Classical Duality Research Articles

Controllability for forward stochastic heat equations with dynamic boundary conditions without extra forces

In this paper, we continue the study of some controllability issues for the forward stochastic heat equation with dynamic boundary conditions. The main novelty in the present paper consists of considering only one control without extra forces in the noise parts. Under a strong measurability condition, and using a spectral inequality, we first establish an appropriate observability inequality for the corresponding adjoint system. Then, by the classical duality approach, the null and approximate controllability results are established.

Equivalent conditions of null controllability for controlled stochastic relaxed system

ABSTRACT This paper focuses on the investigation of null controllability in the context of stochastic controlled relaxed systems with one control, aiming to establish several equivalent conditions for this property. More precisely, by the classical duality analysis, we present the equivalence between an observability inequality for the backward stochastic system associated with the relaxed system and null controllability. Furthermore, we characterize the null controllability through a minimal norm control problem, employing a constructed quadratic functional and leveraging a variational characterization of its minimizer. Finally, we provide an application of our findings and extend the analysis to relaxed systems with two controls.

The [formula omitted]-Adic Schrödinger equation and the two-slit experiment in quantum mechanics

p-Adic quantum mechanics is constructed from the Dirac–von Neumann axioms identifying quantum states with square-integrable functions on the N-dimensional p-adic space, QpN. This choice is equivalent to the hypothesis of the discreteness of the space. The time is assumed to be a real variable. The p-adic quantum mechanics is motivated by the question: what happens with the standard quantum mechanics if the space has a discrete nature? The time evolution of a quantum state is controlled by a nonlocal Schrödinger equation obtained from a p-adic heat equation by a temporal Wick rotation. This p-adic heat equation describes a particle performing a random motion in QpN. The Hamiltonian is a nonlocal operator; thus, the Schrödinger equation describes the evolution of a quantum state under nonlocal interactions. In this framework, the Schrödinger equation admits complex-valued plane wave solutions, which we interpret as p-adic de Broglie waves. These mathematical waves have all wavelength p−1. In the p-adic framework, the double-slit experiment cannot be explained using the interference of the de Broglie waves. The wavefunctions can be represented as convergent series in the de Broglie waves, but the p-adic de Broglie waves are just mathematical objects. Only the square of the modulus of a wave function has a physical meaning as a time-dependent probability density. These probability densities exhibit interference patterns similar to the ones produced by ‘quantum waves’. In the p-adic framework, in the double-slit experiment, each particle goes through one slit only. The p-adic quantum mechanics is an analog (or model) of the standard one under the hypothesis of the existence of a Planck length. The precise connection between these two theories is an open problem. Finally, we propose the conjecture that the classical de Broglie wave-particle duality is a manifestation of the discreteness of space–time.

From Antagonism to Fusion: Military and Industrial Society Fused in Militarist Capitalism

ABSTRACT This article reconsiders the classical duality between military and industrial society and the evolutionary scheme from the first to the second. It argues and theorizes the concept of military-capitalist society, or militarist capitalism, including its theocratic-militarist variant. It elaborates a substantive index of militarist capitalism that is composed of certain indicators and proxies of the latter as its components and describes data and data sources concerning the index components. It reports the substantive findings of an empirical analysis, namely numerical militarist capitalism indexes for Western and comparable societies such as Organisation for Economic Co-operation and Development (OECD) countries. The main results are that military capitalism is primarily a phenomenon of regions outside Western Europe; of unregulated, inegalitarian, and coercive capitalism; and of societies dominated by conservatism and religion-overdetermined cultures, such as the United States and other countries. It discusses these findings with reference to these countries. Lastly, it draws conclusions and theoretical implications.

Duality between the Maxwell-Chern-Simons and self-dual models in very special relativity

This work investigates the classical and quantum duality between the SIM(1)-Maxwell-Chern-Simons (MCS) model and its self-dual counterpart. Initially, we focus on free-field cases to establish equivalence through two distinct approaches: comparing the equations of motion and utilizing the master Lagrangian method. In both instances, the classical correspondence between the self-dual and MCS dual fields undergoes modifications due to very special relativity (VSR). Specifically, the duality is established when the associated VSR-mass parameters are identical, and the dual field is introduced through a non-local VSR correction. Furthermore, we analyze the duality when the self-dual model is minimally coupled to fermions. As a result, we demonstrate that Thirring-like interactions, corrected for non-local VSR contributions, are included in the MCS model. Additionally, we establish the quantum equivalence of the models by performing a functional integration of the fields and comparing the resulting effective Lagrangians.

