Complex Algebra Research Articles

Idempotents in Banach Algebras II

Let A be a complex and unital Banach algebra, and let E denote the collection of idempotents of A. An old paper of J. Zemánek’s, which was really the starting point of studies of idempotents in general Banach algebras, exhibits a multitude of results concerning the set E. More specifically, it was shown that the connected component of E which contains p ∈ E, say Ep , is precisely the set of elements of the form wpw −1 where w runs through the principal component of the invertible group of A. Following Zemánek’s work, a considerable number of papers concerning the form of paths connecting the members of Ep have seen the light of day. In the present paper, we elaborate on the second part of Zemánek’s article; by utilizing modern techniques and results, together with a surprisingly general connection between the spectra of products of idempotents and linear combinations of idempotents, we show that the global algebraic conditions which characterize central idempotents of A can be replaced by significantly weaker local spectral conditions.

Projective relative unification through duality

AbstractUnification problems can be formulated and investigated in an algebraic setting, by identifying substitutions to modal algebra homomorphisms. This opens the door to applications of the notorious duality between Heyting or modal algebras and descriptive frames. Through substantial use of this correspondence, we give a necessary and sufficient condition for formulas to be projective. A close inspection of this characterization will motivate a generalization of standard unification, which we dub relative unification. Applying this result to a number of different logics, we then obtain new proofs of their projective—or non-projective—character. Aside from reproving known results, we show that the projective extensions of $\textbf{K5}$ are exactly the extensions of $\textbf{K45}$. This resolves the open question of whether $\textbf{K5}$ is projective.

Geometrical correspondence of the Miura transformation induced from affine Kac–Moody algebras

AbstractThe Miura transformation plays a crucial role in the study of integrable systems. There have been various extensions of the Miura transformation, which have been used to relate different kinds of integrable equations and to classify the bi‐Hamiltonian structures. In this paper, we are mainly concerned with the geometric aspects of the Miura transformation. The generalized Miura transformations from the mKdV‐type hierarchies to the KdV‐type hierarchies are constructed under both algebraic and geometric settings. It is shown that the Miura transformations not only relate integrable curve flows in different geometries but also induce the transition between different moving frames. Moreover, the Miura transformation gives the factorization of generating operators of constraint Gelfand–Dickey hierarchy. Other geometric formulations are also investigated.

WEAK HEIRS, COHEIRS, AND THE ELLIS SEMIGROUPS

Abstract Assume $G\prec H$ are groups and ${\cal A}\subseteq {\cal P}(G),\ {\cal B}\subseteq {\cal P}(H)$ are algebras of sets closed under left group translation. Under some additional assumptions we find algebraic connections between the Ellis [semi]groups of the G-flow $S({\cal A})$ and the H-flow $S({\cal B})$ . We apply these results in the model theoretic context. Namely, assume G is a group definable in a model M and $M\prec ^* N$ . Using weak heirs and weak coheirs we point out some algebraic connections between the Ellis semigroups $S_{ext,G}(M)$ and $S_{ext,G}(N)$ . Assuming every minimal left ideal in $S_{ext,G}(N)$ is a group we prove that the Ellis groups of $S_{ext,G}(M)$ are isomorphic to closed subgroups of the Ellis groups of $S_{ext,G}(N)$ .

Faculty Performance Evaluation through Multi-Criteria Decision Analysis Using Interval-Valued Fermatean Neutrosophic Sets

