Further results on the covering radius of small codes

The minimum number of codewords in a code with t ternary and b binary coordinates and covering radius R is denoted by K(t,b,R). In the paper, necessary and sufficient conditions for K(t,b,R)=M are given for M=6 and 7 by proving that there exist exactly three families of optimal codes with six codewords and two families of optimal codes with seven codewords. The cases M?5 were settled in an earlier study by the same authors. For binary codes, it is proved that K(0,2b+4,b)?9 for b?1. For ternary codes, it is shown that K(3t+2,0,2t)=9 for t?2. New upper bounds obtained include K(3t+4,0,2t)?36 for t?2. Thus, we have K(13,0,6)?36 (instead of 45, the previous best known upper bound).     

Small time two-sided LIL behavior for Lévy processes at zero

We wish to characterize when a Lévy process X _t crosses boundaries b(t), in a two-sided sense, for small times t, where b(t) satisfies very mild conditions. An integral test is furnished for computing the value of sup _t→0|X _t |/b(t) = c. In some cases, we also specify a function b(t) in terms of the Lévy triplet, such that sup _t→0 |X _t |/b(t) = 1.     

Asymptotic behavior of solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0

The author discusses the asymptotic behavior of the solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0 (1) where x(t) is an n-dimensional column vector and A, B are n × n matrices with complex constant entries. He obtains the following results for the case 0 < λ < 1: (i) If B is diagonalizable with eigenvalues b_i such that Re b_i < 0 for all i, then there is a constant α such that every solution of (1) is O(t^α) as t → ∞. (ii) If B is diagonalizable with eigenvalues b_i such that 0 < Re b₁ ? Re b₂ ? ··· ? Re b_n and λ times Re b_n < Re b₁, then every solution of (1) is O(e^bn^t) as t → ∞. For the case λ>1, he has the following results: (i) If B is diagonalizable with eigenvalues b_i such that Re b_i>0 for all i, then there is a constant α such that no solution x(t) of (1), except the identically zero solution, is 0(t^α) as t → ∞. (ii) If B is diagonalizable with eigenvalues b_i such that Re b₁ ? Re b₂ ? ··· ? Re b_n < 0 and λ Re b_n < Re b₁, then no solution x(t) of (1), except the identically zero solution, is 0(e^b1^t) as t → ∞.     

Extended Jacobson density theorem for rings with derivations and automorphisms

A deterministic version of the Itô calculus is presented. We consider a modelY _t=H(N _t ,t) with a deterministic Brownian N _t and an unknown functionH. We predictY _c from the observation {Y _t;t ∈ [a, b]}, wherea. We prove that there exists an estimatorY _t based on the observation such thatE[(? _t?Y _c)²]=O((c?b)²) asc ↓b.     

Some new spaces of double sequences

In this study, we define the double sequence spaces BS, BS(t), CSp, CSbp, CSr and BV, and also examine some properties of those sequence spaces. Furthermore, we show that these sequence spaces are complete paranormed or normed spaces under some certain conditions. We determine the α-duals of the spaces BS, BV, CSbp and the β(?)-duals of the spaces CSbp and CSr of double series. Finally, we give the conditions which characterize the class of four-dimensional matrix mappings defined on the spaces CSbp, CSr and CSp of double series.     

New Bounds for Perfect Hashing via Information Theory

A set of sequences of length t from a b-element alphabet is called k-separated if for every k-tuple of the sequences there exists a coordinate in which they all differ. The problem of finding, for fixed t, b, and k, the largest size N(t, b, k) of a k-separated set of sequences is equivalent to finding the minimum size of a (b, k)-family of perfect hash functions for a set of a given size. We shall improve the bounds for N(t, b, k) obtained by Fredman and Komlós [1].Körner [2] has shown that the proof in [1] can be reduced to an application of the sub-additivity of graph entropy [3]. He also pointed out that this sub-additivity yields a method to prove non-existence bounds for graph covering problems. Our new non-existence bound is based on an extension of graph entropy to hypergraphs.     

Perturbing non-real eigenvalues of non-negative real matrices

Let σ=(ρ,b+ic,b-ic,λ₄,…,λ_n) be the spectrum of an entry non-negative matrix and t?0. Laffey [T. J. Laffey, Perturbing non-real eigenvalues of nonnegative real matrices, Electron. J. Linear Algebra 12 (2005) 73-76] has shown that σ=(ρ+2t,b-t+ic,b-t-ic,λ₄,…,λ_n) is also the spectrum of some nonnegative matrix. Laffey (2005) has used a rank one perturbation for small t and then used a compactness argument to extend the result to all nonnegative t. In this paper, a rank two perturbation is used to deduce an explicit and constructive proof for all t?0.     

Graded Identities of Some Simple Lie Superalgebras

We study ?₂-graded identities of Lie superalgebras of the type b(t), t?≥?2, over a field of characteristic zero. Our main result is that the n-th codimension is strictly less than \((\dim b(t))^n\) asymptotically. As a consequence we obtain an upper bound for ordinary (non-graded) PI-exponent for each simple Lie superalgebra b(t), t?≥?3.     