Howe Duality of Type P

AbstractWe establish classical and categorical Howe dualities between the Lie superalgebras $$\mathfrak {p}(m)$$ p ( m ) and $$\mathfrak {p}(n)$$ p ( n ) , for $$m,n \ge 1$$ m , n ≥ 1 . We also describe a presentation via generators and relations as well as a Kostant $$\mathbb {Z}$$ Z -form for the universal enveloping superalgebra $$U(\mathfrak {p}(m))$$ U ( p ( m ) ) .

Gravitational wave double copy of radiation from gluon shockwave collisions

In [1], we showed that the gravitational wave spectrum from trans-Planckian shockwave scattering in Einstein gravity is determined by the gravitational Lipatov vertex expressed as the bilinear double copy Γμν=12CμCν−12NμNν where Cμ is the QCD Lipatov vertex and Nμ is the QED soft photon factor. We show here that this result can be directly obtained by careful application of the classical color-kinematic duality to the spectrum obtained in gluon shockwave collisions.

Duality for Nonlinear Filtering II: Optimal Control

This paper is concerned with the development and use of duality theory for a nonlinear filtering model with white noise observations. The main contribution of this paper is to introduce a stochastic optimal control problem as a dual to the nonlinear filtering problem. The mathematical statement of the dual relationship between the two problems is given in the form of a duality principle. The constraint for the optimal control problem is the backward stochastic differential equation (BSDE) introduced in the companion paper. The optimal control solution is obtained from an application of the maximum principle, and subsequently used to derive the equation of the nonlinear filter. The proposed duality is shown to be an exact extension of the classical Kalman-Bucy duality, and different from other types of optimal control and variational formulations given in literature.

Duality for Nonlinear Filtering I: Observability

This paper is concerned with the development and use of duality theory for a hidden Markov model (HMM) with white noise observations. The main contribution of this work is to introduce a backward stochastic differential equation (BSDE) as a dual control system. A key outcome is that stochastic observability (resp. detectability) of the HMM is expressed in dual terms: as controllability (resp. stabilizability) of the dual control system. All aspects of controllability, namely, definition of controllable space and controllability gramian, along with their properties and explicit formulae, are discussed. The proposed duality is shown to be an exact extension of the classical duality in linear systems theory. One can then relate and compare the linear and the nonlinear systems. A side-by-side summary of this relationship is given in a tabular form (Table II).

Threshold-Based Belief Change

In this paper we study changes of beliefs in a ranking-theoretic setting using non-extremal implausibility thresholds for belief. We represent implausibilities as ranks and introduce natural rank changes subject to a minimal change criterion. We show that many of the traditional AGM postulates for revision and contraction are preserved, except for the postulate of Preservation which is invalid. The diagnosis for belief contraction is similar, but not exactly the same. We demonstrate that the one-shot versions of both revision and contraction can be represented as revisions based on semiorders, but in two subtly different ways. We provide sets of postulates that are sound and complete in the sense that they allow us to prove representation theorems. We show that, and explain why, the classical duality between revision and contraction, as exhibited by the Levi and Harper identities, is partly broken by threshold-based belief changes. We also study the logic of iterated threshold-based revision and contraction. The traditional Darwiche-Pearl postulates for iterated revision continue to hold, as well as two additional postulates that characterize ranking-based revision as a restricted `improvement' operator. We investigate the dual notion of iterated threshold-based belief contraction and provide a new set of postulates for it, characterizing contraction as a restricted 'degrading' operator.

Lefschetz duality for local cohomology

Since the 1974 paper by Peskine and Szpiro, liaison theory via complete intersections, and more generally via Gorenstein varieties, has become a standard tool kit in commutative algebra and algebraic geometry, allowing to compare algebraic features of linked varieties. In this paper we develop a liaison theory via quasi-Gorenstein varieties, a much broader class than Gorenstein varieties. As applications, we derive a connectedness property of quasi-Gorenstein subspace arrangements generalizing previous results by Benedetti and the first author, and we deduce the classical topological Lefschetz duality via the Stanley-Reisner correspondence.

Tate Duality in Positive Dimension over Function Fields

We extend the classical duality results of Poitou and Tate for finite discrete Galois modules over local and global fields (local duality, nine-term exact sequence, etc.) to all affine commutative group schemes of finite type, building on the recent work of Česnavičius (“Poitou-Tate without restrictions on the order,” 2015) extending these results to all finite commutative group schemes. We concentrate mainly on the more difficult function field setting, giving some remarks about the number field case along the way.

