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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.

Full-waveform inversion (FWI) is today a standard process for the inverse problem of seismic imaging. PDE-constrained optimization is used to determine unknown parameters in a wave equation that represent geophysical properties. The objective function measures the misfit between the observed data and the calculated synthetic data, and it has traditionally been the least-squares norm. In a sequence of papers, we introduced the Wasserstein metric from optimal transport as an alternative misfit function for mitigating the so-called cycle skipping, which is the trapping of the optimization process in local minima. In this paper, we first give a sharper theorem regarding the convexity of the Wasserstein metric as the objective function. We then focus on two new issues. One is the necessary normalization of turning seismic signals into probability measures such that the theory of optimal transport applies. The other, which is beyond cycle skipping, is the inversion for parameters below reflecting interfaces. For the first, we propose a class of normalizations and prove several favorable properties for this class. For the latter, we demonstrate that FWI using optimal transport can recover geophysical properties from domains where no seismic waves travel through. We finally illustrate these properties by the realistic application of imaging salt inclusions, which has been a significant challenge in exploration geophysics.

This paper concerns the existence of transonic shock solutions to the 2-D steady compressible Euler system in an almost flat finite nozzle ( in the sense that it is a generic small perturbation of a flat one ), under physical boundary conditions proposed by Courant-Friedrichs in \cite{CourantFriedrichs1948}, in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location of the shock-front is one of the most desirable information one would like to determine. However, the location of the normal shock-front in a flat nozzle can be anywhere in the nozzle so that it provides little information on the possible location of the shock-front when the nozzle's boundary is perturbed. So one of the key difficulties in looking for transonic shock solutions is to determine the shock-front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution will be taken as an initial approximation for the transonic shock solution. In this paper, a sufficient condition in terms of the geometry of the nozzle and the given exit pressure is derived which yields the existence of the solutions to the proposed free boundary problem. Once an initial approximation is obtained, a further nonlinear iteration could be constructed and proved to lead to a transonic shock solution., 53 pages

We establish a theory of $Q$-valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents $\mathrm{mod}(p)$ when $p=2Q$, and to establish a first general partial regularity theorem for every $p$ in any dimension and codimension., 37 pages. First part of a two-papers work aimed at establishing a first general partial regularity theory for area minimizing currents modulo p, for any p and in any dimension and codimension. v3 is the final version, to appear on Comm. Pure Appl. Math

It is well known that when the geometry and/or coefficients allow stable trapped rays, the solution operator of the Helmholtz equation (a.k.a. the resolvent of the Laplacian) grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongest-possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numerical methods for solving the Helmholtz equation at high frequency that are based on a polynomial-growth assumption on the solution operator (e.g. $hp$-finite elements, $hp$-boundary elements, certain multiscale methods). The result of this paper shows that this assumption holds, even in the presence of the strongest-possible trapping, for most frequencies.

The first goal of this paper is to prove multiple asymptotic results for a time-discrete and space-continuous polymer model of a random walk in a random potential. These results include: existence of deterministic free energy density in the infinite volume limit for every fixed asymptotic slope; concentration inequalities for free energy implying a bound on its fluctuation exponent; straightness estimates implying a bound on the transversal fluctuation exponent. The culmination of this program is almost sure existence and uniqueness of polymer measures on one-sided infinite paths with given endpoint and slope, and interpretation of these infinite-volume Gibbs measures as thermodynamic limits. Moreover, we prove that marginals of polymer measures with the same slope and different endpoints are asymptotic to each other. The second goal of the paper is to develop ergodic theory of the Burgers equation with positive viscosity and random kick forcing on the real line without any compactness assumptions. Namely, we prove a One Force -- One Solution principle, using the infinite volume polymer measures to construct a family of stationary global solutions for this system, and proving that each of those solutions is a one-point pullback attractor on the initial conditions with the same spatial average. This provides a natural extension of the same program realized for the inviscid Burgers equation with the help of action minimizers that can be viewed as zero temperature limits of polymer measures., Comment: 67 pages. This is an extension of the ergodic program for the Burgers equation in arXiv:1205.6721 and arXiv:1406.5660 to the positive viscosity case. Minor clarifications and additions to bibliography in this version

