The Parseval equality for multiple Fourier-Stieltjes series with respect to the Haar system

Let a function f : \(\Pi ^{ * ^m } \) → ? be Lebesgue integrable on \(\Pi ^{ * ^m } \) and Riemann-Stieltjes integrable with respect to a function G : \(\Pi ^{ * ^m } \) → ? on \(\Pi ^{ * ^m } \). Then the Parseval equality

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holds, where

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(k) = (f, χk) = (L)

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f(x)χk(x) dx and \(\widehat{dG}\)(k) =

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χk(x) dG(x) are Fourier coefficients of the function f and Fourier-Stieltjes coefficients of the function G with respect to the Haar system, respectively; the integrals in the equality and in the definition of the coefficients of the function G are the Riemann-Stieltjes integrals; the series in the right-hand side of the equality converges in the sense of rectangular partial sums; and the overline indicates the complex conjugation. If f : Π ^m → ? is a complex-valued Lebesgue integrable function, G is a complex-valued function of bounded variation on Π ^m ,

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are Fourier-Lebesgue coefficients of the function f _x(t) = f(x ⊕ t), where ⊕ is the group addition, then the Parseval equality

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holds for almost all x ∈ \(\Pi ^{ * ^m } \) in the sense of any summation method with respect to which the Fourier series of Lebesgue integrable functions are summable to these functions almost everywhere (the integral here is interpreted in the sense of Lebesgue-Stieltjes).

     

The bounds for matrices with monotonic rows

The bounds LX,Y (A) and ‖A‖ _X,Y of an operator A = (a _n,k ) _{n, k} ≥0 with monotonic rows are evaluated, where X and Y are quasi-normed real valued sequence spaces. In particular, in the case where X = ℓ _p and Y = ℓ _q , our results give the results when part 0 < p ≤ 1 and 0 < q < ∞ to complement the other results with range p ≥ 1 and 0 < q < ∞. Moreover, we give a partial answer to a problem [1, Problem 7.23] which was posed by Bennett.     

The mixed general routing polyhedron

 In Arc Routing Problems, ARPs, the aim is to find on a graph a minimum cost traversal satisfying some conditions related to the links of the graph. Due to restrictions to traverse some streets in a specified way, most applications of ARPs must be modeled with a mixed graph. Although several exact algorithms have been proposed, no polyhedral investigations have been done for ARPs on a mixed graph. In this paper we deal with the Mixed General Routing Problem which consists of finding a minimum cost traversal of a given link subset and a given vertex subset of a mixed graph. A formulation is given that uses only one variable for each link (edge or arc) of the graph. Some properties of the associated polyhedron and some large families of facet-inducing inequalities are described. A preliminary cutting-plane algorithm has produced very good lower bounds over a set of 100 randomly generated instances of the Mixed Rural Postman Problem. Finally, applications of this study to other known routing problems are described. Received: June 30, 1999 / Accepted: March 2002 Published online: March 21, 2003 Key Words. polyhedral combinatorics – facets – routing – arc routing – rural postman problem – general routing problem – mixed chinese postman problem     

The zero-one knapsack problem with equality constraint

The zero-one knapsack problem is a linear zero-one programming problem with a single inequality constraint. This problem has been extensively studied and many applications and efficient algorithms have been published. In this paper we consider a similar problem, one with an equality instead of the inequality constraint. By replacing the equality by two inequalities one of which is placed in the economic function, a Lagrangean relaxation of the problem is obtained. The relation between the relaxed problem and the original problem is examined and it is shown how the optimal value of the relaxed problem varies with increasing values of the Lagrangean multiplier. Using these results an algorithm for solving the problem is proposed.The paper concludes with a discussion of computational experience.     

The triangle closure is a polyhedron

Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with certain families of lattice-free sets are polyhedra. For a long time, the only result known was the celebrated theorem of Cook, Kannan and Schrijver who showed that the split closure is a polyhedron. Although some fairly general results were obtained by Andersen et al. (Math Oper Res 35(1):233–256, 2010) and Averkov (Discret Optimiz 9(4):209–215, 2012), some basic questions have remained unresolved. For example, maximal lattice-free triangles are the natural family to study beyond the family of splits and it has been a standing open problem to decide whether the triangle closure is a polyhedron. In this paper, we show that when the number of integer variables $m=2$ the triangle closure is indeed a polyhedron and its number of facets can be bounded by a polynomial in the size of the input data. The techniques of this proof are also used to give a refinement of necessary conditions for valid inequalities being facet-defining due to Cornuéjols and Margot (Math Program 120:429–456, 2009) and obtain polynomial complexity results about the mixed integer hull.     

