Algebras Of Type Research Articles

On the ranks and implicit constant fields of valuations induced by pseudo monotone sequences

Given a valued field ( K , v ) and a pseudo monotone sequence E in ( K , v ) , one has an induced valuation v E extending v to K ( X ) . After fixing an extension of v E to a fixed algebraic closure K ( X ) ‾ of K ( X ) , we show that the implicit constant field of the extension ( K ( X ) | K , v E ) is simply the henselization of ( K , v ) . We consider the question: given a value transcendental extension w of v to K ( X ) and a pseudo monotone sequence E in ( K , v ) , under which precise conditions are w induced by E ? The dual nature of pseudo convergent sequences of algebraic type and pseudo divergent sequences is also explored. Further, we provide a complete description of the various possibilities of the rank of the valuation v E , provided that v has finite rank.

Classification of Four Dimensional Train Algebras of Degree 2 and Exponent 4

This paper is devoted to the classification of train algebras of degree 2 and exponent 4. This classification is made in dimension at most four and according to the type of the algebra. We first show that in four dimension, the type of algebra can only be of (2,0,1,1), (2,1,0,1) or (2,1 ,1,0).

Verification of component-based systems with recursive architectures

We study a sound verification method for parametric component-based systems. The method uses a resource logic, a new formal specification language for distributed systems consisting of a finite yet unbounded number of components. The logic allows the description of architecture configurations coordinating instances of a finite number of types of components, by means of inductive definitions similar to the ones used to describe algebraic data types or recursive data structures. For parametric systems specified in this logic, we show that decision problems such as reaching deadlock or violating critical section are undecidable, in general. Despite this negative result, we provide for these decision problems practical semi-algorithms relying on the automatic synthesis of structural invariants allowing the proof of general safety properties. The invariants are defined using the WSκS fragment of the monadic second order logic, known to be decidable by a classical automata-logic connection, thus reducing a verification problem to checking satisfiability of a WSκS formula.

A Friendly Introduction to Path Algebras and Quiver Representations

We give an elementary introduction to the theory of quiver representations, which is the foundation of a dynamic area of research. Relying only on basic concepts from linear and abstract algebra, we construct the path algebra of a quiver, and illustrate the notion of the representation type of a finite-dimensional algebra.

Superconformal Algebras and Holomorphic Field Theories

We show that four-dimensional superconformal algebras admit an infinite-dimensional derived enhancement after performing a holomorphic twist. The type of higher symmetry algebras we find is closely related to algebras studied by Faonte–Hennion–Kapranov, Hennion–Kapranov, and the second author with Gwilliam in the context of holomorphic QFT. We show that these algebras are related to the two-dimensional chiral algebras extracted from four-dimensional superconformal theories by Beem and collaborators; further deforming by a superconformal element induces the Koszul resolution of a plane in mathbb {C}^2 cong {mathbb {R}}^4. The central charges at the level of chiral algebras arise from central extensions of the higher symmetry algebras.

Numerical quadrature for singular integrals on fractals

We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain Gamma subset mathbb {R}^n is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any “invariant” (also known as “balanced” or “self-similar”) measure supported on Gamma, including in particular the Hausdorff measure mathcal {H}^d restricted to Gamma, where d is the Hausdorff dimension of Gamma. Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over Gamma are decomposed into sums of integrals over suitable partitions of Gamma into self-similar subsets. For certain singular integrands of logarithmic or algebraic type, we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals, we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens.

Representations of Munn algebras and related semigroups

We establish representation types (finite, tame or wild) of finite dimensional Munn algebras with semisimple bases. As an application, we establish representation types of finite Rees matrix semigroups, in particular, 0-simple semigroups, and their mutually annihilating unions.

A completely unique account of enumeration

How can we enumerate the inhabitants of an algebraic datatype? This paper explores a datatype generic solution that works for all regular types and indexed families . The enumerators presented here are provably both complete and unique —they will eventually produce every value exactly once—and fair —they avoid bias when composing enumerators. Finally, these enumerators memoise previously enumerated values whenever possible, thereby avoiding repeatedly recomputing recursive results.

