Search

In this paper, we study operator dependent modifiers and we interpret the dual pair of modal operators based on an algebraic definition. It is a known fact that the substantiating and weakening modifier operators can be induced by repeating the arguments of conjunctive and disjunctive operators. We provide the conditions for which these modifier operators satisfy the requirements for a dual pair of necessity and possibility operators. Next, the necessary and sufficient condition for the distributivity of unary operators over conjunctive and disjunctive operators is presented. This also means that setting the distributivity as a requirement results in a unary operator that is identical to the modal operators mentioned above. Using this property, we establish an important connection between modal operators and linguistic hedges. Previously, we demonstrated that the unary operators induced by compositions of two strong negations satisfy the requirements for a dual pair of modal operators. Here, we view the negation operator as a modifier operator. Then, it is shown that (1) the strong negations, (2) the substantiating and weakening modifier operators, modal operators and linguistic hedges mentioned above, and (3) the unary operators, which are distributive over conjunctive and disjunctive operators, may be viewed as special cases of a unified unary operator class.

This paper proposes a new algebraic definition that facilities calculating of Stackelberg stability in a graph model for conflict resolution with two decision makers. Most stability definitions used in the graph model methodology place decision makers at the same level, in the sense that their roles are symmetric. In practice, however, one decision maker may join by forcing the other to respond to his or her decision. So, to be applied, a model must specify the leader and the follower. Stackelberg stability can be defined logically, but an algorithm to implement it has not been developed until now, due to its complicated recursive formula. To permit Stackelberg stability to be calculated efficiently and encoded conveniently, within a decision support system, an algebraic test for the stability is developed. This algebraic representation of Stackelberg stability is easy to implement and interpret. A superpower military confrontation is used to illustrate how Stackelberg stability can be employed to a real-world application using the new approach.

Using the definitions of the notion of point and those of the commutative algebras under consideration, the formal algebraic definition of a smooth manifold is given in this chapter. It is defined as the dual space \(|\mathcal F|\) of any complete geometric algebra \(\mathcal F\), supplied with an open covering \(\{U_k\}\) in the Zariski topology such that each algebras \(\mathcal F|_{U_k}\) is isomorphic to \(C^{\infty }(U_k)\). Numerous concrete examples of how this works are presented.

What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only partial answers to these questions. To make these problems more precise and easier to tackle, we describe an abstract algebraic definition of classical computation by generalizing traditional models to semigroups. This way implementations are morphic relations between semigroups. The mathematical abstraction also allows the investigation of different computing paradigms (e.g. cellular automata, reversible computing) in the same framework. While semigroup theory helps in clarifying foundational issues about computation, at the same time it has several open problems that require extensive computational efforts. This mutually beneficial relationship is the central tenet of the described research.

Let G be a discrete group. It is well known that the (co)homology groups of G have both topological and algebraic definitions. Now, consider a subgroup H of G. In the literature, there are two versions of relative (co)homology groups for the pair (G, H), one generalises in a natural way the topological definition, while the other one generalises in a natural way the algebraic definition. In this article, we give a topological definition for the latter one, we give simple examples that show that these theories do not coincide in general, and we give a sufficient condition on the subgroup H in order that both relative group (co)homologies of the pair (G, H) coincide.

Conventional algebraic estimate of the fiscal multipliers ignores the concept of velocity of money and mistakenly assumes that money changes hands an infinite number of times during a given year while we know money only has a finite velocity. Apart from the velocity of money, fiscal multipliers tend to depend on average propensity to consume and average propensity to import of the economy as a whole and also on average tax rate among other things which are not reflected in the modern SVAR based estimation. Here, in the first place, we amend the algebraic definition of the fiscal multipliers considering the impact of velocity of money, provide a micro-foundation relating fiscal multipliers with money velocity and other macro variables and later propose a modification in the conventional SVAR set up by incorporating aforesaid macro variables arranged in a logical manner. Proposed amendments to the SVAR set up entail relatively stable estimates of the fiscal multipliers as can be seen from empirical estimation of the multiplier values for US and UK data during the period 1972-2018., Fiscal multiplier; Velocity of money; Average propensity to consume; Average propensity to import; Average tax rate; Public finance; Public economics; Economic development; Macroeconomics; Econometrics

