Solutions concentrating at curves for some singularly perturbed elliptic problems

We study positive solutions of the equation ?ε²Δu+u=u^p, where p>1 and ε>0 is small, with Neumann boundary conditions in a three-dimensional domain $\text{Ω}$. We prove the existence of solutions concentrating along some closed curve on $\text{?Ω}$. To cite this article: A. Malchiodi, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Oscillations caused by several retarded and advanced arguments

We consider the equation $\text{u}{}_{\mathrm{t}}\text{+}\text{▽}\text{· [}\text{v}\text{(x)?]=0, x ∈}$R², with $\text{v}$ defined as a divergenceless vector valued function (in the generalized sense) and subject to initial data u₀. We prove the existence of a generalized global solution to this equation under assumptions of bounded variation for $\text{v}$ and u₀, and smoothness for ?.     

On the maximum number of diagonals of a circuit in a graph

For a graph G, let ?(G) denote the maximum number k such that G contains a circuit with k diagonals.Theorem. For any graph G with minimum valency$\text{n? 3, ?(G) ?}\text{1}\text{2}\text{(n+1)(n-2)}$.If the equality holds and G is connected, then either G is isomorphic to K_n+1 or G is separable and each of its terminal blocks is isomorphic to K_n+1, or K_n+1 with one edge subdivided.     

Roundoff-error analysis of a new class of conjugate-gradient algorithms

We perform the rounding-error analysis of the conjugate-gradient algorithms for the solution of a large system of linear equations Ax=b where Ais an hermitian and positive definite matrix. We propose a new class of conjugate-gradient algorithms and prove that in the spectral norm the relative error of the computed sequence {x_k} (in floating-point arithmetic) depends at worst on ζк^{$\text{3}\text{2}$}, where ζ is the relative computer precision and к is the condition number of A. We show that the residual vectors r_k=Ax_k-b are at worst of order ζк?vA?v ?vx_k?v. We p oint out that with iterative refinement these algorithms are numerically stable. If ζк ² is at most of order unity, then they are also well behaved.     

A parabolic free boundary problem with double pinning

We study the Cauchy problem for the equation ∂_tu_ε−Δu_ε=−β_ε(u_ε) in (0,∞)×Rⁿ as $\text{ε→0}$, where the nonlinearity β_ε is assumed to converge to a measure concentrated at $\text{0}$. In this paper we allow for sign changes of β_ε and u_ε. The solutions are uniformly Lipschitz continuous in space and Hölder continuous in time. We show that each limit of u_ε is a solution of the free boundary problem ∂_tu−Δu=0 in {u>0}∩(0,∞)×Rⁿ,|∇u⁺|²−|∇u⁻|²=g on (∂{u>0}∪∂{u<0})∩((0,∞)×Rⁿ) in the sense of domain variations. Depending on the structure of the nonlinearity $\text{β}{}_{\mathrm{\epsilon }}\text{,}$ the function g in the condition on the free boundary need not be a constant.     

Clones satisfying the term condition

The term condition considered here is the property of an operation ? that holds iff ? and all of its variants obtained by permuting the variables satisfy (for all x,y,u₁,…v₁,…)$\text{?(x,u}{}_1\text{,…) = ?(x,v}{}_1\text{…)??(y,u}{}_1\text{,…) = ?(y,v)}{}_1\text{,…)}$. Clones consisting entirely of operations satisfying this term condition are called TC clones; algebras whose clone of term operations is a TC clone are called TC algebras; varieties such that every algebra in the variety is a TC algebra are called TC varieties. The paper is a systematic study of these notions, giving primary attention to operations and algebras on finite base sets, and to varieties generated by finite algebras. It is proved, among other results, that the number of n-ary TC operations on a k-element set is logarithmically asymptotic to k^(k?1)n when n increases without bound and k is held fixed; that there exist only countably many TC clones on any finite set; that the maximal TC clones on a finite set are finite in number (for each set). Some necessary conditions for an algebra to generate a TC variety are given, also some sufficient conditions.     

Simple components of group algebras Q[G] central over real quadratic fields

Let k be a real quadratic field, and $\text{U}$ a central division quaternion algebra over k. In this paper sufficient conditions are given to insure that $\text{U}$ appears in a simple component of the group algebra Q[G] of some finite group G over the rational field Q. In particular, when k is assumed to be Q(√2) or Q(√5), the necessary and sufficient conditions for $\text{U}$ to appear in some Q[G] are given.     

Fourier transforms of smooth functions on certain nilpotent Lie groups

The authors consider irreducible representations $\text{π ?}\text{N}\text{?}$ of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels K_φ(x, y) of the trace class operations π_φ = ∝_Nφ(n)π_ndn, regarding the π as modeled in L²(R^k) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels K_φ(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in Rⁿ, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rⁿ → Rⁿ with the following properties. The Fourier transforms F₁φ = K_φ(x, y, λ) all factor through A to give “rationalized” Fourier transforms Fφ(u) such that Fφ ° A = F₁φ. On the rationalized parameter space a function f(u) is of the form F_φ = f ? f is Schwartz class on Rⁿ. If polynomial operators T?P(N) are transferred to operators $\text{T}\text{?}$ on Rⁿ such that $\text{F(Tφ) =}\text{T}\text{?}\text{(Fφ), P(N)}$ is transformed isomorphically to P(Rⁿ).     

