Associativity of triangular norms in light of web geometry
|
|
- Lucinda Mathews
- 5 years ago
- Views:
Transcription
1 Associativit of triangular norms in light of web geometr Milan Petrík 1,2 Peter Sarkoci 3 1. Institute of Computer Science, Academ of Sciences of the Czech Republic, Prague, Czech Republic 2. Center for Machine Perception, Department of Cbernetics Facult of Electrical Engineering, Czech echnical Universit Prague, Czech Republic 3. Department of Knowledge-Based Mathematical Sstems, Johannes Kepler Universit, Linz, Austria s: {petrik@cs.cas.cz petrikm@cmp.felk.cvut.cz}, peter.sarkoci@{jku.at gmail.com} Abstract he aim of this paper is to promote web geometr and, especiall, the Reidemeister closure condition as a powerful and intuitive tool characterizing associativit of the Archimedean triangular norms. In order to demonstrate its possible applications, we provide the full solution to the problem of conve combinations of nilpotent triangular norms. Kewords Archimedean triangular norm, web geometr, Reidemeister closure condition. 1 riangular norms he notion of triangular norm was originall introduced within the framework of probabilistic metric spaces [12]. Since then, triangular norms have found diverse applications in the theor of fuzz sets, fuzz decision making, in models of certain man-valued logics or in multivariate statistical analsis; for a reference see the books b Alsina, Frank, and Schweizer [5] and b Klement, Mesiar, and Pap [8]. A conjunctor is a function K : [0, 1] 2 [0, 1] which is nondecreasing in both arguments, commutative, and which satisfies the boundar condition (, 1 = for all [0, 1]. A triangular norm (shortl a t-norm, usuall denoted b is a conjunctor which satisfies the associativit equation ( (,,z= (, (, z for all,, z [0, 1]. his paper deals mainl with Archimedean t-norms. Let us recall that for ever t-norm, a number n N {0},and [0, 1] a natural power of, denoted ( (n, is defined b: { 1 ( (n ( if n =0, =, ( (n 1 (1 if n>0. A t-norm is said to be Archimedean if and onl if for ever pair, ]0, 1[, <, there eists a natural number n N such that ( (n <. A t-norm which is continuous and strictl increasing on the half-open square ]0, 1] 2 is said to be strict. A continuous t-norm is called nilpotent if and onl if for ever ]0, 1[ there eists a natural number n N such that ( (n =0. A prototpical eample of a strict and a nilpotent t-norm is the product t-norm, P (, =, and the Łukasiewicz t-norm L (, =ma{ + 1, 0}, Figure 1: Eample of a complete 3-web. respectivel. It is possible to show that a continuous Archimedean t-norm is either strict or nilpotent. A t-norm is said to be cancellative if it satisfies (a, b = (a, c b = c for ever a, b, c [0, 1], a 0. A t-norm is said to be weakl cancellative [9] if it satisfies (a, b = (a, c b = c for ever a, b, c [0, 1] with (a, b 0and (a, c 0. Ever cancellative t-norm is weakl cancellative. Under the assumption of continuit, the set of cancelative t-norms and the set of strict t-norms coincide. Under the same assumption, the set of weakl cancelative t-norms coincides with the set of Archimedean t-norms. 2 Web geometr and local loops In this section, web geometr is eplained as a tool allowing to visualize algebraic identities. In particular, it is shown that it visualizes associativit of Archimedean t-norms. A detailed introduction to the subject is given in the monograph b Blashke and Bol [6]. Also the collection of papers b Aczél, Akivis, and Goldberg [1, 2, 3] can serve as an (English introductor tet. Definition 2.1 A groupoid (or a magma is an algebraic structure G =(G, on a set G where : G G G is a binar operation. A quasigroup is a groupoid in which the 595
2 equations a = b and a = b have unique solutions for ever a and b in G. Finall, a loop is an algebraic structure L =(G,,e where (G, is a quasigroup and e is an identit element, i.e. e = = e for ever G. he definition of a quasigroup, G =(G,, allows to define the left and the right inverse. For this purpose we introduce a prefi notation g(, =.he left inverse is then defined for ever u G as 1 g(u, = such that g(, =u. Similarl, the right inverse is, for ever v G, g 1 (, v = such that g(, =v. We sa that g is invertible in, resp.. Defining a point A =(a, b G G we ma introduce a new operation, : G G G, as u v = g ( 1 g(u, b,g 1 (a, v. (2 It is eas to show that (G, is a loop with a unit element e = g(a, b; this loop is called a local loop of the quasigroup (G, at the point A =(a, b. Definition 2.2 Let M be a non-empt set and let λ 1,λ 2,λ 3 be three families of subsets of M. oelementsofm we refer as to points and to elements of λ α, α {1, 2, 3}, as to lines. Wesathatthesstem(M,λ 1,λ 2,λ 3 is a complete 3-web if and onl if the following conditions hold: g u g U A W V u w = u v v Figure 2: Operation defined on a complete 3-web. g g v 1. An point p M is incident to just one line of each famil λ α, α {1, 2, 3}. 