Interrupted time series (ITS) analysis is a valuable study design for evaluating the effectiveness of population-level health interventions that have been implemented at a clearly defined point in time. It is increasingly being used to evaluate the effectiveness of interventions ranging from clinical therapy to national public health legislation. Whereas the design shares many properties of regression-based approaches in other epidemiological studies, there are a range of unique features of time series data that require additional methodological considerations. In this tutorial we use a worked example to demonstrate a robust approach to ITS analysis using segmented regression. We begin by describing the design and considering when ITS is an appropriate design choice. We then discuss the essential, yet often omitted, step of proposing the impact model a priori. Subsequently, we demonstrate the approach to statistical analysis including the main segmented regression model. Finally we describe the main methodological issues associated with ITS analysis: over-dispersion of time series data, autocorrelation, adjusting for seasonal trends and controlling for time-varying confounders, and we also outline some of the more complex design adaptations that can be used to strengthen the basic ITS design.

The cross product u→×v→ of vectors u→=[a1a2a3], v→=[b1b2b3] in the Euclidean 3-space is defined as follows (see [1, p. 138] or [3, p. 266]):Algebraic definition. u→×v→=[a2b3−a3b2−(a1b3−a3b1)a1b2−a2...

In their daily practices, meteorologists make extensive use of the geostrophic wind properties to explain many weather phenomena such as the meaning and direction of the horizontal winds that take place around the low atmospheric pressures. The biggest challenge that faces the public who is interested in information disseminated by meteorologists is to know exactly what means the geostrophic wind. Besides the literal definitions scattered in very little scientific work, there is unfortunately no book which gives importance to the algebraic definition of the geostrophic wind. Our work shows that to better understand the behavior of natural phenomena, it is essential to combine the theories with based observations. Obviously, observations cannot be relevant without a theory that guides the observers. Conversely, no theory can be validated without experimental verification. Synoptic observations show that in the “free atmosphere!” the wind vectors are very nearly parallel to isobars, and the flow is perpendicular to the horizontal pressure gradient force, at least at any given instant. This kind of information recommends great caution when making geostrophic approximations. Our work also shows that for tornadoes, there is no need to move away from the surface of the oceans to observe the geostrophic balance. Undoubtedly, identification and interpretation of earth’s atmosphere dynamics’ and thermodynamics’ similarities between rogue waves and oceans’ surface geostrophic wind will be an easy exercise to researchers who will give importance to result provided by this paper.

The investigation of the umbral calculus based generalization of Bernstein polynomials and Bezier curves is continued in this paper: First a generalization of the de Casteljau algorithm that uses umbral shift operators is described. Then it is shown that the quite involved umbral shifts can be replaced by a surprisingly simple recursion which in turn can be understood in geometrical terms as an extension of the de Casteljau interpolation scheme. Namely, instead of using only the control points of level r − 1 to generate the points on level r as in the ordinary de Casteljau algorithm, one uses also points on level r − 2 or more previous levels. Thus the unintuitive parameters in the algebraic definition of generalized Bernstein polynomials get geometric meaning. On this basis a new direct method for the design of Bezier curves is described that allows to adapt the control polygon as a whole by moving a point of the associated Bezier curve.

This work presents an element-local algebraic approach to constructing coarse spaces for p -multigrid solvers and preconditioners of high-order discontinuous Galerkin discretizations. The target class of problems is convective systems on unstructured meshes, a class for which traditional p -multigrid typically fails to reach textbook multigrid efficiency due to a mismatch between smoothers and coarse spaces. Smoothers that attempt to alleviate this mismatch, such as line-implicit, incomplete LU, or Gauss–Seidel, deteriorate on grids that are not aligned with the flow, and they rely on sequential operations that do not scale well to distributed-memory architectures. In this work we shift attention from the smoothers to the coarse spaces, and we present an algebraic definition of the coarse spaces within each element based on a singular-value decomposition of the neighbor influence matrix. On each multigrid level, we employ a block-Jacobi smoother, which maintains algorithmic scalability as all elements can be updated in parallel. We demonstrate the performance of our solver on discretizations of advection and the linearized compressible Euler equations.

