DOCSLIB.ORG

Groups for PKE: a Quick Primer Discrete Log and DDH Assumptions, Candidate Trapdoor One-Way Permutations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Groups for PKE: a Quick Primer Discrete Log and DDH Assumptions, Candidate Trapdoor One-Way Permutations Groups for PKE: A Quick Primer Discrete Log and DDH Assumptions, Candidate Trapdoor One-Way Permutations 1 Groups, a primer 2 Groups, a primer A set G (for us finite, unless otherwise specified) 2 Groups, a primer A set G (for us finite, unless otherwise specified) A binary operation *: G×G→G 2 Groups, a primer A set G (for us finite, unless otherwise specified) A binary operation *: G×G→G Abuse of notation: G instead of (G,*) 2 Groups, a primer A set G (for us finite, unless otherwise specified) A binary operation *: G×G→G Abuse of notation: G instead of (G,*) Sometimes called addition, sometimes called multiplication (depending on the set/operation) 2 Groups, a primer A set G (for us finite, unless otherwise specified) A binary operation *: G×G→G Abuse of notation: G instead of (G,*) Sometimes called addition, sometimes called multiplication (depending on the set/operation) Properties: 2 Groups, a primer A set G (for us finite, unless otherwise specified) A binary operation *: G×G→G Abuse of notation: G instead of (G,*) Sometimes called addition, sometimes called multiplication (depending on the set/operation) Properties: Associativity, 2 Groups, a primer A set G (for us finite, unless otherwise specified) A binary operation *: G×G→G Abuse of notation: G instead of (G,*) Sometimes called addition, sometimes called multiplication (depending on the set/operation) Properties: Associativity, Existence of identity, 2 Groups, a primer A set G (for us finite, unless otherwise specified) A binary operation *: G×G→G Abuse of notation: G instead of (G,*) Sometimes called addition, sometimes called multiplication (depending on the set/operation) Properties: Associativity, Existence of identity, Invertibility 2 Groups, a primer A set G (for us finite, unless otherwise specified) A binary operation *: G×G→G Abuse of notation: G instead of (G,*) Sometimes called addition, sometimes called multiplication (depending on the set/operation) Properties: Associativity, Existence of identity, Invertibility Also for us (unless specified otherwise) Commutativity 2 Groups, a primer A set G (for us finite, unless otherwise specified) A binary operation *: G×G→G Abuse of notation: G instead of (G,*) Sometimes called addition, sometimes called multiplication (depending on the set/operation) Properties: Associativity, Existence of identity, Invertibility Also for us (unless specified otherwise) Commutativity Examples: Z (integers, with addition operation; infinite group), n ZN (integers modulo N), G (G group; coordinate-wise op) 2 Groups, a primer 3 Groups, a primer Computation with groups 3 Groups, a primer Computation with groups For us, finite, but typically large: i.e., exponential in k 3 Groups, a primer Computation with groups For us, finite, but typically large: i.e., exponential in k Some compact representation of the elements (poly(k) bits) 3 Groups, a primer Computation with groups For us, finite, but typically large: i.e., exponential in k Some compact representation of the elements (poly(k) bits) Efficient algorithms for: 3 Groups, a primer Computation with groups For us, finite, but typically large: i.e., exponential in k Some compact representation of the elements (poly(k) bits) Efficient algorithms for: Checking if an element (in standard rep.) is in the group 3 Groups, a primer Computation with groups For us, finite, but typically large: i.e., exponential in k Some compact representation of the elements (poly(k) bits) Efficient algorithms for: Checking if an element (in standard rep.) is in the group Group operation (on elements in standard rep.) 3 Groups, a primer Computation with groups For us, finite, but typically large: i.e., exponential in k Some compact representation of the elements (poly(k) bits) Efficient algorithms for: Checking if an element (in standard rep.) is in the group Group operation (on elements in standard rep.) Inverting 3 Groups, a primer Computation with groups For us, finite, but typically large: i.e., exponential in k Some compact representation of the elements (poly(k) bits) Efficient algorithms for: Checking if an element (in standard rep.) is in the group Group operation (on elements in standard rep.) Inverting Sampling a random element (almost) uniformly 3 Groups, a primer Computation with groups For us, finite, but typically large: i.e., exponential in k Some compact representation of the elements (poly(k) bits) Efficient algorithms for: Checking if an element (in standard rep.) is in the group Group operation (on elements in standard rep.) Inverting Sampling a random element (almost) uniformly Group itself represented by these algorithms 3 Groups, a primer Computation with groups For us, finite, but typically large: i.e., exponential in k Some compact representation of the elements (poly(k) bits) Efficient algorithms for: Checking if an element (in standard rep.) is in the group Group operation (on elements in standard rep.) Inverting Sampling a random element (almost) uniformly Group itself represented by these algorithms For collection of groups: GroupGen, an algorithm to select a group 3 Groups, a primer 4 Groups, a primer Additive notation: X+Y, identity denoted as 0, X+X denoted by 2X 4 Groups, a primer Additive notation: X+Y, identity denoted as 0, X+X denoted by 2X Multiplicative notation: XY, identity 1, XX denoted as X2 4 Groups, a primer Additive notation: X+Y, identity denoted as 0, X+X denoted by 2X Multiplicative notation: XY, identity 1, XX denoted as X2 Order of a group G: |G| = number of elements in G 4 Groups, a primer Additive notation: X+Y, identity denoted as 0, X+X denoted by 2X Multiplicative notation: XY, identity 1, XX denoted as X2 Order of a group G: |G| = number of elements in G Cyclic group (finite, in multiplicative notation): there is one element g such that G = {g0, g1, g2, ... g|G|-1} 4 Groups, a primer Additive notation: X+Y, identity denoted as 0, X+X denoted by 2X Multiplicative notation: XY, identity 1, XX denoted as X2 Order of a group G: |G| = number of elements in G Cyclic group (finite, in multiplicative notation): there is one element g such that G = {g0, g1, g2, ... g|G|-1} 4 Groups, a primer Additive notation: X+Y, identity denoted as 0, X+X denoted by 2X Multiplicative notation: XY, identity 1, XX denoted as X2 Order of a group G: |G| = number of elements in G Cyclic group (finite, in multiplicative notation): there is one element g such that G = {g0, g1, g2, ... g|G|-1} g0 4 Groups, a primer Additive notation: X+Y, identity denoted as 0, X+X denoted by 2X Multiplicative notation: XY, identity 1, XX denoted as X2 Order of a group G: |G| = number of elements in G Cyclic group (finite, in multiplicative notation): there is one element g such that G = {g0, g1, g2, ... g|G|-1} 0 g g1 4 Groups, a primer Additive notation: X+Y, identity denoted as 0, X+X denoted by 2X Multiplicative notation: XY, identity 1, XX denoted as X2 Order of a group G: |G| = number of elements in G Cyclic group (finite, in multiplicative notation): there is one element g such that G = {g0, g1, g2, ... g|G|-1} 0 g g1 g2 4 Groups, a primer Additive notation: X+Y, identity denoted as 0, X+X denoted by 2X Multiplicative notation: XY, identity 1, XX denoted as X2 Order of a group G: |G| = number of elements in G Cyclic group (finite, in multiplicative notation): there is one element g such that G = {g0, g1, g2, ... g|G|-1} 0 g g1 g2 g3 . 