Natural Duality Research Articles

Multi-graded Macaulay dual spaces

We describe an algorithm for computing Macaulay dual spaces for multi-graded ideals. For homogeneous ideals, the natural grading is inherited by the Macaulay dual space which has been leveraged to develop algorithms to compute the Macaulay dual space in each homogeneous degree. Our main theoretical result extends this idea to multi-graded Macaulay dual spaces inherited from multi-graded ideals. This natural duality allows ideal operations to be translated from homogeneous ideals to their corresponding operations on the multi-graded Macaulay dual spaces. In particular, we describe a linear operator with a right inverse for computing quotients by a multi-graded polynomial. By using a total ordering on the homogeneous components of the Macaulay dual space, we also describe how to recursively construct a basis for each component. Several examples are included to demonstrate this new approach.

Limitations of Duality

The attempt to rearticulate traditional conceptions of nature can be both a useful strategy and a stumbling block when it comes to feminist examinations of the continuity between the objectification of women’s bodies and the domination of nature. This paper contributes to existing debates by providing a critique of what I term the “duality view” of nature: a view stipulating that nature is primarily characterised by a stable sexual duality, and advancing that the objectification of women’s bodies arises because the specificity of “femaleness” is ignored and duality is therefore neglected. I focus, specifically, on Alison Stone’s interpretation of Luce Irigaray, insofar as the account emerging from Stone’s interpretation clearly outlines the principles that most versions of the duality view should endorse. I problematise this account by showing that it becomes inconsistent with the critique of objectification which grounds it in the first place. I conclude by advancing that, overall, a view insisting on a natural sexual duality because of normative reasons conflicts with the feminist considerations at its basis. I also suggest that while the present analysis is primarily condemnatory, it can contribute to the development of feminist philosophies of nature by fleshing out avoidable pitfalls.

On symplectic Banach spaces

We extend and generalize the result of Kalton and Swanson (Z_2 is a symplectic Banach space with no Lagrangian subspace) by showing that all higher order Rochgberg spaces {mathfrak {R}}^{(n)} are symplectic Banach spaces with no Lagrangian subspaces. The nontrivial symplectic structure on Rochberg spaces of even order is the one induced by the natural duality; while the nontrivial symplectic structure on Rochberg spaces of odd order requires perturbation with a complex structure. We will also study symplectic structures on general Banach spaces and, motivated by the unexpected appearance of complex structures, we introduce and study almost symplectic structures.

Modularized Mutuality Network for Emotion-Cause Pair Extraction

Emotion-cause pair extraction (ECPE) is an emerging task born out of Emotion cause extraction (ECE), which aims to extract the emotion clause and the corresponding cause clause simultaneously. Previous methods decompose ECPE into multiple sub-tasks, namely emotion clause extraction, cause clause extraction, and emotion-cause pair extraction, and employ different modules to address them separately. However, these methods fail to effectively capture the mutuality within the three sub-tasks, which may hinder the information interaction between emotion and cause. In this paper, we revisit and analyze the mutuality between emotion and cause clauses from a linguistic perspective and further propose a novel Modularized Mutuality Network (MMN) to capture the mutuality explicitly. Specifically, the mutuality can be divided into the following categories, including position bias, sentiment consistency, and natural duality. To this end, we design three modules wrapped with various simple but effective mechanisms to address the mutuality, respectively. Extensive experiments demonstrate that MMN achieves state-of-the-art performances on the ECPE task and detailed analyzed the effect of the three modules for capturing the mutuality within sub-tasks.

Lifespan functors and natural dualities in persistent homology

Lifespan functors and natural dualities in persistent homology

The Minor Order of Homomorphisms via Natural Dualities

We study the minor relation for algebra homomorphims in finitely generated quasivarieties that admit a logarithmic natural duality. We characterize the minor homomorphism posets of finite algebras in terms of disjoint unions of dual partition lattices and investigate reconstruction problems for homomorphisms.

