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Hyperbolic systems of conservation laws in multiple spatial dimensions display features absent in the one-dimensional case, such as involutions and non-trivial stationary states. These features need to be captured by numerical methods without excessive grid refinement. The active flux method is an extension of the finite volume scheme with additional point values distributed along the cell boundary. For the equations of linear acoustics, an exact evolution operator can be used for the update of these point values. It incorporates all multi-dimensional information. The active flux method is stationarity preserving, i.e., it discretizes all the stationary states of the PDE. This paper demonstrates the experimental evidence for the discrete stationary states of the active flux method and shows the evolution of setups towards a discrete stationary state.

We show that the solution (in the sense of distribution) to the Cauchy problem with the periodic boundary condition associated with the modified Benjamin–Ono equation is unique in $L^\infty _t(H^s(\mathbb{T} ))$ for $s>1/2$. The proof is based on the analysis of a normal form equation obtained by infinitely many reduction steps using integration by parts in time after a suitable gauge transform.

In this paper we develop a general computer-assisted proof method for periodic solutions to delay differential equations. The class of problems considered includes systems of delay differential equations with an arbitrary number of (forward and backward) delays. When the nonlinearities include nonpolynomial terms we introduce auxiliary variables to first rewrite the problem into an equivalent polynomial one. We then apply a flexible fixed point technique in a space of geometrically decaying Fourier coefficients. We showcase the efficacy of this method by proving periodic solutions in the well-known Mackey–Glass delay differential equation for the classical parameter values.

We study fluid-saturated porous materials that undergo poro-elastic deformations in thin domains. The mechanics in such materials are described using a biphasic model based on the theory of porous media (TPM) and consisting of a system of differential equations for material’s displacement and fluid’s pressure. These equations are in general strongly coupled and nonlinear, such that exact solutions are hard to obtain and numerical solutions are computationally expensive. This paper reduces the complexity of the biphasic model in thin domains with a scale separation between domain’s width and length. Based on standard asymptotic analysis, we derive a reduced model that combines two sub-models. Firstly, a limit model consists of averaged equations that describe the fluid pore pressure and displacement in the longitudinal direction of the domain. Secondly, a corrector model re-captures the mechanics in the transverse direction. The validity of the reduced model is finally tested using a set of numerical examples. These demonstrate the computational efficiency of the reduced model, while maintaining reliable solutions in comparison with original biphasic TPM model in thin domain., Deutsche Forschungsgemeinschaft, Projekt DEAL

Statistical solutions have recently been introduced as an alternative solution framework for hyperbolic systems of conservation laws. In this work, we derive a novel a posteriori error estimate in the Wasserstein distance between dissipative statistical solutions and numerical approximations obtained from the Runge-Kutta Discontinuous Galerkin method in one spatial dimension, which rely on so-called regularized empirical measures. The error estimator can be split into deterministic parts which correspond to spatio-temporal approximation errors and a stochastic part which reflects the stochastic error. We provide numerical experiments which examine the scaling properties of the residuals and verify their splitting., Baden-Württemberg Stiftung, Projekt DEAL, Deutsche Forschungsgemeinschaft

We propose a new three-dimensional formulation based on the mixed tangential-displacement normal-normal-stress method for elasticity. In elastic tangential-displacement normal-normal-stress elements, the tangential component of the displacement field and the normal component of the stress vector are degrees of freedom and continuous across inter-element interfaces. Tangential-displacement normal-normal-stress finite elements have been shown to be locking-free with respect to shear locking in thin elements, which makes them suitable for the discretization of laminates or macro-fiber composites. In the current paper, we extend the formulation to piezoelectric materials by adding the electric potential as degree of freedom.