4 Groups, a primer Additive notation: X+Y, identity denoted as 0, X+X denoted by 2X Multiplicative notation: XY, identity 1, XX denoted as X2 Order of a group G: |G| = number of elements in G Cyclic group (finite, in multiplicative notation): there is one element g such that G = {g0, g1, g2, ... g|G|-1} 0 gN-1 g g1 2 gN-2 g .. g3 .. .. 4 Groups, a primer Additive notation: X+Y, identity denoted as 0, X+X denoted by 2X Multiplicative notation: XY, identity 1, XX denoted as X2 Order of a group G: |G| = number of elements in G Cyclic group (finite, in multiplicative notation): there is one element g such that G = {g0, g1, g2, ... g|G|-1} 0 gN-1 g g1 2 gN-2 g e.g. N, with say g=1 (additive group) 3 Z .. g .. .. 4 Groups, a primer Additive notation: X+Y, identity denoted as 0, X+X denoted by 2X Multiplicative notation: XY, identity 1, XX denoted as X2 Order of a group G: |G| = number of elements in G Cyclic group (finite, in multiplicative notation): there is one element g such that G = {g0, g1, g2, ... g|G|-1} 0 gN-1 g g1 2 gN-2 g e.g. ZN, with say g=1 (additive group) . g3 . or any g s.t. gcd(g,N) = 1 . 4 Groups, a primer Additive notation: X+Y, identity denoted as 0, X+X denoted by 2X Multiplicative notation: XY, identity 1, XX denoted as X2 Order of a group G: |G| = number of elements in G Cyclic group (finite, in multiplicative notation): there is one element g such that G = {g0, g1, g2, ... g|G|-1} 0 gN-1 g g1 2 gN-2 g e.g. ZN, with say g=1 (additive group) . g3 . or any g s.t. gcd(g,N) = 1 . Number of generators of ZN =: φ(N) 4 Groups, by examples 0 gN-1 g g1 gN-2 g2 . 3 . g . 5 Groups, by examples 0 gN-1 g g1 gN-2 g2 . 3 . g . * ZN = (generators of ZN, multiplication mod N) 5 Groups, by examples 0 gN-1 g g1 gN-2 g2 .
Load more

Recommended publications
  • On Stochastic Distributions and Currents
    NISSUNA UMANA INVESTIGAZIONE SI PUO DIMANDARE VERA SCIENZIA S’ESSA NON PASSA PER LE MATEMATICHE DIMOSTRAZIONI LEONARDO DA VINCI vol. 4 no. 3-4 2016 Mathematics and Mechanics of Complex Systems VINCENZO CAPASSO AND FRANCO FLANDOLI ON STOCHASTIC DISTRIBUTIONS AND CURRENTS msp MATHEMATICS AND MECHANICS OF COMPLEX SYSTEMS Vol. 4, No. 3-4, 2016 dx.doi.org/10.2140/memocs.2016.4.373 ∩ MM ON STOCHASTIC DISTRIBUTIONS AND CURRENTS VINCENZO CAPASSO AND FRANCO FLANDOLI Dedicated to Lucio Russo, on the occasion of his 70th birthday In many applications, it is of great importance to handle random closed sets of different (even though integer) Hausdorff dimensions, including local infor- mation about initial conditions and growth parameters. Following a standard approach in geometric measure theory, such sets may be described in terms of suitable measures. For a random closed set of lower dimension with respect to the environment space, the relevant measures induced by its realizations are sin- gular with respect to the Lebesgue measure, and so their usual Radon–Nikodym derivatives are zero almost everywhere. In this paper, how to cope with these difficulties has been suggested by introducing random generalized densities (dis- tributions) á la Dirac–Schwarz, for both the deterministic case and the stochastic case. For the last one, mean generalized densities are analyzed, and they have been related to densities of the expected values of the relevant measures. Ac- tually, distributions are a subclass of the larger class of currents; in the usual Euclidean space of dimension d, currents of any order k 2 f0; 1;:::; dg or k- currents may be introduced.
