The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, .... A natural toy model for further

The purpose of this paper is to provide results like Peixoto’s Theorem inside a class $\mathfrak{X}$ of 3-dimensional reversible systems. We deal with reversible systems associated to an involution $\varphi_1$ such that $\operatorname{Dim}\operatorname{Fix}(\varphi_1)$ is equal to one. We further assume the existence of a first integral for any $X\in\mathfrak{X}$ that leaves invariant any sphere centered

We consider the heat equation with dynamic boundary conditions involving gradient terms in a bounded domain. In this paper we study the cost of approximate controllability for this equation. Combining new developed Carleman estimates and some optimization techniques, we obtain explicit bounds of the minimal norm control. We consider the linear and the semilinear cases.

We study upper bounds and the asymptotic behavior of high order moments for solutions to a family of linear and nonlinear Schrödinger equations.

Let $X$ be a topological Hausdorff space and $\mathcal B o$ be the $\sigma$-algebra of Borel sets in $X$. Let $B(\mathcal B o)$ be the space of all bounded $\mathcal B o$-measurable scalar functions on $X$, equipped with the Mackey topology $\tau(B(\mathcal B o),M(X))$, where $M(X)$ denotes the Banach space of all scalar Radon measures on $X$. It is proved that $(B(\mathcal B o),\tau(B(\mathcal B o)

In this paper, we show that if a vector bundle of rank $n-1$ on projective $(n-1)$-space is very ample and if its top Chern class is greater than one, then the discriminant locus of the vector bundle, the locus of singular sections of the vector bundle, is an irreducible hypersurface and that the degree of the hypersurface can be expressed as a function of invariants of the vector bundle. As applications

This is a short paper describing how a herbrandized functional interpretation can give a new treatment of some issues in classical second-order arithmetic. It is perhaps worthy of note that, in our interpretation, second-order variables are interpreted by finite sets of natural numbers.

In this paper, we introduce new results on evolution algebras obtained by following a relatively new line of research on these objects. It consists of the use of certain properties of graphs to facilitate the study of evolution algebras and reciprocally.

In this article we use techniques of proof mining to analyse a result, due to Yonghong Yao and Muhammad Aslam Noor, concerning the strong convergence of a generalized proximal point algorithm which involves multiple parameters. Yao and Noor's result ensures the strong convergence of the algorithm to the nearest projection point onto the set of zeros of the operator. Our quantitative analysis, guided

In this paper we study the effect of a homoclinic tangency in the variation of the topological entropy. We prove that a diffeomorphism with a homoclinic tangency associated to a basic hyperbolic set with maximal entropy is a point of entropy variation in the $C^{\infty}$-topology. We also prove results about variation of entropy in other topologies and when the tangency does not correspond to a basic

We consider the Einstein equations in T2 symmetry, either for vacuum spacetimes or coupled to the Euler equations for a compressible fluid, and we introduce the notion of T2 areal flows on T3 with finite total energy. By uncovering a hidden structure of the Einstein equations, we establish a compensated compactness framework and solve the global evolution problem for vacuum spacetimes as well as for

In network dynamics, synchrony between nodes defines an equivalence relation, usually represented as a colouring. If the colouring is balanced, meaning that nodes of the same colour have colour-isomorphic inputs, it determines a subspace that is flow-invariant for any ODE compatible with the network structure. Therefore any state lying in such a subspace has the synchrony pattern determined by that

In the setting of 2-uniformly smooth and $q$-uniformly convex Banach spaces, we prove the existence of solutions of the following multivalued differential equation: $$-\frac{d}{dt} J(u(t)) \in N^C(C(t,u(t));u(t)) \mbox{ a.e. in } [0,T]. \:\:\: \mathrm{(SDNSP)}$$ This inclusion is called State Dependent Nonconvex Sweeping Process (SDNSP). Here $N^C(C(t,u(t)); u(t))$ stands for the Clarke normal cone

We study the location of the spectrum of the Laplacian on compact metric graphs with complex Robin-type vertex conditions, also known as $\delta$ conditions, on some or all of the graph vertices. We classify the eigenvalue asymptotics as the complex Robin parameter(s) diverge to $\infty$ in $\mathbb{C}$: for each vertex $v$ with a Robin parameter $\alpha \in \mathbb{C}$ for which $\mathrm{\Re}\,\alpha

We give a new and constructive proof of the existence of a special class of univariate polynomials whose graphs have preassigned shapes. By definition, all the critical points of a Morse polynomial function are real and distinct and all its critical values are distinct. Thus we can associate to it an alternating permutation: the so-called Arnold snake, given by the relative positions of its critical

We study the vanishing viscosity limit of a nonlinear diffusion equation describing chemical reaction interface or the spatial segregation interface of competing species, where the diffusion rate for the negative part of the solution converges to zero. As in the standard one phase Stefan problem, we prove that the positive part of the solution converges uniformly to the solution of a generalized one