In this paper, we investigate the relationships between proper efficiency and the solutions of a general scalarization problem in multi-objective optimization. We provide some conditions under which the solutions of the dealt with scalar program are properly efficient and vice versa. We also show that, under some conditions, if the considered general scalar problem is unbounded, then the original multi-objective problem does not have any properly efficient solution. In another part of the work, we investigate a general transformation of the objective functions which preserves proper efficiency. We show that several important results existing in the literature are direct consequences of the results of the present paper.

This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain under consideration. Unlike previous approaches to this and related problems our starting point is a regularized least squares formulation in a continuous infinite-dimensional setting that is based on stable variational time-space formulations of the parabolic partial differential equation. This allows us to derive a priori as well as a posteriori error bounds for the recovered states with respect to a certain reference solution. In these bounds the regularization parameter is disentangled from the underlying discretization. An important ingredient for the derivation of a posteriori bounds is the construction of suitable Fortin operators which allow us to control oscillation errors stemming from the discretization of dual norms. Moreover, the variational framework allows us to contrive preconditioners for the discrete problems whose application can be performed in linear time, and for which the condition numbers of the preconditioned systems are uniformly proportional to that of the regularized continuous problem. In particular, we provide suitable stopping criteria for the iterative solvers based on the a posteriori error bounds. The presented numerical experiments quantify the theoretical findings and demonstrate the performance of the numerical scheme in relation with the underlying discretization and regularization.

Geological disasters such as slope failure and landslides can cause loss of life and property. Therefore, reproducing their evolution process is of great importance for risk assessment and mitigation. The recently developed SIMSAND critical state sand model combined with the smoothed particle hydrodynamics (SPH) method is adopted in this work to study slope failure under large deformations. To illustrate the efficiency and accuracy of the SIMSAND-SPH approach, a series of slope collapse studies using the discrete element method (DEM) considering various particle shapes (i.e. spherical, tetrahedral and elongated) is adopted as benchmarks. The parameters of the SIMSAND model are calibrated using DEM triaxial tests. In comparison to the DEM simulations, the runout distance and final slope height are well characterized with the SIMSAND-SPH approach with less computational cost. All comparisons show that the SIMSAND-SPH approach is highly efficient and accurate, which can be an alternative numerical tool to simulate real scale granular flow.

Using the approach of so-called c-sequences introduced by the fifth author in his recent work, we give much simpler and shorter proofs of multivalued Perov's type results with respect to the ones presented in the recently published paper by M. Abbas et al. Our proofs improve, complement, unify and enrich the ones from the recent papers. Further, in the last section of this paper, we correct and generalize the well-known Perov's fixed point result. We show that this result is in fact equivalent to Banach's contraction principle.

The Kuramoto–Sinelshchikov–Cahn–Hilliard equation models the spinodal decomposition of phase separating systems in an external field, the spatiotemporal evolution of the morphology of steps on crystal surfaces and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension. In this paper, we prove the well-posedness of the Cauchy problem, associated with this equation.

Many systems in life sciences have been modeled by reaction–diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such that an appropriate formalism like impulsive reaction–diffusion equations is necessary to analyze them. While several works tackled the issue of traveling waves for monotone reaction–diffusion equations and the computation of spreading speeds, very little has been done in the case of monotone impulsive reaction–diffusion equations. Based on vector-valued recursion equations theory, we aim to present in this paper results that address two main issues of monotone impulsive reaction–diffusion equations. Our first result deals with the existence of traveling waves for monotone systems of impulsive reaction–diffusion equations. Our second result tackles the computation of spreading speeds for monotone systems of impulsive reaction–diffusion equations. We apply our methodology to a planar system of impulsive reaction–diffusion equations that models tree–grass interactions in fire-prone savannas. Numerical simulations, including numerical approximations of spreading speeds, are finally provided in order to illustrate our theoretical results and support the discussion.

We study timelike surface, curve pairs by using their curvature properties. We assume that timelike surfaces whose one of the principal curvatures is identically constant, have diagonalizable shape operators in $${\mathbb {E}}_{1}^{3}$$ . Then we give some results about timelike surfaces on which inclined curves lie as geodesic curves.