    [Show full text]
  • SOME ALGEBRAIC DEFINITIONS and CONSTRUCTIONS Definition
    SOME ALGEBRAIC DEFINITIONS AND CONSTRUCTIONS Definition 1. A monoid is a set M with an element e and an associative multipli- cation M M M for which e is a two-sided identity element: em = m = me for all m M×. A−→group is a monoid in which each element m has an inverse element m−1, so∈ that mm−1 = e = m−1m. A homomorphism f : M N of monoids is a function f such that f(mn) = −→ f(m)f(n) and f(eM )= eN . A “homomorphism” of any kind of algebraic structure is a function that preserves all of the structure that goes into the definition. When M is commutative, mn = nm for all m,n M, we often write the product as +, the identity element as 0, and the inverse of∈m as m. As a convention, it is convenient to say that a commutative monoid is “Abelian”− when we choose to think of its product as “addition”, but to use the word “commutative” when we choose to think of its product as “multiplication”; in the latter case, we write the identity element as 1. Definition 2. The Grothendieck construction on an Abelian monoid is an Abelian group G(M) together with a homomorphism of Abelian monoids i : M G(M) such that, for any Abelian group A and homomorphism of Abelian monoids−→ f : M A, there exists a unique homomorphism of Abelian groups f˜ : G(M) A −→ −→ such that f˜ i = f. ◦ We construct G(M) explicitly by taking equivalence classes of ordered pairs (m,n) of elements of M, thought of as “m n”, under the equivalence relation generated by (m,n) (m′,n′) if m + n′ = −n + m′.
    [Show full text]
  • MTH 620: 2020-04-28 Lecture
    MTH 620: 2020-04-28 lecture Alexandru Chirvasitu Introduction In this (last) lecture we'll mix it up a little and do some topology along with the homological algebra. I wanted to bring up the cohomology of profinite groups if only briefly, because we discussed these in MTH619 in the context of Galois theory. Consider this as us connecting back to that material. 1 Topological and profinite groups This is about the cohomology of profinite groups; we discussed these briefly in MTH619 during Fall 2019, so this will be a quick recollection. First things first though: Definition 1.1 A topological group is a group G equipped with a topology, such that both the multiplication G × G ! G and the inverse map g 7! g−1 are continuous. Unless specified otherwise, all of our topological groups will be assumed separated (i.e. Haus- dorff) as topological spaces. Ordinary, plain old groups can be regarded as topological groups equipped with the discrete topology (i.e. so that all subsets are open). When I want to be clear we're considering plain groups I might emphasize that by referring to them as `discrete groups'. The main concepts we will work with are covered by the following two definitions (or so). Definition 1.2 A profinite group is a compact (and as always for us, Hausdorff) topological group satisfying any of the following equivalent conditions: Q G embeds as a closed subgroup into a product i2I Gi of finite groups Gi, equipped with the usual product topology; For every open neighborhood U of the identity 1 2 G there is a normal, open subgroup N E G contained in U.
    [Show full text]
  • Arxiv:2011.03427V2 [Math.AT] 12 Feb 2021 Riae Fmp Ffiiest.W Eosrt Hscategor This Demonstrate We Sets
    HYPEROCTAHEDRAL HOMOLOGY FOR INVOLUTIVE ALGEBRAS DANIEL GRAVES Abstract. Hyperoctahedral homology is the homology theory associated to the hyperoctahe- dral crossed simplicial group. It is defined for involutive algebras over a commutative ring using functor homology and the hyperoctahedral bar construction of Fiedorowicz. The main result of the paper proves that hyperoctahedral homology is related to equivariant stable homotopy theory: for a discrete group of odd order, the hyperoctahedral homology of the group algebra is isomorphic to the homology of the fixed points under the involution of an equivariant infinite loop space built from the classifying space of the group. Introduction Hyperoctahedral homology for involutive algebras was introduced by Fiedorowicz [Fie, Sec- tion 2]. It is the homology theory associated to the hyperoctahedral crossed simplicial group [FL91, Section 3]. Fiedorowicz and Loday [FL91, 6.16] had shown that the homology theory constructed from the hyperoctahedral crossed simplicial group via a contravariant bar construc- tion, analogously to cyclic homology, was isomorphic to Hochschild homology and therefore did not detect the action of the hyperoctahedral groups. Fiedorowicz demonstrated that a covariant bar construction did detect this action and sketched results connecting the hyperoctahedral ho- mology of monoid algebras and group algebras to May’s two-sided bar construction and infinite loop spaces, though these were never published. In Section 1 we recall the hyperoctahedral groups. We recall the hyperoctahedral category ∆H associated to the hyperoctahedral crossed simplicial group. This category encodes an involution compatible with an order-preserving multiplication. We introduce the category of involutive non-commutative sets, which encodes the same information by adding data to the preimages of maps of finite sets.
