Representations Of Algebras Research Articles

A study of the local to normal mode transition in pyramidal molecules and their vibrational description in terms of an algebraic model

First a new perspective to study the local to normal mode transition in the series of pyramidal molecules is presented. Then a full study of the vibrational excitations of the series of pyramidal molecules XH3 with X = P, As and Sb is given in the framework of a polyad-conserving Hamiltonian of a set of interacting Morse oscillators. The model is based on an algebraic representation of the Hamiltonian in terms of su(2)-operators. A simple Hamiltonian including both Fermi and Darling–Dennison interactions is considered. For the PH3 molecule a fit involving 64 experimental energies up to polyad 8 (energies up to 9054 cm−1) provided an rms=1.74cm−1 using a Hamiltonian with 14+3(frozen) parameters. For the AsH3, with 35 experimental energies the deviation obtained was rms=2.50cm−1 with 10+4 parameters. Lastly, for the SbH3 molecule, considering 23 experimental energy levels an rms=0.86cm−1 was reached with 9+6 parameters. In all cases the polyad scheme P=2(ν1+ν3)+ν2+ν4 was considered.

An Explicit Expression of the Unit Step Function

In this paper, an analytical form of the Unit Step Function (or Heaviside Step Function) is presented. In particular, this piecewise – defined function, which constitutes a fundamental concept of Operational Calculus and is also involved in many other areas of applied and engineering mathematics, is explicitly performed in a very simple manner by the aid of purely algebraic representations. The novelty of this work, when compared with other analytical approximations to this discontinuous function, is that the proposed exact formula is not performed in terms of non – elementary special functions, e.g. Gamma function or Error function. In addition, this function does not contain any infinitesimal quantities and also is not the limit of a sequence of functions with a pointwise or uniform convergence. Hence, it may be much more appropriate and useful to the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

Character sheaves for graded Lie algebras: Stable gradings

In this paper we construct full support character sheaves for stably graded Lie algebras. Conjecturally these are precisely the cuspidal character sheaves. Irreducible representations of Hecke algebras associated to complex reflection groups at roots of unity enter the description. We do so by analysing the Fourier transform of the nearby cycle sheaves constructed in [8].

Logical Minimization of Multilevel Representations of Boolean Function Systems

A decisive influence on complexity and speed of a combinational logic circuit of library CMOS elements is exerted by the preliminary stage of technologically independent optimization of the implemented system of Boolean functions. At present, the main methods of such optimization in the logical synthesis of custom CMOS VLSI blocks are methods for minimizing binary decision diagrams — Binary Decision Diagrams (BDD) or their modifications. Graphical representations of BDD are built on the basis of the Shannon expansions of Boolean functions. A BDD graph corresponds to a set of interrelated Shannon expansion formulas that form a multilevel representation of the minimized system of Boolean functions. The efficiency of applying various optimization procedures of minimization for several types of BDD representations of systems of Boolean functions is investigated in the paper. 7hese procedures are used as a technologically independent optimization in the synthesis of multi-output logic circuits of library CMOS elements. In addition to single logical optimization procedures, sequences of such procedures are studied that form various methods of logical optimization of multilevel representations of systems of Boolean functions. The results of experiments on 49 examples of systems of Boolean functions are presented. 25 optimization routes have been studied, efficient routes have been determined for various types of specifications of function systems. The obtained experimental results are compared with the known ones. It has been established that to estimate the complexity of optimized algebraic representations of systems of functions, it is advisable to use such a criterion as the total number of literals (variables or their inversions) of Boolean variables.

Type A DAHA and doubly periodic tableaux

Analogously to the construction of Suzuki and Vazirani, we construct representations of the GLm-type Double Affine Hecke Algebra at roots of unity. These representations are graded and the weight spaces for the X-variables are parametrized by the combinatorial objects we call doubly periodic tableaux. We show that our representations exhaust all graded X-semisimple representations, and the direct sum of all our representations is faithful. Analogously to the construction of Jordan and Vazirani of rectangular DAHA representations, we show that our representations can be interpreted in terms of ribbon fusion categories associated to Uq(glN) at roots of unity. Combining the ribbon structure with faithfulness we deduce a conjecture of Morton and Samuelson about realization of DAHA as a skein algebra of the torus with base string modulo certain local relations.

Persistent Tor-algebra for protein-protein interaction analysis.