International audience; In the first edition of this book, we discussed methods for determining the support of an inhomogeneous media from measured far field data as well as an extensive study of the central role played by the transmission eigenvalue problem in the mathematical development of these methods. In particular, we introduced the generalized linear sampling method (GLSM) and showed that this method provides a mathematical explanation of why it is permissible to use Tikhonov regularization to obtain an approximate solution of the far field equation associated with the linear sampling method.In the five years since the first edition of our book appeared, there has been considerable progress in both the development of GLSM as well as the theory of transmission eigenvalues. In this second edition, in addition to correcting typos in the first edition, we have added several highlights taken from these new developments. In particular, we have included new chapters on 1) the use of modified background media in the nondestructive testing of materials and in particular methods for determining the modified transmission eigenvalues that arise in such applications from measured far field data, 2) a study of a subset of transmission eigenvalues, called non-scattering wave numbers, through the use of techniques taken from the theory of free boundary value problems for elliptic partial differential equations and 3) the duality between scattering poles and transmission eigenvalues which, in addition to their intrinsic mathematical interest, leads to new methods for the numerical computation of scattering poles.

The aim of this paper is to analyze a general class of storage processes, in which the rate at which the storage level increases or decreases is assumed to be an affine function of the current storage level, and, in addition, both upward and downward jumps are allowed. To do so, we first focus on a related insurance risk model, for which we determine the ruin probability at an exponentially distributed epoch jointly with the corresponding undershoot and overshoot, given that the capital level at time 0 is exponentially distributed as well. The obtained results for this insurance risk model can be translated in terms of two types of storage models, in one of those two cases by exploiting a duality relation. Many well-studied models are shown to be special cases of our insurance risk and storage models. We conclude by showing how the exponentiality assumptions can be greatly relaxed.

Space separation within the Proper Generalized Decomposition—PGD—rationale allows solving high dimensional problems as a sequence of lower dimensional ones. In our former works, different geometrical transformations were proposed for addressing complex shapes and spatially non-separable domains. Efficient implementation of separated representations needs expressing the domain as a product of characteristic functions involving the different space coordinates. In the case of complex shapes, more sophisticated geometrical transformations are needed to map the complex physical domain into a regular one where computations are performed. This paper aims at proposing a very efficient route for accomplishing such space separation. A NURBS-based geometry representation, usual in computer aided design—CAD—, is retained and combined with a fully separated representation for allying efficiency (ensured by the fully separated representations) and generality (by addressing complex geometries). Some numerical examples are considered to prove the potential of the proposed methodology.

In this paper we study a class of impulsive systems of stochastic differential equations with infinite Brownian motions. Sufficient conditions for the existence and uniqueness of solutions are established by mean of some fixed point theorems in vector Banach spaces. An example is provided to illustrate the theory.

In this paper, optimal bounds for the sine and hyperbolic tangent means by arithmetic and centroidal means in exponential type are established using the monotone form of L'Hospital's rule and the criterion for the monotonicity of the quotient of power series.

This paper proposes a numerical upscaling procedure for elliptic boundary value problems with diffusion tensors that vary randomly on small scales. The method compresses the random partial differential operator to an effective quasilocal deterministic operator that represents the expected solution on a coarse scale of interest. Error estimates consisting of a priori and a posteriori terms are provided that allow one to quantify the impact of uncertainty in the diffusion coefficient on the expected effective response of the process.

Let Aut(p) denote the space of all self-fibre homotopy equivalences of a principal G-bundle \(p: E\rightarrow X\) of simply connected CW complexes with E finite. When G is a compact connected topological group, we show that there exists an inequality $$\begin{aligned} n-\mathrm{N}(p)\le \mathrm{Hnil}_{\mathbb {Q}}(\mathrm{{Aut}}(p)_0)\le n \end{aligned}$$ for any space X, where n is the number of non-trivial rational homotopy groups of G and \(\mathrm{N}(p)\) is defined in Sect. 2. In particular, \(\mathrm{Hnil}_{\mathbb {Q}}(\mathrm{{Aut}}(p)_{0})=n\) if p is a fibre homotopy trivial bundle and X is finite.