    [Show full text]
  • 10 Heat Equation: Interpretation of the Solution
    Math 124A { October 26, 2011 «Viktor Grigoryan 10 Heat equation: interpretation of the solution Last time we considered the IVP for the heat equation on the whole line u − ku = 0 (−∞ < x < 1; 0 < t < 1); t xx (1) u(x; 0) = φ(x); and derived the solution formula Z 1 u(x; t) = S(x − y; t)φ(y) dy; for t > 0; (2) −∞ where S(x; t) is the heat kernel, 1 2 S(x; t) = p e−x =4kt: (3) 4πkt Substituting this expression into (2), we can rewrite the solution as 1 1 Z 2 u(x; t) = p e−(x−y) =4ktφ(y) dy; for t > 0: (4) 4πkt −∞ Recall that to derive the solution formula we first considered the heat IVP with the following particular initial data 1; x > 0; Q(x; 0) = H(x) = (5) 0; x < 0: Then using dilation invariance of the Heaviside step function H(x), and the uniquenessp of solutions to the heat IVP on the whole line, we deduced that Q depends only on the ratio x= t, which lead to a reduction of the heat equation to an ODE. Solving the ODE and checking the initial condition (5), we arrived at the following explicit solution p x= 4kt 1 1 Z 2 Q(x; t) = + p e−p dp; for t > 0: (6) 2 π 0 The heat kernel S(x; t) was then defined as the spatial derivative of this particular solution Q(x; t), i.e. @Q S(x; t) = (x; t); (7) @x and hence it also solves the heat equation by the differentiation property.
    [Show full text]
  • Chapter 3 Proofs
    Chapter 3 Proofs Many mathematical proofs use a small range of standard outlines: direct proof, examples/counter-examples, and proof by contrapositive. These notes explain these basic proof methods, as well as how to use definitions of new concepts in proofs. More advanced methods (e.g. proof by induction, proof by contradiction) will be covered later. 3.1 Proving a universal statement Now, let’s consider how to prove a claim like For every rational number q, 2q is rational. First, we need to define what we mean by “rational”. A real number r is rational if there are integers m and n, n = 0, such m that r = n . m In this definition, notice that the fraction n does not need to satisfy conditions like being proper or in lowest terms So, for example, zero is rational 0 because it can be written as 1 . However, it’s critical that the two numbers 27 CHAPTER 3. PROOFS 28 in the fraction be integers, since even irrational numbers can be written as π fractions with non-integers on the top and/or bottom. E.g. π = 1 . The simplest technique for proving a claim of the form ∀x ∈ A, P (x) is to pick some representative value for x.1. Think about sticking your hand into the set A with your eyes closed and pulling out some random element. You use the fact that x is an element of A to show that P (x) is true. Here’s what it looks like for our example: Proof: Let q be any rational number.