An algebraic definition of laws is formulated, motivated by analyzing points in Euclidean geometry and from considerations of two physical examples, Ohm’s law and Newton’s second law. Simple algebraic examples constructed over a field are presented.

Still nowadays, no one algebraic definition of heat is taught. Although anybody can easily make is own idea about heat, this non-regular lack, from the point of view of theoretical physics, can be explained by an historical consideration on the creation by Clausius of internal energy. In 1850, Clausius publishes On the Moving Force of Heat in which internal energy and entropy are created in order to make thermodynamics an entirely mathematical science. At the beginning, Clausius deny to Carnot that heat could be a conserved quantity, then, considering an ideal gas in a thermodynamical cycle, he shows that the conserved quantity cannot be heat, but another one, made by Clausius and called later "internal energy" by himself too. The main point of his demonstration is that the elementary quantity δQ cannot be an exact differential. Of course, a so important conclusion has several great consequences about thermodynamics in general. Firstly, internal energy becomes the new main quantity in thermodynamics, secondly, heat can be defined as the internal energy variation not due to mechanical work. Moreover, what is called "first principle of thermodynamics" can be seen as a simple consequence of this heat definition.

Module structures of an algebra on a fixed finite dimensional vector space form an algebraic variety. Isomorphism classes correspond to orbits of the action of an algebraic group on this variety and a module is a degeneration of another if it belongs to the Zariski closure of the orbit. Riedtmann and Zwara gave an algebraic characterisation of this concept in terms of the existence of short exact sequences. Jensen, Su and Zimmermann, as well as independently Yoshino, studied the natural generalisation of the Riedtmann-Zwara degeneration to triangulated categories. The definition has an intrinsic non-symmetry. Suppose that we have a triangulated category in which idempotents split and either for which the endomorphism rings of all objects are artinian, or which is the category of compact objects in an algebraic compactly generated triangulated K-category. Then we show that the non-symmetry in the algebraic definition of the degeneration is inessential in the sense that the two possible choices which can be made in the definition lead to the same concept.

We give an algebraic definition of a Markowitz market and classify markets up to isomorphism. Given this classification, the theory of portfolio optimization in Markowitz markets without short selling constraints becomes trivial. Conversely, this classification shows that, up to isomorphism, there is little that can be said about a Markowitz market that is not already detected by the theory of portfolio optimization. In particular, if one seeks to develop a simplified low-dimensional model of a large financial market using mean--variance analysis alone, the resulting model can be at most two-dimensional., 1 figure

I argue that a version of the quantum-corrected Ryu-Takayanagi formula holds in any quantum error-correcting code. I present this result as a series of theorems of increasing generality, with the final statement expressed in the language of operator-algebra quantum error correction. In AdS/CFT this gives a "purely boundary" interpretation of the formula. I also extend a recent theorem, which established entanglement-wedge reconstruction in AdS/CFT, when interpreted as a subsystem code, to the more general, and I argue more physical, case of subalgebra codes. For completeness, I include a self-contained presentation of the theory of von Neumann algebras on finite-dimensional Hilbert spaces, as well as the algebraic definition of entropy. The results confirm a close relationship between bulk gauge transformations, edge-modes/soft-hair on black holes, and the Ryu-Takayanagi formula. They also suggest a new perspective on the homology constraint, which basically is to get rid of it in a way that preserves the validity of the formula, but which removes any tension with the linearity of quantum mechanics. Moreover they suggest a boundary interpretation of the "bit threads" recently introduced by Freedman and Headrick., 40 pages plus appendix, 11 figures, many subscripts on subscripts. v2: Minor corrections and improvements, section 6.3 revised more substantially for clarity, section 6.4 added to discuss some limitations

We present the linear algebraic definition of QSAT and propose a direct logical characterization of such a definition. We then prove that this logical version of QSAT is not an extension of classical satisfiability problem (SAT). This shows that QSAT does not allow a direct comparison between the complexity classes NP and QMA, for which SAT and QSAT are respectively complete., Comment: In Proceedings LSFA 2011, arXiv:1203.5423

A rigorous algebraic definition of noncommutative coverings is developed. In the case of commutative algebras this definition is equivalent to the classical definition of topological coverings of locally compact spaces. The theory has following nontrivial applications: • Coverings of continuous trace algebras, • Coverings of noncommutative tori, • Coverings of the quantum SU(2) group, • Coverings of foliations, • Coverings of isospectral deformations of Spin – manifolds. The theory supplies the rigorous definition of noncommutative Wilson lines.