Natural and restricted Priestley duality for ternary algebras and their cousins

Natural and restricted Priestley duality for ternary algebras and their cousins

On the existence of topologies compatible with a group duality with predetermined properties

The paper deals with group dualities. A group duality is simply a pair (G,H) where G is an abstract abelian group and H a subgroup of characters defined on G. A group topology τ defined on G is compatible with the group duality (also called dual pair) (G,H) if G equipped with τ has dual group H.A topological group (G,τ) gives rise to the natural duality (G,G∧), where G∧ stands for the group of continuous characters on G. We prove that the existence of a g-barrelled topology on G compatible with the dual pair (G,G∧) is equivalent to the semireflexivity in Pontryagin's sense of the group G∧ endowed with the pointwise convergence topology σ(G∧,G). We also deal with k-group topologies. We prove that the existence of k-group topologies on G compatible with the duality (G,G∧) is determined by a sort of completeness property of its Bohr topology σ(G,G∧) (Theorem 3.3).

Stirling operators in spatial combinatorics

We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol (m)n can be extended from a natural number m∈N to the falling factorials (z)n=z(z−1)⋯(z−n+1) of an argument z from F=R or C, and Stirling numbers of the first and second kinds are the coefficients of the expansions of (z)n through zk, k≤n and vice versa. When taking into account spatial positions of elements in a locally compact Polish space X, we replace N by the space of configurations—discrete Radon measures γ=∑iδxi on X, where δxi is the Dirac measure with mass at xi. The spatial falling factorials (γ)n:=∑i1∑i2≠i1⋯∑in≠i1,…,in≠in−1δ(xi1,xi2,…,xin) can be naturally extended to mappings M(1)(X)∋ω↦(ω)n∈M(n)(X), where M(n)(X) denotes the space of F-valued, symmetric (for n≥2) Radon measures on Xn. There is a natural duality between M(n)(X) and the space CF(n)(X) of F-valued, symmetric continuous functions on Xn with compact support. The Stirling operators of the first and second kind, s(n,k) and S(n,k), are linear operators, acting between spaces CF(n)(X) and CF(k)(X) such that their dual operators, acting from M(k)(X) into M(n)(X), satisfy (ω)n=∑k=1ns(n,k)⁎ω⊗k and ω⊗n=∑k=1nS(n,k)⁎(ω)k, respectively. In the case where X has only a single point, the Stirling operators can be identified with Stirling numbers. We derive combinatorial properties of the Stirling operators, present their connections with a generalization of the Poisson point process and with the Wick ordering under the canonical commutation relations.

Fourier duality in the Brascamp–Lieb inequality

AbstractIt was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.

Perfect extensions of de Morgan algebras

An algebra \(\mathbf{A}\) is called a perfect extension of its subalgebra \(\mathbf{B}\) if every congruence of \(\mathbf{B}\) has a unique extension to \(\mathbf{A}\). This terminology was used by Blyth and Varlet [1994]. In the case of lattices, this concept was described by Grätzer and Wehrung [1999] by saying that \(\mathbf{A}\) is a congruence-preserving extension of \(\mathbf{B}\). Not many investigations of this concept have been carried out so far. The present authors in another recent study faced the question of when a de Morgan algebra \(\mathbf{M}\) is perfect extension of its Boolean subalgebra \(B(\mathbf{M})\), the so-called skeleton of \(\mathbf{M}\). In this note a full solution to this interesting problem is given. The theory of natural dualities in the sense of Davey and Werner [1983] and Clark and Davey [1998], as well as Boolean product representations, are used as the main tools to obtain the solution.