The theme of this paper is the analysis of bootstrap percolation processes on random graphs generated by preferential attachment. This is a class of infection processes where vertices have two states: they are either infected or susceptible. At each round every susceptible vertex which has at least $r\geq 2$ infected neighbours becomes infected and remains so forever. Assume that initially $a(t)$ vertices are randomly infected, where $t$ is the total number of vertices of the graph. Suppose also that $r < m$, where $2m$ is the average degree. We determine a critical function $a_c(t)$ such that when $a(t) \gg a_c(t)$, complete infection occurs with high probability as $t \rightarrow \infty$, but when $a(t) \ll a_c (t)$, then with high probability the process evolves only for a bounded number of rounds and the final set of infected vertices is asymptotically equal to $a(t)$., This paper is significantly different to the previous version

In a paper that initiated the modern study of the stochastic block model (SBM), Decelle, Krzakala, Moore, and Zdeborova, backed by Mossel, Neeman, and Sly, conjectured that detecting clusters in the symmetric SBM in polynomial time is always possible above the Kesten-Stigum (KS) threshold, while it is possible to detect clusters information theoretically (i.e., not necessarily in polynomial time) below the KS threshold when the number of clusters k is at least 4. Massoulie, Mossel et al., and Bordenave, Lelarge, and Massoulie proved that the KS threshold is in fact efficiently achievable for k = 2, while Mossel et al. proved that it cannot be crossed at k = 2. The above conjecture remained open for k ≥ 3. This paper proves the two parts of the conjecture, further extending the results to general SBMs. For the efficient part, an approximate acyclic belief propagation (ABP) algorithm is developed and proved to detect communities for any k down to the KS threshold in quasi-linear time. Achieving this requires showing optimality of ABP in the presence of cycles, a challenge for message-passing algorithms. The paper further connects ABP to a power iteration method on a nonbacktracking operator of generalized order, formalizing the interplay between message passing and spectral methods. For the information-theoretic part, a nonefficient algorithm sampling a typical clustering is shown to break down the KS threshold at k = 4. The emerging gap is shown to be large in some cases, making the SBM a good case study for information-computation gaps. © 2017 Wiley Periodicals, Inc.

The Bogdanov-Takens bifurcations of a Leslie-Gower predator-prey model with Michaelis-Menten–type prey harvesting were studied. In the paper “Diff. Equ. Dyn. Syst. 20(2012), 339-366,” Gupta et al proved that the Leslie-Gower predator-prey model with Michaelis-Menten–type prey harvesting has rich dynamics. Some equilibria of codimension 1 and their bifurcations were discussed. In this paper, we find that the model has an equilibrium of codimensions 2 and 3. We also prove analytically that the model undergoes Bogdanov-Takens bifurcations (cusp cases) of codimensions 2 and 3. Hence, the model can have 2 limit cycles, coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1 as the values of parameters vary. Moreover, several numerical simulations are conducted to illustrate the validity of our results.

This paper studies the dynamics of the generalized Lengyel-Epstein reaction-diffusion model proposed in a recent study by Abdelmalek and Bendoukha. Two main results are shown in this paper. The first of which is sufficient conditions that guarantee the nonexistence of Turing patterns, ie, nonconstant solutions. Second, more relaxed conditions are derived for the stability of the system's unique steady-state solution.

In this paper, a novel simulation methodology based on the reproducing kernels is proposed for solving the fractional order integro-differential transport model for a nuclear reactor. The analysis carried out in this paper thus forms a crucial step in the process of development of fractional calculus as well as nuclear science models. The fractional derivative is described in the Captuo Riemann–Liouville sense. Results are presented graphically and in tabulated forms to study the efficiency and accuracy of method. The present scheme is very simple, effective, and appropriate for obtaining numerical simulation of nuclear science models. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, an iteration process is considered to solve linear ill-posed problems. Based on the randomness of the involved variables, this kind of problems is regarded as simulation problems of the posterior distribution of the unknown variable given the noise data. We construct a new ensemble Kalman filter-based method to seek the posterior target distribution. Despite the ensemble Kalman filter method having widespread applications, there has been little analysis of its theoretical properties, especially in the field of inverse problems. This paper analyzes the propagation of the error with the iteration step for the proposed algorithm. The theoretical analysis shows that the proposed algorithm is convergence. We compare the numerical effect with the Bayesian inversion approach by two numerical examples: backward heat conduction problem and the first kind of integral equation. The numerical tests show that the proposed algorithm is effective and competitive with the Bayesian method. Copyright © 2017 John Wiley & Sons, Ltd.

