Partial Algebras Research Articles

Order Isomorphisms on Order Intervals of Atomic JBW-Algebras

In this paper a full description of order isomorphisms between effect algebras of atomic JBW-algebras is given. We will derive a closed formula for the order isomorphisms on the effect algebra of type I factors by proving that the invertible part of the effect algebra of a type I factor is left invariant. This yields an order isomorphism on the whole cone, for which a characterisation exists. Furthermore, we will show that the obtained formula for the order isomorphism on the invertible part can be extended to the whole effect algebra again. As atomic JBW-algebras are direct sums of type I factors and order isomorphisms factor through the direct sum decomposition, this yields the desired description.

What Is the Sense in Logic and Philosophy of Language

In the paper, various notions of the logical semiotic sense of linguistic expressions – namely, syntactic and semantic, intensional and extensional – are considered and formalised on the basis of a formal-logical conception of any language L characterised categorially in the spirit of certain Husserl's ideas of pure grammar, Leśniewski-Ajdukiewicz's theory of syntactic/semantic categories and, in accordance with Frege's ontological canons, Bocheński's and some of Suszko's ideas of language adequacy of expressions of L. The adequacy ensures their unambiguous syntactic and semantic senses and mutual, syntactic and semantic correspondence guaranteed by the acceptance of a postulate of categorial compatibility of syntactic and semantic (extensional and intensional) categories of expressions of L. This postulate defines the unification of these three logical senses. There are three principles of compositionality which follow from this postulate: one syntactic and two semantic ones already known to Frege. They are treated as conditions of homomorphism of partial algebra of L into algebraic models of L: syntactic, intensional and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, obviously, an idealisation. The syntactic and semantic unambiguity of its expressions is not, of course, a feature of natural languages, but every syntactically and semantically ambiguous expression of such languages may be treated as a schema representing all of its interpretations that are unambiguous expressions.

LEVEL-ZERO VAN DER KALLEN MODULES AND SPECIALIZATION OF NONSYMMETRIC MACDONALD POLYNOMIALS AT t = ∞

Let λ ∈ P+ be a level-zero dominant integral weight, and w the coset representative of minimal length for a coset in W/Wλ, where Wλ is the stabilizer of λ in a finite Weyl group W. In this paper, we give a module $$ {\mathbbm{K}}_w^{-}\left(\uplambda \right) $$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at t = ∞ of the nonsymmetric Macdonald polynomial Ewλ(q, t) multiplied by a certain explicit finite product of rational functions of q of the form (1 − q−r)−1 for a positive integer r. This module $$ {\mathbbm{K}}_w^{-}\left(\uplambda \right) $$ (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module $$ {V}_w^{-}\left(\uplambda \right) $$ by the sum of the submodules $$ {V}_z^{-}\left(\uplambda \right) $$ for all those coset representatives z of minimal length for cosets in W/Wλ such that z > w in the Bruhat order < on W.

2‐adic slopes of Hilbert modular forms over Q(5)

We show that for arithmetic weights with a fixed finite order character, the slopes of $U_p$ (for $p=2$) acting on overconvergent Hilbert modular forms of level $U_0(4)$ are independent of the (algebraic part of the) weight and can be obtained by a simple recipe from the classical slopes in parallel weight $3$.

Partial clones

A set [Formula: see text] of operations defined on a nonempty set [Formula: see text] is said to be a clone if [Formula: see text] is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the [Formula: see text]-ary operations defined on set [Formula: see text] for all natural numbers [Formula: see text] and the operations are the so-called superposition operations [Formula: see text] for natural numbers [Formula: see text] and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set [Formula: see text] and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties.

Implicative algebras: a new foundation for realizability and forcing

AbstractWe introduce the notion of implicative algebra, a simple algebraic structure intended to factorize the model-theoretic constructions underlying forcing and realizability (both in intuitionistic and classical logic). The salient feature of this structure is that its elements can be seen both as truth values and as (generalized) realizers, thus blurring the frontier between proofs and types. We show that each implicative algebra induces a (Set-based) tripos, using a construction that is reminiscent from the construction of a realizability tripos from a partial combinatory algebra. Relating this construction with the corresponding constructions in forcing and realizability, we conclude that the class of implicative triposes encompasses all forcing triposes (both intuitionistic and classical), all classical realizability triposes (in the sense of Krivine), and all intuitionistic realizability triposes built from partial combinatory algebras.

