Time-discretization scheme for quasi-static Maxwell's equations with a non-linear boundary condition

We study a time dependent eddy current equation for the magnetic field H$\mathit{H}$ accompanied with a non-linear degenerate boundary condition (BC), which is a generalization of the classical Silver–Müller condition for a non-perfect conductor. More exactly, the relation between the normal components of electrical E$\mathit{E}$ and magnetic H$\mathit{H}$ fields obeys the following power law ν×E=ν×(|H×ν|^α-1H×ν)$\mathit{\nu }\times \mathit{E}=\mathit{\nu }\times (|\mathit{H}\times \mathit{\nu }{|}^{\alpha -1}\mathit{H}\times \mathit{\nu })$ for some α∈(0,1]$\alpha \in (0,1]$. We establish the existence and uniqueness of a weak solution in a suitable function space under the minimal regularity assumptions on the boundary Γ$\Gamma$ and the initial data _H0${\mathit{H}}_0$. We design a non-linear time discrete approximation scheme based on Rothe's method and prove convergence of the approximations to a weak solution. We also derive the error estimates for the time-discretization.     

Nonlinear diffusion in type-II superconductors

In this paper we study a nonlinear evolution equation _∂t(σ(|E|)E)+∇×∇×E=F${\mathrm{\partial }}_t(\sigma (\left|\mathit{E}\right|)\mathit{E})+\nabla \times \nabla \times \mathit{E}=\mathit{F}$ in a bounded domain subject to appropriate initial and boundary conditions. This governs the evolution of the electric field E$\mathit{E}$ in a conductive medium under the influence of a force F$\mathit{F}$. It is an approximation of Bean's critical-state model for type-II superconductors. We design a nonlinear numerical scheme for the time discretization. We prove the convergence of the proposed method. The proof is based on a generalization of div–curl lemma for transient problems. We also derive some error estimates for the approximate solution.     

Short propositional refutations for dense random 3CNF formulas

Random 3CNF formulas constitute an important distribution for measuring the average-case behavior of propositional proof systems. Lower bounds for random 3CNF refutations in many propositional proof systems are known. Most notable are the exponential-size resolution refutation lower bounds for random 3CNF formulas with Ω(n^1.5−ε)$\mathrm{\Omega }(n^{1.5-\epsilon })$ clauses (Chvátal and Szemerédi [14], Ben-Sasson and Wigderson [10]). On the other hand, the only known non-trivial upper bound on the size of random 3CNF refutations in a non-abstract propositional proof system is for resolution with Ω(n²/log?n)$\mathrm{\Omega }(n^2/\log ?n)$ clauses, shown by Beame et al. [6]. In this paper we show that already standard propositional proof systems, within the hierarchy of Frege proofs, admit short refutations for random 3CNF formulas, for sufficiently large clause-to-variable ratio. Specifically, we demonstrate polynomial-size propositional refutations whose lines are TC⁰${\mathit{T}\mathit{C}}^0$ formulas (i.e., TC⁰${\mathit{T}\mathit{C}}^0$-Frege proofs) for random 3CNF formulas with n   variables and Ω(n^1.4)$\mathrm{\Omega }(n^{1.4})$ clauses.     

Maximum principles and isoperimetric inequalities for some Monge–Ampère-type problems

In this note we derive a maximum principle for an appropriate functional combination of u(x)$u(\mathbf{x})$ and |∇u|²${|\mathrm{\nabla }u|}^2$, where u(x)$u(\mathbf{x})$ is a strictly convex classical solution to a general class of Monge–Ampère equations. This maximum principle is then employed to establish some isoperimetric inequalities of interest in the theory of surfaces of constant Gauss curvature in R^N+1${\mathbb{R}}^{N+1}$.     

Nonuniform continuity of the solution map to the two component Camassa–Holm system

We prove that the solution map of the two-component Camassa–Holm system is not uniformly continuous as a map from a bounded subset of the Sobolev space H^s(T)×H^r(T)$H^s(\mathbb{T})\times H^r(\mathbb{T})$ to C([0,1],H^s(T)×H^r(T))$C([0,1],H^s(\mathbb{T})\times H^r(\mathbb{T}))$ when s?1$s?1$ and r?0$r?0$. We also demonstrate the nonuniform continuous property in the continuous function space C¹(T)×C¹(T)$C^1(\mathbb{T})\times C^1(\mathbb{T})$.     

Liouville type results and a maximum principle for non-linear differential operators on the Heisenberg group

We prove Liouville type results for non-negative solutions of the differential inequality _Δφu?f(u)?(|_∇0u|)${\mathrm{\Delta }}_{\phi }u?f(u)?(|{\mathrm{\nabla }}_{0}u|)$ on the Heisenberg group under a generalized Keller–Osserman condition. The operator _Δφu${\mathrm{\Delta }}_{\phi }u$ is the φ  -Laplacian defined by _div0(|_∇0u|⁻¹φ(|_∇0u|)_∇0u)${\mathrm{div}}_0({|{\mathrm{\nabla }}_{0}u|}^{-1}\phi (|{\mathrm{\nabla }}_{0}u|){\mathrm{\nabla }}_{0}u)$ and φ, f and ? satisfy mild structural conditions. In particular, ? is allowed to vanish at the origin. A key tool that can be of independent interest is a strong maximum principle for solutions of such differential inequality.     

