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Eigenvalues of Euclidean Distance Matrices and Rs-Majorization on R2
Archive of SID 46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University 2 Talk Eigenvalues of Euclidean distance matrices and rs-majorization on R pp.: 1{4 Eigenvalues of Euclidean Distance Matrices and rs-majorization on R2 Asma Ilkhanizadeh Manesh∗ Department of Pure Mathematics, Vali-e-Asr University of Rafsanjan Alemeh Sheikh Hoseini Department of Pure Mathematics, Shahid Bahonar University of Kerman Abstract Let D1 and D2 be two Euclidean distance matrices (EDMs) with correspond- ing positive semidefinite matrices B1 and B2 respectively. Suppose that λ(A) = ((λ(A)) )n is the vector of eigenvalues of a matrix A such that (λ(A)) ... i i=1 1 ≥ ≥ (λ(A))n. In this paper, the relation between the eigenvalues of EDMs and those of the 2 corresponding positive semidefinite matrices respect to rs, on R will be investigated. ≺ Keywords: Euclidean distance matrices, Rs-majorization. Mathematics Subject Classification [2010]: 34B15, 76A10 1 Introduction An n n nonnegative and symmetric matrix D = (d2 ) with zero diagonal elements is × ij called a predistance matrix. A predistance matrix D is called Euclidean or a Euclidean distance matrix (EDM) if there exist a positive integer r and a set of n points p1, . , pn r 2 2 { } such that p1, . , pn R and d = pi pj (i, j = 1, . , n), where . denotes the ∈ ij k − k k k usual Euclidean norm. The smallest value of r that satisfies the above condition is called the embedding dimension. As is well known, a predistance matrix D is Euclidean if and 1 1 t only if the matrix B = − P DP with P = I ee , where I is the n n identity matrix, 2 n − n n × and e is the vector of all ones, is positive semidefinite matrix. -
Polynomial Approximation Algorithms for Belief Matrix Maintenance in Identity Management
Polynomial Approximation Algorithms for Belief Matrix Maintenance in Identity Management Hamsa Balakrishnan, Inseok Hwang, Claire J. Tomlin Dept. of Aeronautics and Astronautics, Stanford University, CA 94305 hamsa,ishwang,[email protected] Abstract— Updating probabilistic belief matrices as new might be constrained to some prespecified (but not doubly- observations arrive, in the presence of noise, is a critical part stochastic) row and column sums. This paper addresses the of many algorithms for target tracking in sensor networks. problem of updating belief matrices by scaling in the face These updates have to be carried out while preserving sum constraints, arising for example, from probabilities. This paper of uncertainty in the system and the observations. addresses the problem of updating belief matrices to satisfy For example, consider the case of the belief matrix for a sum constraints using scaling algorithms. We show that the system with three objects (labelled 1, 2 and 3). Suppose convergence behavior of the Sinkhorn scaling process, used that, at some instant, we are unsure about their identities for scaling belief matrices, can vary dramatically depending (tagged X, Y and Z) completely, and our belief matrix is on whether the prior unscaled matrix is exactly scalable or only almost scalable. We give an efficient polynomial-time algo- a 3 × 3 matrix with every element equal to 1/3. Let us rithm based on the maximum-flow algorithm that determines suppose the we receive additional information that object whether a given matrix is exactly scalable, thus determining 3 is definitely Z. Then our prior, but constraint violating the convergence properties of the Sinkhorn scaling process. -
MOLETRONICS: TOWARDS NEW ERA of NANOTECHNOLOGY 1 Prof
International Journal of Technical Research and Applications e-ISSN: 2320-8163, www.ijtra.com, Volume 5, Issue 2 (March - April 2017), PP. 73-75 MOLETRONICS: TOWARDS NEW ERA OF NANOTECHNOLOGY 1 Prof. Y.D. Kapse, 2Mr.Akash A. Sindhikar 1Assistant professor, 2M.tech 1,2 ENTC,GCOEJ, Jalgaon, India Abstract—The Molecular electronics is the fundamental single molecule logic gate. C. Joachim and J.K experimented building blocks of an emerging technology called the conductance of single molecule in IBM. In 1990 Mark ‘nanoelectronics a field that holds promise for application in all Reed et al added few hundred molecules. In 2000 Shirakawa, kinds of electronic devices, from cell phones to sensors. Heeger and MacDiarmid won the Nobel Prize in physics for Molecular devices could be built a thousand times smaller than Si the demoletronics. The conductive polymers can be used as based devices in use now. Computer industry execs might start “molecular wires”. This will altogether alter the fabrication breathing easier because their biggest fear - that smaller and faster devices will eventually come to an end because silicon industry because a conventional silicon chip houses over 10-50 microchips will get so small that eventually they will contain too million switches over an as large as a postage stamp. few silicon atoms to work - might be lessened as advancements in Moletronics aims at redefining this powerful integration molecular electronics come to the rescue. Molecular electronics is technology to density of over one million times than today’s enabling an area of nanoscience and technology that holds state-of-the-art IC fabrication. -