The Neutrosophic Set (Nset) represents the uncertainty in data with fuzzy attributes beyond true and false values independently. The problem arises when the summation of true (Tr), false (Fa), and indeterminacy In values crosses the membership value of one, that is, Tr+In+Fa<1. It becomes more crucial during decision-making processes like medical diagnoses or any data sets where Tr+In+Fa<1. To achieve this goal, the FNset is recently introduced. This study employs the Interval-Valued Fermatean Neutrosophic Set (IVFNset) as its chosen framework to address instances of partial ignorance within the domains of truth, falsehood, or uncertainty. This selection stands out due to its unique approach to managing such complexities within multi-decision processes when compared to alternative methodologies. Furthermore, the proposed method reduces the propensity for information loss often encountered in other techniques. IVFNS excels at preserving intricate relationships between variables even when dealing with incomplete or vague information. In the present work, we introduce the IVFNset, which deals with partial ignorance in true, false, or uncertain regions independently for multi-decision processes. The IVFNset contains the interval-valued Trmembership value, Inmembership value, and Famembership for knowledge representation. The algebraic properties and set theory between the interval-valued FNset have also been presented with an illustrative example.

Introduction to Algebraic Geometry

Summary A classical algebraic geometry is study of zero points of system of multivariate polynomials [3], [7] and those zero points would be corresponding to points, lines, curves, surfaces in an affine space. In this article we give some basic definition of the area of affine algebraic geometry such as algebraic set, ideal of set of points, and those properties according to [4] in the Mizar system[5], [2]. We treat an affine space as the n-fold Cartesian product k n as the same manner appeared in [4]. Points in this space are identified as n-tuples of elements from the set k. The formalization of points, which are n-tuples of numbers, is described in terms of a mapping from n to k, where the domain n corresponds to the set n = {0, 1, . . ., n − 1}, and the target domain k is the same as the scalar ring or field of polynomials. The same approach has been applied when evaluating multivariate polynomials using n-tuples of numbers [10]. This formalization aims at providing basic notions of the field which enable to formalize geometric objects such as algebraic curves which is used e.g. in coding theory [11] as well as further formalization of the fields [8] in the Mizar system, including the theory of polynomials [6].

(Slow-)twisting inflationary attractors

We explore in detail the dynamics of multi-field inflationary models. We first revisit the two-field case and rederive the coordinate independent expression for the attractor solution with either small or large turn rate, emphasizing the role of isometries for the existence of rapid-turn solutions.Then, for three fields in the slow-twist regime we provide elegant expressions for the attractor solution for generic field-space geometries and potentials and study the behaviour of first order perturbations. For generic 𝒩-field models, our method quickly grows in algebraic complexity. We observe that field-space isometries are common in the literature and are able to obtain the attractor solutions and deduce stability for some isometry classes of 𝒩-field models. Finally, we apply our discussion to concrete supergravity models. These analyses conclusively demonstrate the existence of 𝒩 > 2 dynamical attractors distinct from the two-field case, and provide tools useful for future studies of their phenomenology in the cosmic microwave background and stochastic gravitational wave spectrum.

Systems of Precision: Coherent Probabilities on Pre-Dynkin Systems and Coherent Previsions on Linear Subspaces.

In the literature on imprecise probability, little attention is paid to the fact that imprecise probabilities are precise on a set of events. We call these sets systems of precision. We show that, under mild assumptions, the system of precision of a lower and upper probability form a so-called (pre-)Dynkin system. Interestingly, there are several settings, ranging from machine learning on partial data over frequential probability theory to quantum probability theory and decision making under uncertainty, in which, a priori, the probabilities are only desired to be precise on a specific underlying set system. Here, (pre-)Dynkin systems have been adopted as systems of precision, too. We show that, under extendability conditions, those pre-Dynkin systems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum probability are equivalent to coherence from the imprecise probability literature. On this basis, we spell out a lattice duality which relates systems of precision to credal sets of probabilities. We conclude the presentation with a generalization of the framework to expectation-type counterparts of imprecise probabilities. The analogue of pre-Dynkin systems turns out to be (sets of) linear subspaces in the space of bounded, real-valued functions. We introduce partial expectations, natural generalizations of probabilities defined on pre-Dynkin systems. Again, coherence and extendability are equivalent. A related but more general lattice duality preserves the relation between systems of precision and credal sets of probabilities.

