A generalization of the Lee distance and error correcting codes

In this contribution, we will define an l-dimensional Lee distance which is a generalization of the Lee distance defined only over a prime field, and we will construct 2-error correcting codes for this distance. Our l-dimensional Lee distance can be defined not only over a prime field but also over any finite field. The ordinary Lee distance is just the one-dimensional Lee distance. Also the Mannheim or modular distances introduced by Huber are special cases of our distance.     

l-trace k-Sperner families of sets

A family of sets F⊆X² is defined to be l-trace k-Sperner if for any subset Y of X with size l the trace of F on Y (the restriction of F to Y) does not contain any chain of length k+1. In this paper we investigate the maximum size that an l-trace k-Sperner family (with underlying set [n]={1,2,…,n}) can have for various values of k, l and n.     

Mixed sensitivity minimization problems with rationall 1-optimal solutions

Up to this time, the only known method to solve the discrete-time mixed sensitivity minimization problem inl ¹ has been to use a certain infinite-dimensional linear programming approach, presented by Dahleh and Pearson in 1988 and later modified by Mendlovitz. That approach does not give in general true optimal solutions; only suboptimal ones are obtained. Here, for the first time, the truel ¹-optimal solutions are found for some mixed sensitivity minimization problems. In particular, Dahleh and Pearson construct an 11h order suboptimal compensator for a certain second-order plan with first-order weight functions; it is shown that the unique optimal compensator for that problem is rational and of order two. The author discovered this fact when trying out a new scheme of solving the infinite-dimensional linear programming system. This scheme is of independent interest, because when it is combined with the Dahleh-Pearson-Mendlovitz scheme, it gives both an upper bound and a lower bound on the optimal performance; hence, it provides the missing error bound that enables one to truncate the solution. Of course, truncation is appropriate only if the order of the optimal compensator is too high. This may indeed be the case, as is shown with an example where the order of the optimal compensator can be arbitrarily high.     

The period and base of a reducible sign pattern matrix

A square sign pattern matrix A (whose entries are ) is said to be powerful if all the powers A,A²,A³,…, are unambiguously defined. For a powerful pattern A, if A^l=A^l+p with l and p minimal, then l is called the base of A and p is called the period of Li et al. [On the period and base of a sign pattern matrix, Linear Algebra Appl. 212/213 (1994) 101-120] characterized irreducible powerful sign pattern matrices. In this paper, we characterize reducible, powerful sign pattern matrices and give some new results on the period and base of a powerful sign pattern matrix.     

Generalized spherical and simplicial coordinates

Elementary trigonometric quantities are defined in l2,p analogously to that in l2,2, the sine and cosine functions are generalized for each p>0 as functions sinp and cosp such that they satisfy the basic equation p|cosp(φ)|+p|sinp(φ)|=1. The p-generalized radius coordinate of a point ξ∈Rn is defined for each p>0 as . On combining these quantities, ln,p-spherical coordinates are defined. It is shown that these coordinates are nearly related to ln,p-simplicial coordinates. The Jacobians of these generalized coordinate transformations are derived. Applications and interpretations from analysis deal especially with the definition of a generalized surface content on ln,p-spheres which is nearly related to a modified co-area formula and an extension of Cavalieri's and Torricelli's indivisibeln method, and with differential equations. Applications from probability theory deal especially with a geometric interpretation of the uniform probability distribution on the ln,p-sphere and with the derivation of certain generalized statistical distributions.     

Generalized exponents of primitive two-colored digraphs

A l-colored digraph D^(l) is primitive if there exists a nonnegative integer vector α such that for each ordered pair of vertices x and y (not necessarily distinct), there exists an α-walk in D^(l) from x to y. The exponent of the primitive l-colored digraph D^(l) is defined to be the minimum value of the sum of all coordinates of α taken over all such α. In this paper, we generalize the concept of exponent of a primitive l-colored digraph by introducing three types of generalized exponents. Further, we study the generalized exponents of primitive two-colored Wielandt digraphs.     

