General Duality Theorem Research Articles

Generalized local duality, canonical modules, and prescribed bound on projective dimension

We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well as a general local duality theorem which extends, to a much broader class of rings, previous results by Herzog-Zamani and Suzuki. As an application, we establish a prescribed upper bound for the projective dimension of a module satisfying suitable cohomological conditions, and we derive some freeness criteria and questions of Auslander-Reiten type. Along the way, we prove a new characterization of Cohen-Macaulay modules which truly relies on generalized local cohomology, and in addition we introduce and study a generalization of the notion of canonical module.

A Galois Connection Approach to Wei-Type Duality Theorems

In 1991, Wei proved a duality theorem that established an interesting connection between the generalized Hamming weights of a linear code and those of its dual code. Wei’s duality theorem has since been extensively studied from different perspectives and extended to other settings. In this paper, we re-examine Wei’s duality theorem and its various extensions, henceforth referred to as Wei-type duality theorems, from a new Galois connection perspective. Our approach is based on the observation that the generalized Hamming weights and the dimension/length profiles of a linear code form a Galois connection. The central result of this paper is a general Wei-type duality theorem for two Galois connections between finite subsets of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}$ </tex-math></inline-formula> , from which all the known Wei-type duality theorems can be recovered. As corollaries of our central result, we prove new Wei-type duality theorems for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$w$ </tex-math></inline-formula> -demi-matroids defined over finite sets and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$w$ </tex-math></inline-formula> -demi-polymatroids defined over modules with a composition series, which further allows us to unify and generalize all the known Wei-type duality theorems established for codes endowed with various metrics.

A Duality Principle for Groups II: Multi-frames Meet Super-Frames

The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that {pi _{Lambda }(m, n)g_{1} oplus cdots oplus pi _{Lambda }(m, n)g_{k}}_{m, n in {mathbb {Z}}^{d}} is a frame for L^{2}({mathbb {R}},^{d})oplus cdots oplus L^{2}({mathbb {R}},^{d}) if and only if cup _{i=1}^{k}{pi _{Lambda ^{o}}(m, n)g_{i}}_{m, nin {mathbb {Z}}^{d}} is a Riesz sequence, and cup _{i=1}^{k} {pi _{Lambda }(m, n)g_{i}}_{m, nin {mathbb {Z}}^{d}} is a frame for L^{2}({mathbb {R}},^{d}) if and only if {pi _{Lambda ^{o}}(m, n)g_{1} oplus cdots oplus pi _{Lambda ^{o}}(m, n)g_{k}}_{m, n in {mathbb {Z}}^{d}} is a Riesz sequence, where pi _{Lambda } and pi _{Lambda ^{o}} is a pair of Gabor representations restricted to a time–frequency lattice Lambda and its adjoint lattice Lambda ^{o} in {mathbb {R}},^{d}times {mathbb {R}},^{d}.

Tangle-tree duality in abstract separation systems

We prove a general width duality theorem for combinatorial structures with well-defined notions of cohesion and separation. These might be graphs or matroids, but can be much more general or quite different. The theorem asserts a duality between the existence of high cohesion somewhere local and a global overall tree structure.We describe cohesive substructures in a unified way in the format of tangles: as orientations of low-order separations satisfying certain consistency axioms. These axioms can be expressed without reference to the underlying structure, such as a graph or matroid, but just in terms of the poset of the separations themselves. This makes it possible to identify tangles, and apply our tangle-tree duality theorem, in very diverse settings.Our result implies all the classical duality theorems for width parameters in graph minor theory, such as path-width, tree-width, branch-width or rank-width. It yields new, tangle-type, duality theorems for tree-width and path-width. It implies the existence of width parameters dual to cohesive substructures such as k-blocks, edge-tangles, or given subsets of tangles, for which no width duality theorems were previously known.Abstract separation systems can be found also in structures quite unlike graphs and matroids. For example, our theorem can be applied to image analysis by capturing the regions of an image as tangles of separations defined as natural partitions of its set of pixels. It can be applied in big data contexts by capturing clusters as tangles. It can be applied in the social sciences, e.g. by capturing as tangles the few typical mindsets of individuals found by a survey. It could also be applied in pure mathematics, e.g. to separations of compact manifolds.