2. An two lines of different families are incident to eactl one point of M. 3. wo distinct lines from the same famil λ α are disjoint. Note that Item 3 of the definition is redundant and can be derived from Item 1. We kept the definition in this redundant form in order to keep some of the furher considerations simpler. Usuall M is equipped with a topolog which turns the set into a manifold. In such a case the families λ α are often required to be foliations. Figure 1 shows such an eample of 3-web where M is a two dimmensional domain in plane and λ α are foliations of co-dimension 1. Notice that all the eamples of 3-webs, shown in figures of this paper, are simplified. hat means that onl some particular lines from uncountable sets λ α, α {1, 2, 3}, are drawn. Complete 3-webs are closel connected to quasigroups and, especiall, loops. Ever quasigroup defines a 3-web in the following wa. Let G =(G, be a quasigroup defined on a manifold M = G G and let, G. Let the lines of sets λ 1, λ 2,andλ 3 be given b the equations = a, = b, and = c respectivel for fied a, b, c G. he pair (M,λ α, α {1, 2, 3}, is then a 3-web. Conversel, a 3-web (M,λ α, α {1, 2, 3},onamanifold M, defines a binar operation which is invertible with respect to both operands; et this time the result is not unique. Denote one set of the lines of the 3-web as λ, second as λ, and third as λ z (the lines in the set λ z are called contour lines, let A M and let g λ and g λ be the lines passing through Figure 3: 3-webs given b the product t-norm, P, and b the Łukasiewicz t-norm, L. A; cf. Figure 2. Now we define an operation : λ z λ z λ z. ake u, v λ z, define points U, V M as U = u g and V = v g, respectivel. Let g u λ be the line passing through U and let g v λ be the line passing through V. Denote the intersection of these two lines as W = g u g v ;the line w passing through W is the result of the operation u v. It can be shown easil that the operation is invertible in both variables. Moreover, the line e λ z, passing through the point A, behaves, with respect to the oparation, as the unit element. hus (λ z, is a loop. Moreover, this loop coincides, up to an isomorphism, with the local loop of (G, at the point A. -norms, defined on the unit interval [0, 1], form neither groupoids nor loops. Nevertheless, the 3-web given b a continuous Archimedean (i.e. strict or nilpotent t-norm satisfies all the requirements given b Definition 2.2 if the manifold M is defined as the subset of [0, 1] 2 where the t-norm attains non-zero values. A manifold, induced this wa b a binar operation : [0, 1] 2 [0, 1], will be denoted as Man. hus { } M =Man = (, [0, 1] 2 (, > 0. (3 Figure 3 shows 3-webs given b P and L as eamples of a strict and a nilpotent t-norm respectivel. 596
3 b 4 4 c bc (abc a(bc Figure 4: Eample of a closed Reidemeister figure and a nonclosed Reidemeister figure. 3 Reidemeister closure condition Different tpes of 3-webs are characterized b closure conditions. hese closure conditions have their counterparts in the related loops as algebraic properties of the loop operations. In this tet we are interested in the associativit of t-norms since the associativit is the onl propert of a t-norm which cannot be intuitivel interpreted from its graph. he 3-web counterpart of the associativit is the Reidemeister closure condition. A 3-web satisfies the Reidemeister closure condition if and onl if ever Reidemeister figure in this web is closed; see Figure 4. Described in the terminolog of contour lines, the Reidemeister closure condition is as follows. Let 1, 2, 3, 4 λ and 1, 2, 3, 4 λ. If the points ( 1 2 and ( 2 1 lie on the same contour line (i.e. a line from the set λ z, and if so do the pair of points ( 1 4 and ( 2 3, and the pair of points ( 3 2 and ( 4 1, then the points ( 3 4 and ( 4 3 also lie on the same contour line. he left part of Figure 4 shows a 3-web which satisfies the Reidemeister closure condition whereas the right part shows a 3- web where the condition is violated. Let us now define a relation M M. We sa that two points A, B M are in the relation, and we write A B, if and onl if the are both elements of the same line from the set λ z. Using the language of the Reidemeister condition reads as: ( ( (1 2 ( 2 1 & ( ( 1 4 ( 2 3 & ( ( 3 2 ( 4 1 ( ( 3 4 ( 4 3. (4 4 Reidemeister closure condition and continuous Archimedean t-norms We are going to show that the level set sstem of a continuous Archimedean t-norm alwas satisfies the Reidemeister closure condition. Moreover, the Reidemeister closure condition characterizes the associativit of a continuous Archimedean t-norm. For this purpose we recall that if we rela the requirement of the associativit in the definition of a t-norm, we obtain the definition of a conjunctor. Let K be a continuous Archimedean conjunctor; for the sake of compactness we will use the infi notation: K(, = ab Figure 5: Reidemeister figure on the 3-web given b a continuous Archimedean t-norm a b 2 3 Figure 6: Illustration for the proof: an arbitrar Reidemeister figure.. Let(M,λ α be the corresponding 3-web defined on the manifold M =ManK = {(, [0, 1] 2 >0}. Such a 3-web, as in the case of the continuous Archimedean t-norms, satisfies all the requirements given b Definition 2.2. In view of the previous section, the 3-web given b satisfies the Reidemeister closure condition if and onl if the following condition holds for an 1, 2, 3, 4, 1, 2, 3, 4 M: ( (1 2 = 2 1 & ( 1 4 = & ( 3 2 = 4 1 ( 3 4 = 4 3. (5 In the sequel, this condition will be denoted b R. Figure 5 shows that R implies the associativit of in a rather intuitive wa. For an a, b, c [0, 1] 2, such that (a b c Man and a (b c Man, there can be constructed a Reidemeister figure which, as can be seen in Figure 5, is closed if and onl if (a b c = a (b c. Figure 5 shows also that, in the other wa round, if is associative then all the Reidemeister figures that touch the lines given b =1and =1are closed. In other words, R has to be satisfied for an 2, 3, 4, 2, 3, 4 [0, 1] and, at least, for 1 = 1 =1. Let us denote this weaker condition b R 1. It is now clear that K is associative if and onl if it 597
