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Directions of Arrival for Uniform Linear Arrays represent a widely studied topic in many fields of signal processing, with large attention on computational complexity, estimation accuracy and noise rejection. In this article we study the mathematical model behind Directions of Arrival from the point of view of algebraic geometry, focusing on its relation with the rational normal curve. On this basis, we give a novel interpretation for the root-MUSIC algorithm, that is a widely adopted estimation approach in signal processing. Furthermore, we propose some novel estimation techniques. The first one is based on the computation of the points on the rational normal curve that minimize the distance from the linear subspace defined by the measured Directions of Arrival. The others require the study of the secant varieties of the rational normal curve and the minimization of the distance between the point of the Grassmannian defined by the signal subspace and a certain secant variety. The results obtained from simulations in a noisy scenario show that our estimators are statistically consistent. One algorithm outperforms root-MUSIC over a wide range of scenarios, especially in presence of few snapshots and low Signal to Noise Ratio. [ABSTRACT FROM AUTHOR]

LaSalle’s invariance principle is a commonly used extension of Lyapunov’s second method to study asymptotic stability of nonlinear systems. If the system can be written in polynomial form, the examination can be automated using algebraic geometry and quantifier elimination. This article addresses this automated examination using a method relying on polynomial ideals and applies it on some example systems. In addition, some properties of these special ideals are derived that allow to reduce the computational effort significantly. [ABSTRACT FROM AUTHOR]

• Mutual Constrained Independence (MCI) is a novel notion in probability theory. • MCI allows to objectively estimate frequencies of itemsets given certain constraints. • The mathematical results for the existence and characterization of MCI are proven. • A link between MCI models and Maximum Entropy models is established. • A method for computing MCI models based on algebraic geometry is presented. Dealing with redundancy is one of the main challenges in frequency based data mining and itemset mining in particular. To tackle this issue in the most objective possible way, we introduce the theoretical bases of a new probabilistic concept: Mutual constrained independence (MCI). Thanks to this notion, we describe a MCI model for the frequencies of all itemsets which is the least binding in terms of model hypotheses defined by the knowledge of the frequencies of some of the itemsets. We provide a method for computing MCI models based on algebraic geometry. We establish the link between MCI models and a class of MaxEnt models which has already known to be used in pattern mining. As such, our research presents further insight on the nature of such models and an entirely novel approach for computing them. [ABSTRACT FROM AUTHOR]

In this paper, the path planning problem for an unicycle-like mobile robot is considered. By using some results borrowed from algebraic geometry, a technique is given to determine a dynamical system that is affine in the input and whose trajectories tend to a chosen algebraic set independently of the control input. Since this does not guarantee that the corresponding paths of motion are collision free, an optimal control problem is formulated to enforce this behavior, and its approximate solution is determined via integral reinforcement learning. Finally, it is shown how such results can be used to derive a feedback control law for unicycle-like mobile robots. [ABSTRACT FROM AUTHOR]

In this article, uncertain continuous-time and discrete-time linear time-invariant systems are considered. The uncertainties are assumed to affect polynomially the dynamics of the system and they can be structured. The problem of computing the distance to internal instability of an internally exponentially stable nominal system is solved by using tools from algebraic geometry, thus extending previous results valid in case of unstructured uncertainties. The choice of the nominal system is formulated and solved as the choice of a point in the parameter space that is sufficiently far from the boundary of the domain of stability. A simple example of application to robust control is outlined. [ABSTRACT FROM AUTHOR]