The dB is an important and widely used concept in relation to antennas, propagation, and other areas. However, its usual definitions contain idiosyncrasies that hamper clear understanding and symbolic calculation, thereby reducing general applicability to far below its potential. A novel perspective replaces the common view - namely, the dB as a unit, the logarithm, and ratios - by the dB as a function, the exponential, and numbers. This yields a pithy algebraic definition placing the dB among the elementary functions. Diverse examples show the advantages for calculation and widened applicability. Combination with units and scales as functions allows assigning meaning to ?atomic? affixes like dBm by function composition such that, for instance, 30 dBm = 30 dBmW = (((30 d) B) m) W = 1 W. Unlike the usual definitions, this also enables dimensional analysis, which is crucial in physics and engineering. The approach is presented in the wider context of the principle that tools used by engineers should themselves be engineered for effectiveness, and mathematical tools are no exception. This is important for practice as well as for education.

International audience; We study the algebraic structure of a programming language with higher-order store, in the style of ML references. Instead of working directly on the operational semantics of the language, we consider its fully abstract game semantics defined by Abramsky, Honda and McCusker one decade ago. This alternative description of the language is nice and conceptual, except on one significant point: the interactive behavior of the higher-order memory cell is reflected in the model by a strategy $\mathtt{cell}$ whose definition remains slightly enigmatic. The purpose of our work is precisely to clarify this point, by providing a neat algebraic definition of the strategy. This conceptual reconstruction of the memory cell is based on the idea that a general reference behaves essentially as a linear feedback (or trace operator) in an ambient category of Conway games and strategies. This analysis leads to a purely axiomatic proof of soundness of the model, based on a natural refinement of the replication modality of tensor logic.

In his 1965 paper C. T. C. Wall demonstrated that if a CW complex Y is finitely dominated, then the reduced projective class group of Y contains an obstruction which vanishes if and only if Y is homotopy equivalent to a finite CW complex. Wall also demonstrated that such an obstruction is invariant under homotopy equivalences. Subsequently Sum and Product Theorems for this obstruction were proved by L. C. Siebenmann. ¶ In his second paper on the subject Wall gives an algebraic definition of the relative finiteness obstruction. If a CW complex Y is finitely dominated rel. a subcomplex X, then the reduced projective class group of Y contains an obstruction which vanishes if and only if Y is homotopy equivalent to a finite complex rel. X. ¶ In this paper we will use a geometric construction to reduce the relative finiteness obstruction to the non-relative version. We will demonstrate that the relative finiteness obstruction is invariant under certain types of homotopy equivalences. We will also prove the relative versions of the Sum and the Product Theorems.

What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only partial answers to these questions. In order to make these problems more precise, we describe an abstract algebraic definition of classical computation, generalizing traditional models to semigroups. The mathematical abstraction also allows the investigation of different computing paradigms (e.g. cellular automata, reversible computing) in the same framework. Here we summarize the main questions and recent results of the research of finite computation., Comment: 12 pages, 3 figures, will be presented at CANDAR'16 and final version published by IEEE Computer Society

Topology is the branch of mathematics that studies shapes and maps among them. From the algebraic definition of topology a new set of algorithms have been derived. These algorithms are identified with “computational topology” or often pointed out as Topological Data Analysis (TDA) and are used for investigating high-dimensional data in a quantitative manner. Persistent homology appears as a fundamental tool in Topological Data Analysis. It studies the evolution of k−dimensional holes along a sequence of simplicial complexes (i.e. a filtration). The set of intervals representing birth and death times of k−dimensional holes along such sequence is called the persistence barcode. k−dimensional holes with short lifetimes are informally considered to be topological noise, and those with a long lifetime are considered to be topological feature associated to the given data (i.e. the filtration). In this paper, we derive a simple method for separating topological noise from topological features using a novel measure for comparing persistence barcodes called persistent entropy. Ministerio de Economía y Competitividad MTM2015-67072-P

International audience; This paper presents a formal framework allowing to extend the simplification of static fault trees to fault trees built with gates PRIORITY AND. The laws which make these simplifications possible have been demonstrated thanks to a homogeneous algebraic definition of each gate studied. These definitions use a mathematical model of events able to take into account their order of appearance. The processing of an example points out the possibilities offered by this algebraic framework dedicated to non-repairable faults.

