On the enumeration of chains in regular chain-groups

Let N be a regular chain-group on E (see W. T. Tutte, Canad. J. Math.8 (1956), 13–28); for instance, N may be the group of integer flows or tensions of a directed graph with edge-set E). It is known that the number of proper Z_λ-chains of N (λ ∈ Z, λ ≥ 2) is given by a polynomial in λ, P(N, λ) (when N is the chain-group of integer tensions of the connected graph G, λP(N, λ) is the usual chromatic polynomial of G). We prove the formula: P(N, λ) = Σ_{[E′]∈O(N)⁺/～}Q(R_[E′](N), λ), where O(N)⁺ is the set of orientations of N with a proper positive chain, ～ is a simple equivalence relation on O(N)⁺ (sequence of reversals of positive primitive chains), and Q(R_[E′](N), λ) is the number of chains with values in [1, λ ? 1] in any reorientation of N associated to an element of [E′]. Moreover, each term Q(R_[E′](N), λ) is a polynomial in λ. As applications we obtain: P(N, 0) = (?1)^r(N)∥O(N)⁺/～∥; P(N, ?1) = (?1)^r(N)∥O(N)⁺∥ (a result first proved by Brylawski and Lucas); P(N, λ + 1) ≥ P(N, λ) for λ ≥ 2, λ ∈ Z. Our result can also be considered as a refinement of the following known fact: A regular chain-group N has a proper Z_λ-chain iff it has a proper chain in [?λ + 1, λ ? 1].     

The Distribution of the Bit Error Rate for an M Branch Antenna with N Interferers

Given an antenna with M branches, the bit error rate (BER) and mean squared error (MSE) for choosing the antenna weights (to approximately cancel M???1 interferers), are given by $$ \mathit{BER} \approx C \;\exp \left(-\alpha-\alpha Z_N\right) \mbox{ and } \mathit{MSE}=1/\left(1+Z_N\right), $$ where Z _N is the signal-to-interference plus noise ratio and C, α are some fixed parameters. So, obtaining the distribution of Z _N allows one to obtain the distribution of the MSE and to approximate that of the BER. Three cases are presented:

• the case of fixed powers for the interferers, say Q ₁, ..., Q _N , and for the wanted signal, say Q ₀;
• the case of fixed power for the wanted signal and random powers for the interferers;
• the case of random powers for both the wanted signal and the interferers.

We assume that Q ₀,...,Q _N are independent with different distributions. We show that to magnitude 1/N, the distribution of Z is just that of Q ₀ g _M /T, where g _M is a gamma random variable with mean M and T is the average of the total interferer power: $$ T = \mathbb{E} \ \left\{ \sum\limits_{j=1}^N Q_j\right\}. $$ We also show how to obtain an expansion in powers of 1/N for the distribution of $\mathit{TZ}$ about that of Q ₀ g _M . For example, to get the distribution of $\mathit{TZ}$ up to magnitude 1/N ², one requires only the means of Q ₁,...,Q _N and $Q_1^2,\ldots,Q_N^2$ and the distribution of Q ₀.     

An explicit model of default time with given survival probability

For a given filtered probability space (Ω,F,P), an F-adapted continuous increasing process Λ and a positive P-F local martingale N such that Λ₀=0 and N_te^−Λ_t≤1, we construct a probability measure Q^Z and a random time τ such that Q|_F∞=P|_F∞ and Q[τ>t|F_t]=Z_t. The probability Q^Z is linked with the well-known Cox model by an explicit density function. Various properties exist, which characterize Q^Z from others. Let G=(G_t)_t≥0 with G_t=F_t∨σ({τ≤s}:s≤t). We establish the (H^′)-property between the filtrations F and G, and we provide the enlargement of filtration formula.     

Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation

For a singularly perturbed parabolic convection-diffusion equation, the conditioning and stability of finite difference schemes on uniform meshes are analyzed. It is shown that a convergent standard monotone finite difference scheme on a uniform mesh is not ?-uniformly well conditioned or ?-uniformly stable to perturbations of the data of the grid problem (here, ? is a perturbation parameter, ? ∈ (0, 1]). An alternative finite difference scheme is proposed, namely, a scheme in which the discrete solution is decomposed into regular and singular components that solve grid subproblems considered on uniform meshes. It is shown that this solution decomposition scheme converges ?-uniformly in the maximum norm at an O(N ^?1lnN + N ₀ ^?1 ) rate, where N + 1 and N ₀ + 1 are the numbers of grid nodes in x and t, respectively. This scheme is ?-uniformly well conditioned and ?-uniformly stable to perturbations of the data of the grid problem. The condition number of the solution decomposition scheme is of order O(δ^?2lnδ^?1 + δ ₀ ^?1 ); i.e., up to a logarithmic factor, it is the same as that of a classical scheme on uniform meshes in the case of a regular problem. Here, δ = N ^?1lnN and δ₀ = N ₀ ^?1 are the accuracies of the discrete solution in x and t, respectively.     

Necessary conditions for ɛ-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers

A grid approximation of the boundary value problem for a singularly perturbed parabolic reaction-diffusion equation is considered in a domain with the boundaries moving along the axis x in the positive direction. For small values of the parameter ? (this is the coefficient of the higher order derivatives of the equation, ? ∈ (0, 1]), a moving boundary layer appears in a neighborhood of the left lateral boundary S ₁ ^L . In the case of stationary boundary layers, the classical finite difference schemes on piece-wise uniform grids condensing in the layers converge ?-uniformly at a rate of O(N ^?1lnN + N ₀), where N and N ₀ define the number of mesh points in x and t. For the problem examined in this paper, the classical finite difference schemes based on uniform grids converge only under the condition N ^?1 + N ₀ ^?1 ? ?. It turns out that, in the class of difference schemes on rectangular grids that are condensed in a neighborhood of S ₁ ^L with respect to x and t, the convergence under the condition N ^?1 + N ₀ ^?1 ≤ ?^1/2 cannot be achieved. Examination of widths that are similar to Kolmogorov’s widths makes it possible to establish necessary and sufficient conditions for the ?-uniform convergence of approximations of the solution to the boundary value problem. These conditions are used to design a scheme that converges ?-uniformly at a rate of O(N ^?1lnN + N ₀).     

Trace au bord latéral des solutions positives d'équations paraboliques non linéaires

Let us consider the equation (PE): in Q_T B x (0, T), where B is the open unit ball in R^N. We show that every positive solution u possesses a uniquely defined trace on ∂₁Q_T = ∂B x (0, T), given by a positive, regular Borel measure (not necessarily a Radon measure). If, there exists a one-to-one correspondence between the set of measures as above and the set of positive solutions of (PE) vanishing for t = 0.     

A generalization of N-matrices

Ky Fan defines an N-matrix to be a matrix of the form A = tI ? B, B ? 0, λ < t < ?(B), where ?(B) is the spectral radius of B and λ is the maximum of the spectral radii of all principal submatrices of B of order n ? 1. In this paper, we define the closure (N₀-matrices) of N-matrices by letting λ ? t. It is shown that if A ∈ Z and A^-1 < 0, then A ∈ N. Certain inequalities of N-matrices are shown to hold for N₀-matrices, and a method for constructing an N-matrix from an M-matrix is given.     

The Evolution of Random Subgraphs of the Cube

be a random Qⁿ”-process, that is let Q₀ be the empty spanning subgraph of the cube Qⁿ and, for 1 ? t ? M = nN/2 = n2^n?1, let the graph Q_t be obtained from Q_t?1 by the random addition of an edge of Q_n not present in Q_t?1. When t is about N/2, a typical Q_t undergoes a certain “phase transition'': the component structure changes in a sudden and surprising way. Let t = (1 + ?) N/2 where ? is independent of n. Then all the components of a typical Q_t have o(N) vertices if ? < 0, while if ? > 0 then, as proved by Ajtai, Komlós, and Szemerédi, a typical Q_t has a “giant” component with at least α(?)N vertices, where α(?) > 0. In this note we give essentially best possible results concerning the emergence of this giant component close to the time of phase transition. Our results imply that if η > 0 is fixed and t ? (1 ? n^?η) N/2, then all components of a typical Q_t have at most n^β(η) vertices, where β(η) > 0. More importantly, if 60(log n)³/n ? ? = ?n = o(1), then the largest component of a typical Q_t has about 2?N vertices, while the second largest component has order O(n?^?2). Loosely put, the evolution of a typical Qⁿ process is such that shortly after time N/2 the appearance of each new edge results in the giant component acquiring 4 new vertices.     

