Duality Formula Research Articles

An interpolation of the generalized duality formula for the Schur multiple zeta values to complex functions

An interpolation of the generalized duality formula for the Schur multiple zeta values to complex functions

Some Symmetry and Duality Theorems on Multiple Zeta(-Star) Values

In this paper, we provide a symmetric formula and a duality formula relating multiple zeta values and zeta-star values. We find that the summation ∑a+b=r−1(−1)aζ★(a+2,{2}p−1)ζ★({1}b+1,{2}q) equals ζ★({2}p,{1}r,{2}q)+(−1)r+1ζ★({2}q,r+2,{2}p−1). With the help of this equation and Zagier’s ζ★({2}p,3,{2}q) formula, we can easily determine ζ★({2}p,1,{2}q) and several interesting expressions.

X-y duality in topological recursion for exponential variables via quantum dilogarithm

For a given spectral curve, the theory of topological recursion generates two different families \omega_{g,n}ωg,n and \omega_{g,n}^\veeωg,n∨ of multi-differentials, which are for algebraic spectral curves related via the universal x-yx−y duality formula. We propose a formalism to extend the validity of the x-yx−y duality formula of topological recursion from algebraic curves to spectral curves with exponential variables of the form e^x=F(e^y)ex=F(ey) or e^x=F(y)e^{a y}ex=F(y)eay with FF rational and aa some complex number, which was in principle already observed in [Commun. Number Theory Phys. 13, 763 (2019); J. Lond. Math. Soc. 109, e12946 (2024)]. From topological recursion perspective the family \omega_{g,n}^\veeωg,n∨ would be trivial for these curves. However, we propose changing the n=1n=1 sector of \omega_{g,n}^\veeωg,n∨ via a version of the Faddeev’s quantum dilogarithm which will lead to the correct two families \omega_{g,n}ωg,n and \omega_{g,n}^\veeωg,n∨ related by the same x-yx−y duality formula as for algebraic curves. As a consequence, the x-yx−y symplectic transformation formula extends further to important examples governed by topological recursion including, for instance, Gromov-Witten invariants of \mathbb{C}^3ℂ3 (or, equivalently, triple Hodge integrals), orbifold Hurwitz numbers, and stationary Gromov-Witten invariants of \mathbb{P}^1ℙ1. The proposed formalism is related to the issue topological recursion encounters for specific choices of framings for the topological vertex curve.

Duality in non-Hermitian random matrix theory

We consider 9 Gaussian matrix ensembles characterized by single symmetry among the 38-fold symmetry classification classes of non-Hermitian random matrices, and establish exact duality formulae of certain observables between them. Particularly, averaged products of K characteristic polynomials in an N×N matrix ensemble can be expressed in terms of another K×K matrix ensemble. Our method is to combine matrix-valued heat equations and differential identities for determinants and Pfaffians and has more possible applications.

Coupling capacity in C*-algebras

Given two unital C*-algebras equipped with states and a positive operator in the enveloping von Neumann algebra of their minimal tensor product, we define three parameters that measure the capacity of the operator to align with a coupling of the two given states. Further, we establish a duality formula that shows the equality of two of the parameters for operators in the minimal tensor product of the relevant C*-algebras. In the context of abelian C*-algebras, our parameters are related to quantitative versions of Arveson's null set theorem and to dualities considered in the theory of optimal transport. On the other hand, restricting to matrix algebras we recover and generalize quantum versions of Strassen's theorem. We show that in the latter case our parameters can detect maximal entanglement and separability.

On the exponential growth rates of lattice animals and interfaces

AbstractWe introduce a formula for translating any upper bound on the percolation threshold of a lattice$G$into a lower bound on the exponential growth rate of lattice animals$a(G)$and vice versa. We exploit this in both directions. We obtain the rigorous lower bound${\dot{p}_c}({\mathbb{Z}}^3)\gt 0.2522$for 3-dimensional site percolation. We also improve on the best known asymptotic bounds on$a({\mathbb{Z}}^d)$as$d\to \infty$. Our formula remains valid if instead of lattice animals we enumerate certain subspecies called interfaces. Enumerating interfaces leads to functional duality formulas that are tightly connected to percolation and are not valid for lattice animals, as well as to strict inequalities for the percolation threshold.Incidentally, we prove that the rate of the exponential decay of the cluster size distribution of Bernoulli percolation is a continuous function of$p\in (0,1)$.

Duality for finite/symmetric multiple zeta values of fixed weight, depth, and height

In this paper, we prove a duality formula for the sum of the finite/symmetric multiple zeta-star values of fixed weight, depth, and height, which was conjectured by Kaneko. This result is a generalization of “height-one duality” proved by Hoffman and an analogue of the duality formula for the sum of the multiple zeta-star values proved by Li.