    [Show full text]
  • Structure Theory of Set Addition
    Structure Theory of Set Addition Notes by B. J. Green1 ICMS Instructional Conference in Combinatorial Aspects of Mathematical Analysis, Edinburgh March 25 { April 5 2002. 1 Lecture 1: Pl¨unnecke's Inequalities 1.1 Introduction The object of these notes is to explain a recent proof by Ruzsa of a famous result of Freiman, some significant modifications of Ruzsa's proof due to Chang, and all the background material necessary to understand these arguments. Freiman's theorem concerns the structure of sets with small sumset. Let A be a subset of an abelian group G, and define the sumset A + A to be the set of all pairwise sums a + a0, where a; a0 are (not necessarily distinct) elements of A. If A = n then A + A n, and equality can occur (for example if A is a subgroup of G).j Inj the otherj directionj ≥ we have A + A n(n + 1)=2, and equality can occur here too, for example when G = Z and j j ≤ 2 n 1 A = 1; 3; 3 ;:::; 3 − . It is easy to construct similar examples of sets with large sumset, but ratherf harder to findg examples with A + A small. Let us think more carefully about this problem in the special case G = Z. Proposition 1 Let A Z have size n. Then A + A 2n 1, with equality if and only if A is an arithmetic progression⊆ of length n. j j ≥ − Proof. Write A = a1; : : : ; an where a1 < a2 < < an. Then we have f g ··· a1 + a1 < a1 + a2 < < a1 + an < a2 + an < < an + an; ··· ··· which amounts to an exhibition of 2n 1 distinct elements of A + A.
    [Show full text]
  • Delta Functions and Distributions
    When functions have no value(s): Delta functions and distributions Steven G. Johnson, MIT course 18.303 notes Created October 2010, updated March 8, 2017. Abstract x = 0. That is, one would like the function δ(x) = 0 for all x 6= 0, but with R δ(x)dx = 1 for any in- These notes give a brief introduction to the mo- tegration region that includes x = 0; this concept tivations, concepts, and properties of distributions, is called a “Dirac delta function” or simply a “delta which generalize the notion of functions f(x) to al- function.” δ(x) is usually the simplest right-hand- low derivatives of discontinuities, “delta” functions, side for which to solve differential equations, yielding and other nice things. This generalization is in- a Green’s function. It is also the simplest way to creasingly important the more you work with linear consider physical effects that are concentrated within PDEs, as we do in 18.303. For example, Green’s func- very small volumes or times, for which you don’t ac- tions are extremely cumbersome if one does not al- tually want to worry about the microscopic details low delta functions. Moreover, solving PDEs with in this volume—for example, think of the concepts of functions that are not classically differentiable is of a “point charge,” a “point mass,” a force plucking a great practical importance (e.g. a plucked string with string at “one point,” a “kick” that “suddenly” imparts a triangle shape is not twice differentiable, making some momentum to an object, and so on.
    [Show full text]
  • Diagrams of Affine Permutations and Their Labellings Taedong
    Diagrams of Affine Permutations and Their Labellings by Taedong Yun Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2013 c Taedong Yun, MMXIII. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. Author.............................................................. Department of Mathematics April 29, 2013 Certified by. Richard P. Stanley Professor of Applied Mathematics Thesis Supervisor Accepted by . Paul Seidel Chairman, Department Committee on Graduate Theses 2 Diagrams of Affine Permutations and Their Labellings by Taedong Yun Submitted to the Department of Mathematics on April 29, 2013, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract We study affine permutation diagrams and their labellings with positive integers. Bal- anced labellings of a Rothe diagram of a finite permutation were defined by Fomin- Greene-Reiner-Shimozono, and we extend this notion to affine permutations. The balanced labellings give a natural encoding of the reduced decompositions of affine permutations. We show that the sum of weight monomials of the column-strict bal- anced labellings is the affine Stanley symmetric function which plays an important role in the geometry of the affine Grassmannian. Furthermore, we define set-valued balanced labellings in which the labels are sets of positive integers, and we investi- gate the relations between set-valued balanced labellings and nilHecke words in the nilHecke algebra. A signed generating function of column-strict set-valued balanced labellings is shown to coincide with the affine stable Grothendieck polynomial which is related to the K-theory of the affine Grassmannian.
