Theorem Of Algebra Research Articles

Algorithm for Accumulating Part Mating Gaps to Evaluate Solid and Fluid Performances

Abstract Part mating gaps need to be effectively adjusted during the assembly of products and greatly affect the working performance. This paper develops an algorithm to calculate the dimensional and aerodynamic indexes affected by part mating gaps. Mating gaps are expressed by displacements of one contact surface, and indexes are evaluated by displacements of key surface, when the surface is ready to compare with the other. A graph that denotes contacts as nodes and related paths through the physical domain as lines is proposed to express assembly sequences and hierarchies. A variable is defined to combine the time set with the displacement set. Boolean algebraic theorems are extended to derive a compact expression for the contact graph that supports the organization of accumulations at different surfaces with propagations through physical domains. Demonstrations of this method using three products exhibit the general applicability, and the application shows that the performance deviations of the centrifugal fan and axial turbine are apparent. In particular, the isentropic efficiency is good with a certain probability, despite turbines having mating gaps. The algorithm benefits both design and assembly: design can be performed through the fluid domain, which is affected by the mating gaps, and when the parts are being adjusted, the selected tolerance limit allows engineers to monitor key surfaces to ensure good aerodynamic performance.

The Cramer's rule for the parametrization of phenol and its hydroxylated byproducts: UV spectroscopy vs. high performance liquid chromatography.

A linear algebra theorem like Cramer's rule was used for the analysis of a system of equations obtained from UV spectroscopy, and results were compared against those obtained from HPLC analysis. This parametrization allowed to quantify the concentration of the main intermediate products detected along the photodegradation of phenol under UV-Vis irradiation of TiO2. UV spectroscopy data for phenol, hydroquinone, and benzoquinone were analyzed using the Cramer's rule. The overlapping interference of the intermediate products in the UV spectra was corrected. It can be concluded that the Cramer's rule can be used for the parametrization of the UV absorbance data of phenol and its main intermediate products. This methodology permitted to obtain the concentration of phenol and their intermediate products by UV-visible with a high precision in comparison of HPLC. The parametrization showed a correlation coefficient of ca. 0.9775 between the phenol concentration obtained by UV spectroscopy and values obtained from HPLC analysis. In this sense, results can be considered with good precision, and accordingly, it can be concluded that the methodology is reliable, and UV-visible spectroscopy can be selected instead of HPLC in much of the experiments concerning with aqueous-phase reactions.

On Kuiper type theorems for uniform Roe algebras

Generalizing the case of an infinite discrete metric space of finite diameter, we say that a discrete metric space (X,d) is a Kuiper space, if the group of invertible elements of its uniform Roe algebra is norm-contractible. Various sufficient conditions on (X,d) to be or not to be a Kuiper space are obtained.

The Singular Value Decomposition over Completed Idempotent Semifields

In this paper, we provide a basic technique for Lattice Computing: an analogue of the Singular Value Decomposition for rectangular matrices over complete idempotent semifields (i-SVD). These algebras are already complete lattices and many of their instances—the complete schedule algebra or completed max-plus semifield, the tropical algebra, and the max-times algebra—are useful in a range of applications, e.g., morphological processing. We further the task of eliciting the relation between i-SVD and the extension of Formal Concept Analysis to complete idempotent semifields (K-FCA) started in a prior work. We find out that for a matrix with entries considered in a complete idempotent semifield, the Galois connection at the heart of K-FCA provides two basis of left- and right-singular vectors to choose from, for reconstructing the matrix. These are join-dense or meet-dense sets of object or attribute concepts of the concept lattice created by the connection, and they are almost surely not pairwise orthogonal. We conclude with an attempt analogue of the fundamental theorem of linear algebra that gathers all results and discuss it in the wider setting of matrix factorization.

Double centralizer properties related to (co)triangular Hopf coquasigroups

Let H be a triangular Hopf coquasigroup with bijective antipode and B a cotriangular Hopf coquasigroup with bijective antipode. Under some favorable conditions, the aim of this paper is to find some objects satisfying the double centralizer property for any Hopf coquasigroup A which can be written as a tensor product Hopf coquasigroup As a consequence of our theory, both Schur’s double centralizer theorems for triangular and cotriangular Hopf algebras can be obtained. Our main result provides a new approach to construct more objects which have double centralizer property too.