Protein-protein interactions (PPIs) play crucial roles in almost all biological processes from cell-signaling and membrane transport to metabolism and immune systems. Efficient characterization of PPIs at the molecular level is key to the fundamental understanding of PPI mechanisms. Even with the gigantic amount of PPI models from graphs, networks, geometry and topology, it remains as a great challenge to design functional models that efficiently characterize the complicated multiphysical information within PPIs. Here we propose persistent Tor-algebra (PTA) model for a unified algebraic representation of the multiphysical interactions. Mathematically, our PTA is inherently algebraic data analysis. In our PTA model, protein structures and interactions are described as a series of face rings and Tor modules, from which PTA model is developed. The multiphysical information within/between biomolecules are implicitly characterized by PTA and further represented as PTA barcodes. To test our PTA models, we consider PTA-based ensemble learning for PPI binding affinity prediction. The two most commonly used datasets, i.e. SKEMPI and AB-Bind, are employed. It has been found that our model outperforms all the existing models as far as we know. Mathematically, our PTA model provides a highly efficient way for the characterization of molecular structures and interactions.

Real Representation of the Polarimetric Scattering Matrix for Monostatic Radar

Synthetic aperture radar with polarimetric diversity is a powerful tool in remote sensing. Each pixel is described by the scattering matrix corresponding to the emission/reception polarization states (usually horizontal and vertical). The algebraic real representation, a block symmetric matrix form, is introduced to adopt a more comprehensive framework (non-restricted by reciprocity assumptions) in mapping the scattering matrix by the consimilarity equivalence relation. The proposed representation can reveal potentially new information. For example, its eigenvalue decomposition, which is itself a necessary step in obtaining the consimilarity transformation products, may be useful in characterizing the degree of reciprocity/nonreciprocity. As a consequence, it can be employed in testing the reciprocity compliance assumed with monostatic PolSAR data. Full-wave simulated polarimetric data confirm that oriented scatterers can present complex eigenvalues, even with the monostatic geometry.

Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $\mathfrak{g}$, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian $Y(\mathfrak{g})$. These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter $Q$-operators, arising from our previous study (arXiv:2001.04929, arXiv:2104.14518) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional $\mathfrak{g}$-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety $G/P$ stratified by $B_{-}$-orbits.

Sheaf representation of monoidal categories

Every small monoidal category with universal finite joins of central idempotents is monoidally equivalent to the category of global sections of a sheaf of local monoidal categories on a topological space. Every small stiff monoidal category monoidally embeds into such a category of global sections. An infinitary version of these theorems also holds in the spatial case. These representation results are functorial and subsume the Lambek–Moerdijk–Awodey sheaf representation for toposes, the Stone representation of Boolean algebras, and the Takahashi representation of Hilbert modules as continuous fields of Hilbert spaces. Many properties of a monoidal category carry over to the stalks of its sheaf, including having a trace, having exponential objects, having dual objects, having limits of some shape, and the central idempotents forming a Boolean algebra.

Rational Singularities of Nested Hilbert Schemes

Abstract The Hilbert scheme of points $\textrm {Hilb}^{n}(S)$ of a smooth surface $S$ is a well-studied parameter space, lying at the interface of algebraic geometry, commutative algebra, representation theory, combinatorics, and mathematical physics. The foundational result is a classical theorem of Fogarty, stating that $\textrm {Hilb}^{n}(S)$ is a smooth variety of dimension $2n$. In recent years there has been growing interest in a natural generalization of $\textrm {Hilb}^{n}(S)$, the nested Hilbert scheme$\textrm {Hilb}^{(n_{1}, n_{2})}(S)$, which parametrizes nested pairs of zero-dimensional subschemes $Z_{1} \supseteq Z_{2}$ of $S$ with $\deg Z_{i}=n_{i}$. In contrast to Fogarty’s theorem, $\textrm {Hilb}^{(n_{1}, n_{2})}(S)$ is almost always singular, and very little is known about its singularities. In this paper, we aim to advance the knowledge of the geometry of these nested Hilbert schemes. Work by Fogarty in the 70’s shows that $\textrm {Hilb}^{(n,1)}(S)$ is a normal Cohen–Macaulay variety, and Song more recently proved that it has rational singularities. In our main result, we prove that the nested Hilbert scheme $\textrm {Hilb}^{(n,2)}(S)$ has rational singularities. We employ an array of tools from commutative algebra to prove this theorem. Using Gröbner bases, we establish a connection between $\textrm {Hilb}^{(n,2)}(S)$ and a certain variety of matrices with an action of the general linear group. This variety of matrices plays a central role in our work, and we analyze it by various algebraic techniques, including the Kempf–Lascoux–Weyman technique of calculating syzygies, square-free Gröbner degenerations, and the Stanley–Reisner correspondence. Along the way, we also obtain results on classes of irreducible and reducible nested Hilbert schemes, dimension of singular loci, and $F$-singularities in positive characteristic.