Ren and Zeng (2013) introduced a new kind of q-Bernstein-Schurer operators and studied some approximation properties. Acu etal. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q-Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K-functional. Next, we introduce the bivariate case of q-Bernstein-Schurer-Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K-functional. Finally, we define the generalized Boolean sum operators of the q-Bernstein-Schurer-Kantorovich type and investigate the approximation of the Bogel continuous and Bogel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright (c) 2017 John Wiley & Sons, Ltd.

This paper studies a higher dimensional generalization of Frieze's $\zeta(3)$-limit theorem in the Erd\"os-R\'enyi graph process. Frieze's theorem states that the expected weight of the minimum spanning tree converges to $\zeta(3)$ as the number of vertices goes to infinity. In this paper, we study the $d$-Linial-Meshulam process as a model for random simplicial complexes, where $d=1$ corresponds to the Erd\"os-R\'enyi graph process. First, we define spanning acycles as a higher dimensional analogue of spanning trees, and connect its minimum weight to persistent homology. Then, our main result shows that the expected weight of the minimum spanning acycle behaves in $O(n^{d-1})$., Comment: 24 pages

The k-core, defined as the maximal subgraph of minimum degree at least k, of the random graph G(n,p) has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [J Combin Theory Ser B 67 (1996), 111–151] determined the threshold dk for the appearance of an extensive k-core. The aim of the present paper is to describe how the k-core is “embedded” into the random graph in the following sense. Let k≥3 and fix d=np>dk. Colour each vertex that belongs to the k-core of G(n,p) in black and all remaining vertices in white. Here we derive a multi-type branching process that describes the local structure of this coloured random object as n tends to infinity. This generalises prior results on, e.g., the internal structure of the k-core. In the physics literature it was suggested to characterize the core by means of a message passing algorithm called Warning Propagation. Ibrahimi, Kanoria, Kraning and Montanari [Ann Appl Probab 25 (2015), 2743–2808] used this characterization to describe the 2-core of random hypergraphs. To derive our main result we use a similar approach. A key observation is that a bounded number of iterations of this algorithm is enough to give a good approximation of the k-core. Based on this the study of the k-core reduces to the analysis of Warning Propagation on a suitable Galton-Watson tree. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 00, 000–000, 2017

We consider consecutive random subdivision of polygons described as follows. Given an initial convex polygon with $d\ge 3$ edges, we choose a point at random on each edge, such that the proportions in which these points divide edges are i.i.d. copies of some random variable $\xi$. These new points form a new (smaller) polygon. By repeatedly implementing this procedure we obtain a sequence of random polygons. The aim of this paper is to show that under very mild non-degenerateness conditions on $\xi$, the shapes of these polygons eventually become "flat" The convergence rate to flatness is also investigated; in particular, in the case of triangles ($d=3$), we show how to calculate the exact value of the rate of convergence, connected to Lyapunov exponents. Using the theory of products of random matrices our paper greatly generalizes the results of Volkov (2013) which are achieved mostly by using ad hoc methods., Comment: 35 pages, 2 figures

This paper discusses uncertainty principles of images defined on the square, or, equivalently, uncertainty principles of signals on the 2-torus. Means and variances of time and frequency for signals on the 2-torus are defined. A set of phase and amplitude derivatives are introduced. Based on the derivatives, we obtain three comparable lower bounds of the product of variances of time and frequency, of which the largest lower bound corresponds to the strongest uncertainty principles known for periodic signals. Examples, including simulations, are provided to illustrate the obtained results. To the authors' knowledge, it is in the present paper, and for the first time, that uncertainty principles on the torus are studied. Copyright © 2016 John Wiley & Sons, Ltd.

