Dual Statement Research Articles

Estimates of thermoelastic characteristics of composites reinforced by short anisotropic fibers

We construct a mathematical model describing thermomechanical interaction between composite structure elements (isotropic particles of the matrix and anisotropic short fibers) and the macroscopically isotropic elastic medium with desired thermoelastic characteristics. At the first stage of this model, the self-consistency method is used to obtain relations determining the elasticity moduli of the composite, and at the second stage, the model permits determining its linear thermal expansion coefficient. The dual variational statement of the linear thermoelasticity problem in an inhomogeneous solid permits obtaining two-sided estimates for the bulk elasticity modulus, shear modulus, and linear thermal expansion coefficient of the composite under study. The calculated dependencies presented in the paper permit predicting the thermoelastic characteristics of a composite reinforced by anisotropic short fibers (including those in the form of nanostructure elements).

Processing of defective ring data by “super-interpolators”

We develop an analytical-numerical scheme to make up for full or partial sensor failures in a ring array. The technique’s goal is to restore missing information to a recorded signal in the form of a Hermitian covariance matrix for an arbitrary dependent variable. Our analysis breaks the problem down into two: first, by considering the special case of an imperfect array in which the positions and degrees of failure of the flawed sensors are known, and, second, by generalizing that approach to one in which neither piece of information is presumed nor in fact is the knowledge of how many sensors are affected required. An important application of the first part is the retrieval of the perfect signal at equispaced points that have been left un-instrumented for reasons of economy or lack of physical access (a missing sensor becomes equivalent to one with a known null gain). The problem’s more general second part has led to a fundamental relationship for a periodic array’s total number of sensors, the bandwidth of the ideal signal being restored, and the rank of an integral equation developed from one of Fourier series’ dual statements of orthogonality. We call the products of our two-tier technique “super-interpolators” even though neither engages in that activity mathematically.

An axiomatic foundation of Cayley-Klein geometries

We present a common axiomatic characterization of Cayley-Klein geometries over fields of characteristic \({\neq 2}\). To this end the axiom system of Bachmann (Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd edn. Springer, Heidelberg 1973) for plane absolute geometry, which allows a common axiomatization of Euclidean, hyperbolic and elliptic geometry, is generalized. The notion of plane absolute geometry is broadened in several aspects. The most important one is that the principle of duality holds: the dual of a Cayley-Klein geometry is also a Cayley-Klein geometry. The various Cayley-Klein geometries are singled out by additional axioms like the Euclidean or hyperbolic parallel axiom or their dual statements.

Functoriality and duality in Morse–Conley–Floer homology

In [J. Topol. Anal. 6 (2014), 305–338], we have developed a homology theory (Morse–Conley–Floer homology) for isolated invariant sets of arbitrary flows on finite-dimensional manifolds. In this paper, we investigate functoriality and duality of this homology theory. As a preliminary, we investigate functoriality in Morse homology. Functoriality for Morse homology of closed manifolds is known, but the proofs use isomorphisms to other homology theories. We give direct proofs by analyzing appropriate moduli spaces. The notions of isolated map and flow map allow the results to generalize to local Morse homology and Morse–Conley–Floer homology. We prove Poincaré-type duality statements for local Morse homology and Morse–Conley–Floer homology.

A Note on the Multiple-Unicast Network Coding Conjecture

Network coding encourages in-network information mixing for enhanced network capacity. An important open problem in network coding, the multiple-unicast network coding (MUNC) conjecture, claims that network coding is equivalent to routing for multiple unicast sessions in an undirected network. Simple and intuitive as it appears, the MUNC conjecture has remained open for a decade. This work studies a special case of MUNC, where intra-session connections are better than inter-session connections. In the cost domain, we prove that if each source is closer to its own receiver than to other receivers, then network coding is equivalent to routing. In the throughput domain, an intuitive dual statement is that if each source has a larger max-flow to its own receiver than to other receivers, then network coding is equivalent to routing. We prove that this statement is equivalent to the original MUNC conjecture. An interesting consequence is that an equivalence between the two seemingly dual statements studied in this work, if true, would imply the original MUNC conjecture.