Norm attaining operators and renormings of Banach spaces

We define two geometric concepts of a Banach space, property α and β, which generalize in a certain way the geometric situation ofl andc _o. These properties have been used by J. Lindenstrauss and J. Partington in the study of norm attaining operators. J. Partington has shown that every Banach space may (3+ε)-equivalently be renormed to have property β. We show that many Banach spaces (e.g., every WCG space) may (3+ε)-equivalently be renormed to have property α. However, an example due to S. Shelah shows that not every Banach space is isomorphic to a Banach space with property α.     

Matroid matching with Dilworth truncation

Let H=(V,E) be a hypergraph and let k?1 and l?0 be fixed integers. Let M be the matroid with ground-set E s.t. a set F⊆E is independent if and only if each X⊆V with k|X|-l?0 spans at most k|X|-l hyperedges of F. We prove that if H is dense enough, then M satisfies the double circuit property, thus Lovász’ min-max formula on the maximum matroid matching holds for M. Our result implies the Berge-Tutte formula on the maximum matching of graphs (k=1, l=0), generalizes Lovász’ graphic matroid (cycle matroid) matching formula to hypergraphs (k=l=1) and gives a min-max formula for the maximum matroid matching in the two-dimensional rigidity matroid (k=2, l=3).     

Un semi-groupe régularisé pour un modèle de Lebowitz–RubinowA regularized semigroup for a Lebowitz–Rubinow's model

In this Note, we complete [1] and we study the Lebowitz–Rubinow's model with the biological law of perfect memory. In this model, each cell is characterized by its cell cycle length l (0?l₁<l<l₂<∞) and its age a (0<a<l). If l₁>0, a complete study of this model can be found in [1]. Here we show that if l₁=0 then this model becomes ill-posed. We use the theory of generalized semigroups to remedy to this model. To cite this article: M. Boulanouar, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 865–868.     

A methodology for sensitivity analysis of models fitted to data using statistical methods

A simple methodology is presented for sensitivity analysis ofmodels that have been fitted to data by statistical methods.Such analysis is a decision support tool that can focus theeffort of a modeller who wishes to further refine a model and/orto collect more data. A formula is given for the calculationof the proportional reduction in the variance of the model ‘output’that would be achievable with perfect knowledge of a subsetof the model parameters. This is a measure of the importanceof the set of parameters, and is shown to be asymptoticallyequal to the squared correlation between the model output andits best predictor based on the omitted parameters. The methodology is illustrated with three examples of OR problems,an age-based equipment replacement model, an ARIMA forecastingmodel and a cancer screening model. The sampling error of thecalculated percentage of variance reduction is studied theoretically,and a simulation study is then used to exemplify the accuracyof the method as a function of sample size.     

Toeplitz Operators on Generalized Bergman Spaces

We consider the weighted Bergman spaces HL²(\mathbb B^d, m_l){\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}, where we set dm_l(z) = c_l(1-|z|²)^l dt(z){d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}, with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.     

Multiplicativity of the gamma factors of Rankin–Selberg integrals for SO 2l × GL n

Let SO _2l be the special even orthogonal group, split or quasi–split, defined over a local non–Archimedian field. The Rankin–Selberg method for a pair of generic representations of SO _2l × GL _n constructs a family of integrals, which are used to define γ and L-factors. Here we prove full multiplicative properties for the γ-factor, namely that it is multiplicative in each variable. As a corollary, the γ-factor is identical with Shahidi’s standard γ-factor.     

Moving least squares based sensitivity analysis for models with dependent variables

For models with dependent input variables, sensitivity analysis is often a troublesome work and only a few methods are available. Mara and Tarantola in their paper (“Variance-based sensitivity indices for models with dependent inputs”) defined a set of variance-based sensitivity indices for models with dependent inputs. We in this paper propose a method based on moving least squares approximation to calculate these sensitivity indices. The new proposed method is adaptable to both linear and nonlinear models since the moving least squares approximation can capture severe change in scattered data. Both linear and nonlinear numerical examples are employed in this paper to demonstrate the ability of the proposed method. Then the new sensitivity analysis method is applied to a cantilever beam structure and from the results the most efficient method that can decrease the variance of model output can be determined, and the efficiency is demonstrated by exploring the dependence of output variance on the variation coefficients of input variables. At last, we apply the new method to a headless rivet model and the sensitivity indices of all inputs are calculated, and some significant conclusions are obtained from the results.     

What Is the Complexity of Surface Integration?