USE OF LINEAR ALGEBRA AND PARTIAL DERIVATIVES IN SUPERVISED LEARNING (ARTIFICIAL INTELLIGENCE AND MACHINE LEARNING)

When we talk about new technologies and the advancement in the field of Computer Science, the first thing that comes to our mind is Artificial Intelligence and Machine Learning. Artificial Intelligence has seen resurgence in the 21st century because of its ability to mimic functions done by human intelligence like “problem solving” and “learning”. It is slowly becoming the area of interest of the new generation because of its modern capabilities which even human intelligence struggle to perform like competing at highest level in strategic game systems, intelligent routing, operating cars autonomously and simulations. Artificial Intelligence may look easy but there are several tools involved in making it successful. One of the main tool is “Statistical Methods”. Linear algebra and Partial Differential Equations have become the base of this field. The objective of our paper is to throw light on how Statistical Methods and Mathematical optimization provide the base for the working of Supervised Learning. Over years, algorithms inspired by Partial Differential Equations (PDE) and Linear Algebra have had an immense impact on many processing and autonomously performed tasks that involve speech, image and video data. Image processing tasks and intelligent routing done using PDE models has lead to ground-breaking contributions. The reinterpretation of many modern machine capabilities like artificial neural networks through PDE lens has been creating multiple celebrated approaches that benefit a vast area. In this paper, we have established some working of these methods in different subfields of Artificial Intelligence. Guided by well-established theories we demonstrate new insights and algorithms for Supervised Learning and demonstrate the competitiveness of different numerical experiments used in the sub-fields. Not only will we see the wide application of Artificial intelligence but also its ability to slowly replace human works leading to unemployment which are part of its limitation. This research will provide wider insights into the multiple mathematical processes which acts as roots to make the field of Computer Science interesting and successful.

Notes on superconformal representations in two dimensions

We study global subalgebras of superconformal algebras in two dimensions and their unitary representations. Global superconformal multiplets are decomposed into conformal multiplets using Racah-Speiser algorithm, revealing many essential aspects of superconformal theories such as stress-energy tensor, conserved current, supersymmetric deformation and supersymmetry enhancement. Character formulae for the representations are presented. We further find a collection of conserved charges that are k-forms under the R-symmetry, which must be part of the super Virasoro algebra with N≥3 supersymmetries.

ON “PARTIAL ALGEBRAS AND THEIR THEORIES”BY HANS-JÜRGEN HOEHNKE AND JÜRGEN SCHRECKENBERGER

and concrete Mal’cev clones are commonly denoted as Mal’cev clones. (Of course, each concrete Mal’cev clone is an abstract one.) Let T be any partial theory and K be an arbitrary dht-symmetric category. Then the monoidal dht-symmetric functors F : T → K can be considered as the natural functorial description of the notion of a T -algebra over K. Especially for K = Par one obtains thus the natural functorial description of a partial T -algebra. For any functorial T -algebra over K, F : T → K, the authors introduce the arityinterpretation (“typization”) of T F ⊆ K, which is again a partial theory, Ary Im F . One observes, that each C-clone G by assumption generates a partial theory GC(Ko J ) = T ∈ |T hJ |, and then G is also called the (Mal’cev) clone-part of T . One may consider the problem to characterize the clone-part of a partial theory T independently of its connection to T . If T ∈ |T hJ | and F : T → K is a T -algebra over K, then we define Mcl F : = {(A, fF,B); f : A → B ∈ T , B ∈ J ∪ {I,O}}. This is the clone-part of the partial theory AryImF . It is also called the Mal’cev clone of the T -algebra F . In particular this notion applies to the case K = Par . In this case, Mcl F is obviously a concrete Mal’cev clone. Observe, that F maps the clone-part Mcl IdT of T onto Mcl F . The connection between the structures of partial theories T and their cloneparts Mcl T can be derived from the clone-part functor Cp = (|Cp|, Cp) : T hJ → MclJ , which is defined by: T |Cp| = Mcl T (T ∈ |T hJ |), F Cp = F ∣∣ MclT (F = (|F |, F ) : T → T ′ ∈ T hJ ). The clone-part functor is fully faithful. This has several consequences which are described in section 7.2 of the book. In section 8 the authors study clones which are derived from free algebras. 11 Strong Varieties, Regular Hyperidentities and Solid Strong Varieties Strong varieties of partial algebras of the same type are model classes of partial algebras defined by so-called strong identities. A strong identity in a partial algebra A is a pair (s, t) ∈

Regulation of heat transfer in Rayleigh–Bénard convection in Newtonian liquids and Newtonian nanoliquids using gravity, boundary temperature and rotational modulations

