Canonical Dual Research Articles

Duality relations associated with weak g-R-duals

Recently, duality principles in g-frame theory were studied. This paper addresses a relaxation of g-R-duals. We introduce the notion of weak g-R-dual in g-frame theory setting, and present the link between frame properties of a bounded operator sequence and its weak g-R-duals. Using weak g-R-duals, we characterize g-frames and (unitary) equivalence between g-frames. And using pseudo-inverse operators, we represent the canonical duals of weak g-R-duals. Some examples are also provided.

Homological mirror symmetry of CPn and their products via Morse homotopy

We propose a way of understanding homological mirror symmetry when a complex manifold is a smooth compact toric manifold. So far, in many examples, the derived category Db(coh(X)) of coherent sheaves on a toric manifold X is compared with the Fukaya–Seidel category of the Milnor fiber of the corresponding Landau–Ginzburg potential. We instead consider the dual torus fibration π: M → B of the complement of the toric divisors in X, where B̄ is the dual polytope of the toric manifold X. A natural formulation of homological mirror symmetry in this setup is to define Fuk(M̄) a variant of the Fukaya category and show the equivalence Db(coh(X))≃Db(Fuk(M̄)). As an intermediate step, we construct the category Mo(P) of weighted Morse homotopy on P≔B̄ as a natural generalization of the weighted Fukaya–Oh category proposed by Kontsevich and Soibelman [Symplectic geometry and mirror symmetry (Seoul, 2000) (World Science Publishing, 2001), p. 203]. We then show that a full subcategory MoE(P) of Mo(P) generates Db(coh(X)) for the cases X is a complex projective space and their products.

Optimal time trajectory and coordination for connected and automated vehicles

In this paper, we provide a decentralized theoretical framework for coordination of connected and automated vehicles (CAVs) at different traffic scenarios. The framework includes: (1) an upper-level optimization that yields for each CAV its optimal time trajectory and lane to pass through a given traffic scenario while alleviating congestion; and (2) a low-level optimization that yields for each CAV its optimal control input (acceleration/deceleration). We provide a complete, analytical solution of the low-level optimization problem that includes the rear-end, speed-dependent safety constraint. Furthermore, we provide a problem formulation for the upper-level optimization in which there is no duality gap. The latter implies that the optimal time trajectory for each CAV does not activate any of the state, control, and safety constraints of the low-level optimization, thus allowing for online implementation. Finally, we present a geometric duality framework with hyperplanes to derive the condition under which the optimal solution of the upper-level optimization always exists. We validate the effectiveness of the proposed theoretical framework through simulation.

Duality relations for a class of generalized weak R-duals

Since the introduction of R-duals by Casazza, Kutyniok and Lammers with the motivation to obtain a general version of duality principle in Gabor analysis, various R-duals and some relaxations of the R-dual setup have been introduced and studied by some mathematicians. They provide a powerful tool in the analysis of duality relations in general frame theory, and are far beyond the duality principle in Gabor analysis. In this paper, we introduce the concept of generalized weak R-dual based on a pair of frames which is a relaxation of the R-dual setup. Using generalized weak R-duals, we characterize the frame properties of a sequence and the equivalence between two frames, prove that the generalized weak R-duals of frames (Riesz bases) are frame sequences (frames), and present a coefficient expression corresponding to the canonical duals of generalized weak R-duals. Some examples are provided to illustrate the generality of the theory.

On extraction of P-configurations

Consider a parallelotope P and its dual polytope P∗. The paral- lelotope configuration or the p-configuration is a system of lines and points projected vertices and edges of the polytope P∗ from the center of the polytope P∗ to a special (n − 1)-dimensional hyperplane. The extraction of the parallelotope defines an ex- traction of the p-configuration, as well. In this paper we examine properties of the extraction of the p-configuration.