On the number of generators of an algebra over a commutative ring

A theorem of O. Forster says that if R R is a noetherian ring of Krull dimension d d , then every projective R R -module of rank n n can be generated by d + n d+n elements. S. Chase and R. Swan subsequently showed that this bound is sharp: there exist examples that cannot be generated by fewer than d + n d+n elements. We view rank- n n projective R R -modules as R R -forms of the non-unital R R -algebra R n R^n where the product of any two elements is 0 0 . The first two authors generalized Forster’s theorem to forms of other algebras (not necessarily commutative, associative or unital); A. Shukla and the third author then showed that this generalized Forster bound is optimal for finite étale algebras. In this paper, we prove new upper and lower bounds on the number of generators of an R R -form of a k k -algebra, where k k is an infinite field and R R is a finitely generated k k -ring of Krull dimension d d . In particular, we show that, contrary to expectations, for most types of algebras, the generalized Forster bound is far from optimal. Our results are particularly detailed in the case of Azumaya algebras. Our proofs are based on reinterpreting the problem as a question about approximating the classifying stack B G BG , where G G is the automorphism group of the algebra in question, by algebraic spaces of a certain type.

Simulation of hyperbolic type solitary waves based on velocity parameter for (3 + 1)-dimensional the B-type Kadomtsev–Petviashvili–Boussinesq equation

In this paper, we investigate solutions to the [Formula: see text]-dimensional B-type Kadomtsev–Petviashvili–Boussinesq (B-KPB) equation, which has strong nonlinearity feature by using sub-equation and the modified [Formula: see text]-expansion methods. Using these analytical methods, dark soliton, trigonometric oscillation, singular soliton, algebraic and hyperbolic type oscillating traveling wave solutions are presented. By giving special values to the parameters in the obtained traveling wave solutions, 3D, 2D and contour graphs representing the solitary waves are drawn. The advantages and disadvantages of both analytical methods are presented in comparison with each other. In addition, the changes of the solutions of the B-KPB equation according to the velocity parameter and the refraction phenomenon are supported by simulation. In addition, the solutions of the equation discussed are first discussed in detail, physical interpretations. Secondly, the geometric interpretations of the traveling wave solutions are analyzed and it is shown that the graphs of the solutions correspond to flat hyper-surfaces. It is argued that these hyper-surfaces can be leveled on a hyperplane without distortion.

Degree counting formula for non-symmetric Toda systems of rank two

In this paper, we consider B2 and G2 Toda systems on a compact Riemann surface M. We investigate the relation between the topological property of M and the Leray-Schauder degree of the Toda systems by computing the degree jump caused by multi-bubbling phenomena of the Toda system with non-symmetric Cartan matrix. The bubbling phenomena can reduce the computations of this degree jump to the calculations of the topological degree for some mean field equation together with an additional condition, which is called the shadow system in this paper. The main purpose of this paper is to compute the topological degree of this shadow system via a suitable deformation from a “decoupled” shadow system to our shadow system, and to show the a priori bound along this deformation. The proof of a priori bounds is very subtle, which depends on the type of Lie algebra.

On the type of the von Neumann algebra of an open subgroup of the Neretin group

The Neretin group N d , k \mathcal {N}_{d, k} is the totally disconnected locally compact group consisting of almost automorphisms of the tree T d , k \mathcal {T}_{d, k} . This group has a distinguished open subgroup O d , k \mathcal {O}_{d, k} . We prove that this open subgroup is not of type I. This gives an alternative proof of the recent result of P.-E. Caprace, A. Le Boudec and N. Matte Bon which states that the Neretin group is not of type I, and answers their question whether O d , k \mathcal {O}_{d, k} is of type I or not.