Staircase tableaux are combinatorial objects that were first introduced due to a connection with the asymmetric simple exclusion process (ASEP) and Askey-Wilson polynomials. Since their introduction, staircase tableaux have been the object of study in many recent papers. Relevant to this paper, Hitczenko and Janson proved that distribution of parameters on the first diagonal is asymptotically normal. In addition, they conjectured that other diagonals would be asymptotically Poisson. Since then, only the second and the third diagonal were proven to follow the conjecture. This paper builds upon those results to prove the conjecture for the kth diagonal where k is fixed. In particular, we prove that the distribution of the number of α's (β's) on the kth diagonal, k > 1, is asymptotically Poisson with parameter 1/2. In addition, we prove that symbols on the kth diagonal are asymptotically independent and thus, collectively follow the Poisson distribution with parameter 1. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 795–818, 2016

This is the first of a series of papers devoted to the initial value problem for the one-dimensional Euler system of compressible fluids and augmented versions containing higher-order terms. In the present paper, we are interested in dispersive shock waves and analyze the zero viscosity-capillarity limit associated with the Navier-Stokes-Korteweg system. Specifically, we establish the existence of finite energy solutions as well as their convergence toward entropy solutions to the Euler system. Our method of proof combines energy and effective energy estimates, a nonlinear Sobolev inequality, high-integrability properties for the mass density and for the velocity, and compactness properties based on entropies.© 2015 Wiley Periodicals, Inc.

The regression-based Monte Carlo methods for backward stochastic differential equations (BSDEs) have been the object of considerable research, particularly for solving nonlinear partial differential equations (PDEs). Unfortunately, such methods often become unstable when implemented with small time steps because the variance of gradient estimates is inversely proportional to the time step (σ2∼ 1/Δ t). Recently new variance reduction techniques were introduced to address this problem in~a paper by the author and Avellaneda. The purpose of this paper is to provide a rigorous justification for these techniques in the context of the discrete-time BSDE scheme of Bouchard and Touzi. We also suggest a new higher-order scheme that makes the variance proportional to the time step (σ2∼Δ t). These techniques are easy to implement. Numerical examples strongly indicate that they render the regression-based Monte Carlo methods stable for small time steps and thus viable for numerical solution of nonlinear PDEs.© 2016 Wiley Periodicals, Inc.

This paper applies Nevanlinna theory of value distribution to discuss existence of solutions of certain types of non-linear differential-difference equations such as (5) and (8) given in the succeeding paragraphs. Existence of solutions of differential equations and difference equations can be said to have been well studied, that of differential-difference equations, on the other hand, have been paid little attention. Such mixed type equations have great significance in applications. This paper, in particular, generalizes the Rellich–Wittich-type theorem and Malmquist-type theorem about differential equations to the case of differential-difference equations. Copyright © 2015 John Wiley & Sons, Ltd.

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most C|VG|. In this paper, we show that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph Gn,p, which improves upon existing results showing that asymptotically almost surely the cop number of Gn,p is Onlogn provided that pni¾?2+elogn for some e>0. We do this by first showing that the conjecture holds for a general class of graphs with some specific expansion-type properties. This will also be used in a separate paper on random d-regular graphs, where we show that the conjecture holds asymptotically almost surely when d=dni¾?3. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 396-421, 2016

The global existence of weak solutions of the incompressible viscoelastic flows in two spatial dimensions has been a longstanding open problem, and it is studied in this paper. We show global existence if the initial deformation gradient is close to the identity matrix in L2 ∩ L∞ and the initial velocity is small in L2 and bounded in Lp for some p > 2. While the assumption on the initial deformation gradient is automatically satisfied for the classical Oldroyd-B model, the additional assumption on the initial velocity being bounded in Lp for some p > 2 may due to techniques we employed. The smallness assumption on the L2 norm of the initial velocity is, however, natural for global well-posedness. One of the key observations in the paper is that the velocity and the “ effective viscous flux” G are sufficiently regular for positive time. The regularity of G leads to a new approach for the pointwise estimate for the deformation gradient without using L∞ bounds on the velocity gradients in spatial variables. © 2015 Wiley Periodicals, Inc.

In this paper, we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves. We construct such spacetimes as solutions to the characteristic initial value problem of the Einstein vacuum equations with a data curvature delta singularity. We show that in the resulting spacetime, the delta singularity propagates along a characteristic hypersurface, while away from that hypersurface the spacetime remains smooth. Unlike the known explicit examples of impulsive gravitational spacetimes, this work in particular provides the first construction of an impulsive gravitational wave of compact extent and does not require any symmetry assumptions. The arguments in the present paper also extend to the problem of existence and uniqueness of solutions to a larger class of nonregular characteristic data. © 2015 Wiley Periodicals, Inc.