In this work, we present the package NeatIBP , which automatically generates small-size integration-by-parts (IBP) identities for Feynman integrals. Based on the syzygy and module intersection techniques, the generated IBP identities' propagator degree is controlled and thus the size of the system of IBP identities is shorter than that generated by the standard Laporta algorithm. This package is powered by the computer algebra systems Mathematica and Singular , and the library SpaSM. It is parallelized on the level of Feynman integral sectors. The generated small-size IBP identities can subsequently be used for either finite field reduction or analytic reduction. We demonstrate the capabilities of this package on several multi-loop IBP examples. Program Title: NeatIBP CPC Library link to program files: https://doi.org/10.17632/ms85fpfm7b.1 Developer's repository link: https://github.com/yzhphy/NeatIBP Licensing provisions: GPLv3 Programming language: Mathematica , Singular , C Nature of problem: The difficulty of the IBP reduction of Feynman integrals stems from the large size of the IBP system. Solution method: We apply the module intersection method to generate IBP relations by algebraic geometry methods, and compare the smaller IBP system we generate with that from Laporta's algorithm. NeatIBP generates the module intersection in Singular [1] and selects relevant and independent IBP relations via an algorithm relying on the library SpaSM [2]. The workflow is parallelized by a task manager written in Mathematica [3]. [1] W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 4-3-1 – A computer algebra system for polynomial computations, http://www.singular.uni-kl.de , 2022. [2] The SpaSM group, SpaSM : a Sparse direct Solver Modulo p , http://github.com/cbouilla/spasm , 2017. [3] Wolfram Research, Mathematica 13.2, https://www.wolfram.com/mathematica/. [ABSTRACT FROM AUTHOR]

Born: July 1978 Occupation: Mathematician Caucher Birkar was one of four mathematicians who won the 2018 Fields Medal, an award considered the highest honor in mathematical achievement, equivalent to the more famous Nobel Prize. From 2001 to 2004, Birkar studied advanced mathematics as a doctoral student at the University of Nottingham, working under the direction of Professor Ivan Fesenko. While chatting, Shokurov began telling Birkar how he was working in birational geometry, a subfield of algebraic geometry on which few mathematicians were then focused. With birational geometry, Birkar focused on classifying polynomial equations and sets of solutions known as algebraic varieties. [Extracted from the article]

The article offers information on triangulation; a simplicial complex that models a topological or algebraic object. It mentions use of the van Kampen obstruction and topological techniques; and also mentions observations that relate the biased pairing property and the Lefschetz property, setting up a way to prove the Lefschetz property inductively.

Born: December 11, 1987 Occupation: Mathematician GLO:cub/01nov20:77n1.jpg PHOTO (COLOR): Photo by Kat Baulu via Wikimedia Commons gl " possesses a type of mathematical talent that emerges only rarely", Allyn Jackson wrote in a 2018 paper posted on the website of the International Mathematical Union (IMU). Scholze's concept of perfectoid spaces allows mathematicians to bridge the gap between p-adic arithmetic and geometry, and its development was hailed as a key innovation in mathematics. Scholze's award citation mentioned "the revolution that he launched in arithmetic geometry", and former professor and mentor Michael Rapoport delivered the lecture in his honor at the event. Throughout his career Scholze often fielded questions about how his field of extraordinarily complex mathematics benefits society. [Extracted from the article]

• We reconsider the use of algebraic geometry codes in cryptography. • Applying list decoding algorithms to get smaller key size. • An algorithm to generate secure elliptic codes which can resist known structure attacks is presented. • An IND-CPA variant of post-quantum McEliece cryptosystem is proposed. Code-based cryptography has aroused wide public concern as one of the main candidates for post quantum cryptography to resist attacks against cryptosystems from quantum computation. However, the large key size becomes a drawback that prevents it from wide practical applications although it performs pretty well on the speed of both encryption and decryption. The use of algebraic geometry codes is considered to be a good solution to reduce the key size, but the special structures of algebraic geometry codes results in lots of attacks including Minder's attack. To cope with the barriers of large key size as well as attacks from the special structures of algebraic codes, we propose a code-based encryption system using elliptic codes. The special structure of elliptic codes helps us to effectively reduce the size of secret key. By choosing the rational points carefully, we build elliptic codes whose minimum weight codeword is hard to sample. Such codes are used in constructing encryption systems such that Minder's attacks can be resisted. More importantly, we apply the list decoding algorithm in the decryption process thus more errors beyond half of the minimum distance of the code could be corrected, which is the key point to resist other known attacks for algebraic geometry codes based cryptosystems. Our implementation shows that the proposed encryption system performs well on the key size and ciphertext expansion rate. [ABSTRACT FROM AUTHOR]

