniques of Lagrange interpolation in time direction decouple time and space variables, the method proposed in this paper has the advantages of reducing calculation and simplifying theoretical analysis. The space and time convergence orders are illustrated in the first numerical example with smooth solutions. A comparison between the traditional STFE scheme and the constructed scheme for the problem having exponential boundary layers is presented in the second numerical example. The simulation results show that the novel method can greatly reduce nonphysical oscillations. And the influences of the stabilization parameters on the behavior of the approximate solution are discussed by some numerical results.

Image restoration is an ill-conditioned problem, so the regularization method is often applied to stabilize the solution. In this paper, we consider an additive half-quadratic (HQ) regularized image restoration problem and use the Newton method to solve it. At each Newton iteration step, a structured system of linear equations with symmetric positive definite coefficient matrix is needed to be solved. By taking an approximate Schur complement in the coefficient matrix, we construct a restrictive preconditioner and combine it into the conjugate gradient method in order to solve the linear system. The spectral properties of the preconditioned matrix are also analyzed. The numerical experiments demonstrate the effectiveness of the proposed method for image restoration.

This paper proposes a method based on two-dimensional Euler polynomials combined with Gauss-Jacobi quadrature formula. The method is used to solve two-dimensional Volterra integral equations with fractional order weakly singular kernels. Firstly, we prove the existence and uniqueness of the original equation by Gronwall inequality and mathematical induction method. Secondly, we use two-dimensional Euler polynomials to approximate the unknown function of the original equation, and the Gauss-Jacobi quadrature formula is used to approximate the integrals in the original equation. Thirdly, we prove the existence and uniqueness of the solution of approximate equation, and the error analysis of the proposed method is given. Finally, some numerical examples illustrate the efficiency of the method.

In this paper, we deal with the existence and stability of an equilibrium age distribution of the linearly implicit Euler-Riemann method for nonlinear age-structured population model with density dependence, i.e., the Gurtin-MacCamy models. It is shown that a dynamical invariance is replicated by numerical solutions for a long time. With the help of infinite-dimensional Leslie operators, the numerical processes are embedded into a nonlinear dynamical process in an infinite dimensional space, which provides a numerical basic reproduction function and numerical endemic equilibrium distributions. As an application to the Logistic model, a numerical reproduction number R 0 h ensures the global stability of disease-free equilibrium whenever R 0 h 1 and the existence of the numerical endemic equilibrium for R 0 h > 1 . Moreover, instead of the convergence of numerical solutions, it is much more interesting that the numerical solutions preserve the existence of endemic equilibrium for small stepsize, since the numerical reproduction numbers, numerical endemic equilibrium and distribution converge to the exact ones with accuracy of order 1. Finally, some numerical experiments illustrate the verification and the efficiency of our results.

Model tests have been conducted to study the performance of slack, catenary and hybrid mooring systems for a point absorber in survival sea states. The snap loads can be found in mooring tension time series of all the tested mooring configurations. The classical Weibull distribution is applied to model the extreme dynamic mooring tensions. In addition to the Weibull distribution, the Bayesian inference method with a new Markov chain Monte Carlo sampling procedure is proposed to study the short term extreme mooring tensions based on model test results. Before applying this method to analyse extreme mooring tension, its accuracy is validated. The probability densities as well as short term extreme dynamic tension estimated by a mixture of Gamma and Generalised Pareto Distributions models are compared with the two-parameter Weibull model results. In the end, an extensive discussion of about the mixture of Gamma components, peak definitions, threshold and Markov chain Monte Carlo iteration on Bayesian inference is carried out. It is validated that the mixture model proposed in this paper shows good performance in predicting the extreme mooring tensions even when the snap loads occur, while the Weibull distribution fails to give accuracy predictions.