Using artificial reaction force to design compliant mechanism with multiple equality displacement constraints

Monolithic compliant mechanisms are elastic workpieces which transmit force and displacement from an input position to an output position. Continuum topology optimization is suitable to generate the optimized topology, shape and size of such compliant mechanisms. The optimization strategy for a single input single output compliant mechanism under volume constraint is known to be best implemented using an optimality criteria or similar mathematical programming method. In this standard form, the method appears unsuitable for the design of compliant mechanisms which are subject to multiple outputs and multiple constraints. Therefore an optimization model that is subject to multiple design constraints is required. With regard to the design problem of compliant mechanisms subject to multiple equality displacement constraints and an area constraint, we here present a unified sensitivity analysis procedure based on artificial reaction forces, in which the key idea is built upon the Lagrange multiplier method. Because the resultant sensitivity expression obtained by this procedure already compromises the effects of all the equality displacement constraints, a simple optimization method, such as the optimality criteria method, can then be used to implement an area constraint. Mesh adaptation and anisotropic filtering method are used to obtain clearly defined monolithic compliant mechanisms without obvious hinges. Numerical examples in 2D and 3D based on linear small deformation analysis are presented to illustrate the success of the method.     

The equality betweenp-capacity andp-modulus

The work was financially supported by the Russian Foundation for Fundamental Research (code 3-011-277).     

Matrices with prescribed rows and invariant factors

In this paper, we solve the problem of the existence of an n × n matrix over an arbitrary field when its invariant polynomials and either some rows or columns are prescribed. The solution is given in terms of invariant factor inequalities and of majorization inequalities involving controllability indices and the degrees of the invariant polynomials.     

The bending of a polygon into a polyhedron with a given boundary

Siberian Mathematical Journal - The papers [1, 2] were devoted to the realization of nonconvex polyhedral evolutes. Yu. D. Burago and V. A. Zalgaller [1] proved the imbeddability (realizability...     

The stable marriage problem with master preference lists

We study variants of the classical stable marriage problem in which the preferences of the men or the women, or both, are derived from a master preference list. This models real-world matching problems in which participants are ranked according to some objective criteria. The master list(s) may be strictly ordered, or may include ties, and the lists of individuals may involve ties and may include all, or just some, of the members of the opposite sex. In fact, ties are almost inevitable in the master list if the ranking is done on the basis of a scoring scheme with a relatively small range of distinct values. We show that many of the interesting variants of stable marriage that are NP-hard remain so under very severe restrictions involving the presence of master lists, but a number of special cases can be solved in polynomial time. Under this master list model, versions of the stable marriage problem that are already solvable in polynomial time typically yield to faster and/or simpler algorithms, giving rise to simple new structural characterisations of the solutions in these cases.     

Constructions of optimal orthogonal arrays with repeated rows

We construct orthogonal arrays ${\mathsf{OA}}_{\lambda }(k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m∕\lambda$ is as large as possible; these OAs are termed optimal. We provide constructions of optimal OAs for any $k\ge n+1$, albeit with large $\lambda$. We also study basic OAs; these are optimal OAs in which $gcd(m,\lambda )=1$. We construct a basic OA with $n=2$ and $k=4t+1$, provided that a Hadamard matrix of order $8t+4$ exists. This completely solves the problem of constructing basic OAs with $n=2$, modulo the Hadamard matrix conjecture.     

On deformation with constant Milnor number and Newton polyhedron

We show that every \(\mu \)-constant family of isolated hypersurface singularities satisfying a nondegeneracy condition in the sense of Kouchnirenko, is topologically trivial, also is equimultiple.     

The skorokhod oblique reflection problem in a convex polyhedron

The Skorokhod oblique reflection problem is studied in the case ofn-dimensional convex polyhedral domains. The natural sufficient condition on the reflection directions is found, which together with the Lipschitz condition on the coefficients gives the existence and uniqueness of the solution. The continuity of the corresponding solution mapping is established. This property enables one to construct in a direct way the reflected (in a convex polyhedral domain) diffusion processes possessing the nice properties.