Relative Entropy of Fermion Excitation States on the CAR Algebra

The relative entropy of certain states on the algebra of canonical anticommutation relations (CAR) is studied in the present work. The CAR algebra is used to describe fermionic degrees of freedom in quantum mechanics and quantum field theory. The states for which the relative entropy is investigated are multi-excitation states (similar to multi-particle states) with respect to KMS states defined with respect to a time-evolution induced by a unitary dynamical group on the one-particle Hilbert space of the CAR algebra. If the KMS state is quasifree, the relative entropy of multi-excitation states can be explicitly calculated in terms of 2-point functions, which are defined entirely by the one-particle Hilbert space defining the CAR algebra and the Hamilton operator of the dynamical group on the one-particle Hilbert space. This applies also in the case that the one-particle Hilbert space Hamilton operator has a continuous spectrum so that the relative entropy of multi-excitation states cannot be defined in terms of von Neumann entropies. The results obtained here for the relative entropy of multi-excitation states on the CAR algebra can be viewed as counterparts of results for the relative entropy of coherent states on the algebra of canonical commutation relations which have appeared recently. It turns out to be useful to employ the setting of a self-dual CAR algebra introduced by Araki.

With or Without You: Programming with Effect Exclusion

Type and effect systems have been successfully used to statically reason about effects in many different domains, including region-based memory management, exceptions, and algebraic effects and handlers. Such systems’ soundness is often stated in terms of the absence of effects. Yet, existing systems only admit indirect reasoning about the absence of effects. This is further complicated by effect polymorphism which allows function signatures to abstract over arbitrary, unknown sets of effects. We present a new type and effect system with effect polymorphism as well as union, intersection, and complement effects. The effect system allows us to express effect exclusion as a new class of effect polymorphic functions: those that permit any effects except those in a specific set. This way, we equip programmers with the means to directly reason about the absence of effects. Our type and effect system builds on the Hindley-Milner type system, supports effect polymorphism, and preserves principal types modulo Boolean equivalence. In addition, a suitable extension of Algorithm W with Boolean unification on the algebra of sets enables complete type and effect inference. We formalize these notions in the λ ∁ calculus. We prove the standard progress and preservation theorems as well as a non-standard effect safety theorem: no excluded effect is ever performed. We implement the type and effect system as an extension of the Flix programming language. We conduct a case study of open source projects identifying 59 program fragments that require effect exclusion for correctness. To demonstrate the usefulness of the proposed type and effect system, we recast these program fragments into our extension of Flix.

Semantical investigations on non-classical logics with recovery operators: negation

Abstract We investigate mathematical structures that provide natural semantics for families of (quantified) non-classical logics featuring special unary connectives, known as recovery operators, that allow us to ‘recover’ the properties of classical logic in a controlled manner. These structures are known as topological Boolean algebras, which are Boolean algebras extended with additional operations subject to specific conditions of a topological nature. In this study, we focus on the paradigmatic case of negation. We demonstrate how these algebras are well-suited to provide a semantics for some families of paraconsistent Logics of Formal Inconsistency and paracomplete Logics of Formal Undeterminedness. These logics feature recovery operators used to earmark propositions that behave ‘classically’ when interacting with non-classical negations. Unlike traditional semantical investigations, which are carried out in natural language (extended with mathematical shorthand), our formal meta-language is a system of higher-order logic (HOL) for which automated reasoning tools exist. In our approach, topological Boolean algebras are encoded as algebras of sets via their Stone-type representation. We use our higher-order meta-logic to define and interrelate several transformations on unary set operations, which naturally give rise to a topological cube of opposition. Additionally, our approach enables a uniform characterization of propositional, first-order and higher-order quantification, including restrictions to constant and varying domains. With this work, we aim to make a case for the utilization of automated theorem proving technology for conducting computer-supported research in non-classical logics. All the results presented in this paper have been formally verified, and in many cases obtained, using the Isabelle/HOL proof assistant.

