Explicit Basis Research Articles

PBW bases of irreducible Ising modules

To every h+N-graded module M over an N-graded conformal vertex algebra V, we associate an increasing filtration (GpM)p∈Z, which is compatible with the filtrations introduced by Haisheng Li. The associated graded vector space grG(M) is naturally a module over the vertex Poisson algebra grG(V). We study grG(M) for the three irreducible modules over the Ising model Vir3,4, namely Vir3,4=L(1/2,0), L(1/2,1/2) and L(1/2,1/16). We obtain an explicit PBW basis of each of these modules and a formula for their refined characters, which are related to Nahm sums for the matrix (8332).

Well-Rounded ideal lattices of cyclic cubic and quartic fields

In this paper, we find criteria for when cyclic cubic and cyclic quartic fields have well-rounded ideal lattices. We show that every cyclic cubic field has at least one well-rounded ideal. We also prove that there exist families of cyclic quartic fields which have well-rounded ideals and explicitly construct their minimal bases. In addition, for a given prime number $p$, if a cyclic quartic field has a unique prime ideal above $p$, then we provide the necessary and sufficient conditions for that ideal to be well-rounded. Moreover, in cyclic quartic fields, we provide the prime decomposition of all odd prime numbers and construct an explicit integral basis for every prime ideal.

A Comparative Study of the Right to Choose in the Legal Systems of Afghanistan and Iran with a Focus on Islamic Jurisprudence

The right of choice is not defined in the law; however, it can be stated that when the subject of a transaction is more than one item and at most three items, and the customer has the right to choose one of these items, this choice is recognized as the right of choice. The period for the right of choice is three days, and the holder of the right of choice must select one item among the maximum of three items as the subject of the transaction. The fundamental question is whether the right of choice falls under the category of cases that grant the right of rescission to the opposing party, and what is its position in the legal systems of Iran and Afghanistan?
 The recognition of the position, source, and nature of the inclusion of this right in contracts in the legal system of Afghanistan and its elucidation in the legal system of Iran, considering that this institution has not been explicitly accepted in the legal system of Iran, are the main objectives of this research.
 The findings of this study indicate that there is no explicit textual basis for the right of choice in the Quran and Hadith, and it is solely derived from the jurisprudential interpretations of Hanafi and Maliki scholars. This institution bears many similarities with the recognized institution of conditional obligations in international law. Although the right of choice is not explicitly stipulated in the legal system of Iran, it can be established in contracts under certain conditions using some provisions of the Iranian Civil Code. It appears that the right of choice can be accommodated within contracts by invoking Article 216 of the Civil Code, where the principle of the supremacy of will is well-recognized. This research follows an analytical and descriptive approach and relies on library research and documentary analysis.

Settler Colonial Strategic Culture: Australia, AUKUS, and the Anglosphere

Australia's history as a settler colony within the British Empire fundamentally shapes its sense of security within the Indo‐Pacific region. Australia has consistently looked outside of its region for security and sought partners on the explicit basis of political, cultural, and ethnic similarity. What role does Australia's history play in shaping its foreign policy? We argue that these choices in foreign policy are inextricable from Australia's history as a settler colony on the farthest reaches of the British Empire. The AUKUS Agreement (AUKUS) is an example of how Australia operates to preserve racial hegemony in the face of non‐white threat — real or perceived. This research utilises critical discourse analysis to interrogate elite‐level discourse around AUKUS to ascertain the dominant narratives that inform its creation, the issues it seeks to address in Australian security policy, how it is structured by historical narratives of security, and how it functions to structure those narratives going forward. This article seeks to participate in the growing push to decolonise International Relations by illuminating the way Australia is ontologically and epistemologically invested in the preservation of racial hegemony.

Graded dimensions and monomial bases for the cyclotomic quiver Hecke superalgebras

In this paper we derive a closed formula for the (Z×Z2)-graded dimension of the cyclotomic quiver Hecke superalgebra RΛ(β) associated to an arbitrary Cartan superdatum (A,P,Π,Π∨), polynomials (Qi,j(x1,x2))i,j∈I, β∈Qn+ and Λ∈P+. As applications, we obtain a necessary and sufficient condition for which e(ν)≠0 in RΛ(β). We construct an explicit monomial basis for the bi-weight space e(ν˜)RΛ(β)e(ν˜), where ν˜ is a certain specific n-tuple defined in (1.4). In particular, this gives rise to a monomial basis for the cyclotomic odd nilHecke algebra. Finally, we consider the case when β=α1+α2+⋯+αn with α1,⋯,αn distinct. We construct an explicit monomial basis of RΛ(β) and show that it is indecomposable in this case.

