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The paper alluded to in the title contains the following striking result: Let $I$ be the unit interval and $\Delta$ the Cantor set. If $X$ is a quasi Banach space containing no copy of $c_0$ which is isomorphic to a closed subspace of a space with a basis and $C(I, X)$ is linearly homeomorphic to $C(\Delta, X)$, then $X$ is locally convex, i.e., a Banach space. It is shown that Kalton result is sharp by exhibiting non locally convex quasi Banach spaces X with a basis for which $C(I, X)$ and $C(\Delta, X)$ are isomorphic. Our examples are rather specific and actually in all cases X is isomorphic to $C(\Delta, X)$ if $K$ is a metric compactum of finite covering dimension., Comment: 4 pages

The aim of this paper is to study a new equivalent form of the weak maximum principle for a large class of differential operators on Riemannian manifolds. This new form has been inspired by the work of Berestycki, Hamel and Rossi for trace operators, and allows us to shed new light on it and to introduce a new sufficient bounded Khas’minskii type condition for its validity. We show its effectiveness by applying it to obtain some uniqueness results in a geometric setting.

The aim of this paper is to discuss the main result in the paper by D.Y. Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective function has not local extrema with the mentioned constraints for $p\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer. We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995]., 12 pages; in this version we added the forgotten condition $F(x) \ne 0$ for $x\in (a,b)$ on page 3

We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freist\"uhler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations, Comment: to appear in Proceedings of the AMS

The spectral gap conjecture of M. van den Berg [2, formula (65)] asserts that λ2 − λ1 ≥ 3π for all convex euclidean domains of diameter 1, where λ1 and λ2 denote the first two eigenvalues of the Dirichlet Laplacian. Notice that equality holds for the 1-dimensional unit interval, which can be regarded also as a degenerate n-dimensional rectangular box. The gap estimate is conjectured to hold more generally for Schrodinger operators with convex potentials, under Dirichlet boundary conditions; see the work of S.-T. Yau and collaborators [9, 11]. This Schrodinger gap conjecture was proved some time ago in 1 dimension by R. Lavine [8], and more recently in all dimensions by B. Andrews and J. Clutterbuck [1]. The proof in this journal by H. Donnelly [3] of the original gap conjecture in 2 dimensions (for the Dirichlet Laplacian with zero potential) is not correct. The Editors of Mathematische Zeitschrift have asked us to describe the flaws in the proof, in order to clarify the state of the literature. Donnelly’s approach to the problem is a natural one: first perform a shape optimization to rule out a non-degenerate minimizing domain, and then analyze the spectral gap for a sequence of domains degenerating to an interval, with the help of results by D. Jerison [5]. (For some history on this approach, and on the gap conjecture more generally, see the report on the AIM meeting “Low Eigenvalues of Laplace and Schrodinger Operators” [10], especially page 12 of the open problems list.) The error lies in the proof of the shape optimization step, as we now explain. Donnelly wishes to prove that no minimizing domain can exist for

Recently, Saleh claimed to have solved `a long standing open question' in Topology; namely, he proved that every almost continuous function is closure continuous (= $\theta$-continuous). Unfortunately, this problem was settled long time ago and even a better result is known., Comment: 2 pages, to appear in "Glasgow Math. J."

The preeminent target of present study is to reveal the speed characteristic of ongoing outbreak COVID-19 due to novel coronavirus. On January 2020, the novel coronavirus infection (COVID-19) detected in India, and the total statistic of cases continuously increased to 7 128 268 cases including 109 285 deceases to October 2020, where 860 601 cases are active in India. In this study, we use the Hermite wavelets basis in order to solve the COVID-19 model with time- arbitrary Caputo derivative. The discussed framework is based upon Hermite wavelets. The operational matrix incorporated with the collocation scheme is used in order to transform arbitrary-order problem into algebraic equations. The corrector scheme is also used for solving the COVID-19 model for distinct value of arbitrary order. Also, authors have investigated the various behaviors of the arbitrary-order COVID-19 system and procured developments are matched with exiting developments by various techniques. The various illustrations of susceptible, exposed, infected, and recovered individuals are given for its behaviors at the various value of fractional order. In addition, the proposed model has been also supported by some numerical simulations and wavelet-based results.