Advances in the Theory of Compact Groups and Pro-Lie Groups in the Last Quarter Century

This article surveys the development of the theory of compact groups and pro-Lie groups, contextualizing the major achievements over 125 years and focusing on some progress in the last quarter century. It begins with developments in the 18th and 19th centuries. Next is from Hilbert’s Fifth Problem in 1900 to its solution in 1952 by Montgomery, Zippin, and Gleason and Yamabe’s important structure theorem on almost connected locally compact groups. This half century included profound contributions by Weyl and Peter, Haar, Pontryagin, van Kampen, Weil, and Iwasawa. The focus in the last quarter century has been structure theory, largely resulting from extending Lie Theory to compact groups and then to pro-Lie groups, which are projective limits of finite-dimensional Lie groups. The category of pro-Lie groups is the smallest complete category containing Lie groups and includes all compact groups, locally compact abelian groups, and connected locally compact groups. Amongst the structure theorems is that each almost connected pro-Lie group G is homeomorphic to RI×C for a suitable set I and some compact subgroup C. Finally, there is a perfect generalization to compact groups G of the age-old natural duality of the group algebra R[G] of a finite group G to its representation algebra R(G,R), via the natural duality of the topological vector space RI to the vector space R(I), for any set I, thus opening a new approach to the Hochschild-Tannaka duality of compact groups.

Algebraic characterizations of some relative notions of size

We obtain algebraic characterizations of relative notions of size in a discrete semigroup that generalize the usual combinatorial notions of syndetic, thick, and piecewise syndetic sets. “Filtered” syndetic and piecewise syndetic sets were defined and applied earlier by Shuungula et al. (Semigroup Forum 79:531–539, 2009). Other instances of these relative notions of size have appeared explicitly (and more often implicitly) in the literature related to the algebraic structure of the Stone–Čech compactification. Building on this prior work, we observe a natural duality and demonstrate how these notions of size may be composed to characterize previous notions of size (like piecewise syndetic sets) and serve as a convenient description for new notions of size.

Langlands duality and Poisson–Lie duality via cluster theory and tropicalization

Let G be a connected semisimple Lie group. There are two natural duality constructions that assign to G: its Langlands dual group \(G^\vee \), and its Poisson–Lie dual group \(G^*\), respectively. The main result of this paper is the following relation between these two objects: the integral cone defined by the cluster structure and the Berenstein–Kazhdan potential on the double Bruhat cell \(G^{\vee ; w_0, e} \subset G^\vee \) is isomorphic to the integral Bohr–Sommerfeld cone defined by the Poisson structure on the partial tropicalization of \(K^* \subset G^*\) (the Poisson–Lie dual of the compact form \(K \subset G\)). By Berenstein and Kazhdan (in: Contemporary mathematics, vol. 433. American Mathematical Society, Providence, pp 13–88, 2007), the first cone parametrizes the canonical bases of irreducible G-modules. The corresponding points in the second cone belong to integral symplectic leaves of the partial tropicalization labeled by the highest weight of the representation. As a by-product of our construction, we show that symplectic volumes of generic symplectic leaves in the partial tropicalization of \(K^*\) are equal to symplectic volumes of the corresponding coadjoint orbits in \({{\,\mathrm{Lie}\,}}(K)^*\). To achieve these goals, we make use of (Langlands dual) double cluster varieties defined by Fock and Goncharov (Ann Sci Ec Norm Supér (4) 42(6):865–930, 2009). These are pairs of cluster varieties whose seed matrices are transpose to each other. There is a naturally defined isomorphism between their tropicalizations. The isomorphism between the cones described above is a particular instance of such an isomorphism associated to the double Bruhat cells \(G^{w_0, e} \subset G\) and \(G^{\vee ; w_0, e} \subset G^\vee \).

Semigroup quasivarieties: Two lattices and a reopened problem

We determine all quasivarieties of aperiodic semigroups that are contained in some residually finite variety. This endeavor was initially motivated by a problem in natural dualities, but our work here also serves as a partial correction to an error found in a result of Sapir from the 1980s.

Calculi for Many-Valued Logics

We present a number of equivalent calculi for many-valued logics and prove soundness and strong completeness theorems. The calculi are obtained from the truth tables of the logic under consideration in a straightforward manner and there is a natural duality among these calculi. We also prove the cut elimination theorems for the sequent-like systems.

POISSON - JENSEN FORMULAS AND BALAYAGE OF MEASURES

Our main results are certain developments of the classical Poisson--Jensen formula for subharmonic functions. The basis of the classical Poisson--Jensen formula is the natural duality between harmonic measures and Green's functions. Our generalizations use some duality between the balayage of measures and and their potentials.