G1 spline functions for point cloud fitting

In this work we present a new construction of basis functions that generate the space of geometrically smooth splines on an unstructured quadrilateral mesh. The basis is represented in terms of biquintic Bézier polynomials on each quadrilateral face. The gluing along the face boundaries is achieved using quadratic gluing data functions, leading to globally G1–smooth spaces. We analyze the latter space and provide a combinatorial formula for its dimension as well as an explicit basis construction. Moreover, we assess the use of this basis in point cloud fitting problems. To apply G1 least squares fitting, a quadrilateral structure as well as parameters in each quadrilateral is required. Even though the general problem of segmenting and parametrizing point clouds is beyond the focus of the present work, we describe a procedure that produces such a structure as well as patch-local parameters. Our experiments demonstrate the accuracy and smoothness of the obtained reconstructed models in several challenging instances.

Resonant vibrations produce quantum bridge over high-energy states in heterogeneous antenna.

Photosynthetic light-harvesting complexes usually contain several pools of molecules with a big difference in transition energies, for example, chlorophylls a and b in plant antennas. Some pathways of the excitation energy transfer may include pigments from the low-energy pool separated by a site occupied by a high-energy molecule. We demonstrate that such pathways may be functional if high-frequency intramolecular vibrations fall in resonance with the energy gap between the neighboring molecules belonging to different pools. In this case, a vibration-assisted mixing of the excited states can produce delocalized vibronic states playing a role of 'quantum bridge' that facilitates a passage over high-energy barrier. We perform calculations of the excitation dynamics in the model three-state system with the parameters emerging from our previous studies of real antennas. Simulation of the dynamics in an explicit electron-vibrational basis demonstrates that the rate of transfer between the two chlorophylls a through the chlorophyll b intermediate is increased by a factor of 1.7-2 in the presence of resonant vibration. A possible influence of energetic disorder and other (non-resonant) vibrations on this effect is discussed.

Polyhedral compactifications, I

Abstract In this work we describe horofunction compactifications of metric spaces and finite-dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods, that is, by ultrapowers of the spaces at hand. The polyhedral compactifications of the vector spaces carry the structure of stratified spaces with the strata indexed by dual faces of the polyhedral unit ball. Explicit neighborhood bases and descriptions of the horofunctions are provided.

A basis- and integral-free representation of time-dependent perturbation theory via the Omega matrix calculus

We obtain a basis- and integral-free representation of the n -term in time-dependent perturbation theory of the quantum time-evolution operator. The main technical tool to construct the aforementioned representation comprises the Omega matrix calculus; that is, an extension of MacMahon’s partition analysis to the realm of matrix calculus. In particular, we show that if we specialize our formulation to the time-independent and finite-dimensional case, we recover Putzer (1966) original formulation to compute the matrix exponential avoiding the Jordan canonical form. Furthermore, if we use an explicit basis for the time-independent part of the generator of evolution, then our work implies a representation of the perturbation expansion similar to an approach discussed in the recent work of Kalev and Hen (2021) using divided differences. Finally, we obtain closed form expressions which generalize and unify all the perturbation calculations considered in Kalev and Hen (2021) and advance some considerations regarding degeneracy associated with the unperturbed Hamiltonian in the context of adiabatic switching.

Evidence-inclusive communication: steering crisis leadership outcomes in Portugal, Brazil, New Zealand, and the US

ABSTRACT While the essence of leadership communication is recognized as a linchpin in crisis management, the distinct characteristics that define successful communication are often unknown. Recent perspectives on the topic highlight the importance of incorporating political and value-based considerations to achieve a balanced communication style during a crisis. This article empirically tests these claims by examining whether an evidence-inclusive communication style of leadership tends to yield more successful outcomes in crisis management. We compare speeches by leaders from Portugal, Brazil, the United States, and New Zealand, to assess the main explicit knowledge and value bases of decision-making. Our findings suggest that success of a crisis response is associated with a leader’s ability to effectively communicate an evidence-based narrative, inclusive of different evidence sources, when tackling pressing and complex issues such as the COVID-19 pandemic.

Triangular decomposition of $\mathrm{SL}_3$ skein algebras

We give an \mathrm{SL}_3 analogue of the triangular decomposition of the Kauffman bracket stated skein algebras described by Lê. To any punctured bordered surface, we associate an \mathrm{SL}_3 stated skein algebra which contains the \mathrm{SL}_3 skein algebra of closed webs. These algebras admit natural algebra morphisms associated to the splitting of surfaces along ideal arcs. We give an explicit basis for the \mathrm{SL}_3 stated skein algebra and show that the splitting morphisms are injective and describe their images. By splitting a surface along the edges of an ideal triangulation, we see that the \mathrm{SL}_3 stated skein algebra of any ideal triangulable surface embeds into a tensor product of stated skein algebras of triangles. As applications, we prove that the stated skein algebras do not have zero divisors, we construct Frobenius maps at roots of unity, and we obtain a new proof that Kuperberg's web relations generate the kernel of the Reshetikhin–Turaev functor.