On Steinberg algebras of Hausdorff ample groupoids over commutative semirings

We investigate the algebra of a Hausdorff ample groupoid , introduced by Steinberg, over a commutative semiring S . In particular, we obtain a complete characterization of congruence-simpleness for such Steinberg algebras , extending the well-known characterizations when S is a field or a commutative ring . We also provide a criterion for the Steinberg algebra A S ( G E ) of the graph groupoid G E associated to an arbitrary graph E to be congruence-simple. Motivated by a result of Clark and Sims, we show that the natural homomorphism from the Leavitt path algebra L B ( E ) to the Steinberg algebra A B ( G E ) , where B is the Boolean semifield, is an isomorphism if and only if E is row-finite. Moreover, we establish the Reduction Theorem and Uniqueness Theorems for Leavitt path algebras of row-finite graphs over the Boolean semifield B .

Central orders in simple finite dimensional superalgebras

The well-known Formanek's module finiteness theorem states that every unital prime PI-algebra (i.e. a central order in a matrix algebra by Posner's theorem) embeds into a finitely generated module over its center. An analogue of this theorem for alternative and Jordan algebras was earlier proved by V.N.Zhelyabin and the author. In this paper we discuss this problem for associative, classical Jordan and some alternative superalgebras.

A duality theorem for vertex operator algebras

In this paper, we prove that (V#H)#H⁎≅M(n,V) as S-local vertex operator algebras, where V is a simple vertex operator algebra, and H is the group algebra of a finite subgroup of Aut(V), and n=dim⁡H.

A dynamic dimensional accumulation with propagations of part mating gaps for series-parallel assembly operations

The changing spring-back effects of part mating gaps restrict the control of the overall dimensional error for the parts assembled by many sequential and parallel processes. So far, it has been merely adjusted by component positions or compensated by some joining processes, which is still limited and lack of a theory and tool for the general dynamic process. To obtain the dimensional response of current assembly when the part contacts with the other, this paper proposes algebraic modelling for dimensional error accumulations that organize the propagation through every part. A new contact graph firstly expresses the part liaisons. Then, a variable array encodes the displacements and the time when each contact finishes, which yields algebraic theorems for the conversion of contact graph to algebraic expression, and for estimating dimensional distortions. If propagations take finite element methods, the algebraic expression outputs the coupled spring-back effects for the current assembled parts that support both variation and tolerance analyses. The dynamic accumulation with propagations is validated by case studies, and the comparison indicates that the assembly with the largest number of parallel modes has the highest nonlinear variations caused by locating errors.

Gelfand theory for real Banach algebras

We are concerned with the development of the more general real case of the classical theorem of Gelfand on representation of a complex commutative unital Banach algebra as an algebra of continuous functions defined on a compact Hausdorff space. To that end, we use only intrinsic methods which do not depend on the complexification of the algebra, and obtain two representation theorems for commutative unital real Banach algebras as algebras of continuous real (respectively, complex) functions on the compact space of real-valued (respectively, complex-valued) $$\mathbb R$$ -algebra homomorphisms.

An Algebraic Analogue of Exel–Pardo C∗-Algebras

We introduce an algebraic version of the Katsura C∗-algebra of a pair A,B of integer matrices and an algebraic version of the Exel–Pardo C∗-algebra of a self-similar action on a graph. We prove a Graded Uniqueness Theorem for such algebras and construct a homomorphism of the latter into a Steinberg algebra that, under mild conditions, is an isomorphism. Working with Steinberg algebras over non-Hausdorff groupoids we prove that in the unital case, our algebraic version of Katsura C∗-algebras are all isomorphic to Steinberg algebras.

The minimal degree standard identity on MnE2 and MnE3

We prove an Amitsur–Levitzki-type theorem for Grassmann algebras, stating that the minimal degree of a standard identity that is a polynomial identity of the ring of n × n matrices over the m-generated Grassmann algebra is at least \(2\lfloor\frac{m}{2}\rfloor+4n-4\) for all n, m ≥ 2 and this bound is sharp for m = 2,3 and any n ≥ 2. The arguments are purely combinatorial, based on computing sums of signs corresponding to Eulerian trails in directed graphs.

Degeneracy second main theorem for meromorphic mappings and moving hypersurfaces with truncated counting functions and applications

In this paper, we establish a new second main theorem for meromorphic mappings of [Formula: see text] into [Formula: see text] and moving hypersurfaces with truncated counting functions in the case, where the meromorphic mappings may be algebraically degenerate. A version of the second main theorem with weighted counting functions is also given. Our results improve the recent results on this topic. As an application, an algebraic dependence theorem for meromorphic mappings sharing moving hypersurfaces is given.