Study of the Lennard-Jones and H2 potentials in the framework of the Morse based algebraic DVR approach

We propose a method based on an algebraic discrete variable representation (DVR) of the coordinate and momentum in the scheme of complete basis related to the 1D Morse potential, with this approach, a Hamiltonian associated to a 1D system can be expressed in terms of diagonal matrices using the transformation coefficients and the diagonalization of the matrix representation of the coordinate and momentum. We probe the scope of our method obtaining the solutions associated of the Lennard-Jones potential, also we apply the method to a practical example considering the ab initio potential of the H2 molecule. For both cases, the vibrational energies and the wave functions were obtained with good accuracy and low computational cost.

Symmetry Analysis of the Square Well Potential

Symmetry considerations are taken into account when a particle in a square well potential is studied. This system may display natural degeneracy, accidental degeneracy or systematic accidental degeneracy depending on the depth of the potential. In order to obtain the solutions associated with an arbitrary potential an algebraic discrete variable representation approach based on Pöschl-Teller functions is proposed. It is proved that the geometrical group C 4v is the symmetry group of the system for the case of a finite potential barrier. A similar analysis is carried out for the rectangular square well potential with commensurate sides. In both cases the symmetry projection is crucial to simplify the calculations.

Shuffle Algebras and Non-Commutative Probability for Pairs of Faces

One can build an operatorial model for freeness by considering either the right-handed or the left-handed representation of algebras of operators acting on the free product of the underlying pointed Hilbert spaces. Considering both at the same time, that is, computing distributions of operators in the algebra generated by the left- and right-handed representations, led Voiculescu in 2013 to define and study bifreeness and, in the sequel, triggered the development of an extension of noncommutative probability now frequently referred to as multi-faced (two-faced in the example given above). Many examples of two-faced independences emerged these past years. Of great interest to us are biBoolean, bifree and type I bimonotone independences. In this paper, we extend the preLie calculus pertaining to free, Boolean, and monotone moment-cumulant relations initiated by K. Ebrahimi-Fard and F. Patras to their above-mentioned two-faced equivalents.

Klein-Gordon Theory in Noncommutative Phase Space

We extend the three-dimensional noncommutative relations of the position and momentum operators to those in the four dimension. Using the Seiberg-Witten (SW) map, we give the Heisenberg representation of these noncommutative algebras and endow the noncommutative parameters associated with the Planck constant, Planck length and cosmological constant. As an analog with the electromagnetic gauge potential, the noncommutative effect can be interpreted as an effective gauge field, which depends on the Plank constant and cosmological constant. Based on these noncommutative relations, we give the Klein-Gordon (KG) equation and its corresponding current continuity equation in the noncommutative phase space including the canonical and Hamiltonian forms and their novel properties beyond the conventional KG equation. We analyze the symmetries of the KG equations and some observables such as velocity and force of free particles in the noncommutative phase space. We give the perturbation solution of the KG equation.

Carrollian conformal scalar as flat-space singleton

We show that, in any space-time dimension, the on-shell (electric) conformal Carrollian scalar can be interpreted as the flat-space limit of the singleton representation of the conformal algebra. In fact, a recently proposed higher-spin algebra for Minkowski spacetime amounts to the Poincaré enveloping algebra on the corresponding module. This higher-spin algebra is a contraction of that entering Vasiliev's equations, which can be constructed analogously from the singleton representation of the conformal algebra. We also show that the higher-spin extension of the Poincaré algebra we consider is a subalgebra of all symmetries of the conformal Carrollian scalar, given by a higher-spin version of the (extended) BMS algebra.