ON STRONGLY C2 MODULES AND D2 MODULES

Let R be a ring, MR be a right R-module, n be a positive integer and S = End (MR) be the endomorphism ring of MR. MR is called a strongly C2 module if [Formula: see text] is C2 for every positive integer m. MR is called an n-C2 module if the annihilator rM(K) ≠ 0 for any n-generated proper left ideal K of S. We prove that MR is strongly C2 if and only if M is n-C2 for every positive integer n, if and only if the annihilator rM(K) is not zero for every finitely generated proper left ideal K of S, and then we get some characterizations of right n-C2 rings and strongly right C2 rings. Also we obtain some dual statements of n-D2 module and strongly D2 module.

Vector duality for convex vector optimization problems by means of the quasi-interior of the ordering cone

We define weakly minimal elements of a set with respect to a convex cone by means of the quasi-interior of the cone and characterize them via linear scalarization, generalizing the classical weakly minimal elements from the literature. Then we attach to a general vector optimization problem, a dual vector optimization problem with respect to (generalized) weakly efficient solutions and establish new duality results. By considering particular cases of the primal vector optimization problem, we derive vector dual problems with respect to weakly efficient solutions for both constrained and unconstrained vector optimization problems and the corresponding weak, strong and converse duality statements.

Asymptotic Stability of Two-Dimensional Discrete Systems With Saturation Nonlinearities

This paper investigates the asymptotic stability of discrete dynamical systems in a class of two-dimensional (2-D) systems whose dynamical parts are described in the Fornasini-Marchesini model along with a standard saturation operator on the state space. Under the assumption that the stability of the nominal system is ensured by the solvability of a linear matrix inequality, two techniques are introduced for checking the tolerance of the system against saturation effects. One is a quadratic method that accounts for the stability margin of a system. Another is a non-quadratic method that uses the asymptotic property of a majorant nonnegative system. These techniques are useful to improve upon previously known results. Two theorems are introduced in this paper in different manners. The first theorem provides a plain interpretation on the stability condition; however, it requires a two-step process to search for a solution. The second theorem is expressed as a linear matrix inequality, which is the dual statement of the first theorem. These results can be naturally modified for 1-D systems, 2-D systems in the Roesser's model, and multidimensional systems. Illustrative examples show that the two techniques adopted in this paper have different effectiveness in enlarging the scope of application.

Duality for [formula omitted]-variational inequalities via the subdifferential calculus

Based on the properties of the (convex) ε-subdifferential calculus, we introduce to a general ε-variational inequality (formulated with the help of a set valued operator and a perturbation function) a dual one, expressed by making use of the (Fenchel) conjugate of the perturbation function. Under convexity hypotheses, we show that the fulfillment of a regularity condition guarantees that the primal ε-variational inequality is solvable if and only if its dual one is solvable. By particularizing the perturbation function, we obtain several dual statements and we succeed to generalize and improve a duality scheme recently given by Kum, Kim and Lee. An example justifying this generalization is also provided. Among the special instances of the general result, we rediscover also the duality scheme concerning variational inequalities due to Mosco.

Duality for vector optimization problems via a general scalarization

Considering a vector optimization problem to which properly efficient solutions are defined by using convex cone-monotone scalarization functions, we attach to it, by means of perturbation theory, new vector duals. When the primal problem, the scalarization function and the perturbation function are particularized, different dual vector problems are obtained, some of them are already known in the literature. Weak and strong duality statements are delivered in each case.

Limits and colimits of Hopf algebras

It is shown that for any commutative unital ring R the category Hopf R of R-Hopf algebras is locally presentable and a coreflective subcategory of the category Bialg R of R-bialgebras, admitting cofree Hopf algebras over arbitrary R-algebras. The proofs are based on an explicit analysis of the construction of colimits of Hopf algebras, which generalizes an observation of Takeuchi. Essentially be a duality argument also the dual statement, namely that Hopf R is closed in Bialg R under limits, is shown to hold, provided that the ring R is von Neumann regular. It then follows that Hopf R is reflective in Bialg R and admits free Hopf algebras over arbitrary R-coalgebras, for any von Neumann regular ring R. Finally, Takeuchi's free Hopf algebra construction is analysed and shown to be simply a composition of standard categorical constructions. By simple dualization also a construction of the Hopf coreflection is provided.

