Local existence and uniqueness theorems for a free boundary problem

Free boundary problems are considered, where the tangential and normal components u_t and u_n of an otherwise unknown plane harmonic vector field are prescribed along the unknown boundary curve as a function of the coordinates x, y and the tangent angle θ. The vector field is required to exist either in the interior region G⁺ or in the exterior G^?. In each case the free boundary is characterized by a nonlinear integral equation. A linearised version of this equation is a one-dimensional singular integral equation. Under rather general hypotheses which are easy to check, the properties of the linear equation are described by Noether's theorems. The regularity of the solution is studied and the effect of the nonlinear terms is estimated. A variant of the Nash-Moser implicit-function theorem can be applied. This yields local existence and uniqueness theorems for the free boundary problem in Hölder-classes H^2+μ. The boundary curve depends continuously on the defining data. Finally some examples are given, where the linearised equation can be completely discussed.     

The generalized dock problem

We show existence and uniqueness for a linearized water wave problem in a two dimensional domain G with corner, formed by two semi-axes Γ₁ and Γ₂ which intersect under an angle α?∈?(0,?π]. The existence and uniqueness of the solution is proved by considering an auxiliary mixed problem with Dirichlet and Neumann boundary conditions. The latter guarantees the existence of the Dirichlet to Neumann map. The water wave boundary value problem is then shown to be equivalent to an equation like v_tt ?+?gΛv?=?P_t with initial conditions, where t stands for time, g is the gravitational constant, P means pressure and Λ is the Dirichlet to Neumann map. We then prove that Λ is a positive self-adjoint operator.     

Local solvability of a nonstationary leakage problem for an ideal incompressible fluid, 3

In this paper we prove the existence and uniqueness of solutions of the leakage problem for the Euler equations in bounded domain Ω C R³ with corners π/n, n = 2, 3… We consider the case where the tangent components of the vorticity vector are given on the part S₁ of the boundary where the fluid enters the domain. We prove the existence of an unique solution in the Sobolev space W_p^l(Ω), for arbitrary natural l and p > 1. The proof is divided on three parts: (1) the existence of solutions of the elliptic problem in the domain with corners where v – velocity vector, ω – vorticity vector and n is an unit outward vector normal to the boundary, (2) the existence of solutions of the following evolution problem for given velocity vector (3) the method of successive approximations, using solvability of problems (1) and (2).     

Extremal results for rooted minor problems

In this article, we consider the following problem. Given four distinct vertices v₁,v₂,v₃,v₄. How many edges guarantee the existence of seven connected disjoint subgraphs X_i for i = 1,…, 7 such that X_j contains v_j for j = 1, 2, 3, 4 and for j = 1, 2, 3, 4, X_j has a neighbor to each X_k with k = 5, 6, 7. This is the so called “rooted K_{3, 4}‐minor problem.” There are only few known results on rooted minor problems, for example, [15,6]. In this article, we prove that a 4‐connected graph with n vertices and at least 5n ? 14 edges has a rooted K_3,4‐minor. In the proof we use a lemma on graphs with 9 vertices, proved by computer search. We also consider the similar problems concerning rooted K_3,3‐minor problem, and rooted K_3,2‐minor problem. More precisely, the first theorem says that if G is 3‐connected and e(G) ≥ 4|G| ? 9 then G has a rooted K_3,3‐minor, and the second theorem says that if G is 2‐connected and e(G) ≥ 13/5|G| ? 17/5 then G has a rooted K_3,2‐minor. In the second case, the extremal function for the number of edges is best possible. These results are then used in the proof of our forthcoming articles 7 , 8 . © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 191–207, 2007     