Infinite-Dimensional Algebras as Extensions of Kinematic Algebras

Kinematic algebras can be realised on geometric spaces and constrain the physical models that can live on these spaces. Different types of kinematic algebras exist and we consider the interplay of these algebras for non-relativistic limits of a relativistic system, including both the Galilei and the Carroll limit. We develop a framework that captures systematically the corrections to the strict non-relativistic limit by introducing new infinite-dimensional algebras, with emphasis on the Carroll case. One of our results is to highlight a new type of duality between Galilei and Carroll limits that extends to corrections as well. We realise these algebras in terms of particle models. Other applications include curvature corrections and particles in a background electro-magnetic field.

Verifying Catamorphism-Based Contracts using Constrained Horn Clauses

AbstractWe address the problem of verifying that the functions of a program meet their contracts, specified by pre/postconditions. We follow an approach based on constrained Horn clauses (CHCs) by which the verification problem is reduced to the problem of checking satisfiability of a set of clauses derived from the given program and contracts. We consider programs that manipulate algebraic data types (ADTs) and a class of contracts specified by catamorphisms, that is, functions defined by simple recursion schemata on the given ADTs. We show by several examples that state-of-the-art CHC satisfiability tools are not effective at solving the satisfiability problems obtained by direct translation of the contracts into CHCs. To overcome this difficulty, we propose a transformation technique that removes the ADT terms from CHCs and derives new sets of clauses that work on basic sorts only, such as integers and booleans. Thus, when using the derived CHCs there is no need for induction rules on ADTs. We prove that the transformation is sound, that is, if the derived set of CHCs is satisfiable, then so is the original set. We also prove that the transformation always terminates for the class of contracts specified by catamorphisms. Finally, we present the experimental results obtained by an implementation of our technique when verifying many non-trivial contracts for ADT manipulating programs.

Realizations of inner automorphisms of order four and fixed points subgroups by them on the connected compact exceptional Lie group E8, Part III

The compact simply connected Riemannian 4-symmetric spaces were classified by J. A. Jiménez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form G/H, where G is a connected compact simple Lie group with an automorphism γ˜ of order four on G and H is a fixed points subgroup Gγ of G. According to the classification by J. A. Jiménez, there exist seven compact simply connected Riemannian 4-symmetric spaces G/H in the case where G is of type E8. In the present article, we give the explicit form of automorphisms ω˜4, κ˜4 and ε˜4 of order four on E8 induced by the C-linear transformations ω4, κ4 and ε4 of the 248-dimensional C-vector space e8C, respectively. Further, we determine the structure of these fixed points subgroups (E8)ω4, (E8)κ4 and (E8)ε4 of E8. These amount to the global realizations of three spaces among seven Riemannian 4-symmetric spaces G/H above corresponding to the Lie algebras h=su(2)⊕iR⊕e6, iR⊕so(14) and h=su(2)⊕iR⊕so(12), where h=Lie(H). With this article, the all realizations of inner automorphisms of order four and fixed points subgroups by them have been completed in E8.

A Brief Note On QS/BP/BOI-Algebras

In this paper, the three types of the logical algebra of interest are QS-algebras, BOI-algebras, and BP-algebras, which are continuously studied by algebraists. Since the terms for each of the above logical algebras appear to be independent of each other, it may be understood that they are different. Therefore, the main purpose of this brief note is to show that QS-algebras, BOI-algebras, and BP-algebras satisfying (BM) as one.

STUDY OF STATISTICAL STABILITY AND FAST CHARACTERISTICS OF THE PROPOSED HASHING FUNCTION OF THE IMPROVED CRYPTOGRAPHIC MODULE IN INFORMATION AND COMMUNICATION SYSTEMS