Measure-Based Extension of Continuous Functions and p-Average-Slope-Minimizing Regression

This work is inspired by some recent developments on the extension of Lipschitz real functions based on the minimization of the maximum value of the slopes of a reference set for this function. We propose a new method in which an integral p–average is optimized instead of its maximum value. We show that this is a particular case of a more general theoretical approach studied here, provided by measure-valued representations of the metric spaces involved, and a duality formula. For p=2, explicit formulas are proved, which are also shown to be a particular case of a more general class of measure-based extensions, which we call ellipsoidal measure extensions. The Lipschitz-type boundedness properties of such extensions are shown. Examples and concrete applications are also given.

Applications of Strassen’s theorem and Choquet theory to optimal transport problems, to uniformly convex functions and to uniformly smooth functions

We provide a unifying interpretation of various optimal transport problems as a minimisation of a linear functional over the set of all Choquet representations of a given pair of probability measures ordered with respect to a certain convex cone of functions. This allows us to provide novel proofs of duality formulae. Among our tools is Strassen’s theorem.We provide new formulations of the primal and the dual problem for martingale optimal transport employing a novel representation of the set of extreme points of probability measures in convex order on Euclidean space. We exhibit a link to uniformly convex and uniformly smooth functions and provide a new characterisation of such functions.We introduce a notion of martingale triangle inequality. We show that Kantorovich–Rubinstein duality bears an analogy in the martingale setting employing the cost functions that satisfy the inequality.

Another proof of Kaneko–Ohno’s duality formula for multiple zeta star values

Another proof of Kaneko–Ohno’s duality formula for multiple zeta star values

Matrix group integrals, surfaces, and mapping class groups II: $$\textrm{O}\left( n\right) $$ and $$\textrm{Sp}\left( n\right) $$

Let w be a word in the free group on r generators. The expected value of the trace of the word in r independent Haar elements of \(\textrm{O}(n)\) gives a function \({\mathcal {T}}r_{w} ^{\textrm{O}}(n)\) of n. We show that \({\mathcal {T}}r_{w} ^{\textrm{O}}(n)\) has a convergent Laurent expansion at \(n=\infty \) involving maps on surfaces and \(L^{2}\)-Euler characteristics of mapping class groups associated to these maps. This can be compared to known, by now classical, results for the GUE and GOE ensembles, and is similar to previous results concerning \({\textrm{U}}\left( n\right) \), yet with some surprising twists. A priori to our result, \({\mathcal {T}}r_{w} ^{\textrm{O}}(n)\) does not change if w is replaced with \(\alpha (w)\) where \(\alpha \) is an automorphism of the free group. One main feature of the Laurent expansion we obtain is that its coefficients respect this symmetry under \(\textrm{Aut}({\textbf{F}} _{r})\). As corollaries of our main theorem, we obtain a quantitative estimate on the rate of decay of \({\mathcal {T}}r _{w}^{\textrm{O}}(n)\) as \(n\rightarrow \infty \), we generalize a formula of Frobenius and Schur, and we obtain a universality result on random orthogonal matrices sampled according to words in free groups, generalizing a theorem of Diaconis and Shahshahani. Our results are obtained more generally for a tuple of words \(w_{1},\ldots ,w_{\ell }\), leading to functions \({\mathcal {T}}r_{w_{1},\ldots ,w_{\ell }} ^{\textrm{O}}\). We also obtain all the analogous results for the compact symplectic groups \(\textrm{Sp}(n)\) through a rather mysterious duality formula.

A duality theorem for the ic-resurgence of edge ideals

The aim of this work is to use linear programming and polyhedral geometry to prove a duality formula for the ic-resurgence of edge ideals. We show that the ic-resurgence of the edge ideal I of a clutter C and the ic-resurgence of the edge ideal I∨ of the blocker C∨ of C coincide. If C is the clutter of bases of certain uniform matroids, we recover a formula for the resurgence of I, and if C is a connected non-bipartite graph with a perfect matching, we show a formula for the Waldschmidt constant of I∨.

On the Serre Functor in the Category of Strict Polynomial Functors

AbstractWe study a Serre functor in functor categories related to the category $\mathcal {P}_{d}$ P d of strict polynomial functors over a field of positive characteristic. Our main result shows that the derived category of the category of affine strict polynomial functors in some cases carries the structure of Calabi–Yau category. We also re-obtain the Poincaré duality formulas for Ext groups in $\mathcal {P}_{d}$ P d and construct a certain recollement diagram relating the derived categories of affine and ordinary strict polynomial functors.

A Note on Riccati Matrix Difference Equations

Discrete algebraic Riccati equations and their fixed points are well understood and arise in a variety of applications, however, the time-varying equations have not yet been fully explored in the literature. In this article we provide a self-contained study of discrete time Riccati matrix difference equations. In particular, we provide a novel Riccati semigroup duality formula and a new Floquet-type representation for these equations. Due to the aperiodicity of the underlying flow of the solution matrix, conventional Floquet theory does not apply in this setting and thus further analysis is required. We illustrate the impact of these formulae with an explicit description of the solution of time-varying Riccati difference equations and its fundamental-type solution in terms of the fixed point of the equation and an invertible linear matrix map, as well as uniform upper and lower bounds on the Riccati maps. These are the first results of this type for time varying Riccati matrix difference equations.

