Search

Abstract: We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials. [Copyright &y& Elsevier]

We present the basic analytical and algebraic properties of triangular norms. We discuss continuity as well as the important classes of Archimedean, strict and nilpotent t-norms. Triangular conorms and De Morgan triples are also mentioned. Finally, a brief historical survey on triangular norms is given. [Copyright &y& Elsevier]

Abstract: Ahn, Lee, Park and Yoo proved that the best constrained degree reduction of a polynomial f in -norm equals the best weighted Euclidean approximation of the Bernstein–Bézier coefficients of f in a paper, published in the journal, Computer Aided Geometric Design 21 (2) (2004) 181–191. In this note, we point out an error in their paper and give the correct result. [Copyright &y& Elsevier]

Abstract: In this paper, we present some sixth-order class of modified Ostrowski’s methods for solving nonlinear equations. Per iteration each class member requires three function and one first derivative evaluations, and is shown to be at least sixth-order convergent. Several numerical examples are given to illustrate the performance of some of the presented methods. [Copyright &y& Elsevier]

In Ref. [1] , the κ -Dirac equation, based on the κ -deformed Poincaré-Hopf algebra, have been studied. In particular, solutions of the κ -Dirac oscillator (DO), in a three-dimensional space, were obtained by deriving the associated radial equations. We point out, however, a miscalculation in treating these equations, which had led to erroneous conclusions, particularly, about the energy eigenvalues and the breaking of their infinite degeneracy by the κ -deformation. By the way, we present a simple alternative method to solve the problem using an algebraic procedure. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero. By a ⁎-superalgebra we mean an algebra A with graded involution over F. Recently, algebras with graded involution have been extensively studied in PI-theory and the sequence of ⁎-graded codimensions { c n gri (A) } n ≥ 1 has been investigated by several authors. In this paper, we classify varieties generated by unitary ⁎-superalgebras having quadratic growth of ⁎-graded codimensions. As a result we obtain that a unitary ⁎-superalgebra with quadratic growth is T 2 ⁎ -equivalent to a finite direct sum of minimal unitary ⁎-superalgebras with at most quadratic growth, where at least one ⁎-superalgebra of this sum has quadratic growth. Furthermore, we provide a method to determine explicitly the factors of those direct sums. [ABSTRACT FROM AUTHOR]

For two given idempotents p and q from an associative algebra A , in this paper, we offer a comprehensive classification of algebras spanned by the idempotents p and q. This classification is based on the condition that p and q are not tightly coupled and satisfy (p q) m − 1 = (p q) m but (p q) m − 2 p ≠ (p q) m − 1 p for some m (≥ 2) ∈ N. Subsequently, we categorize all the group invertible elements and establish an upper bound for the Drazin index of any elements in these algebras spanned by p , q. Moreover, we formulate a new representation for the Drazin inverse of α p + q under two different assumptions, (p q) m − 1 = (p q) m and λ (p q) m − 1 = (p q) m , where α is a non-zero and λ is a non-unit real or complex number. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero and let m ≥ 2 be an integer. In this paper, we prove that if a group grading on U T m (F) admits a graded involution then this grading is a coarsening of a Z ⌊ m 2 ⌋ -grading on U T m (F) and the graded involution is equivalent to the reflection or symplectic involution on U T m (F) , this grading is called the finest grading on U T m (F). Furthermore, if m ≤ 4 the algebra U T m (F) with the finest grading satisfies no non-trivial monomial identities. For the finest grading, a finite basis for the (Z ⌊ m 2 ⌋ , ⁎) -identities is exhibited with the reflection and symplectic involutions and the asymptotic growth of the (Z ⌊ m 2 ⌋ , ⁎) -codimensions is determined. As a consequence, we prove that for any G -grading on U T m (F) and any graded involution the (G , ⁎) -exponent is m. Finally, we study the algebra U T 3 (F). For this algebra, there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical Z -grading and the Z 2 -grading induced by (0 , 1 , 0). We determine a basis for the (Z 2 , ⁎) -identities and we compute the codimension sequence for the (Z 2 , ⁎) -graded identities for U T 3 (F). [ABSTRACT FROM AUTHOR]