WOS: 000354282000007, The aim of this study was to observe the development process of the concept of a parabola in Taxicab geometry. The study was carried out in two stages. First, some activities related to Euclidean geometry and Taxicab geometry were designed based on concept development and real-life applications, and they were administered to a ninth-grade student. According to the findings, once the student learnt the definition of a parabola in Euclidean geometry, she was able to define a Taxicab parabola using the distance function in Taxicab geometry. Also, she came up with an algebraic definition of a Taxicab parabola based on this geometric definition of the concept of a parabola. Moving from algebraic definition to geometric representation, she configured the concept of a parabola in Taxicab geometry. By means of this application activity, the student had the opportunity to observe and practise the concept of a parabola in a real-life situation based on Euclidean geometry and Taxicab geometry.

There are many different ways of defining a sequence in terms of solutions to difference equations. In fact, if a sequence satisfies one recurrence then it satisfies an infinite number of recurrences. Arithmetic properties of an integral sequence are often studied by direct methods based on the combinatorial or algebraic definition of the numbers or using their generating function. The rational generating function is the main tool in obtaining various difference equations with coefficients and initial values exhibiting divisibility patterns that can imply particular arithmetic properties of the solutions. In this process, we face the challenging task of finding difference equations that are relevant to the divisibility properties by transforming the original rational generating function. As a matter of fact, it is not necessarily the simple difference equation that helps the most in proving the properties. We illustrate this process on several examples and a sequence involving a p -sected binomial sum of ...

The aim of this paper is to develop a suitable calculus of L -fuzzy relations. We show that for this purpose the theory of Dedekind categories is too weak. Therefore, we introduce the notion of a Goguen category as a suitable algebraic definition and show several properties of this kind of a relational category.

This paper is an analysis of equilibrium evaporation and its role in the energy balance of a terrestrial surface, as described by combination theory. Three themes are covered: first, a brief historical review identifies multiple definitions of the concept of equilibrium evaporation. Second, these are formalized by developing the basic principles of combination theory with minimum approximation. Several measures are utilized to do this: linearization is avoided, radiative and storage coupling are incorporated systematically, and actual and linearized saturation deficits are distinguished. The formalism is used to analyse several algebraically defined states and limits for the surface energy balance. Third, the thermodynamic foundation of equilibrium evaporation is analysed by studying surface-atmosphere feedbacks in arbitrary closed and open evaporating systems. It is shown that under steady energy supply any closed evaporating system evolves towards a quasi-steady state in which the Bowen ratio takes the equilibrium value 1/ev, where ev is the ratio of the latent- and sensible-heat contents of saturated air with temperature, evaluated at the volume-averaged temperature in the closed system. This applies whether the system is well-mixed or imperfectly mixed, and whatever the internal distribution of surface fluxes and surface and aerodynamic resistances. In contrast, open systems cannot reach such an equilibrium. This evolutionary definition of equilibrium evaporation differs from an alternative algebraic definition, the fully decoupled limit. The differences between the two definitions are identified, and the evolutionary definition is shown to be more fundamental. Thus, the correct temperature for evaluating e in determining equilibrium evaporation is the volume-averaged temperature in a closed region, which in the case of a convective boundary layer is well approximated by the mixed-layer temperature.

We give a more algebraic definition of algebraic supergroup via its Hopf algebra. The Hopf algebra of the supergroup SL (m|n) is given together with its quantization. The example of supercoadjoint orbits is examined at the end.

