Product Algebra Research Articles

Mathematical Modelling and Analysis of the Simplest Fuzzy Two-Input Two-Output Two-Term Controller of Takagi−Sugeno Type

Mathematical modelling of the simplest Takagi−Sugeno fuzzy Two-Input Two-Output (TITO) Proportional-Integral/Proportional-Derivative (PI/PD) controller is presented in this paper. Mathematical model of fuzzy PI/PD controller is proposed using Algebraic Product (AP) t-norm, Bounded Sum (BS) t-co-norm and Center of Gravity (CoG) defuzzifier. The inputs are fuzzified by fuzzy sets having <i>L</i> and <inline-formula id="M2"> <math id="mathml_M2" display="inline" overflow="scroll"><mi>Γ</mi></math></inline-formula> type membership functions. The rule base consists of five rules with different linear models in the consequent parts. Both static and dynamic coupling is taken into account while deriving the models. The model of the fuzzy PI/PD controller reveals that it is a variable gain/structure controller. Also, each output of the TITO fuzzy controller is the sum of two nonlinear PI or PD controllers with variable gains. The properties and the gain variations of the controllers are investigated.

K-theory of noncommutative Bernoulli shifts

For a large class of C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras A, we calculate the K-theory of reduced crossed products A⊗G⋊rG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^{\\otimes G}\\rtimes _rG$$\\end{document} of Bernoulli shifts by groups satisfying the Baum–Connes conjecture. In particular, we give explicit formulas for finite-dimensional C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras, UHF-algebras, rotation algebras, and several other examples. As an application, we obtain a formula for the K-theory of reduced C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras of wreath products H≀G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H\\wr G$$\\end{document} for large classes of groups H and G. Our methods use a generalization of techniques developed by the second named author together with Joachim Cuntz and Xin Li, and a trivialization theorem for finite group actions on UHF algebras developed in a companion paper by the third and fourth named authors.

Neutrosophic Near Algebra Over Neutrosophic Field

This piece of paper aims to learn neutrosophic near algebra and neutrosophic sub near algebra. This paper is summarized with the suitable definitions and theorems of neutrosophic near algebra and neutrosophic sub near algebra. It has also been demonstrated that the direct product of neutrosophic near algebra is a neutrosophic near algebra and the intersection of neutrosophic sub near algebra is a neutrosophic near algebra on a neutrosophic field. It also examined the union of couple of neutrosophic near algebras is a neutrosophic near algebra.

A new proof of Stanley’s theorem on the strong Lefschetz property

A standard graded artinian monomial complete intersection algebra $A=\Bbbk [x_1,\ldots ,x_n]/(x_1^{a_1},\ldots ,x_n^{a_n})$, with $\Bbbk $ a field of characteristic zero, has the strong Lefschetz property defined by Stanley in 1980. In this paper, we give

On the derived Lusztig correspondence

Abstract Let G be a connected reductive group, T a maximal torus of G, N the normalizer of T and $W=N/T$ the Weyl group of G. Let ${\mathfrak {g}}$ and ${\mathfrak {t}}$ be the Lie algebras of G and T. The affine variety $\mathfrak {car}={\mathfrak {t}}/\!/W$ of semisimple G-orbits of ${\mathfrak {g}}$ has a natural stratification $$ \begin{align*} \mathfrak{car}=\coprod_{\lambda}\mathfrak{car}_{\lambda} \end{align*} $$ indexed by the set of G-conjugacy classes of Levi subgroups: the open stratum is the set of regular semisimple orbits and the closed stratum is the set of central orbits. In [17], Rider considered the triangulated subcategory $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\mathrm {nil}}/G])^{\mathrm {Spr}}$ of $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\mathrm {nil}}/G])$ generated by the direct summand of the Springer sheaf. She proved that it is equivalent to the derived category of finitely generated dg modules over the smash product algebra ${\overline {\mathbb {Q}}_{\ell }}[W]\# H^{\bullet }_G(G/B)$ where $H^{\bullet }_G(G/B)$ is the G-equivariant cohomology of the flag variety. Notice that the later derived category is $D_{\mathrm {c}}^{\mathrm {b}}(\mathrm {B}(N))$ where $\mathrm {B}(N)=[\mathrm {Spec}(k)/N]$ is the classifying stack of N-torsors. The aim of this paper is to understand geometrically and generalise Rider’s equivalence of categories: For each $\lambda $ we construct a cohomological correspondence inducing an equivalence of categories between $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {t}}_{\lambda }/N])$ and $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\lambda }/G])^{\mathrm {Spr}}$ .

On approximately biprojective and approximately biflat Banach algebras

In this paper, we study the approximate biprojectivity and the approximate biflatness of a Banach algebra A and find some relations between theses concepts with ?-amenability and ? -contractibility, where ? is a character on A. Among other things, we show that ?-Lau product algebra L1(G) ?? A(G) is approximately biprojective if and only if G is finite, where L1(G) and A(G) are the group algebra and the Fourier algebra of a locally compact group G, respectively. We also characterize approximately biprojective and approximately biflat semigroup algebras associated with the inverse semigroups.