This study presents a fair performance comparison of the continuous finite element method, the symmetric interior penalty discontinuous Galerkin method, and the hybridized discontinuous Galerkin method. Modern implementations of high-order methods with state-of-the-art multigrid solvers for the Poisson equation are considered, including fast matrix-free implementations with sum factorization on quadrilateral and hexahedral elements. For the hybridized discontinuous Galerkin method, a multigrid approach that combines a grid transfer from the trace space to the space of linear finite elements with algebraic multigrid on further levels is developed. Despite similar solver complexity of the matrix-based HDG solver and matrix-free geometric multigrid schemes with continuous and discontinuous Galerkin finite elements, the latter offer up to order of magnitude faster time to solution, even after including the superconvergence effects. This difference is because of vastly better performance of matrix-free operator evaluation as compared to sparse matrix-vector products. A roofline performance model confirms the advantage of the matrix-free implementation.

We propose a novel class of convex risk measures, based on the concept of the Fr\'echet mean, designed in order to handle uncertainty which arises from multiple information sources regarding the risk factors of interest. The proposed risk measures robustly characterize the exposure of the firm, by filtering out appropriately the partial information available in individual sources into an aggregate model for the risk factors of interest. Importantly, the proposed risks can be expressed in closed analytic forms allowing for interesting qualitative interpretations as well as comparative statics and thus facilitate their use in the everyday risk management process of the insurance firms. The potential use of the proposed risk measures in insurance is illustrated by two concrete applications, capital risk allocation and premia calculation under uncertainty., Comment: 32 pages

The paper is devoted to the simulation of maritime two-phase flows of air and water. Emphasis is put on an extension of the classical Volume-of-Fluid (VoF) method by a diffusive contribution derived from a Cahn-Hilliard (CH) model and its benefits for simulating immiscible, incompressible two-phase flows. Such flows are predominantly simulated with implicit VoF schemes, which mostly employ heuristic downwind-biased approximations for the concentration transport to mimic a sharp interface. This strategy introduces a severe time step restriction and requires pseudo time-stepping of steady flows. Our overall goal is a sound description of the free-surface region that alleviates artificial time-step restrictions, facilitates an efficient and robust numerical framework and inherently includes surface tension effects when needed. The approach is verified for an analytical Couette-flow example and the bubble formation under the influence of surface tension forces. 2D Validation examples are concerned with laminar standing waves reaching from gravity to capillary scale as well as a submerged hydrofoil flow. The final application refers to the 3D flow around an experimentally investigated container vessel at fixed floatation for Re=1.4E+07 and Fn=0.26. Results are compared with data obtained from VoF approaches, supplemented by analytical solutions and measurements. The study indicates the superior efficiency, resharpening capability and wider predictive realm of the CH-based extension for free surface flows with a confined spatial range of interface Courant numbers., Submitted to Computers & Mathematics With Application

We provide a quantitative lower bound to the Cheeger constant of a set $\Omega$ in both the Euclidean and the Gaussian settings in terms of suitable asymmetry indexes. We provide examples which show that these quantitative estimates are sharp., Comment: 18 pages, 3 figures

In this paper, we introduce weighted Morrey-Herz spaces $ M\dot K^{\alpha, \lambda}_{q, p(\cdot)}(w~^{p(\cdot)}) $ with variable exponent $ p(\cdot) $. Then we prove the boundedness of multilinear Calderón-Zygmund singular operators on weighted Lebesgue spaces and weighted Morrey-Herz spaces with variable exponents.