Surjective morphisms from affine space to its Zariski open subsets

We prove constructively the existence of surjective morphisms from affine space onto certain open subvarieties of affine space of the same dimension. For any algebraic set [Formula: see text], we construct an endomorphism of [Formula: see text] with [Formula: see text] as its image. By Noether’s normalization lemma, these results extend to give surjective maps from any [Formula: see text]-dimensional affine variety [Formula: see text] to [Formula: see text].

Euclidean distance degree and limit points in a Morsification

Motivated by finding an effective way to compute the algebraic complexity of the nearest point problem for algebraic models, we introduce an efficient method for detecting the limit points of the stratified Morse trajectories in a small perturbation of any polynomial function on a complex affine variety. We compute the multiplicities of these limit points in terms of vanishing cycles. In the case of functions with only isolated stratified singularities, we express the local multiplicities in terms of polar intersection numbers.

Graphs Isomorphisms Under Edge-Replacements and the Family of Amoebas

This paper offers a systematic study of a family of graphs called amoebas. Amoebas recently emerged from the study of forced patterns in $2$-colorings of the edges of the complete graph in the context of Ramsey-Turan theory and played an important role in extremal zero-sum problems. Amoebas are graphs %with a unique behavior with regards to defined by means of the following operation: Let $G$ be a graph and let $e\in E(G)$ and $e'\in E(\overline{G})$. If the graph $G'=G-e+e'$ is isomorphic to $G$, we say $G'$ is obtained from $G$ by performing a feasible edge-replacement. We call $G$ a local amoeba if, for any two copies $G_1$, $G_2$ of $G$ on the same vertex set, $G_1$ can be transformed into $G_2$ by a chain of feasible edge-replacements. On the other hand, $G$ is called global amoeba if there is an integer $t_0 \ge 0$ such that $G \cup tK_1$ is a local amoeba for all $t \ge t_0$. To model the dynamics of the feasible edge-replacements of $G$, we define a group $\rm{Fer}(G)$ that satisfies that $G$ is a local amoeba if and only if $\rm{Fer}(G) \cong S_n$, where $n$ is the order of $G$. Via this algebraic setting, a deeper understanding of the structure of amoebas and their intrinsic properties comes into light. Moreover, we present different constructions that prove the richness of these graph families showing, among other things, that any connected graph can be a connected component of a global amoeba, that global amoebas can be very dense and that they can have, in proportion to their order, large clique and chromatic numbers. Also, a family of global amoeba trees with a Fibonacci-like structure and with arbitrary large maximum degree is constructed.

Jacobson’s theorem on derivations of primitive rings with nonzero socle: a proof and applications

We provide a proof of Jacobson’s theorem on derivations of primitive rings with nonzero socle. Both Jacobson’s theorem and its formulation (in terms of the so-called differential operators on left vector spaces over a division ring) underlie our paper. We apply Jacobson’s theorem to describe derivations of standard operator rings on real, complex, or quaternionic left normed spaces. Indeed, when the space is infinite-dimensional, every derivation of such a standard operator ring is of the form Arightarrow AB-BA for some continuous linear operator B on the space. Our quaternionic approach allows us to generalize Rickart’s theorem on representation of primitive complete normed associative complex algebras with nonzero socle to the case of primitive real or complex associative normed Q-algebras with nonzero socle. We prove that additive derivations of the Jordan algebra of a continuous nondegenerate symmetric bilinear form on any infinite-dimensional real or complex Banach space are in a one-to-one natural correspondence with those continuous linear operators on the space which are skew-adjoint relative to the form. Finally we prove that additive derivations of a real or complex (possibly non-associative) H^*-algebra with no nonzero finite-dimensional direct summand are linear and continuous.