Semilinear fractional elliptic equations involving measures

We study the existence of weak solutions to (E) (−Δ)^αu+g(u)=ν${(-\mathrm{\Delta })}^{\alpha }u+g(u)=\nu$ in a bounded regular domain Ω   in R^N(N≥2)${\mathbb{R}}^N(N\ge 2)$ which vanish in R^N?Ω${\mathbb{R}}^N?\Omega$, where (−Δ)^α${(-\mathrm{\Delta })}^{\alpha }$ denotes the fractional Laplacian with α∈(0,1)$\alpha \in (0,1)$, ν is a Radon measure and g is a nondecreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of weak solution for problem (E) for any measure. In the case where ν   is a Dirac measure, we characterize the asymptotic behavior of the solution. When g(r)=|r|^k−1r$g(r)={|r|}^{k-1}r$ with k supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved.     

Generic global rigidity of body–bar frameworks

A basic geometric question is to determine when a given framework G(p)$G(\mathbf{p})$ is globally rigid in Euclidean space R^d${\mathbb{R}}^d$, where G is a finite graph and p is a configuration of points corresponding to the vertices of G  . G(p)$G(\mathbf{p})$ is globally rigid in  R^d${\mathbb{R}}^d$ if for any other configuration q for G   such that the edge lengths of G(q)$G(\mathbf{q})$ are the same as the corresponding edge lengths of G(p)$G(\mathbf{p})$, then p is congruent to q. A framework G(p)$G(\mathbf{p})$ is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G.     

Classification of invariant valuations on the quaternionic plane

We describe the orbit space of the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$ on the real Grassmann manifolds _Grk(H²)${\mathrm{Gr}}_k({\mathbb{H}}^2)$ in terms of certain quaternionic matrices of Moore rank not larger than 2. We then give a complete classification of valuations on the quaternionic plane H²${\mathbb{H}}^2$ which are invariant under the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$.     

Maximal linear groups induced on the Frattini quotient of a p-group

Let $p>3$ be a prime. For each maximal subgroup $H?\mathrm{GL}(d,p)$ with $|H|?p^{3d+1}$, we construct a d-generator finite p-group G with the property that $\mathrm{Aut}(G)$ induces H on the Frattini quotient $G/\mathrm{\Phi }(G)$ and $|G|?p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on $G/\mathrm{\Phi }(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.     

Approximation of maximum of Gaussian random fields

This contribution is concerned with Gumbel limiting results for supremum ${\mathbb{M}}_{\mathit{n}}={\sup }_{\mathit{t}\in [\mathbf{0},{\mathit{T}}_{\mathit{n}}]}?|X_{\mathit{n}}(\mathit{t})|$ with $X_{\mathit{n}},\mathit{n}\in {\mathbb{N}}^2$ centered Gaussian random fields with continuous trajectories. We show first the convergence of a related point process to a Poisson point process thereby extending previous results obtained in [8] for Gaussian processes. Furthermore, we derive Gumbel limit results for ${\mathbb{M}}_{\mathit{n}}$ as $\mathit{n}\to \infty$ and show a second-order approximation for $\mathbb{E}{\{{\mathbb{M}}_{\mathit{n}}^p\}}^{1/p}$ for any $p\ge 1$.     

A computational matrix method for solving systems of high order fractional differential equations

In this paper, we introduced an accurate computational matrix method for solving systems of high order fractional differential equations. The proposed method is based on the derived relation between the Chebyshev coefficient matrix A of the truncated Chebyshev solution u(t)$\mathbf{u}(\mathbf{t})$ and the Chebyshev coefficient matrix A^(ν)${\mathbf{A}}^{(\nu )}$ of the fractional derivative u^(ν)${\mathbf{u}}^{(\nu )}$. The fractional derivatives are presented in terms of Caputo sense. The matrix method for the approximate solution for the systems of high order fractional differential equations (FDEs) in terms of Chebyshev collocation points is presented. The systems of FDEs and their conditions (initial or boundary) are transformed to matrix equations, which corresponds to system of algebraic equations with unknown Chebyshev coefficients. The remaining set of algebraic equations is solved numerically to yield the Chebyshev coefficients. Several numerical examples for real problems are provided to confirm the accuracy and effectiveness of the present method.     

Regularity properties of the solutions to the 3D Maxwell–Landau–Lifshitz system in weighted Sobolev spaces

This paper is concerned with special regularity properties of the solutions to the Maxwell–Landau–Lifshitz (MLL) system describing ferromagnetic medium. Besides the classical results on the boundedness of _∂tm,_∂tE${\mathrm{\partial }}_t\mathbf{m},{\mathrm{\partial }}_t\mathbf{E}$ and _∂tH${\mathrm{\partial }}_t\mathbf{H}$ in the spaces L^∞(I,L²(Ω))$L^{\infty }(I,L^2(\Omega ))$ and L²(I,W^1,2(Ω))$L^2(I,W^{1,2}(\Omega ))$ we derive also estimates in weighted Sobolev spaces. This kind of estimates can be used to control the Taylor remainder when estimating the error of a numerical scheme.