The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples, the first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data.

Funder: John Simon Guggenheim Memorial Foundation; doi: https://doi.org/10.13039/100005851, Funder: Einstein Visiting Fellow Program, The Rayleigh–Bénard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the L 2 norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy–palinstrophy plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations.

COVID-19 pandemic has affected all aspects of people's lives and disrupted the economy. Forecasting the number of cases infected with this virus can help authorities make accurate decisions on the interventions that must be implemented to control the pandemic. Investigation of the studies on COVID-19 forecasting indicates that various techniques such as statistical, mathematical, and machine and deep learning have been utilized. Although deep learning models have shown promising results in this context, their performance can be improved using auxiliary features. Therefore, in this study, we propose two hybrid deep learning methods that utilize the statistical features as auxiliary inputs and associate them with their main input. Specifically, we design a hybrid method of the multihead attention mechanism and the statistical features (ATT_FE) and a combined method of convolutional neural network and the statistical features (CNN_FE) and apply them to COVID-19 data of 10 countries with the highest number of confirmed cases. The results of experiments indicate that the hybrid models outperform their conventional counterparts in terms of performance measures. The experiments also demonstrate the superiority of the hybrid ATT_FE method over the long short-term memory model.

The first symptomatic infected individuals of coronavirus (Covid-19) was confirmed in December 2020 in the city of Wuhan, China. In India, the first reported case of Covid-19 was confirmed on 30 January 2020. Today, coronavirus has been spread out all over the world. In this manuscript, we studied the coronavirus epidemic model with a true data of India by using Predictor-Corrector scheme. For the proposed model of Covid-19, the numerical and graphical simulations are performed in a framework of the new generalised Caputo sense non-integer order derivative. We analysed the existence and uniqueness of solution of the given fractional model by the definition of Chebyshev norm, Banach space, Schauder's second fixed point theorem, Arzel's-Ascoli theorem, uniform boundedness, equicontinuity and Weissinger's fixed point theorem. A new analysis of the given model with the true data is given to analyse the dynamics of the model in fractional sense. Graphical simulations show the structure of the given classes of the non-linear model with respect to the time variable. We investigated that the mentioned method is copiously strong and smooth to implement on the systems of non-linear fractional differential equation systems. The stability results for the projected algorithm is also performed with the applications of some important lemmas. The present study gives the applicability of this new generalised version of Caputo type non-integer operator in mathematical epidemiology. We compared that the fractional order results are more credible to the integer order results.

We show that bounded self-contracted curves are rectifiable in metric spaces with weak lower curvature bound in a sense we introduce in this article. This class of spaces is wide and includes, for example, finite-dimensional Alexandrov spaces of curvature bounded below and Berwald spaces of nonnegative flag curvature. (To be more precise, our condition is regarded as a strengthened doubling condition and holds also for a certain class of metric spaces with upper curvature bound.) We also provide the non-embeddability of large snowflakes into (balls in) metric spaces in the same class. We follow the strategy of the last author's previous paper based on the small rough angle condition, where spaces with upper curvature bound are considered. The results in this article show that such a strategy applies to spaces with lower curvature bound as well., Comment: A goofy mistake in formulations of Theorem 2 and 4 is fixed

In this work we introduce the r-Hunter-Saxton equation, a generalisation of the Hunter-Saxton equation arising as extremals of an action principle posed in L_r. We characterise solutions to the Cauchy problem, quantifying the blow-up time and studying various symmetry reductions. We construct piecewise linear functions and show that they are weak solutions to the r-Hunter-Saxton equation., Revised after referee comments