4 a b 1 a b Figure 7: Illustration for the proof. c 1 Figure 9: Illustration for the proof. c Figure 8: Illustration for the proof. Figure 10: Illustration for the proof. satisfies R 1. In order to show that is associative if and onl if it satisfies R, we need to show that R 1 R. Obviousl, R R 1. he inverse implication, R 1 R, isgiventhe following wa. Let us have a continuous Archimedean conjunctor K which satisfies R 1. Let us have a Reidemeister figure drawn on its 3-web for arbitrar 1, 2, 3, 4, 1, 2, 3, 4 [0, 1], see Figure 6. We are going to show that this figure shall be alwas closed. hanks to R 1,the Reidemeister figures, shown in Figure 7-a and Figure 7-b, are closed. Combining these two figures together it can be concluded that the Reidemeister figure in Figure 8-c is closed as well. B the same deduction, from the closedness of the Reidemeister figures in Figure 9-a and Figure 9-b it can be concluded that the Reidemeister figure in Figure 10-c is closed. Now, combining the closed Reidemeister figures in Figure 8-c and Figure 10-c, the closedness of the Reidemeister figure in Figure 6 is proven. Corollar 4.1 A continuous Archimedean conjunctor is associative (and, thus, a t-norm if and onl if it satisfies R. 5 Applications 5.1 Problem of conve combinations of t-norms he question was introduced b Alsina, Frank, and Schweizer [4]. he approach of web geometr allows to give the following, recentl published [11], answers: heorem 5.1 Let 1 and 2 be two continuous Archimedean t-norms such that Man 1 Man 2. hen no non-trivial conve combination F of 1 and 2 is a t-norm. Corollar 5.2 Combining the result of heorem 5.1 with the result given b Jenei [7], a non-trivial conve combination of two distinct nilpotent t-norms is never a t-norm. Corollar 5.3 A non-trivial conve combination of a strict and a nilpotent t-norm is never a t-norm. his is an alternative proof of the result given earlier b Ouang, Fang, and Li [10]. Acknowledgment he first author was supported b the Grant Agenc of the Czech Republic under Project 401/09/H007. References [1] J. Aczél. Quasigroups, nets and nomograms. Advances in Mathematics, 1: , [2] M. A. Akivis and V. V. Goldberg. Algebraic aspects of web geometr. Commentationes Mathematicae Universitatis Carolinae, 41(2: , [3] M. A. Akivis and V. V. Goldberg. Local algebras of a differential quasigroup. Bulletin of the American Mathematical Societ, 43(2: , [4] C. Alsina, M. J. Frank, and B. Schweizer. Problems on associative functions. Aequationes Mathematicae, 66(1 2: , [5] C. Alsina, M. J. Frank, and B. Schweizer. Associative Functions: riangular Norms and Copulas. World Scientific, Singapore, [6] W. Blaschke and G. Bol. Geometrie der Gewebe, topologische Fragen der Differentialgeometrie. Springer, Berlin, [7] S. Jenei. On the conve combination of left-continuous t-norms. Aequationes Mathematicae, 72(1 2:47 59,
5 [8] E.P.Klement,R.Mesiar,andE.Pap.riangular Norms, vol. 8 of rends in Logic. Kluwer Academic Publishers, Dordrecht, Netherlands, [9] F.Montagna, C. Noguera, and R. Horčík. On Weakl Cancellative Fuzz Logics. Journal of Logic and Computation, 16(4: , August [10] Y. Ouang and J. Fang. Some observations about the conve combinations of continuous triangular norms. Nonlinear Analsis, [11] M. Petrík, P. Sarkoci. Conve combinations of nilpotent triangular norms. Journal of Mathematical Analsis and Applications, 350(1: , Februar DOI: /j.jmaa [12] B. Schweizer, A. Sklar. Probabilistic Metric Spaces. North-Holland, Amsterdam 1983; 2nd edition: Dover Publications, Mineola, NY,
8. BOOLEAN ALGEBRAS x x
8. BOOLEAN ALGEBRAS 8.1. Definition of a Boolean Algebra There are man sstems of interest to computing scientists that have a common underling structure. It makes sense to describe such a mathematical
More informationFuzzy Topology On Fuzzy Sets: Regularity and Separation Axioms
wwwaasrcorg/aasrj American Academic & Scholarl Research Journal Vol 4, No 2, March 212 Fuzz Topolog n Fuzz Sets: Regularit and Separation Aioms AKandil 1, S Saleh 2 and MM Yakout 3 1 Mathematics Department,
More informationThe Domination Relation Between Continuous T-Norms
The Domination Relation Between Continuous T-Norms Susanne Saminger Department of Knowledge-Based Mathematical Systems, Johannes Kepler University Linz, Altenbergerstrasse 69, A-4040 Linz, Austria susanne.saminger@jku.at
More informationContinuous R-implications
Continuous R-implications Balasubramaniam Jayaram 1 Michał Baczyński 2 1. Department of Mathematics, Indian Institute of echnology Madras, Chennai 600 036, India 2. Institute of Mathematics, University
More informationA PARTIAL CHARACTERIZATION OF THE COCIRCUITS OF A SPLITTING MATROID
DEPARTMENT OF MATHEMATICS TECHNICAL REPORT A PARTIAL CHARACTERIZATION OF THE COCIRCUITS OF A SPLITTING MATROID DR. ALLAN D. MILLS MAY 2004 No. 2004-2 TENNESSEE TECHNOLOGICAL UNIVERSITY Cookeville, TN 38505
More informationThe approximation of piecewise linear membership functions and Łukasiewicz operators
Fuzz Sets and Sstems 54 25 275 286 www.elsevier.com/locate/fss The approimation of piecewise linear membership functions and Łukasiewicz operators József Dombi, Zsolt Gera Universit of Szeged, Institute
More informationOn Range and Reflecting Functions About the Line y = mx
On Range and Reflecting Functions About the Line = m Scott J. Beslin Brian K. Heck Jerem J. Becnel Dept.of Mathematics and Dept. of Mathematics and Dept. of Mathematics and Computer Science Computer Science
More informationTRIANGULAR NORMS WITH CONTINUOUS DIAGONALS
Tatra Mt. Math. Publ. 6 (999), 87 95 TRIANGULAR NORMS WITH CONTINUOUS DIAGONALS Josef Tkadlec ABSTRACT. It is an old open question whether a t-norm with a continuous diagonal must be continuous [7]. We
More informationYou don't have to be a mathematician to have a feel for numbers. John Forbes Nash, Jr.
