Spectral properties of quantum graphs via pseudo orbits
|
|
- Kristian Little
- 6 years ago
- Views:
Transcription
1 Spectral properties of quantum graphs via pseudo orbits 1, Ram Band 2, Tori Hudgins 1, Christopher Joyner 3, Mark Sepanski 1 1 Baylor, 2 Technion, 3 Queen Mary University of London Cardiff 12/6/17
2 Outline Quantum graphs 1 Quantum graphs 2 3
3 Quantum graphs Metric graph: associate each edge e to interval [0, l e ]. Total length L = E e=1 l e. Laplace equation on [0, l e ], d2 dxe 2 f e (x e ) = k 2 f e (x e ). (1) Hilbert space H := E e=1 L2( [0, l e ] ). Vertex conditions chosen s.t. op. self-adjoint.
4 Applications Free electron model in organic molecules (1930 s) Anderson localization Nanotechnology Photonic crystals Quantum chaos Wave guides
5 Scattering approach f e (x e ) = a in e e ikxe + a out e e ikxe (2) At vertex v, d v incoming and outgoing plane waves related by vertex scattering matrix, a v out = σ (v) a v in. Example: Neumann conditions f is continuous at v and e v f e (v) = 0. [σ (v) ] ij = 2 d v δ ij
6 Scattering approach Alternatively, to quantize graph assign a unitary vertex scattering matrix σ (v) to each vertex v. Example: A democratic choice is the discrete Fourier transform matrix, σ (v) e,e = 1 1 ω ω 2 dv... ω 1 1 ω 2 ω 4 2(dv... ω 1) dv ω dv 1 ω 2(dv 1) (dv 1)(dv... ω 1) ω = e 2πi dv a primitive d v root of unity.
7 Spectrum Quantum graphs Combine vertex scattering matrices into an E E matrix Σ, { σ (v) Σ e,e = e,e v = t(e ) = o(e), (3) 0 otherwise Let L = diag{l 1,..., l E }, the quantum evolution op., U (k) = e ikl Σ. (4) Spectrum { k 2 det (I U (k)) = 0 }
8 Characteristic polynomial Characteristic polynomial of U(k) F ξ (k) = det (ξi U (k)) = 2B n=0 a n ξ 2B n Note: spectrum corresponds to roots of F 1 (k) = 0.
9 Riemann-Siegel lookalike formula Unitarity of U connects pairs of coefficients of characteristic polynomial. ( ) F ξ (k) = det ( ξu (k)) det ξ 1 I U (k) (5) = det ( ξu (k)) = det (U (k)) E anξ n E (6) n=0 E anξ n (7) n=0 Compare with F ξ (k) = E n=0 a nξ 2B n and use a E = det (U (k)). a n = a E a E n. (8)
10 Periodic orbits on a quantum graph To periodic orbit γ = (e 1,..., e m ) on a quantum graph associate, topological length E γ = m. metric length l γ = e j γ l e j. stability amplitude A γ = Σ e2 e 1 Σ e3 e 2... Σ enen 1 Σ e1 e m.
11 Pseudo orbits on a quantum graph To pseudo orbit γ = {γ 1,..., γ M } associate, m γ = M no. of periodic orbits in γ. topological length E γ = M j=1 E γ j. metric length l γ = M j=1 l γ j. stability amplitude A γ = M j=1 A γ j.
12 Theorem 1 (Band,H.,Joyner) Coefficients of the characteristic polynomial F ξ (k) are given by, a n = γ E γ=n ( 1) m γ A γ exp (ikl γ ), where the finite sum is over all primitive pseudo orbits topological length n. Idea Expand det (ξi U (k)) as a sum over permutations. A permutation ρ S E can contribute iff ρ(e) is connected to e for all e in ρ. Representing ρ as a product of disjoint cycles each cycle is a primitive periodic orbit.