Part of the work was done while the authors were visiting the South China University of Science and Technology (SUSTech), Shenzhen, in 2018, and we are grateful (in particular to Professor Cai Heng Li) for the hospitality that we received.The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, representations and applications: new perspectives (supported by EPSRC grant no.EP/R014604/1), where further work on this paper was undertaken. In particular we acknowledge a Simons Fellowship (Cameron) and a Kirk Distinguished Visiting Fellowship (Praeger) during this programme. Schneider thanks the Centre for the Mathematics of Symmetry and Computation of The University of Western Australia and Australian Research Council Discovery Grant DP160102323 for hosting his visit in 2017 and acknowledges the support of the CNPq projects Produtividade em Pesquisa (project no.: 308212/2019-3) and Universal (project no.:421624/2018-3). Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied in general. The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan-Scott theorem, which are defined over finite characteristically simple groups, our construction works over arbitrary groups, finite or infinite. A diagonal semilattice depends on a dimension m and a group T. For m=2, it is a Latin square, the Cayley table of T, though in fact any Latin square satisfies our combinatorial axioms. However, for m≥3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in great profusion but higher-dimensional structures are coordinatised by an algebraic object, a division ring.) A diagonal semilattice is contained in the partition lattice on a set Ω, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics. We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions turn out to be equivalent for diagonal groups). Associated with the diagonal semilattice is a graph, the diagonal graph, which has the same automorphism group as the diagonal semilattice except in four small cases with m

In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages

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.

A question of Conca, Rossi, and Valla asks whether every quadratic Gorenstein ring $R$ of regularity three is Koszul. In a previous paper, we use idealization to answer their question, proving that in nine or more variables there exist quadratic Gorenstein rings of regularity three which are not Koszul. In this paper, we study the analog of the Conca-Rossi-Valla question when the regularity of $R$ is four or more. Let $R$ be a quadratic Gorenstein ring having $\mathrm{codim}\, R = c$ and $\mathrm{reg}\, R = r \ge 4$. We prove that if $c = r+1$ then $R$ is always Koszul, and for every $c \geq r+2$, we construct quadratic Gorenstein rings that are not Koszul, answering questions of Matsuda and Migliore-Nagel concerning the $h$-vectors of quadratic Gorenstein rings., Comment: v2 - Minor changes based on referee comments

In this paper, we generalize the classical Freidlin-Wentzell’s theorem for random perturbations of Hamiltonian systems. In (Probability Theory and Related Fields 128 (2004) 441–466), M.Freidlin and M.Weber generalized the original result in the sense that the coefficient for the noise term is no longer the identity matrix but a state-dependent matrix and taking the drift term into consideration. In this paper, We generalize the result by adding a state-dependent matrix that converges uniformly to 0 on any compact sets as ϵ tends to 0 to a state-dependent noise and considering the drift term which contains two parts, the state-dependent mapping and a state-dependent mapping that converges uniformly to 0 on any compact sets as ϵ tends to 0. In the proof, we adapt a new way to prove the weak convergence inside the edge by constructing an auxiliary process and modify the proof in (Probability Theory and Related Fields 128 (2004) 441–466) when proving gluing condition.

Starting with work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of $p$-adic and $\bmod p$ modular forms and Galois representations. This paper achieves two main results for theta operators on automorphic forms on PEL-type Shimura varieties: 1) the analytic continuation at unramified primes $p$ to the whole Shimura variety of the $\bmod p$ reduction of $p$-adic Maass--Shimura operators {\it a priori} defined only over the $\mu$-ordinary locus, and 2) the construction of new $\bmod p$ theta operators that do not arise as the $\bmod p$ reduction of Maass--Shimura operators. While the main accomplishments of this paper concern the geometry of Shimura varieties and consequences for differential operators, we conclude with applications to Galois representations. Our approach involves a careful analysis of the behavior of Shimura varieties and enables us to obtain more general results than allowed by prior techniques, including for arbitrary signature, vector weights, and unramified primes in CM fields of arbitrary degree., Comment: Accepted for publication in IMRN. 42 pages

Diffusion equations with Riemann–Liouville fractional derivatives are Volterra integro-partial differential equations with weakly singular kernels and present fundamental challenges for numerical computation. In this paper, we make a convergence analysis of the Schwarz waveform relaxation (SWR) algorithms with Robin transmission conditions (TCs) for these problems. We focus on deriving good choice of the parameter involved in the Robin TCs, at the continuous and fully discretized level. Particularly, at the space-time continuous level, we show that the derived Robin parameter is much better than the one predicted by the well-understood equioscillation principle. At the fully discretized level, the problem of determining a good Robin parameter is studied in the convolution quadrature framework, which permits us to precisely capture the effects of different temporal discretization methods on the convergence rate of the SWR algorithms. The results obtained in this paper will be preliminary preparations for our further study of the SWR algorithms for integro-partial differential equations.