The individual effect of time-periodic gravity modulation, in-phase and out-of-phase temperature modulations and rotational modulation on Rayleigh–Benard convection in twenty-eight nanoliquids is studied in the paper using the two-phase description of the generalized Buongiorno model. The generalized Lorenz model for each modulation problem is derived using the truncated Fourier series representation. The method of multiscales is employed to arrive at the Ginzburg–Landau equations from the Lorenz models, and the solution of Ginzburg–Landau equations is used to quantify the heat transport. The modulation amplitude is considered to be small (of order less than unity) and low frequencies of modulation are considered. The coefficient of the linear term of the algebraic part of the Ginzburg–Landau equations is shown to exclusively hold the information on the amplitude and the frequency of modulation. The influence of nanoparticles (nanotubes) on heat transport in the presence/absence of various modulations is explained. The study reveals that the frequency of modulation is a dominant factor in the case of gravity and rotational modulations whereas in the case of boundary temperature modulation in addition to the frequency of modulation, the phase difference plays an important role. Effect of these three modulations is to enhance/diminish heat transport but depends strongly on the choice of values of frequency of modulation and amplitude. For fixed values of frequency ( $$\omega ^*=5$$ ) and amplitude ( $$\delta _2=0.1$$ ) of various modulations, it is shown that the maximum percentage of heat transport enhancement achieved in glycerin due to 5% of $$\hbox {SWCNTs}$$ is 21.86% for gravity modulation, 17.36% for rotational modulation and 15.63% for boundary temperature modulation (out of phase). The reason for highest heat transport in gravity modulation is explained by finding area under the curves of three modulations. The study reveals that the modulation whose area under the curve is maximum transports maximum heat. The results pertaining to the single-phase model are recovered as a limiting case of the present study. The study shows that the single-phase model under-predicts heat transport compared to the two-phase model. The results obtained in the present study are compared with those of previous investigations and qualitatively good agreement is found.

The rank of Mazur’s Eisenstein ideal

We use pseudodeformation theory to study Mazur’s Eisenstein ideal. Given prime numbers N and p>3, we study the Eisenstein part of the p-adic Hecke algebra for Γ0(N). We compute the rank of this Hecke algebra (and, more generally, its Newton polygon) in terms of Massey products in Galois cohomology, thereby answering a question of Mazur and generalizing a result of Calegari and Emerton. We also give new proofs of Merel’s result on this rank and of Mazur’s results on the structure of the Hecke algebra.

NeutroAlgebra is a Generalization of Partial Algebra

In this paper we recall, improve, and extend several definitions, properties and applications of our previous 2019 research referred to NeutroAlgebras and AntiAlgebras (also called NeutroAlgebraic Structures and respectively AntiAlgebraic Structures). Let be an item (concept, attribute, idea, proposition, theory, etc.). Through the process of neutrosphication, we split the nonempty space we work on into three regions {two opposite ones corresponding to and , and one corresponding to neutral (indeterminate) (also denoted ) between the opposites}, which may or may not be disjoint – depending on the application, but they are exhaustive (their union equals the whole space). A NeutroAlgebra is an algebra which has at least one NeutroOperation or one NeutroAxiom (axiom that is true for some elements, indeterminate for other elements, and false for the other elements). A Partial Algebra is an algebra that has at least one Partial Operation, and all its Axioms are classical (i.e. axioms true for all elements). Through a theorem we prove that NeutroAlgebra is a generalization of Partial Algebra, and we give examples of NeutroAlgebras that are not Partial Algebras. We also introduce the NeutroFunction (and NeutroOperation).

ON THE COHOMOLOGY OF TORELLI GROUPS

We completely describe the algebraic part of the rational cohomology of the Torelli groups of the manifolds$\#^{g}S^{n}\times S^{n}$relative to a disc in a stable range, for$2n\geqslant 6$. Our calculation is also valid for$2n=2$assuming that the rational cohomology groups of these Torelli groups are finite-dimensional in a stable range.

On the conical Novikov homology

Let $\omega$ be a Morse form on a manifold $M$. Let $p:\hat M\to M$ be a regular covering with structure group $G$, such that $p^*([\omega])=0$. Let $\xi:G\to\mathbf{R}$ be the corresponding period homomorphism. Denote by ${\hat \Lambda}_\xi$ the Novikov completion of the group ring $\mathbf{Z} G$. Choose a transverse $\omega$-gradient $v$. Counting the flow lines of $v$ one defines the Novikov complex $\mathcal{N}_*$ freely generated over ${\hat \Lambda}_\xi$ by the set of zeroes of $\omega$. In this paper we introduce a refinement of this construction. We define a subring $\hat\Lambda_\Gamma$ of ${\hat \Lambda}_\xi$ and show that the Novikov complex $\mathcal{N}_*$ is defined actually over $\hat\Lambda_\Gamma$ and computes the homology of the chain complex $C_*(\hat M)\underset{\Lambda}{\otimes}\hat\Lambda_\Gamma $. When $G\approx\mathbf{Z}^2$, and the irrationality degree of $\xi$ equals 2, the ring $\hat\Lambda_\Gamma$ is isomorphic to the ring of series in $2$ variables $x, y$ of the form $\sum_{r\in\mathbf{N}} a_r x^{n_r}y^{m_r}$ where $a_r, n_r, m_r\in\mathbf{Z}$ and both $n_r, \ m_r$ converge to $\infty$ when $r\to \infty$. The algebraic part of the proof is based on a suitable generalization of the classical algorithm of approximating irrational numbers by rationals. The geometric part is a straightforward generalization of the author's proof of the particular case of this theorem concerning the circle-valued Morse maps. In Appendix 1 we give an overview of E. Pitcher's work on circle-valued Morse theory (1939). We show that Pitcher's lower bounds for the number of critical points of a circle-valued Morse map coincide with the torsion-free part of the Novikov inequalities. In Appendix 2 we construct a circle-valued Morse map and its gradient such that its unique Novikov incidence coefficient is a power series in one variable with an arbitrarily small convergence radius.