Elliptic Systems on Riemann Surfaces
Lecture Notes of TICMI, Vol.13, 2012 Giorgi Akhalaia, Grigory Giorgadze, Valerian Jikia, Neron Kaldani, Giorgi Makatsaria, Nino Manjavidze ELLIPTIC SYSTEMS ON RIEMANN SURFACES © Tbilisi University Press Tbilisi Abstract. Systematic investigation of Elliptic systems on Riemann surface and new results from the theory of generalized analytic functions and vectors are pre- sented. Complete analysis of the boundary value problem of linear conjugation and Riemann-Hilbert monodromy problem for the Carleman-Bers-Vekua irregular systems are given. 2000 Mathematics Subject Classi¯cation: 57R45 Key words and phrases: Generalized analytic function, Generalized analytic vector, irregular Carleman-Bers-Vekua equation, linear conjugation problem, holo- morphic structure, Beltrami equation Giorgi Akhalaia, Grigory Giorgadze*, Valerian Jikia, Neron Kaldani I.Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State University 2 University st., Tbilisi 0186, Georgia Giorgi Makatsaria St. Andrews Georgian University 53a Chavchavadze Ave., Tbilisi 0162, Georgia Nino Manjavidze Georgian Technical University 77 M. Kostava st., Tbilisi 0175, Georgia * Corresponding Author. email: [email protected] Contents Preface 5 1 Introduction 7 2 Functional spaces induced from regular Carleman-Bers-Vekua equations 15 2.1 Some functional spaces . 15 2.2 The Vekua-Pompej type integral operators . 16 2.3 The Carleman-Bers-Vekua equation . 18 2.4 The generalized polynomials of the class Ap;2(A; B; C), p > 2 . 20 2.5 Some properties of the generalized power functions . 22 2.6 The problem of linear conjugation for generalized analytic functions . 26 3 Beltrami equation 28 4 The pseudoanalytic functions 36 4.1 Relation between Beltrami and holomorphic disc equations . 38 4.2 The periodicity of the space of generalized analytic functions . -
A 1 Case-PR/ }*Rciofft.;Is Report
.A 1 case-PR/ }*rciofft.;is Report (a) This eruption site on Mauna Loa Volcano was the main source of the voluminous lavas that flowed two- thirds of the distance to the town of Hilo (20 km). In the interior of the lava fountains, the white-orange color indicates maximum temperatures of about 1120°C; deeper orange in both the fountains and flows reflects decreasing temperatures (<1100°C) at edges and the surface. (b) High winds swept the exposed ridges, and the filter cannister was changed in the shelter of a p^hoehoc (lava) ridge to protect the sample from gas contamination. (c) Because of the high temperatures and acid gases, special clothing and equipment was necessary to protect the eyes. nose, lungs, and skin. Safety features included military flight suits of nonflammable fabric, fuil-face respirators that are equipped with dual acidic gas filters (purple attachments), hard hats, heavy, thick-soled boots, and protective gloves. We used portable radios to keep in touch with the Hawaii Volcano Observatory, where the area's seismic activity was monitored continuously. (d) Spatter activity in the Pu'u O Vent during the January 1984 eruption of Kilauea Volcano. Magma visible in the circular conduit oscillated in a piston-like fashion; spatter was ejected to heights of 1 to 10 m. During this activity, we sampled gases continuously for 5 hours at the west edge. Cover photo: This aerial view of Kilauea Volcano was taken in April 1984 during overflights to collect gas samples from the plume. The bluish portion of the gas plume contained a far higher density of fine-grained scoria (ash). -
Arxiv:1711.06300V1
EXPLICIT BLOCK-STRUCTURES FOR BLOCK-SYMMETRIC FIEDLER-LIKE PENCILS∗ M. I. BUENO†, M. MARTIN ‡, J. PEREZ´ §, A. SONG ¶, AND I. VIVIANO k Abstract. In the last decade, there has been a continued effort to produce families of strong linearizations of a matrix polynomial P (λ), regular and singular, with good properties, such as, being companion forms, allowing the recovery of eigen- vectors of a regular P (λ) in an easy way, allowing the computation of the minimal indices of a singular P (λ) in an easy way, etc. As a consequence of this research, families such as the family of Fiedler pencils, the family of generalized Fiedler pencils (GFP), the family of Fiedler pencils with repetition, and the family of generalized Fiedler pencils with repetition (GFPR) were con- structed. In particular, one of the goals was to find in these families structured linearizations of structured matrix polynomials. For example, if a matrix polynomial P (λ) is symmetric (Hermitian), it is convenient to use linearizations of P (λ) that are also symmetric (Hermitian). Both the family of GFP and the family of GFPR contain block-symmetric linearizations of P (λ), which are symmetric (Hermitian) when P (λ) is. Now the objective is to determine which of those structured linearizations have the best numerical properties. The main obstacle for this study is the fact that these pencils are defined implicitly as products of so-called elementary matrices. Recent papers in the literature had as a goal to provide an explicit block-structure for the pencils belonging to the family of Fiedler pencils and any of its further generalizations to solve this problem. -