In order to solve the complicated process and low efficiency and low accuracy of solving a class of matrix equations, this paper introduces the linear saturated system model neural network architecture to solve the bisymmetric solution of a class of matrix equations. Firstly, a class of matrix equations is constructed to determine the key problems of solving the equations. Secondly, the linear saturated system model neural network structure is constructed to determine the characteristic parameters in the process of bisymmetric solution. Then, the matrix equations is solved by using backpropagation neural network topology. Finally, the class normalization is realized by using the objective function of bisymmetric solution, and the bisymmetric solution of a class of matrix equations is realized. In order to verify the solving effect of the method in this paper, three indexes (accuracy, correction accuracy, and solving time) are designed in the experiment. The experimental results show that the proposed method can effectively reduce the solving time, can improve the accuracy and correction effect of the bisymmetric solution, and has high practicability.

The present paper aims to find some new midpoint-type inequalities for twice quantum differentiable convex functions. The consequences derived in this paper are unification and generalization of the comparable consequences in the literature on midpoint inequalities. © 2021 Muhammad Aamir Ali et al., published by De Gruyter. National Natural Science Foundation of China, NSFC: 11301127, 11601485, 11626101, 11701176, 11971241, 61673169 Funding information : The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485, 11971241). 2-s2.0-85105011594

This paper presents some new formulas and properties of the generalized fractional derivative with non-singular kernel that covers various types of fractional derivatives such as the Caputo–Fabrizio fractional derivative, the Atangana–Baleanu fractional derivative, and the weighted Atangana–Baleanu fractional derivative. These new properties extend many recent results existing in the literature. Furthermore, the paper proposes some interesting inequalities that estimate the generalized fractional derivatives of some specific functions. These inequalities can be used to construct Lyapunov functions with the aim to study the global asymptotic stability of several fractional-order systems arising from diverse fields of science and engineering.

The paper focuses on the analysis of the Euler projection Galerkin finite element method (FEM) for the dynamics of magnetization in ferromagnetic materials, described by the Landau–Lifshitz equation with the point-wise constraint $|{\textbf{m}}|=1$. The method is based on a simple sphere projection that projects the numerical solution onto a unit sphere at each time step, and the method has been used in many areas in the past several decades. However, error analysis for the commonly used method has not been done since the classical energy approach cannot be applied directly. In this paper we present an optimal $\textbf{L}^2$ error analysis of the backward Euler sphere projection method by using quadratic or higher order finite elements under a time step condition $\tau =O(\epsilon _0 h)$ with some small $\epsilon _0>0$. The analysis is based on more precise estimates of the extra error caused by the sphere projection in both $\textbf{L}^2$ and $\textbf{H}^1$ norms, and the classical estimate of dual norm. Numerical experiment is provided to confirm our theoretical analysis.

For a separable finite diffuse measure space $${\mathcal {M}}$$ M and an orthonormal basis $$\{\varphi _n\}$$ { φ n } of $$L^2({\mathcal {M}})$$ L 2 ( M ) consisting of bounded functions $$\varphi _n\in L^\infty ({\mathcal {M}})$$ φ n ∈ L ∞ ( M ) , we find a measurable subset $$E\subset {\mathcal {M}}$$ E ⊂ M of arbitrarily small complement $$|{\mathcal {M}}{\setminus } E| | M \ E | < ϵ , such that every measurable function $$f\in L^1({\mathcal {M}})$$ f ∈ L 1 ( M ) has an approximant $$g\in L^1({\mathcal {M}})$$ g ∈ L 1 ( M ) with $$g=f$$ g = f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of $${\mathcal {M}}=G/H$$ M = G / H being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.