On groups acting freely on a tree

Suppose we are given a group G\mit\Gamma and a tree X on which G\mit\Gamma acts. Let d be the distance in the tree. Then we are interested in the asymptotic behavior of the numbers a_d: = # {w ? vertX : w=gv, g ? G , d(v₀,w)=d }a_d:= \# \{w\in {\rm {vert}}X : w=\gamma {v}, \gamma \in {\mit\Gamma} , d({v}_0,w)=d \} if d? ￥d\rightarrow \infty , where v, v_o are some fixed vertices in X.¶ In this paper we consider the case where G\mit\Gamma is a finitely generated group acting freely on a tree X. The growth function ?a_d x^d\textstyle\sum\limits a_d x^d is a rational function [3], which we describe explicitely. From this we get estimates for the radius of convergence of the series. For the cases where G\mit\Gamma is generated by one or two elements, we look a little bit closer at the denominator of this rational function. At the end we give one concrete example.     

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a penetrable bounded obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity T H^-\frac12(div_G,G){\mathsf T \mathsf H^{-\frac{1}{2}}({\rm div}_{\Gamma},\Gamma)}. Using Helmholtz decomposition, we can base their analysis on the study of pseudo-differential integral operators in standard Sobolev spaces, but we then have to study the Gateaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.     

New upper bounds for the greatest number of proper colorings of a (V,E)-graph

Let ?? denote the family of simple undirected graphs on v vertices having e edges ((v, e)-graphs) and P(G; λ) be the chromatic polynomial of a graph G. For the given integers v, e, and λ, let f(v, e, λ) denote the greatest number of proper colorings in λ or less colors that a (v, e)-graph G can have, i.e., f(v, e, λ) = max{P(G; λ): G ∈ ??}. In this paper we determine some new upper bounds for f(v, e, λ).     

Local and global average degree in graphs and multigraphs

A vertex v of a graph G is called groupie if the average degree t_v of all neighbors of v in G is not smaller than the average degree t_G of G. Every graph contains a groupie vertex; the problem of whether or not every simple graph on ≧2 vertices has at least two groupie vertices turned out to be surprisingly difficult. We present various sufficient conditions for a simple graph to contain at least two groupie vertices. Further, we investigate the function f(n) = max min_v (t_v/t_G), where the maximum ranges over all simple graphs on n vertices, and prove that f(n) = 1/42n + o(1). The corresponding result for multigraphs is in sharp contrast with the above. We also characterize trees in which the local average degree t_v is constant.     

A Riemann-Hilbert Problem for the Moisil-Teodorescu System

In a bounded domain with smooth boundary in ?³ we consider the stationary Maxwell equations for a function u with values in ?³ subject to a nonhomogeneous condition (u, v)_x = u₀ on the boundary, where v is a given vector field and u₀ a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.     

On the centralizer ofK in the universal enveloping algebra of SO(n,1) and SU(n,1)

LetG _o be a non compact real semisimple Lie group with finite center, and letU U(g) ^K denote the centralizer inU U(g) of a maximal compact subgroupK _o ofG _o. To study the algebraU U(g) ^K , B. Kostant suggested to consider the projection mapP:U U(g)→U(k)⊗U(a), associated to an Iwasawa decompositionG _o=K _o A _o N _o ofG _o, adapted toK _o. WhenP is restricted toU U(g) ^K J. Lepowsky showed thatP becomes an injective anti-homomorphism ofU U(g) ^K intoU(k) ^M ⊗U(a). HereU(k) ^M denotes the centralizer ofM _o inU(k),M _o being the centralizer ofA _o inK _o. To pursue this idea further it is necessary to have a good characterization of the image ofU U(g) ^K inU(k)^M×U(a). In this paper we describe such image whenG _o=SO(n,1)_e or SU(n,1). This is acomplished by establishing a (minimal) set of equations satisfied by the elements in the image ofU U(g) ^K , and then proving that they are enough to characterize such image. These equations are derived on one hand from the intertwining relations among the principal series representations ofG _o given by the Kunze-Stein interwining operators, and on the other hand from certain imbeddings among Verma modules. This approach should prove to be useful to attack the general case. Supported in part by Fundación Antorchas     

Classification of Some Graded not Necessarily Associative Division Algebras I

We study not necessarily associative (NNA) division algebras over the reals. We classify in this paper series those that admit a grading over a finite group G, and have a basis {v _g |g ∈ G} as a real vector space, and the product of these basis elements respects the grading and includes a scalar structure constant with values only in {1, ? 1}. We classify here those graded by an abelian group G of order |G| ≤8 with G non–isomorphic to ?/8?. We will find the complex, quaternion, and octonion algebras, but also a remarkable set of novel non–associative division algebras.     