Given measurable functions $u, \sigma$ on an interval $(0,b)$ and a kernel function $k(x,y)$ on $(0,b)^2$ satisfying Oinarov condition, the supremum-involving Hardy-type operators $$Rf(x)=\sup_{x\leq\tau < b}u(\tau)\int_0^\tau k(\tau,y)\sigma (y)f(y)dy, x > 0$$ in Orlicz-Lorentz spaces are investigated. We obtain sufficient conditions of boundedness of $R: \Lambda_{u_0}^{G_0}(w_0)\\ \rightarrow \L

In this paper, we put emphasis on discussing a full discrete finite element scheme for the Korteweg–de Vries equation, where nonlinear term is dealt with a semi-implicit scheme and temporal term is discreted by the Euler scheme. Theoretical analysis is based on error splitting technique, i.e., error function is split as temporal error function plus spatial error function, and then unconditionally optimal

We study the initial value problem for a kind of Euler equation with a source term. Our main result is the existence of a globally-in-time weak solution whose total variation is bounded on the domain of definition, allowing the existence of shock waves. Our proof relies on a well-balanced random choice method called Glimm method which preserves the fluid equilibria and we construct a sequence of approximate

In a previous paper, the authors introduced and studied the Bruhat order in the class of Latin squares of order n. In this paper, we investigate the restriction of the Bruhat order in a class of isotopic Latin squares. We present equivalent conditions for two Latin squares be related by the Bruhat order when one of them is obtained from the other by interchanging rows or columns or symbols. The cover

Let $T$ be an algebraic torus over an algebraically closed field, let $X$ be a smooth closed subvariety of a $T$-toric variety such that $U = X \cap T$ is not empty, and let $\mathscr{L}(X)$ be the arc scheme of $X$. We define a tropicalization map on $\mathscr{L}(X) \setminus \mathscr{L}(X \setminus U)$, the set of arcs of $X$ that do not factor through $X \setminus U$. We show that each fiber of

The purpose of this paper is to investigate the existence and approximation of fixed points of generalized Bregman nonspreading mapping which are also the zero points of maximal monotone operator in a real reflexive Banach space. Without assuming the ‘AKTT’ condition, we prove a strong convergence theorem for approximating a common element in the set of fixed points of system of generalized Bregman

This paper addresses the problem of solving block tridiagonal quasi-Toeplitz linear systems. Inspired by [10], we propose a more general algorithm for such systems. The algorithm is based on a block decomposition for block tridiagonal quasi-Toeplitz matrices and the Sherman–Morrison–Woodbury inversion formula. We also compare the proposed approach to the standard block $LU$ decomposition method and

This paper establishes a generalization of the celebrated Stein–Weiss inequalities for the fractional integral operators on radial functions in local Morrey spaces. We find that some conditions can be relaxed for the Stein–Weiss inequalities on radial local Morrey spaces.

In this paper we study the existence of solution to the problem \begin{equation*} \left\{\begin{array}{l} u\in H_{0}^{1}(\Omega), \\[4pt] -\textrm{div}\,(A(x)Du)=H(x,u,Du)+f(x)+a_{0}(x)\, u\quad \text{in} \quad\mathcal{D}'(\Omega), \end{array} \right. \end{equation*} where $\Omega$ is an open bounded set of $\mathbb{R}^{2}$, $A(x)$ a coercive matrix with coefficients in $L^\infty(\Omega)$, $H(x,s,\xi)$

The aim of this paper is to formulate a local systolic inequality for odd-symplectic forms (also known as Hamiltonian structures) and to establish it in some basic cases. Let $\Omega$ be an odd-symplectic form on an oriented closed manifold $\Sigma$ of odd dimension. We say that $\Omega$ is Zoll if the trajectories of the flow given by $\Omega$ are the orbits of a free $S^1$-action. After defining

We revisit the Bohnenblust--Hille multilinear and polynomial inequalities and prove some new properties. Our main result is a multilinear version of a recent result on polynomials whose monomials have a uniformly bounded number of variables.