The study of statistical stability and speed characteristics of the proposed hashing function of the advanced module of cryptographic protection of information, which by recording information about the user ID, session ID, sending time, message length and its serial number, as well as using a new session key for encryption , allows to ensure the confidentiality and integrity of data in information and communication systems management processes. The object of research is the process of ensuring the confidentiality of data in information and communication systems management systems based on cloud technologies. The subject is the study of statistical stability and speed characteristics of the proposed hashing function of the advanced module of cryptographic protection in information and communication systems. The purpose of this work is to study the statistical stability and speed characteristics of the proposed hashing function of the advanced module of cryptographic protection in information and communication systems for process control based on cloud technologies. To use this module effectively, it is important to choose crypto-resistant encryption and hashing methods, as well as secret key synchronization. Cryptoalgorithms resistant to linear, differential, algebraic, quantum and other known types of cryptanalysis can be used as encryption and hashing functions. An experimental study was conducted, which confirmed the crypto-resistance of the advanced algorithm to linear and differential cryptanalysis and determined that the speed of cryptographic data processing by the advanced method using the developed encryption function is 1.98 times higher than analogues, and the key generation speed 1.17 times compared to the generators used in particular in the selected prototype.

Real simple symplectic triple systems

The simple symplectic triple systems over the real numbers are classified up to isomorphism, and linear models of all of them are provided. Besides the split cases, there are two kinds of non-split real simple symplectic triple systems with classical enveloping algebra, called unitarian and quaternionic types, and five non-split real simple symplectic triple systems with exceptional enveloping algebra.

The least dimonoid congruences on relatively free trioids

When Loday and Ronco studied ternary planar trees, they introduced types of algebras,called trioids and trialgebras. A trioid is a nonempty set equipped with three binary associativeoperations satisfying additional eight axioms relating these operations, while a trialgebra is justa linear analog of a trioid. If all operations of a trioid (trialgebra) coincide, we obtain the notionof a semigroup (associative algebra), and if two concrete operations of a trioid (trialgebra)coincide, we obtain the notion of a dimonoid (dialgebra) and so, trioids (trialgebras) are ageneralization of semigroups (associative algebras) and dimonoids (dialgebras). Trioids andtrialgebras have close relationships with the Hopf algebras, the Leibniz 3-algebras, the Rota-Baxter operators, and the post-Jordan algebras. Originally, these structures arose in algebraictopology. One of the most useful concepts in algebra is the free object. Every variety containsfree algebras and free objects in any variety of algebras are important in the study of thatvariety. Loday and Ronco constructed the free trioid of rank 1 and the free trialgebra. Recently,the free trioid of an arbitrary rank, the free commutative trioid, the free n-nilpotent trioid, thefree rectangular triband, the free left n-trinilpotent trioid and the free abelian trioid wereconstructed and the least dimonoid congruences as well as the least semigroup congruence onthe first four free algebras were characterized. However, just mentioned congruences on freeleft (right) n-trinilpotent trioids and free abelian trioids were not considered. In this paper, wecharacterize the least dimonoid congruences and the least semigroup congruence on free left(right) n-trinilpotent trioids and free abelian trioids.

On the Stabilization of a Network of a Class of SISO Coupled Hybrid Linear Subsystems via Static Linear Output Feedback

This paper deals with the closed-loop stabilization of a network which consists of a set of coupled hybrid single-input single-output (SISO) subsystems. Each hybrid subsystem involves a continuous-time subsystem together with a digital (or, eventually, discrete-time) one being subject to eventual mutual couplings of dynamics and also to discrete delayed dynamics. The stabilizing controller is static and based on linear output feedback. The controller synthesis method is of algebraic type and based on the use of a linear algebraic system, whose unknown is a vector equivalent form of the controller gain matrix, which is obtained from a previous algebraic problem version which is based on the ad hoc use of the matrix Kronecker product of matrices. As a first step of the stabilization, an extended discrete-time system is built by discretizing the continuous parts of the hybrid system and to unify them together with its digital/discrete-time ones. The stabilization study via static linear output feedback contains several parts as follows: (a) stabilizing controller existence and controller synthesis for a predefined targeted closed-loop dynamics, (b) stabilizing controller existence and its synthesis under necessary and sufficient conditions based on the statement of an ad hoc algebraic matrix equation for this problem, (c) achievement of the stabilization objective under either partial or total decentralized control so that the whole controller has only a partial or null information about couplings between the various subsystems and (d) achievement of the objective under small coupling dynamics between subsystems.