On the size of maximal antichains and the number of pairwise disjoint maximal chains

Fix integers n and k with n ≥ k ≥ 3 . Duffus and Sands proved that if P is a finite poset and n ≤ | C | ≤ n + ( n − k ) / ( k − 2 ) for every maximal chain in P , then P must contain k pairwise disjoint maximal antichains. They also constructed a family of examples to show that these inequalities are tight. These examples are two-dimensional which suggests that the dual statement may also hold. In this paper, we show that this is correct. Specifically, we show that if P is a finite poset and n ≤ | A | ≤ n + ( n − k ) / ( k − 2 ) for every maximal antichain in P , then P has k pairwise disjoint maximal chains. Our argument actually proves a somewhat stronger result, and we are able to show that an analogous result holds for antichains.

Approximated approximations for SAO

We propose to replace a number of popular approximations by their diagonal quadratic Taylor series expansions. The resulting separable quadratic approximations are easily convexified, and are well suited for use in dual sequential approximate optimization (SAO) algorithms. Global convergence of the resulting SAO algorithms may be enforced in a natural way using conservatism. The approximated approximation approach is explicitly illustrated for (i) reciprocal and exponential intervening variables, (ii) the intervening variables used in the method of moving asymptotes (MMA), (iii) the intervening variables used in CONLIN, and (iv) the TANA-3 approximations. The use of intermediate responses for use in, for example, truss and frame-like structures, is also discussed. Key advantages of replacing nonlinear approximations by their diagonal quadratic approximations are that these approximated approximations can all be used simultaneously in a single dual statement; the dual does not depend on the form of the original approximations. In addition, in a dual setting, the resulting subproblems yield simple analytical relationships between the primal and dual variables, which is often not the case with the original nonlinear approximations. An important example hereof is the exponential approximation. Although the diagonal quadratic approximations may differ notably from their original counterparts, they typically are quite similar in a sufficiently small search subregion, which relates to the move limits commonly used in SAO anyway.

Finding paths through narrow and wide trees

AbstractWe consider two axioms of second-order arithmetic. These axioms assert, in two different ways, that infinite but narrow binary trees always have infinite paths. We show that both axioms are strictly weaker than Weak König's Lemma, and incomparable in strength to the dual statement (WWKL) that wide binary trees have paths.

A duality theorem for generalized local cohomology

We prove a duality theorem for graded algebras over a field that implies several known duality results : graded local duality, versions of Serre duality for local cohomology and of Suzuki duality for generalized local cohomology, and Herzog-Rahimi bigraded duality. Several duality results for local cohomology (notably Grothendieck and Serre dualities) and for generalized local cohomology (Suzuki duality) are classical results commonly used in commutative algebra. A more recent duality theorem was established by Herzog and Rahimi in the context of bigraded algebras. We here present a duality result that encapsulates all these duality statements. Before stating this theorem, we need to fix some notation. Let be an abelian group and R be a graded polynomial ring over a commutative Noetherian ring R0.

Around the Gysin triangle. II.

The notions of orientation and duality are well understood in algebraic topology in the framework of the stable homotopy category. In this work, we follow these lines in algebraic geometry, in the framework of motivic stable homotopy, introduced by F. Morel and V. Voevodsky. We use an axiomatic treatment which allows us to consider both mixed motives and oriented spectra over an arbitrary base scheme. In this context, we introduce the Gysin triangle and prove several formulas extending the traditional panoply of results on algebraic cycles modulo rational equivalence. We also obtain the Gysin morphism of a projective morphism and prove a duality theorem in the (relative) pure case. These constructions involve certain characteristic classes (Chern classes, fundamental classes, cobordism classes) together with their usual properties. They imply statements in motivic cohomology, algebraic K-theory (assuming the base is regular) and “abstract” algebraic cobordism as well as the dual statements in the corresponding homology theories. They apply also to ordinary cohomology theories in algebraic geometry through the notion of a mixed Weil cohomology theory, introduced by D.-C. Cisinski and the author in [CD06], notably rigid cohomology.

