A free boundary problem for $\infty$–Laplace equation

We consider a free boundary problem for the p-Laplacian describing nonlinear potential flow past a convex profile K with prescribed pressure on the free stream line. The main purpose of this paper is to study the limit as of the classical solutions of the problem above, existing under certain convexity assumptions on a(x). We show, as one can expect, that the limit solves the corresponding potential flow problem for the -Laplacian in a certain weak sense, strong enough however, to guarantee uniqueness. We show also that in the special case the limit is given by the distance function. Received: 10 October 2000 / Accepted: 23 February 2001 / Published online: 19 October 2001     

Oscillation constant of second-order non-linear self-adjoint differential equations

Our concern is to solve the oscillation problem for the non-linear self-adjoint differential equation (a(t)x’)’+b(t)g(x)=0, where g(x) satisfies the signum condition xg(x)>0 if x≠0, but is not assumed to be monotone. Sufficient conditions and necessary conditions are given for all non-trivial solutions to be oscillatory. The obtained results show that the number 1/4 is a critical value for this problem. This paper takes a different approach from most of the previous research. Proof is given by means of phase plane analysis of systems of Liénard type. Examples are included to illustrate the relation between our theorems and results which were given by Cecchi, Marini and Villari. Received: January 5, 2001?Published online: June 11, 2002     

Maps preserving numerical radius or cross norms of products of self-adjoint operators

Let H be a complex Hilbert space with dimH ≥3, Bs（H） the （real） Jordan algebra of all self-adjoint operators on H. Every surjective map Ф ： Bs（H）→13s（H） preserving numerical radius of operator products （respectively, Jordan triple products） is characterized. A characterization of surjective maps on Bs （H） preserving a cross operator norm of operator products （resp. Jordan triple products of operators） is also given.     

On the hot spots of a catalytic super-Brownian motion

Consider the catalytic super-Brownian motion X ^ϱ (reactant) in ℝ ^d , d≤3, which branching rates vary randomly in time and space and in fact are given by an ordinary super-Brownian motion ϱ (catalyst). Our main object of study is the collision local time L = L _[ϱ,Xϱ] (d(s,x) )of catalyst and reactant. It determines the covariance measure in themartingale problem for X ^ϱ and reflects the occurrence of “hot spots” of reactant which can be seen in simulations of X ^ϱ. In dimension 2, the collision local time is absolutely continuous in time, L(d(s,x) ) = ds K _s (dx). At fixed time s, the collision measures K _s (dx) of ϱ _s and X _s ^ϱ have carrying Hausdorff dimension 2. Spatial marginal densities of L exist, and, via self-similarity, enter in the long-term randomergodic limit of L (diffusiveness of the 2-dimensional model). We alsocompare some of our results with the case of super-Brownian motions withdeterministic time-independent catalysts. Received: 2 December 1998 / Revised version: 2 February 2001 / Published online: 9 October 2001     

Polar factorization of maps on Riemannian manifolds

Let (M,g) be a connected compact manifold, C³ smooth and without boundary, equipped with a Riemannian distance d(x,y). If s : M ? M s : M \to M is merely Borel and never maps positive volume into zero volume, we show s = t °u s = t \circ u factors uniquely a.e. into the composition of a map t(x) = exp_x[-?y(x)] t(x) = {\rm exp}_x[-\nabla\psi(x)] and a volume-preserving map u : M ? M u : M \to M , where y: M ? \bold R \psi : M \to {\bold R} satisfies the additional property that (y^c)^c = y (\psi^c)^c = \psi with y^c(y) :=inf{c(x,y) - y(x) | x ? M} \psi^c(y) :={\rm inf}\{c(x,y) - \psi(x)\,\vert\,x \in M\} and c(x,y) = d²(x,y)/2. Like the factorization it generalizes from Euclidean space, this non-linear decomposition can be linearized around the identity to yield the Hodge decomposition of vector fields.¶The results are obtained by solving a Riemannian version of the Monge--Kantorovich problem, which means minimizing the expected value of the cost c(x,y) for transporting one distribution f 3 0 f \ge 0 of mass in L¹(M) onto another. Parallel results for other strictly convex cost functions c(x,y) 3 0 c(x,y) \ge 0 of the Riemannian distance on non-compact manifolds are briefly discussed.     