The main goal of this article is first to introduce a new generalization of the fractional integral operators with a certain modified Mittag-Leffler kernel and then investigate the Chebyshev inequality via this general family of fractional integral operators. We improve our results and we investigate the Chebyshev inequality for more than two functions. We also derive some inequalities of this type for functions whose derivatives are bounded above and bounded below. In addition, we establish an estimate for the Chebyshev functional by using the new fractional integral operators. Finally, we find similar inequalities for some specialized fractional integrals keeping some of the earlier results in view.

He (J. Inequal. Appl. 2012:Article ID 162 2012) introduced the proximal point CQ algorithm (PPCQ) for solving the split equilibrium problem (SEP). However, the PPCQ converges weakly to a solution of the SEP and is restricted to monotone bifunctions. In addition, the step-size used in the PPCQ is a fixed constant μ in the interval $(0, \frac{1}{ \| A \|^{2} } )$ ( 0 , 1 ∥ A ∥ 2 ) . This often leads to excessive numerical computation in each iteration, which may affect the applicability of the PPCQ. In order to overcome these intrinsic drawbacks, we propose a robust step-size $\{ \mu _{n} \}_{n=1}^{\infty }$ { μ n } n = 1 ∞ which does not require computation of $\| A \|$ ∥ A ∥ and apply the adaptive step-size rule on $\{ \mu _{n} \}_{n=1}^{\infty }$ { μ n } n = 1 ∞ in such a way that it adjusts itself in accordance with the movement of associated components of the algorithm in each iteration. Then, we introduce a self-adaptive extragradient-CQ algorithm (SECQ) for solving the SEP and prove that our proposed SECQ converges strongly to a solution of the SEP with more general pseudomonotone equilibrium bifunctions. Finally, we present a preliminary numerical test to demonstrate that our SECQ outperforms the PPCQ.

The definition of a Green’s function of a Cauchy–Dirichlet problem for the hyperbolic equation in a quarter plane is given. Its existence and uniqueness have been proven. Representation of the Green’s function is given. It is shown that the Green’s function can be represented by the Riemann–Green function.

We consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP-convergence, i.e. convergence in the sense of the energy-dissipation principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics. Deutsche Forschungsgemeinschafthttps://doi.org/10.13039/501100001659

In this paper, we devoted to deal with the rotational periodic problem of some fractional iterative systems in the sense of Caputo fractional derivative. Under one sided-Lipschtiz condition on nonlinear term, the existence and uniqueness of solution for a fractional iterative equation is proved by applying the Leray-Schauder fixed point theorem and topological degree theory. Furthermore, the well posedness for a nonlinear control system with iteration term and a multivalued disturbance is established by using set-valued theory. The existence of solutions for a iterative neural network system is demonstrated at the end.

Lassa fever is a zoonotic disease spread by infected rodents known as multimammate rats. The disease has posed a significant and major health challenge in West African countries, including Nigeria. To have a deeper understanding of Lassa fever epidemiology in Nigeria, we present a deterministic dynamical model to study its dynamical transmission behavior in the population. To mimic the disease’s biological history, we divide the population into two groups: humans and rodents. We established the quantity known as reproduction number $${\mathcal {R}}_{0}$$ R 0 . The results show that if $${\mathcal {R}}_{0} R 0 < 1 then the system is stable, otherwise it is unstable. The model fitting was performed using the nonlinear least square method on cumulative reported cases from Nigeria between 2018 and 2020 to obtain the best fit that describes the dynamics of this disease in Nigeria. In addition, sensitivity analysis was performed, and the numerical solution of the system was derived using an iterative scheme, the fifth-order Runge–Kutta method. Using different numeric values for each parameter, we investigate the effect of all highest sensitivity indices’ parameters on the population of infected humans and infected rodents. Our findings indicate that any control strategies and methods that reduce rodent populations and the risk of transmission from rodents to humans and rodents would aid in the population’s control of Lassa fever.

