Dual Statement Research Articles

In-situ stress field simulation and fracture distribution prediction of middle lower Ordovician in Zone B of Tahe Oilfield, Tarim Basin

This study constructed a heterogeneous rock mechanics model through well seismic joint inversion, set up a dual cycle statement to optimize the application of boundary loads, and obtained the current stress field distribution in Zone B. At the same time, we preferred the rock fracture criterion, carried out the quantitative characterization of different natures of fractures, and used the integrated fracture development rate to portray the integrated fracture development situation. Finally, the horizontal stress difference, stress heterogeneity coefficient, and comprehensive fracture rate were proposed to define the development index of fractured reservoirs. The results show that (1) the maximum horizontal principal stress is 102–144 MPa, and the minimum horizontal principal stress is 85–125 MPa; (2) the development rate of tension fractures is 0.03–0.23, the development rate of shear fractures is 0.15–0.54, and the integrated fracture development rate is 0.28–0.79; (3) the development index of fracture-type reservoirs is >2.3, Class I; 1.8–2.3, Class II; When it is lower than 1.8, Grade III. This article attempts to apply the stress field simulation results to the quantitative prediction of reservoir fractures, adding new ideas for the practical application of stress field results.

Lorentz symmetry and IR structure of the BFSS matrix model

The BFSS matrix model relates flat space M-theory to a large N limit of matrix quantum mechanics describing N non-relativistic D0-branes. M-theory, being a theory of gravity in flat space, has a rich infrared structure that includes various soft theorems and an infinite set of conserved charges associated to asymptotic symmetries. In this work, we ask: to what extent is this infrared structure present in BFSS? We find that all the salient features concerning the infrared structure of M-theory carry over naturally to the quantum mechanics dual. Moreover, we demonstrate that the dual statement of the soft graviton theorem in the matrix model implies that D0-brane scattering amplitudes in BFSS enjoy the full 11d Lorentz symmetry of M-theory, a claim which has been long anticipated. We also offer several first-principle consistency checks for our findings, including a computation of the soft theorem which does not presuppose the BFSS duality and a non-trivial match between several known symmetries of M-theory and BFSS that appear naturally in this formalism. These calculations give non-perturbative evidence in support of the BFSS duality as a model of flat space holography.

Disjoint dijoins for classes of dicuts in finite and infinite digraphs

A dicut in a directed graph is a cut for which all of its edges are directed to a common side of the cut. A famous theorem of Lucchesi and Younger states that in every finite digraph the least size of a set of edges meeting every non-empty dicut equals the maximum number of disjoint dicuts in that digraph. Such sets are called dijoins. Woodall conjectured a dual statement. He asked whether the maximum number of disjoint dijoins in a directed graph equals the minimum size of a non-empty dicut. We study a modification of this question where we restrict our attention to certain classes of non-empty dicuts, i.e. whether for a class \(\mathfrak{B}\) of dicuts of a directed graph the maximum number of disjoint sets of edges meeting every dicut in \(\mathfrak{B}\) equals the size of a minimum dicut in \(\mathfrak{B}\). In particular, we verify this questions for nested classes of finite dicuts, for the class of dicuts of minimum size, and for classes of infinite dibonds, and we investigate how this generalised setting relates to a capacitated version of this question.Mathematics Subject Classifications: 05C20, 05C70 (primary); 05C65, 05C63 (secondary)Keywords: Woodall's conjecture, digraphs, directed cuts, dijoins, dijoin packing

Constructing certain families of 3‐polytopal graphs

AbstractLet and be a 3‐polytopal graph such that for every , has at least one vertex of degree . We find the minimal vertex count for . We then describe an algorithm to construct the graphs . A dual statement may be formulated for faces of 3‐polytopes. The ideas behind the algorithm generalise readily to solve related problems. Moreover, given a 3‐polytope comprising a vertex of degree for all , fixed, we define an algorithm to output for a 3‐polytope comprising a vertex of degree , for all , and such that the initial is a subgraph of . The vertex count of is asymptotically optimal, in the sense that it matches the aforementioned minimal vertex count up to order of magnitude, as gets large. In fact, we only lose a small quantity on the coefficient of the second highest term, and this quantity may be taken as small as we please, with the tradeoff of first constructing an accordingly large auxiliary graph.

