Further results on the existence of nested orthogonal arrays

Nested orthogonal arrays provide an option for designing an experimental setup consisting of two experiments, the expensive one of higher accuracy being nested in a larger and relatively less expensive one of lower accuracy. We denote by OA_(λ, μ)(t, k, (v, w)) (or OA(t, k, (v, w)) if λ = μ = 1) a (symmetric) orthogonal array OA _λ (t, k, v) with a nested OA _μ (t, k, w) (as a subarray). It is proved in this article that an OA(t, t + 1,(v, w)) exists if and only if v ≥ 2w for any positive integers v, w and any strength t ≥ 2. Some constructions of OA_(λ, μ)(t, k, (v, w))′s with λ ≠ μ and k ? t > 1 are also presented.     

Limit Property for Regular and Weak Generalized Convolution

We denote by ? \((\mathcal{P_{+}})\) the set of all probability measures defined on the Borel subsets of the real line (the positive half-line [0,∞)). K. Urbanik defined the generalized convolution as a commutative and associative ?₊-valued binary operation ? on ? ₊ ² which is continuous in each variable separately. This convolution is distributive with respect to convex combinations and scale changes T _a (a>0) with δ ₀ as the unit element. The key axiom of a generalized convolution is the following: there exist norming constants c _n and a measure ν other than δ ₀ such that \(T_{c_{n}}\delta_{1}^{\bullet n}\to\nu\).In Sect. 2 we discuss basic properties of the generalized convolution on ? which hold for the convolutions without the key axiom. This rather technical discussion is important for the weak generalized convolution where the key axiom is not a natural assumption. In Sect. 4 we show that if the weak generalized convolution defined by a weakly stable measure μ has this property, then μ is a factor of strictly stable distribution.     

Semigroups of Distributions with Linear Jacobi Parameters

We show that a convolution semigroup {?? _t } of measures has Jacobi parameters polynomial in the convolution parameter t if and only if the measures come from the Meixner class. Moreover, we prove the parallel result, in a more explicit way, for the free convolution and the free Meixner class. We then construct the class of measures satisfying the same property for the two-state free convolution. This class of two-state free convolution semigroups has not been considered explicitly before. We show that it also has Meixner-type properties. Specifically, it contains the analogs of the normal, Poisson, and binomial distributions, has a Laha?CLukacs-type characterization, and is related to the q=0 case of quadratic harnesses.     

Finite determinacy and Whitney equisingularity of map germs from {\mathbb{C}^n} to {\mathbb{C}^{2n-1}}

We show that a holomorphic map germ ${f : (\mathbb{C}^n,0)\to(\mathbb{C}^{2n-1},0)}$ is finitely determined if and only if the double point scheme D(f) is a reduced curve. If n ≥ 3, we have that μ(D ²(f)) = 2μ(D ²(f)/S ₂)+C(f)?1, where D ²(f) is the lifting of the double point curve in ${(\mathbb{C}^n\times \mathbb{C}^n,0)}$ , μ(X) denotes the Milnor number of X and C(f) is the number of cross-caps that appear in a stable deformation of f. Moreover, we consider an unfolding F(t, x) = (t, f _t (x)) of f and show that if F is μ-constant, then it is excellent in the sense of Gaffney. Finally, we find a minimal set of invariants whose constancy in the family f _t is equivalent to the Whitney equisingularity of F. We also give an example of an unfolding which is topologically trivial, but it is not Whitney equisingular.     

Determination of the Unknown Cauchy Data in a Linear Parabolic Problem from the Measured Data at the Final Time

The problem of determining the initial value u(x, 0) = μ ₀(x) in the parabolic equation u _t = (k(x)u _x (x, t)) _x + F(x, t) from the final overdetermination μ _T (x) = u(x, T) is formulated. It is proved that the Fréchet derivative of the cost functional ${{J(\mu_0) = \|\mu_T(x) - u(x, T)\|_0^2}}$ can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. The existence of a quasisolution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method.     

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.     

On the perturbation of the observability equation in linear control systems

The aim of this paper is to investigate stability and sensitivity of the observability variable in linear control systems, (LCS) for short. We first present two results of Hölder continuity in the abstract framework of the ordinary differential equation initial-value problem x′(t) = f(t,x(t)),x(t ₀) = x ₀. Afterwards, we apply our results to automatic systems, providing henceforth the sharpest bounds for the parametric input-output relation in LCS.     

