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 Re$\alpha \to -\infty$

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

We study the vanishing viscosity limit of a nonlinear di¤usion equation describing chemical reaction interface or the spatial segregation interface of competing species, where the di¤usion 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

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 consider 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

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

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

In this paper we study the existence of solution to the problem \begin{equation*} \left\{\begin{array}{l} u\in H_{0}^{1}(\Omega), \\ -\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 $L^\infty(\Omega)$, $H(x,s,\xi)$

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.

We identify several classes of complex projective plane 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

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 $$\begin{cases} -\div(a(x)|\nabla u|^{p-2} \nabla u) + u|u|^{r-1} = \dfrac{f(x)}{u^{\theta}} & \mbox{in $\Omega$,} \\ u > 0 & \mbox{in $\Omega$,} \\ u = 0 & \mbox{on $\partial\Omega$} \\ \end{cases}$$ proving that the lower order term $ u|u|^{r-1}$ has some regularizing effects on the solutions.