Existence of Dλ-cycles and Dλ-paths

A cycle of C of a graph G is called a D_λ-cycle if every component of G ? V(C) has order less than λ. A D_λ-path is defined analogously. In particular, a D₁-cycle is a hamiltonian cycle and a D₁-path is a hamiltonian path. Necessary conditions and sufficient conditions are derived for graphs to have a D_λ-cycle or D_λ-path. The results are generalizations of theorems in hamiltonian graph theory. Extensions of notions such as vertex degree and adjacency of vertices to subgraphs of order greater than 1 arise in a natural way.     

The distribution of Brownian motion in Rn at a natural stopping time

For a measure μ on Rⁿ let ((B_t, P^μ) be Brownian motion in Rⁿ with initial distribution μ. Let D be an open subset of Rⁿ with exit time ζ ≡ inf {t > 0: B_t ? D}. In the case where D is a Green region with Green function G and μ is a measure in D such that G_μ is not identically infinite on any component of D, we have given necessary and sufficient conditions for a measure ν in D to be of the form ν(dx) = P^μ(B_T ? dx, T <ζ), where T is some natural stopping time for (B_t), and we have applied this characterization to show that a measure ν in D satisfies G_ν ? G_μ iff ν is of the form ν(dx) = P^α(B_T ? dx, T <ζ) + β(dx), where T is some natural stopping time for (B_t) and α and β are measures in D such that α + β = μ and β lives on a polar set. We have proved analogous results in the case where D = R² and μ is a finite measure on R² such that ∫ log⁺ ∥x∥ du(x) < ∞, and applied this to give a characterization of the stopping times T for Brownian motion in R² such that (log⁺ ∥B_T∧t∥)_0<t<∞ is P^μ-uniformly integrable.     

A necessary condition for the local solvability of the operator Pm2(x, D) + P2m − 1(x, D)

It is shown that a necessary condition for the local solvability of the operator P(x, D) = P_m²(x, D) + P_{2m ? 1}(x, D), where P_m(x, D) is an mth-order homogeneous differential operator of principal type with real coefficients, is that along any null-bicharacteristic strip of P_m(x, ξ) the imaginary part of the sub-principal symbol cannot have an odd-order zero where its real part does not vanish.     

On the spectrum of the Dirichlet Laplacian for horn-shaped regions in Rn with infinite volume

An inequality for trace (e^tΔD) is proven, where ?Δ_D is the Dirichlet Laplacian for horn-shaped regions D in Rⁿ. The results of Rozenbljum and Simon for the leading asymptotics for the growth of the number of eigenvalues of the two-dimensional Dirichlet Laplacian in the regions {$\text{(x, y):¦x¦}{}^{\mathrm{\mu }}\text{· ¦y¦ ? 1, μ > 0}$} are easily recovered. An example of a horn-shaped region in R² where that asymptotics is exponential is given.     

Order estimates for irreducible projections in L2 of a nilmanifold

Let π be an irreducible representation occurring in $\text{L}$²(Г?N), where N is a nilpotent Lie group and Γ is a discrete, cocompact subgroup. The projection onto the π-equivariant subspace is given by convolution against a distribution D_π. For certain π, we obtain an estimate on the order of D_π. The condition on π involves an extension of the “canonical objects” associated to elements of the Kirillov orbit of π; there does not appear to be an example in the literature where it is not satisfied.     

New discoveries of domination between traffic matrices

In this paper the definition of domination is generalized to the case that the elements of the traffic matrices may have negative values. It is proved that D₃ dominates D₃ + λ(D₂ ? D₁) for any λ ? 0 if D₁ dominates D₂. Let U(D) be the set of all the traffic matrices that are dominated by the traffic matrix D. It is shown that U(D_∞) and U(D_∈) are isomorphic. Besides, similar results are obtained on multi-commodity flow problems. Furthermore, the results are the generalized to integral flows.     

On simple modules of Hecke algebras of type Dn

This is a continuation of our previous work. We classify all the simple ?_q(D _n )-modules via an automorphismh defined on the set { λ | D^λ ≠ 0}. Whenf _n(q) ≠ 0, this yields a classification of all the simple ? ^q (D _n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ⁽¹⁾ = λ⁽²⁾,wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ?_q,1(B _n )- module D^λ remains irreducible on restriction to ?_q(D _n ).     