Rigid inner forms over local function fields

We generalize the concept of rigid inner forms, defined by Kaletha in [12], to the setting of a local function field F in order state the local Langlands conjectures for arbitrary connected reductive groups over F. To do this, we define for a connected reductive group G over F a new cohomology set H1(E,Z→G)⊂Hfpqc1(E,G) for a gerbe E attached to a class in Hfppf2(F,u) for a certain canonically-defined profinite commutative group scheme u, building up to an analogue of the classical Tate-Nakayama duality theorem. We define a relative transfer factor for an endoscopic datum serving a connected reductive group G over F, and use rigid inner forms to extend this to an absolute transfer factor, enabling the statement of endoscopic conjectures relating stable virtual characters and s˙-stable virtual characters for a semisimple s˙ associated to a tempered Langlands parameter.

Lattice-free and point-free: Vickers duality for subbases of stably locally compact spaces

Inspired by classic work of Wallman and more recent work of Jung-Kegelmann-Moshier and Vickers, we show how to encode general subbases of stably locally compact spaces via certain entailment relations. We further build this up to a categorical duality encompassing the classic Priestley-Stone duality and its various extensions to stably locally compact spaces by Shirota, De Vries, Hofmann-Lawson (in the stable case), Jung-Sünderhauf, Hansoul-Poussart, Bezhanishvili-Jansana, van Gool and Bice-Starling.

The Value of Data Records

Abstract Many e-commerce platforms use buyers’ personal data to intermediate their transactions with sellers. How much value do such intermediaries derive from the data record of each single individual? We characterize this value and find that one of its key components is a novel externality between records, which arises when the intermediary pools some records to withhold the information they contain. Our analysis has several implications about compensating individuals for the use of their data, guiding companies’ investments in data acquisition, and more broadly studying the demand side of data markets. Our methods combine modern information design with classic duality theory and apply to a large class of principal-agent problems.

AdS/BCFT with brane-localized scalar field

In this paper, we study the dynamics of end-of-the-world (EOW) branes in AdS with scalar fields localized on the branes as a new class of gravity duals of CFTs on manifolds with boundaries. This allows us to construct explicit solutions dual to boundary RG flows. We also obtain a variety of annulus-like or cone-like shaped EOW branes, which are not possible without the scalar field. We also present a gravity dual of a CFT on a strip with two different boundary conditions due to the scalar potential, where we find the confinement/deconfinement-like transition as a function of temperature and the scalar potential. Finally, we point out that this phase transition is closely related to the measurement-induced phase transition, via a Wick rotation.

Stone duality for spectral sheaves and the patch monad

We establish a duality between global sheaves on spectral spaces and right distributive bands. This is a sheaf-theoretical extension of classical Stone duality between spectral spaces and bounded distributive lattices.The topology of a spectral space admits a refinement, the so-called patch topology, giving rise to a patch monad on sheaves over a fixed spectral space. Under the duality just mentioned the algebras of this patch monad are shown to correspond to distributive skew lattices.

Topological dualities in the Ising model

We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in $2$ dimensions, with electromagnetic duality for finite gauge theories in $3$ dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.

Classical vertex model dualities in a family of two-dimensional frustrated quantum antiferromagnets

We study a general class of easy-axis spin models on a lattice of corner sharing even-sided polygons with all-to-all interactions within a plaquette. The low energy description corresponds to a quantum dimer model on a dual lattice of even coordination number with a multi dimer constraint. At an appropriately constructed frustration-free Rokhsar-Kivelson (RK) point, the ground state wavefunction can be exactly mapped onto a classical vertex model on the dual lattice. When the dual lattice is bipartite, the vertex models are bonded and are self dual under Wegner's duality, with the self dual point corresponding to the RK point of the original multi-dimer model. We argue that the self dual point is a critical point based on known exact solutions to some of the vertex models. When the dual lattice is non-bipartite, the vertex model is arrowed, and we use numerical methods to argue that there is no phase transition as a function of the vertex weights. Motivated by these wavefunction dualities, we construct two other distinct families of frustration-free Hamiltonians whose ground states can be mapped onto these vertex models. Many of these RK Hamiltonians provably host $\mathbb{Z}_2$ topologically ordered phases.

A dual equivalence for cofinal quantum B-algebras

This paper introduces weak logical algebraic domains and a special class of quantum B-algebras, namely, cofinal quantum B-algebras. It proves that there is a dual equivalence between the category of cofinal quantum B-algebras and that of weak logical algebraic domains. In particular, the category of residuated posets is shown to be dual equivalent to that of logical algebraic domains. This correspondence generalizes the classical duality of posets and algebraic domains.