A Survey on the Pseudo-process Driven by the High-order Heat-type Equation \boldsymbol{\partial/\partial t=\pm\partial^N\!/\partial x^N} Concerning the Hitting and Sojourn Times

Fix an integer N?>?2 and let X?=?(X(t)) _t????0 be the pseudo-process driven by the high-order heat-type equation $\partial/\partial t=\pm\partial^N\!/\partial x^N$ . The denomination ??pseudo-process?? means that X is related to a signed measure (which is not a probability measure) with total mass equal to one. In this survey, we present several explicit results and discuss some problems concerning the pseudo-distributions of various functionals of the pseudo-process X: the first or last overshooting times of a single barrier {a} or a double barrier {a, b} by X; the sojourn times of X in the intervals [a,?+????) and [a, b] up to a fixed time; the maximum or minimum of X up to a fixed time.     

A physical basis for Krein's prediction formula

A prediction problem of the following variety is considered. A stationary random process w(·) of known spectrum is observed over |t|?a. Using these observed values, w(b) is to be predicted for some b with |b|>a. We present a physical interpretation of a solution to this problem due to Krein, which used the theory of inverse Sturm-Liouville problems. Our physical model involves a nonuniform lossless transmission line excited at one end by white noise. The signal at the other end is the process w(t), and the prediction is found by calculating as intermediate quantities the voltage and current stored on the line at t=0. These quantities are spatially uncorrelated and provide a spatial representation at t=0 of the innovations of w(t) over |t|?a.     

On the maximal energy tree with two maximum degree vertices

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacent matrix.For Δ?3 and t?3, denote by T_a(Δ,t) (or simply T_a) the tree formed from a path P_t on t vertices by attaching Δ-1P₂’s on each end of the path P_t, and T_b(Δ,t) (or simply T_b) the tree formed from P_t+2 by attaching Δ-1P₂’s on an end of the P_t+2 and Δ-2P₂’s on the vertex next to the end.In Li et al.(2009) [16] proved that among trees of order n with two vertices of maximum degree Δ, the maximal energy tree is either the graph T_a or the graph T_b, where t=n+4-4Δ?3.However, they could not determine which one of T_a and T_b is the maximal energy tree.This is because the quasi-order method is invalid for comparing their energies.In this paper, we use a new method to determine the maximal energy tree.It turns out that things are more complicated.We prove that the maximal energy tree is T_b for Δ?7 and any t?3, while the maximal energy tree is T_a for Δ=3 and any t?3.Moreover, for Δ=4, the maximal energy tree is T_a for all t?3 but one exception that t=4, for which T_b is the maximal energy tree.For Δ=5, the maximal energy tree is T_b for all t?3 but 44 exceptions that t is both odd and 3?t?89, for which T_a is the maximal energy tree.For Δ=6, the maximal energy tree is T_b for all t?3 but three exceptions that t=3,5,7, for which T_a is the maximal energy tree.One can see that for most cases of Δ, T_b is the maximal energy tree,Δ=5 is a turning point, and Δ=3 and 4 are exceptional cases, which means that for all chemical trees (whose maximum degrees are at most 4) with two vertices of maximum degree at least 3, T_a has maximal energy, with only one exception T_a(4,4).     

Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation

This paper discusses a randomized non-autonomous logistic equation , where B(t) is a 1-dimensional standard Brownian motion. In [D.Q. Jiang, N.Z. Shi, A note on non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl. 303 (2005) 164-172], the authors show that E[1/N(t)] has a unique positive T-periodic solution E[1/Np(t)] provided a(t), b(t) and α(t) are continuous T-periodic functions, a(t)>0, b(t)>0 and . We show that this equation is stochastically permanent and the solution Np(t) is globally attractive provided a(t), b(t) and α(t) are continuous T-periodic functions, a(t)>0, b(t)>0 and mint∈[0,T]a(t)>maxt∈[0,T]α²(t). By the way, the similar results of a generalized non-autonomous logistic equation with random perturbation are yielded.     

Optimal control of capital injections by reinsurance in a diffusion approximation

In this paper we consider a diffusion approximation to a classical risk process, where the claims are reinsured by some reinsurance with deductible b ∈ [0,b?], where b = b? means “no reinsurance” and b = 0 means “full reinsurance”. The cedent can choose an adapted reinsurance strategy (b _t ) _{t ≥0}, i.?e. the deductible can be changed continuously. In addition, the cedent has to inject fresh capital in order to keep the surplus positive. The problem is to minimise the expected discounted cost over all admissible reinsurance strategies. We find an explicit expression for the value function and the optimal strategy using the Hamilton–Jacobi–Bellman approach. Some examples illustrate the method.     

Positive solutions for the one-dimensional p-Laplacian

In this paper, the criterion for the existence of at least one positive solution of the one-dimensional p-Laplacian (b(t)Φ(u)′ + c(t)f(u) = 0, are obtained, where Φ(u) = |u|^p−1u, p > 0 is a constant, and b(t) > 0 for t > 0. The method used in this paper is shooting method.     

