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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.

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.

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.

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.

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

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.

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 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.

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.

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.

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.

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"

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.

An algebraic definition of Gardner's deformations for completely integrable bi-Hamiltonian evolutionary systems is formulated. The proposed approach extends the class of deformable equations and yields new integrable evolutionary and hyperbolic Liouville-type systems. An exactly solvable two-component extension of the Liouville equation is found., Comment: Proc. conf. "Nonlinear Physics: Theory and Experiment IV" (Gallipoli, 2006); Theor. Math. Phys. (2007) 151:3/152:1-2, 16p. (to appear)

The submanifold quantum mechanics was opened by Jensen and Koppe and has been studied for more than three decades. This article gives its more algebraic definition and show the essential aspects of the submanifold quantum mechanics from an algebraic viewpoint.

Algorithms for finding arbitrary sets of Kochen-Specker (KS) qunits (n-level systems) as well as all the remaining vectors in a space of an arbitrary dimension are presented. The algorithms are based on linear MMP diagrams which generate orthogonalities of KS qunits, on an algebraic definition of states on the diagrams, and on nonlinear equations corresponding to MMP diagrams whose solutions are either KS qunits or the remaining vectors of a chosen space depending on whether the diagrams allow 0-1 states or not. The complexity of the algorithms is polynomial. New results obtained with the help of the algorithms are presented., Comment: 4 pages, 4 figures, appeared in Barnett, S.M., Andersson, E., Jeffers, J., Ohberg, P., and Hirota, O., (Eds), QCMC 2004. Quantum Communication, Measurement and Computing: The Seventh International Conference on Quantum Communication, Measurement and Computing held in Glasgow, United Kingdom, 25-29 July 2004, American Institute of Physics Conference Proceedings 734, 2004, pp. 195-198, under the title "Kochen-Specker algorithms for qubits"; qubits changed into qunits here; 3 typos corrected; Web page: http://m3k.grad.hr/pavicic

The algebraic definition of charges for symmetry-preserving D-branes in Wess-Zumino-Witten models is shown to coincide with the geometric definition, for all simple Lie groups. The charge group for such branes is computed from the ambiguities inherent in the geometric definition., 12 pages, fixed typos, added references and a couple of remarks

We introduce new motivic invariants of arbitrary varieties over a perfect field. These cohomological invariants take values in the category of one-motives (considered up to isogeny in positive characteristic). The algebraic definition of these invariants presented here proves a conjecture of Deligne. Applications include some cases of conjectures of Serre, Katz and Jannsen on the independence of $\ell$ of parts of the \'etale cohomology of arbitrary varieties over number fields and finite fields., Comment: 43 pages, latex file, uses amsart, amscd packages (Major revisions; final version). Journal of Algebraic Geometry (to appear)

We show that the relations which define the algebras of the quantum Euclidean planes $\b{R}^N_q$ can be expressed in terms of projections provided that the unique central element, the radial distance from the origin, is fixed. The resulting reduced algebras without center are the quantum Euclidean spheres $S^{N-1}_q$. The projections $e=e^2=e^*$ are elements in $\Mat_{2^n}(S^{N-1}_q)$, with N=2n+1 or N=2n, and can be regarded as defining modules of sections of q-generalizations of monopoles, instantons or more general twisted bundles over the spheres. We also give the algebraic definition of normal and cotangent bundles over the spheres in terms of canonically defined projections in $\Mat_{N}(S^{N-1}_q)$., 14 pages, latex. Additional minor changes; final version for the journal

In this study, the authors present a new strategy to implement an illumination compensation-based contrast enhancement. Different from the traditional pixel-to-pixel transformation, the proposed method offers a level-to-level framework by generating the reference intensity level and the given target intensity level. Fundamentally, the traditional illumination compensation algorithms such as Histogram equalisation, log and gamma transformation are trade-off strategies and face the same dilemma. For example, the log function compresses the dynamic range of image with large variations in pixel values. The proposed method is an integration algorithm, the illumination compensation could be considered as the transformation from reference intensity to the chosen target intensity, and the contrast is enhanced by the given algebraic definition of the characteristic values of each pixel. The advantages are the adjusted intensity would be accomplished in the native phase according to the variant characteristic intensity values, clarifying the details of the darker areas as well as preserving the highlight areas, reducing HALO, colour constancy, removing colour cast and colour restoration. Shown as the experimental results, the proposed method can obtain better performance compared with other methods in subjective evaluation with contour plot and the objective evaluations by universal quality index, structural similarity index and average of standard deviations.

An algebraic definition of time differentiation is presented and used to relate independent-measurements of position and velocity. With this, students can grasp certain essential physical, geometric, and algebraic properties of motion and differentiation before undertaking the study of limits.