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In this paper, by using Schwarz rearrangement and isoperimetric inequalities, we prove comparison results for the solutions of Poisson equations on complete Riemannian manifolds with Ric ≥ (n − 1) κ , κ = 0 or 1 , which extends the results in [A. Alvino, C. Nitsch and C. Trombetti, A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions, Comm. Pure Appl. Math. 76(3) (2023) 585–603]. Furthermore, as applications of our comparison results, we obtain the Saint-Venant inequality and Bossel–Daners inequality for Robin Laplacian. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show how the modular method together with different techniques can be used to prove non-existence of primitive non-trivial solutions of the equation x 2 + d y 6 = z p for square-free values 1 ≤ d ≤ 2 0. The key ingredients are: the approach presented in [A. Pacetti and L. V. Torcomian, ℚ -curves, Hecke characters and some Diophantine equations, Math. Comp. 91(338) (2022) 2817–2865] (in particular its recipe for the space of modular forms to be computed) together with the use of the symplectic method (as developed in [E. Halberstadt and A. Kraus, Courbes de Fermat: Résultats et problèmes, J. Reine Angew. Math. 548 (2002) 167–234], although we give a variant over ramified extensions needed in our applications) to discard solutions and the use of a second Frey curve, aiming to prove large image of residual Galois representations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem: − div (A (x , ∇ u)) = μ in Ω , u = 0 on ∂ Ω , in Lorentz–Morrey spaces, where Ω ⊂ ℝ n (n ≥ 2), μ is a finite Radon measure, A is a monotone Carathéodory vector-valued function defined on W 0 1 , p (Ω) and the p -capacity uniform thickness condition is imposed on the complement of our domain Ω. It is remarkable that the local gradient estimates have been proved first by Mingione in [Gradient estimates below the duality exponent, Math. Ann.346 (2010) 571–627] at least for the case 2 ≤ p ≤ n , where the idea for extending such result to global ones was also proposed in the same paper. Later, the global Lorentz–Morrey and Morrey regularities were obtained by Phuc in [Morrey global bounds and quasilinear Riccati type equations below the natural exponent, J. Math. Pures Appl.102 (2014) 99–123] for regular case p > 2 − 1 n . Here in this study, we particularly restrict ourselves to the singular case 3 n − 2 2 n − 1 < p ≤ 2 − 1 n . The results are central to generalize our technique of good- λ type bounds in the previous work [M.-P. Tran, Good- λ type bounds of quasilinear elliptic equations for the singular case, Nonlinear Anal.178 (2019) 266–281], where the local gradient estimates of solution to this type of equation were obtained in the Lorentz spaces. Moreover, the proofs of most results in this paper are formulated globally up to the boundary results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional p -Laplacian: (− Δ) p α u = f (x , u , ∇ u) , u > 0 in  Ω , u ≡ 0 in  ℝ n ∖ Ω , where Ω is a bounded or an unbounded domain which is convex in x 1 -direction, and (− Δ) p α is the fractional p -Laplacian operator defined by (− Δ) p α u (x) = C n , α , p P. V. ∫ ℝ n | u (x) − u (y) | p − 2 [ u (x) − u (y) ] | x − y | n + α p d y. Under some mild assumptions on the nonlinearity f (x , u , ∇ u) , we establish the monotonicity and symmetry of positive solutions to the nonlinear equations involving the fractional p -Laplacian in both bounded and unbounded domains. Our results are extensions of Chen and Li [Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math.  335 (2018) 735–758] and Cheng et al. [The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math.  19(6) (2017) 1750018]. [ABSTRACT FROM AUTHOR]

Dispersive SQG equation have been studied by many works (see, e.g., [M. Cannone, C. Miao and L. Xue, Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. Londen. Math. Soc. 106 (2013) 650–674; T. M. Elgindi and K. Widmayer, Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684; A. Kiselev and F. Nazarov, Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation, Nonlinearity23 (2010) 549–554; R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys.67 (2016) 104]), which is very similar to the 3D rotating Euler or Navier–Stokes equations. Long time stability for the dispersive SQG equation without dissipation was obtained by Elgindi–Widmayer [Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal.47 (2015) 4672–4684], where the initial condition 𝜃 0 ∈ W 3 + μ , 1 (μ > 0) plays a important role in their proof. In this paper, by using the Strichartz estimate, we can remove this initial condition. Namely, we only assume the initial data is in the Sobolev space like H s . As an application, we can also obtain similar result for the 2D Boussinesq equations with the initial data near a nontrivial equilibrium. [ABSTRACT FROM AUTHOR]