In a previous paper [4] we tried to build the basic theory of unbounded Tomita's observable algebras called T †-algebras which are related to unbounded operator algebras, especially unbounded Tomita-Takesaki theory, operator algebras on Krein spaces, studies of positive linear functionals on *-algebras and so on. And we defined the notions of regularity, semisimplicity and singularity of T †-algebras and characterized them. In this paper we shall proceed further with our studies of T †-algebras and investigate whether a T †-algebra is decomposable into a regular part and a singular part. [ABSTRACT FROM AUTHOR]

This paper enriches a pre-existing decision algorithm, which in turn augmented a fragment of Tarski's elementary algebra with one-argument real functions endowed with a continuous first derivative. In its present (still quantifier-free) version, our decidable language embodies the addition of functions and multiplication of functions by scalars; the issue we address is the one of satisfiability. As regards real numbers, individual variables and constructs designating the basic arithmetic operations are available, along with comparison relators. As regards functions, we have variables of another sort, out of which compound terms are formed by means of constructs designating addition and differentiation. An array of predicates designates various relationships between functions, as well as function properties, that may hold over intervals of the real line; those are: function comparisons, strict and non-strict monotonicity / convexity / concavity, comparisons between the derivative of a function and a real-valued term. Our decision method consists in preprocessing the given formula into an equi-satisfiable quantifier-free formula of the elementary algebra of real numbers, whose satisfiability can then be checked by means of Tarski's decision method. No direct reference to functions will appear in the target formula, each function variable having been superseded by a collection of stub real variables; hence, in order to prove that the proposed translation is satisfiability-preserving, we must figure out a flexible-enough family of interpolating C 1 functions that can accommodate a model for the source formula whenever the target formula turns out to be satisfiable. With respect to the results announced in earlier papers of the same stream, a significant effort went into designing the family of interpolating functions so that it could meet the new constraints stemming from the presence of function addition (along with differentiation) among the constructs of our fragment of mathematical analysis. • A formal language RDF* concerned with differentiable real functions is proposed. • The class of differentiable functions treated is closed under addition. • The expressive power of RDF* is illustrated through a gallery of examples. • A satisfiability-preserving algorithm reducing RDF* formulas into an existential sentence of Tarskian algebra is presented. • The correctness of the reduction algorithm is reported. [ABSTRACT FROM AUTHOR]

In 2013 Benkart, Lopes and Ondrus introduced and studied in a series of papers the infinite-dimensional unital associative algebra A h generated by elements x , y , which satisfy the relation y x − x y = h for some 0 ≠ h ∈ F [ x ]. In this paper we investigate the standard polynomial identities and minimal identities for certain subspaces of A h over an infinite field of arbitrary characteristic. [ABSTRACT FROM AUTHOR]

Let F be any field of characteristic different from two and let E be the Grassmann algebra of an infinite dimensional F -vector space L. In this paper we will provide a condition for a Z 2 -grading on E to behave like the natural Z 2 -grading E c a n. More specifically, our aim is to prove the validity of a weak version of a conjecture presented in [10]. The conjecture poses that every Z 2 -grading on E has at least one non-zero homogeneous element of L. As a consequence, we obtain a characterization of E c a n by means of its Z 2 -graded polynomial identities. Furthermore we construct a Z 2 -grading on E that gives a negative answer to the conjecture. [ABSTRACT FROM AUTHOR]

Let n be an integer greater than 1, and M n (C) be the algebra of all n × n -complex matrices. Let x 0 ∈ C n be a nonzero vector, and Φ be a linear map on M n (C) such that Φ (I) is invertible. For any matrix T ∈ M n (C) , let γ (T , x 0) denote the local reduced minimum modulus of T at x 0. In this paper, we show that Φ satisfies γ (T , x 0) = 0 ⇔ γ (Φ (T) , x 0) = 0 , (T ∈ M n (C)) , if and only if there are two invertible matrices A , B ∈ M n (C) such that A x 0 = A ⁎ x 0 = x 0 and Φ (T) = B T A for all T ∈ M n (C). When n = 2 , we show that the invertibility hypothesis of Φ (I) is redundant. [ABSTRACT FROM AUTHOR]