Course Title: Real Analsis Course Code: MTH3 Course instructor: Dr. Atiq ur Rehman Class: MSc-II Course URL: www.mathcit.org/atiq/fa5-mth3 You don't have to be a mathematician to have a feel for numbers.
More informationChapter 4 Analytic Trigonometry
Analtic Trigonometr Chapter Analtic Trigonometr Inverse Trigonometric Functions The trigonometric functions act as an operator on the variable (angle, resulting in an output value Suppose this process
More informationLeft-continuous t-norms in Fuzzy Logic: an Overview
Left-continuous t-norms in Fuzzy Logic: an Overview János Fodor Dept. of Biomathematics and Informatics, Faculty of Veterinary Sci. Szent István University, István u. 2, H-1078 Budapest, Hungary E-mail:
More informationTensor products in Riesz space theory
Tensor products in Riesz space theor Jan van Waaij Master thesis defended on Jul 16, 2013 Thesis advisors dr. O.W. van Gaans dr. M.F.E. de Jeu Mathematical Institute, Universit of Leiden CONTENTS 2 Contents
More informationCCSSM Algebra: Equations
CCSSM Algebra: Equations. Reasoning with Equations and Inequalities (A-REI) Eplain each step in solving a simple equation as following from the equalit of numbers asserted at the previous step, starting
More information3.7 InveRSe FUnCTIOnS
CHAPTER functions learning ObjeCTIveS In this section, ou will: Verif inverse functions. Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
More informationNew types of bipolar fuzzy sets in -semihypergroups
Songlanaarin J. Sci. Technol. 38 (2) 119-127 Mar. - pr. 2016 http://www.sjst.psu.ac.th Original rticle New tpes of bipolar fu sets in -semihpergroups Naveed Yaqoob 1 * Muhammad slam 2 Inaatur Rehman 3
More informationRough neutrosophic set in a lattice
nternational Journal of ppld esearch 6; 5: 43-5 SSN Print: 394-75 SSN Online: 394-5869 mpact actor: 5 J 6; 5: 43-5 wwwallresearchjournalcom eceived: -3-6 ccepted: -4-6 rockiarani Nirmala College for Women
More informationREMARKS ON TWO PRODUCT LIKE CONSTRUCTIONS FOR COPULAS
K Y B E R N E T I K A V O L U M E 4 3 2 0 0 7 ), N U M B E R 2, P A G E S 2 3 5 2 4 4 REMARKS ON TWO PRODUCT LIKE CONSTRUCTIONS FOR COPULAS Fabrizio Durante, Erich Peter Klement, José Juan Quesada-Molina
More informationINTRODUCTION TO DIFFERENTIAL EQUATIONS
INTRODUCTION TO DIFFERENTIAL EQUATIONS. Definitions and Terminolog. Initial-Value Problems.3 Differential Equations as Mathematical Models CHAPTER IN REVIEW The words differential and equations certainl
More informationUNIVERSIDAD CARLOS III DE MADRID MATHEMATICS II EXERCISES (SOLUTIONS )
UNIVERSIDAD CARLOS III DE MADRID MATHEMATICS II EXERCISES (SOLUTIONS ) CHAPTER : Limits and continuit of functions in R n. -. Sketch the following subsets of R. Sketch their boundar and the interior. Stud
More informationFlows and Connectivity
Chapter 4 Flows and Connectivit 4. Network Flows Capacit Networks A network N is a connected weighted loopless graph (G,w) with two specified vertices and, called the source and the sink, respectivel,
More informationLines and Planes 1. x(t) = at + b y(t) = ct + d
1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. Will we call this the ABC form. Recall that the slope-intercept
More information12.1 Systems of Linear equations: Substitution and Elimination
. Sstems of Linear equations: Substitution and Elimination Sstems of two linear equations in two variables A sstem of equations is a collection of two or more equations. A solution of a sstem in two variables
More informationAn analytic proof of the theorems of Pappus and Desargues
Note di Matematica 22, n. 1, 2003, 99 106. An analtic proof of the theorems of Pappus and Desargues Erwin Kleinfeld and Tuong Ton-That Department of Mathematics, The Universit of Iowa, Iowa Cit, IA 52242,
More informationGeneral Vector Spaces
CHAPTER 4 General Vector Spaces CHAPTER CONTENTS 4. Real Vector Spaces 83 4. Subspaces 9 4.3 Linear Independence 4.4 Coordinates and Basis 4.5 Dimension 4.6 Change of Basis 9 4.7 Row Space, Column Space,
More informationRELATIONS AND FUNCTIONS through
RELATIONS AND FUNCTIONS 11.1.2 through 11.1. Relations and Functions establish a correspondence between the input values (usuall ) and the output values (usuall ) according to the particular relation or
More informationOn the Intersections of QL-Implications with (S, N)- and R-Implications