Density Functional Theory and Optimal Transportation with Coulomb Cost

We present here novel insight into exchange‐correlation functionals in density functional theory, based on the viewpoint of optimal transport. We show that in the case of two electrons and in the semiclassical limit, the exact exchange‐correlation functional reduces to a very interesting functional that depends on an optimal transport map T associated with a given density ρ. The limit problem has been suggested, on grounds of formal arguments, in the physics literature, but it appears that it has not hitherto been interpreted as an optimal transport problem. Since the above limit is strongly correlated, the limit functional yields insight into electron correlations. We prove the existence and uniqueness of such an optimal map for any number of electrons and each ρ and determine the map explicitly in the case when ρ is radially symmetric. © 2012 Wiley Periodicals, Inc.     

Approximation of singularly perturbed parabolic equations in unbounded domains subject to piecewise smooth boundary conditions in the case of solutions that grow at infinity

The initial-boundary value problem in a domain on a straight line that is unbounded in x is considered for a singularly perturbed reaction-diffusion parabolic equation. The higher order derivative in the equation is multiplied by a parameter ɛ², where ɛ ∈ (0, 1]. The right-hand side of the equation and the initial function grow unboundedly as x → ∞ at a rate of O(x ²). This causes the unbounded growth of the solution at infinity at a rate of O(Ψ(x)), where Ψ(x) = x ² + 1. The initialboundary function is piecewise smooth. When ɛ is small, a boundary and interior layers appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristics of the reduced equation passing through the discontinuity points of the initial function. In the problem under examination, the error of the grid solution grows unboundedly in the maximum norm as x → ∞ even for smooth solutions when ɛ is fixed. In this paper, the proximity of solutions of the initial-boundary value problem and its grid approximations is considered in the weighted maximum norm ∥·∥ ^w with the weighting function Ψ⁻¹(x); in this norm, the solution of the initial-boundary value problem is ɛ-uniformly bounded. Using the method of special grids that condense in a neighborhood of the boundary layer or in neighborhoods of the boundary and interior layers, special finite difference schemes are constructed and studied that converge ɛ-uniformly in the weighted norm. It is shown that the convergence rate considerably depends on the type of nonsmoothness in the initial-boundary conditions. Grid approximations of the Cauchy problem with the right-hand side and the initial function growing as O(Ψ(x)) that converge ɛ-uniformly in the weighted norm are also considered.     

The Cendian of a Tree

The eccentricity e(x) and the distance sum s(x) of a vertex x of a connected graph G are well-known functions which measure the centrality of the vertex x in G. The set of vertices which minimize e(x) is called the center of G and the set of vertices which minimize s(x) is known as the median. In this paper we introduce the idea of the so-called cendian of a graph, which unifies the concepts of center and median, and study its structure in trees.     

Estimates for the -Neumann problem and nonexistence of <Emphasis Type="Italic">C</Emphasis><Superscript>2</Superscript> Levi-flat hypersurfaces in

The Markov moment problem is to characterize sequences admitting the representation s_n=₀¹x^nf(x)dx, where f(x) is a probability density on [0,1] and 0f(x)c for almost all x. There are well-known characterizations through complex systems of non-linear inequalities on {s_n}. Necessary and sufficient linear conditions are the following: s₀=1, and for all and . Here, is the forward difference operator. This result is due to Hausdorff. We give a new proof with some ancillary results, for example, characterizing monotone densities. Then we make the connection to de Finettis theorem, with characterizations of the mixing measure.in final form: 18 June 2003     

Stellar permutations of the two-element subsets of a finite set

Let X be a finite set and denote by X⁽²⁾ the set of 2-element subsets of X. A permutation ϕ of X⁽²⁾ is called stellar if, for each x in X, the image under ϕ of the star St(x) = {{x, y}: x ≠ y ∈ X} is a 2-regular graph spanning X − {x}. Several constructions of stellar permutations are given; in particular, there is a natural direct construction using self-orthogonal Latin squares, and a simple recursive construction using linear spaces having all line sizes at least four. Apart from some intrinsic interest, stellar permutations arise in the construction of certain designs. For example, applying such a map ϕ to each of the stars St(x) yields a double covering of the complete graph on X by near 2-factors. We also study stellar groups, that is groups {ϕ₁, …, ϕ_s} of permutations of X⁽²⁾ such that each ϕ_i is stellar (or the identity map). It is elementary to prove that s ≤ for any stellar group; when equality holds, one can construct an associated elation semibiplane. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 381–387, 1998     