Duality Notions in Real Projective Plane

Summary In this article, we check with the Mizar system [1], [2], the converse of Desargues’ theorem and the converse of Pappus’ theorem of the real projective plane. It is well known that in the projective plane, the notions of points and lines are dual [11], [9], [15], [8]. Some results (analytical, synthetic, combinatorial) of projective geometry are already present in some libraries Lean/Hott [5], Isabelle/Hol [3], Coq [13], [14], [4], Agda [6], . . . . Proofs of dual statements by proof assistants have already been proposed, using an axiomatic method (for example see in [13] - the section on duality: “[...] For every theorem we prove, we can easily derive its dual using our function swap [...]2”). In our formalisation, we use an analytical rather than a synthetic approach using the definitions of Leończuk and Prażmowski of the projective plane [12]. Our motivation is to show that it is possible by developing dual definitions to find proofs of dual theorems in a few lines of code. In the first part, rather technical, we introduce definitions that allow us to construct the duality between the points of the real projective plane and the lines associated to this projective plane. In the second part, we give a natural definition of line concurrency and prove that this definition is dual to the definition of alignment. Finally, we apply these results to find, in a few lines, the dual properties and theorems of those defined in the article [12] (transitive, Vebleian, at_least_3rank, Fanoian, Desarguesian, 2-dimensional). We hope that this methodology will allow us to continued more quickly the proof started in [7] that the Beltrami-Klein plane is a model satisfying the axioms of the hyperbolic plane (in the sense of Tarski geometry [10]).

The Kechris–Pestov–Todorčević Correspondence from the Point of View of Category Theory

The Kechris–Pestov–Todorcevic correspondence (KPT-correspondence for short) is a surprising correspondence between model theory, combinatorics and topological dynamics. In this paper we present a categorical re-interpretation of (a part of) the KPT-correspondence with the aim of proving a dual statement. Our strategy is to take a “direct” result and then analyze the necessary infrastructure that makes the result true by providing a purely categorical proof of the categorical version of the result. We can then capitalize on the Duality Principle to obtain the dual statements almost for free. We believe that the dual version of the KPT-correspondence can not only provide the new insights into the interplay of combinatorial, model-theoretic and topological phenomena this correspondence binds together, but also explores the limits to which categorical treatment of combinatorial phenomena can take us.

English

We discuss dual Ramsey statements for several classes of finite relational structures (such as finite linearly ordered graphs, finite linearly ordered metric spaces and finite posets with a linear extension) and conclude the paper with another rendering of the Nesetřil-Rodl Theorem for relational structures. Instead of embeddings which are crucial for ``direct'' Ramsey results, for each class of structures under consideration we propose a special class of quotient maps and prove a dual Ramsey theorem in such a setting. Although our methods are based on reinterpreting the (dual) Ramsey property in the language of category theory, all our results are about classes of finite structures.

Weak Vopěnka's Principle does not imply Vopěnka's Principle

Vopěnka's Principle says that the category of graphs has no large discrete full subcategory, or equivalently that the category of ordinals cannot be fully embedded into it. Weak Vopěnka's Principle is the dual statement, which says that the opposite category of ordinals cannot be fully embedded into the category of graphs. It was introduced in 1988 by Adámek, Rosický, and Trnková, who showed that it follows from Vopěnka's Principle and asked whether the two statements are equivalent. We show that they are not. However, we show that Weak Vopěnka's Principle is equivalent to the generalization of itself known as Semi-Weak Vopěnka's Principle, introduced by Adámek and Rosický in 1993.

Mixed Hodge structures and formality of symmetric monoidal functors

We use mixed Hodge theory to show that the functor of singular chains with rational coefficients is formal as a lax symmetric monoidal functor, when restricted to complex schemes whose weight filtration in cohomology satisfies a certain purity property. This has direct applications to the formality of operads or, more generally, of algebraic structures encoded by a colored operad. We also prove a dual statement, with applications to formality in the context of rational homotopy theory. In the general case of complex schemes with non-pure weight filtration, we relate the singular chains functor to a functor defined via the first term of the weight spectral sequence.

