Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space H the Cauchy problem (_P0$P_0$): u^′(t)+Au(t)+Bu(t)=f(t)$u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, where −A   is the infinitesimal generator of a _C0$C_0$-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (_P0$P_0$) the following regularization (_Pε$P_{\epsilon }$): −εu^″(t)+u^′(t)+Au(t)+Bu(t)=f(t)$-\epsilon u^{″}(t)+u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, u^′(T)=_uT$u^{\prime }(T)=u_T$, where ε>0$\epsilon >0$ is a small parameter. We investigate existence, uniqueness and higher regularity for problem (_Pε$P_{\epsilon }$). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (_Pε$P_{\epsilon }$). Problem (_Pε$P_{\epsilon }$) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H)$C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of L²(0,T;H)$L^2(0,T;H)$.     

Singular points of cyclic weighted shift matrices

Let S be an n-by-n   cyclic weighted shift matrix, and _FS(t,x,y)=det(tI+xℜ(S)+yℑ(S))$F_S(t,x,y)=\det (tI+x\Re (S)+y\Im (S))$ be a ternary form associated with S  . We investigate the number of singular points of the curve _FS(t,x,y)=0$F_S(t,x,y)=0$, and show that the number of singular points of _FS(t,x,y)=0$F_S(t,x,y)=0$ associated with a cyclic weighted shift matrix whose weights are neither 1-periodic nor 2-periodic is less than or equal to n(n−3)/2$n(n-3)/2$. Furthermore, we verify the upper bound n(n−3)/2$n(n-3)/2$ is sharp for 4?n?7$4?n?7$.     

Decomposing a graph into bistars

Bárat and the present author conjectured that, for each tree T  , there exists a natural number _kT$k_T$ such that the following holds: If G   is a _kT$k_T$-edge-connected graph such that |E(T)|$|E(T)|$ divides |E(G)|$|E(G)|$, then G has a T-decomposition, that is, a decomposition of the edge set into trees each of which is isomorphic to T  . The conjecture has been verified for infinitely many paths and for each star. In this paper we verify the conjecture for an infinite family of trees that are neither paths nor stars, namely all the bistars S(k,k+1)$S(k,k+1)$.     

An approximation method for strictly pseudocontractive mappings

Let K$K$ be a closed convex subset of a q$q$-uniformly smooth separable Banach space, T:K→K$T:K\to K$ a strictly pseudocontractive mapping, and f:K→K$f:K\to K$ an L$L$-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)$t\in (0,1)$, let _xt$x_t$ be the unique fixed point of tf+(1-t)T$\mathit{tf}+(1-t)T$. We prove that if T$T$ has a fixed point, then {_xt}$\{x_t\}$ converges to a fixed point of T$T$ as t$t$ approaches to 0.     

On the norming constants for normal maxima

Given n   independent standard normal random variables, it is well known that their maxima _Mn$M_n$ can be normalized such that their distribution converges to the Gumbel law. In a remarkable study, Hall proved that the Kolmogorov distance _dn$d_n$ between the normalized _Mn$M_n$ and its associated limit distribution is less than 3/log?n$3/\log ?n$. In the present study, we propose a different set of norming constants that allow this upper bound to be decreased with _dn≤C(m)/log?n$d_n\le C(m)/\log ?n$ for n≥m≥5$n\ge m\ge 5$. Furthermore, the function C(m)$C(m)$ is computed explicitly, which satisfies C(m)≤1$C(m)\le 1$ and _limm→∞?C(m)=1/3${\lim }_{m\to \infty }?C(m)=1/3$. As a consequence, some new and effective norming constants are provided using the asymptotic expansion of a Lambert W type function.     

On multilinear polynomials in four variables evaluated on matrices

Let K   be an algebraically closed field of characteristic 0 and let _Mn(K)$M_n(K)$, n?3$n?3$, be the matrix ring over K  . We will show that the image of any multilinear polynomial in four variables evaluated on _Mn(K)$M_n(K)$ contains all matrices of trace 0.     