This paper is the continuation of a previous paper (H. Knupfer and C. B. Muratov, Comm. Pure Appl. Math. 66 (2013), 1129‐1162). We investigate the classical isoperimetric problem modified by an addition of a nonlocal repulsive term generated by a kernel given by an inverse power of the distance. In this work, we treat the case of a general space dimension. We obtain basic existence results for minimizers with sufficiently small masses. For certain ranges of the exponent in the kernel, we also obtain nonexistence results for sufficiently large masses, as well as a characterization of minimizers as balls for sufficiently small masses and low spatial dimensionality. The physically important special case of three space dimensions and Coulombic repulsion is included in all the results mentioned above. In particular, our work yields a negative answer to the question if stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter. In all cases the minimal energy scales linearly with mass for large masses, even if the infimum of energy cannot be attained. © 2014 Wiley Periodicals, Inc.

We consider connectivity properties of certain i.i.d. random environments on i¾?d, where at each location some steps may not be available. Site percolation and oriented percolation are examples of such environments. In these models, one of the quantities most often studied is the random set of vertices that can be reached from the origin by following a connected path. More generally, for the models we consider, multiple different types of connectivity are of interest, including: the set of vertices that can be reached from the origin; the set of vertices from which the origin can be reached; the intersection of the two. As with percolation models, many of the models we consider admit, or are expected to admit phase transitions. Among the main results of the paper is a proof of the existence of phase transitions for some two-dimensional models that are non-monotone in their underlying parameter, and an improved bound on the critical value for oriented site percolation on the triangular lattice. The connectivity of the random directed graphs provides a foundation for understanding the asymptotic properties of random walks in these random environments, which we study in a second paper. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 111-137, 2014

In this paper, we first propose a concept of weighted pseudo-almost periodic functions on time scales and study some basic properties of weighted pseudo-almost periodic functions on time scales. Then, we establish some results about the existence of weighted pseudo-almost periodic solutions to linear dynamic equations on time scales. Finally, as an application of our results, we study the existence and global exponential stability of weighted pseudo-almost periodic solutions for a class of cellular neural networks with discrete delays on time scales. The results of this paper are completely new. Copyright © 2016 John Wiley & Sons, Ltd.

This paper [1], which was published online on 1 June 2011, has been retracted by agreement between the authors, the journal’s Editor-in-Chief Derek Holt, the London Mathematical Society and Cambridge University Press. The retraction was agreed to prevent other authors from using incorrect mathematical results. (In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no. 1, 50–58; J. Number Theory 130 (2010) no. 4, 1109–1114. Furthermore, we formulate a criterion for the partial Riemann hypothesis and we provide some numerical evidence for it using new formulas for the Li coefficients.)

The rotation correspondence is a map that sends the set of plane trees onto the set of binary trees. In this paper, we first show that when n goes to + ∞, the image by the rotation correspondence of a uniformly chosen random plane tree τ with n nodes is close to 2τ (in a sense to be defined). The second part of the paper is devoted to the right and left depth of nodes in binary trees. We show that the empiric measure (suitably normalized) associated with the difference of the right depth and the left depth processes converges to the integrated super Brownian excursion.

Szemeredi's Regularity Lemma is a well-known and powerful tool in modern graph theory. This result led to a number of interesting applications, particularly in extremal graph theory. A regularity lemma for 3-uniform hypergraphs developed by Frankl and Rodl [8] allows some of the Szemeredi Regularity Lemma graph applications to be extended to hypergraphs. An important development regarding Szemeredi's Lemma showed the equivalence between the property of e-regularity of a bipartite graph G and an easily verifiable property concerning the neighborhoods of its vertices (Alon et al. [1]; cf. [6]). This characterization of e-regularity led to an algorithmic version of Szemeredi's lemma [1]. Similar problems were also considered for hypergraphs. In [2], [9], [13], and [18], various descriptions of quasi-randomness of k-uniform hypergraphs were given. As in [1], the goal of this paper is to find easily verifiable conditions for the hypergraph regularity provided by [8]. The hypergraph regularity of [8] renders quasi-random "blocks of hyperedges" which are very sparse. This situation leads to technical difficulties in its application. Moreover, as we show in this paper, some easily verifiable conditions analogous to those considered in [2] and [18] fail to be true in the setting of [8]. However, we are able to find some necessary and sufficient conditions for this hypergraph regularity. These conditions enable us to design an algorithmic version of a hypergraph regularity lemma in [8]. This algorithmic version is presented by the authors in [5].