Classically, connectivity is a topological notion for sets, often introduced by means of arcs. An algebraic definition, called connection, has been proposed by Serra to extend the notion of connectivity to complete sup-generated lattices. A connection turns out to be characterized by a family of openings parameterized by the sup-generators, which partition each element of the lattice into maximal components. Starting from a first connection, several others may be constructed; e.g., by applying dilations. The present paper applies this theory to numerical functions. Every connection leads to segmenting the support of the function under study into regions. Inside each region, the function is r-continuous, for a modulus of continuity r given a priori, and characteristic of the connection. However, the segmentation is not unique, and may be particularized by other considerations (self-duality, large or low number of point components, etc.). These variants are introduced by means of examples for three different connections: flat zone connections, jump connections, and smooth path connections. They turn out to provide remarkable segmentations, depending only on a few parameters. In the last section, some morphological filters are described, based on flat zone connections, namely openings by reconstruction, flattenings and levelings.

Algebraic spinors can be defined as minimal left ideals of Clifford algebras. We consider gauge transformations which are two-sided equivalence transformations of a complete algebra, including the spinors. These transformations of the spinors introduce new interaction terms which appear hard to interpret. We establish algebraic theorems which allow these new interaction terms to be evaluated and use these ideas to provide a new formulation of Glashow's electroweak interactions of leptons. The theorems also lead us to propose a new Clifford algebraic definition of spinors based on nilpotents, rather than idempotents.

In this paper we provide an algebraic definition for the Schouten product and give a decomposition for homogeneous Poisson structures in any n-dimensional vector space. A large class of n-homogeneous Poisson structures in ℝk is also characterised.

An individual’s inclusive fitness is derived by augmenting its traditional Darwinian fitness by certain components, and stripping it of others (Hamilton 1964). The component to be added is the sum of all eVects of the individual on his neighbours, weighted by the coeYcient of genetic relationship (r) between them. In the original derivation of inclusive fitness, the component to be subtracted was verbally defined as ‘all components which can be considered as due to the individual’s social environment’ (Hamilton 1964). It is not widely appreciated that this verbal definition of the component to be subtracted does not agree with its algebraic definition. Creel (1990a) used Hamilton’s algebraic definition of inclusive fitness to show that the component to be subtracted, e 0 , is actually equal to the average eVect

Exchange effects in rearrangement reactions are studied in the framework of a general algebraic theory of identical-particle scattering. An algebraic definition of reaction mechanisms proposed previously is used to distinguish the contributions to scattering amplitudes made by different exchange processes. Previous statements concerning the number of possible exchange mechanisms and their relative weight factors are rigorously proved. The results of this paper differ from those obtained in previous studies, as here identical clusters of particles are also treated as indistinguishable.

International audience; We tackle the problem of graph transformation with particular focus on node cloning. We propose a new approach to graph rewriting, called polarized node cloning, where a node may be cloned together with either all its incident edges or with only its outgoing edges or with only its incoming edges or with none of its incident edges. We thus subsume previous works such as the sesqui-pushout, the heterogeneous pushout and the adaptive star grammars approaches. We first define polarized node cloning algorithmically, then we propose an algebraic definition. We use polarization annotations to declare how a node must be cloned. For this purpose, we introduce the notion of polarized graphs as graphs endowed with some annotations on nodes and we define graph transformations with polarized node cloning by means of sesqui-pushouts in the category of polarized graphs.

This paper states the purpose and significance to research spatial data cube, and explains relationship between with spatial data cube and spatial data warehouse, and introduces the concept and structure of nonspatial dimension, spatial dimension, digital measure and spatial measure, and designs conceptual model for spatial data cube, i.e. extended star/sonwflake model. At last, based on mathematical tool — algebraic system, this paper gives algebraic definition of spatial data cube.

We give both a topological and an algebraic definition of the order of a global branch of a variety along a subvariety. Then we show that these two definitions agree. Finally we compare this order with the orders of the related local branches.

Hecke operators on rational period functions on the modular group have beend defined based on modular integrals. We give a purely algebraic definition of Hecke operators of rational period functions on the Hecke groups \( G({\sqrt 2}) \) and \( G({\sqrt 3}) \) and investigate their main properties. This is done with the help of description on the action of Hecke operators on rational period functions on the modular group.

The determinant is a “machine” which “processes” the column (or row) vectors of a square matrix A, producing a number denoted by det A. As a function of the column vectors of A, det is (1) linear, (2) skew-symmetric, and (3) normalized. The purpose of this paper is to show how these three properties, which completely characterize the determinant, can be deduced from a geometric definition of the determinant as a “calculator of areas and volumes.” I believe that this approach has a pedagogical advantage over the traditional algebraic definition because many of the familiar properties of the determinant are more easily conceptualized geometrically than algebraically.