Three convolution inequalities on the real line with connections to additive combinatorics

We discuss three convolution inequalities that are connected to additive combinatorics. Cloninger and the second author showed that for nonnegative f∈L1(−1/4,1/4),max−1/2≤t≤1/2⁡∫Rf(t−x)f(x)dx≥1.28(∫−1/41/4f(x)dx)2 which is related to g−Sidon sets (1.28 cannot be replaced by 1.52). We prove a dual statement, related to difference bases, and show that for f∈L1(R),min0≤t≤1⁡∫Rf(x)f(x+t)dx≤0.42‖f‖L12, where the constant 1/2 is trivial, 0.42 cannot be replaced by 0.37. This suggests a natural conjecture about the asymptotic structure of g−difference bases. Finally, we show for all functions f∈L1(R)∩L2(R),∫−1212∫Rf(x)f(x+t)dxdt≤0.91‖f‖L1‖f‖L2.

Brief note on equality constraints in pure dual SAO settings

We reflect on using equality constraints in a pure dual setting, to augment the predominant use of inequality constraints only in pure dual statements in structural optimization. Notwithstanding the fact that equalities have previously not been used in pure dual statements, their inclusion is sound from a theoretical point of view. This was already shown by Falk as early as 1965, but not really exploited in the structural optimization community. Hence, we herein mainly elaborate on some possible interesting numerical results. To overcome the difficulties associated with dual variables going to infinity if a subproblem is infeasible, we rely on the so-called bounded dual approach to achieve convergence. While the theme of this brief note is a pure dual method which can accommodate equality constraints, we depart with a sequential approximate optimization (SAO) algorithm, which provides for both equality and inequality constraints. The SAO algorithm in essence solves a sequence of approximate quadratic programs (QPs), but the approximate Hessian is diagonal. This means that the separable QP subproblems can then easily be solved using a pure dual method, which is attractive if the number of design variables n ≫ m, with m the total number of constraints. When m > n, the subproblems can efficiently be solved using a sequential quadratic programming (SQP)-like approach with an approximate diagonal Hessian matrix, but the resulting iteration paths are identical to that of the dual formulation; only the cost of solving the subproblems themselves is affected.

Continuous functions with given super-additive and sub-additive transformations

ABSTRACTResults of F. Kouchakinejad and the authors [Internat. J. General Systems 46 (2017), 225–234] imply that if are continuous functions with zero value at the origin, such that , f is strictly super-additive and g is sub-additive but not linear, then there is no continuous aggregation function such that f and g are super-additive and sub-additive transformations of h, respectively. In this paper we extend this result to all continuous functions , as well as to a corresponding dual statement obtained by swapping the roes of super- and sub-additivity. We also show that the continuity assumption here cannot be dropped even at a single point.

Rivalry and fan aggression: why acknowledging conflict reduces tension between rival fans and downplaying makes things worse

ABSTRACTResearch question: In order to reduce fan aggression surrounding rivalry games, team sport organizations often try to placate fans by downplaying the importance of the game (e.g. ‘the derby is not a war’). Drawing on the intergroup conflict literature, this research derives dual identity statements and examines their effectiveness in reducing fan aggressiveness compared to the managerial practice of downplaying rivalry.Research methods: Three field experimental studies (one face-to-face survey and two online surveys) tested the hypotheses. Established rivalries in the German soccer league Bundesliga served as the empirical setting of the studies. The data were analyzed using ANCOVA and linear regression analyses.Results and findings: Dual identity statements reduce fan aggressiveness compared to both downplay statements and a no-statement control condition, independent of team identification and trait aggression. Importantly, the managerial practice of downplaying rivalry appears to be counterproductive. It produces even higher levels of fan aggressiveness than making no statement, an effect caused by psychological reactance.Implications: Sport organizations should not alienate their fan base by attempting to play down the importance of rivalry, which is an integral part of fan identity. Instead, they should strengthen the supporters’ unique identity (as fans of a particular team) while at the same time facilitating identification with the rival at a superordinate level (e.g. as joint fans of a region).

Duality results for extended multifacility location problems

Duality statements are presented for multifacility location problems as suggested by Drezner Hiu 1991, where for each given point the sum of weighted distances to all facilities plus set-up costs is determined and the maximal value of these sums is to be minimized. We develop corresponding dual problems for the cases with and without set-up costs and present associated optimality conditions. In the concluding part of this note we use these optimality conditions for a geometrical characterization of the set of optimal solutions and consider for an illustration corresponding examples.