Comparison between different duals in multiobjective fractional programming

The present paper is a continuation of [2] where we deal with the duality for a multiobjective fractional optimization problem. The basic idea in [2] consists in attaching an intermediate multiobjective convex optimization problem to the primal fractional problem, using an approach due to Dinkelbach ([6]), for which we construct then a dual problem expressed in terms of the conjugates of the functions involved. The weak, strong and converse duality statements for the intermediate problems allow us to give dual characterizations for the efficient solutions of the initial fractional problem. The aim of this paper is to compare the intermediate dual problem with other similar dual problems known from the literature. We completely establish the inclusion relations between the image sets of the duals as well as between the sets of maximal elements of the image sets.

On the equivalence of optimality criterion and sequential approximate optimization methods in the classical topology layout problem

AbstractWe study the ‘classical’ topology optimization problem, in which minimum compliance is sought, subject to linear constraints. Using a dual statement, we propose two separable and strictly convex subproblems for use in sequential approximate optimization (SAO) algorithms.Respectively, the subproblems use reciprocal and exponential intermediate variables in approximating the non‐linear compliance objective function. Any number of linear constraints (or linearly approximated constraints) are provided for. The relationships between the primal variables and the dual variables are found in analytical form.For the special case when only a single linear constraint on volume is present, we note that application of the ever‐popular optimality criterion (OC) method to the topology optimization problem, combined with arbitrary values for the heuristic numerical damping factor η proposed by Bendsøe, results in an updating scheme for the design variables that is identical to the application of a rudimentary dual SAO algorithm, in which the subproblems are based on exponential intermediate variables. What is more, we show that the popular choice for the damping factor η=0.5 is identical to the use of SAO with reciprocal intervening variables.Finally, computational experiments reveal that subproblems based on exponential intervening variables result in improved efficiency and accuracy, when compared to SAO subproblems based on reciprocal intermediate variables (and hence, the heuristic topology OC method hitherto used). This is attributed to the fact that a different exponent is computed for each design variable in the two‐point exponential approximation we have used, using gradient information at the previously visited point. Copyright © 2007 John Wiley & Sons, Ltd.

Zero-Free Intervals for Flow Polynomials of Near-Cubic Graphs

Let $P(G,t)$ and $F(G,t)$ denote the chromatic and flow polynomials of a graph $G$. G. D. Birkhoff and D C. Lewis showed that, if $G$ is a plane near-triangulation, then the only zeros of $P(G,t)$ in $(-\infty,2]$ are 0, 1 and 2. We will extend their theorem by showing that a stronger result to the dual statement holds for both planar and non-planar graphs: if $G$ is a bridge graph with at most one vertex of degree other than three, then the only zeros of $F(G,t)$ in $(-\infty,\alpha]$ are 1 and 2, where $\alpha\approx 2.225\cdots$ is the real zero in $(2,3)$ of the polynomial $t^4-8t^3+22t^2-28t+17$. In addition we construct a sequence of ‘near-cubic’ graphs whose flow polynomials have zeros converging to $\alpha$ from above.

THE EXISTENCE OF CERTAIN (CO-)LIMITS IN THE CATEGORY OF CHAIN FUNCTORS

Chain functors A*, B* have been introduced for calculating generalized homology theories by using chains and cycles, as one is doing for ordinary homology theories by means of chain complexes. Like chain complexes these chain functors form a category ℭh displaying interesting properties by themselves. The present paper emphasizes more the algebraic aspescts of this category ℭh. Although not every morphism f ∈ ℭh(A*, B*) in the category of chain functors admits a kernel or a cokernel, it turns out that: (1) all cofibrations have a cokernel, (2) all regular fibrations have a kernel, (3) the pushout of a cofibration along a cofibration exists in ℭh, respectively the dual statement for fibrations, (4) there are interesting results about exact sequences involving (co-)fibrations.