The pro-p-Iwahori invariants of the universal quotient representation for GL2(F)

Let [Formula: see text] be a finite extension of [Formula: see text]. Every supersingular mod [Formula: see text] representation of [Formula: see text] is a quotient of a certain universal module, say [Formula: see text] The space of pro-[Formula: see text]-Iwahori invariants of [Formula: see text] is instrumental in classifying the supersingular mod [Formula: see text] representations of [Formula: see text]. For [Formula: see text] the space of pro-[Formula: see text]-Iwahori invariants of [Formula: see text] has been determined by Schein and Hendel using the spherical Hecke algebra. An alternative way to study mod [Formula: see text] representations of [Formula: see text] is introduced by Anandavardhanan and Borisagar using the Iwahori–Hecke algebra. In this paper, we give a canonical isomorphism between the universal modules for [Formula: see text] obtained from the spherical approach and the Iwahori–Hecke approach. We also compute an explicit basis of the pro-[Formula: see text]-Iwahori invariant space of [Formula: see text] for [Formula: see text] using the Iwahori–Hecke approach. This method allows us to extend Hendel’s computation slightly in the range of [Formula: see text]-adic digits from [Formula: see text] to [Formula: see text]

자동적 처분에 대한 통제 수단으로서의 알고리즘 영향평가

The Framework Act on Administration, scheduled to take effect in March 2021, provides an explicit basis for fully automated administrative decisions by introducing the concept of ‘automatic disposition’ in Article 20. Traditionally, the ‘automation of administration’ was significant as an auxiliary administrative procedure consisting of automatic mechanical devices such as traffic signal transmission, tax, and utility bill calculation. Moreover, since automated administrative decisions using artificial intelligence (AI) technology are also included in this automatic disposition, there is room for legal acceptance of automated administrative decisions based on artificial intelligence algorithms that exceed the level of conventional computerization or partial automation. However, it is still a long way from implementing a fully automated administrative decision system based on current law and having technical and institutional safety devices for it. On the other hand, the Framework Act on Intelligent Informatization, which took effect in December 2020, introduced a new ‘social impact assessment’ for intelligent information services that have far-reach effects on citizens’ lives. According to Article 56 of the Act, the main goal of this impact assessment system is to investigate and evaluate the impact of intelligent information services on society, economy, culture, etc., disclose the results, reflect them in policies, and directly recommend appropriate measures. Since the scope of intelligent information services subject to evaluation is not limited to the private sector, it should be considered that intelligent information services in the administrative sector can also be subject to social impact assessment in this article. It is reasonable to assume that automatic administrative services by artificial intelligence system belong to a wide range of intelligent information services prescribed by law, and that automatic disposition issued as a fully automated system based on intelligent information technology can also be subject to social impact assessment. Although no provisions have been found in law to assess the impact of automated administrative decisions, the government of Canada has implemented the “Directive on Automated Decision-Making” since April 2019, which systematically regulates the automated administrative decision-making system. The requirement that constitutes an important axis of this directive is the ‘algorithmic impact assessment'. For an automated administrative decision-making system that bears relatively stricter legality and public responsibility than private companies' services, it is necessary to process and operate an impact assessment that reviews safety, transparency, and solidity in advance. In this paper, referring to the Canadian case, as a control system of the algorithm-based administrative decision-making system, I propose the institutionalization of the “algorithmic impact assessment” in consideration of the connection with the social impact assessment under the current law.