Decomposition of idempotent 2-cocycles

Let $L$ be a finite Galois field extension of $K$ with Galois group $G$. We decompose any idempotent 2-cocycle $f$ using finite sequences of descending two-sided ideals of the corresponding weak crossed product algebra $A_f$. We specialize the results in

T-HGSP: Hypergraph Signal Processing Using t-Product Tensor Decompositions

Graph signal processing (GSP) techniques are powerful tools that model complex relationships within large datasets, being now used in a myriad of applications in different areas including data science, communication networks, epidemiology, and sociology. Simple graphs can only model pairwise relationships among data which prevents their application in modeling networks with higher-order relationships. For this reason, some efforts have been made to generalize well-known graph signal processing techniques to more complex graphs such as hypergraphs, which allow capturing higher-order relationships among data. In this paper, we provide a new hypergraph signal processing framework (t-HGSP) based on a novel tensor-tensor product algebra that has emerged as a powerful tool for preserving the intrinsic structures of tensors. The proposed framework allows the generalization of traditional GSP techniques while keeping the dimensionality characteristic of the complex systems represented by hypergraphs. To this end, the core elements of the t-HGSP framework are introduced, including the shifting operators and the hypergraph signal. The hypergraph Fourier space is also defined, followed by the concept of bandlimited signals and sampling. In our experiments, we demonstrate the benefits of our approach in applications such as clustering and denoising.

Simple modules over and at root of unity

In this article the smash product algebras as in the title due to V. V. Bavula and T. Lu have been studied at the root of unity. A full classification of simple modules over along with a class of finite dimensional indecomposable modules are presented. Also simple X, Y-torsionfree modules over are classified.

Subregular J-rings of Coxeter systems via quiver path algebras

We use quivers and their representations to bring new perspectives on the subregular J-ring JC of a Coxeter system (W,S), a subring of Lusztig's J-ring. We prove that JC is isomorphic to a suitable quotient of the path algebra of the double quiver of (W,S). Up to Morita equivalence, such quotients include the group algebras of all free products of finite cyclic groups. We then use quiver representations to study the category mod-AK of finite dimensional right modules of the algebra AK=K⊗ZJC over an algebraically closed field K of characteristic zero. Our results include classifications of Coxeter systems for which mod-AK is semisimple, has finitely many simple modules up to isomorphism, or has a bound on the dimensions of simple modules.

Structural Completeness in Many-Valued Logics with Rational Constants

The logics RŁ, RP, and RG have been obtained by expanding Łukasiewicz logic Ł, product logic P, and Gödel–Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining results that stand in contrast to the known situation in Ł, P, and G. Namely, RŁ is hereditarily structurally complete. RP is algebraized by the variety of rational product algebras that we show to be Q-universal. We provide a base of admissible rules in RP, show their decidability, and characterize passive structural completeness for extensions of RP. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of RP, and this is also the case for extensions of RG, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of RG, we provide a base of admissible rules. We leave the problem open whether the variety of rational Gödel algebras is Q-universal.

Two results on double crossed biproducts

Let HH be an algebra with a distinguished element εH∈H∗εH∈H∗ and C,DC,D two coalgebras. Based on the construction of Brzeziński’s crossed coproduct, under some suitable conditions, we introduce a coassociative coalgebra C×GTHβR×DC×TGHRβ×D which is a more general two-sided coproduct structure including two-sided smash coproduct. Necessary and sufficient conditions for C×GTHβR×DC×TGHRβ×D equipped with two-sided tensor product algebra C⊗H⊗DC⊗H⊗D to be a bialgebra (Hopf algebra) are provided. On the other hand, we obtain an improved version of the double crossed biproduct C⋆αHβ⋆DC⋆αHβ⋆D in [An extended form of Majid's double biproduct, J. Algebra Appl. 16 (4), 1760061, 2017] which induces a description of C⋆αHβ⋆DC⋆αHβ⋆D similar to Majid double biproduct C⋆H⋆DC⋆H⋆D and also present some related structures.

Origamis associated to minimally intersecting filling pairs

Let $S_{g}$ denote the closed orientable surface of genus $g$. In joint work with Huang, the first author constructed exponentially-many (in $g$) mapping class group orbits of pairs of simple closed curves whose complement is a single topological disk. Using different techniques, we improve on this result by constructing factorially-many (again in $g$) such orbits. These new orbits are chosen so that the absolute value of the algebraic intersection number is equal to the geometric intersection number, implying that each pair naturally gives rise to an origami. We collect some rudimentary experimental data on the corresponding $SL(2, \mathbb{Z})$-orbits and suggest further study and conjectures.