The article provides an overview of geometric complexity theory (GCT), an approach to the P versus NP problems which uses representation theory and algebraic geometry. The author examines topics such as the strong flip theorem, the geometric obstruction hypothesis (GOH), the derandomization hypothesis, self-referential difficulty, and the formal definition of varieties. Diagrams are presented which provide information on lower bounds, ellipsoids, and geometric obstructions. The author concludes that GCT reveals positivity problems and explicit construction problems in complexity theory. INSETS: Formal Definition of the Varities;The Strong Flip Theorem;Self-Referential Difficulty;The Decomposition

In college I majored in math, and became curious about theoretical physicist Eugene Wigner's question about the "unreasonable effectiveness" of mathematics: Why should our universe be so readily governed by mathematical laws? Nonetheless, the possible benefits to physics of reducing topology to algebra are what got me so excited about Grothendieck's work. A student of physics once asked a famous expert how much mathematics a physicist needs to know. [Extracted from the article]

Abstract In the framework of t -norm based logics, Gödel propositional logic G and drastic product logic DP are strictly connected. In this paper we explore the even stricter relation between DP and the logic G Δ , the expansion of G with Baaz–Monteiro connective Δ. In particular we provide functional representations of free algebras in the corresponding algebraic semantics. We use then these functional representations to develop a theory of states, that is, finitely additive probability measures, for both G Δ and DP. Finally, we use dual equivalences for the algebraic semantics of both G Δ and DP, to provide a completely combinatorial characterization of states. [ABSTRACT FROM AUTHOR]

A list of a number of natural developments for the field of algebraic manipulation is given. Then the prospects for a general theory of functions defined by ordinary differential equations are discussed. The claim is made that recent developments in mathematics indicate that it should be possible to algorithmically generate many properties of solutions to differential equations. Such a theory is preferable to a less general effort to make algebraic manipulation systems knowledgeable about the usual special functions (e g exponential, hyper geometric) [ABSTRACT FROM AUTHOR]

An interview with University of Utah mathematician Christopher Hacon is presented. He describes the two phases of breakthroughs in mathematics, based on his own experience, discusses the subject of algebraic geometry and its applications to string theory and differential geometry, and tackles the rise in collaborations among mathematicians. Hacon also cites the idea about the extra six dimensions to the universe as the big question in string theory.

This paper presents an investigation of a relatively unstudied approach to modeling lower and upper subjective probabilities. It is based on the fact that every cumulative distribution function (CDF) with support (0,1) has a “dual” CDF that obeys the conjugacy relation between coherent lower and upper probabilities. A new 2-parameter family of “CDF-Quantile” distributions with support (0,1) is extended via a third parameter for the purpose of modeling lower–upper probabilities via this approach. The extension exploits certain properties of the CDF-Quantile family, and the fact that continuous CDFs on (0,1) random variables form an algebraic group that is closed under composition. This extension also yields methods for testing specific models of lower–upper probability assignments. Finally, the new models are applied to real data-sets, and compared with alternative approaches for their relative advantages and drawbacks. [ABSTRACT FROM AUTHOR]

This paper, by means of complex representation of a split quaternion matrix, studies the problems of right split quaternion eigenvalues and eigenvectors of a split quaternion matrix, and derives algebraic techniques for the right split quaternion eigenvalues and eigenvectors of the split quaternion matrix in split quaternionic mechanics. [ABSTRACT FROM AUTHOR]

Multivariate residues appear in many different contexts in theoretical physics and algebraic geometry. In theoretical physics, they for example give the proper definition of generalized-unitarity cuts, and they play a central role in the Grassmannian formulation of the S -matrix by Arkani-Hamed et al. In realistic cases their evaluation can be non-trivial. In this paper we provide a Mathematica package for efficient evaluation of multivariate residues based on methods from computational algebraic geometry. Program summary Program Title: MultivariateResidues Program Files doi: http://dx.doi.org/10.17632/7z87sf98xs.1 Licensing provisions: GNU General Public License (GPLv3) Programming language: Wolfram Mathematica version 7.0 or higher Nature of problem: Evaluation of multivariate complex residues Solution method: Mathematica implementation [ABSTRACT FROM AUTHOR]