In this article we make an improvement in the Banach contraction using a controlled function in controlled metric like spaces, which generalizes many results in the literature. Moreover, we present an application on Fredholm type integral equation.

This paper gives a new formulation to design adaptive structures through total energy optimization (TEO). This methodology enables the design of truss as well as tensegrity configurations that are equipped with linear actuators to counteract the effect of loading through active control. The design criterion is whole-life energy minimization which comprises an embodied part in the material and an operational part for structural adaptation during service. The embodied energy is minimized through simultaneous optimization of element sizing and actuator placement, which is formulated as a mixed-integer nonlinear programming problem. Optimization variables include element cross-sectional areas, actuator positions, element forces, and node displacements. For tensegrity configurations, the actuators are not only employed to counteract the effect of loading but also to apply appropriate prestress which is included in the optimization variables. Actuator commands during service are obtained through minimization of the operational energy that is required to control the state of the structure within required limits, which is formulated as a nonlinear programming problem. Embodied and operational energy minimization problems are nested within a univariate optimization process that minimizes the structure’s whole-life energy (embodied + operational). TEO has been applied to design a roof and a high-rise adaptive tensegrity structure. The adaptive tensegrity solutions are benchmarked with equivalent passive tensegrity as well as adaptive truss solutions, which are also designed through TEO. Results have shown that since cables can be kept in tension through active control, adaptive tensegrity structures require low prestress, which in turn reduces mass, embodied energy, and construction costs compared to passive tensegrity structures. However, while adaptive truss solutions achieve significant mass and energy savings compared to passive solutions, adaptive tensegrity solutions are not efficient configurations in whole-life energy cost terms. Since cable elements must be kept in tension, significant operational energy is required to maintain stable equilibrium for adaptation to loading. Generally, adaptive tensegrity solutions are not as efficient as their equivalent adaptive truss configurations in mass and energy cost terms.

In this paper, we firstly give improvement of Hermite-Hadamard type and Fej$ \acute{e} $r type inequalities. Next, we extend Hermite-Hadamard type and Fej$ \acute{e} $r types inequalities to a new class of functions. Further, we give bounds for newly defined class of functions and finally presents refined estimates of some already proved results. Furthermore, we obtain some new discrete inequalities for univariate harmonic convex functions on linear spaces related to a variant most recently presented by Baloch et al. of Jensen-type result that was established by S. S. Dragomir.

This paper presents a numerical scheme based on Haar wavelet for the solutions of higher order linear and nonlinear boundary value problems. In nonlinear cases, quasilinearization has been applied to deal with nonlinearity. Then, through collocation approach computing solutions of boundary value problems reduces to solve a system of linear equations which are computationally easy. The performance of the proposed technique is portrayed on some linear and nonlinear test problems including tenth, twelfth, and thirteen orders. Further convergence of the proposed method is investigated via asymptotic expansion. Moreover, computed results have been matched with the existing results, which shows that our results are comparably better. It is observed from convergence theoretically and verified computationally that by increasing the resolution level the accuracy also increases.

We give a new non-smooth Clark’s theorem without the global symmetric condition. The theorem can be applied to generalized quasi-linear elliptic equations with small continous perturbations. Our results improve the abstract results about a semi-linear elliptic equation in Kajikiya [10] and Li-Liu [11].

The paper deals with the coexistence between the oscillatory dynamics and the nonoscillatory one for a generalized super-linear Emden-Fowler differential equation. In particular, the coexistence of infinitely many oscillatory solutions with unbounded positive solutions are proved. The asymptotics of the unbounded positive solutions are described as well.

The major goal of this work is investigating sufficient conditions for the existence and uniqueness of solutions for implicit impulsive coupled system ofφ-Hilfer fractional differential equations (FDEs) with instantaneous impulses and terminal conditions. First, we derive equivalent fractional integral equations of the proposed system. Next, by employing some standard fixed point theorems such as Leray–Schauder alternative and Banach, we obtain the existence and uniqueness of solutions. Further, by mathematical analysis technique we investigate the Ulam–Hyers (UH) and generalized UH (GUH) stability of solutions. Finally, we provide a pertinent example to corroborate the results obtained.