Singular perturbations in the interaction representation,II

Perturbations of $\text{(?Δ)}{}^{\mathrm{<mtext>m</mtext><mtext>2</mtext>}}\text{in}\text{L}{}^2\text{(R}{}^{\mathrm{k}}\text{),}\text{for}\text{k ? 1}$ and suitable m, by distributions V for which $\text{V}\text{?}\text{(k) = 0(¦k¦}{}^{\mathrm{\alpha }}\text{),}\text{where}\text{α =}\text{(m + 1 ? ε)}\text{2 ? k}\text{, 0 < ε ? m + 1 ? 2k}$, are shown to correspond to self-adjoint operators H_v, in such a way that H_v depends continuously on V, and agrees with H + V when V is sufficiently regular. These results extend joint work with Irving E. Segal [J. Functional Analysis38 (1980), 71–98], in which perturbations of $\text{(}\text{?id}\text{dx}\text{)}{}^{\mathrm{m}}$ by distributions V with bounded Fourier transforms in L²(R¹) were considered.     

The asymptotic distribution of weighted empirical distribution functions

Let G_n denote the empirical distribution based on n independent uniform (0, 1) random variables. The asymptotic distribution of the supremum of weighted discrepancies between G_n(u) and u of the forms 6w_v(u)D_n(u)6 and 6w_v(G_n(u))D_n(u)6, where D_n(u) = G_n(u)?u, w_v(u) = (u(1?u))^?1+v and 0 ? v < $\text{1}\text{2}$ is obtained. Goodness-of-fit tests based on these statistics are shown to be asymptotically sensitive only in the extreme tails of a distribution, which is exactly where such statistics that use a weight function w_v with $\text{1}\text{2}$ ? v ? 1 are insensitive. For this reason weighted discrepancies which use the weight function w_v with 0 ? v < $\text{1}\text{2}$ are potentially applicable in the construction of confidence contours for the extreme tails of a distribution.     

Nonlinear eigenvalue problems with positively convex operators

We consider the equation u = λAu (λ > 0), where A is a forced isotone positively convex operator in a partially ordered normed space with a complete positive cone K. Let Λ be the set of positive λ for which the equation has a solution u?K, and let Λ₀ be the set of positive λ for which a positive solution—necessarily the minimum one—can be obtained by an iteration u_n = λAu_n?1, u₀ = 0. We show that if K is normal, and if Λ is nonempty, then Λ₀ is nonempty, and each set Λ₀, Λ is an interval with $\text{inf}\text{(Λ}{}_0\text{) =}\text{inf}\text{(Λ) = 0}\text{and sup}\text{(Λ}{}_0\text{) =}\text{sup}\text{(Λ) (= λ1,}\text{say}\text{)}$; but we may have λ1 ? Λ₀ and λ1 ? Λ. Furthermore, if A is bounded on the intersection of K with a neighborhood of 0, then Λ₀ is nonempty. Let u⁰(λ) = lim_n→∞(λA)ⁿ(0) be the minimum positive fixed point corresponding to λ ? Λ₀. Then u⁰(λ) is a continuous isotone convex function of λ on Λ₀.     

Residue properties of certain quadratic units

Explicit formulas are given for the quadratic and quartic characters of units of certain quadratic fields in terms of representations by positive definite binary quadratic forms, as conjectured by Leonard and Williams (Pacific J. Math.71 (1977), Rocky Mountain J. Math.9 (1979)), and by Lehmer (J. Reine Angew. Math.268/69 (1974)). For example, if p and a are primes such that p≡1 (mod 8), q≡5 (mod 8) and the Legendre symbol $\text{(}\text{q}\text{p}\text{)=1}$, and if ε is the fundamental unit of $\text{Q}$(√q), then $\text{(}\text{ε}\text{p}\text{)}{}_4\text{=(?1)}{}^{\mathrm{b+d}}$, where p=a²+16b² and p^k=c²+16qd² with k odd.     