An additive eigenvalue problem of physics related to linear programming

A model of recent interest in theoretical physics concerns an infinite elastic chain of atoms placed in a periodic potential field with period 1. Let λ be the energy per atom when the system attains a state of minimum energy. Robert B. Griffiths was led to the following novel equation for λ:min[K(s,t)+x(t)]=λ+x(s) Here K(s, t) is a given periodic function of period 1 in s and t. The problem is to find a periodic function x(s) and a constant λ to satisfy equation (G). In this note a fixed point theorem is used to show that a solution of (G) exists. The same proof shows that the eigenvalue λ is unique. To obtain an approximate solution Eq. (G) is discretized. Then the kernel function K(s, t) becomes a finite square matrix K_ij. It is then shown that the resulting finite system can be solved by two linear programs. The first program has maximum value λ. Then the second linear program furnishes a corresponding eigenvector x.     

Overlaying Classifiers: A Practical Approach to Optimal Scoring

The Receiver Operating Characteristic (ROC) curve is one of the most widely used visual tools to evaluate performance of scoring functions regarding their capacities to discriminate between two populations. It is the goal of this paper to propose a statistical learning method for constructing a scoring function with nearly optimal ROC curve. In this bipartite setup, the target is known to be the regression function up to an increasing transform, and solving the optimization problem boils down to recovering the collection of level sets of the latter, which we interpret here as a continuum of imbricated classification problems. We propose a discretization approach, consisting of building a finite sequence of N classifiers by constrained empirical risk minimization and then constructing a piecewise constant scoring function s _N (x) by overlaying the resulting classifiers. Given the functional nature of the ROC criterion, the accuracy of the ranking induced by s _N (x) can be conceived in a variety of ways, depending on the distance chosen for measuring closeness to the optimal curve in the ROC space. By relating the ROC curve of the resulting scoring function to piecewise linear approximates of the optimal ROC curve, we establish the consistency of the method as well as rate bounds to control its generalization ability in sup -norm. Eventually, we also highlight the fact that, as a byproduct, the algorithm proposed provides an accurate estimate of the optimal ROC curve.     

Explanation of Variability and Removal of Confounding Factors from Data through Optimal Transport

A methodology based on the theory of optimal transport is developed to attribute variability in data sets to known and unknown factors and to remove such attributable components of the variability from the data. Denoting by x the quantities of interest and by z the explanatory factors, the procedure transforms x into filtered variables y through a z‐dependent map, so that the conditional probability distributions ρ(x|z) are pushed forward into a target distribution μ(y), independent of z. Among all maps and target distributions that achieve this goal, the procedure selects the one that minimally distorts the original data: the barycenter of the ρ(x|z). Connections are found to unsupervised learning and to fundamental problems in statistics such as conditional density estimation and sampling. Particularly simple instances of the methodology are shown to be equivalent to k‐means and principal component analysis. An application is shown to a time series of ground temperature hourly data across the United States.© 2017 Wiley Periodicals, Inc.     

Über Homologiegruppen von Faserbündeln

Let L be a distributive lattice with 0 and C (L) be its lattice of congruences. The skeleton, SC (L), of C (L) consists of all those congruences which are the pseudocomplements of members of C (L), and is a complete BOOLEan lattice. An ideal is the kernel of a skeletal congruence if and only if it is an intersection of relative annihilator ideals, i.e. ideals of the form <r, s>j={x∈L: xΔr≤s} for suitable r, s∈L. The set KSC (L) of all such kernels forms an upper continuous distributive lattice and the map a ? (a={x∈L: x≤a} is a lower regular joindense embedding of L into KSC (L). The relationship between SC (L) and KSC (L) leads to numerous characterizations of disjunctive and generalized BOOLEan lattices. In particular, a distributive lattice L is disjunctive (generalized Boolean) if and only if the map Θ ? ker Θ is a lattice-isomorphism of SC (L) onto KSC (L), whose inverse is the map J ? Θ (J)** (the map J ? Θ(J)). In addition, a study of KSC (L) leads to new simple proofs of results on the completions of special classes of lattices.     