This note obtains a new regularity criterion for the three-dimensional magneto-micropolar fluid flows in terms of one velocity component and the gradient field of the magnetic field. The authors prove that the weak solution $(u,\omega,b)$ ( u , ω , b ) to the magneto-micropolar fluid flows can be extended beyond time $t=T$ t = T , provided if $u_{3}\in L^{\beta }(0,T;L^{\alpha }(R^{3}))$ u 3 ∈ L β ( 0 , T ; L α ( R 3 ) ) with $\frac{2}{\beta }+\frac{3}{\alpha }\leq \frac{3}{4}+\frac{1}{2\alpha },\alpha > \frac{10}{3}$ 2 β + 3 α ≤ 3 4 + 1 2 α , α > 10 3 and $\nabla b\in L^{\frac{4p}{3(p-2)}}(0,T;\dot{M}_{p,q}(R^{3}))$ ∇ b ∈ L 4 p 3 ( p − 2 ) ( 0 , T ; M ˙ p , q ( R 3 ) ) with $1< q\leq p 1 < q ≤ p < ∞ and $p\geq 3$ p ≥ 3 .

In this paper, we introduce the notions of regularly invariant convergence, regularly strongly invariant convergence, regularly p-strongly invariant convergence, regularly $(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$ ( I σ , I 2 σ ) -convergence, regularly $(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$ ( I σ ∗ , I 2 σ ∗ ) -convergence, regularly $(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2} )$ ( I σ , I 2 σ ) -Cauchy double sequence, regularly $(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$ ( I σ ∗ , I 2 σ ∗ ) -Cauchy double sequence and investigate the relationship among them.

In this paper, our aim is to build generalized homogeneous q-difference equations for q-polynomials. We also consider their applications to generating functions and fractional q-integrals by using the perspective of q-difference equations. In addition, we also reveal relevant relations of various special cases of our main results involving some known results.

In this paper, we study the inviscid and zero Froude number limits of the viscous shallow water system. We prove that the limit system is represented by the incompressible Euler equations on the whole space. Furthermore, the rate of convergence is also obtained.

Studying wild relatives of cultivated plants in each specific region is an essential component in assessing the state of national plant genetic resources. This is especially true for the regions of the Russian Caucasus, with its tremendous diversity of plant species. This paper presents the results of exploring crop wild relatives (CWR) in natural plant communities of the Karachay-Cherkess Republic. The author conducted an inventory of CWR and analysis of their species composition in Karachay-Cherkessia. The CWR list includes 516 species belonging to 134 genera of 36 families; in this list, 449 species from 107 genera of 33 families are native to this region. The Dzheguta floristic district is the richest in CWR species diversity (391 species). There are 350 and 346 species in the Arkhyz and Uchkulan districts, respectively, and 301 species in the Cherkessk district. Species were ranked according to their agricultural value and economic importance: the 1st rank was assigned to 149 plant species, 2nd rank to 17, 3rd rank to 32, 4th rank to 97, and 5th rank to 222 species. In the context of their economic use, the species for feed (158) and food (136) purposes dominated over those for medicinal (60), melliferous (54), ornamental (53), industrial (51), and revegetation (5) applications. The list of priority conservation species was compiled for CWR of Karachay-Cherkessia according to the criteria of rarity and vulnerability, based on the analysis of the CWR species distribution over the studied area and their assessment for useful economic and biological traits. A map showing localities of these species within the Republic is presented, and recommendations are given for their effective conservation.

This article is devoted to a study of the blow-up result for a system of coupled viscoelastic wave equations. By establishing a new auxiliary function and using the reduction to absurdity method, we obtain some sufficient conditions on initial data such that the solution blows up in finite time at arbitrarily high initial energy. This work generalizes and improves earlier results in the literature.