Max-plus algebra is an algebraic structure in which classical addition and multiplication are replaced by maximum and addition, respectively. We say that the columns of a real matrix A are strongly independent if the max-plus linear system A ⊗ x = b has a unique solution for at least one real vector b. A square matrix A with strongly independent columns is called strongly regular. The investigation of the properties of regularity is important for applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. The present paper studies three versions of the regularity of matrices and interval matrices, namely, strong regularity, von Neumann regularity and Gondran-Minoux regularity. For each concept of regularity we will present equivalent conditions which can be verified in polynomial time. [ABSTRACT FROM AUTHOR]

In geometric tolerancing, functional specification is becoming challenging as the complex mutual situations of geometrical features increase and the requisites for complex assembly accuracy escalate. A computational representation of sets of situation features is proposed in this paper to manage functional specification of complex assemblies. A minimum triplet set of a point, a straight line and a plane called ToLiP, is introduced to represent situation features. Functional specification operations can thus be mathematically established using projective geometric algebra (PGA). The effectiveness of the proposed approach is highlighted through a case study using quadrupole magnets of a particle accelerator. [ABSTRACT FROM AUTHOR]

In this paper we study identities and images of polynomials on null-filiform Leibniz algebras. If L n is an n -dimensional null-filiform Leibniz algebra, we exhibit a finite minimal basis for Id (L n) , the polynomial identities of L n , and we explicitly compute the images of multihomogeneous polynomials on L n. We present necessary and sufficient conditions for the image of a multihomogeneous polynomial f to be a subspace of L n. For the particular case of multilinear polynomials, we prove that the image is always a vector space, showing that the analogue of the L'vov-Kaplansky conjecture holds for L n. We also prove similar results for an analog of null-filiform Leibniz algebras in the infinite-dimensional case. [ABSTRACT FROM AUTHOR]

The problem of discretization of Darboux integrable equations is considered. Given a Darboux integrable continuous equation, one can obtain a Darboux integrable differential-discrete equation, using the integrals of the continuous equation. In the present paper, the discretization of the differential-discrete equations is done using the corresponding characteristic algebras. New examples of integrable discrete equations are obtained. [ABSTRACT FROM AUTHOR]

The question we address in this paper is: which monomial ideals have minimal cellular resolutions, that is, minimal resolutions obtained from homogenizing the chain maps of CW-complexes? Velasco gave families of examples of monomial ideals that do not have minimal cellular resolutions, but those examples have large minimal generating sets. In this paper, we show that if a monomial ideal has at most four generators, then the ideal and its (monomial) Artinian reductions have minimal cellular resolutions. When the ideal is generated by two monomials, we can give a precise description of the CW-complex supporting minimal free resolution of the ideal and its Artinian reduction. Also, in this case, we compute the multigraded Betti numbers, Cohen-Macaulay type and determine when the corresponding algebra is a level algebra. [ABSTRACT FROM AUTHOR]

Let G be a group and F an infinite field. Assume that A is a finite dimensional F -algebra with an elementary G -grading. In this paper, we study the graded identities satisfied by the tensor product grading on the F -algebra A ⊗ C , where C is an H -graded colour β -commutative algebra. More precisely, under a technical condition, we provide a basis for the T G -ideal of graded polynomial identities of A ⊗ C , up to graded monomial identities. Furthermore, the F -algebra of upper block-triangular matrices U T (d 1 , ... , d n) , as well as the matrix algebra M n (F) , with an elementary grading such that the neutral component corresponds to its diagonal, are studied. As a consequence of our results, a basis for the graded identities, up to graded monomial identities of degrees ≤ 2 d − 1 , for M d (E) and M q (F) ⊗ U T (d 1 , ... , d n) , with a tensor product grading, is exhibited. In this latter case, d = d 1 + ... + d n. Here E denotes the infinite dimensional Grassmann algebra with its natural Z 2 -grading, and the grading on M q (F) is Pauli grading. The results presented in this paper generalize results from [14] and from other papers which were obtained for fields of characteristic zero. [ABSTRACT FROM AUTHOR]