Cohomology, deformations and extensions of Rota-Baxter Leibniz algebras

A Rota-Baxter Leibniz algebra is a Leibniz algebra $(\mathfrak{g},[~,~]_{\mathfrak{g}})$ equipped with a Rota-Baxter operator $T : \mathfrak{g} \rightarrow \mathfrak{g}$. We define representation and dual representation of Rota-Baxter Leibniz algebras. Next, we define a cohomology theory of Rota-Baxter Leibniz algebras. We also study the infinitesimal and formal deformation theory of Rota-Baxter Leibniz algebras and show that our cohomology is deformation cohomology. Moreover, We define an abelian extension of Rota-Baxter Leibniz algebras and show that equivalence classes of such extensions are related to the cohomology groups.

Representations of the rank two Racah algebra and orthogonal multivariate polynomials

The algebraic structure of the rank two Racah algebra is studied in detail. We provide an automorphism group of this algebra, which is isomorphic to the permutation group of five elements. This group can be geometrically interpreted as the symmetry of a folded icosidodecahedron. It allows us to study a class of equivalent irreducible representations of this Racah algebra. They can be chosen symmetric so that their transition matrices are orthogonal. We show that their entries can be expressed in terms of Racah polynomials. This construction gives an alternative proof of the recurrence, difference and orthogonal relations satisfied by the Tratnik polynomials, as well as their expressions as a product of two monovariate Racah polynomials. Our construction provides a generalization of these bivariate polynomials together with their properties.

Optimal Quadratic Binding for Relational Reasoning in Vector Symbolic Neural Architectures.

Binding operation is fundamental to many cognitive processes, such as cognitive map formation, relational reasoning, and language comprehension. In these processes, two different modalities, such as location and objects, events and their contextual cues, and words and their roles, need to be bound together, but little is known about the underlying neural mechanisms. Previous work has introduced a binding model based on quadratic functions of bound pairs, followed by vector summation of multiple pairs. Based on this framework, we address the following questions: Which classes of quadratic matrices are optimal for decoding relational structures? And what is the resultant accuracy? We introduce a new class of binding matrices based on a matrix representation of octonion algebra, an eight-dimensional extension of complex numbers. We show that these matrices enable a more accurate unbinding than previously known methods when a small number of pairs are present. Moreover, numerical optimization of a binding operator converges to this octonion binding. We also show that when there are a large number of bound pairs, however, a random quadratic binding performs, as well as the octonion and previously proposed binding methods. This study thus provides new insight into potential neural mechanisms of binding operations in the brain.

Harmonic Oscillator Coherent States from the Standpoint of Orbit Theory

We study the known coherent states of a quantum harmonic oscillator from the standpoint of the originally developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry properties of the Schrödinger equation and on the orbit geometry of the coadjoint representation of Lie groups. We have shown that analogs of coherent states constructed by the noncommutative integration can be expressed in terms of the solution to a system of differential equations on the Lie group of the oscillatory Lie algebra. The solutions constructed are directly related to irreducible representation of the Lie algebra on the Hilbert space functions on the Lagrangian submanifold to the orbit of the coadjoint representation.

A study on multiple representation and self-efficacy perception in systems of linear equations

This study was performed to determine preservice elementary mathematics teachers’ solution performance of systems of linear equations using geometric representation, algebraic representation and transition from geometric representation to the algebraic representation and the predictive role of preservice elementary mathematics teachers’ performance of different representation usage on their systems of linear equations self-efficacy perceptions. The research follows the prediction model, which is one of the relational screening models. The data were obtained by using the ‘Systems of Linear Equations Self-Efficacy Test' developed by Kardes [Kardeş, D. (2010). Matematik öğretmen adaylarının lineer denklem sistemleri çözüm süreçlerinin öz-yeterlik algısı ve çoklu temsil bağlamında incelenmesi [An investigation of the processes of pre-services mathematics teachers? Solving systems of linear equations within the context of self-efficacy and multiple representations] [Master's thesis]. Marmara University. Turkish Higher Education Council thesis repository (#264144). https://tez.yok.gov.tr/UlusalTezMerkezi/] and the ‘Systems of Linear Equations Performance Test' developed by the researchers. Because of the research, it was found out that preservice teachers’ algebraic representation performances and their self-efficacy perceptions were at the maximum level when the number of equations and unknown variables was equal (m = n). Additionally, due to the multiple regression analysis, preservice teachers’ performances on geometric representation, the transition from geometric representation to algebraic representation and systems of linear equations in R 3 were determined to be significant predictors of their self-efficacy perception.