In this paper we generalize an equation studied by Mossino and Temam, to the fully nonlinear case. This equation arises in plasma physics as an approximation to Grad equations, which were introduced by Harold Grad, to model the behavior of plasma confined in a toroidal vessel. We prove existence of a $W^{2,p}$-viscosity solution and regularity up to $C^{1,\alpha}(\overline{\Omega})$ for any $\alpha

The aim of this paper is to construct a numerical scheme for solving a class of bilateral free boundaries problem. First, using a shape functional and some regularization terms, an optimal control problem is formulated, in addition, we prove its solution existence's. The first optimality conditions and the shape gradient are computed. the proposed numerical scheme is a genetic algorithm guided conjugate gradient combined with the finite element method, at each mesh regeneration, we perform a mesh refinement in order to avoid any domain singularities. Some numerical examples are shown to demonstrate the validity of the theoretical results, and to prove the robustness of the proposed scheme.

This paper is about quantitative linearization results for the Monge-Ampere equation with rough data. We develop a large-scale regularity theory and prove that if a measure µ is close to the Lebesgue measure in Wasserstein distance at all scales, then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding Poisson equation. The main ingredient we use is a harmonic approximation result for the optimal transport plan between arbitrary measures. This is used in a Campanato iteration which transfers the information through the scales.

In this paper, a generalized fractional $(q,h)$-Gronwall inequality is investigated. Based on this inequality, we derive the uniqueness theorem and the finite-time stability criterion of nonlinear fractional delay $(q,h)$-difference systems. Several examples are given to illustrate our theoretical result.

This paper is devoted to the study of flows associated to non-smooth vector fields. We prove the well-posedness of regular Lagrangian flows associated to vector fields $\mathbf{B}=(\mathbf{B}^1,...,\mathbf{B}^d)\in L^1(\mathbb{R}_+;L^1(\mathbb{R}^d)+L^\infty(\mathbb{R}^d))$ satisfying $ \mathbf{B}^i=\sum_{j=1}^{m}\mathbf{K}_j^i*b_j,$ $b_j\in L^1(\mathbb{R}_+,BV(\mathbb{R}^d))$ and $\operatorname{div}(\mathbf{B})\in L^1(\mathbb{R}_+;L^\infty(\mathbb{R}^d))$ for $d,m\geq 2$, where $(\mathbf{K}_j^i)_{i,j}$ are singular kernels in $\mathbb{R}^d$. Moreover, we also show that there exist an autonomous vector-field $\mathbf{B}\in L^1(\mathbb{R}^2)+L^\infty(\mathbb{R}^2)$ and singular kernels $(\mathbf{K}_j^i)_{i,j}$, singular Radon measures $\mu_{ijk}$ in $\mathbb{R}^2$ satisfying $\partial_{x_k} \mathbf{B}^i=\sum_{j=1}^{m}\mathbf{K}_j^i\star\mu_{ijk}$ in distributional sense for some $m\geq 2$ and for $k,i=1,2$ such that regular Lagrangian flows associated to vector field $\mathbf{B}$ are not unique.

The research contained in this paper belongs to the qualitative theory of dynamic equations on time scales. Via the detailed analysis of solutions of the associated Riccati equation and an advanced averaging technique, we provide the description of domain of nonoscillation of very general equations. The results are formulated and proved for half-linear equations (i.e., equations connected to PDEs with one dimensional p-Laplacian) on time scales. Nevertheless, we obtain new results also for linear difference equations. Moreover, the combination of the presented results with previous ones shows that many useful equations are conditionally oscillatory. Such equations are ideal as testing and comparison equations in real-world models which are beyond capabilities of known oscillation and nonoscillation criteria often.

This paper proves the validity of the joint mean-field and classical limit of the quantum $N$-body dynamics leading to the pressureless Euler-Poisson system for factorized initial data whose first marginal has a monokinetic Wigner measure. The interaction potential is assumed to be the repulsive Coulomb potential. The validity of this derivation is limited to finite time intervals on which the Euler-Poisson system has a smooth solution that is rapidly decaying at infinity. One key ingredient in the proof is an inequality from [S. Serfaty, with an appendix of M. Duerinckx arXiv:1803.08345v3 [math.AP]].