13 Variance of coefficients of the characteristic polynomial a n k = a n 2 k = γ E γ=n = ( 1) m γ A γ γ, γ E γ=e γ =n γ, γ E γ=e γ =n Diagonal contribution lim K { 1 K 1 n = 0 e ikl γ dk = K 0 0 otherwise ( 1) m γ+m γ A γ Ā γ lim K 1 K e ik(l γ l γ ) dk K 0 ( 1) m γ+m γ A γ Ā γ δ l γ,l γ (9) a n 2 diag = γ E γ=n A γ 2
14 Background Variance of coeffs of the characteristic polynomial of graphs Kottos and Smilansky (1999). Spectral statistics of binary graphs Tanner (2000)&(2001). Variance of coeffs of the characteristic polynomial of binary graphs studied via permanent of transition matrix Tanner (2002) Random matrix variance a n 2 COE = 1 + a n 2 CUE = 1 n(e n) E + 1
15 q-nary graphs V = q m vertices, labeled by words length m on A. E = q m+1 directed edges e, each labeled by word w length m + 1. Origin vertex o(e), first m letters of w. Terminal vertex t(e), last m letters of w. Each vertex has q incoming and q outgoing edges. Spectral gap: adjacency matrix has simple eval. 1 and eval. 0 with multiplicity V 1, i.e. maximal spectral gap and maximally mixing.
16 Example: binary graph with 2 3 vertices
17 Example: binary graph with 2 3 vertices
18 Example: binary graph with 2 3 vertices
19 Example: binary graph with 2 3 vertices
20 Example: ternary graph with 3 2 vertices
21 Lyndon words Lexicographic order w = a 1 a 2... a l (10) w = b 1 b 2... b k (11) with a i, b j A a totally ordered alphabet of q letters. w > lex w iff there exists i st a 1 = b 1,..., a i 1 = b i 1 and a i > b i or l > k and a 1 = b 1,..., a k = b k. A word is a Lyndon word if it is strictly smaller in lexicographic order than all its cyclic shifts. Example: binary Lyndon words length 3, 0 < lex 001 < lex 01 < lex 011 < lex 1.
22 The standard decomposition Theorem 2 (Chen, Fox, Lyndon) Every word w can be uniquely written as a concatenation of Lyndon words in non-increasing lexicographic order, the standard decomposition of w. Example: standard decompositions of binary words length 3, (0)(0)(0) (01)(0) (1)(0)(0) (1)(1)(0) (001) (011) (1)(01) (1)(1)(1) A standard decomposition w = v 1 v 2... v k with v j a Lyndon word and v j lex v j+1 is strictly decreasing if v j > lex v j+1.
23 The standard decomposition Theorem 2 (Chen, Fox, Lyndon) Every word w can be uniquely written as a concatenation of Lyndon words in non-increasing lexicographic order, the standard decomposition of w. Example: standard decompositions of binary words length 3, (0)(0)(0) (01)(0) (1)(0)(0) (1)(1)(0) (001) (011) (1)(01) (1)(1)(1) A standard decomposition w = v 1 v 2... v k with v j a Lyndon word and v j lex v j+1 is strictly decreasing if v j > lex v j+1.