This paper is concerned with the critical threshold phenomenon for one-dimensional damped, pressureless Euler-Poisson equations with electric force induced by a constant background, originally studied in [S. Engelberg and H. Liu and E. Tadmor, Indiana Univ. Math. J. 50 (2001) 109-157]. A simple transformation is used to linearize the characteristic system of equations, which allows us to study the geometrical structure of critical threshold curves for three damping cases: overdamped, underdamped and borderline damped through phase plane analysis. We also derive the explicit form of these critical curves. These sharp results state that if the initial data is within the threshold region, the solution will remain smooth for all time, otherwise it will have a finite time breakdown. Finally, we apply these general results to identify critical thresholds for a non-local system subjected to initial data on the whole line. [ABSTRACT FROM AUTHOR]

The goal of this paper is three-fold. First, we prove that the Cauchy problem for a generalized KP-I equation u t + | D x | α ∂ x u + ∂ x − 1 ∂ y 2 u + 1 2 ∂ x (u 2) = 0 , α ≥ 4 is locally well-posed in the anisotropic Sobolev spaces H s 1 , s 2 ( R 2) with s 1 > − α − 1 4 and s 2 ≥ 0. Second, we prove that the Cauchy problem is globally well-posed in H s 1 , 0 ( R 2) with s 1 > − (α − 1) (3 α − 4) 4 (5 α + 3) if 4 ≤ α ≤ 5. Finally, we show that the Cauchy problem is globally well-posed in H s 1 , 0 ( R 2) with s 1 > − α (3 α − 4) 4 (5 α + 4) if α > 5. Our result improves the result of Saut and Tzvetkov [The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl.79 (2000) 307–338] and Li and Xiao [Well-posedness of the fifth order Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces with nonnegative indices, J. Math. Pures Appl.90 (2008) 338–352]. [ABSTRACT FROM AUTHOR]

We consider an Oldroyd-B model which is derived in Ref. 4 [J. W. Barrett, Y. Lu and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci.15 (2017) 1265–1323] via micro–macro-analysis of the compressible Navier–Stokes–Fokker–Planck system. The global well posedness of strong solutions as well as the associated time-decay estimates in Sobolev spaces for the Cauchy problem are established near an equilibrium state. The terms related to η , in the equation for the extra stress tensor and in the momentum equation, lead to new technical difficulties, such as deducing L t 2 L x 2 -norm dissipative estimates for the polymer number density and its spatial derivatives. One of the main objectives of this paper is to develop a way to capture these dissipative estimates via a low–medium–high-frequency decomposition. [ABSTRACT FROM AUTHOR]

In this paper, I generalize Holzer's theorem for semi-diagonal equation. [ABSTRACT FROM AUTHOR]

Let Nq be the number of solutions to the equation \[ (x_1+ \cdots + x_n)^2 = ax_1 \cdots x_n \] over the finite field 픽q = 픽ps. L. Carlitz found formulas for Nq when n = 3 or 4. In an earlier paper, we found formulas for Nq when d = gcd(n - 2, q - 1) = 1 or 2 or 3 or 4; and when there exists an ℓ such that pℓ ≡ -1 (mod d). In our other paper, the cases d = 7 or 14, p ≡ 2 or 4 (mod 7) were considered. Recently, we obtained formulas for Nq when d = 8. In this paper, we find formulas for Nq when d = 2t, t ≥ 4, p ≡ 3 or 5 (mod 8) or p ≡ 9 (mod 16). [ABSTRACT FROM AUTHOR]

In this paper, we extend the local gradient estimates established in [8, 10] to the conformal quotient equations. An existence of solutions of the conformal quotient equations follows from the local gradient estimates and the paper [14] of Gursky and Viaclovsky. [ABSTRACT FROM AUTHOR]

