Partial particle and wave information and weak duality games

Duality games are a way of looking at wave-particle duality. In these games. Alice and Bob together are playing against the House. The House specifies, at random, which of two sub-games Alice and Bob will play. One game, Ways, requires that they obtain path information about a particle going through an N-path interferometer and the other, Phases, requires that they obtain phase information. In general, because of wave-particle duality, Alice and Bob cannot always win the overall game. However, if the required amount of path and phase information is not too great, for example specifying a set of paths or phases, one of which is the right one, then they can always win. Here we study examples of duality games that can always be won, and develop a wave-particle duality relation expressed only in terms of mutual information to help analyze these games.

Correlator correspondences for subregular $$ \mathcal{W} $$-algebras and principal $$ \mathcal{W} $$-superalgebras

We examine a strong/weak duality between a Heisenberg coset of a theory with $$ \mathfrak{sl} $$ sl n subregular $$ \mathcal{W} $$ W -algebra symmetry and a theory with a $$ \mathfrak{sl} $$ sl n|1-structure. In a previous work, two of the current authors provided a path integral derivation of correlator correspondences for a series of generalized Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality. In this paper, we derive correlator correspondences in a similar way but for a different series of generalized duality. This work is a part of the project to realize the duality of corner vertex operator algebras proposed by Gaiotto and Rapčák and partly proven by Linshaw and one of us in terms of two dimensional conformal field theory. We also examine another type of duality involving an additional pair of fermions, which is a natural generalization of the fermionic FZZ-duality. The generalization should be important since a principal $$ \mathcal{W} $$ W -superalgebra appears as its symmetry and the properties of the superalgebra are less understood than bosonic counterparts.

3d large $N$ vector models at the boundary

We consider a 4d scalar field coupled to large NN free or critical O(N)O(N) vector models, either bosonic or fermionic, on a 3d boundary. We compute the \betaβ function of the classically marginal bulk/boundary interaction at the first non-trivial order in the large NN expansion and exactly in the coupling. Starting with the free (critical) vector model at weak coupling, we find a fixed point at infinite coupling in which the boundary theory is the critical (free) vector model and the bulk decouples. We show that a strong/weak duality relates one description of the renormalization group flow to another one in which the free and the critical vector models are exchanged. We then consider the theory with an additional Maxwell field in the bulk, which also gives decoupling limits with gauged vector models on the boundary.

Self-dualities and renormalization dependence of the phase diagram in 3d O(N) vector models

In the classically unbroken phase, 3d O(N) symmetric ϕ4 vector models admit two equivalent descriptions connected by a strong-weak duality closely related to the one found by Chang and Magruder long ago. We determine the exact analytic renormalization dependence of the critical couplings in the weak and strong branches as a function of the renormalization scheme (parametrized by κ) and for any N. It is shown that for κ = κ∗ the two fixed points merge and then, for κ < κ∗, they move into the complex plane in complex conjugate pairs, making the phase transition no longer visible from the classically unbroken phase. Similar considerations apply in 2d for the N = 1 ϕ4 theory, where the role of classically broken and unbroken phases is inverted. We verify all these considerations by computing the perturbative series of the 3d O(N) models for the vacuum energy and for the mass gap up to order eight, and Borel resumming the series. In particular, we provide numerical evidence for the self-duality and verify that in renormalization schemes where the critical couplings are complex the theory is gapped. As a by-product of our analysis, we show how the non-perturbative mass gap at large N in 2d can be seen as the analytic continuation of the perturbative one in the classically unbroken phase.

Special Class of Second-Order Non-Differentiable Symmetric Duality Problems with (G,αf)-Pseudobonvexity Assumptions

In this paper, we introduce the various types of generalized invexities, i.e., α f -invex/ α f -pseudoinvex and ( G , α f ) -bonvex/ ( G , α f ) -pseudobonvex functions. Furthermore, we construct nontrivial numerical examples of ( G , α f ) -bonvexity/ ( G , α f ) -pseudobonvexity, which is neither α f -bonvex/ α f -pseudobonvex nor α f -invex/ α f -pseudoinvex with the same η . Further, we formulate a pair of second-order non-differentiable symmetric dual models and prove the duality relations under α f -invex/ α f -pseudoinvex and ( G , α f ) -bonvex/ ( G , α f ) -pseudobonvex assumptions. Finally, we construct a nontrivial numerical example justifying the weak duality result presented in the paper.

A Kind of New Higher-Order Mond-Weir Type Duality for Set-Valued Optimization Problems

In this paper, we introduce the notion of higher-order weak adjacent epiderivative for a set-valued map without lower-order approximating directions and obtain existence theorem and some properties of the epiderivative. Then by virtue of the epiderivative and Benson proper efficiency, we establish the higher-order Mond-Weir type dual problem for a set-valued optimization problem and obtain the corresponding weak duality, strong duality and converse duality theorems, respectively.

Symmetric duality results for second-order nondifferentiable multiobjective programming problem

In this article, we study the existence of Gf-bonvex/Gf -pseudo-bonvex functions and construct various nontrivial numerical examples for the existence of such type of functions. Furthermore, we formulate Mond-Weir type second-order nondifferentiable multiobjective programming problem and give a nontrivial concrete example which justify weak duality theorem present in the paper. Next, we prove appropriate duality relations under aforesaid assumptions.

Duality relations for second-order programming problem under (G,αf)-bonvexity assumptions

In this paper, we introduce the definition of [Formula: see text]-bonvex/[Formula: see text]-pseudobonvex functions and to show the existence of such functions, we construct nontrivial numerical examples. In the next section, we formulate a pair of second-order symmetric dual model in optimization problem and proved the duality results under [Formula: see text]-bonvexity/[Formula: see text]-pseudobonvexity assumptions. Further, we also construct nontrivial concrete examples which justifying definitions as well as the weak duality theorem presented in the paper.

Duality and Dimensionality Reduction Discrete Line Generation Algorithm for a Triangular Grid

Vectors are a key type of geospatial data, and their discretization, which involves solving the problem of generating a discrete line, is particularly important. In this study, we propose a method for constructing a discrete line mathematical model for a triangular grid based on a “weak duality” hexagonal grid, to overcome the drawbacks of existing discrete line generation algorithms for a triangular grid. First, a weak duality relationship between triangular and hexagonal grids is explored. Second, an equivalent triangular grid model is established based on the hexagonal grid, using this weak duality relationship. Third, the two-dimensional discrete line model is solved by transforming it into a one-dimensional optimal wandering path model. Finally, we design and implement the dimensionality reduction generation algorithm for a discrete line in a triangular grid. The results of our comparative experiment indicate that the proposed algorithm has a computation speed that is approximately 10 times that of similar existing algorithms; in addition, it has better fitting effectiveness. Our proposed algorithm has broad applications, and it can be used for real-time grid transformation of vector data, discrete global grid system (DGGS), and other similar applications.