We study the worst case complexity of computing ε-approximations of surface integrals. This problem has two sources of partial information: the integrand f and the function g defining the surface. The problem is nonlinear in its dependence on g. Here, f is an r times continuously differentiable scalar function of l variables, and g is an s times continuously differentiable injective function of d variables with l components. We must have d⩽l and s⩾1 for surface integration to be well-defined. Surface integration is related to the classical integration problem for functions of d variables that are min{r, s−1} times continuously differentiable. This might suggest that the complexity of surface integration should be Θ((1/ε)^{d/min{r, s−1}}). Indeed, this holds when d<l and s=1, in which case the surface integration problem has infinite complexity. However, if d⩽l and s⩾2, we prove that the complexity of surface integration is O((1/ε)^d/min{r, s}). Furthermore, this bound is sharp whenever d<l.     

A class of primary Banach spaces

It is shown that if X is an infinite-dimensional Banach space with a boundedly complete subsymmetric basis, then the infinite l^∞ direct sum l^∞(X⊕X⊕X⊕?) is primary.     

Almost splitting sets in integral domains, II

Let D be an integral domain. A saturated multiplicative subset S of D is an almost splitting set if, for each 0≠d∈D, there exists a positive integer n=n(d) such that dⁿ=st for some s∈S and t∈D which is v-coprime to each element of S. We show that every upper to zero in D[X] contains a primary element if and only if D?{0} is an almost splitting set in D[X], if and only if D is a UMT-domain and Cl(D[X]) is torsion. We also prove that D[X] is an almost GCD-domain if and only if D is an almost GCD-domain and Cl(D[X]) is torsion. Using this result, we construct an integral domain D such that Cl(D) is torsion, but Cl(D[X]) is not torsion.     

Implicit compact difference schemes for the fractional cable equation

In this paper, we propose two implicit compact difference schemes for the fractional cable equation. The first scheme is proved to be stable and convergent in l_∞-norm with the convergence order O(τ + h⁴) by the energy method, where new inner products defined in this paper gives great convenience for the theoretical analysis. Numerical experiments are presented to demonstrate the accuracy and effectiveness of the two compact schemes. The computational results show that the two new schemes proposed in this paper are more accurate and effective than the previous.     

The effect of a small background inhomogeneity on the asymptotic properties of linear perturbations

General regularities in the evolution of one-dimensional unstable linear perturbations on a weakly inhomogeneous background are studied when, at the initial instant, the perturbations are concentrated in the δ-neighbourhood of a certain point. Times are considered when these perturbations do not fall outside the limits of a certain domain of size l such that δ ? l ? L, where L is the large characteristic size of the background inhomogeneity. With contain assumptions, the effect of the background inhomogeneity on the asymptotic behaviour of the perturbations at long times is taken into account in a general form. The first corrections to the well known asymptotic relation for the evolution of perturbations on a homogeneous background, that arise because of background inhomogeneity, are obtained using Hamilton's method. An example of the use of the proposed approximate method is considered and the error in the approximation is estimated.     

Sum formulas for reductive algebraic groups

Let V be a Weyl module either for a reductive algebraic group G or for the corresponding quantum group Uq. If G is defined over a field of positive characteristic p, respectively if q is a primitive lth root of unity (in an arbitrary field) then V has a Jantzen filtration V=V⁰⊃V¹⊃?⊃Vr=0. The sum of the positive terms in this filtration satisfies a well-known sum formula.If T denotes a tilting module either for G or Uq then we can similarly filter the space HomG(V,T), respectively HomUq(V,T) and there is a sum formula for the positive terms here as well.We give an easy and unified proof of these two (equivalent) sum formulas. Our approach is based on an Euler type identity which we show holds without any restrictions on p or l. In particular, we get rid of previous such restrictions in the tilting module case.     

Hilbert's 16th problem for classical Liénard equations of even degree

Classical Liénard equations are two-dimensional vector fields, on the phase plane or on the Liénard plane, related to scalar differential equations . In this paper, we consider f to be a polynomial of degree 2l−1, with l a fixed but arbitrary natural number. The related Liénard equation is of degree 2l. We prove that the number of limit cycles of such an equation is uniformly bounded, if we restrict f to some compact set of polynomials of degree exactly 2l−1. The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most l.