On rigid origami I: piecewise-planar paper with straight-line creases

Origami (paper folding) is an effective tool for transforming two-dimensional materials into three-dimensional structures, and has been widely applied to robots, deployable structures, metamaterials, etc. Rigid origami is an important branch of origami where the facets are rigid, focusing on the kinematics of a panel-hinge model. Here, we develop a theoretical framework for rigid origami, and show how this framework can be used to connect rigid origami and its cognate areas, such as the rigidity theory, graph theory, linkage folding and computer science. First, we give definitions regarding fundamental aspects of rigid origami, then focus on how to describe the configuration space of a creased paper. The shape and 0-connectedness of the configuration space are analysed using algebraic, geometric and numeric methods. In the algebraic part, we study the tangent space and generic rigid-foldability based on the polynomial nature of constraints for a panel-hinge system. In the geometric part, we analyse corresponding spherical linkage folding and discuss the special case when there is no cycle in the interior of a crease pattern. In the numeric part, we review methods to trace folding motion and avoid self-intersection. Our results will be instructive for the mathematical and engineering design of origami structures.

Globalization of group cohomology in the sense of Alvares-Alves-Redondo

Recently E.R. Alvares, M.M. Alves and M.J. Redondo introduced a cohomology for a group G with values in a module over the partial group algebra Kpar(G), which is different from the partial group cohomology defined earlier by the first two named authors of the present paper. Given a unital partial action α of G on a (unital) algebra A we consider A as a Kpar(G)-module in a natural way and study the globalization problem for the cohomology in the sense of Alvares-Alves-Redondo with values in A. The problem is reduced to an extendibility property of cocycles. Furthermore, assuming that A is a product of blocks, we prove that any cocycle is globalizable, and globalizations of cohomologous cocycles are also cohomologous. As a consequence we obtain that the Alvares-Alves-Redondo cohomology group Hparn(G,A) is isomorphic to the usual cohomology group Hn(G,M(B)), where M(B) is the multiplier algebra of B and B is the algebra under the enveloping action of α.

A lower bound result for the central L-values of elliptic curves

In the present paper, we prove a stronger lower bound for the 2-adic valuation of the algebraic part of the central value of the complex L-series for the family of certain quadratic twists of elliptic curves with analytic rank zero and non-trivial rational 2-torsion.

Inequalities for the extended positive part of a von Neumann algebra related to operator-monotone and operator-convex functions

We extend inequalities for operator monotone and operator convex functions onto elements of the extended positive part of a von Neumann algebra. In particular, this provides an opportunity to extend the inequalities onto unbounded positive self-adjoint operators.

Free fermions in disguise

I solve a quantum chain whose Hamiltonian is comprised solely of local four-fermi operators by constructing free-fermion raising and lowering operators. The free-fermion operators are both non-local and highly non-linear in the local fermions. This construction yields the complete spectrum of the Hamiltonian and an associated classical transfer matrix. The spatially uniform system is gapless with dynamical critical exponent z = 3/2, while staggering the couplings gives a more conventional free-fermion model with an Ising transition. The Hamiltonian is equivalent to that of a spin-1/2 chain with next-nearest-neighbour interactions, and has a supersymmetry generated by a sum of fermion trilinears. The supercharges are part of a large non-abelian symmetry algebra that results in exponentially large degeneracies. The model is integrable for either open or periodic boundary conditions but the free-fermion construction only works for the former, while for the latter the extended symmetry is broken and the degeneracies split.

Dynamics and control of singular boolean networks

AbstractConsider a class of singular Boolean networks, which consist of two parts: difference part and algebraic part. Using the truth matrix of the algebraic part, the trajectories of the networks are obtained and certain properties are investigated. Then the results are extended to Boolean control networks and the controllability of singular Boolean control systems is investigated. Necessary and sufficient conditions are obtained.