Sum formulas and duality theorems of multiple zeta values

Multiple zeta values or r-ford Euler sums are defined byζ(α1,α2,…,αr)=∑1≤n1<n2<⋯<nrn1−α1n2−α2⋯nr−αr with α1,α2,…,αr are positive integers and αr≥2. Let |α|=α1+α2+⋯+αr, the sum formula then asserted that∑|α|=mζ(α1,α2,…,αr+1)=ζ(m+1) for any positive integer m≥r.In this paper, we investigate the multiple zeta values with a parameter a>−1 given by∑1≤n1<n2<⋯<np+r1n1n2⋯np(np+1+a)⋯(np+r+a)np+r as well as their vector versions. Among other things, we obtain that the above multiple zeta values are equal to∑|α|=p+r∑1≤k1<k2<⋯<kp+1k1−α1k2−α2⋯kp+1−αp+1(kp+1+a)−1. By differentiations with respect to a and then set a=0, we obtain some interesting sum formulas including the aforementioned sum formula and the restricted sum formula∑|α|=mζ({1}p,α1,…,αr+1)=∑|c|=p+rζ(c1,c2,…,cp+1+(m−r)+1). Also, a generalized duality theorem is given in the final section.

Dualities Induced by Topological Semirings

In this paper we establish a general duality theorem for compact Hausdorff spaces being recognizable over certain pairs consisting of a commutative unital topological semiring and a closed proper prime ideal. Indeed, we utilize the concept of blueprints and their localization to prove that the category of compact Hausdorff spaces generated by such a pair can be dually embedded into the category of commutative unital semirings if the pair possesses sufficiently many covering polynomials.

Calculating quotient algebras of generic embeddings

Many consistency results in set theory involve forcing over a universe V0 that contains a large cardinal to get a model V1. The original large cardinal embedding is then extended generically using a further forcing by a partial ordering ℚ. Determining the properties of ℚ is often the crux of the consistency result. Standard techniques can usually be used to reduce to the case where ℚ is of the form P(Z)/J for appropriately chosen Z and countably complete ideal J. This paper proves a general algebraic Duality Theorem that exactly characterizes the Boolean algebra P(Z)/J. The Duality Theorem is general enough that it applies even if the original embedding in V0 was itself generic. Thus it has as corollaries the theorems of Kakuda, Baumgartner, Laver and others about preservation properties of precipitous and saturated ideals. A corollary is drawn showing that precipitous ideals are indestructible under small proper forcing.

A general duality theorem for the Monge–Kantorovich transport problem

The duality theory of the Monge-Kantorovich transport problem is analyzed in a general setting. The spaces $X, Y$ are assumed to be polish and equipped with Borel probability measures $\mu$ and $\nu$. The transport cost function $c:X\times Y \to [0,\infty]$ is assumed to be Borel. Our main result states that in this setting there is no duality gap, provided the optimal transport problem is formulated in a suitably relaxed way. The relaxed transport problem is defined as the limiting cost of the partial transport of masses $1-\varepsilon$ from $(X,\mu)$ to $(Y, \nu)$, as $\varepsilon >0$ tends to zero. The classical duality theorems of H. Kellerer, where $c$ is lower semi-continuous or uniformly bounded, quickly follow from these general results.

무선망에서의 상하향 링크 쌍대성 성질을 활용한 분산적 수율 최대화 기법

We consider the throughput-maximization problem for both the up- and downlink in a wireless network with interference channels. For this purpose, we design an iterative and distributive uplink algorithm based on Lagrangian relaxation. Using the uplink power prices and network duality, we achieve throughput-maximization in the dual downlink that has a symmetric channel and an equal power budget compared to the uplink. The network duality we prove here is a generalized version of previous research [10], [11]. Computational tests show that the performance of the up- and downlink throughput for our algorithms is close to the optimal value for the channel orthogonality factor, (0.5, 1]. On the other hand, when the channels are slightly orthogonal ((0, 0.5]), we observe some throughput degradation in the downlink. We have extended our analysis to the real downlink that has a nonsymmetric channel and an unequal power budget compared to the uplink. It is shown that the modified duality-based approach is thoroughly applied to the real downlink. Considering the complexity of the algorithms in [6] and [18], we conclude that these results are quite encouraging in terms of both performance and practical applicability of the generalized duality theorem.

Normal Factor Graphs and Holographic Transformations

This paper stands at the intersection of two distinct lines of research. One line is “holographic algorithms,” a powerful approach introduced by Valiant for solving various counting problems in computer science; the other is “normal factor graphs,” an elegant framework proposed by Forney for representing codes defined on graphs. We introduce the notion of holographic transformations for normal factor graphs, and establish a very general theorem, called the generalized Holant theorem, which relates a normal factor graph to its holographic transformation. We show that the generalized Holant theorem on the one hand underlies the principle of holographic algorithms, and on the other hand reduces to a general duality theorem for normal factor graphs, a special case of which was first proved by Forney. In the course of our development, we formalize a new semantics for normal factor graphs, which highlights various linear algebraic properties that potentially enable the use of normal factor graphs as a linear algebraic tool.