Polynomials with Dp as Galois group

A useful criterion characterizing a monic irreducible polynomial over $\text{Q}$ with Galois group D_p (the dihedral group of order 2p, p: prime) is given by making use of the geometry of D_p, i.e., D_p is the symmetry group of the regular p-gon. We derive explicit numerical examples of polynomials with dihedral Galois groups D₅ and D₇.     

A decomposition of R into two homeomorphic rigid parts

Let X be separable, completely metrizable, and dense in itself. We show that if X admits a triple (D₁, D₂, h) of two countable dense subsets D₁ and D₂ and a homeomorphism h: X?D₁ → X?D₂, satisfying some special properties, then there is a rigid subspace A of X such that A is homeomorphic to X?A = h[A]; for $\text{X =}\text{R}$, such atriple is shown to exist.     

Weak similarities of metric and semimetric spaces

Let (X,d _X ) and (Y,d _Y ) be semimetric spaces with distance sets D(X) and D(Y), respectively. A mapping F:?X→Y is a weak similarity if it is surjective and there exists a strictly increasing f:?D(Y)→D(X) such that d _X =f°d _Y °(F?F). It is shown that the weak similarities between geodesic spaces are usual similarities and every weak similarity F:?X→Y is an isometry if X and Y are ultrametric and compact with D(X)=D(Y). Some conditions under which the weak similarities are homeomorphisms or uniform equivalences are also found.     

Maximum modulus sets in pseudoconvex boundaries

LedD be a strictly pseudoconvex domain in ? ⁿ withC ^∞ boundary. We denote byA ^∞(D) the set of holomorphic functions inD that have aC ^∞ extension to \(\bar D\) . A closed subsetE of ?D is locally a maximum modulus set forA ^∞(D) if for everyp∈E there exists a neighborhoodU ofp andf∈A ^∞(D∩U) such that |f|=1 onE∩U and |f|<1 on \(\bar D \cap U\backslash E\) . A submanifoldM of ?D is an interpolation manifold ifT _p (M)?T _p ^c (?D) for everyp∈M, whereT _p ^c (?D) is the maximal complex subspace of the tangent spaceT _p (?D). We prove that a local maximum modulus set forA ^∞ (D) is locally contained in totally realn-dimensional submanifolds of ?D that admit a unique foliation by (n?1)-dimensional interpolation submanifolds. LetD =D ₁ x ... xD _r ? ? ⁿ whereD _i is a strictly pseudoconvex domain withC ^∞ boundary in ? ⁿ _i ,i=1,…,r. A submanifoldM of ?D ₁×…×?D _r verifies the cone condition if \(II_p (T_p (M)) \cap \bar C[Jn_1 (p),...,Jn_r (p)] = \{ 0\} \) for everyp∈M, wheren _i (p) is the outer normal toD _i atp, J is the complex structure of ? ⁿ , \(\bar C[Jn_1 (p),...,Jn_r (p)]\) is the closed positive cone of the real spaceV _p generated byJ _{n 1}(p),…,J _{n r}(p), and II _p is the orthogonal projection ofT _p (?D) onV _p . We prove that a closed subsetE of ?D ₁×…×?D _r which is locally a maximum modulus set forA ^∞(D) is locally contained inn-dimensional totally real submanifolds of ?D ₁×…×?D _r that admit a foliation by (n?1)-dimensional submanifolds such that each leaf verifies the cone condition at every point ofE. A characterization of the local peak subsets of ?D ₁×…×?D _r is also given.     

Limit of the spectra of the interior Neumann problems when a solid domain shrinks to a plane one

Let (Δ + λ) u = 0 in $\text{D}$c$\text{R}$^d, $\text{?u}\text{?N}\text{=0}$ on ?$\text{D}$. How do the eigenvalues λ_j behave when $\text{D}$ shrinks to a domain Δ ? R^{d ? 1} ? The answer depends not only on Δ but on the way $\text{D}$ shrinks to Δ. The limit of λ_j is found. Examples are given.     

Asymptotic properties of matrix differential operators

For given matrices A(s) and B(s) whose entries are polynomials in s, the validity of the following implication is investigated: ?_ylim_{t → ∞}A(D) y(t) = 0 ? lim_{t → ∞}B(D) y(t) = 0. Here D denotes the differentiation operator and y stands for a sufficiently smooth vector valued function. Necessary and sufficient conditions on A(s) and B(s) for this implication to be true are given. A similar result is obtained in connection with an implication of the form ?_yA(D) y(t) = 0, lim_{t → ∞}B(D) y(t) = 0, C(D) y(t) is bounded ? lim_{t → ∞}E(D) y(t) = 0.     