In this paper we study the existence and regularity of solutions to the following Dirichlet problem        −div(a(x)|∇u| p−2 ∇u) + u|u| r−1 = f (x) u θ in Ω, u > 0 in Ω, u = 0 on ∂Ω proving that the lower order term u|u| r−1 has some regularizing effects on the solutions. Mathematics Subject Classification (2010). 35J66, 35J75

We identify several classes of curves $C:f=0$, for which the Hilbert vector of the Jacobian module $N(f)$ can be completely determined, namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having only rational irreducible components. A result due to Hartshorne, on the cohomology of some rank 2 vector bundles on $\mathbb{P}^2$, is used to get a sharp lower bound for the initial

We use moving frames to construct and classify the joint invariants and joint differential invariants of binary and ternary forms. In particular, we prove that the differential invariant algebra of ternary forms is generated by a single third order differential invariant. To connect our results with earlier analysis of Kogan, we develop a general method for relating differential invariants associated

In this paper we give a streamlined proof of an inequality recently obtained by the author: For every $\alpha \in (0,1)$ there exists a constant $C=C(\alpha,d)>0$ such that \begin{align*} \|u\|_{L^{d/(d-\alpha),1}(\mathbb{R}^d)} \leq C \| D^\alpha u\|_{L^1(\mathbb{R}^d;\mathbb{R}^d)} \end{align*} for all $u \in L^q(\mathbb{R}^d)$ for some $1 \leq q

We consider generic degenerate subvarieties $X_i\subset\mathbb{P}^n$. We determine an integer $N$, depending on the varieties, and for $n\geq N$ we compute dimension and degree formulas for the Hadamard product of the varieties $X_i$. Moreover, if the varieties $X_i$ are smooth, their Hadamard product is smooth too. For $n

We obtain large deviations estimates for systems with stretched exponential decay of correlations, which improve the ones obtained in \cite{AFLV11}. As a consequence we obtain better large deviations estimates for Viana maps and get large deviations estimates for a class of intermittent maps with stretched exponential loss of memory.

This paper is devoted to the determination of the cases where there is equality in Courant's nodal domain theorem in the case of a Robin boundary condition. For the square, we partially extend the results that were obtained by Pleijel, B\'erard--Helffer, Helffer--Persson--Sundqvist for the Dirichlet and Neumann problems. After proving some general results that hold for any value of the Robin parameter

We consider the scalar second order ODE u + |u | $\alpha$ u + |u| $\beta$ u = 0, where $\alpha$, $\beta$ are two positive numbers and the non-linear semi-group S(t) generated on IR 2 by the system in (u, u). We prove that S(t)IR 2 is bounded for all t > 0 whenever 0 0, u (t) 2 + |u(t)| $\beta$+2 $\le$ C max{t -- 2 $\alpha$ , t -- ($\alpha$+1)($\beta$+2) $\beta$--$\alpha$ }.

We define a Gaussian invariant measure for the two-dimensional averaged-Euler equation and show the existence of its solution with initial conditions on the support of the measure. An invariant surface measure on the level sets of the energy is also constructed, as well as the corresponding flow. Poincare recurrence theorem is used to show that the flow returns infinitely many times in a neighbourhood

In this note, we develop Fourier approximation methods for the solutions of first-order nonlocal mean-field games (MFG) systems. Using Fourier expansion techniques, we approximate a given MFG system by a simpler one that is equivalent to a convex optimization problem over a finite-dimensional subspace of continuous curves. Furthermore, we perform a time-discretization for this optimization problem

We review some classical results in symmetrization theory, some recent progress in understanding rigidity, and indicate some open problems.

We derive optimal a priori and a posteriori error estimates for Nitsche's method applied to unilateral contact problems. Our analysis is based on the interpretation of Nitsche's method as a stabilised finite element method for the mixed Lagrange multiplier formulation of the contact problem wherein the Lagrange multiplier has been eliminated elementwise. To simplify the presentation, we focus on the

We study the homogenization of obstacle problems in Orlicz-Sobolev spaces for a wide class of monotone operators (possibly degenerate or singular) of the $p(\cdot)$-Laplacian type. Our approach is based on the Lewy-Stampacchia inequalities, which then give access to a compactness argument. We also prove the convergence of the coincidence sets under non-degeneracy conditions.

In this paper, we consider a coupling mixed finite element and continuous Galerkin finite element formulation for a coupled flow and geomechanics model. We use the lowest order Raviart-Thomas space for the spatial approximation of the flow variables and continuous piecewise linear finite elements for the deformation variable while we consider the backward Euler method for the time discretization. This

A new scheme for proving pseudoidentities from a given set {\Sigma} of pseudoidentities, which is clearly sound, is also shown to be complete in many instances, such as when {\Sigma} defines a locally finite variety, a pseudovariety of groups, more generally, of completely simple semigroups, or of commutative monoids. Many further examples when the scheme is complete are given when {\Sigma} defines

In these notes we describe heuristics to predict computational-to-statistical gaps in certain statistical problems. These are regimes in which the underlying statistical problem is information-theoretically possible although no efficient algorithm exists, rendering the problem essentially unsolvable for large instances. The methods we describe here are based on mature, albeit non-rigorous, tools from

We study bitangents of non-smooth tropical plane quartics. Our main result is that with appropriate multiplicities, every such curve has 7 equivalence classes of bitangent lines. Moreover, the multiplicity of bitangent lines varies continuously in families of tropical plane curves.

