The quantitative Faber–Krahn inequality for the Robin Laplacian

We prove a quantitative form of the Faber–Krahn inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions. The asymmetry term involves the square power of the Fraenkel asymmetry, multiplied by a constant depending on the Robin parameter, the dimension of the space and the measure of the set.     

Spectral and modulational stability of periodic wavetrains for the nonlinear Klein–Gordon equation

This paper is a detailed and self-contained study of the stability properties of periodic traveling wave solutions of the nonlinear Klein–Gordon equation _utt−_uxx+V^′(u)=0$u_{tt}-u_{xx}+V^{\prime }(u)=0$, where u is a scalar-valued function of x and t  , and the potential V(u)$V(u)$ is of class C²$C^2$ and periodic. Stability is considered both from the point of view of spectral analysis of the linearized problem (spectral stability analysis) and from the point of view of wave modulation theory (the strongly nonlinear theory due to Whitham as well as the weakly nonlinear theory of wave packets). The aim is to develop and present new spectral stability results for periodic traveling waves, and to make a solid connection between these results and predictions of the (formal) modulation theory, which has been developed by others but which we review for completeness.     

Spectral Asymptotics for Laplacian with Mixed Boundary-value Condition

Let Ω be a non-empty bounded open set in Rn(n ≥1) with boundary (?)Ω=Γ1∪Γ2. WedefineIn this paper, we consider the following variational eigenvalue problem:where △ denotes the Laplacian in Ω. We say that the scalar λ is an eigenvalue of (P) if     

On the Aleksandrov–Bakelman–Pucci estimate for the infinity Laplacian

We prove L ^∞ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and p-Laplacian, namely $$\begin{array}{ll} -\Delta_p^N u=f\quad{\rm for } \; n < p \leq\infty.\end{array}$$ We are able to provide a stable family of results depending continuously on the parameter p. We also prove the failure of the classical Alexandrov–Bakelman–Pucci estimate for the normalized infinity Laplacian and propose alternate estimates.     

The landscape of the proximal point method for nonconvex–nonconcave minimax optimization

Mathematical Programming - Minimax optimization has become a central tool in machine learning with applications in robust optimization, reinforcement learning, GANs, etc. These applications are...     

The Moore–Penrose inverse of the normalized graph Laplacian

We prove a formula that relates the Moore–Penrose inverses of two matrices A,B$A,B$ such that A=N⁻¹BM⁻¹$A=N^{-1}B{M}^{-1}$ and discuss some applications, in particular to the representation of the Moore–Penrose inverse of the normalized Laplacian of a graph. The Laplacian matrix of an undirected graph is symmetric and is strictly related to its connectivity properties. However, our formula applies to asymmetric matrices, so that we can generalize our results for asymmetric Laplacians, whose importance for the study of directed graphs is increasing.     

Precise Arrhenius Law for p-forms: The Witten Laplacian and Morse–Barannikov Complex

Accurate asymptotic expressions are given for the exponentially small eigenvalues of Witten Laplacians acting on p-forms. The key ingredient, which replaces explicit formulas for global quasimodes in the case p = 0, is Barannikov’s presentation of Morse theory in Barannikov (Adv Soviet Math 21:93–115, 1994).     

The Karush–Kuhn–Tucker optimality conditions for fuzzy optimization problems

This paper considers optimization problems with fuzzy-valued objective functions. For this class of fuzzy optimization problems we obtain Karush–Kuhn–Tucker type optimality conditions considering the concept of generalized Hukuhara differentiable and pseudo-invex fuzzy-valued functions.     

The stability problem and special solutions for the 5-components Maxwell–Bloch equations

For the 5-components Maxwell–Bloch system the stability problem for the isolated equilibria is completely solved. Using the geometry of the symplectic leaves, a detailed construction of the homoclinic orbits is given. Studying the problem of invariant sets for the system, we discover a rich family of periodic solutions in explicit form.     