In this paper, we found a common fixed point for several multivalued mappings on proximinal sets in regular modular metric space. Also, we introduced the notions of conjoint F-proximinal contraction as well as conjoint F-proximinal contraction of Hardy–Rogers-type for several multivalued mappings. Furthermore, we enhanced our results by giving an application in integral equations.

In this paper, we investigate the following tropical climate model with fractional diffusion \begin{document}$ \begin{eqnarray} \left\{\begin{array}{ll} u_t+u\cdot\nabla u+\nabla p+\Lambda^{2\alpha}u+{\rm div}(v\otimes v) = 0,\\[1ex] v_t+u\cdot\nabla v+\nabla\theta+\Lambda^{2\beta}v+v\cdot\nabla u = 0,\\[1ex] \theta_t+u\cdot\nabla\theta+\Lambda^{2\gamma}\theta+{\rm div} v = 0,\\[1ex] {\rm div} u = 0,\\[1ex] ( u, v, \theta)(x,0) = ( u_0, v_0, \theta_0), \end{array} \right. \end{eqnarray} $\end{document} where $ (u_0, v_0, \theta_0) \in H^s(R^n) $ with $ s\geq 1, n\geq 3 $ and $ {\rm div} u_0 = 0 $. When the nonnegative constants $ \alpha, \beta $ and $ \gamma $ satisfy $ \alpha\geq\frac{1}{2}+\frac{n}{4}, \ \alpha+\beta\geq 1+\frac{n}{2}, \ \alpha+\gamma\geq1+\frac{n}{2} $, by using the energy methods, we obtain the global existence and uniqueness of solution for the system. In the special case $ \theta = 0 $, we could obtain the global solution provide that $ \alpha\geq\frac{1}{2}+\frac{n}{4}, \alpha+\beta\geq1+\frac{n}{2} $ and $ (u_0, v_0)\in H^s(s\geq1) $, which generalizes the existing result.

In this paper, we discuss the solution of the inhomogeneous conformable abstract Cauchy problem. The homogeneous problem is also studied. The analysis of conformable fractional calculus and fractional semigroups is also given. Existence, uniqueness and regularity of a mild solution for the conformable abstract Cauchy problem are studied. Applications illustrating our main abstract results are also given.

Lane-level road cluster is a most representative phenomenon in road networks and is vital to spatial data mining, cartographic generalization, and data integration. In this article, a lane-level road cluster recognition method was proposed. First, the conception of lane-level road cluster and our motivation were addressed and the spatial characteristics were given. Second, a region growing cluster algorithm was defined to recognize lane-level road clusters, where constraints including distance and orientation were used. A novel moving distance (MD) metric was proposed to measure the distance of two lines, which can effectively handle the non-uniformly distributed vertexes, heterogeneous length, inharmonious spatial alignment, and complex shape. Experiments demonstrated that the proposed method can effectively recognize lane-level road clusters with the agreement to human spatial cognition.

Fractional calculus is the field of mathematical analysis that investigates and applies integrals and derivatives of arbitrary order. Fractionalq-calculus has been investigated and applied in a variety of research subjects including the fractionalq-trapezoid andq-midpoint type inequalities. Fractional$(p,q)$(p,q)-calculus on finite intervals, particularly the fractional$(p,q)$(p,q)-integral inequalities, has been studied. In this paper, we study two identities for continuous functions in the form of fractional$(p,q)$(p,q)-integral on finite intervals. Then, the obtained results are used to derive some fractional$(p,q)$(p,q)-trapezoid and$(p,q)$(p,q)-midpoint type inequalities.

Using the (p, q)-derivative operator we introduce new subclasses of analytic and bi-univalent functions, we obtain estimates on coefficients and the Fekete-Szegö functional.