Apparent Entropy of Cellular Automata
Apparent Entropy of Cellular Automata Bruno Martin Laboratoire d’Informatique et du Parallelisme,´ Ecole´ Normale Superieure´ de Lyon, 46, allee´ d’Italie, 69364 Lyon Cedex 07, France We introduce the notion of apparent entropy on cellular automata that points out how complex some configurations of the space-time diagram may appear to the human eye. We then study, theoretically, if possible, but mainly experimentally through natural examples, the relations between this notion, Wolfram’s intuition, and almost everywhere sensitivity to initial conditions. Introduction A radius-r one-dimensional cellular automaton (CA) is an infinite se- quence of identical finite-state machines (indexed by Ÿ) called cells. Each finite-state machine is in a state and these states change simultane- ously according to a local transition function: the following state of the machine is related to its own state as well as the states of its 2r neigh- bors. A configuration of an automaton is the function which associates to each cell a state. We can thus define a global transition function from the set of all the configurations to itself which associates the following configuration after one step of computation. Recently, a lot of articles proposed classifications of CAs [5, 8, 13] but the canonical reference is still Wolfram’s empirical classification [14] which has resisted numerous attempts of formalization. Among the lat- est attempts, some are based on the mathematical definitions of chaos for dynamical systems adapted to CAs thanks to Besicovitch topol- ogy [2, 6] and [11] introduces the almost everywhere sensitivity to initial conditions for this topology and compares this notion with information propagation formalization. -
Reactivity Landscape of Pyruvate Under Simulated Hydrothermal Vent
Reactivity landscape of pyruvate under simulated SEE COMMENTARY hydrothermal vent conditions Yehor Novikova and Shelley D. Copleyb,c,1 aDepartment of Chemistry and Biochemistry, bDepartment of Molecular, Cellular, and Developmental Biology, and cCooperative Institute for Research in Environmental Sciences, University of Colorado Boulder, Boulder, CO 80309 Edited by Paul G. Falkowski, Rutgers, The State University of New Jersey, New Brunswick, NJ, and approved June 14, 2013 (received for review March 14, 2013) Pyruvate is an important “hub” metabolite that is a precursor for concentrations of many components (4). Fig. 1 shows an example in amino acids, sugars, cofactors, and lipids in extant metabolic net- which the availability of catalysts for different steps in a network works. Pyruvate has been produced under simulated hydrother- results in significantly different network topologies and accumu- mal vent conditions from alkyl thiols and carbon monoxide in the lation of different products. Network topology also depends on the presence of transition metal sulfides at 250 °C [Cody GD et al. set of reagents available and the concentrations of those reagents. K (2000) Science 289(5483):1337–1340], so it is plausible that pyru- For example, the network depicted in Fig. 1 would form only and M H J M vate was formed in hydrothermal systems on the early earth. We if no were available, and would form only and if the concentration of H were very high (assuming equal rate constants report here that pyruvate reacts readily in the presence of transi- D tion metal sulfide minerals under simulated hydrothermal vent for the partitioning of between the two possible pathways). -
Introduction to Ionic Mechanisms Part I: Fundamentals of Bronsted-Lowry Acid-Base Chemistry
INTRODUCTION TO IONIC MECHANISMS PART I: FUNDAMENTALS OF BRONSTED-LOWRY ACID-BASE CHEMISTRY HYDROGEN ATOMS AND PROTONS IN ORGANIC MOLECULES - A hydrogen atom that has lost its only electron is sometimes referred to as a proton. That is because once the electron is lost, all that remains is the nucleus, which in the case of hydrogen consists of only one proton. The large majority of organic reactions, or transformations, involve breaking old bonds and forming new ones. If a covalent bond is broken heterolytically, the products are ions. In the following example, the bond between carbon and oxygen in the t-butyl alcohol molecule breaks to yield a carbocation and hydroxide ion. H3C CH3 H3C OH H3C + OH CH3 H3C A tertiary Hydroxide carbocation ion The full-headed curved arrow is being used to indicate the movement of an electron pair. In this case, the two electrons that make up the carbon-oxygen bond move towards the oxygen. The bond breaks, leaving the carbon with a positive charge, and the oxygen with a negative charge. In the absence of other factors, it is the difference in electronegativity between the two atoms that drives the direction of electron movement. When pushing arrows, remember that electrons move towards electronegative atoms, or towards areas of electron deficiency (positive, or partial positive charges). The electron pair moves towards the oxygen because it is the more electronegative of the two atoms. If we examine the outcome of heterolytic bond cleavage between oxygen and hydrogen, we see that, once again, oxygen takes the two electrons because it is the more electronegative atom. -