A universal algebra A with underlying set A is said to be a matroid algebra if (A , 〈 ⋅ 〉) , where 〈 ⋅ 〉 denotes the operator subalgebra generated by , is a matroid. A matroid algebra is said to be an independence algebra if every mapping α : X → A defined on a minimal generating X of A can be extended to an endomorphism of A. These algebras are particularly well-behaved generalizations of vector spaces, and hence they naturally appear in several branches of mathematics, such as model theory, group theory, and semigroup theory. It is well known that matroid algebras have a well-defined notion of dimension. Let A be any independence algebra of finite dimension n , with at least two elements. Denote by End (A) the monoid of endomorphisms of A. In the 1970s, Głazek proposed the problem of extending the matrix theory for vector spaces to a class of universal algebras which included independence algebras. In this paper, we answer that problem by developing a theory of matrices for (almost all) finite-dimensional independence algebras. In the process of solving this, we explain the relation between the classification of independence algebras obtained by Urbanik in the 1960s, and the classification of finite independence algebras up to endomorphism-equivalence obtained by Cameron and Szabó in 2000. (This answers another question by experts on independence algebras.) We also extend the classification of Cameron and Szabó to all independence algebras. The paper closes with a number of questions for experts on matrix theory, groups, semigroups, universal algebra, set theory or model theory. [ABSTRACT FROM AUTHOR]

The n -th Schrödinger algebra sch n defined in [14] is the semi-direct product of the Lie algebra sl 2 with the n -th Heisenberg Lie algebra h n , which generalizes the Schrödinger algebra sl 2 ⋉ h 1. Let ϕ : h n → C be a nonzero Lie algebra homomorphism. A sch n -module V is called quasi-Whittaker of type ϕ if V = U (sch n) v , where U (sch n) is the universal enveloping algebra of sch n , v is a nonzero vector such that x v = ϕ (x) v for any x ∈ h n. In this paper, we prove that a simple sch n -module V is a quasi-Whittaker module if and only if V is a locally finite h n -module. Then we classify the simple quasi-Whittaker modules of ϕ , according to the rank of ϕ. Furthermore, we characterize arbitrary quasi-Whittaker modules through the rank of ϕ. [ABSTRACT FROM AUTHOR]

Let A be a finite dimensional associative algebra with derivations over a field of characteristic zero, i.e., an algebra whose structure is enriched by the action of a Lie algebra L by derivations, and let c n L (A) , n ≥ 1 , be its differential codimension sequence. Such sequence is exponentially bounded and exp L ⁡ (A) = lim n → ∞ ⁡ c n L (A) n is an integer that can be computed, called differential PI-exponent of A. In this paper we prove that for any Lie algebra L , exp L ⁡ (A) coincides with exp ⁡ (A) , the ordinary PI-exponent of A. Furthermore, in case L is a solvable Lie algebra, we apply such result to classify varieties of L -algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth. [ABSTRACT FROM AUTHOR]

Let A be a unital C⁎-algebra equipped with its natural order. As usual the effect algebra of A is the interval { a ∈ A : 0 ≤ a ≤ I A } , where I A denotes the unit of A. In this paper, we give a complete description of order isomorphisms between effect algebras of C⁎-algebras of type C (X) ⊗ B (H) , where C (X) stands for the algebra of all continuous complex valued functions on a (non pathological) Hausdorff compact space X and B (H) denotes the algebra of all bounded linear operators on a complex Hilbert space H. Our results generalize some works by L. Molnár and P. Šemrl. [ABSTRACT FROM AUTHOR]

In this paper we construct a graded universal enveloping algebra of a G -graded Lie algebra, where G is not necessarily an abelian group. If the grading group is abelian, then it coincides with the classical construction. We prove the existence and uniqueness of the graded enveloping algebra. As consequences, we prove a graded variant of Witt's Theorem on the universal enveloping algebra of the free Lie algebra, and the graded version of Ado's Theorem, which states that every finite-dimensional Lie algebra admits a faithful finite dimensional representation. Furthermore we investigate if a Lie grading is equivalent to an abelian grading. [ABSTRACT FROM AUTHOR]

Let M be a type II 1 von Neumann algebra, S (M) be the Murray–von Neumann algebra associated with M and let A be a ⁎-subalgebra in S (M) with M ⊆ A. We prove that any ring derivation D from A into S (M) is necessarily inner. Further, we prove that if A is an E W ⁎ -algebra such that its bounded part A b is a W ⁎ -algebra without finite type I direct summands, then any ring derivation D from A into L S (A b) — the algebra of all locally measurable operators affiliated with A b , is an inner derivation. We also give an example showing that the condition M ⊆ A is essential. At the end of this paper, we provide several criteria for an abelian extended W ⁎ -algebra such that all ring derivations on it are linear. [ABSTRACT FROM AUTHOR]