We find fundamental solutions in closed form for a family of parabolic equations with two spatial variables, whose symmetry groups had been determined in an earlier paper by Finkel [12]. We show how these results can be applied in finance to yield closed form solutions for special affine and quadratic two factor term structure models as well as a new class of models with inverse square behavior. The latter can be considered a partial extension to two factors of pricing models related to the Bessel process devised by Albanese and Campolieti [3] and Albanese et al. [2]. A by-product of our results is that Lie's reduction method in this setting leads only to fundamental solutions that can be factorized as products of functions that depend jointly on time and on one spatial coordinate. Thus all the results in this paper extend immediately to n factor models. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Einstein vacuum equations under the additional assumption of T3-Gowdy symmetry. We prove that there is a generic set of initial data such that the corresponding solutions exhibit curvature blow up on a dense subset of the singularity. By generic, we mean a countable intersection of open sets (i.e. a Gδ set) which is also dense. Furthermore, the set of initial data is given the C∞ topology. This result was presented at a conference in Miami 2004. Recently, we have obtained a stronger result, but the argument to prove it is different and much longer. Therefore, we here wish to present the original argument. Finally, combining the results presented here with a paper by Chruściel and Lake, one obtains strong cosmic censorship for T3-Gowdy spacetimes. [ABSTRACT FROM AUTHOR]

We show how to prescribe the initial data of a characteristic problem satisfying the constraints, the smallness, the regularity and the asymptotic decay suitable to prove a global existence result. In this paper, the first of two, we show in detail the construction of the initial data and give a sketch of the existence result. This proof, which mimicks the analogous one for the non-characteristic problem in [19], will be the content of a subsequent paper. [ABSTRACT FROM AUTHOR]

A system that consists of a damped oscillator impacting two immovable fenders has been considered in this paper. In the first part a method of analytical determination of the existence of periodic solutions to the equations of motion and a method of analysis of the stability of these solutions have been presented. The results of the computations carried out by means of these methods have been illustrated by a few examples. In the second part of the paper, the results of some numerical investigations have been presented. The goal of these studies was to determine, in which regions of parameters characterizing the system, the motion of the oscillator is periodic and stable. [ABSTRACT FROM AUTHOR]

An emulated digital CNN-UM (CASTLE) architecture was published few years ago.1 Different emulated digital CNN-UM architectures are analyzed in the paper. These new modified architectures are optimized according to the silicon area, operating speed or dissipated power. A reconfigurable arithmetic core will also be shown in the paper, by which solution of the neighborhood size can be changed. An advanced CASTLE with pipe-lining is presented. The operation frequency is increased by using this solution in approximately 10 times. [ABSTRACT FROM AUTHOR]

By using the method of dynamical systems, this paper studies the exact traveling wave solutions and their bifurcations in the Gardner equation. Exact parametric representations of all wave solutions as well as the explicit analytic solutions are given. Moreover, several series of exact traveling wave solutions of the Gardner-KP equation are obtained via an auxiliary function method. [ABSTRACT FROM AUTHOR]

The rearward propagation of tonal noise from the main fan and the engine core of modern high bypass aeroengines is one of the current demanding applications of CAA methods. One of the main features of this problem is the radiation of tones from main fan and turbine through the shear layers of core and bypass jets. This can approximately be described by a solution of the linearized perturbed Euler equations over a sheared turbulent averaged base flow field. However, these equations not only describe sound propagation, but also provide a stability analysis for the sheared base flow. Three techniques with the potential to calculate an acoustic solution and at the same time to suppress the instability are compared in this paper. The radiation of a source from a two-dimensional hot jet, chosen from a CAA workshop on benchmark problems, is considered first. Then, the techniques are adopted for the simulation of a single azimuthal mode radiating from the bypass duct of a turbofan engine, as an example for the realistic application. The first technique is based on filtering the mean flow field, over which the perturbation equations are solved. A low-order filter is applied. Subsequently, an adaption of this method, which considers a filtering of the mean flow derivatives in addition, is proved to be very beneficial. The result then reflects the analytical solution of the benchmark problem very well. The second technique filters the source terms in the governing equations. In a first attempt, all mean flow derivatives are neglected to suppress the instability. A more physical motivated variant of the approach neglects only source terms in the momentum equations. However, both provide unsatisfactory predictions of the acoustic field for the benchmark. Finally, a third technique is implemented, which considers the modification of the velocity derivatives in the momentum equations, as this method has demonstrated one of the best predictions for the benchmark problem. Nevertheless, the latter technique has no 3D extension and thus fails in suppressing the instability waves in the turbofan application. [ABSTRACT FROM AUTHOR]