Robust alternative theorem for linear inequalities with applications to robust multiobjective optimization

We first establish a new alternative theorem for a robust linear inequality system, where the dual statement is expressed in terms of linear matrix inequalities and thus, it can be verified by solving a semidefinite linear program. We then apply the established alternative theorem to derive a characterization of optimality for weakly Pareto solutions of a robust linear multiobjective optimization problem, and to examine weak, strong and converse duality relations in robust linear multiobjective optimization.

Duality results for nonlinear single minimax location problems via multi-composed optimization

In the framework of conjugate duality we discuss nonlinear and linear single minimax location problems with geometric constraints, where the gauges are defined by convex sets of a Frechet space. The version of the nonlinear location problem is additionally considered with set-up costs. Associated dual problems for this kind of location problems will be formulated as well as corresponding duality statements. As conclusion of this paper, we give a geometrical interpretation of the optimal solutions of the dual problem of an unconstraint linear single minimax location problem when the gauges are a norm. For an illustration, an example in the Euclidean space will follow.

An Elementary Exposition of Topological Overlap in the Plane

The aim of this text is to provide an elementary and self-contained exposition of Gromov’s argument on topological overlap (the presentation is based on Gromov’s work, as well as two follow-up papers of Matousek and Wagner, and of Dotterrer, Kaufman and Wagner). We also discuss a simple generalization in which the vertices are weighted according to some probability distribution. This allows to use von Neumann’s minimax theorem to deduce a dual statement.

Constructions of triangular norms on lattices by means of irreducible elements

Constructions and characterizations of triangular norms(t-norms) have been discussed in many different contexts. In this paper, we present two methods to construct t-norms on lattices from given partial information. Since the idea is to follow from part-to-whole, we especially consider the lattices with ∨-decompositions. Firstly, we investigate the structure of the partially ordered set of ∨-irreducible elements. Then, for a given t-norm T on the complete distributive lattice [0, 1]2, we study the restriction of T to the poset of ∨-irreducible elements of [0, 1]2. Furthermore, we give a method for generating t-norms on a finite distributive lattice L by means of ∨-irreducible elements in L. We show that a t-norm on a finite distributive lattice is idempotent if and only if it is idempotent on the set of ∨-irreducible elements. Finally, we introduce a formula to obtain t-norms on L[n] from given t-norms on ∨-irreducible elements of L[n]. We present the dual statements for t-conorms.

A generalized Farkas lemma with a numerical certificate and linear semi-infinite programs with SDP duals

In this paper, we establish a new non-homogeneous Farkas lemma for a linear semi-infinite inequality system, where the dual statement is given in terms of linear matrix inequalities and thus, it can be numerically verified by solving a semidefinite linear program. It is achieved by way of employing a special variable transformation together with a strong separation theorem. Consequently, we establish a duality theorem for a class of parametric linear semi-infinite programs which admit semi-definite linear programming (SDP) dual problems. A numerical example is given to show how a parametric linear semi-infinite optimization problem can be solved by way of solving its SDP dual problem using the Matlab toolbox CVX.

On sequential approximate simultaneous analysis and design in classical topology optimization

SummaryWe study the simultaneous analysis and design (SAND) formulation of the ‘classical’ topology optimization problem subject to linear constraints on material density variables. Based on a dual method in theory, and a primal‐dual method in practice, we propose a separable and strictly convex quadratic Lagrange–Newton subproblem for use in sequential approximate optimization of the SAND‐formulated classical topology design problem.The SAND problem is characterized by a large number of nonlinear equality constraints (the equations of equilibrium) that are linearized in the approximate convex subproblems. The availability of cheap second‐order information is exploited in a Lagrange–Newton sequential quadratic programming‐like framework. In the spirit of efficient structural optimization methods, the quadratic terms are restricted to the diagonal of the Hessian matrix; the subproblems have minimal storage requirements, are easy to solve, and positive definiteness of the diagonal Hessian matrix is trivially enforced.Theoretical considerations reveal that the dual statement of the proposed subproblem for SAND minimum compliance design agrees with the ever‐popular optimality criterion method – which is a nested analysis and design formulation. This relates, in turn, to the known equivalence between rudimentary dual sequential approximate optimization algorithms based on reciprocal (and exponential) intervening variables and the optimality criterion method. Copyright © 2016 John Wiley & Sons, Ltd.