This paper introduces a logical characterization of multi-adjoint algebras with a twofold contribution. On the one hand, the study of multi-adjoint algebras, from a logical perspective, will allow us to discover both the core and new features of these algebras. On the other hand, the axiomatization of multi-adjoint algebras will be useful to take advantage of the properties of the logical connectives considered in the corresponding deductive system. The mechanism considered to carry out the mentioned axiomatization follows the one given by Petr Hájek for residuated lattices. Specifically, the paper presents the bounded poset logic (BPL) as an axiomatization of a bounded poset, since this algebraic structure is the most simple structure from which a multi-adjoint algebra is defined. In the following, the language of BPL is enriched with a family of pairs, composed of a conjunctor and an implication, and its axiomatic system is endowed with new axioms, giving rise to the multi-adjoint logic (ML). The soundness and completeness of BPL and ML are proven. Finally, a comparison between the axiomatization of the multi-adjoint logic and the one given for the BL logic is introduced. [ABSTRACT FROM AUTHOR]

Let B (X) be the algebra of all bounded linear operators on a complex or real Banach space X with dim ⁡ X ≥ 3. In this paper, we characterize the maps from B (X) into itself which preserve the ascent of product of operators or, they preserve the descent of product of operators. It turns out that both problems are connected with preservers of the rank-one nilpotency of the product. [ABSTRACT FROM AUTHOR]

The present paper introduces a new definition of entropy on sequential effect algebras. Unlike the logarithmic behaviour of the entropy defined in the literature, the entropy considered here is of exponential nature. Conditional entropy and entropy on the dynamical system are also introduced and studied. It is also proved that entropy of the dynamical system is invariant under isomorphism. [ABSTRACT FROM AUTHOR]

In this paper we consider the generalized eigenproblem in max-min (fuzzy) algebra, i.e. given matrices A , B find a vector x and a constant λ such that A x = λ B x where the standard pair of operations, plus and times, have been replaced by the operations maximum and minimum. The entries of the vector or matrix are, in practice, usually not exact numbers and can rather be considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval) entries are studied and complete solutions of the strong, the universal, the L-controllable and the R-controllable generalized eigenproblems in max-min (fuzzy) algebra are presented. As a consequence of the obtained results, efficient algorithms for checking all equivalent conditions are introduced. [ABSTRACT FROM AUTHOR]

For a frame Q , this paper studies the Kleisli monoids and Eilenberg-Moore algebras with respect to the Q -Scott open set monad. Firstly, we show that the Q -Scott open set functor can form a monad, and prove that the Kleisli monoids of this monad are exactly the algebraic Q -ordered closure spaces or equivalently the strong Q -convex spaces. Then we show that the Eilenberg-Moore algebras of Q -Scott open set monad are precisely the algebraic Q -modules or equivalently the fuzzy frames. Finally, we construct the strong Q -Scott open set monad, and study the corresponding Kleisli monoids and the Eilenberg-Moore algebras. [ABSTRACT FROM AUTHOR]

Let A be a unital C ⁎ algebra and for every a ∈ A , r (a) denote the numerical radius of a ∈ A. The power inequality for numerical radius states that for every polynomial P (z) = z n and a ∈ A the inequality P (r (a)) ≥ r (P (a)) holds. In this paper, we get a characterization of polynomials with real coefficients that satisfy the power inequality on all 2 × 2 matrices with real entries. We also characterize all polynomials that satisfy the power inequality on every commutative unital C ⁎ algebra. [ABSTRACT FROM AUTHOR]

An algebra 〈 A , ∧ , ∨ , → , □ , 0 , 1 〉 of type (2 , 2 , 2 , 1 , 0 , 0) is said to be a modal weak Heyting algebra if 〈 A , ∧ , ∨ , → , 0 , 1 〉 is a weak Heyting algebra and the following conditions are satisfied for every a , b ∈ A : M1) □ (1) = 1 , M2) □ (a ∧ b) = □ (a) ∧ □ (b) and M3) □ (a → b) ≤ □ (a) → □ (b). If this algebra satisfies the inequality a ∧ (a → b) ≤ b then it is called modal RWH-algebra. In this paper we study the variety of modal RWH-algebras, which is denoted by KRWH, and some of its subvarieties. We focus our attention on the study of the lattice of congruences of any member of KRWH and some related properties. In particular, we give an equational basis for the subvariety of KRWH generated by the class of their totally ordered members. [ABSTRACT FROM AUTHOR]