Rectangular <Emphasis Type="Italic">R</Emphasis>-Transform as the Limit of Rectangular Spherical Integrals

In this paper, we connect rectangular free probability theory and spherical integrals. We prove the analogue, for rectangular or square non-Hermitian matrices, of a result that Guionnet and Maïda proved for Hermitian matrices in (J. Funct. Anal. 222(2):435–490, 2005). More specifically, we study the limit, as n and m tend to infinity, of \(\frac{1}{n}\log\mathbb{E}\{\exp[\sqrt{nm}\theta X_{n}]\}\), where θ∈?, X _n is the real part of an entry of U _n M _n V _m and M _n   is a certain n×m deterministic matrix and U _n and V _m are independent Haar-distributed orthogonal or unitary matrices with respective sizes n×n and m×m. We prove that when the singular law of M _n converges to a probability measure μ, for θ small enough, this limit actually exists and can be expressed with the rectangular R-transform of μ. This gives an interpretation of this transform, which linearizes the rectangular free convolution, as the limit of a sequence of log-Laplace transforms.     

An Efficient Algorithm to Identify DNA Motifs

We consider the problem of identifying motifs that abstracts the task of finding short conserved sites in genomic DNA. The planted (l, d)-motif problem, PMP, is the mathematical abstraction of this problem, which consists of finding a substring of length l that occurs in each s _i in a set of input sequences S = {s ₁, s ₂, . . . ,s _t } with at most d substitutions. Our propose algorithm combines the voting algorithm and pattern matching algorithm to find exact motifs. The combined algorithm is achieved by running the voting algorithm on t′ sequences, t′ < t. After that we use the pattern matching on the output of the voting algorithm and the reminder sequences, t ? t′. Two values of t′ are calculated. The first value of t′ makes the running time of our proposed algorithm less than the running time of voting algorithm. The second value of t′ makes the running time of our proposed algorithm is minimal. We show that our proposed algorithm is faster than the voting algorithm by testing both algorithms on simulated data from (9, d ≤ 2) to (19, d ≤ 7). Finally, we test the performance of the combined algorithm on realistic biological data.     

Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay

In this paper, we consider initial-boundary value problem of viscoelastic wave equation with a delay term in the interior feedback. Namely, we study the following equation $$u_{tt}(x, t) - \Delta {u}(x, t) + \int_{0}^{t} g(t - s)\,\Delta {u}(x, s){\rm d}s + \mu_{1} u_{t}(x, t) + \mu_{2} u_{t}(x, t -\tau) = 0$$ together with initial-boundary conditions of Dirichlet type in Ω × (0, + ∞) and prove that for arbitrary real numbers  μ ₁ and μ ₂, the above-mentioned problem has a unique global solution under suitable assumptions on the kernel g. This improve the results of the previous literature such as Nicaise and Pignotti (SIAM J. Control Optim 45:1561–1585, 2006) and Kirane and Said-Houari (Z. Angew. Math. Phys. 62:1065–1082, 2011) by removing the restriction imposed on μ ₁ and μ ₂. Furthermore, we also get an exponential decay results for the energy of the concerned problem in the case μ ₁ = 0 which solves an open problem proposed by Kirane and Said-Houari (Z. Angew. Math. Phys. 62:1065–1082, 2011).     

Small Data Scattering for the One-Dimensional Nonlinear Dirac Equation with Power Nonlinearity

We study scattering problems for the one-dimensional nonlinear Dirac equation (?_t + α?_x + iβ)Φ = λ|Φ|^p?1Φ. We prove that if p > 3 (resp. p > 3 + 1/6), then the wave operator (resp. the scattering operator) is well-defined on some 0-neighborhood of a weighted Sobolev space. In order to prove these results, we use linear operators D(t)xD(?t) and t?_x + x?_t ? α/2, where {D(t)}_t∈? is the free Dirac evolution group. For the reader's convenience, in an appendix we list and prove fundamental properties of D(t)xD(?t) and t?_x + x?_t ? α/2.     

Measures as convolution operators on H1 and Lip α

For any measure μ, let T_μ denote the operator defined as convolution by μ. The spectral theory of the operators T_μ is studied. We focus our attention on certain infinite Bernoulli convolutions of the form

, where the sequence {t_n} is subject to certain arithmetic constraints. It is shown that the corresponding convolution operators do not have “natural” spectra on the spaces H¹ and Lip α.     

On Unicyclic Graphs with Uniquely Restricted Maximum Matchings

A graph is called unicyclic if it owns only one cycle. A matching M is called uniquely restricted in a graph G if it is the unique perfect matching of the subgraph induced by the vertices that M saturates. Clearly, μ _r (G) ≤ μ(G), where μ _r (G) denotes the size of a maximum uniquely restricted matching, while μ(G) equals the matching number of G. In this paper we study unicyclic bipartite graphs enjoying μ _r (G) = μ(G). In particular, we characterize unicyclic bipartite graphs having only uniquely restricted maximum matchings. Finally, we present some polynomial time algorithms recognizing unicyclic bipartite graphs with (only) uniquely restricted maximum matchings.     