Girsanov functionals and optimal bang-bang laws for final value stochastic control

Girsanov's theorem is a generalization of the Cameron-Martin formula for the derivative of a measure induced by a translation in Wiener space. It states that for ? a nonanticipative Brownian functional with ∫|?|² ds < ∞ a.s. and $\text{d}\text{P}\text{?}\text{=exp[ζ(?)] d}\text{P}$ with $\text{E}\text{?}\text{{1}=1}$, where $\text{ζ(?) = ∫?}\text{d}\text{w-}\text{1}\text{2}\text{∫|?|}{}^2\text{d}\text{s}$, the translated functions $\text{(Tw)(t) = w}{}_{\mathrm{t}}\text{-}\text{∫}\text{0}\text{t}\text{?}\text{d}\text{s}$ are a Wiener process under P?. The Girsanov functionals exp [ζ(?)] have been used in stochastic control theory to define measures corresponding to solutions of stochastic DEs with only measurable control laws entering the right-hand sides. The present aim is to show that these same concepts have direct practical application to final value problems with bounded control. This is done here by an example, the noisy integrator: Make E{x²₁}∣small, subject to dx_t = u_t dt + dw_t, |u|? 1, x_t observed. For each control law there is a definite cost v(1?t, x) of starting at x, t and using that law till t = 1, expressible as an integral with respect to (a suitable) P?. By restricting attention to a dense set of smooth laws, using Itô's lemma, Kac's theorem, and the maximum principle for parabolic equations, it is possible to calculate sgn v_x for a critical class of control laws, then to compare control laws, “solve” the Bellman-Hamilton-Jacobi equation, and thus justify selection of the obvious bang-bang law as optimal.     

Capacitary estimates of solutions of a class of nonlinear elliptic equations

Let $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$ and K a compact subset of $\text{?Ω}$. Assume that q?(N+1)/(N?1) and denote by U_K the maximal solution of ?Δu+u^q=0 in $\text{Ω}$ which vanishes on $\text{?Ω?K}$. We obtain sharp upper and lower estimates for U_K in terms of the Bessel capacity C_2/q,q′ and prove that U_K is σ-moderate. In addition we relate the strong ‘blow-up’ points of U_K on $\text{?Ω}$ to the ‘thick’ points of K in the fine topology associated with C_2/q,q′ and characterize these points by a path integral condition on U_K. To cite this article: M. Marcus, L. Véron, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Existence and comparison theorems for nonlinear diffusion systems

In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v¹(x), v²(x),…, v^m(x)) from $\text{Ω ? R}{}^{\mathrm{n}}$ into R^m which satisfy ψⁱ(x, t) ? vⁱ(x) ? θⁱ(x, t) for all $\text{x ? Ω}\text{and}\text{1 ? i ? m}$, where ψⁱand θⁱ are extended real-valued functions on $\text{}\text{?}\text{gW × [0, T)}$. We find conditions which will ensure that a solution U(x, t) ≡ (u¹(x, t), u²(x, t),…, u^m(x, t)) which satisfies U(x, 0) ?K(0) will also satisfy U(x, t) ?K(t) for all 0 ? t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem.     

Resolvable path designs

A resolvable (balanced) path design, RBPD(v, k, λ) is the decomposition of λ copies of the complete graph on v vertices into edge-disjoint subgraphs such that each subgraph consists of $\text{v}\text{k}$ vertex-disjoint paths of length k ? 1 (k vertices). It is shown that an RBPD(v, 3, λ) exists if and only if v ≡ 9 (modulo 12/gcd(4, λ)). Moreover, the RBPD(v, 3, λ) can have an automorphism of order $\text{v}\text{3}$. For k > 3, it is shown that if v is large enough, then an RBPD(v, k, 1) exists if and only if v ≡ k² (modulo lcm(2k ? 2, k)). Also, it is shown that the categorical product of a k-factorable graph and a regular graph is also k-factorable. These results are stronger than two conjectures of P. Hell and A. Rosa     

Graphical t-wise balanced designs

A pair (X, $\text{B}$) will be a t-wise balanced design (tBD) of type t?(v, K, λ) if $\text{B}\text{= (B}{}_{\mathrm{i}}\text{: i ? I)}$ is a family of subsets of X, called blocks, such that: (i) $\text{|X| = v ?}\text{N}$, where $\text{N}$ is the set of positive integers; (ii) $\text{1?t?|B}{}_{\mathrm{i}}\text{|?K?}\text{N}$, for every i ? I; and (iii) if T ? X, |T| = t, then there are λ ? $\text{N}$ indices i ? I where T ? B_i. Throughout this paper we make three restrictions on our tBD's: (1) there are no repeated blocks, i.e. $\text{B}$ will be a set of subsets of X; (2) t ? K or there are no blocks of size t; and (3) $\text{P}{}_{\mathrm{k}}\text{(X)?}\text{B}$ or $\text{B}$ does not contain all k-subsets of X for any t<k?v. Note then that X ? $\text{B}$. Also, if we give the parameters of a specific tBD, then we will choose a minimal K.We focus on the $\text{t?((}\text{p}\text{2}\text{), K, λ)}$ designs with the symmetric group S_p as automorphism group, i.e. X will be the set of $\text{v = (}\text{p}\text{2}\text{)}$ labelled edges of the undirected complete graph K_p and if B ? $\text{B}$ then all subgraphs of K_p isomorphic to B are also in $\text{B}$. Call such tBD's ‘graphical tBD's’. We determine all graphical tBD's with λ = 1 or 2 which will include one with parameters 4?(15,{5,7},1).