In this paper, a novel scheme is presented for forming the matrix equations of multilevel adaptive cross approximation (MLACA) algorithm. The main idea of the proposed technique is to use the directional grouping scheme to subdivide the far-field domain of MLACA algorithm. By using the grouping scheme, the far-field interaction domain can be divided into many cone structures. The matrix between the observation group and far-field group in the cone structure is low-rank, which meets the directional far-field requirement. At the same time, the near-field interaction matrices are formed by the SVD(T) method to further reduce the total memory requirements. With the given techniques, the memory requirement of the novel grouping scheme for the far-field is much less than half of traditional MLACA algorithm. Meanwhile, the memory requirement of the SVD(T) method for the near-field is only about one-third of direct filling. [ABSTRACT FROM AUTHOR]

For any given Feynman graph, the set of integrals with all possible powers of the propagators spans a vector space of finite dimension. We introduce the package Azurite ( A ZUR ich-bred method for finding master I n TE grals), which efficiently finds a basis of this vector space. It constructs the needed integration-by-parts (IBP) identities on a set of generalized-unitarity cuts. It is based on syzygy computations and analyses of the symmetries of the involved Feynman diagrams and is powered by the computer algebra systems Singular and Mathematica . It can moreover analytically calculate the part of the IBP identities that is supported on the cuts. In some cases, the basis obtained by Azurite may be slightly overcomplete. Program summary Program Title: Azurite Licensing provisions: GNU General Public License (GPL) Programming language: Wolfram Mathematica version 10.0 or higher Supplementary material: A manual in the form of a Mathematica notebook Nature of problem: Determination of a basis of the space of loop integrals spanned by a given Feynman diagram and all of its subdiagrams Solution method: Mathematica implementation [ABSTRACT FROM AUTHOR]

Random intersection graphs have received much attention recently and been used in a wide range of applications ranging from key predistribution in wireless sensor networks to modeling social networks. For these graphs, each node is equipped with a set of objects in a random manner, and two nodes have an undirected edge in between if they have at least one object in common. In this paper, we investigate connectivity and robustness in a general random intersection graph model. Specifically, we establish sharp asymptotic zero-one laws for $k$-connectivity and $k$-robustness, as well as the asymptotically exact probability of $k$-connectivity, for any positive integer $k$. The $k$ -connectivity property quantifies how resilient is the connectivity of a graph against node or edge failures, while $k$-robustness measures the effectiveness of local-information-based consensus algorithms (which do not use global graph topology information) in the presence of adversarial nodes. In addition to presenting the results under the general random intersection graph model, we consider two special cases of the general model, a binomial random intersection graph and a uniform random intersection graph, which both have numerous applications as well. For these two specialized graphs, we present asymptotically exact probabilities of $k$-connectivity and asymptotic zero-one laws for $k$-robustness. [ABSTRACT FROM PUBLISHER]

We state a formula for the Pythagorean theorem that is valid in Euclidean, spherical, and hyperbolic geometries and give a proof using only properties the geometries have in common. [ABSTRACT FROM AUTHOR]

A precise mathematical definition is given for spiral plane tilings. It is not restricted to monohedral tilings and is tested on a series of examples from the literature. Unwanted cases from regular tilings can be excluded. In case of one single arm a modified definition can be applied. Also the special case of locally infinite tilings with one singular point can be treated with any number of spiral arms. The question whether such a definition in mathematical terms could be given was posed by Grünbaum and Shephard in the late 1970s. [ABSTRACT FROM AUTHOR]

A novel approach to estimation of structured uncertainties is proposed for a class of nonlinear systems based on algebraic geometry tools. The notion of Polynomial Partition of Unity (PPU) is introduced and it constitutes a key feature of the proposed estimation technique, whose main advantage is the potentiality of being executed offline. Provided that the system dynamics satisfy some PPU and gradient properties, and collecting a sufficient number of measurements at different time steps, the values of unknown parameters are proved to be included in the finite set of common solutions to a family of polynomial equations. An optimization method based on multiple-models is then proposed to further refine the estimation. The case-study of sensorless permanent magnet synchronous motors is presented as a support to theoretical developments. [ABSTRACT FROM AUTHOR]