Duality theorem for motives

A general duality theorem for the category of motives is established, with a short, simple, and self-contained proof. Introduction Recently, due to the active study of cohomological invariants in algebraic geometry, “transplantation” of classical topological constructions to the algebraic-geometrical “soil” seems to be rather important. In particular, it is very interesting to study topological properties of the category of motives. The concept of a motive was introduced by Alexander Grothendieck in 1964 in order to formalize the notion of universal (co-)homology theory (see the detailed exposition of Grothendieck’s ideas in [5]). For us, the principal example of this type is the category of motives DM−, constructed by Voevodsky [12] for algebraic varieties. The Poincare duality is a classical and fundamental result in algebraic topology that initially appeared in Poincare’s first topological memoir “Analysis Situs” [9] (as a part of the Betti numbers symmetry theorem proof). The proof of the general duality theorem for extraordinary cohomology theories apparently belongs to Adams [1]. Our purpose in this paper is to establish a general duality theorem for the category of motives. Essentially, we extend the main statement of [8] to this category. Many known results can easily be interpreted in these terms. In particular, we get a generalization of the Friedlander–Voevodsky duality theorem [4] to the case of the ground field of arbitrary characteristic. The proof of this fact, involving the main result of [8], was kindly conveyed to the authors by Andrěi Suslin in a private communication. Being inspired by his work and Dold–Puppe’s category approach [2] to the duality phenomenon in topology, we decided to present a short, simple, and self-contained proof of a similar result for the category of motives. Our result might be viewed as a purely abstract theorem and rewritten in the spirit of “abstract nonsense” as a statement about some category with a distinguished class of morphisms. Essentially, what is required for the proof is the existence of finite fiber products and the terminal object in the category of varieties, a small part of the tensor triangulated category structure for motives, and finally, the existence of transfers for the class of morphisms generated by graphs of a special type (of projective morphisms). However, rather, we preferred to formulate all statements for motives of algebraic varieties in order to clarify the geometric nature of the construction and make possible applications easier. This led, in particular, to the appearence of the second (co)homology index responsible for twist with the Tate object Z(1) (see Voevodsky [12]). The only exception is the classical Example 2. 2000 Mathematics Subject Classification. Primary 14F42.

Noncommutative supergeometry, duality and deformations

Following Lett. Math. Phys. 50 (1999) 309, we introduce a notion of Q-algebra that can be considered as a generalization of the notion of Q-manifold (a supermanifold equipped with an odd vector field obeying { Q, Q}=0). We develop the theory of connections on modules over Q-algebras and prove a general duality theorem for gauge theories on such modules. This theorem containing as a simplest case SO(d,d, Z ) -duality of gauge theories on noncommutative tori can be applied also in more complicated situations. We show that Q-algebras appear naturally in Fedosov construction of formal deformation of commutative algebras of functions and that similar Q-algebras can be constructed also in the case when the deformation parameter is not formal.

A Duality Theorem for Étale p-Torsion Sheaves on Complete Varieties over a Finite Field

Let X be an arbitrary variety over a finite field k and p=char k,n∈ N. We will construct a complex of etale sheaves on X together with trace isomorphism from the highest etale cohomology group of this complex onto Z/pnZ such that for every constructible Z/pnZ-sheaf on X the Yoneda pairing is a nondegenerate pairing of finite groups. If X is smooth, this complex is the Gersten resolution of the logarithmic de Rham–Witt sheaf introduced by Gros and Suwa. The proof is based on the special case proven by Milne when the sheaf is constant and X is smooth, as well as on a purity theorem which in turn follows from a theorem about the cohomological dimension of Ci-fields due to Kato and Kuzumaki. If the existence of the Lichtenbaum complex is proven, the theorem will be the p-part of a general duality theorem for varieties over finite fields.

Noncommutative supergeometry and duality.

We introduce a notion of Q-algebra that can be considered as a generalization of the notion of Q-manifold (a supermanifold equipped with an odd vector field obeying {Q,Q} =0). We develop the theory of connections on modules over Q-algebras and prove a general duality theorem for gauge theories on such modules. This theorem contains as a simplest case SO(d,d,Z)-duality of gauge theories on noncommutative tori.

Algebraic morphisms into grassmannians:differential geometry and frobenius

Let Xbe a projective variety (over Spec(K)) and f:X→G(r,v) a morphism to a Grassmannian, i.e. a pair (E,V) where E is a rank r vector bundle on V⊆HO(X,E) is a subspace spanning E with dim(V) = v. Here we study the differential properties of f and their relations to a sequence of quotient bundles E→E1→E2→of E called the derived bundles of (E,V). In the first 5 sections we study the case X a smooth curve, char(K) >0 (the case char(K) = 0, being due to D. Perkinson). Then we give a general duality theorem for the derived bundles when Xis any normal variety.