Introduction to Game Theory
Introduction to Game Theory June 8, 2021 In this chapter, we review some of the central concepts of Game Theory. These include the celebrated theorem of John Nash which gives us an insight into equilibrium behaviour of interactions between multiple individuals. We will also develop some of the mathematical machinery which will prove useful in our understanding of learning algorithms. Finally, we will discuss why learning is a useful concept and the real world successes that multi-agent learning has achieved in the past few years. We provide many of the important proofs in the Appendix for those readers who are interested. However, these are by no means required to understand the subsequent chapters. 1 What is a Game? What is the first image that comes to mind when you think of a game? For most people, it would be something like chess, hide and seek or perhaps even a video game. In fact, all of these can be studied under Game Theoretical terms, but we can extend our domain of interest much further. We do this by understanding what all of our above examples have in common I There are multiple players II All players want to win the game III There are rules which dictate who wins the game IV Each player's behaviour will depend on the behaviour of the others With these in place, we can extend our analysis to almost any interaction between mul- tiple players. Let us look at some examples, noticing in each, that components of a game that we discussed above show up in each of the following. -
On the Topological Convergence of Multi-Rule Sequences of Sets and Fractal Patterns
Soft Computing https://doi.org/10.1007/s00500-020-05358-w FOCUS On the topological convergence of multi-rule sequences of sets and fractal patterns Fabio Caldarola1 · Mario Maiolo2 © The Author(s) 2020 Abstract In many cases occurring in the real world and studied in science and engineering, non-homogeneous fractal forms often emerge with striking characteristics of cyclicity or periodicity. The authors, for example, have repeatedly traced these characteristics in hydrological basins, hydraulic networks, water demand, and various datasets. But, unfortunately, today we do not yet have well-developed and at the same time simple-to-use mathematical models that allow, above all scientists and engineers, to interpret these phenomena. An interesting idea was firstly proposed by Sergeyev in 2007 under the name of “blinking fractals.” In this paper we investigate from a pure geometric point of view the fractal properties, with their computational aspects, of two main examples generated by a system of multiple rules and which are enlightening for the theme. Strengthened by them, we then propose an address for an easy formalization of the concept of blinking fractal and we discuss some possible applications and future work. Keywords Fractal geometry · Hausdorff distance · Topological compactness · Convergence of sets · Möbius function · Mathematical models · Blinking fractals 1 Introduction ihara 1994; Mandelbrot 1982 and the references therein). Very interesting further links and applications are also those The word “fractal” was coined by B. Mandelbrot in 1975, between fractals, space-filling curves and number theory but they are known at least from the end of the previous (see, for instance, Caldarola 2018a; Edgar 2008; Falconer century (Cantor, von Koch, Sierpi´nski, Fatou, Hausdorff, 2014; Lapidus and van Frankenhuysen 2000), or fractals Lévy, etc.). -
Tiling Problems: from Dominoes, Checkerboards, and Mazes to Discrete Geometry
TILING PROBLEMS: FROM DOMINOES, CHECKERBOARDS, AND MAZES TO DISCRETE GEOMETRY BERKELEY MATH CIRCLE 1. Looking for a number Consider an 8 × 8 checkerboard (like the one used to play chess) and consider 32 dominoes that each may cover two adjacent squares (horizontally or vertically). Question 1 (***). What is the number N of ways in which you can cover the checkerboard with the dominoes? This question is difficult, but we will answer it at the end of the lecture, once we will have familiarized with tilings. But before we go on Question 2. Compute the number of domino tilings of a n × n checkerboard for n small. Try n = 1; 2; 3; 4; 5. Can you do n = 6...? Question 3. Can you give lower and upper bounds on the number N? Guess an estimate of the order of N. 2. Other settings We can tile other regions, replacing the n × n checkerboard by a n × m rectangle, or another polyomino (connected set of squares). We say that a region is tileable by dominos if we can cover entirely the region with dominoes, without overlap. Question 4. Are all polyominoes tileable by dominoes? Question 5. Can you find necessary conditions that polyominoes have to satisfy in order to be tileable by dominos? What if we change the tiles? Try the following questions. Question 6. Can you tile a 8×8 checkerboard from which a square has been removed with triminos 1 × 3 (horizontal or vertical)? Question 7 (*). If you answered yes to the previous question, can you tell what are the only possible squares that can be removed so that the corresponding board is tileable? Date: February 14th 2012; Instructor: Adrien Kassel (MSRI); for any questions, email me.