This paper is concerned with the qualitative analysis of solutions to the following class of quasilinear problems $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{\Phi }u=f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$ - Δ Φ u = f ( x , u ) in Ω , u = 0 on ∂ Ω , where $$\Delta _{\Phi }u=\mathrm{div}\,(\varphi (x,|\nabla u|)\nabla u)$$ Δ Φ u = div ( φ ( x , | ∇ u | ) ∇ u ) and $$\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds$$ Φ ( x , t ) = ∫ 0 | t | φ ( x , s ) s d s is a generalized N-function. We assume that $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N is a smooth bounded domain that contains two open regions $$\Omega _N,\Omega _p$$ Ω N , Ω p with $${\overline{\Omega }}_N \cap {\overline{\Omega }}_p=\emptyset $$ Ω ¯ N ∩ Ω ¯ p = ∅ . The features of this paper are that $$-\Delta _{\Phi }u$$ - Δ Φ u behaves like $$-\Delta _N u $$ - Δ N u on $$\Omega _N$$ Ω N and $$-\Delta _p u $$ - Δ p u on $$\Omega _p$$ Ω p , and that the growth of $$f:\Omega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$$ f : Ω × R → R is like that of $$e^{\alpha |t|^{\frac{N}{N-1}}}$$ e α | t | N N - 1 on $$\Omega _N$$ Ω N and as $$|t|^{p^{*}-2}t$$ | t | p ∗ - 2 t on $$\Omega _p$$ Ω p when |t| is large enough. The main result establishes the existence of solutions in a suitable Musielak–Sobolev space in the case of high perturbations with respect to the values of a positive parameter.

In this paper, generalized versions of Hadamard and Fejér–Hadamard type fractional integral inequalities are obtained. By using generalized fractional integrals containing Mittag-Leffler functions, some well-known results for convex and harmonically convex functions are generalized. The results of this paper are connected with various published fractional integral inequalities.

PurposeIn this paper, Picard–S hybrid iterative process is defined, which is a hybrid of Picard and S-iterative process. This new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid and Picard–Ishikawa hybrid iterative processes for contraction mappings and to find the solution of delay differential equation, using this hybrid iteration also proved some results for Picard–S hybrid iterative process for nonexpansive mappings.Design/methodology/approachThis new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid, Picard–Ishikawa hybrid iterative processes for contraction mappings.FindingsShowed the fastest convergence of this new iteration and then other iteration defined in this paper. The author finds the solution of delay differential equation using this hybrid iteration. For new iteration, the author also proved a theorem for nonexpansive mapping.Originality/valueThis new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid, Picard–Ishikawa hybrid iterative processes for contraction mappings and to find the solution of delay differential equation, using this hybrid iteration also proved some results for Picard–S hybrid iterative process for nonexpansive mappings.

Based on the basic theory of the material point method (MPM), the factors affecting the computational efficiency are analyzed and discussed, and the problem of improving calculation efficiency is studied. This paper introduces a mirror reflection boundary condition to the MPM to solve axisymmetric problems; to improve the computational efficiency of solving large deformation problems, the concept of "dynamic background domain (DBD)" is also proposed in this paper. Taking the explosion and/or shock problems as an example, the numerical simulation are calculated, and the typical characteristic parameters and the CPU time are compared. The results show that the processing method introducing mirror reflection boundary condition and MPM with DBD can improve the calculation efficiency of the corresponding problems, which, under the premise of ensuring its calculation accuracy, provide useful reference for further promoting the engineering application of this method.

This paper concerns on two types of integral boundary value problems of a nonlinear fractional differential system, i . e ., nonlocal strip integral boundary value problems and coupled integral boundary value problems. With the aid of the monotone iterative method combined with the upper and lower solutions, the existence of extremal system of solutions for the above two types of differential systems is investigated. In addition, a new comparison theorem for fractional differential system is also established, which is crucial for the proof of the main theorem of this paper. At the end, an example explaining how our studies can be used is also given.

The main aim of this paper is to give refinement of bounds of fractional integral operators involving extended generalized Mittag-Leffler functions. A new definition, namely, strongly α , m -convex function is introduced to obtain improvements of bounds of fractional integral operators for convex, m -convex, and α , m -convex functions. The results of this paper will provide simultaneous generalizations as well as refinements of various published results.