Duality theorem for a generalized Fermat-Weber problem

The classical Fermat-Weber problem is to minimize the sum of the distances from a point in a plane tok given points in the plane. This problem was generalized by Witzgall ton-dimensional space and to allow for a general norm, not necessarily symmetric; he found a dual for this problem. The authors generalize this result further by proving a duality theorem which includes as special cases a great variety of choices of norms in the terms of the Fermat-Weber sum. The theorem is proved by applying a general duality theorem of Rockafellar. As applications, a dual is found for the multi-facility location problem and a nonlinear dual is obtained for a linear programming problem with a priori bounds for the variables. When the norms concerned are continuously differentiable, formulas are obtained for retrieving the solution for each primal problem from the solution of its dual.

The theory of truth tabular connectives, both truth functional and modal

Briefly, the following things are done in this paper. An informal exposition of the significance of partial truth tables in modal logic is presented. The intuitive concept of a truth tabular sentence connective is examined and made precise. From among a vast assortment of truth tabular connectives, the semantically well-behaved ones (called regular connectives) are singled out for close investigation. The familiar concept of functional completeness is generalized to all sets of regular connectives, and the functional completeness (or incompleteness) of selected sets of connectives is established. Some modal analogues of Sheffer's stroke are presented, i.e., single connectives are introduced which serve to define not only the truth functional connectives but all the regular modal connectives as well. The notion of duality is extended to all regular connectives and a general duality theorem is proved. Lastly, simplified proofs are given of several metatheorems about the system S5.

Duality and perfect probability spaces

Given probability spaces ( X i , A i , P i ) , i = 1 , 2 , (X_i,\mathcal {A}_i,P_i), i=1,2, let M ( P 1 , P 2 ) \mathcal {M}(P_1,P_2) denote the set of all probabilities on the product space with marginals P 1 P_1 and P 2 P_2 and let h h be a measurable function on ( X 1 × X 2 , A 1 ⊗ A 2 ) . (X_1 \times X_2,\mathcal {A}_1 \otimes \mathcal {A}_2). Continuous versions of linear programming stemming from the works of Monge (1781) and Kantorovich-Rubinštein (1958) for the case of compact metric spaces are concerned with the validity of the duality a m p ; sup { ∫ h d P : P ∈ M ( P 1 , P 2 ) } a m p ; = inf { ∑ i = 1 2 ∫ h i d P i : h i ∈ L 1 ( P i ) a n d h ≤ ⊕ i h i } \begin{align*} &amp;\sup \{ \int h dP: P \in \mathcal {M}(P_1,P_2) \} \ &amp;\qquad = \: \inf \{ \sum _{i=1}^{2} \int h_i dP_i : h_i \in \mathcal {L}^1 (P_i) \; \; and \; \; h \leq {\oplus }_i h_i\} \end{align*} (where M ( P 1 , P 2 ) \mathcal {M}(P_1,P_2) is the collection of all probability measures on ( X 1 × X 2 , A 1 ⊗ A 2 ) (X_1 \times X_2, \mathcal {A}_1 \otimes \mathcal {A}_2) with P 1 P_1 and P 2 P_2 as the marginals). A recently established general duality theorem asserts the validity of the above duality whenever at least one of the marginals is a perfect probability space. We pursue the converse direction to examine the interplay between the notions of duality and perfectness and obtain a new characterization of perfect probability spaces.

A general duality theorem for marginal problems

Given probability spaces (Xi,Ai,Pi),i=1, 2 letM(P1,P2) denote the set of all probabilities on the product space with marginalsP1 andP2 and leth be a measurable function on (X1×X2,A1 ⊗A2). In order to determine supfh dP where the supremum is taken overP inM(P1,P2), a general duality theorem is proved. Only the perfectness of one of the coordinate spaces is imposed without any further topological or tightness assumptions. An example without any further topological or tightness assumptions. An example is given to show that the assumption of perfectness is essential. Applications to probabilities with given marginals and given supports, stochastic order and probability metrics are included.

Lagrangian function and duality theory in multiobjective programming with set functions

Using the concept of vector-valued Lagrangian functions, we characterize a special class of solutions,D-solutions, of a multiobjective programming problem with set functions in which the domination structure is described by a closed convex coneD. Properties of two perturbation functions, primal map and dual map, are also studied. Results lead to a general duality theorem.