We describe some group theory which is useful in the classification of combinatorial objects having given groups of automorphisms. In particular, we show the usefulness of the concept of a friendly subgroup: a subgroup H of a group K is a friendly subgroup of K if every subgroup of K isomorphic to H is conjugate in K to H. We explore easy-to-test sufficient conditions for a subgroup H to be a friendly

This article describes a practical approach for determining the lattice of subgroups U < V < G between given subgroups U and G, provided the total number of such subgroups is not too large. It builds on existing functionality for element conjugacy, double cosets and maximal subgroups.

In this paper, we present two algorithms based on the Froidure-Pin Algorithm for computing the structure of a finite semigroup from a generating set. As was the case with the original algorithm of Froidure and Pin, the algorithms presented here produce the left and right Cayley graphs, a confluent terminating rewriting system, and a reduced word of the rewriting system for every element of the semigroup

In this paper, we give a correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the torus $ \mathbb{T}^2$. To achieve this, we use the correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the complex plane and a relation between the Berezin-Toeplitz quantization of a periodic symbol on the real phase space $\mathbb{R}^2$ and the Berezin-Toeplitz

In this note we prove a selection of commutativity theorems for various classes of semigroups. For instance, if in a separative or completely regular semigroup $S$ we have $x^p y^p = y^p x^p$ and $x^q y^q = y^q x^q$ for all $x,y\in S$ where $p$ and $q$ are relatively prime, then $S$ is commutative. In a separative or inverse semigroup $S$, if there exist three consecutive integers $i$ such that $(xy)^i

Within classical propositional logic, assigning probabilities to formulas is shown to be equivalent to assigning probabilities to valuations. A novel notion of probabilistic entailment enjoying desirable properties of logical consequence is proposed and shown to collapse into the classical entailment when the language is left unchanged. Motivated by this result, a decidable conservative enrichment

Recently Peter Keevash solved asymptotically the existence question for Steiner systems by showing that $S(t,k,n)$ exists whenever the necessary divisibility conditions on the parameters are satisfied and $n$ is sufficiently large in terms of $k$ and $t$. The purpose of this paper is to make a conjecture which if true would be a significant extension of Keevash's theorem, and to give some theoretical

Suppose we count the positive integer lattice points beneath a convex decreasing curve in the first quadrant having equal intercepts. Then stretch in the coordinate directions so as to preserve the area under the curve, and again count lattice points. Which choice of stretch factor will maximize the lattice point count? We show the optimal stretch factor approaches $1$ as the area approaches infinity

We prove that, on a Riemann surface, the functor $\mathrm{RH}^S$ constructed in a previous work as a right quasi-inverse of the solution functor from the bounded derived category of regular relative holonomic modules to that of relative constructible complexes satisfies the left quasi-inverse property in a generic sense.

The moduli space of Higgs bundles has two stratifications. The Bialynicki-Birula stratification comes from the action of the non-zero complex numbers by multiplication on the Higgs field, and the Shatz stratification arises from the Harder-Narasimhan type of the vector bundle underlying a Higgs bundle. While these two stratification coincide in the case of rank two Higgs bundles, this is not the case

Recently, the use of special local test functions other than polynomials in Discontinuous Galerkin (DG) approaches has attracted a lot of attention and became known as DG-Trefftz methods. In particular, for the 2D Helmholtz equation plane waves have been used in [10] to derive an Interior Penalty (IP) type Plane Wave DG (PWDG) method and to provide an a priori error analysis of its p-version with respect

We present a class of adaptive multilevel trust-region methods for the efficient solution of optimization problems governed by time–dependent nonlinear partial differential equations with control constraints. The algorithm is based on the ideas of the adaptive multilevel inexact SQP-method from [26, 27]. It is in particular well suited for problems with time–dependent PDE constraints. Instead of the

For two general polytopal complexes the set of face-wise affine maps between them is shown to be a polytopal complex in an algorithmic way. The resulting algorithm for the affine hom-complex is analyzed in detail. There is also a natural tensor product of polytopal complexes, which is the left adjoint functor for Hom(-,-). This extends the corresponding facts from single polytopes, systematic study

We show that the polynomial decay rate of the heat semigroup of the Dirichlet Laplacian in curved planar wedges equals the sum of the usual dimensional decay rate and a multiple of the reciprocal value of the opening angle. To prove the result, we develop the method of self-similar variables for the associated heat equation and study the asymptotic behaviour of the transformed non-autonomous parabolic

We provide an example of a trivalent, 3-connected graph G such that, for any choice of metric on G, the resulting metric graph is Brill-Noether special.

Differential forms are a rich source of invariants in algebraic geometry. This approach was very successful for smooth varieties, but the singular case is less wellunderstood. We explain how the use of the h-topology (introduced by Suslin and Voevodsky in order to study motives) gives a very good object also in the singular case, at least in characteristic 0. We also explain problems and solutions