In this paper, with the study of combinatorial Dyson–Schwinger equations at the level of the universal Hopf algebra of renormalization and with the extension of the universality of this specific Hopf algebra and also category of flat equisingular vector bundles to the level of these equations, we are going to consider the notion of a geometric description from nonperturbative theory. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of a smooth solution to a tamed 3D Navier–Stokes equation in the whole space. In particular, if there exists a bounded smooth solution to the classical 3D Navier–Stokes equation, then this solution satisfies our tamed equation. Moreover, using this tamed equation we can give a new construction for a suitable weak solution of the classical 3D Navier–Stokes equation introduced in Refs. 16 and 2. [ABSTRACT FROM AUTHOR]

In this paper, a new technique is presented for structural acoustic analysis in the case of nonconforming acoustic–solid interface meshes. We first describe a simple method for coupling nonconforming acoustic–acoustic meshes, and then show that a similar approach, together with the coupling operators from conforming analysis, can also be applied to nonconforming structural acoustics. In the case of acoustic–acoustic interfaces, the continuity of acoustic pressure is enforced with a set of linear constraint equations. For structural acoustic interfaces, the same set of linear constraints is used, in conjunction with the weak formulation and the coupling operators that are commonly used in conforming structural acoustics. The constraint equations are subsequently eliminated using a static condensation procedure. We show that our method is equally applicable to time domain, frequency domain, and coupled eigenvalue analysis for structural acoustics. Numerical examples in both the time and frequency domains are presented to verify the methods. [ABSTRACT FROM AUTHOR]

Differential modes provide examples of modes that do not embed as subreducts into semimodules over commutative semirings. The current paper studies differential modes, so-called Szendrei differential modes, which actually do embed into semimodules. These algebras form a variety. The main result states that the lattice of nontrivial subvarieties is dually isomorphic to the (nonmodular) lattice of congruences of the free commutative monoid on two generators. Consequently, all varieties of Szendrei differential modes are finitely based. [ABSTRACT FROM AUTHOR]

In this paper we consider the Kaluza–Klein field equations in the presence of a generic 5D matter tensor which is governed by a conservation equation due to 5D Bianchi identities. Following a previous work, we provide a consistent approach to matter where the problem of huge massive modes is removed, without relaxing the compactification hypotheses; therefore we perform the dimensional reduction either for metric fields and for matter, thus identifying a pure 4D tensor term, a 4D vector term and a scalar one. Hence we are able to write down a consistent set of equations for the complete dynamics of matter and fields; with respect to the pure Einstein–Maxwell system we now have two additional scalar fields: the usual dilaton one plus a scalar source term. Some significant scenarios involving these terms are discussed and perspectives for cosmological applications are suggested. [ABSTRACT FROM AUTHOR]

In this paper, based on a talk given at the Oberwolfach research centre in May 2008 I will describe how biquandles and their big brother, biracks, can be used to differentiate isotopy classes of virtual (and welded) knots and links. [ABSTRACT FROM AUTHOR]

We consider the problem of minimizing the total late work (Σj=1n Vj) on an unbounded batch processing machine, where Vj = min{Tj, pj} and Tj = max{Cj - dj, O}. The batch processing machine can process up to B (B ≥ n) jobs simultaneously. The jobs that are processed together form a batch, and all jobs in a batch start and complete at the same time, respectively. For a batch of jobs, the processing time of the batch is equal to the largest processing time among the jobs in this batch. In this paper, we prove that the problem 1∣B ≥ n∣ Σj=1nVj is NP-hard. [ABSTRACT FROM AUTHOR]

In this paper we present the analytical solutions of the one-dimensional Dirac equation for the Scarf-type potential with equal scalar and vector potentials. Using Nikiforov–Uvarov mathematical method, spinor wave function and the corresponding exact energy equation are obtained for the s-wave bound state. It has been shown that the results for this potential reduce to the well-known potentials in the special cases. [ABSTRACT FROM AUTHOR]

In this paper we investigate, via numerical simulations, the parameter space of the set of autonomous differential equations of a Chua oscillator, where the piecewise-linear function usually taken to describe the nonlinearity of the Chua diode was replaced by a cubic polynomial. As far as we know, we are the first to report that this parameter-space presents islands of periodicity embedded in a sea of chaos, scenario typically observed only in discrete-time models until recently. We show that these islands are self-similar, and organize themselves in period-adding bifurcation cascades. [ABSTRACT FROM AUTHOR]