Integral Equation Method for the First and Second Problems of the Stokes System

Using the integral equation method we study solutions of boundary value problems for the Stokes system in Sobolev space H ¹(G) in a bounded Lipschitz domain G with connected boundary. A solution of the second problem with the boundary condition $\partial {\bf u}/\partial {\bf n} -p{\bf n}={\bf g}$ is studied both by the indirect and the direct boundary integral equation method. It is shown that we can obtain a solution of the corresponding integral equation using the successive approximation method. Nevertheless, the integral equation is not uniquely solvable. To overcome this problem we modify this integral equation. We obtain a uniquely solvable integral equation on the boundary of the domain. If the second problem for the Stokes system is solvable then the solution of the modified integral equation is a solution of the original integral equation. Moreover, the modified integral equation has a form f?+?S f?=?g, where S is a contractive operator. So, the modified integral equation can be solved by the successive approximation. Then we study the first problem for the Stokes system by the direct integral equation method. We obtain an integral equation with an unknown ${\bf g}=\partial {\bf u}/\partial {\bf n} -p{\bf n}$ . But this integral equation is not uniquely solvable. We construct another uniquely solvable integral equation such that the solution of the new eqution is a solution of the original integral equation provided the first problem has a solution. Moreover, the new integral equation has a form ${\bf g}+\tilde S{\bf g}={\bf f}$ , where $\tilde S$ is a contractive operator, and we can solve it by the successive approximation.     

On the greatest number of 2 and 3 colorings of a (v,e)-graph

Let F denote the family of simple undirected graphs on v vertices having e edges ((v, e)-graphs) and P(λ, G) be the chromatic polynomial of a graph G. For the given integers v, e, Δ, let f(v, e, Δ) denote the greatest number of proper colorings in Δ or less colors that a (v, e)-graph G can have, i.e., f(v, e, Δ) = max{P(Δ, G): G ∈ F}. In this paper we determine f(v, e, 2) and describe all graphs G for which P(2, G) = f(v, e, 2). For f(v, e, 3), a lower bound and an upper bound are found.     

Boundary integral equations for the Helmholtz equation: The third boundary value problem

In this paper we study the application of boundary integral equation methods for the solution of the third, or Robin, boundary value problem for the exterior Helmholtz equation. In contrast to earlier work, the boundary value problem is interpreted here in a weak sense which allows data to be specified in L_∞ (?D), ?D being the boundary of the exterior domain which we assume to be Lyapunov of index 1. For this exterior boundary value problem, we employ Green's theorem to derive a pair of boundary integral equations which have a unique simultaneous solution. We then show that this solution yields a solution of the original exterior boundary value problem.     

Rooted induced trees in triangle‐free graphs

For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of Gthat is a tree. Further, for a vertex v∈V(G), let t(G, v) denote the maximum number of vertices in an induced subgraph of Gthat is a tree, with the extra condition that the tree must contain v. The minimum of t(G) (t(G, v), respectively) over all connected triangle‐free graphs G(and vertices v∈V(G)) on nvertices is denoted by t₃(n) (t(n)). Clearly, t(G, v)?t(G) for all v∈V(G). In this note, we solve the extremal problem of maximizing |G| for given t(G, v), given that Gis connected and triangle‐free. We show that and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^{\ast}(n) = \lceil {\frac{1}{2}}(1+{\sqrt{8n-7}})\rceilFor a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of Gthat is a tree. Further, for a vertex v∈V(G), let t(G, v) denote the maximum number of vertices in an induced subgraph of Gthat is a tree, with the extra condition that the tree must contain v. The minimum of t(G) (t(G, v), respectively) over all connected triangle‐free graphs G(and vertices v∈V(G)) on nvertices is denoted by t₃(n) (t(n)). Clearly, t(G, v)?t(G) for all v∈V(G). In this note, we solve the extremal problem of maximizing |G| for given t(G, v), given that Gis connected and triangle‐free. We show that and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^{\ast}(n) = \lceil {\frac{1}{2}}(1+{\sqrt{8n-7}})\rceil$, improving a recent result by Fox, Loh and Sudakov. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 206–209, 2010     