CONNECTION OF THE WORLDVIEW EXPECTATIONS OF CITIZENS TO THE HOUSING POLICY OF THE STATE

The development of worldview expectations of citizens is a natural, orderly and inevitable changes in the perception of the subject, to have the right to the object, due to the tendency of constant social transformations. The worldview expectations of citizens for the realization of their right to housing combine the theory of the nature of things, the philosophy of values and the theory of forms of being as phenomenological concepts of law and consist of the legal consciousness of each individual separately and society as a whole. When analysing studies of the worldview expectations of citizens with any state policy, including housing, one should consider the natural duality of each person, which requires an integrated approach to considering the problem of the functioning of interrelated factors, namely: society, the system of legal norms, personality, human internal perception of real and the actual actions of the subjects of the relevant regulations. In modern conditions, the worldview and legal thinking in Ukrainian society is undergoing a process of transformation from a system of socialist perception of mutual obligations to democratic principles of the development of political perception, considering the European integration course of the country, therefore, it is at a transformational stage. Insufficient housing has traditionally been a powerful potential source of dissatisfaction for citizens living in post-Soviet countries. This is largely due to the legacy of the Soviet system when the state provided housing to families based on non-market principles. Another reason is the traditional shortage of housing in the Soviet era. One way or another, the housing affordability situation remains a pressing political issue. People who are dissatisfied with their housing situation are more likely to be critical of the country's government, while home and apartment owners are more susceptible to government support. Among the potential sources of instability in post-Soviet Eurasia, researchers have traditionally paid much more attention to general economic problems and political conflicts. However, the tension created by the unresolved housing issue is a long-term challenge, which is dangerous to ignore. The paternalistic hopes of a large part of the population of Ukraine (as well as the entire post-Soviet space) for the decisive role of the state in providing citizens with housing are still preserved. To some extent, such sentiments are supported by the social orientation of the current Constitution of Ukraine, adopted in this form in 1996 as a political compromise. Nevertheless, the norms of the Basic Law are the norms of direct action. Whether the state creates such conditions in practice can be asked of families who for decades have been almost without advancing in the queues for housing in local authorities.

Observation of Bose-Einstein condensates in an Earth-orbiting research lab.

Quantum mechanics governs the microscopic world, where low mass and momentum reveal a natural wave-particle duality. Magnifying quantum behaviour to macroscopic scales is a major strength of the technique of cooling and trapping atomic gases, in which low momentum is engineered through extremely low temperatures. Advances in this field have achieved such precise control over atomic systems that gravity, often negligible when considering individual atoms, has emerged as a substantial obstacle. In particular, although weaker trapping fields would allow access to lower temperatures1,2, gravity empties atom traps that are too weak. Additionally, inertial sensors based on cold atoms could reach better sensitivities if the free-fall time of the atoms after release from the trap could be made longer3. Planetary orbit, specifically the condition of perpetual free-fall, offers to lift cold-atom studies beyond such terrestrial limitations. Here we report production of rubidium Bose-Einstein condensates (BECs) in an Earth-orbiting research laboratory, the Cold Atom Lab. We observe subnanokelvin BECs in weak trapping potentials with free-expansion times extending beyond one second, providing an initial demonstration of the advantages offered by a microgravity environment for cold-atom experiments and verifying the successful operation of this facility. With routine BEC production, continuing operations will support long-term investigations of trap topologies unique to microgravity4,5, atom-laser sources6, few-body physics7,8 and pathfinding techniques for atom-wave interferometry9-12.

Bitopological duality for algebras of Fitting’s logic and natural duality extension

In this paper, we investigate a bitopological duality for algebras of Fitting’s multi-valued logic. We also extend the natural duality theory for $$\mathbb {ISP_I}(\mathcal {L})$$ by developing a duality for $$\mathbb {ISP}(\mathcal {L})$$ , where $$\mathcal {L}$$ is a finite algebra in which underlying lattice is bounded distributive.