Mixed problems for the time-dependent telegraph equation: Continuous numerical solutions with a priori error bounds

This paper deals with the construction of continuous numerical solutions of mixed problems described by the time-dependent telegraph equation u_tt + c(t)u_t + b(t)u = a(t)u_xx, 0 < x < d, t > 0. Here a(t), b(t), and c(t) are positive functions with appropiate additional alternative hypotheses. First, using the separation of variables technique a theoretical series solution is obtained. Then, after truncation using one-step matrix methods and interpolating functions, a continuous numerical solution with a prefixed accuracy in a bounded subdomain is constructed.     

On the signless Laplacian spectral characterization of the line graphs of T-shape trees

A graph is determined by its signless Laplacian spectrum if no other nonisomorphic graph has the same signless Laplacian spectrum (simply G is DQS). Let T (a, b, c) denote the T-shape tree obtained by identifying the end vertices of three paths P _a+2, P _b+2 and P _c+2. We prove that its all line graphs L(T(a, b, c)) except L(T(t, t, 2t+1)) (t ? 1) are DQS, and determine the graphs which have the same signless Laplacian spectrum as L(T(t, t, 2t + 1)). Let µ₁(G) be the maximum signless Laplacian eigenvalue of the graph G. We give the limit of µ₁(L(T(a, b, c))), too.     

The fine triangle intersections for maximum kite packings

In this paper, the fine triangle intersection problem for a pair of maximum kite packings is investigated. Let Fin(v) = {(s,t) : a pair of maximum kite packings of order v intersecting in s blocks and s+t triangles}. Let Adm(v) = {(s, t) : s + t ≤ bv , s,t are non-negative integers}, where b v = v(v 1)/8 . It is established that Fin(v) = Adm(v)\{(bv-1, 0), (bv-1,1)} for any integer v ≡ 0, 1 (mod 8) and v ≥ 8; Fin(v) = Adm(v) for any integer v ≡ 2, 3, 4, 5, 6, 7 (mod 8) and v ≥ 4.     

Large time behavior of differential equations with drifted periodic coefficients modeling carbon storage in soil

This paper is concerned with the linear ODE in the form y′(t) = λρ(t)y(t) + b(t), λ < 0 which represents a simplified storage model of the carbon in the soil. In the first part, we show that, for a periodic function ρ(t), a linear drift in the coefficient b(t) involves a linear drift for the solution of this ODE. In the second part, we extend the previous results to a classical heat non-homogeneous equation. The connection with an analytic semi-group associated to the ODE equation is considered in the third part. Numerical examples are given.     

Metric and ultrametric spaces of resistances

Consider an electrical circuit, each edge e of which is an isotropic conductor with a monomial conductivity function . In this formula, y_e is the potential difference and current in e, while μ_e is the resistance of e; furthermore, r and s are two strictly positive real parameters common for all edges. In particular, the case r=s=1 corresponds to the standard Ohm’s law.In 1987, Gvishiani and Gurvich [A.D. Gvishiani, V.A. Gurvich, Metric and ultrametric spaces of resistances, in: Communications of the Moscow Mathematical Society, Russian Math. Surveys 42 (6 (258)) (1987) 235-236] proved that, for every two nodes a,b of the circuit, the effective resistance μ_a,b is well-defined and for every three nodes a,b,c the inequality holds. It obviously implies the standard triangle inequality μ_a,b≤μ_a,c+μ_c,b whenever s≥r. For the case s=r=1, these results were rediscovered in the 1990s. Now, after 23 years, I venture to reproduce the proof of the original result for the following reasons:

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It is more general than just the case r=s=1 and one can get several interesting metric and ultrametric spaces playing with parameters r and s. In particular, (i) the effective Ohm resistance, (ii) the length of a shortest path, (iii) the inverse width of a bottleneck path, and (iv) the inverse capacity (maximum flow per unit time) between any pair of terminals a and b provide four examples of the resistance distances μ_a,b that can be obtained from the above model by the following limit transitions: (i) r(t)=s(t)≡1, (ii) r(t)=s(t)→∞, (iii) r(t)≡1,s(t)→∞, and (iv) r(t)→0,s(t)≡1, as t→∞. In all four cases the limits μ_a,b=lim_t→∞μ_a,b(t) exist for all pairs a,b and the metric inequality μ_a,b≤μ_a,c+μ_c,b holds for all triplets a,b,c, since s(t)≥r(t) for any sufficiently large t. Moreover, the stronger ultrametric inequality μ_a,b≤max(μ_a,c,μ_c,b) holds for all triplets a,b,c in examples (iii) and (iv), since in these two cases s(t)/r(t)→∞, as t→∞.

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Communications of the Moscow Math. Soc. in Russ. Math. Surveys were (and still are) strictly limited to two pages; the present paper is much more detailed.Although a translation in English of the Russ. Math. Surveys is available, it is not free in the web and not that easy to find.

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The last but not least: priority.