Sur la répartition des puissances modulo 1

For almost all x>1$x>1$, (xⁿ)$(x^n)$(n=1,2,…)$(n=1,2,\dots )$ is equidistributed modulo 1, a classical result. What can be said on the exceptional set? It has Hausdorff dimension one. Much more: given an (b_n)$(b_n)$ in [0,1[$[0,1[$ and ε>0$\epsilon >0$, the x  -set such that |xⁿ−b_n|<ε$|x^n-b_n|<\epsilon$ modulo 1 for n   large enough has dimension 1. However, its intersection with an interval [1,X]$[1,X]$ has a dimension <1, depending on ε and X. Some results are given and a question is proposed.     

On an open problem concerning total domination critical graphs

A graph G   with no isolated vertex is total domination vertex critical if for any vertex v$v$ of G   that is not adjacent to a vertex of degree one, the total domination number of G-v$G-v$ is less than the total domination number of G  . We call these graphs _γt${\gamma }_t$-critical. If such a graph G has total domination number k, we call it k  -_γt${\gamma }_t$-critical. We verify an open problem of k  -_γt${\gamma }_t$-critical graphs and obtain some results on the characterization of total domination critical graphs of order n=Δ(G)(_γt(G)-1)+1$n=\Delta (G)({\gamma }_t(G)-1)+1$.     

A new characterization of comonotonicity and its application in behavioral finance

It is well-known that an Rⁿ${\mathbb{R}}^n$-valued random vector (_X1,_X2,?,_Xn)$(X_1,X_2,?,X_n)$ is comonotonic if and only if (_X1,_X2,?,_Xn)$(X_1,X_2,?,X_n)$ and (_Q1(U),_Q2(U),?,_Qn(U))$(Q_1(U),Q_2(U),?,Q_n(U))$ coincide in distribution, for any random variable U   uniformly distributed on the unit interval (0,1)$(0,1)$, where _Qk(⋅)$Q_k(\cdot )$ are the quantile functions of _Xk$X_k$, k=1,2,?,n$k=1,2,?,n$. It is natural to ask whether (_X1,_X2,?,_Xn)$(X_1,X_2,?,X_n)$ and (_Q1(U),_Q2(U),?,_Qn(U))$(Q_1(U),Q_2(U),?,Q_n(U))$ can coincide almost surely for some special U. In this paper, we give a positive answer to this question by construction. We then apply this result to a general behavioral investment model with a law-invariant preference measure and develop a universal framework to link the problem to its quantile formulation. We show that any optimal investment output should be anti-comonotonic with the market pricing kernel. Unlike previous studies, our approach avoids making the assumption that the pricing kernel is atomless, and consequently, we overcome one of the major difficulties encountered when one considers behavioral economic equilibrium models in which the pricing kernel is a yet-to-be-determined unknown random variable. The method is applicable to general models such as risk sharing model.     

Consistency,optimality, and incompleteness

Assume that the problem _P0$P_0$ is not solvable in polynomial time. Let T   be a first-order theory containing a sufficiently rich part of true arithmetic. We characterize T∪{_ConT}$T\cup \{{\mathit{Con}}_T\}$ as the minimal extension of T   proving for some algorithm that it decides _P0$P_0$ as fast as any algorithm B$\mathbb{B}$ with the property that T   proves that B$\mathbb{B}$ decides _P0$P_0$. Here, _ConT${\mathit{Con}}_T$ claims the consistency of T. As a byproduct, we obtain a version of Gödel?s Second Incompleteness Theorem. Moreover, we characterize problems with an optimal algorithm in terms of arithmetical theories.     

Layered viscosity solutions of nonautonomous Hamilton–Jacobi equations: Semiconvexity and relations to characteristics

We construct an explicit representation of viscosity solutions of the Cauchy problem for the Hamilton–Jacobi equation (H,σ)$(H,\sigma )$ on a given domain Ω=(0,T)×Rⁿ$\Omega =(0,T)\times {\mathbb{R}}^n$. It is known that, if the Hamiltonian H=H(t,p)$H=H(t,p)$ is not a convex (or concave) function in p  , or H(⋅,p)$H(\cdot ,p)$ may change its sign on (0,T)$(0,T)$, then the Hopf-type formula does not define a viscosity solution on Ω  . Under some assumptions for H(t,p)$H(t,p)$ on the subdomains (_ti,_ti+1)×Rⁿ⊂Ω$(t_i,t_{i+1})\times {\mathbb{R}}^n\subset \Omega$, we are able to arrange “partial solutions” given by the Hopf-type formula to get a viscosity solution on Ω. Then we study the semiconvexity of the solution as well as its relations to characteristics.