In this paper, we consider a version of energy minimisation technique applied to images of a 2D fluid flow. The two Navier–Stokes equations describe the static flow of a 2D fluid in terms of velocity field, (u, v), pressure field, p and forcing field, f. Apart from these two Navier–Stokes equations, we have the incompressibility condition to evaluate the three parameters. While implementing this system, random noise (usually non-Gaussian) creeps into the random force field $$\underline{f}(x,y)$$ f ̲ ( x , y ) . We denote this random field by $$\delta \underline{f}(x,y)$$ δ f ̲ ( x , y ) having zero mean and non-trivial second and third moments. We assume that these two moments are known except for some unknown parameters $$\underline{\theta }$$ θ ̲ (like mean, variance, co-variance, skewness, etc.) which we wish to estimate. In the proposed technique, we first calculate the approximate shift in the average fluid energy defined as a quadratic function of the velocity field. The energy method then requires that $$\underline{\theta }$$ θ ̲ should be such that this average increases in the energy due to the random forcing component be minimised. We should, however, note that the standard statistical approach to force field estimation is to calculate the velocity field as a function of the force field and then adopt the statistical moment matching technique. Such an approach assumes spatial ergodicity of the velocity field. This approach to force field estimation is more accurate from the statistical moment matching view point but works only if velocity measurements are made. The former technique of energy minimisation does not require any velocity measurements. Both of these techniques are discussed in this paper and MATLAB simulations presented.

The study in this paper is made on the nonlinear fractional differential equation whose nonlinearity involves the explicit fractional order D0+β u(t). The corresponding Green's function is derived first, and then the completely continuous operator is proved. Besides, based on the Schauder's fixed point theorem and the Krasnosel'skii's fixed point theorem, the sufficient conditions for at least one or two existence of positive solutions are established. Furthermore, several other sufficient conditions for at least three, n or 2n-1 positive solutions are also obtained by applying the generalized AveryHenderson fixed point theorem and the Avery-Peterson fixed point theorem. Finally, several simulation examples are provided to illustrate the main results of the paper. In particularly, a novel efficient iterative method is employed for simulating the examples mentioned above, that is, the interesting point of this paper is that the approximation graphics for the solutions are given by using the iterative method.

The aim of this paper is to characterize the solutions Φ : G → M 2(ℂ) of the following matrix functional equations Φ ( x y ) + Φ ( σ ( y ) x ) 2 = Φ ( x ) Φ ( y ) , x , y , ∈ G , {{\Phi \left( {xy} \right) + \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G, and Φ ( x y ) − Φ ( σ ( y ) x ) 2 = Φ ( x ) Φ ( y ) , x , y , ∈ G , {{\Phi \left( {xy} \right) - \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G, where G is a group that need not be abelian, and σ : G → G is an involutive automorphism of G. Our considerations are inspired by the papers [13, 14] in which the continuous solutions of the first equation on abelian topological groups were determined.

The paper aims to establish new results on Tauberian theorem for Cesàro summability of double sequences of fuzzy numbers, and thus to extend and unify several results in the available literature. Further, a number of special cases, corollaries and illustrative example in support of the investigation of this paper are also presented.

The motivation of this paper is to study the uniqueness problems of meromorphic functions concerning differential polynomials that share a small function. The results of the paper improve and generalize the recent results due to Fengrong Zhang and Linlin Wu [13]. We also solve an open problem as posed in the last section of [13].

PurposeThe purpose of this current paper is to deal with the study of non-constant entire solutions of some non-linear complex differential equations in connection to Brück conjecture, by using the theory of complex differential equation. The results generalize the results due to Pramanik et al.Design/methodology/approach39B32, 30D35.FindingsIn the current paper, we mainly study the Brück conjecture and the various works that confirm this conjecture. In our study we find that the conjecture can be generalized for differential monomials under some additional conditions and it generalizes some works related to the conjecture. Also we can take the complex number a in the conjecture to be a small function. More precisely, we obtain a result which can be restate in the following way: Let f be a non-constant entire function such that σ2(f)<∞, σ2(f) is not a positive integer and δ(0, f)>0. Let M[f] be a differential monomial of f of degree γM and α(z), β(z)∈S(f) be such that max{σ(α), σ(β)} <σ(f). If M[f]+β and fγM−α share the value 0 CM, then M[f]+βfγM−α=c,where c≠0 is a constant.Originality/valueThis is an original work of the authors.