We study some classical uniqueness and existence results, such as Peano's or Osgood's uniqueness criteria, in the context of Stieltjes differential equations. This type of equation is based on derivatives with respect to monotone functions, and enables the investigation of discrete and continuous problems from a common standpoint. We compare our results with previous work on the topic and illustrate the advantages of the theorems presented in this paper with an example. Finally, we make some remarks regarding analogous uniqueness results which can be derived for measure differential equations.

Orthogonal systems in $\mathrm{L}_2(\mathbb{R})$, once implemented in spectral methods, enjoy a number of important advantages if their differentiation matrix is skew-symmetric and highly structured. Such systems, where the differentiation matrix is skew-symmetric, tridiagonal and irreducible, have been recently fully characterised. In this paper we go a step further, imposing the extra requirement of fast computation: specifically, that the first $N$ coefficients {of the expansion} can be computed to high accuracy in $\mathcal{O}(N\log_2N)$ operations. We consider two settings, one approximating a function $f$ directly in $(-\infty,\infty)$ and the other approximating $[f(x)+f(-x)]/2$ and $[f(x)-f(-x)]/2$ separately in $[0,\infty)$. In each setting we prove that there is a single family, parametrised by $\alpha,\beta > -1$, of orthogonal systems with a skew-symmetric, tridiagonal, irreducible differentiation matrix and whose coefficients can be computed as Jacobi polynomial coefficients of a modified function. The four special cases where $\alpha, \beta= \pm 1/2$ are of particular interest, since coefficients can be computed using fast sine and cosine transforms. Banded, Toeplitz-plus-Hankel multiplication operators are also possible for representing variable coefficients in a spectral method. In Fourier space these orthogonal systems are related to an apparently new generalisation of the Carlitz polynomials.

International audience; In this paper we examine a control variate estimator for a quantity that can be expressed as the expectation of a functional of a random process, that is itself the solution of a differential equation driven by fast mean-reverting ergodic forces. The control variate is the expectation of the same functional for the limit diffusion process that approximates the original process when the mean-reversion time goes to zero. To get an efficient control variate estimator, we propose a coupling method to build the original process and the limit diffusion process. We show that the correlation between the two processes indeed goes to one when the mean reversion time goes to zero and we quantify the convergence rate, which makes it possible to characterize the variance reduction of the proposed control variate method. The efficiency of the method is illustrated on a few examples.

Motivated by gradient methods in optimization theory, we give methods based on $\psi$-fractional derivatives of order $\alpha$ in order to solve unconstrained optimization problems. The convergence of these methods is analyzed in detail. This paper also presents an Adams-Bashforth-Moulton (ABM) method for the estimation of solutions to equations involving $\psi$-fractional derivatives. Numerical examples using the ABM method show that the fractional order $\alpha$ and weight $\psi$ are tunable parameters, which can be helpful for improving the performance of gradient descent methods., Comment: 27 pages

Main aim of the present paper is explore certain physical phenomena by means of $p$-adic probability theory. To overcome this study, we deal with a more general setting to define $p$-adic Gibbs measures. For the sake of simplicity of explanations, we restrict ourselves to the Ising model on the Cayley tree, since such a model has broad theoretical and practical applications. To study $p$-adic quasi Gibbs measures, we reduce the problem to the description of the fixed points of the Ising-Potts mapping. Finding fixed points is not an easy job as in the real setting. Furthermore, the phase transition for the model is established. In the real case, the phase transition yields the the singularity of the limiting Gibbs measures. However, we show that the $p$-adic quasi Gibbs measures do not exhibit the mentioned type of singularity, such kind of phenomena is called strong phase transition. Finally, we deal with the solvability and the number of solutions of ceratin $p$-adic equation depending on several parameters. Such a description allows us to find all possible translation-invariant $p$a-adic quasi Gibbs measures.

In this paper, we consider the generalized approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls. We analyse the relationship between the generalized approximate boundary synchronization and the generalized exact boundary synchronization, give a sufficient condition to realize the generalized approximate boundary synchronization and a necessary condition in terms of Kalman’s matrix, and show the meaning of the number of total controls. Besides, by the generalized synchronization decomposition, we define the generalized approximately synchronizable state, and obtain its properties and a sufficient condition for it to be independent of applied boundary controls.