Quasi-free states on a class of algebras of multicomponent commutation relations

Multicomponent commutations relations (MCRs) describe plektons, i.e. multicomponent quantum systems with a generalized statistics. In such systems, exchange of quasiparticles is governed by a unitary matrix [Formula: see text] that depends on the position of quasiparticles. For such an exchange to be possible, the matrix must satisfy several conditions, including the functional Yang–Baxter equation. The aim of the paper is to give an appropriate definition of a quasi-free state on an MCR algebra, and construct such states on a class of MCR algebras. We observe a significant difference between the classical setting for bosons and fermions and the setting of MCR algebras. We show that the developed theory is applicable to systems that contain quasiparticles of opposite type. An example of such a system is a two-component system in which two quasiparticles, under exchange, change their respective types to the opposite ones ([Formula: see text], [Formula: see text]). Fusion of quasiparticles means intuitively putting several quasiparticles in an infinitely small box and identifying the statistical behavior of the box. By carrying out fusion of an odd number of particles from the two-component system as described above, we obtain further examples of quantum systems to which the developed theory is applicable.

A Comparison of Sets of Recognizable Weighted Tree Languages Over Specific Sets of Bounded Lattices

We consider weighted tree automata (wta) over bounded lattices, locally finite bounded lattices, distributive bounded lattices, and finite chains and we consider the weighted tree languages defined by their run semantics and by their initial algebra semantics. Due to these eight combinations, there are eight sets of weighted tree languages. Moreover, we consider the four sets of weighted tree languages which are recognized by crisp-deterministic wta over the above mentioned sets of weight algebras. We give all inclusion relations among all these twelve sets of weighted tree languages. More precisely, we prove that eleven of the twelve sets are equal, and this set is a proper subset of the set of weighted tree languages recognized by wta over bounded lattices with initial algebra semantics.

Divisibility on point counting over finite Witt rings

Let Fq denote the finite field of q elements with characteristic p. Let Zq denote the unramified extension of the p-adic integers Zp with residue field Fq. In this paper, we investigate the q-divisibility for the number of solutions of a polynomial system in n variables over the finite Witt ring Zq/pmZq, where the n variables of the polynomials are restricted to run through a box lifting Fqn. It turns out that in general the answers do depend upon the box chosen. Based on the addition operation of Witt vectors, we prove a q-divisibility theorem for any box of low algebraic complexity, including the simplest Teichmüller box. This extends the classical Ax-Katz theorem over finite field Fq (the case m=1). Taking q=p to be a prime, our result extends and improves a recent theorem of Grynkiewicz for the unweighted case.

On weak annihilators and nilpotent associated primes of skew PBW extensions

We investigate the notions of weak annihilator and nilpotent associated prime defined by Ouyang and Birkenmeier (Weak annihilator over extension rings, 2012) in the setting of skew PBW extensions. We extend several results formulated in the literature concerning annihilators and associated primes of commutative rings and skew polynomial rings to a more general setting of algebras not considered before. We exemplify our results with families of algebras appearing in the theory of enveloping algebras, noncommutative algebraic geometry, and theoretical physics. Finally, we present some ideas for future research.

A framework for the finite series method of the generalized Lorenz-Mie theory and its application to freely-propagating Laguerre-Gaussian beams

In face of a revival of interest in the finite series (FS) method due to recent developments upon generalized Lorenz-Mie theories (GLMTs), a more general, understandable, and systematic formulation is proposed. Possibly due to an apparent lack of flexibility in the FS method’s earlier statements, there has been a void in its use since the 1990s. Particularly, the method demands some degree of mathematical labor each time it is applied to a different kind of field profile. Furthermore, the algebraic complexity of its earlier occurrences might also have weighted upon the method’s historical shunning. Dissecting the later works reclaiming the FS, several possibilities for generalization, simplification, and organization were found. Accordingly, with the intent to render the method more approachable and to encourage its use, this work derives an alternative path suitable for both understanding and implementation. Applying the procedure, expressions for the beam shape coefficients of freely propagating Laguerre-Gaussian beams are obtained in closed form in a more straightforward manner when compared to previous formulations – this time not relying on recursions.