On the Center Conjecture for the Cyclotomic KLR Algebras

Abstract The center conjecture for the cyclotomic KLR algebras $\mathscr{R}_{\beta }^{\Lambda }$ asserts that the center of $\mathscr{R}_{\beta }^{\Lambda }$ consists of symmetric elements in its KLR $x$ and $e(\nu )$ generators. In this paper, we show that this conjecture is equivalent to the injectivity of some natural map $\overline{\iota }_{\beta }^{\Lambda ,i}$ from the cocenter of $\mathscr{R}_{\beta }^{\Lambda }$ to the cocenter of $\mathscr{R}_{\beta }^{\Lambda +\Lambda _{i}}$ for all $i\in I$ and $\Lambda \in P^{+}$. We prove that the map $\overline{\iota }_{\beta }^{\Lambda ,i}$ is given by multiplication with a center element $z(i,\beta )\in \mathscr{R}_{\beta }^{\Lambda +\Lambda _{i}}$ and we explicitly calculate the element $z(i,\beta )$ in terms of the KLR $x$ and $e(\nu )$ generators. We present explicit monomial bases for certain bi-weight spaces of the defining ideal of $\mathscr{R}_{\beta }^{\Lambda }$. For $\beta =\sum _{j=1}^{n}\alpha _{i_{j}}$ with $\alpha _{i_{1}},\cdots , \alpha _{i_{n}}$ pairwise distinct, we construct an explicit monomial basis of $\mathscr{R}_{\beta }^{\Lambda }$, prove the map $\overline{\iota }_{\beta }^{\Lambda ,i}$ is injective, and thus verify the center conjecture for these $\mathscr{R}_{\beta }^{\Lambda }$.

Examples of exponential bases on union of intervals

AbstractIn this paper, we construct explicit exponential bases of unions of segments of total measure one. Our construction applies to finite or infinite unions of segments, with some conditions on the gaps between them. We also construct exponential bases on finite or infinite unions of cubes in $\mathbb {R}^d$ and prove a stability result for unions of segments that generalize Kadec’s $\frac 14$ -theorem.

Symmetry and Invariant Bases in Finite Element Exterior Calculus

We study symmetries of bases and spanning sets in finite element exterior calculus, using representation theory. We want to know which vector-valued finite element spaces have bases invariant under permutation of vertex indices. The permutations of vertex indices correspond to the symmetry group of the simplex. That symmetry group is represented on simplicial finite element spaces by the pullback action. We determine a natural notion of invariance and sufficient conditions on the dimension and polynomial degree for the existence of invariant bases. We conjecture that these conditions are necessary too. We utilize Djoković and Malzan’s classification of monomial irreducible representations of the symmetric group and show new symmetries of the geometric decomposition and canonical isomorphisms of the finite element spaces. Explicit invariant bases with complex coefficients are constructed in dimensions two and three for different spaces of finite element differential forms.

Speculative hybrids: Investigating the generation of conceptual architectural forms through the use of 3D generative adversarial networks

The process of architectural design aims at solving complex problems that have loosely defined formulations, no explicit basis for terminating the problem-solving activity, and where no ideal solution can be achieved. This means that design problems, as wicked problems, sit in a space between incompleteness and precision. Applying digital tools in general and artificial intelligence in particular to design problems will then mediate solution spaces between incompleteness and precision. In this paper, we present a study where we employed machine learning algorithms to generate conceptual architectural forms for site-specific regulations. We created an annotated dataset of single-family homes and used it to train a 3D Generative Adversarial Network that generated annotated point clouds complying with site constraints. Then, we presented the framework to 23 practitioners of architecture in an attempt to understand whether this framework could be a useful tool for early-stage design. We make a three-fold contribution: First, we share an annotated dataset of architecturally relevant 3D point clouds of single-family homes. Next, we present and share the code for a framework and the results from training the 3D generative neural network. Finally, we discuss machine learning and creative work, including how practitioners feel about the emergence of these tools as mediators between incompleteness and precision in architectural design.

Approximation via gradients on the ball. The Zernike case

In this work, we deal in a d dimensional unit ball equipped with an inner product constructed by adding a mass point at zero to the classical ball inner product applied to the gradients of the functions. Apart from determining an explicit orthogonal polynomial basis, we study approximation properties of Fourier expansions in terms of this basis. In particular, we deduce relations between the partial Fourier sums in terms of the new orthogonal polynomials and the partial Fourier sums in terms of the classical ball polynomials. We also give an estimate of the approximation error by polynomials of degree at most n in the corresponding Sobolev space, proving that we can approximate a function by using its gradient. Numerical examples are given to illustrate the approximation behavior of the Sobolev basis.

Explicit basis for admissible rules in K-saturated tabular logics

Abstract We construct an explicit finite basis for admissible rules in K-saturated tabular logics that extend the logic Grz.

Centers of Hecke algebras of complex reflection groups

We provide a dual version of the Geck–Rouquier Theorem (Geck and Rouquier in Finite Reductive Groups (Luminy, 1994), Progr. Math., vol. 141, Birkhäuser Boston, Boston, pp. 251–272, 1997) on the center of an Iwahori–Hecke algebra, which also covers the complex case. For the eight complex reflection groups of rank 2, for which the symmetrising trace conjecture is known to be true, we provide a new faithful matrix model for their Hecke algebra H. These models enable concrete calculations inside H. For each of the eight groups, we compute an explicit integral basis of the center of H.