A polar dual to the momentum of toric Fano manifolds

Abstract We introduce an invariant on the Fano polytope of a toric Fano manifold as a polar dual counterpart to the momentum of its polar dual polytope. Moreover, we prove that if the momentum of the polar dual polytope is equal to zero, then the dual invariant on a Fano polytope vanishes.

Strongly self-dual polytopes and distance graphs in the unit sphere

Lovasz proved that the chromatic number of the graph formed by the principal diagonals of an $$n$$ -dimensional strongly self-dual polytope is greater than or equal to $$n+1$$ . There is equality if the length of the principal diagonals is greater than the Euclidean diameter of the monochromatic parts of that coloring of the unit sphere which is based on a partition of $$n+1$$ congruent spherical regular simplices. We determine this quantity for all $$n$$ and prove that in dimension three all such graphs can be colored by four colors.

Pseudo Wavelet Frames

We seek to alleviate the unpleasantness of the fact that the class of wavelet frames is not closed under canonical duals by introducing and studying a new class of frames that we call pseudo wavelet frames, which includes all wavelet frames and is ‘self-dual’ in terms of canonical duals. The class also has the ‘extension property’ and is invariant under the Fourier transform.

Gravity dual of Navier-Stokes equation in a rotating frame through parallel transport

The fluid-gravity correspondence documents a precise mathematical map between a class of dynamical spacetime solutions of the Einstein field equations of gravity and the dynamics of its corresponding dual fluid flows governed by the Navier-Stokes (NS) equations of hydrodynamics. This striking connection has been explored in several dynamics-based approaches and has surfaced in various forms over the past four decades. In a recent construction, it has been shown that the manifold properties of geometric duals are in fact intimately connected to the dynamics of incompressible fluids, thus bypassing the conventional on-shell standpoints. Following such a prescription, we construct the geometrical description that effectively captures the dynamics of an incompressible NS fluid with respect to a uniformly rotating frame. We propose the gravitational dual(s) described by bulk metric(s) in $(p+2)$-dimensions such that the equations of parallel transport of an appropriately defined bulk velocity vector field when projected onto an induced timelike hypersurface require that the incompressible NS equation of a fluid relative to a uniformly rotating frame be satisfied at the relevant perturbative order in $(p+1)$-dimensions. We argue that free fluid flows on manifold(s) described by the proposed metric(s) can be effectively considered as an equivalent theory of non-relativistic viscous fluid dynamics with respect to (w.r.t) a uniform rotating frame. We also present suggestive insights as to how space-time rotation parameters encode information pertaining to the inertial effects in the corresponding fluid dual.

An FPTAS for the volume of some [formula omitted]-polytopes — It is hard to compute the volume of the intersection of two cross-polytopes

Given an n-dimensional convex body by a membership oracle in general, it is known that any polynomial-time deterministic algorithm cannot approximate its volume within ratio (n/log⁡n)n. There is a substantial progress on randomized approximation such as Markov chain Monte Carlo for a high-dimensional volume, and for many #P-hard problems, while only a few #P-hard problems are known to yield deterministic approximation. Motivated by the problem of deterministically approximating the volume of a V-polytope, that is a polytope with a small number of vertices and (possibly) exponentially many facets, this paper investigates the problem of computing the volume of a “knapsack dual polytope,” which is known to be #P-hard due to Khachiyan (1989) [16]. We reduce an approximate volume of a knapsack dual polytope to that of the intersection of two cross-polytopes in a short distance, and give FPTASs for those volume computations. Interestingly, computing the volume of the intersection of two cross-polytopes (i.e., L1-balls) is #P-hard, unlike the cases of L∞-balls or L2-balls.