On the Intersections of QL-Implications with (S, N)- and R-Implications Balasubramaniam Jayaram Dept. of Mathematics and Computer Sciences, Sri Sathya Sai Institute of Higher Learning, Prasanthi Nilayam,
More informationThe Sigmoidal Approximation of Łukasiewicz Operators
The Sigmoidal Approimation of Łukasiewicz Operators József Dombi, Zsolt Gera Universit of Szeged, Institute of Informatics e-mail: {dombi gera}@inf.u-szeged.hu Abstract: In this paper we propose an approimation
More informationResearch Article On Sharp Triangle Inequalities in Banach Spaces II
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 323609, 17 pages doi:10.1155/2010/323609 Research Article On Sharp Triangle Inequalities in Banach Spaces
More informationChapter 5: Systems of Equations
Chapter : Sstems of Equations Section.: Sstems in Two Variables... 0 Section. Eercises... 9 Section.: Sstems in Three Variables... Section. Eercises... Section.: Linear Inequalities... Section.: Eercises.
More informationIntroduction to Differential Equations. National Chiao Tung University Chun-Jen Tsai 9/14/2011
Introduction to Differential Equations National Chiao Tung Universit Chun-Jen Tsai 9/14/011 Differential Equations Definition: An equation containing the derivatives of one or more dependent variables,
More information10 Back to planar nonlinear systems
10 Back to planar nonlinear sstems 10.1 Near the equilibria Recall that I started talking about the Lotka Volterra model as a motivation to stud sstems of two first order autonomous equations of the form
More information14.1 Systems of Linear Equations in Two Variables
86 Chapter 1 Sstems of Equations and Matrices 1.1 Sstems of Linear Equations in Two Variables Use the method of substitution to solve sstems of equations in two variables. Use the method of elimination
More informationASSOCIATIVE n DIMENSIONAL COPULAS
K Y BERNETIKA VOLUM E 47 ( 2011), NUMBER 1, P AGES 93 99 ASSOCIATIVE n DIMENSIONAL COPULAS Andrea Stupňanová and Anna Kolesárová The associativity of n-dimensional copulas in the sense of Post is studied.
More informationMAT389 Fall 2016, Problem Set 2
MAT389 Fall 2016, Problem Set 2 Circles in the Riemann sphere Recall that the Riemann sphere is defined as the set Let P be the plane defined b Σ = { (a, b, c) R 3 a 2 + b 2 + c 2 = 1 } P = { (a, b, c)
More informationSystems of Linear Equations: Solving by Graphing
8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From
More informationCycle Structure in Automata and the Holonomy Decomposition
Ccle Structure in Automata and the Holonom Decomposition Attila Egri-Nag and Chrstopher L. Nehaniv School of Computer Science Universit of Hertfordshire College Lane, Hatfield, Herts AL10 9AB United Kingdom
More informationDecision tables and decision spaces
Abstract Decision tables and decision spaces Z. Pawlak 1 Abstract. In this paper an Euclidean space, called a decision space is associated with ever decision table. This can be viewed as a generalization
More informationS-MEASURES, T -MEASURES AND DISTINGUISHED CLASSES OF FUZZY MEASURES
K Y B E R N E T I K A V O L U M E 4 2 ( 2 0 0 6 ), N U M B E R 3, P A G E S 3 6 7 3 7 8 S-MEASURES, T -MEASURES AND DISTINGUISHED CLASSES OF FUZZY MEASURES Peter Struk and Andrea Stupňanová S-measures
More information2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a:
SECTION.5 CONTINUITY 9.5 CONTINUITY We noticed in Section.3 that the it of a function as approaches a can often be found simpl b calculating the value of the function at a. Functions with this propert
More informationFunctions. Introduction
Functions,00 P,000 00 0 70 7 80 8 0 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of work b MeasuringWorth)
More information4 Inverse function theorem
Tel Aviv Universit, 2013/14 Analsis-III,IV 53 4 Inverse function theorem 4a What is the problem................ 53 4b Simple observations before the theorem..... 54 4c The theorem.....................
More informationLESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II
LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,
More informationConstant 2-labelling of a graph
Constant 2-labelling of a graph S. Gravier, and E. Vandomme June 18, 2012 Abstract We introduce the concept of constant 2-labelling of a graph and show how it can be used to obtain periodic sphere packing.