Chua's circuit is very suitable as a programmable chaos generator because of its robust nonlinearity. In addition to exhibiting a rich variety of bifurcation and chaos phenomenon, this circuit can be modeled and realized with a fixed main system block and many different nonlinear function blocks such as piecewise-linear function, cubic-like function, piecewise-quadratic function and other trigonometric functions. This paper presents a FPAA (Field Programmable Analog Array) based programmable implementation of Chua's circuit. Nonlinear function blocks used in Chua's circuit are modeled with an FPAA and hence a model can be rapidly changed for realization of other nonlinear functions. In this study, four FPAA-based reconfigurable implementations of Chua's circuit have been realized. Experimental results agree with numerical simulation and results obtained from discrete electronic implementations of Chua's circuit. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of non-trivial weak solutions in $W_0^{1,p} (\Omega)$, 1 < p < ∞, to a class of non-uniformly elliptic equations of the form \[ - {\rm div}({a({x,\nabla u})}) = \lambda f(u) + \mu g(u) \] in a bounded domain Ω of ℝN. Here a satisfies \[ |{a({x,\xi})}| \leqq c_0 ({h_0 (x) + h_1 (x)| \xi |^{p - 1}}) \] for all ξ ∈ ℝN, a.e. x ∈ Ω, $h_0 \in L^{\frac{p}{{p - 1}}} (\Omega)$, $h_1 \in L_{\rm loc}^1 (\Omega)$, h0(x) ≧ 0, h1(x) ≧ 1 for a.e. x in Ω. [ABSTRACT FROM AUTHOR]

Equilibrium manifold is a manifold that consists of equilibrium points. Planar systems with one-dimensional equilibrium manifolds are considered in this paper. First, for such planar systems, a unified equation with the horizontal axis as the equilibrium curve is formulated. Then, according to the corresponding linearized systems, different cases are discussed: For the nondegenerate case, the simplest normal form of a system with simplified Bogdanov–Takens singularities is obtained; for the general first-order degenerative case, the simplest normal forms are completely characterized; finally, for the general higher-order degenerative case, deduction of the simplest normal form is illustrated. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give new dispersive tools for certain kinetic equations. As an application, we study the three-dimensional Vlasov–Poisson equation for initial data having strictly less than six moments in $L_{x,\xi}^{1}$ where the nonlinear term E is a priori unbounded. We prove via new dispersive effects that in fact the force field E is smooth in space at the cost of a localization in a ball and an averaging in time. We deduce new conditions to bound the density ρ in L∞ and to have existence and uniqueness of global weak solutions of the Vlasov–Poisson equation with bounded density for initial data strictly less than six moments in $L^{1}_{x,\xi}$. The proof is based on a new approach which consists in establishing a priori dispersion estimates (moment effects) on the one hand for linear transport equations with unbounded force fields and on the other hand along the trajectories of the Vlasov–Poisson equation. [ABSTRACT FROM AUTHOR]

Nonlinear congruential pseudorandom number generators can have unexpectedly short periods. Shamir and Tsaban introduced the class of counter-dependent generators which admit much longer periods. In this paper, using a technique developed by Niederreiter and Shparlinski, we present discrepancy bounds for sequences of s-tuples of successive pseudorandom numbers generated by counter-dependent generators modulo a composite M. [ABSTRACT FROM AUTHOR]

In this paper, we study the initial value problem for a class of nonlinear equations of Burgers type in the following form: \[ \frac{\partial}{\partial t}u + \nu q(x,D)u + (b \cdot \nabla)f(u) = 0 \] for u:(t,x) ∈ (0,∞) × ℝn ↦ ℝ, where q(x,D) is a pseudo-differential operator with negative definite symbol. We solve the initial value problem for the equation on ℝn by utilising a fix point argument based upon a combination of semigroup approach and Hoh's symbolic calculus. [ABSTRACT FROM AUTHOR]

This paper proves a conjecture proposed by Chen and Ru in [1] on the finiteness of the number of integer solutions to decomposable form inequalities. Let k be a number field and let F(X1,...,Xm) be a non-degenerate decomposable form with coefficients in k. We show that for every finite set of places S of k containing the archimedean places of k, for each real number λ < 1 and each constant c > 0, the inequality $0 < \Pi_{\upsilon \in S}\|F(x_1, \ldots, x_m)\|_{\upsilon}\leq cH_S^{\lambda}(x_1,\ldots,x_m)$ has only finitely many $\mathcal{O}_{S}^{*}$-non-proportional solutions, where HS(x1,...,xm) = Πυ∈Smax1≤i≤m ||xi||υ is the S-height. [ABSTRACT FROM AUTHOR]