D1AD2 theorems for multidimensional matrices

A new technique for proving D₁AD₂ theorems is given. Also a generalization of D₁AD₂ theorems to multidimensional matrices is indicated.     

Large Deviations Principle for a Large Class of?One-Dimensional Markov Processes

We study the large deviations principle for one-dimensional, continuous, homogeneous, strong Markov processes that do not necessarily behave locally as a Wiener process. Any strong Markov process X _t in ? that is continuous with probability one, under some minimal regularity conditions, is governed by a generalized elliptic operator D _v D _u , where v and u are two strictly increasing functions, v is right-continuous and u is continuous. In this paper, we study large deviations principle for Markov processes whose infinitesimal generator is ??D _v D _u where 0<???1. This result generalizes the classical large deviations results for a large class of one-dimensional ??classical?? stochastic processes. Moreover, we consider reaction-diffusion equations governed by a generalized operator D _v D _u . We apply our results to the problem of wavefront propagation for these type of reaction-diffusion equations.     

Lipschitz equivalence of fractal sets in ?

Let T(q,D) be a self-similar (fractal) set generated by $ \left\{ {fi(x) = \frac{1} {q}(x + d_i )} \right\}_{i = 1}^N $ where integer q > 1 and D = {d ₁, d ₂, ??, d _N } ? ?. To show the Lipschitz equivalence of T(q,D) and a dust-like T(q,C), one general restriction is D ? ? by Peres et al. [Israel J Math, 2000, 117: 353?C379]. In this paper, we obtain several sufficient criterions for the Lipschitz equivalence of two self-similar sets by using dust-like graph-directed iterating function systems and combinatorial techniques. Several examples are given to illustrate our theory.     

On global regular solutions of third order partial differential equations

Diffusion in the presence of high-diffusivity paths is an important issue of current technology. In metals high-diffusivity paths are identified with dislocations, grain boundaries, free surfaces and internal microcracks. Diffusion in a media with two distinct families of diffusion paths is modelled by two coupled linear partial differential equations of parabolic type with diffusivities D₁ and D₂. Physically the situation D₂ ? D₁ is of some considerable interest and previously established results, for D₂ non-zero, for the solution of boundary value problems, are not applicable to the idealized theory characterized by D₂ vanishing. An integral equation, which arises in the solution of boundary value problems for this idealized theory, is formally solved.     

A minimum principle for superharmonic functions subject to interface conditions

Let D be a bounded domain in R² with smooth boundary. Let B₁, …, B_m be non-intersecting smooth Jordan curves contained in D, and let D′ denote the complement of ∪_{i ? 1}^mB_i respect to D. Suppose that $\text{u ? C}{}^2\text{(D′) ∩ C(}\text{D}\text{?}$) and Δu ? 0 in D′ (where Δ is the Laplacian), while across each “interface” B_i, i = 1,…, m, there is “continuity of flux” (as suggested by the theory of heat conduction). It is proved here that the presence of the interfaces does not alter the conclusions of the classical minimum principle (for Δu ? 0 in D). The result is extended in several regards. Also it is applied to an elliptic free boundary problem and to the proof of uniqueness for steady-state heat conduction in a composite medium. Finally this minimum principle (which assumes “continuity of flux”) is compared with one due to Collatz and Werner which employs an alternative interface condition.     

Interval hypergraphs and D-interval hypergraphs

A hypergraph $\text{H = (V,}\text{E}\text{)}$ is called an interval hypergraph if there exists a one-to-one function ? mapping the elements of V to points on the real line such that for each edge E, there is an interval I, containing the images of all elements of E, but not the images of any elements not in E₁. The difference hypergraph D(H) determined by H is formed by adding to $\text{E}$ all nonempty sets of the form E₁ ? E₁, where E₁ and E₁ are edges of HH is said to be a D-interval hypergraph if D(H) is an interval hypergraph. A forbidden subhypergraph characterization of D-interval hypergraphs is given. By relating D-interval hypergraphs to dimension theory for posets, we determine all 3-irreducible posets of length one.