More informationMAT 1275: Introduction to Mathematical Analysis. Graphs and Simplest Equations for Basic Trigonometric Functions. y=sin( x) Function
MAT 275: Introduction to Mathematical Analsis Dr. A. Rozenblum Graphs and Simplest Equations for Basic Trigonometric Functions We consider here three basic functions: sine, cosine and tangent. For them,
More information8.1 Exponents and Roots
Section 8. Eponents and Roots 75 8. Eponents and Roots Before defining the net famil of functions, the eponential functions, we will need to discuss eponent notation in detail. As we shall see, eponents
More informationf x, y x 2 y 2 2x 6y 14. Then
SECTION 11.7 MAXIMUM AND MINIMUM VALUES 645 absolute minimum FIGURE 1 local maimum local minimum absolute maimum Look at the hills and valles in the graph of f shown in Figure 1. There are two points a,
More informationDominance in the family of Sugeno-Weber t-norms
Dominance in the family of Sugeno-Weber t-norms Manuel KAUERS Research Institute for Symbolic Computation Johannes Kepler University Linz, Altenbergerstrasse 69, A-4040 Linz, Austria e-mail: mkauers@risc.jku.at
More informationChapter 6. Self-Adjusting Data Structures
Chapter 6 Self-Adjusting Data Structures Chapter 5 describes a data structure that is able to achieve an epected quer time that is proportional to the entrop of the quer distribution. The most interesting
More informationStrain Transformation and Rosette Gage Theory
Strain Transformation and Rosette Gage Theor It is often desired to measure the full state of strain on the surface of a part, that is to measure not onl the two etensional strains, and, but also the shear
More informationSection 1.2: A Catalog of Functions
Section 1.: A Catalog of Functions As we discussed in the last section, in the sciences, we often tr to find an equation which models some given phenomenon in the real world - for eample, temperature as
More information(T; N) and Residual Fuzzy Co-Implication in Dual Heyting Algebra with Applications
Preprints (www.preprints.org) O PEER-REVIEWED Posted: 3 August 2016 Article (; ) Residual Fuzzy Co-Implication in Dual Heyting Algebra with Applications Iqbal H. ebril Department of Mathematics cience
More informationSolutions to Two Interesting Problems
Solutions to Two Interesting Problems Save the Lemming On each square of an n n chessboard is an arrow pointing to one of its eight neighbors (or off the board, if it s an edge square). However, arrows
More informationProperties of T Anti Fuzzy Ideal of l Rings
International Journal of Computational Science and Mathematics. ISSN 0974-3189 Volume 8, Number 1 (2016, pp. 9-17 International esearch Publication House http://www.irphouse.com Properties of T nti Fuzz
More informationComparison of two versions of the Ferrers property of fuzzy interval orders
Comparison of two versions of the Ferrers property of fuzzy interval orders Susana Díaz 1 Bernard De Baets 2 Susana Montes 1 1.Dept. Statistics and O. R., University of Oviedo 2.Dept. Appl. Math., Biometrics
More informationIntuitionistic Fuzzy Sets - An Alternative Look
Intuitionistic Fuzzy Sets - An Alternative Look Anna Pankowska and Maciej Wygralak Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87, 61-614 Poznań, Poland e-mail: wygralak@math.amu.edu.pl
More information11.4 Polar Coordinates
11. Polar Coordinates 917 11. Polar Coordinates In Section 1.1, we introduced the Cartesian coordinates of a point in the plane as a means of assigning ordered pairs of numbers to points in the plane.
More informationThe Steiner Ratio for Obstacle-Avoiding Rectilinear Steiner Trees
The Steiner Ratio for Obstacle-Avoiding Rectilinear Steiner Trees Anna Lubiw Mina Razaghpour Abstract We consider the problem of finding a shortest rectilinear Steiner tree for a given set of pointn the
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 2, Issue 2, Article 15, 21 ON SOME FUNDAMENTAL INTEGRAL INEQUALITIES AND THEIR DISCRETE ANALOGUES B.G. PACHPATTE DEPARTMENT
More informationA sharp Blaschke Santaló inequality for α-concave functions
Noname manuscript No will be inserted b the editor) A sharp Blaschke Santaló inequalit for α-concave functions Liran Rotem the date of receipt and acceptance should be inserted later Abstract We define
More informationLimits and Continuous Functions. 2.2 Introduction to Limits. We first interpret limits loosely. We write. lim f(x) = L
2 Limits and Continuous Functions 2.2 Introduction to Limits We first interpret limits loosel. We write lim f() = L and sa the limit of f() as approaches c, equals L if we can make the values of f() arbitraril
More informationInvariant monotone vector fields on Riemannian manifolds
Nonlinear Analsis 70 (2009) 850 86 www.elsevier.com/locate/na Invariant monotone vector fields on Riemannian manifolds A. Barani, M.R. Pouraevali Department of Mathematics, Universit of Isfahan, P.O.Box
More informationUltra neutrosophic crisp sets and relations
New Trends in Neutrosophic Theor and Applications HEWAYDA ELGHAWALBY 1, A. A. SALAMA 2 1 Facult of Engineering, Port-Said Universit, Egpt. E-mail: hewada2011@eng.psu.edu.eg 2 Facult of Science, Port-Said
More informationd max (P,Q) = max{ x 1 x 2, y 1 y 2 }. Check is it d max a distance function. points in R 2, and let d : R 2 R 2 R denote a
2 Metric geometr At this level there are two fundamental approaches to the tpe of geometr we are studing. The first, called the snthetic approach, involves deciding what are the important properties of
More informationFUZZY relation equations were introduced by E. Sanchez
Position Papers of the Federated Conference on Computer Science and Information Systems pp. 19 23 DOI: 1.15439/216F564 ACSIS, Vol. 9. ISSN 23-5963 Computing the minimal solutions of finite fuzzy relation
More informationGreen s Theorem Jeremy Orloff
Green s Theorem Jerem Orloff Line integrals and Green s theorem. Vector Fields Vector notation. In 8.4 we will mostl use the notation (v) = (a, b) for vectors. The other common notation (v) = ai + bj runs
More informationAggregation and Non-Contradiction
Aggregation and Non-Contradiction Ana Pradera Dept. de Informática, Estadística y Telemática Universidad Rey Juan Carlos. 28933 Móstoles. Madrid. Spain ana.pradera@urjc.es Enric Trillas Dept. de Inteligencia
More informationQUADRATIC AND CONVEX MINIMAX CLASSIFICATION PROBLEMS
Journal of the Operations Research Societ of Japan 008, Vol. 51, No., 191-01 QUADRATIC AND CONVEX MINIMAX CLASSIFICATION PROBLEMS Tomonari Kitahara Shinji Mizuno Kazuhide Nakata Toko Institute of Technolog
More informationOPTIMAL CONTROL FOR A PARABOLIC SYSTEM MODELLING CHEMOTAXIS
Trends in Mathematics Information Center for Mathematical Sciences Volume 6, Number 1, June, 23, Pages 45 49 OPTIMAL CONTROL FOR A PARABOLIC SYSTEM MODELLING CHEMOTAXIS SANG UK RYU Abstract. We stud the
More informationMAXIMA & MINIMA The single-variable definitions and theorems relating to extermals can be extended to apply to multivariable calculus.