    [Show full text]
  • Quantum Mechanical Observables Under a Symplectic Transformation
    Quantum Mechanical Observables under a Symplectic Transformation of Coordinates Jakub K´aninsk´y1 1Charles University, Faculty of Mathematics and Physics, Institute of Theoretical Physics. E-mail address: [email protected] March 17, 2021 We consider a general symplectic transformation (also known as linear canonical transformation) of quantum-mechanical observables in a quan- tized version of a finite-dimensional system with configuration space iso- morphic to Rq. Using the formalism of rigged Hilbert spaces, we define eigenstates for all the observables. Then we work out the explicit form of the corresponding transformation of these eigenstates. A few examples are included at the end of the paper. 1 Introduction From a mathematical perspective we can view quantum mechanics as a science of finite- dimensional quantum systems, i.e. systems whose classical pre-image has configuration space of finite dimension. The single most important representative from this family is a quantized version of the classical system whose configuration space is isomorphic to Rq with some q N, which corresponds e.g. to the system of finitely many coupled ∈ harmonic oscillators. Classically, the state of such system is given by a point in the arXiv:2007.10858v2 [quant-ph] 16 Mar 2021 phase space, which is a vector space of dimension 2q equipped with the symplectic form. One would typically prefer to work in the abstract setting, without the need to choose any particular set of coordinates: in principle this is possible. In practice though, one will often end up choosing a symplectic basis in the phase space aligned with the given symplectic form and resort to the coordinate description.
    [Show full text]
  • Affine Permutations and Rational Slope Parking Functions
    AFFINE PERMUTATIONS AND RATIONAL SLOPE PARKING FUNCTIONS EUGENE GORSKY, MIKHAIL MAZIN, AND MONICA VAZIRANI ABSTRACT. We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund’s bijection ζ exchanging the pairs of statistics (area, dinv) and (bounce, area) on Dyck paths, and the Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions. We also relate our combinatorial constructions to representation theory. We derive new formulas for the Poincar´epolynomials of certain affine Springer fibers and describe a connection to the theory of finite dimensional representations of DAHA and nonsymmetric Macdonald polynomials. 1. INTRODUCTION Parking functions are ubiquitous in the modern combinatorics. There is a natural action of the symmetric group on parking functions, and the orbits are labeled by the non-decreasing parking functions which correspond natu- rally to the Dyck paths. This provides a link between parking functions and various combinatorial objects counted by Catalan numbers. In a series of papers Garsia, Haglund, Haiman, et al. [18, 19], related the combinatorics of Catalan numbers and parking functions to the space of diagonal harmonics. There are also deep connections to the geometry of the Hilbert scheme. Since the works of Pak and Stanley [29], Athanasiadis and Linusson [5] , it became clear that parking functions are tightly related to the combinatorics of the affine symmetric group.
    [Show full text]
  • Paper in Its Early Stages and Gave Valuable Suggestions
    International J. Soc. Sci. & Education 2012 Vol. 3 Issue 1, ISSN: 2223-4934 E and 2227-393X Print How does the Use of Mathematical Symbols Influence Understanding of Mathematical Concepts by Secondary School Students? By Silvanos Chirume Department of Mathematics and Statistics, Faculty of Science and Technology Zimbabwe Open University, Midlands Region. PO Box 1810 Gweru, Zimbabwe [[email protected]] Abstract If students understand and properly communicate using math symbols and notation, their achievement in math might improve (Rubenstein & Thompson, 2001). This study investigated how the use of mathematical symbols influences understanding of math concepts by secondary school students in Shurugwi District (Zimbabwe). Convenience sampling (for the district), simple random sampling (for the schools), judgemental sampling (for the teachers) and stratified random sampling with proportional allocation (for the students) were used. The sample included six schools, 120 “O” level students, 27 “A” level students and 7 teachers of those students. The “O” level students completed questionnaires and were interviewed as a group at each of the 6 selected schools. “A” level students wrote a diagnostic test and were later interviewed as a group while the teachers were individually interviewed. It was found that most students fail to understand or interpret the meaning of math symbols due to the way they are taught to read, pronounce and use them. This misuse (and also abuse) of symbols may considerably hinder formation, understanding and communication of concepts and might affect achievement; the final outcome desired. Teachers are therefore sensitized on appropriate strategies to take to overcome students’ difficulties on the use of symbols.
    [Show full text]