Invertibility of generalized g-frame multipliers in Hilbert spaces

ABSTRACT In this paper, we investigate the invertibility of generalized g-Bessel multipliers. Sufficient and necessary conditions for invertibility are determined depending on the optimal g-frame bounds. Moreover, we show that, for semi-normalized symbols, the inverse of any invertible generalized g-frame multiplier can be represented as a generalized g-frame multiplier with the reciprocal symbol and dual g-frames of the given ones. Furthermore, we investigate some equivalent conditions for the special case, when both dual g-frames can be chosen to be the canonical duals. Finally, we give several approaches for constructing invertible generalized g-frame multipliers from the given ones. It is worth mentioning that some of our results are quite different from those studied in the previous literatures on this topic.

Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals

We continue the study of positive geometries underlying the Grassmannian string integrals, which are a class of “stringy canonical forms”, or stringy integrals, over the positive Grassmannian mod torus action, G+(k, n)/T . The leading order of any such stringy integral is given by the canonical function of a polytope, which can be obtained using the Minkowski sum of the Newton polytopes for the regulators of the integral, or equivalently given by the so-called scattering-equation map. The canonical function of the polytopes for Grassmannian string integrals, or the volume of their dual polytopes, is also known as the generalized bi-adjoint ϕ3 amplitudes. We compute all the linear functions for the facets which cut out the polytope for all cases up to n = 9, with up to k=4 and their parity conjugate cases. The main novelty of our computation is that we present these facets in a manifestly gauge-invariant and cyclic way, and identify the boundary configurations of G+(k, n)/T corresponding to these facets, which have nice geometric interpretations in terms of n points in (k−1)-dimensional space. All the facets and configurations we discovered up to n = 9 directly generalize to all n, although new types are still needed for higher n.

Weighted Zak transforms and the dual tiling condition

For T>0 and a periodic complex-valued sequence c, we introduce the weighted Zak transform ZcT and study its properties. As our main result, we give characterizations for mapping properties and unitarity of ZcT:L2(R)→L2(S) where functions in the image are understood as restrictions to a set S⊂R2. In particular, we show that for almost every choice of L-periodic sequences c, the mapping properties of ZcT:L2(R)→L2(S) simplify to statements on the geometry of S. They involve a dual tiling condition on S with respect to the lattices TZ×ΩZ and LTZ×LΩZ where Ω=1/(LT).

Topological Sweep for Multi-Target Detection of Geostationary Space Objects

Conducting surveillance of the geocenric orbits is a key task towards achieving space situational awareness (SSA). Our work focuses on the optical detection of man-made objects (e.g., satellites, space debris) in Geostationary orbit (GEO), which is home to major space assets such as telecommunications and Earth observing satellites. GEO object detection is challenging due to the distance of the targets, which appear as small dim point-like objects among a background of streak-like objects. In this paper, we propose a novel multi-target detection technique based on topological sweep, to find GEO objects from a short sequence of optical images. Our topological sweep technique exploits the geometric duality that underpins the approximately linear trajectory of target objects across the sequence, to extract the targets from significant clutter and noise. Unlike standard multi-target methods, our algorithm deterministically solves a combinatorial problem to ensure high-recall rates without requiring accurate initializations. The usage of geometric duality also yields an algorithm that is computationally efficient and suitable for online processing.

On a class of weak R-duals and the duality relations

The concept of R-duals of a sequence was first introduced with the motivation to obtain a general version of duality principle in Gabor analysis. Since then, various R-duals (types II, III, IV) and some relaxations of the R-dual setup have been introduced and studied by some mathematicians. All these “R-duals” provide a powerful tool in the analysis of duality relations in general frame theory. It is of independent interest in mathematics and far beyond the duality principle in Gabor analysis. Observe that the underlying sequences of a R-dual are a pair of orthonormal bases. In this paper we introduce the concept of weak R-duals based on a pair of Parseval frames. It is a new relaxation of the R-dual setup. We obtain a characterization of frames based on their weak R-duals, and prove that the weak R-dual of a frame (Riesz basis) is a frame sequence (frame). We also characterize (unitarily) equivalent frames in terms of weak R-duals. Finally, we present an explicit expression of the canonical duals of weak R-duals.