In this note it is shown how, starting with the geometric definition of cross-product, one can prove the distributivity of cross-multiplication over vector addition, thereby recovering the usual algebraic definition of cross-product. This approach gives direct access to geometric problems and enhances geometric and conceptual understanding.

The aim of this paper is to offer an algebraic definition of infinite determinants of finite potent endomorphisms using linear algebra techniques. It generalizes Grothendieck's determinant for finite rank endomorphisms and is equivalent to the classic analytic definitions. The theory can be interpreted as a multiplicative analogue to Tate's formalism of abstract residues in terms of traces of finite potent linear operators on infinite-dimensional vector spaces, and allows us to relate Tate's theory to the Segal-Wilson pairing in the context of loop groups., Version 3. Minor changes

The characterization of physical systems requires a comprehensive understanding of quantum effects. One aspect is a proper quantification of the strength of such quantum phenomena. Here, a general convex ordering of quantum states will be introduced which is based on the algebraic definition of classical states. This definition resolves the ambiguity of the quantumness quantification using topological distance measures. Classical operations on quantum states will be considered to further generalize the ordering prescription. Our technique can be used for a natural and unambiguous quantification of general quantum properties whose classical reference has a convex structure. We apply this method to typical scenarios in quantum optics and quantum information theory to study measures which are based on the fundamental quantum superposition principle., 9 pages, 2 figures, revised version; published in special issue "150 years of Margarita and Vladimir Man'ko"

Many researchers are working on developing fast data mining methods for processing huge data sets efficiently, but some current reduction algorithms based on rough sets still have some disadvantages. In this paper, we indicated their limitations for reduct generation, then a new measure to knowledge was introduced to discuss the roughness of rough sets, and we developed an efficient algorithm for knowledge reduction based on rough sets. So, we modified the mean decision power, and proposed to use the algebraic definition of decision power. To select optimal attribute reduction, the judgment criterion of decision with an inequality was presented and some important conclusions were obtained. A complete algorithm for the attribute reduction was designed. Finally, through analyzing the given example, it is shown that the proposed heuristic information is better and more efficient than the others, and the presented method in the paper reduces time complexity and improves the performance. We report experimental results with several data sets from UCI Machine Learning Repository, and we compare the results with some other methods. The results prove that the proposed method is promising, which enlarges the application areas of rough sets.

We propose a definition of the topological W gravity using some properties of the principal three-dimensional subalgebra of a simple Lie algebra due to Kostant. In our definition, structures of the two-dimensional topological gravity are naturally embedded in the extended theories. In accordance with the definition, we will present some explicit calculations for the W3 gravity.

The main objective of this paper is to put forward Granular Matrix (GrM). By the definition of GrM, all the fundamental algebraic definitions of Rough Set Theory (RST) are redefined by simple matrix operation. Furthermore, two different definitions (algebraic definition and rough membership function definition) for rough inclusion and rough equivalence are united by GrM. GrM help us understand the essence of RST and provide a simple method for boundary computation and knowledge reduction. Concrete computation examples are given to illustrate the new definition.

This paper is mainly devoted to the study of the differentiation index and the order for quasi-regular implicit ordinary differential algebraic equation (DAE) systems. We give an algebraic definition of the differentiation index and prove a Jacobi-type upper bound for the sum of the order and the differentiation index. Our techniques also enable us to obtain an alternative proof of a combinatorial bound proposed by Jacobi for the order. As a consequence of our approach we deduce an upper bound for the Hilbert–Kolchin regularity and an effective ideal membership test for quasi-regular implicit systems. Finally, we prove a theorem of existence and uniqueness of solutions for implicit differential systems.

This article describes a new interaction model for artistic shape generation. Instead of using a metaphor of the classical sculpting tools we propose a new approach based on a two level editing which both uses genetic programming methods: objects are created by means of subjective selection and generative evolution. We develop a new way to define these objects combining algebraic definition of implicit surfaces and CSG-like composition of these shapes. Along with these we propose a software tool that implements a simple way to manage this new approach in three dimensional objects design.