MAXIMA & MINIMA The single-variable definitions and theorems relating to etermals can be etended to appl to multivariable calculus. ( ) is a Relative Maimum if there ( ) such that ( ) f(, for all points
More informationSection 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y
Paul J. Bruillard MATH 0.970 Problem Set 6 An Introduction to Abstract Mathematics R. Bond and W. Keane Section 3.1: 3b,c,e,i, 4bd, 6, 9, 15, 16, 18c,e, 19a, 0, 1b Section 3.: 1f,i, e, 6, 1e,f,h, 13e,
More informationVECTORS IN THREE DIMENSIONS
1 CHAPTER 2. BASIC TRIGONOMETRY 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW VECTORS IN THREE DIMENSIONS 1 Vectors in Two Dimensions A vector is an object which has magnitude
More informationSEPARABLE EQUATIONS 2.2
46 CHAPTER FIRST-ORDER DIFFERENTIAL EQUATIONS 4. Chemical Reactions When certain kinds of chemicals are combined, the rate at which the new compound is formed is modeled b the autonomous differential equation
More information2.2 SEPARABLE VARIABLES
44 CHAPTER FIRST-ORDER DIFFERENTIAL EQUATIONS 6 Consider the autonomous DE 6 Use our ideas from Problem 5 to find intervals on the -ais for which solution curves are concave up and intervals for which
More informationIntroduction to Vector Spaces Linear Algebra, Spring 2011
Introduction to Vector Spaces Linear Algebra, Spring 2011 You probabl have heard the word vector before, perhaps in the contet of Calculus III or phsics. You probabl think of a vector like this: 5 3 or
More informationAnalysis of additive generators of fuzzy operations represented by rational functions
Journal of Physics: Conference Series PAPER OPEN ACCESS Analysis of additive generators of fuzzy operations represented by rational functions To cite this article: T M Ledeneva 018 J. Phys.: Conf. Ser.
More informationResearch Article Chaotic Attractor Generation via a Simple Linear Time-Varying System
Discrete Dnamics in Nature and Societ Volume, Article ID 836, 8 pages doi:.//836 Research Article Chaotic Attractor Generation via a Simple Linear Time-Varing Sstem Baiu Ou and Desheng Liu Department of
More informationMinimal reductions of monomial ideals
ISSN: 1401-5617 Minimal reductions of monomial ideals Veronica Crispin Quiñonez Research Reports in Mathematics Number 10, 2004 Department of Mathematics Stocholm Universit Electronic versions of this
More informationInterval Valued Fuzzy Sets from Continuous Archimedean. Triangular Norms. Taner Bilgic and I. Burhan Turksen. University of Toronto.