Consider n ducts having a common origin and filled with a fluid. Along each duct, the full Euler system describes the evolution of the fluid. At the junction, suitable physical conditions couple the n Euler systems. In this paper we prove the well posedness of the Cauchy problem for the model so obtained, provided the total varaiatio of the initial data is sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper, we numerically study the semiclassical limit of the Schrödinger–Poisson equations as a selection principle for the weak solution of the Vlasov–Poisson in one space dimension. Our numerical results show that this limit gives the weak solution that agrees with the zero diffusion limit of the Fokker–Planck equation. We also numerically justify the multivalued solution given by a moment system of the Vlasov–Poisson equations as the semiclassical limit of the Schrödinger–Poisson equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Hamilton–Jacobi (HJ) equations of multidimensional space variables with convex Hamiltonians. Using Hopf's formula (I), we will study the differentiability of the HJ solutions. For any given point, we give a sufficient and necessary condition under which the solutions are Ck smooth in some neighborhood of the point. We also study the characteristics of the HJ equations. It is shown that there are only two kinds of characteristics, one never touches the singularity point, and the other touches the singularity point in a finite time. The sufficient and necessary condition under which the characteristic never touches the singularity point is given. Based on these results, we study the global structure of the set of singularity points for the HJ solutions. It is shown that there exists a one-to-one correspondence between the path connected components of the set of singularity points and the path connected components of a set on which the initial function does not attain its minimum. A path connected component of the set of singularity points never terminates at a finite time. Our results are independent of the particular forms of the equations as long as the Hamiltonians are convex. [ABSTRACT FROM AUTHOR]

We discuss the existence of weak solutions with moving phase boundaries in thermoelasticity related to dynamic phase transitions. One of the goals is to study the dynamical consequence of the stable and metastable states defined in this paper. We use the entropy condition and the kinetic relation as the main admissibility criteria to study the above goals for the Euler equations with nonmonotone constitutive relation. We discuss the case where there are two noninteracting phase boundaries moving in the opposite directions. A modification to treat the case where the two phase boundaries collide is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of nonlocal stochastic Kuramoto–Sivashinsky equations driven by compensated Poisson random measures and show the existence and uniqueness of the weak solution to the equation. Furthermore, we prove that an invariant measure of the equation indeed exists under some appropriate assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we construct special radicals using class pairs of near-rings. We establish necessary conditions for a class pair to be a special radical class. We then define Jacobson-type near-rings and show that in most cases the class of all near-rings of this type is a special radical class. Subsequently, we investigate the relationship between each Jacobson-type near-ring and the corresponding matrix near-ring. [ABSTRACT FROM AUTHOR]

In this paper, we classify the fine representations and R-groups of quasi-split simple groups, and describe the set of good roots with the help of Yan's method. Part of the results are not new, but the method used here is new and efficient. [ABSTRACT FROM AUTHOR]

In this paper, δ-lifting and δ-supplemented modules are defined as generalizations of lifting and supplemented modules. Several properties of these modules are proved. New characterizations of δ-semiperect and δ-perfect rings studied in [9] are obtained using δ-lifting and δ-supplemented modules. [ABSTRACT FROM AUTHOR]

The notion of xst-rings was introduced by García and Marín in 1999. In this paper, we characterize Morita-like equivalences for right xst-rings, obtain the universal theory of Morita equivalences, and prove that two right xst-rings R and T are Morita-like equivalent if and only if there is a Morita context between R and T. We also prove that Morita-like equivalences can be realized by the covariant functors Hom and ⊗ for these rings. [ABSTRACT FROM AUTHOR]

The Held group He discovered by Held [10] is a sporadic simple group of order 4030387200 = 210.33.52.73.17. The group He has 11 conjugacy classes of maximal subgroups as determined by Butler [5] and listed in the 픸핋핃픸핊. Held himself determined much of the local structure of He as well as the conjugacy classes of its elements. Thompson calculated the character table of He. In the present paper, we determine the Fischer–Clifford matrices and hence compute the character table of the non-split extension 3·S7, which is a maximal subgroups of He of index 226560 using the technique of Fischer–Clifford matrices. Most of the computations were carried out with the aid of the computer algebra system 픾픸ℙ. [ABSTRACT FROM AUTHOR]