A result on quasi-periodic solutions of a nonlinear beam equation with a quasi-periodic forcing term

In this paper, a quasi-periodically forced nonlinear beam equation \({u_{tt}+u_{xxxx}+\mu u+\varepsilon\phi(t)h(u)=0}\) with hinged boundary conditions is considered, where μ > 0, \({\varepsilon}\) is a small positive parameter, \({\phi}\) is a real analytic quasi-periodic function in t with a frequency vector ω = (ω ₁,ω ₂ . . . , ω _m ), and the nonlinearity h is a real analytic odd function of the form \({h(u)=\eta_1u+\eta_{2\bar{r}+1}u^{2\bar{r}+1}+\sum_{k\geq \bar{r}+1}\eta_{2k+1}u^{2k+1},\eta_1,\eta_{2\bar{r}+1} \neq0, \bar{r} \in {\mathbb {N}}.}\) The above equation admits a quasi-periodic solution.     

Properties of derivations on some convolution algebras

For all convolution algebras L ¹[0, 1); L _loc ¹ and A(ω) = ∩ _n L ¹(ω _n ), the derivations are of the form D _μ f = Xf * μ for suitable measures μ, where (Xf)(t) = tf(t). We describe the (weakly) compact as well as the (weakly) Montel derivations on these algebras in terms of properties of the measure μ. Moreover, for all these algebras we show that the extension of D _μ to a natural dual space is weak-star continuous.     

The optimal solution set of the interval linear programming problems

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ _j , μ _ij , μ _i  ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ _ij  = μ _i  = λ _j  = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ _j , μ _ij and μ _i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.     

On HNBUE class after specific age

In this article we introduce new classes of life distributions namely harmonic new better (worse) than used in expectation after specific age t₀ HNBUE¹t₀ (HNWUE¹t₀). The closure properties under various reliability operations such as convolution, mixture, mixing and the homogeneous Poisson shock model of these classes are studied. Furthermore, nonparametric test is proposed to test exponentiality vs. the HNBUE¹t₀ class. The critical values and the powers of this test are calculated to assess the performance of the test. It is shown that the proposed test have high efficiencies for LFR and Weibull distributions. Sets of real data are used as examples to elucidate the use of the proposed test for practical problems.     

Viscosity Approximation Methods for Multivalued Mappings in Banach Spaces

Let K be a nonempty closed and convex subset of a real reflexive Banach space X that has weakly sequentially continuous duality mapping J. Let T: K → K be a multivalued non-expansive non-self-mapping satisfying the weakly inwardness condition as well as the condition T(y) = {y} for any y ∈ F(T) and such that for a contraction f: K → K and any t ∈ (0, 1), there exists x _t  ∈ K satisfying x _t  ∈ tf(x _t ) + (1 ? t)Tx _t . Then it is proved that {x _t } ? K converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, the convergence of two explicit methods are also investigated.     

Calculating the numbers of representations and the Garsia entropy in linear numeration systems

Given a ≥ b, let G ₀ = 1, G ₁ = a + 1, and G _n+2 = aG _n+1 + bG _n for n ≥ 0. For each choice of a and b, we have a linear recurrence that defines a numeration system. Every positive integer n may be written as the sum of the G _n , with alphabet A = {0,1, . . . , a}, in one or more different ways. Let R _(a,b)(n) be the function that counts the number of distinct representations of an integer as a sum of the G _n . We extend results of Berstel, Kocábová, Masáková, and Pelantová, and Edson and Zamboni and give two distinct methods for calculating R _(a,b)(n). One formula involves products of 2 × 2 matrices and the other sums of binomial coefficients modulo 2. For the main result, we consider the limiting measure μ _β of a convergent infinite convolution of measures (Bernoulli convolutions), where β is the dominating root of the characteristic equation of the recurrence above. We study the Garsia entropy of these measures and calculate explicitly the limiting entropy associated with μ _β . This result extends those of Alexander and Zagier, and Grabner, Kirschenhofer, and Tichy. We then see that all these results can be generalized further to confluent numeration systems.     

Commutation properties of automorphism groups and mappings of von Neumann algebras

We prove a commutation theorem for point ultraweakly continuous oneparameter groups of automorphisms of von Neumann algebras. If α_t, is such a group in Aut($\text{R}$) for a von Neumann algebra $\text{R}$, we show the equivalence of the following three conditions on an ultraweakly continuous linear transformation μ: $\text{R}$ → $\text{R}$: (a) μ commutes weakly with the infinitesimal generator for α_t; (b) μ ° α_t = α_t ° μ, t ∈ R; and (c) μ leaves invariant each of the spectral subspaces associated with α_t. A simple condition which is applicable when μ is an automorphism is pointed out.