Sharp Moser–Trudinger inequalities for the Laplacian without boundary conditions

We derive a sharp Moser–Trudinger inequality for the borderline Sobolev imbedding of $W^{2,n/2}(B_n)$ into the exponential class, where $B_n$ is the unit ball of ${\mathbb{R}}^n$. The corresponding sharp results for the spaces $W_0^{d,n/d}(\Omega )$ are well known, for general domains Ω, and are due to Moser and Adams. When the zero boundary condition is removed the only known results are for $d=1$ and are due to Chang–Yang, Cianchi and Leckband. The proof of our result is based on a new integral representation formula for the “canonical” solution of the Poisson equation on the ball, that is, the unique solution of the equation $\mathrm{\Delta }u=f$ which is orthogonal to the harmonic functions on the ball. The main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation, expressed in terms of its distribution function.     

The Signless Laplacian Spectral Radius of Tricyclic Graphs with k Pendant Vertices

In this paper,we determine the unique graph with the largest signless Laplacian spectral radius among all the tricyclic graphs with n vertices and k pendant vertices.     

The Cauchy problem for the wave equation with Lévy Laplacian

We present the solution of the Cauchy problem (the initial-value problem in the whole space) for the wave equation with infinite-dimensional Lévy Laplacian Δ _L , $$ \frac{{\partial ^2 U(t,x)}} {{\partial t^2 }} = \Delta _L U(t,x) $$ in two function classes, the Shilov class and the Gâteaux class.     

Using the Black–Derman–Toy interest rate model for portfolio optimization

No-arbitrage interest rate models are designed to be consistent with the current term structure of interest rates. The diffusion of the interest rates is often approximated with a tree, in which the scenario-dependent fair price of any security is calculated as the present value of the risk-neutral expectation by backward induction. To use this tree in a portfolio optimization context it is necessary to account for the so-called “market price of risk”. In this paper we present a method to change the conditional probabilities in the Black–Derman–Toy model to the physical (or real) measure, including the market price of risk, and explore the economic implications for expected spot rates and for expected bond returns.     

A sharp stability criterion for the Vlasov–Maxwell system

We consider the linear stability problem for a 3D cylindrically symmetric equilibrium of the relativistic Vlasov–Maxwell system that describes a collisionless plasma. For an equilibrium whose distribution function decreases monotonically with the particle energy, we obtained a linear stability criterion in our previous paper [24]. Here we prove that this criterion is sharp; that is, there would otherwise be an exponentially growing solution to the linearized system. We also treat the considerably simpler periodic D case. The new formulation introduced here is applicable as well to the non-relativistic case, to other symmetries, and to general equilibria.     

On Levitin–Polyak well-posedness and stability in set optimization

Positivity - In this paper, Levitin–Polyak (in short LP) well-posedness in the set and scalar sense are defined for a set optimization problem and a relationship between them is found....     

The unbounded solution of a periodic mixed Sturm–Liouville problem in an infinite strip for the Laplacian

The unbounded solution, at the points where the boundary conditions change, for a mixed Sturm–Liouville problem of the Dirichlet–Neumann type can be obtained using the method of the integral equation formulation. Since this formulation is usually reduced to an infinite algebraic system in which the unknowns are the Fourier coefficients of the unknown unbounded entity, a study of ?_p-solutions imposes itself concerning the influence of the truncation on such systems. This study is achieved and the well-known theorem on the ?₂-solutions of the infinite algebraic systems is generalized.     

Asymptotic stability of the rarefaction wave for the generalized KdV–Burgers–Kuramoto equation

In this paper, we will study the Cauchy problem for the generalized KdV–Burgers–Kuramoto equation, which represents a dissipative, stroboscopic and unstable system in physics. When the initial data is a small disturbance of a rarefaction wave of the inviscid Burgers equation, we prove the global existence of the solution to the corresponding Cauchy problem and asymptotic stability of the rarefaction wave. The analysis is based on a priori estimates and the L²$L^2$-energy method.     

Spectral corrections for Sturm–Liouville problems

The numerical solution of the Sturm–Liouville problem can be achieved using shooting to obtain an eigenvalue approximation as a solution of a suitable nonlinear equation and then computing the corresponding eigenfunction. In this paper we use the shooting method both for eigenvalues and eigenfunctions. In integrating the corresponding initial value problems we resort to the boundary value method. The technique proposed seems to be well suited to supplying a general formula for the global discretization error of the eigenfunctions depending on the discretization errors arising from the numerical integration of the initial value problems. A technique to estimate the eigenvalue errors is also suggested, and seems to be particularly effective for the higher-index eigenvalues. Numerical experiments on some classical Sturm–Liouville problems are presented.