Aspace X is said to have the absolutely strongly star -𝒤-Hurewicz (ASS𝒤H) property if for each sequence (𝒰 n : n ∈ 𝕅)of opencovers of X and each dense subset Y of X, there is a sequence (Fn : n ∈ 𝕅) of finite subsets of Y such that for each x ∈ X, {n ∈ 𝕅 : x ∉ St(Fn , 𝒰 n )}∈ 𝒤, where 𝒤 is the proper admissible ideal of 𝕅. In this paper, we investigate the relationship between the ASS𝒤H property and other related properties and study the topological properties of the ASS𝒤H property. This paper generalizes several results of Song [25] to the larger class of spaces having the ASS𝒤H properties.

The paper deals with the existence of solutions for(p,Q)(p,Q)coupled elliptic systems in the Heisenberg group, with critical exponential growth at infinity and singular behavior at the origin. We derive existence of nonnegative solutions with both components nontrivial and different, that is solving an actual system, which does not reduce into an equation. The main features and novelties of the paper are the presence of a general coupled critical exponential term of the Trudinger-Moser type and the fact that the system is set inℍn{{\mathbb{H}}}^{n}.

In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn-Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3-24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn-Hilliard equation for the tumor cell fraction, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, well-posedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding well-posedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases., Comment: Key words: fractional operators, Cahn-Hilliard systems, well-posedness, regularity of solutions, tumor growth models, asymptotic analysis

In this paper, an alternative approach is provided in the well-posedness analysis of elliptic variational–hemivariational inequalities in real Hilbert spaces. This includes the unique solvability and continuous dependence of the solution on the data. In most of the existing literature on elliptic variational–hemivariational inequalities, well-posedness results are obtained by using arguments of surjectivity for pseudomonotone multivalued operators, combined with additional compactness and pseudomonotonicity properties. In contrast, following (Han in Nonlinear Anal B Real World Appl 54:103114, 2020; Han in Numer Funct Anal Optim 42:371–395, 2021), the approach adopted in this paper is based on the fixed point structure of the problems, combined with minimization principles for elliptic variational–hemivariational inequalities. Consequently, only elementary results of functional analysis are needed in the approach, which makes the theory of elliptic variational–hemivariational inequalities more accessible to applied mathematicians and engineers. The theoretical results are illustrated on a representative example from contact mechanics.

This paper is concerned with optimal error estimates and recovery technique of a classical mixed finite element method for the thermistor problem, which is governed by a parabolic/elliptic system with strong nonlinearity and coupling. The method is based on a popular combination of the lowest-order Raviart–Thomas mixed approximation for the electric potential/field $(\phi , \boldsymbol{\theta })$ and the linear Lagrange approximation for the temperature $u$. A common question is how the first-order approximation influences the accuracy of the second-order approximation to the temperature in such a strongly coupled system, while previous work only showed the first-order accuracy $O(h)$ for all three components in a traditional way. In this paper, we prove that the method produces the optimal second-order accuracy $O(h^2)$ for $u$ in the spatial direction, although the accuracy for the potential/field is in the order of $O(h)$. And more importantly, we propose a simple one-step recovery technique to obtain a new numerical electric potential/field of second-order accuracy. The analysis presented in this paper relies on an $H^{-1}$-norm estimate of the mixed finite element methods and analysis on a nonclassical elliptic map. We provide numerical experiments in both two- and three-dimensional spaces to confirm our theoretical analyses.

In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrodinger operator $${\cal L} = \Delta + {\bf{V}}$$ in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Holder’s inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrodinger equations in the new Morrey space.

In the present paper, we obtain a Halanay inequality on time scales with unbounded coefficient for a dynamic problem, which extends a result of Wen et al. (J. Math. Anal. Appl., 347 (2008), 169–178.) to the inequality of integral type on time scales. Moreover, we list two dynamic problems to which the Halanay inequality obtained above can be applied and prove the zero solution of two delay dynamic problems are asymptotically stable. Moreover, it is worth mentioning that the Halanay inequality obtained in the present paper is more precise than the results in [3, 14, 17].