Inverse problem for N × N hyperbolic systems on the plane and the N-wave interactions

We study regiorously the solvability of the direct and inverse problems associated with Ψ_x–JΨ_y = QΨ,(x,y) ∈ ?², where (i) Ψ is an N × N-matrix-valued function on ?² (N ≦ 2), (ii) J is a constant, real, diagonal N × N matrix with entries, J₁ > J₂ > …? > J_N and (iii) Q is off-diagonal with rapidly decreasing (Schwartz) component functions. In particular we show that the direct problem is always solvable and give a small norm condition for the solvability of the inverse problem. In the particular case that Q is skew Hermitian the inverse problem is solvable without the small norm assumption. Furthermore we show how these results can be used to solve certain Cauchy problems for the associated nonlinear evolution equations. For concreteness we consider the N-wave interactions and show that if a certain norm of Q(x, y, 0) is smallor if Q(x, y, 0) is skew Hermitian the N-wave interations equation has a unique global solution.     

Finite difference schemes for the singularly perturbed reaction-diffusion equation in the case of spherical symmetry

The boundary value problem for the singularly perturbed reaction-diffusion parabolic equation in a ball in the case of spherical symmetry is considered. The derivatives with respect to the radial variable appearing in the equation are written in divergent form. The third kind boundary condition, which admits the Dirichlet and Neumann conditions, is specified on the boundary of the domain. The Laplace operator in the differential equation involves a perturbation parameter ?², where ? takes arbitrary values in the half-open interval (0, 1]. When ? → 0, the solution of such a problem has a parabolic boundary layer in a neighborhood of the boundary. Using the integro-interpolational method and the condensing grid technique, conservative finite difference schemes on flux grids are constructed that converge ?-uniformly at a rate of O(N ^?2ln² N + N ₀ ^?1 ), where N + 1 and N ₀ + 1 are the numbers of the mesh points in the radial and time variables, respectively.     

Viscosity analysis on the Boltzmann equation

This paper is devoted to investigating the asymptotic properties of the renormalized solution to the viscosity equation tfε + v ·▽xfε = Q (fε,fε ) + εΔvfε as ε→ 0+ . We deduce that the renormalized solution of the viscosity equation approaches to the one of the Boltzmann equation in L1 ((0 , T ) × RN × RN ). The proof is based on compactness analysis and velocity averaging theory.     

Integral Kirillov model

We prove that an irreducible cuspidal Q̄_ℓ-representation of GL(n, Q_p) with a central character with values in Z̄_ℓ^* has a unique Z̄_ℓ-integral structure, given by the Kirillov Z̄_ℓ-representation.     

Locating peaks of a Schrödinger equation with sign-changing nonlinearity

In this paper, we study the concentration phenomenon of a positive ground state solution of a nonlinear Schrödinger equation on R^N. The coefficient of the nonlinearity of the equation changes sign. We prove that the solution has a maximum point at x₀∈Ω⁺={x∈R^N:Q(x)>0} where the energy attains its minimum.     

Existence of log canonical flips and a special LMMP

Let (X/Z,B+A) be a Q-factorial dlt pair where B,A??0 are Q-divisors and K _X +B+A?? _Q 0/Z. We prove that any LMMP/Z on K _X +B with scaling of an ample/Z divisor terminates with a good log minimal model or a Mori fibre space. We show that a more general statement follows from the ACC for lc thresholds. An immediate corollary of these results is that log flips exist for log canonical pairs.     

Polynomial sums over automorphs of a positive definite binary quadratic form

Let P(X) be a homogeneous polynomial in X = (x, y), Q(X) a positive definite integral binary quadratic form, and G the group of integral automorphs of Q(X). Let A(m) = {N ∈ $\text{Z}$ × $\text{Z}$ : Q(N) = m}. It is shown that if Σ_N∈A(m)P(N) = 0 for each m = 1, 2, 3,… then Σ_U∈GP(UX) ≡ 0.     