English

English

First Integral Method for Constructing New Exact Solutions of The important Nonlinear Evolution Equations in Physics

In this paper, some new exact solutions of the important nonlinear partial differential equations in physics as Gardener’s equation and Sharma-Tasso-Over equation are formally derived by utilizing the first integral method, where it is equipment us with many exact solutions by using the travelling wave transform, then deduce a system of ordinary differential equations which is solved by depending on theorem in commutative algebra and with helping the mathematical software like Maple and Wolfram Mathematica.

A Chekanov–Eliashberg algebra for Legendrian graphs

We define a differential graded algebra for Legendrian graphs and tangles in the standard contact Euclidean three space. This invariant is defined combinatorially by using ideas from Legendrian contact homology. The construction is distinguished from other versions of Legendrian contact algebra by the vertices of Legendrian graphs. A set of countably many generators and a generalized notion of equivalence are introduced for invariance. We show a van Kampen type theorem for the differential graded algebras under the tangle replacement. Our construction recovers many known algebraic constructions of Legendrian links via suitable operations at the vertices.

How Fundamental is the Fundamental Theorem of Algebra?

SummaryWe give several non-obvious proofs of the fundamental theorem of algebra, coming from different parts of mathematics. These include the idea of homotopy, Hurwitz’s theorem, Rouché’s theorem, the open mapping principle, and the argument principle.

Coquasi-Bialgebras with Preantipode and Rigid Monoidal Categories

By a theorem of Majid, every monoidal category with a neutral quasi-monoidal functor to finitely-generated and projective $\Bbbk$-modules gives rise to a coquasi-bialgebra. We prove that if the category is also rigid, then the associated coquasi-bialgebra admits a preantipode, providing in this way an analogue for coquasi-bialgebras of Ulbrich's reconstruction theorem for Hopf algebras. When $\Bbbk$ is field, this allows us to characterize coquasi-Hopf algebras as well in terms of rigidity of finite-dimensional corepresentations.

Formal Verification of Hardware Components in Critical Systems

Hardware components, such as memory and arithmetic units, are integral part of every computer-controlled system, for example, Unmanned Aerial Vehicles (UAVs). The fundamental requirement of these hardware components is that they must behave as desired; otherwise, the whole system built upon them may fail. To determine whether or not a component is behaving adequately, the desired behaviour of the component is often specified in the Boolean algebra. Boolean algebra is one of the most widely used mathematical tools to analyse hardware components represented at gate level using Boolean functions. To ensure reliable computer-controlled system design, simulation and testing methods are commonly used to detect faults; however, such methods do not ensure absence of faults. In critical systems’ design, such as UAVs, the simulation-based techniques are often augmented with mathematical tools and techniques to prove stronger properties, for example, absence of faults, in the early stages of the system design. In this paper, we define a lightweight mathematical framework in computer-based theorem prover Coq for describing and reasoning about Boolean algebra and hardware components (logic circuits) modelled as Boolean functions. To demonstrate the usefulness of the framework, we (1) define and prove the correctness of principle of duality mechanically using a computer tool and all basic theorems of Boolean algebra, (2) formally define the algebraic manipulation (step-by-step procedure of proving functional equivalence of functions) used in Boolean function simplification, and (3) verify functional correctness and reliability properties of two hardware components. The major advantage of using mechanical theorem provers is that the correctness of all definitions and proofs can be checked mechanically using the type checker and proof checker facilities of the proof assistant Coq.

Cluster Consensus of Multiagent Systems With Weighted Antagonistic Interactions.

This article addresses the problem of cluster consensus for multiagent systems (MASs) associated with weighted antagonistic interactions. Compared with some existing results, a general communication topology among agents is introduced in this article, where there is no need to confine directed cycles to the root of a spanning tree. By taking into consideration the structures of directed cycles and multiple paths, the judgment of structural balance is simplified significantly. A novel cluster consensus protocol is also proposed for structurally unbalanced digraphs. Moreover, two necessary and sufficient conditions are derived, by which cluster consensus can be achieved in an MAS if and only if its communication topology contains a directed spanning tree. Then, by employing an algebra theorem, a sufficient criterion for the unstable system under a directed cycle is obtained, whether the number of agents on this cycle is odd or even. Some illustrative examples are given to demonstrate the effectiveness of theoretical results.