In this paper we study the lattice of restricted subalgebras of a restricted Lie algebra. In particular, we consider those algebras in which this lattice is dually atomistic, lower or upper semimodular, or in which every restricted subalgebra is a quasi-ideal. The fact that there are one-dimensional subalgebras which are not restricted results in some of these conditions being weaker than for the corresponding conditions in the non-restricted case. [ABSTRACT FROM AUTHOR]

In this paper, we consider a generalization of the classical Skjelbred–Sund method, used to classify nilpotent low-dimensional Lie algebras, in order to classify Lie-Yamaguti algebras with non-trivial annihilator. We develop this method with the purpose of classifying nilpotent Lie-Yamaguti algebras, and we obtain from it the algebraic classification of the nilpotent Lie-Yamaguti algebras up to dimension four. [ABSTRACT FROM AUTHOR]

This paper proposes a novel approach to the solution of optimal control problems under uncertainty (OCPUUs). OCPUUs are first cast in a general formulation that allows the treatment of uncertainties of different nature, and then solved with a new direct transcription method that combines multiple shooting with generalised polynomial algebra to model and propagate extended sets. The continuity conditions on extended sets at the boundary of two adjacent segments are directly satisfied by a bounding approach. The Intrusive Polynomial Algebra aNd Multiple shooting Approach (IPANeMA) developed in this work can handle optimal control problems under a wide range of uncertainty models, including nonparametric, epistemic, and imprecise probability ones. In this paper, the approach is applied to the design of a robust low-thrust trajectory to a Near-Earth Object with uncertain initial conditions. It is shown that the new method provides more robust and reliable trajectories than the solution of an analogous deterministic optimal control problem. • A general formulation of optimal control problems under uncertainty is proposed. • A generalised multiple shooting transcription approach is developed. • The novel transcription relies on polynomial algebra for uncertainty propagation. • The approach is applied to the robust optimisation of a low-thrust trajectory. • The robust control results greatly more reliable than the deterministic one. [ABSTRACT FROM AUTHOR]

• Geometric algebra framework is applied to solve non sinusoidal and nonlinear circuits. • Power flow direction and magnitude for interharmonics is solved. • The geometric algebra framework applied to power systems is improved with the inclusion of interharmonic representation. • This new method is validated through some examples with linear and non-linear loads. The calculation of power flow in power systems with the presence of harmonics has been properly studied in the scientific literature. However, power flow calculation considering interharmonic components is still an open question. Traditional methods based on the IEEE1459 standard have proven to be valid and accurate only for linear and sinusoidal systems, but have been criticized for non-linear and non-sinusoidal systems because they are not able to explain correctly the current and voltage interactions beyond the active power. This paper proposes the use of a novel mathematical framework called geometric algebra (GA) to study the power flow considering the interaction of current and voltage harmonics and interharmonics. The use of GA enables the precise determination of the direction and magnitude of the total and single active power flow for each component, as well as other power elements related to the non-active power due to cross interaction. Moreover, this paper makes a novel contribution to the definition of interharmonics in geometric algebra space that has not been done before. To test the validity of the method, both linear and non-linear circuits are proposed and solved by applying voltages and currents with harmonic and interharmonic components. The results obtained show that power flow can be analyzed under the prism of the principle of energy conservation (PoCoE) in a way that allows a better understanding of the power spectrum due to the interaction of harmonics and interharmonics of voltage and current. [ABSTRACT FROM AUTHOR]

Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and Gödel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed under the corresponding rules for finite types over N. It is also shown that the existence property (or existential definability property) holds for statements of the form ∃ y σ φ (y) , where the variable y ranges over objects of finite type σ. This applies in particular to CZF (Constructive Zermelo-Fraenkel set theory) and IZF (Intuitionistic Zermelo-Fraenkel set theory), two systems known not to have the general existence property. On the technical side, the paper uses a method that amalgamates generic realizability for set theory with truth, whereby the underlying partial combinatory algebra is required to contain all objects of finite type. [ABSTRACT FROM AUTHOR]