Approximate duals and morphisms of Hilbert $C^{\ast}$ -modules

In the following we show that, under some conditions, φ-morphisms preserve duals and approximate duals of frames in Hilbert C*-modules. Moreover, using φ-morphisms and some concepts related to frame theory such as modular Riesz bases, canonical duals, tensor products, and Bessel multipliers, we construct new approximate duals, considering in particular the approximate duals constructed by compact operators and morphisms.

Valence-Shell Electron-Pair Repulsion Theory Revisited: An Explanation for Core Polarization.

Valence-shell electron-pair repulsion (VSEPR) theory constitutes one of the pillars of theoretical predictive chemistry. It was proposed even before the advent of the concept of "spin", and it is still a very useful tool in chemistry. In this article we propose an extension of VSEPR theory to understand the core structure and predict core polarization in the main-group elements. We show from first principles (Electron Localization Function analysis) how the inner- and outer-core shells are organized. In particular, electrons in these regions are structured following the shape of the dual polyhedron of the valence shell (3rd period) or the equivalent polyhedron (4th and 5th periods). We interpret these results in terms of "hard" and "soft" core character. All the studied systems follow this trend, providing a framework for predicting electron distribution in the core. We also show that lone pairs behave as "standard ligands" in terms of core polarization. The predictive character of the model was tested by proposing the core polarization in different systems not included in the original set (such as XeF4 and [Fe(CN)6 ]3- ) and checking the hypothesis by means of a posteriori calculations. From the experimental point of view, the extension of VSEPR to the core region has consequences for current crystallography research. In particular, it explains the core polarization revealed by high resolution X-ray experiments.

On C–D controlled frames and their duality

In this paper, we introduce controlled frames with two bounded operators when the controller operators are not invertible and positive, in general. We also describe such controlled frames by using their synthesis operators. Next, we define the canonical and alternate duals for them. Finally, several dual characterizations are given.

Sampling and Reconstruction of Multiple-Input Multiple-Output Channels

Based on the recent development of sampling and reconstruction results for slowly time-varying single-input single-output channel operators, we derive sampling results in the multiple-input multiple-output setting where all subchannels satisfy an underspread condition, that is, their spreading functions are supported on individual sets of small measure. At the center of our work is the extension of the single-input single-output dual tiling condition to this setting; it characterizes which periodic weighted delta trains can be used to identify a given class of multiple-input multiple-output channel operators satisfying a spreading support constraint. Building on the dual tiling condition, we compute reconstruction formulas for the operator's symbol in closed form and discuss the problem of identifying multiple-input multiple-output operators where only restrictions in size, but not on location and geometry, of the subchannel spreading supports are known.

Information Geometric Duality of ϕ-Deformed Exponential Families.

In the world of generalized entropies—which, for example, play a role in physical systems with sub- and super-exponential phase space growth per degree of freedom—there are two ways for implementing constraints in the maximum entropy principle: linear and escort constraints. Both appear naturally in different contexts. Linear constraints appear, e.g., in physical systems, when additional information about the system is available through higher moments. Escort distributions appear naturally in the context of multifractals and information geometry. It was shown recently that there exists a fundamental duality that relates both approaches on the basis of the corresponding deformed logarithms (deformed-log duality). Here, we show that there exists another duality that arises in the context of information geometry, relating the Fisher information of -deformed exponential families that correspond to linear constraints (as studied by J.Naudts) to those that are based on escort constraints (as studied by S.-I. Amari). We explicitly demonstrate this information geometric duality for the case of -entropy, which covers all situations that are compatible with the first three Shannon–Khinchin axioms and that include Shannon, Tsallis, Anteneodo–Plastino entropy, and many more as special cases. Finally, we discuss the relation between the deformed-log duality and the information geometric duality and mention that the escort distributions arising in these two dualities are generally different and only coincide for the case of the Tsallis deformation.