24 Periodic orbits A path length l is labeled by a word w = a 1,..., a l+m. A closed path length l is labeled by w = a 1,..., a l. A periodic orbit γ is the equivalence class of closed paths under cyclic shifts. A primitive periodic orbit is a periodic orbit that is not a repartition of a shorter orbit. Primitive periodic orbits length l are in 1-to-1 correspondence with Lyndon words length l. Example: 0011 is a primitive periodic orbit length
25 Pseudo orbits A pseudo orbit γ = {γ 1,..., γ M } is a set of periodic orbits. A primitive pseudo orbit γ is a set of primitive periodic orbits where no periodic orbit appears more than once. Note: there is a bijection between primitive pseudo orbits and strictly decreasing standard decompositions. Example: has strictly decreasing standard decomposition (011)(01)(0)
26 Diagonal contribution Transition probability σ (v) e,e 2 = 1 q. No. of primitive pseudo orbits length n equal to no. of strictly decreasing standard decompositions of words length n, denoted Str q (n). Diagonal contribution a n 2 diag = γ E γ=n = γ E γ=n = Str q(n) q n A γ 2 1 q n
27 Binary words length 4 and
28 Binary words length 4 and 5. (0)(0)(0)(0) (01)(0)(0) (1)(0)(0)(0) (1)(1)(0)(0) (0001) (01)(01) (1)(001) (1)(1)(01) (001)(0) (011)(0) (1)(01)(0) (1)(1)(1)(0) (0011) (0111) (1)(011) (1)(1)(1)(1) (0)(0)(0)(0)(0) (01)(0)(0)(0) (1)(0)(0)(0)(0) (1)(1)(0)(0)(0) (00001) (01)(001) (1)(0001) (1)(1)(001) (0001)(0) (01)(01)(0) (1)(001)(0) (1)(1)(01)(0) (00011) (01011) (1)(0011) (1)(1)(011) (001)(0)(0) (011)(0)(0) (1)(01)(0)(0) (1)(1)(1)(0)(0) (00101) (011)(01) (1)(01)(01) (1)(1)(1)(01) (0011)(0) (0111)(0) (1)(011)(0) (1)(1)(1)(1)(0) (00111) (01111) (1)(0111) (1)(1)(1)(1)(1)
29 Theorem 3 (Band, H., Sepanski) For words of length n 2 the no. of strictly decreasing standard decompositions is, Str q (n) = (q 1)q n 1. Note: Hence the proportion of words length n with strictly decreasing standard decompositions is q 1 q. i.e. half of binary words have strictly decreasing standard decompositions.
30 Comparison with RMT Diagonal contribution a n 2 diag = Str q(n) (q 1) q n = q RMT without time-reversal symmetry a n 2 CUE = 1 Diagonal contribution deviates from RMT as seen in numerical results for binary graphs (Tanner). For a sequence of graphs with increasing connectivity q the diagonal contribution would approach the random matrix result.
31 Counting Lyndon words Let L q (n) be the no. of Lyndon words length n. Lemma 4 A classic result about Lyndon words is, ll q (l) = q m l m Proof. Every word length l is a Lyndon word, a cyclic shift of a Lyndon word, or a repartition of such a word.
32 Proof of Theorem 3 Let Str q (0) = 1, Str q (1) = q and define two functions, p(x) = Str q (n) x n (12) n=0 f (x) = qx 2 1 qx 1 = 1 + qx + (q 1)q n 1 x n (13) We show p = f. As p(0) = f (0) = 1 note p = f if n=2 d dx log p = d dx log f = q 1 qx 2qx 1 qx 2 = 1 (qx) m 2 (qx 2 ) m x x m=1 m=1
33 Proof of Theorem 3 Let Str q (0) = 1, Str q (1) = q and define two functions, p(x) = Str q (n) x n = (1 + x l ) Lq(l) (14) n=0 l=1 f (x) = qx 2 1 qx 1 = 1 + qx + (q 1)q n 1 x n (15) We show p = f. As p(0) = f (0) = 1 note p = f if n=2 d dx log p = d dx log f = q 1 qx 2qx 1 qx 2 = 1 (qx) m 2 (qx 2 ) m x x m=1 m=1
34 log p = ( 1) j+1 L q (l) j l=1 d dx log p = 1 x = 1 x = 1 x = 1 x j=1 l=1 j=1 x lj ( 1) j ll q (l)x lj ( 1) m l ll q (l)x m m=1 l m ll q (l)x m 1 x m=1 l m ll q (n)x m 2 x m=1 l m m=1 l 2m ( 2m ) 1 + ( 1) l llq (n)x 2m ll q (n)x 2m m=1 l m Then applying Lemma 4 proves the result.