In this paper we propose a semantic analysis of Lewis' counterfactuals. By exploiting the structural properties of the recently introduced boolean algebras of conditionals, we show that counterfactuals can be expressed as formal combinations of a conditional object and a normal necessity modal operator. Specifically, we introduce a class of algebras that serve as modal expansions of boolean algebras of conditionals, together with their dual relational structures. Moreover, we show that Lewis' semantics based on sphere models can be reconstructed in this framework. As a consequence, we establish the soundness and completeness of a slightly stronger variant of Lewis' logic for counterfactuals with respect to our algebraic models. In the second part of the paper, we present a novel approach to the probability of counterfactuals showing that it aligns with the uncertainty degree assigned by a belief function, as per Dempster-Shafer theory, to its associated conditional formula. Furthermore, we characterize the probability of a counterfactual in terms of Gärdenfors' imaging rule for the probabilistic update. [ABSTRACT FROM AUTHOR]

Given a unital algebra A of linear transformations on a finite-dimensional complex vector space V , in this paper we study the set Col (A) consisting of those invertible linear transformations S on V which map every subspace M ∈ Lat (A) to a subspace S M ∈ Lat (A). We show that Col (A) is the normalizer of the group of invertible linear transformations in the reflexive cover of A. For the unital algebra (A) which is generated by a linear transformation A , we give the complete description of Col (A). By using the primary decomposition of A , we first reduce the problem of characterizing Col (A) to the problem of characterizing the group Col (N) of a given nilpotent linear transformation N. While Col (N) always contains all invertible linear transformations of the commutant (N) ′ of N , it is always contained in the reflexive cover of (N) ′. We prove that Col (N) is a proper subgroup of (Alg Lat (N) ′) − 1 if and only if at least two Jordan blocks in the Jordan decomposition of N are of dimension 2 or more. We also determine the group Col (J 2 ⊕ J 2). [ABSTRACT FROM AUTHOR]

Although there have been many fuzzy or probabilistic relational database models proposed for representing and handling imprecise and uncertain information of objects in real-world applications, models combining the relevance and strength of both fuzzy set theory and probability theory appear sporadic. In this paper, we propose a new fuzzy and probabilistic relational database model where the imprecision of an attribute value is represented by a fuzzy set and the uncertainty of a relational tuple is represented by a probability interval. The mass assignment theory is employed to deal with the challenge of integration and computation of both fuzzy sets and probabilities in the same model. The conjunction and disjunction strategies to combine imprecise and uncertain information are introduced. Then the fundamental concepts of the classical relational database model are extended and generalized in this new model. The syntax and semantics of the selection operation are formally defined. Finally, the other important algebraic operations on imprecise attributes and uncertain tuples are developed. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study the algebra of almost strongly fuzzy bounded operators on fuzzy normed linear spaces. Some connections with the algebra of strongly fuzzy bounded operators are presented. The completeness of these algebras is also established. A characterization of bounded elements of the algebra of almost strongly fuzzy bounded operators is given. We also introduce two fuzzy spectral radii for almost strongly fuzzy bounded operators. For bounded elements of the considered algebra the equality between these fuzzy spectral radii is proved. [ABSTRACT FROM AUTHOR]

In this paper, the notions of regularity and normality of (L , M) -fuzzy topological spaces are introduced by using residual implication, where L and M are completely distributive De Morgan algebras. It is shown that (L , M) -fuzzy interior operator and (L , M) -fuzzy closure operator can be used to characterize regularity and normality. The relationships among separation axioms of an (L , M) -fuzzy topological space are discussed. Moreover, it is proved that the four separation axioms are equivalent to one another in an (L , M) -fuzzy metric space. [ABSTRACT FROM AUTHOR]

In the present paper we investigate reflexive modules over the endomorphism algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local ring. The main theorem generalizes both of the results of S. Goto, N. Matsuoka, and T. T. Phuong ([20, Theorem 5.1]) and T. Kobayashi ([30, Theorem 1.3]) concerning the endomorphism algebra of its maximal ideal. We also explore the question of when the category of reflexive modules is of finite type, i.e., the base ring has only finitely many isomorphism classes of indecomposable reflexive modules. We show that, if the category is of finite type, the ring is analytically unramified and has only finitely many Ulrich ideals. As a consequence, Arf local rings contain only finitely many Ulrich ideals once the normalization is a local ring. [ABSTRACT FROM AUTHOR]