The existence of soliton metrics for nilpotent Lie groups

We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N, g) satisfy the matrix equation U v = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Kac–Moody algebras to analyze the solution spaces for such linear systems. An application to the existence of soliton metrics on certain filiform Lie groups is given.     

On point-halfspace graphs

The following definition is motivated by the study of circle orders and their connections to graphs. A graphs G is called a point-halfspace graph (in R ^k) provided one can assign to each vertex v ? (G) a point p _v R ^k and to each edge e ? E(G) a closed halfspace H_e ? R ^k so that v is incident with e if and only if p _v ? H_e. Let H ^k denote the set of point-halfspace graphs (in R ^k). We give complete forbidden subgraph and structural characterizations of the classes H ^k for every k. Surprisingly, these classes are closed under taking minors and we give forbidden minor characterizations as well. © 1996 John Wiley & Sons, Inc.     

Geodesics and almost geodesic cycles in random regular graphs

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d_G(u, v) is at least d_C(u, v)?e(n). Let ω(n) be any function tending to infinity with n. We consider a random d‐regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n) = log_d?1log_d?1n+ ω(n) and |C| = 2log_d?1n+ O(ω(n)). Along the way, we obtain results on near‐geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph. Copyright © 2010 John Wiley & Sons, Ltd. J Graph Theory 66:115‐136, 2011     

The node-edge weighted 2-edge connected subgraph problem: Linear relaxation, facets and separation

Let G=(V,E) be a undirected k-edge connected graph with weights c_e on edges and w_v on nodes. The minimum 2-edge connected subgraph problem, 2ECSP for short, is to find a 2-edge connected subgraph of G, of minimum total weight. The 2ECSP generalizes the well-known Steiner 2-edge connected subgraph problem. In this paper we study the convex hull of the incidence vectors corresponding to feasible solutions of 2ECSP. First, a natural integer programming formulation is given and it is shown that its linear relaxation is not sufficient to describe the polytope associated with 2ECSP even when G is series-parallel. Then, we introduce two families of new valid inequalities and we give sufficient conditions for them to be facet-defining. Later, we concentrate on the separation problem. We find polynomial time algorithms to solve the separation of important subclasses of the introduced inequalities, concluding that the separation of the new inequalities, when G is series-parallel, is polynomially solvable.     

List total colorings of series-parallel graphs

A total coloring of a graph G is a coloring of all elements of G, i.e., vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Let L(x) be a set of colors assigned to each element x of G. Then a list total coloring of G is a total coloring such that each element x receives a color contained in L(x). The list total coloring problem asks whether G has a list total coloring. In this paper, we first show that the list total coloring problem is NP-complete even for series-parallel graphs. We then give a sufficient condition for a series-parallel graph to have a list total coloring, that is, we prove a theorem that any series-parallel graph G has a list total coloring if |L(v)|min{5,Δ+1} for each vertex v and |L(e)|max{5,d(v)+1,d(w)+1} for each edge e=vw, where Δ is the maximum degree of G and d(v) and d(w) are the degrees of the ends v and w of e, respectively. The theorem implies that any series-parallel graph G has a total coloring with Δ+1 colors if Δ4. We finally present a linear-time algorithm to find a list total coloring of a given series-parallel graph G if G satisfies the sufficient condition.