35 Summary Quantum graphs Graph spectrum encoded in finite number of short primitive pseudo orbits. Investigating characteristic polynomial led to new questions for Lyndon words. Obtained proportion of standard decompositions that are strictly decreasing. Suggests RMT behavior of coefficients may be obtained under stronger conditions. R. Band, J. M. Harrison and C. H. Joyner, Finite pseudo orbit expansions for spectral quantities of quantum graphs, J. Phys. A: Math. Theor. 45 (2012) arxiv: R. Band, J. M. Harrison and M. Sepanski, Lyndon word decompositions and pseudo orbits on q-nary graphs, arxiv:1610:03808
The effect of spin in the spectral statistics of quantum graphs
The effect of spin in the of quantum graphs J. M. Harrison and J. Bolte Baylor University, Royal Holloway Banff 28th February 08 Outline Random matrix theory 2 Pauli op. on graphs 3 Motivation Bohigas-Gianonni-Schmit
More informationQuantum chaos on graphs
Baylor University Graduate seminar 6th November 07 Outline 1 What is quantum? 2 Everything you always wanted to know about quantum but were afraid to ask. 3 The trace formula. 4 The of Bohigas, Giannoni
More informationTopology and quantum mechanics
Topology, homology and quantum mechanics 1, J.P. Keating 2, J.M. Robbins 2 and A. Sawicki 2 1 Baylor University, 2 University of Bristol Baylor 9/27/12 Outline Topology in QM 1 Topology in QM 2 3 Wills
More informationResonance asymptotics on quantum graphs and hedgehog manifolds
Resonance asymptotics on quantum graphs and hedgehog manifolds Jiří Lipovský University of Hradec Králové, Faculty of Science jiri.lipovsky@uhk.cz joint work with P. Exner and E.B. Davies Warsaw, 3rd December
More informationGEOMETRIC PROPERTIES OF QUANTUM GRAPHS AND VERTEX SCATTERING MATRICES. Pavel Kurasov, Marlena Nowaczyk
Opuscula Mathematica Vol. 30 No. 3 2010 http://dx.doi.org/10.7494/opmath.2010.30.3.295 GEOMETRIC PROPERTIES OF QUANTUM GRAPHS AND VERTEX SCATTERING MATRICES Pavel Kurasov, Marlena Nowaczyk Abstract. Differential
More informationGraph fundamentals. Matrices associated with a graph
Graph fundamentals Matrices associated with a graph Drawing a picture of a graph is one way to represent it. Another type of representation is via a matrix. Let G be a graph with V (G) ={v 1,v,...,v n
More informationLaplacians of Graphs, Spectra and Laplacian polynomials
Mednykh A. D. (Sobolev Institute of Math) Laplacian for Graphs 27 June - 03 July 2015 1 / 30 Laplacians of Graphs, Spectra and Laplacian polynomials Alexander Mednykh Sobolev Institute of Mathematics Novosibirsk
More informationarxiv: v2 [nlin.cd] 30 May 2013
A sub-determinant approach for pseudo-orbit expansions of spectral determinants in quantum maps and quantum graphs Daniel Waltner Weizmann Institute of Science, Physics Department, Rehovot, Israel, Institut
More informationQMI PRELIM Problem 1. All problems have the same point value. If a problem is divided in parts, each part has equal value. Show all your work.
QMI PRELIM 013 All problems have the same point value. If a problem is divided in parts, each part has equal value. Show all your work. Problem 1 L = r p, p = i h ( ) (a) Show that L z = i h y x ; (cyclic
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More informationEntanglement in Topological Phases
Entanglement in Topological Phases Dylan Liu August 31, 2012 Abstract In this report, the research conducted on entanglement in topological phases is detailed and summarized. This includes background developed
More informationMIT Algebraic techniques and semidefinite optimization May 9, Lecture 21. Lecturer: Pablo A. Parrilo Scribe:???