The aim of this paper is to analyze a model for chemotaxis based on a local sensing mechanism instead of the gradient sensing mechanism used in the celebrated minimal Keller-Segel model. The model we study has the same entropy as the minimal Keller-Segel model, but a different dynamics to minimize this entropy. Consequently, the conditions on the mass for the existence of stationary solutions or blow-up are the same, however we make the interesting observation that with the local sensing mechanism the blow-up in the case of supercritical mass is delayed to infinite time. Our observation is made rigorous from a mathematical point via a proof of global existence of weak solutions for arbitrary large masses and space dimension. The key difference of our model to the minimal Keller-Segel model is that the structure of the equation allows for a duality estimate that implies a bound on the $(H^1)'$-norm of the solutions, which can only grow with a square-root law in time. This additional $(H^1)'$-bound implies a lower bound on the entropy, which contrasts markedly with the minimal Keller-Segel model for which it is unbounded from below in the supercritical case. Besides, regularity and uniqueness of solutions are also studied.

In this paper, we study Hegselmann-Krause models with a time-variable time delay. Under appropriate assumptions, we show the exponential asymptotic consensus when the time delay satisfies a suitable smallness assumption. Our main strategies for this are based on Lyapunov functional approach and careful estimates on the trajectories. We then study the mean-field limit from the many-individual Hegselmann-Krause equation to the continuity-type partial differential equation as the number N of individuals goes to infinity. For the limiting equation, we prove global-in-time existence and uniqueness of measure-valued solutions. We also use the fact that constants appearing in the consensus estimates for the particle system are independent of N to extend the exponential consensus result to the continuum model. Finally, some numerical tests are illustrated.

In this paper, we look at a probabilistic approach to a non-local quadratic form that has lately attracted some interest. This form is related to a recently introduced non-local normal derivative. The goal is to construct two Markov process: one corresponding to that form and the other which is related to a probabilistic interpretation of the Neuman problem. We also study the Dirichlet-to-Neumann operator for non-local operators., Comment: 21 pages

In this paper, we study the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds number $\text{Re}$. It was proved that if the initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0\text{Re}^{-1}$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow. This result confirms the transition threshold conjecture in physical literatures.

We provide the first nontrivial upper bound for the chemical distance exponent in two-dimensional critical percolation. Specifically, we prove that the expected length of the shortest horizontal crossing path of a box of side length $n$ in critical percolation on $\mathbb{Z}^2$ is bounded by $Cn^{2-\delta}\pi_3(n)$, for some $\delta>0$, where $\pi_3(n)$ is the "three-arm probability to distance $n$." This implies that the ratio of this length to the length of the lowest crossing is bounded by an inverse power of $n$ with high probability. In the case of site percolation on the triangular lattice, we obtain a strict upper bound for the exponent of $4/3$. The proof builds on the strategy developed in our previous paper, but with a new iterative scheme, and a new large deviation inequality for events in annuli conditional on arm events, which may be of independent interest.

Every now and then, there has been natural or man-made calamities Such adversities instigate various institutions to find solutions for them The current study attempts to explore the disaster caused by the micro enemy called coronavirus for the past few months and aims at finding the solution for the system of nonlinear ordinary differential equations to which q?homotopy analysis transform method (q?HATM) has been applied to arrive at effective results In this paper, there are eight nonlinear ordinary differential equations considered and to solve them the advanced fractional operator Atangana-Baleanu (AB) fractional derivative has been applied to produce better understanding The outcomes have been presented in terms of plots Ultimately, the present study assists in examining the real-world models and aids in predicting their behavior corresponding to the parameters considered in the models

Let $A$ be a $W^{1,2}$-connection on a principle $\text{SU}(2)$-bundle $P$ over a compact $4$-manifold $M$ whose curvature $F_A$ satisfies $\|F_A\|_{L^2(M)}\le \Lambda$. Our main result is the existence of a global section $\sigma: M\to P$ with finite singularities on $M$ such that the connection form $\sigma^*A$ satisfies the Coulomb equation $d^*(\sigma^*A)=0$ and admits a sharp estimate $\|\sigma^*A\|_{\mathcal{L}^{4,\infty}(M)}\le C(M,\Lambda)$. Here $\mathcal{L}^{4,\infty}$ is a new function space we introduce in this paper that satisfies $L^4(M)\subsetneq \mathcal{L}^{4,\infty}(M)\subsetneq L^{4-\epsilon}(M)$ for all $\epsilon>0$.