This paper proposes a new class of difference methods with intrinsic parallelism for solving the Burgers–Fisher equation. A new class of parallel difference schemes of pure alternating segment explicit-implicit (PASE-I) and pure alternating segment implicit-explicit (PASI-E) are constructed by taking simple classical explicit and implicit schemes, combined with the alternating segment technique. The existence, uniqueness, linear absolute stability, and convergence for the solutions of PASE-I and PASI-E schemes are well illustrated. Both theoretical analysis and numerical experiments show that PASE-I and PASI-E schemes are linearly absolute stable, with 2-order time accuracy and 2-order spatial accuracy. Compared with the implicit scheme and the Crank–Nicolson (C-N) scheme, the computational efficiency of the PASE-I (PASI-E) scheme is greatly improved. The PASE-I and PASI-E schemes have obvious parallel computing properties, which show that the difference methods with intrinsic parallelism in this paper are feasible to solve the Burgers–Fisher equation.

In the present paper we study the so-called sampling Kantorovich operators in the very general setting of modular spaces. Here, modular convergence theorems are proved under suitable assumptions, together with a modular inequality for the above operators. Further, we study applications of such approximation results in several concrete cases, such as Musielak–Orlicz and Orlicz spaces. As a consequence of these results we obtain convergence theorems in the classical and weighted versions of the $$L^p$$ L p and Zygmund (or interpolation) spaces. At the end of the paper examples of kernels for the above operators are presented.

In the present paper, we consider the nonparametric regression model with random design based on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbf{X}_\mathrm{t},\mathbf{Y}_\mathrm{t})_{\mathrm{t}\ge 0}$$\end{document}(Xt,Yt)t≥0 a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{d}\times \mathbb {R}^{q}$$\end{document}Rd×Rq-valued strictly stationary and ergodic continuous time process, where the regression function is given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(\mathbf{x},\psi ) = \mathbb {E}(\psi (\mathbf{Y}) \mid \mathbf{X} = \mathbf{x}))$$\end{document}m(x,ψ)=E(ψ(Y)∣X=x)), for a measurable function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi : \mathbb {R}^{q} \rightarrow \mathbb {R}$$\end{document}ψ:Rq→R. We focus on the estimation of the location \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Theta }}$$\end{document}Θ (mode) of a unique maximum of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(\cdot , \psi )$$\end{document}m(·,ψ) by the location \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widehat{{\varvec{\Theta }}}_\mathrm{T}$$\end{document}Θ^T of a maximum of the Nadaraya–Watson kernel estimator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{m}_\mathrm{T}(\cdot , \psi )$$\end{document}m^T(·,ψ) for the curve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(\cdot , \psi )$$\end{document}m(·,ψ). Within this context, we obtain the consistency with rate and the asymptotic normality results for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widehat{{\varvec{\Theta }}}_\mathrm{T}$$\end{document}Θ^T under mild local smoothness assumptions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(\cdot , \psi )$$\end{document}m(·,ψ) and the design density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\cdot )$$\end{document}f(·) of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{X}$$\end{document}X. Beyond ergodicity, any other assumption is imposed on the data. This paper extends the scope of some previous results established under the mixing condition. The usefulness of our results will be illustrated in the construction of confidence regions.

The interpolating boundary element-free method (IBEFM) is a direct solution method of the meshless boundary integral equation method, which has high efficiency and accuracy. The IBEFM is developed based on the interpolating moving least-squares (IMLS) method and the boundary integral equation method. Since the shape function of the IMLS method satisfies the interpolation characteristics, the IBEFM can directly and accurately impose the essential boundary conditions, which overcomes the shortcomings of the original boundary element-free method in enforcing the essential boundary approximately. This paper will study the error estimations of the IBEFM for two-dimensional potential problems and the relationship between the errors and the influence radius and the condition number of the coefficient matrix. Two numerical examples are presented to verify the correctness of the theoretical results in this paper.