MIT 6.972 Algebraic techniques and semidefinite optimization May 9, 2006 Lecture 2 Lecturer: Pablo A. Parrilo Scribe:??? In this lecture we study techniques to exploit the symmetry that can be present
More informationLaplacians of Graphs, Spectra and Laplacian polynomials
Laplacians of Graphs, Spectra and Laplacian polynomials Lector: Alexander Mednykh Sobolev Institute of Mathematics Novosibirsk State University Winter School in Harmonic Functions on Graphs and Combinatorial
More informationquantum statistics on graphs
n-particle 1, J.P. Keating 2, J.M. Robbins 2 and A. Sawicki 1 Baylor University, 2 University of Bristol, M.I.T. TexAMP 11/14 Quantum statistics Single particle space configuration space X. Two particle
More informationCombinatorial Enumeration. Jason Z. Gao Carleton University, Ottawa, Canada
Combinatorial Enumeration Jason Z. Gao Carleton University, Ottawa, Canada Counting Combinatorial Structures We are interested in counting combinatorial (discrete) structures of a given size. For example,
More informationA quantum walk based search algorithm, and its optical realisation
A quantum walk based search algorithm, and its optical realisation Aurél Gábris FJFI, Czech Technical University in Prague with Tamás Kiss and Igor Jex Prague, Budapest Student Colloquium and School on
More informationdynamical zeta functions: what, why and what are the good for?
dynamical zeta functions: what, why and what are the good for? Predrag Cvitanović Georgia Institute of Technology November 2 2011 life is intractable in physics, no problem is tractable I accept chaos
More informationFunctional Analysis Review
Outline 9.520: Statistical Learning Theory and Applications February 8, 2010 Outline 1 2 3 4 Vector Space Outline A vector space is a set V with binary operations +: V V V and : R V V such that for all
More informationLecture 3. Mathematical methods in communication I. REMINDER. A. Convex Set. A set R is a convex set iff, x 1,x 2 R, θ, 0 θ 1, θx 1 + θx 2 R, (1)
3- Mathematical methods in communication Lecture 3 Lecturer: Haim Permuter Scribe: Yuval Carmel, Dima Khaykin, Ziv Goldfeld I. REMINDER A. Convex Set A set R is a convex set iff, x,x 2 R, θ, θ, θx + θx
More informationFrom continuous to the discrete Fourier transform: classical and q
From continuous to the discrete Fourier transform: classical and quantum aspects 1 Institut de Mathématiques de Bourgogne, Dijon, France 2 Sankt-Petersburg University of Aerospace Instrumentation (SUAI),
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationMP463 QUANTUM MECHANICS
MP463 QUANTUM MECHANICS Introduction Quantum theory of angular momentum Quantum theory of a particle in a central potential - Hydrogen atom - Three-dimensional isotropic harmonic oscillator (a model of
More informationLecture 16 Symbolic dynamics.
Lecture 16 Symbolic dynamics. 1 Symbolic dynamics. The full shift on two letters and the Baker s transformation. 2 Shifts of finite type. 3 Directed multigraphs. 4 The zeta function. 5 Topological entropy.
More informationGroup theoretical methods in Machine Learning
Group theoretical methods in Machine Learning Risi Kondor Columbia University Tutorial at ICML 2007 Tiger, tiger, burning bright In the forests of the night, What immortal hand or eye Dare frame thy fearful
More informationSection Summary. Relations and Functions Properties of Relations. Combining Relations
Chapter 9 Chapter Summary Relations and Their Properties n-ary Relations and Their Applications (not currently included in overheads) Representing Relations Closures of Relations (not currently included
More informationOn Boolean functions which are bent and negabent
On Boolean functions which are bent and negabent Matthew G. Parker 1 and Alexander Pott 2 1 The Selmer Center, Department of Informatics, University of Bergen, N-5020 Bergen, Norway 2 Institute for Algebra
More informationQuadrature for the Finite Free Convolution
Spectral Graph Theory Lecture 23 Quadrature for the Finite Free Convolution Daniel A. Spielman November 30, 205 Disclaimer These notes are not necessarily an accurate representation of what happened in
More informationQuantum Field theories, Quivers and Word combinatorics