Approximating the group algebra of the lamplighter by infinite matrix products

AbstractIn this paper, we introduce a new technique in the study of the*{*}-regular closure of some specific group algebrasKGinside𝒰⁢(G){{\mathcal{U}}(G)}, the*{*}-algebra of unbounded operators affiliated to the group von Neumann algebra𝒩⁢(G){{\mathcal{N}}(G)}. The main tool we use for this study is a general approximation result for a class of crossed product algebras of the formCK⁢(X)⋊Tℤ{C_{K}(X)\rtimes_{T}{\mathbb{Z}}}, whereXis a totally disconnected compact metrizable space,Tis a homeomorphism ofX, andCK⁢(X){C_{K}(X)}stands for the algebra of locally constant functions onXwith values on an arbitrary fieldK. The connection between this class of algebras and a suitable class of group algebras is provided by the Fourier transform. Utilizing this machinery, we study an explicit approximation for the lamplighter group algebra. This is used in another paper by the authors to obtain a whole family ofℓ2{\ell^{2}}-Betti numbers arising from the lamplighter group, most of them transcendental.

Affine wreath product algebras with trace maps of generic parity

The goal of this article is to study the structure and representation theory of affine wreath product algebras and its cyclotomic quotients These algebras appear naturally in Heisenberg categorification and generalize many important algebras (degenerate affine Hecke algebras, affine Sergeev algebras and wreath Hecke algebras). The whole class was introduced by D. Rosso and A. Savage in [16]. In [19], Savage studied both structure and representations under the condition that the trace map of A is even. In this paper we extend the definition for the case of odd trace. Since our approach is analogous to Savage’s, we consider the trace map being of arbitrary parity and unify statements and proofs. We also use an approach based on string diagrams, in the spirit of [5]. For simplicity of exposition, we assume A to be symmetric.

Quantized nilradicals of parabolic subalgebras of and algebras of coinvariants

Let PJ be the standard parabolic subgroup of SLn associated to a subset J of simple roots, and let be the standard Levi decomposition. We let and denote the quantized algebras of regular functions on PJ and LJ , respectively. Following work of the first author, we study the quantum analogue of an induced coaction and the corresponding subalgebra of coinvariants. It was previously shown that the smash product algebra is isomorphic to In view of this, – while it is not a Hopf algebra – can be viewed as a quantum analogue of the coordinate ring In this article we prove that when is nonzero and not a root of unity, is isomorphic to a quantum Schubert cell algebra associated to a certain parabolic element w in the Weyl group of In this setting, the quantum Schubert cell algebra is a q-deformation of the universal enveloping algebra of the nilradical of a parabolic subalgebra of We also compute explicit commutation relations among the Lusztig root vectors for these particular quantized nilradicals and we give explicit algebra isomorphisms from to

UNDECIDABILITY AND NON-AXIOMATIZABILITY OF MODAL MANY-VALUED LOGICS

AbstractIn this work we study the decidability of a class of global modal logics arising from Kripke frames evaluated over certain residuated lattices, known in the literature as modal many-valued logics. We exhibit a large family of these modal logics which are undecidable, in contrast with classical modal logic and propositional logics defined over the same classes of algebras. This family includes the global modal logics arising from Kripke frames evaluated over the standard Łukasiewicz and Product algebras. We later refine the previous result, and prove that global modal Łukasiewicz and Product logics are not even recursively axiomatizable. We conclude by closing negatively the open question of whether each global modal logic coincides with its local modal logic closed under the unrestricted necessitation rule.

Partial actions of a Hopf algebra on its base field and the corresponding partial smash product algebra

In this paper, we introduce the concept of a [Formula: see text]-Hopf algebra as a Hopf algebra obtained as the partial smash product algebra of a Hopf algebra and its base field, and show that every Hopf algebra is a [Formula: see text]-Hopf algebra. Moreover, a method to compute partial actions of a given Hopf algebra on its base field is developed and, as an application, we exhibit all partial actions of such type for some families of Hopf algebras.

Double crossed biproducts and related structures

Let H be a bialgebra. Let be a linear map, where A is a left H-comodule coalgebra, and an algebra with a left H-weak action Let be a linear map, where B is a right H-comodule coalgebra, and an algebra with a right H-weak action In this paper, we improve the necessary conditions for the two-sided crossed product algebra and the two-sided smash coproduct coalgebra to form a bialgebra (called double crossed biproduct) such that the condition in Majid’s double biproduct (or double-bosonization) is one of the necessary conditions. On the other hand, we provide a more general two-sided crossed product algebra structure via Brzezński’s crossed product and give some applications.

A procedure for efficient generation and behavioral evaluation of ultra‐packed ellipsoidal particle systems

AbstractMany engineering applications demand ultra‐packing of particles into a matrix. In general, efficiently simulating the random close packing of non‐spherical particles, and with 60% or greater volume loading remains a challenge. In this article, we describe an efficient procedure that achieves ultra‐packing of ellipsoidal particle systems of the order of – particles on a desktop computer. A significant challenge to ellipsoidal particle packing problem is particle‐particle contact detection. We describe an algebraic intersection detection method for ellipsoidal particles and couple it with a heuristic constructive algorithm to generate densely packed microstructures with ellipsoidal fillers. The effective behavior of the generated microstructures is then numerically estimated using an efficient random‐network model that is valid for high‐contrast particulate composites in near percolation arrangement. Finally, the algorithms are applied to evaluate the effective thermal conductivity of ellipsoidal particle filled composites used as thermal interface materials. Simulations of lognormally distributed particle systems with varying aspect ratios of ellipsoidal fillers are systematically carried out to draw conclusions on impact of particle shapes.