Localization of the waves of the sine-Gordon equation depends on the shape of the initial condition. It is shown how initially motionless Gaussian distribution may be modified to obtain propagation of localized waves in both directions. However, the resulting localized wave profile is described neither by an asymptotic envelope- wave solution to the sine-Gordon equation nor by its exact traveling breather solution. The distributed control algorithms are developed to achieve wave localization independent of the shape of the initial condition. It is shown that localization of the waves in both directions is achieved by means of a feedforward (nonfeedback) control. The waves are similar to the envelope wave solution. The feedback distributed algorithm is shown to provide both localized waves according to analytical solutions and their unidirectional propagation. [ABSTRACT FROM AUTHOR]

Finite-time consensus for continuous-time multi-agent systems with disturbance is studied in this paper. By employing the Hölder continuous Lyapunov function, algebraic graph theory and control theory, some sufficient conditions for the finite-time consensus are presented. An example and simulation results are given to illustrate the effectiveness of the presented method and the correctness of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

The goal of this technical note is to design a switching signal estimator for a class of elementary continuous-time switching or switched systems. First, the elementary system is recast into a polynomial form and, secondly, some tools borrowed from Algebraic Geometry are used to express the switching signal as a function of the time derivatives of the output and of the input. [ABSTRACT FROM AUTHOR]

The article discusses a purely combinatorial formula derived from using properties of binomial coefficients. Topics covered include are the Fibonacci numbers, the unexpected symmetry property called algebraic centrosymmetry, and the matrix multiplication. Also mentioned are the applications to combinatorial identities.

This technical note applies methods from Algebraic Geometry, namely polynomial ideals and sub-modules, to find parameterized sets of state-feedback stabilizing control laws for polynomial, continuous-time, affine nonlinear systems. The proposed method is illustrated by simple examples. [ABSTRACT FROM AUTHOR]

A biography of mathematician Yitang Zhang is presented. Zhang was born in Shanghai, China 1995 and he earned his doctorate in algebraic geometry from Purdue University in 1991. He was married to Yaling, who goes by Helen. He won the Ostrowski Prize for outstanding achievement in pure mathematics in 2013 and received the MacArthur Fellowship, as well as the Frank Nelson Cole Prize in Number Theory.

The article offers information on the July 2014 Institute of Mathematics Pedagogy (IMP) with the theme "Symmetry, Complementarity and Conjugacy." Topics include the use of grids and the concept of duality in geometry education, the role of symmetry in mathematics education, and a task presented by speaker Malcolm Swan that involved algebraic geometry.

When fixing a covariant gauge, most popularly the Landau gauge, on the lattice, one encounters the Neuberger 0/0 problem, which prevents one from formulating a Becchi-Rouet-Stora-Tyutin symmetry on the lattice. Following the interpretation of this problem in terms of Witten-type topological field theory and using the recently developed Morse theory for orbifolds, we propose a modification of the lattice Landau gauge via orbifolding of the gauge-fixing group manifold and show that this modification circumvents the orbit-dependence issue and hence can be a viable candidate for evading the Neuberger problem. Using algebraic geometry, we also show that though the previously proposed modification of the lattice Landau gauge via stereographic projection relies on delicate departure from the standard Morse theory due to the noncompactness of the underlying manifold, the corresponding gauge-fixing partition function turns out to be orbit independent for all the orbits except in a region of measure zero. [ABSTRACT FROM AUTHOR]

We consider reachability of discrete-time polynomial systems. The reachability of such systems can be interpreted as the surjectivity of the polynomial mappings defined by the systems. For a polynomial mapping between suitable spaces of the same dimensions, it has been shown that the injectivity of the mapping also implies the surjectivity. By utilizing this fact, we derive sufficient conditions for the reachability of the polynomial systems, which essentially check the injectivity of the polynomial mappings. [ABSTRACT FROM PUBLISHER]

Abstract: A process in a Euclidean space is called an additive process if it has independent increments. We recall the classical Lévy–Itô representation for additive processes without fixed jumps, and describe how fixed jumps were handled in the classical literature. Our main result is an extended Lévy–Itô formula in which the fixed jumps are expressed in a canonical and convenient form. [Copyright &y& Elsevier]

The article discusses the special points in the Symmedian Point and how the Navy practically uses the point in the World War II. It defines the Symmedian Point as the intersection of the isogonal lines to the centroid, which is determined by the medians. It states that the point is used by the Navy to analyze classified data from a test that the Navy was conducting and to track the descending parachute and record its angle from the station to the surface of the water.