The second and the third smallest distances on the sphere

Let s₁ (n) denote the largest possible minimal distance amongn distinct points on the unit sphere . In general, let s_k(n) denote the supremum of thek-th minimal distance. In this paper we prove and disprove the following conjecture of A. Bezdek and K. Bezdek: s₂(n) = s₁([n/3]). This equality holds forn > n₀ however s₂(12) > s₁(4).We set up a conjecture for s_k(n), that one can always reduce the problem of thek-th minimum distance to the function s₁. We prove this conjecture in the casek=3 as well, obtaining that s₃(n) = s₁([n/5]) for sufficiently largen.The optimal construction for the largest second distance is obtained from a point set of size [n/3] with the largest possible minimal distance by replacing each point by three vertices of an equilateral triangle of the same size . If 0, then s₂ tends to s₁([n/3]). In the case of the third minimal distance, we start with a point set of size [n/5] and replace each point by a regular pentagon.     

An initial value problem for a functional equation

We consider the functional equationf(A(x,y))=B(f(x),f(y)), whereA andB are averages. It is known that such a functional equation has exactly one continuous solution satisfying a given two-point condition. By analogy with the theory of differential equations, we may regard the functional equation, together with a two-point condition, as a boundary value problem. (Then each boundary value problem has a unique continuous solution.) If we replace the two-point condition with the specification of a value and derivative at just one point, we obtain an initial value problem.Consider the initial value problemsf(A(x,y))=B(f(x),f(y)),f(a)=s,f(a)=, obtained by fixinga ands and allowing to vary through the set of positive real numbers. The main result of this paper gives a necessary and sufficient condition for each of the initial value problems to have a unique continuous solution, under the hypothesis that at least one of the problems has a continuous solution. This is a partial answer to the problem of determining conditions which are sufficient for the existence of a unique continuous solution of a given initial value problem.     

On a nonlinear wave equation with boundary damping

This paper is concerned with the existence and decay of solutions of the mixed problem for the nonlinear wave equation with boundary conditions Here, Ω is an open bounded set of with boundary Γ of class C²; Γ is constituted of two disjoint closed parts Γ₀ and Γ₁ both with positive measure; the functions μ(t), f(s), g(s) satisfy the conditions μ(t) ≥ μ₀ > 0, f(s) ≥ 0, g(s) ≥ 0 for t ≥ 0, s ≥ 0 and h(x,s) is a real function where x ∈ Γ₁, ν(x) is the unit outward normal vector at x ∈ Γ₁ and α, β are non‐negative real constants. Assuming that h(x,s) is strongly monotone in s for each x ∈ Γ₁, it is proved the global existence of solutions for the previous mixed problem. For that, it is used in the Galerkin method with a special basis, the compactness approach, the Strauss approximation for real functions and the trace theorem for nonsmooth functions. The exponential decay of the energy is derived by two methods: by using a Lyapunov functional and by Nakao's method. Copyright © 2013 John Wiley & Sons, Ltd.     

On the numerical solution of nonlinear Volterra integro-differential equations

Piecewise polynomialss(x) of degreem2 and of continuity classC ¹ are used to obtain approximating functions to the exact solution of a given (ordinary) integro-differential equation of Volterra type. The unknown coefficients ofs(x) are computed recursively, by requiring thats(x) satisfy the integro-differential equation on a finite set of suitably chosen points. Results on the order of convergence of this method are given, together with a numerical illustration.This research was supported by the National Research of Canada (Grant No. A-4805). Received March 13, 1973. Revised July 2, 1973.     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

Multiparameter acoustic imaging in the born approximation

Samples of biological tissue are modelled as inhomogeneous fluids with density ?(X) and sound speed c(x) at point x. The samples are contained in the sphere |x| ? δ and it is assumed that ?(x) ? ?₀ = 1 and c(x) ? c₀ = 1 for |x| ? δ, and |γ_n(x)| ? 1 and |?γ?(x)| ? 1 where γ?(x) = ?(x) ? 1 and γ_n(x) = c^?2(x) ? 1. The samples are insonified by plane pulses s(x · θ₀ – t) where x = |θ₀| = 1 and the scattered pulse is shown to have the form |x|^?1 e_s(|x| – t, θ, θ₀) in the far field, where x = |x| θ. The response e_s(τ, θ, θ₀) is measurable. The goal of the work is to construct the sample parameters γn and γ? from e_s(τ, θ, θ₀) for suitable choiches of s, θ and θ₀. In the limiting case of constant density: γ?(x)_? 0 it is shown that Where δ represents the Dirac δ and S² is the unit sphere |θ| = 1. Analogous formulas, based on two sets of measurements, are derived for the case of variable c(x) and ?(x).