Interval Valued Fuzzy Sets from Continuous Archimedean Triangular Norms Taner Bilgic and I. Burhan Turksen Department of Industrial Engineering University of Toronto Toronto, Ontario, M5S 1A4 Canada bilgic@ie.utoronto.ca,
More informationAffine transformations
Reading Optional reading: Affine transformations Brian Curless CSE 557 Autumn 207 Angel and Shreiner: 3., 3.7-3. Marschner and Shirle: 2.3, 2.4.-2.4.4, 6..-6..4, 6.2., 6.3 Further reading: Angel, the rest
More informationFuzzy Sets, Fuzzy Logic, and Fuzzy Systems
Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems SSIE 617 Fall 2008 Radim BELOHLAVEK Dept. Systems Sci. & Industrial Eng. Watson School of Eng. and Applied Sci. Binghamton University SUNY Radim Belohlavek (SSIE
More informationCoordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general
A Sketch graphs of = a m b n c where m = or and n = or B Reciprocal graphs C Graphs of circles and ellipses D Graphs of hperbolas E Partial fractions F Sketch graphs using partial fractions Coordinate
More information6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
More informationKybernetika. Margarita Mas; Miquel Monserrat; Joan Torrens QL-implications versus D-implications. Terms of use:
Kybernetika Margarita Mas; Miquel Monserrat; Joan Torrens QL-implications versus D-implications Kybernetika, Vol. 42 (2006), No. 3, 35--366 Persistent URL: http://dml.cz/dmlcz/3579 Terms of use: Institute
More information15. Eigenvalues, Eigenvectors
5 Eigenvalues, Eigenvectors Matri of a Linear Transformation Consider a linear ( transformation ) L : a b R 2 R 2 Suppose we know that L and L Then c d because of linearit, we can determine what L does
More informationA Note on Clean Matrices in M 2 ( Z)
International Journal of Algebra Vol. 3 2009 no. 5 241-248 A Note on Clean Matrices in M 2 ( Z) K. N. Rajeswari 1 School of Mathematics Vigan Bhawan Khandwa Road Indore 452 017 India Rafia Aziz Medi-Caps
More information( 7, 3) means x = 7 and y = 3. ( 7, 3) works in both equations so. Section 5 1: Solving a System of Linear Equations by Graphing
Section 5 : Solving a Sstem of Linear Equations b Graphing What is a sstem of Linear Equations? A sstem of linear equations is a list of two or more linear equations that each represents the graph of a
More informationResearch Article Coupled Fixed Point Theorems for a Pair of Weakly Compatible Maps along with CLRg Property in Fuzzy Metric Spaces
Applied Mathematics Volume 2012, Article ID 961210, 13 pages doi:10.1155/2012/961210 Research Article Coupled Fixed Point Theorems for a Pair of Weakly Compatible Maps along with CLRg Property in Fuzzy
More information8.7 Systems of Non-Linear Equations and Inequalities
8.7 Sstems of Non-Linear Equations and Inequalities 67 8.7 Sstems of Non-Linear Equations and Inequalities In this section, we stud sstems of non-linear equations and inequalities. Unlike the sstems of
More informationPath Embeddings with Prescribed Edge in the Balanced Hypercube Network
S S smmetr Communication Path Embeddings with Prescribed Edge in the Balanced Hpercube Network Dan Chen, Zhongzhou Lu, Zebang Shen, Gaofeng Zhang, Chong Chen and Qingguo Zhou * School of Information Science
More informationSpace frames. b) R z φ z. R x. Figure 1 Sign convention: a) Displacements; b) Reactions
Lecture notes: Structural Analsis II Space frames I. asic concepts. The design of a building is generall accomplished b considering the structure as an assemblage of planar frames, each of which is designed
More informationABOUT THE WEAK EFFICIENCIES IN VECTOR OPTIMIZATION
ABOUT THE WEAK EFFICIENCIES IN VECTOR OPTIMIZATION CRISTINA STAMATE Institute of Mathematics 8, Carol I street, Iasi 6600, Romania cstamate@mail.com Abstract: We present the principal properties of the
More informationLinear algebra in turn is built on two basic elements, MATRICES and VECTORS.
M-Lecture():.-. Linear algebra provides concepts that are crucial to man areas of information technolog and computing, including: Graphics Image processing Crptograph Machine learning Computer vision Optimiation
More informationDirectional Monotonicity of Fuzzy Implications
Acta Polytechnica Hungarica Vol. 14, No. 5, 2017 Directional Monotonicity of Fuzzy Implications Katarzyna Miś Institute of Mathematics, University of Silesia in Katowice Bankowa 14, 40-007 Katowice, Poland,
More information18.02 Review Jeremy Orloff. 1 Review of multivariable calculus (18.02) constructs
18.02 eview Jerem Orloff 1 eview of multivariable calculus (18.02) constructs 1.1 Introduction These notes are a terse summar of what we ll need from multivariable calculus. If, after reading these, some
More informationThe Entropy Power Inequality and Mrs. Gerber s Lemma for Groups of order 2 n
The Entrop Power Inequalit and Mrs. Gerber s Lemma for Groups of order 2 n Varun Jog EECS, UC Berkele Berkele, CA-94720 Email: varunjog@eecs.berkele.edu Venkat Anantharam EECS, UC Berkele Berkele, CA-94720
More informationFebruary 11, JEL Classification: C72, D43, D44 Keywords: Discontinuous games, Bertrand game, Toehold, First- and Second- Price Auctions
Eistence of Mied Strateg Equilibria in a Class of Discontinuous Games with Unbounded Strateg Sets. Gian Luigi Albano and Aleander Matros Department of Economics and ELSE Universit College London Februar
More informationFunctions of Several Variables
Chapter 1 Functions of Several Variables 1.1 Introduction A real valued function of n variables is a function f : R, where the domain is a subset of R n. So: for each ( 1,,..., n ) in, the value of f is
More informationLecture 5. Equations of Lines and Planes. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.
Lecture 5 Equations of Lines and Planes Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 Universit of Massachusetts Februar 6, 2018 (2) Upcoming midterm eam First midterm: Wednesda Feb. 21, 7:00-9:00
More informationOn Some I-Convergent Double Sequence Spaces Defined by a Modulus Function
Engineering, 03, 5, 35-40 http://dxdoiorg/0436/eng0355a006 Published Online Ma 03 (http://wwwscirporg/journal/eng) On Some -Convergent Double Sequence Spaces Deined b a Modulus Function Vakeel A Khan,
More information4.3 Exercises. local maximum or minimum. The second derivative is. e 1 x 2x 1. f x x 2 e 1 x 1 x 2 e 1 x 2x x 4
SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH 297 local maimum or minimum. The second derivative is f 2 e 2 e 2 4 e 2 4 Since e and 4, we have f when and when 2 f. So the curve is concave downward
More information