It has been claimed in a recent paper [J.D. Barrow, A.B. Batista, J.C. Fabris, M. J.S. Houndjo, and G. Dito, Phys. Rev. D 84, 123518 (2011).] that sudden singularities will survive in semiclassical gravity. This issue is here carefully reviewed, pointing out that such conclusion, even if valid under some specific conditions, does not stand in other cases. An explicit example is studied in detail to support our statement, stemming from these other situations, that quantum effects may in fact drastically modify the behavior of sudden singularities. [ABSTRACT FROM AUTHOR]

Delay-independent stability (DIS) of a general class of linear time-invariant (LTI) multiple time-delay system (MTDS) is investigated in the entire delay-parameter space. Stability of such systems may be lost only if their spectrum lies on the imaginary axis for some delays. We build an analytical approach that requires the inspection of the roots of finite number of single-variable polynomials in order to detect if the spectrum ever lies on the imaginary axis for some delays, excluding infinite delays. The approach enables to test the necessary and sufficient conditions of DIS of LTI-MTDS, technically known as weak DIS, as well as the robust stability of single-delay systems against all variations in delay ratios. The proposed approach, which does not require any parameter sweeping and graphical display, becomes possible by establishing a link between the infinite spectrum and algebraic geometry. Case studies are provided to show the effectiveness of the approach. [ABSTRACT FROM PUBLISHER]

Abstract: The paper presents a new meshless numerical technique for solving one and two-dimensional Stefan problems. The technique presented is based on the use of the delta-shaped functions and the method of approximate fundamental solutions (MAFS) first suggested for solving elliptic problems and heat equations in domains with fixed boundaries. The one-dimensional problems in the plane and cylindrical geometries are considered. The numerical examples are presented and the results are compared with the analytical solutions. The comparison shows that the method presented provides a very high precision in determining the position of the moving boundary even for degenerate and singular problems when a region initially has zero thickness. The same technique was developed for 2D Stefan problems with completely or partially unknown boundary. [Copyright &y& Elsevier]

The article presents a research study which introduces a family of new mathematical sequences that can illustrate the relation of polygonal numbers to centered polygonal numbers. It notes that the study involves geometry, linear algebra, and Pellian equations. It concludes through the identification of geometrical numbers which are both polygonal and centered polygonal for same number of sides.

In ordinary quantum theory, any mixed state can be purified in an enlarged Hilbert space by adding an ancillary system. The purified state does not depend on the state of any extraneous system with which the mixed state is going to interact or on the physical interaction. Here, we prove that it is not possible to purify a mixed state that traverses a closed timelike curve (CTC) and is allowed to interact in a consistent way with a causality-respecting (CR) quantum system in the same manner. In other words, if a CTC quantum system with a mixed state has to undergo an arbitrary interaction with an arbitrary CR system then it has to be in a "proper" mixture, as it cannot be regarded as a subsystem of a pure entangled state. Thus, in general for arbitrary interactions between CR and CTC systems, there is no universal purification by embedding in a larger Hilbert space for mixed states with CTCs. This shows that in quantum theory with CTCs there can exist proper and improper mixtures. [ABSTRACT FROM AUTHOR]

Abstract: We present a complete characterization of subdirectly irreducible MV-algebras with internal states (SMV-algebras). This allows us to classify subdirectly irreducible state morphism MV-algebras (SMMV-algebras) and describe single generators of the variety of SMMV-algebras, and show that we have a continuum of varieties of SMMV-algebras. [Copyright &y& Elsevier]