The Positivity of Peace Intra-and Inter-State: Strategizing the Balance of Peace-State in Multi-Level Conflict Management Approach

This study develops two hypergame-models of intra-and inter-state conflicts within one coherent conflict management approach, modeling conflicts while considering the complexity of real-world circumstances. The aim is to apply theories of IR and media fields, composing a merged diplomatic-strategic approach for achieving peace rather than war within and inter-nations. Using the hypergame theory as a theoretical-methodological basis for modeling, we stand firmly at some equilibria positions, specifying the modality of achieving positive peace as a “win-win” formula nationally, regionally, and globally, thence strategizing a balance of peace-state of intra-and-inter-state conflicts. It can be argued that this study’s (Positivity of Peace Hypergame Model)’s fundamental equilibrium is equivalently equal to the dual formula of (Global Hegemony of Peace Versus Realist Hegemony of War’s Inevitability) as proved strategically, applicably through our provided equations. Also, a (Balance of Peace-State Hypergame Model)’s equilibria of intra-state conflicts occur at the (Mini-Maximum Equilibrium) and (Nash Equilibrium) positions. Simultaneously, this model’s solution points of the inter-state conflicts are achieved parallelly with the occurrence of three Nash equilibria, where the success of a peace process might be “limited, normal, or multiplied” as given within its relevant equations.

A Dual Formula for the Noncommutative Transport Distance

In this article we study the noncommutative transport distance introduced by Carlen and Maas and its entropic regularization defined by Becker and Li. We prove a duality formula that can be understood as a quantum version of the dual Benamou–Brenier formulation of the Wasserstein distance in terms of subsolutions of a Hamilton–Jacobi–Bellmann equation.

Coxeter combinatorics for sum formulas in the representation theory of algebraic groups

Let G G be a simple algebraic group over an algebraically closed field F \mathbb {F} of characteristic p ≥ h p \geq h , the Coxeter number of G G . We observe an easy ‘recursion formula’ for computing the Jantzen sum formula of a Weyl module with p p -regular highest weight. We also discuss a ‘duality formula’ that relates the Jantzen sum formula to Andersen’s sum formula for tilting filtrations and we give two different representation theoretic explanations of the recursion formula. As a corollary, we also obtain an upper bound on the length of the Jantzen filtration of a Weyl module with p p -regular highest weight in terms of the length of the Jantzen filtration of a Weyl module with highest weight in an adjacent alcove.

On the validity of variational inequalities for obstacle problems with non-standard growth


 The aim of the paper is to show that the solutions to variational problems with non-standard growth conditions satisfy a corresponding variational inequality expressed in terms of a duality formula between the constrained minimizers and the corresponding dual maximizers, without any smallness assumptions on the gap between growth and coercitivity exponents. Our results rely on techniques based on Convex Analysis that consist in establishing pointwise relations that are preserved passing to the limit. We point out that we are able to deal with very general obstacle quasi-continuous up to a subset of zero capacity.

Applications of a duality between generalized trigonometric and hyperbolic functions II

Generalized trigonometric functions and generalized hyperbolic functions can be converted to each other by the duality formulas previously discovered by the authors. In this paper, we apply the duality formulas to prove dual pairs of Wilker-type inequalities, Huygens-type inequalities, and (relaxed) Cusa-Huygens-type inequalities for the generalized functions. In addition, multiple- and double-angle formulas not previously obtained are also given.

Variants of multiple zeta values with even and odd summation indices

In this paper, we define and study a variant of multiple zeta values of level 2 (which is called multiple mixed values or multiple $M$-values, MMVs for short), which forms a subspace of the space of alternating multiple zeta values. This variant includes both Hoffman's multiple $t$-values and Kaneko-Tsumura's multiple $T$-values as special cases. We set up the algebra framework for the double shuffle relations (DBSFs) of the MMVs, and exhibits nice properties such as duality, integral shuffle relation, series stuffle relation, etc., similar to ordinary multiple zeta values. Moreover, we study several $T$-variants of Kaneko-Yamamoto type multiple zeta values by establishing some explicit relations between these $T$-variants and Kaneko-Tsumura $\psi$-values. Furthermore, we prove that all Kaneko-Tsumura $\psi$-values can be expressed in terms of Kaneko-Tsumura multiple $T$-values by using multiple associated integrals, and find some duality formulas for Kaneko-Tsumura $\psi$-values. We also discuss the explicit evaluations for a kind of MMVs of depth two and three by using the method of contour integral and residue theorem. Finally, we investigate the dimensions of a few interesting subspaces of MMVs for small weights.