Alexandrov’s inequality and conjectures on some Toeplitz matrices

We study determinant inequalities for certain Toeplitz-like matrices over C. For fixed n and N ? 1, let Q be the n × (n + N − 1) zero-one Toeplitz matrix with Q_ij = 1 for 0 ? j − i ? N − 1 and Q_ij = 0 otherwise. We prove that det(QQ^∗) is the minimum of det(RR^∗) over all complex matrices R with the same dimensions as Q satisfying ∣R_ij∣ ? 1 whenever Q_ij = 1 and R_ij = 0 otherwise. Although R has a Toeplitz-like band structure, it is not required to be actually Toeplitz. Our proof involves Alexandrov’s inequality for polarized determinants and its generalizations. This problem is motivated by Littlewood’s conjecture on the minimum 1-norm of N-term exponential sums on the unit circle. We also discuss polarized Bazin-Reiss-Picquet identities, some connections with k-tree enumeration, and analogous conjectured inequalities for the elementary symmetric functions of QQ^∗.     

Asymptotic Behaviour for a Nonlinear Schrödinger Equation in Domains with Moving Boundaries

We consider a nonlinear Schrödinger equation in a time-dependent domain Q _τ of ?² given by $$u_{\tau} - i u_{\varepsilon\varepsilon} + |u|^{2} u + \gamma v=0. $$ We prove the well-posedness of the above model and analyze the behaviour of the solution as t→+∞. We consider two situations: the conservative case (γ=0) and the dissipative case (γ>0). In both situations the existence of solutions are proved using the Galerkin method and the stabilization of solutions are obtained considering multiplier techniques.     

The use of solutions on embedded grids for the approximation of singularly perturbed parabolic convection-diffusion equations on adapted grids

The Dirichlet problem on a closed interval for a parabolic convection-diffusion equation is considered. The higher order derivative is multiplied by a parameter ? taking arbitrary values in the semi-open interval (0, 1]. For the boundary value problem, a finite difference scheme on a posteriori adapted grids is constructed. The classical approximations of the equation on uniform grids in the main domain are used; in some subdomains, these grids are subjected to refinement to improve the grid solution. The subdomains in which the grid should be refined are determined using the difference of the grid solutions of intermediate problems solved on embedded grids. Special schemes on a posteriori piecewise uniform grids are constructed that make it possible to obtain approximate solutions that converge almost ?-uniformly, i.e., with an error that weakly depends on the parameter ?: |u(x, t) ? z(x, t)| ≤ M[N ₁ ^?1 ln² N ₁ + N ₀ ^?1 lnN ₀ + ?^?1 N ₁ ^?K ln ^K?1 N ₁], (x, t) ε ? _h , where N ₁ + 1 and N ₀ + 1 are the numbers of grid points in x and t, respectively; K is the number of refinement iterations (with respect to x) in the adapted grid; and M = M(K). Outside the σ-neighborhood of the outflow part of the boundary (in a neighborhood of the boundary layer), the scheme converges ?-uniformly at a rate O(N ₁ ^?1 ln² N ₁ + N ₀ ^?1 lnN ₀), where σ ≤ MN ₁ ^{?K + 1} ln ^K?1 N ₁ for K ≥ 2.     

Similarity classes of3 × 3 matrices over a discrete valuation ring

Let Q be a complete discrete valuation ring. Let Π be a prime element in Q. Write P = ΠQ. For n = 1,2,…, letQ_n be the factor ring Q | Pⁿ. Let G = G1₃(Q_n. Denote by M?_n the G-module of 3 × 3 matrices over Q_n modulo scalar matrices. Canonical forms are found for every element in M?_n, and it is shown that there exist five sets of similarity classes. Some results about the general case of NxN matrices over Q also are proved.     

On solutions of three-dimensional systems describing the transition from an unstable equilibrium to a stable cycle

Given a three-dimensional dynamical system on the interval t ₀ < t < +∞, the transition from the neighborhood of an unstable equilibrium to a stable limit cycle is studied. In the neighbor-hood of the equilibrium, the system is reduced to a normal form. The matrix of the linearized system is assumed to have a complex eigenvalue λ = ? + iβ, with β ? ? > 0 and a real eigenvalue with δ < 0 with |δ| ? ?. On the arbitrary interval [t ₀, +∞), an approximate solution is sought as a polynomial P _N (?) in powers of the small parameter with coefficients from Hölder function spaces. It is proved that there exist ? _N and C _N depending on the initial data such that, for 0 < ? < ? _N , the difference between the exact and approximate solutions does not exceed C _N ? ^N+1.