We readdress the problems associated with bulk Higgs and the gauge fields in a five-dimensional Randall-Sundrum model by extending the model to six dimensions with double warping along the two extra spatial dimensions. In this six-dimensional model, we have a freedom of two moduli scales as against one modulus in the five-dimensional model. With a little hierarchy between these moduli, we can obtain the right magnitude for W and Z boson masses from the Kaluza-Klein modes of massive bulk gauge fields where the spontaneous symmetry breaking is triggered by bulk Higgs. We also have determined the gauge couplings of the standard model fermions with Kaluza-Klein modes of the gauge fields. Unlike the case of the five-dimensional model with a massless bulk gauge field, here, we have shown that the gauge couplings and the masses of the Kaluza-Klein gauge fields satisfy the precision electroweak constraints and also obey the Tevatron bounds. [ABSTRACT FROM AUTHOR]

Abstract: A computational method for numerical solution of a nonlinear Volterra integro-differential equation of fractional (arbitrary) order which is based on CAS wavelets and BPFs is introduced. The CAS wavelet operational matrix of fractional integration is derived and used to transform the main equation to a system of algebraic equations. Some examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

To serve as a dispersion relation, a cotangent bundle function must satisfy three simple algebraic properties. These conditions are derived from the inescapable physical requirements that local matter field dynamics must be predictive and allow for an observer-independent notion of positive energy. Possible modifications of the standard relativistic dispersion relation are thereby severely restricted. For instance, the dispersion relations associated with popular deformations of Maxwell theory by Gambini-Pullin or Myers-Pospelov are not admissible. Dispersion relations passing the simple algebraic checks derived here correspond to physically admissible Finslerian refinements of Lorentzian geometry. [ABSTRACT FROM AUTHOR]

A new methodology, advanced clustering with frequency sweeping (ACFS), is introduced for the stability analysis of the most general class of linear time-invariant time-delay systems with multiple delays. ACFS is a cross fertilization of algebraic geometry, frequency sweeping technique and the root clustering paradigm, and it can directly extract the 2-D cross sections of the stability views in any two-delay domain. In this domain, ACFS can also test the delay-independent stability based on necessary and sufficient conditions, and can compute, for the delay-dependent case, the precise lower and upper bounds of the only parameter, the frequency, that ACFS sweeps. Case studies that are prohibitive to analyze with the existing analytical methods are presented to demonstrate the strengths of the method. [ABSTRACT FROM AUTHOR]

Abstract: An inequality related to Minkowski type for the Sugeno integral on abstract spaces is studied in a rather general form. Some previous results on Chebyshev type inequality obtained by the authors are generalized. Several examples are given to illustrate the validity of this inequality. The conditions such that this inequality becomes an equality are also discussed. Finally, conclusions and some problems for further investigations are included. [Copyright &y& Elsevier]

Abstract: In this paper, the convexity properties of fuzzy nets on the Euclidean space is investigated. [Copyright &y& Elsevier]

Abstract: This paper is devoted to introduce the concepts of transfer closed and transfer open multimaps in minimal spaces. Also, some characterizations of them are considered. Further, the notion of minimal local intersection property will be introduced and characterized. Moreover, some maximal element theorems via minimal transfer closed multimaps and minimal local intersection property are given. [Copyright &y& Elsevier]

Most linear control problems lead directly to matrix inequalities (MIs). Many of these are badly behaved but a classical core of problems are expressible as linear matrix inequalities (LMIs). In many engineering systems problems convexity has all of the advantages of a LMI. Since LMIs have a structure which is seemingly much more rigid than convex MIs, there is the hope that a convexity based theory will be less restrictive than LMIs. How much more restrictive are LMIs than convex MIs? There are two fundamentally different classes of linear systems problems: dimension free and dimension dependent. A dimension free MI is a MI where the unknowns are g-tuples of matrices and appear in the formulas in a manner which respects matrix multiplication. Most of the classic MIs of control theory are dimension free. Dimension dependent MIs have unknowns which are tuples of numbers. The two classes behave very differently and this survey describes what is known in each case about the relation between convex MIs and LMIs. The proof techniques involve and necessitate new developments in the field of semialgebraic geometry. [ABSTRACT FROM AUTHOR]