The geometrically exact quasilinear evolution equations governing the spatial motion of incompressible rods and their specializations to the hyperbolic equations governing nonlinearly elastic rods have novel mathematical structures strikingly different from those for the equations governing the motion of compressible rods. The main objectives of this paper are to formulate the governing equations, an exercise requiring the solution of a sequence of semilinear hyperbolic equations of first order, and to derive a priori estimates for certain strain variables, ensuring that they cannot reach geometrically prohibited ranges in finite time. This process exhibits a new system of physically important quasilinear equations worthy of careful analysis. [ABSTRACT FROM AUTHOR]

We study the initial-boundary value problem for a general class of nonlinear pseudo-differential equations on a half-line \[ \hspace*{5pc} \left\{\begin{array}{ll}u_{t}+\mathcal{N}(u,u_{x})+\mathcal{L}u=f,&\quad (x,t)\in{\mathbf{R}^{+}}\times{\mathbf{R}^{+}},\\[5pt] u(x,0)=u_{0}(x),&\quad x\in{\mathbf{R}}^{+},\\[5pt] \partial_{x}^{j-1}u(0,t)=h_{j}(t)&\quad\mbox{for}\ j=1,\ldots,M, \end{array} \right.\hspace*{5.3pc}(0.1) \label{W} \] where the number M depends on the order of the pseudo-differential operator $\mathcal{L}$ on a half-line. The nonlinear term $\mathcal{N}(u,u_{x})$ is such that $|\mathcal{N}(u,v)| \leq C|u|^{\rho}|v|^{\sigma}$ as u, v → 0, with ρ, σ > 0. Pseudo-differential operator $\mathcal{L}$ is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) and to find the main term of the asymptotic representation of solutions taking into account the influence of inhomogeneous boundary data and a source on the asymptotic properties of solutions. [ABSTRACT FROM AUTHOR]

Let 픛r(M), r ≥ 1, denote the space of all Cr vector fields over a compact, smooth and boundaryless Riemannian manifold M of finite dimension; let $\mathcal F_{\ell}^{\sharp}, 1 ≤ ℓ ≤ dim M, be the bundle of orthonormal ℓ-frames of the tangent space TM of M. For any V ∈ 픛r(M), Liao defined functions $\omega_k(\vec{\gamma}_x, V), k = 1, ..., ℓ, on $\mathcal F_{\ell}^{\sharp}$, which are qualitatively equivalent to the Lyapunov exponents of the differential system V. In this paper, the author shows that every $\omega_k(\vec{\gamma}_x, V)$ depends Cr-1-continuously upon $(\vec{\gamma}_x, V)\in\mathcal F_{\ell}^{\sharp}\times\mathfrak X^{r}(M)$ and Cr-continuously on $\vec{\gamma}_x$ for any given V. In addition, applying the qualitative functions, the author generalizes Liao's global linearization along a given orbit of V and considers the stochastic stability of Lyapunov spectra of linear skew-product flows based on a given ergodic system. [ABSTRACT FROM AUTHOR]

This paper studies the problem -Δu + λu = up in nonsmooth domains with mixed boundary conditions. Special attention will be given here to the critical case and to domains which have no further regularity than a Lipschitzian boundary. For such domains, we obtain a generalized version of Cherrier's inequality and prove an existence result. This was achieved by using an extended definition of the manifold. [ABSTRACT FROM AUTHOR]

This paper is a sequel to Part I (Trabelsi [11]) in which a new two-dimensional membrane model was derived via a formal asymptotic procedure for a family of hyperelastic nonlinear materials proposed by Ciarlet and Geymonat whose stored energy function is polyconvex and becomes infinite when the determinant of the deformation gradient tends to zero, and can be adjusted to arbitrary Lamé constants. Here, we continue the asymptotic analysis by making legitimate assumptions on the data to produce an inextensional two-dimensional model. [ABSTRACT FROM AUTHOR]

We prove the existence of infinitely many geometrically distinct two bump solutions of periodic superlinear Schrödinger equations of the type -Δu + V(x)u = f(x,u), where x ∈ ℝN and lim|x| → ∞u(x) = 0. The solutions we construct change sign and have exactly two nodal domains. The usual multibump constructions for these equations rely on strong non-degeneracy assumptions. We present a new approach that only requires a weak splitting condition. In the second part of the paper we exhibit classes of potentials V for which this splitting condition holds. [ABSTRACT FROM AUTHOR]