Quantum Field theories, Quivers and Word combinatorics Sanjaye Ramgoolam Queen Mary, University of London 21 October 2015, QMUL-EECS Quivers as Calculators : Counting, correlators and Riemann surfaces,
More informationMATRIX ANALYSIS HOMEWORK
MATRIX ANALYSIS HOMEWORK VERN I. PAULSEN 1. Due 9/25 We let S n denote the group of all permutations of {1,..., n}. 1. Compute the determinants of the elementary matrices: U(k, l), D(k, λ), and S(k, l;
More informationChapter 1 Divide and Conquer Algorithm Theory WS 2016/17 Fabian Kuhn
Chapter 1 Divide and Conquer Algorithm Theory WS 2016/17 Fabian Kuhn Formulation of the D&C principle Divide-and-conquer method for solving a problem instance of size n: 1. Divide n c: Solve the problem
More informationI. PLATEAU TRANSITION AS CRITICAL POINT. A. Scaling flow diagram
1 I. PLATEAU TRANSITION AS CRITICAL POINT The IQHE plateau transitions are examples of quantum critical points. What sort of theoretical description should we look for? Recall Anton Andreev s lectures,
More informationRecall that any inner product space V has an associated norm defined by
Hilbert Spaces Recall that any inner product space V has an associated norm defined by v = v v. Thus an inner product space can be viewed as a special kind of normed vector space. In particular every inner
More informationMATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5
MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5.. The Arzela-Ascoli Theorem.. The Riemann mapping theorem Let X be a metric space, and let F be a family of continuous complex-valued functions on X. We have
More informationIsomorphisms between pattern classes
Journal of Combinatorics olume 0, Number 0, 1 8, 0000 Isomorphisms between pattern classes M. H. Albert, M. D. Atkinson and Anders Claesson Isomorphisms φ : A B between pattern classes are considered.
More informationLectures 2 3 : Wigner s semicircle law
Fall 009 MATH 833 Random Matrices B. Való Lectures 3 : Wigner s semicircle law Notes prepared by: M. Koyama As we set up last wee, let M n = [X ij ] n i,j= be a symmetric n n matrix with Random entries
More informationMaximizing the numerical radii of matrices by permuting their entries
Maximizing the numerical radii of matrices by permuting their entries Wai-Shun Cheung and Chi-Kwong Li Dedicated to Professor Pei Yuan Wu. Abstract Let A be an n n complex matrix such that every row and
More informationAdvanced Combinatorial Optimization Feb 13 and 25, Lectures 4 and 6
18.438 Advanced Combinatorial Optimization Feb 13 and 25, 2014 Lectures 4 and 6 Lecturer: Michel X. Goemans Scribe: Zhenyu Liao and Michel X. Goemans Today, we will use an algebraic approach to solve the
More informationSOFTWARE FOR MULTI DE BRUIJN SEQUENCES. This describes Maple software implementing formulas and algorithms in
SOFTWARE FOR MULTI DE BRUIJN SEQUENCES GLENN TESLER gptesler@math.ucsd.edu http://math.ucsd.edu/ gptesler/multidebruijn Last updated: July 16, 2017 This describes Maple software implementing formulas and
More informationUndirected Graphical Models
Undirected Graphical Models 1 Conditional Independence Graphs Let G = (V, E) be an undirected graph with vertex set V and edge set E, and let A, B, and C be subsets of vertices. We say that C separates
More informationUnitarity, Dispersion Relations, Cutkosky s Cutting Rules
Unitarity, Dispersion Relations, Cutkosky s Cutting Rules 04.06.0 For more information about unitarity, dispersion relations, and Cutkosky s cutting rules, consult Peskin& Schröder, or rather Le Bellac.
More informationThe critical group of a graph
The critical group of a graph Peter Sin Texas State U., San Marcos, March th, 4. Critical groups of graphs Outline Laplacian matrix of a graph Chip-firing game Smith normal form Some families of graphs
More informationA. S. Kechris: Classical Descriptive Set Theory; Corrections and Updates (October 1, 2018)
A. S. Kechris: Classical Descriptive Set Theory; Corrections and Updates (October 1, 2018) Page 3, line 9-: add after spaces,with d i < 1, Page 8, line 11: D ϕ D(ϕ) Page 22, line 6: x 0 e x e Page 22:
More informationDefinition: A binary relation R from a set A to a set B is a subset R A B. Example:
Chapter 9 1 Binary Relations Definition: A binary relation R from a set A to a set B is a subset R A B. Example: Let A = {0,1,2} and B = {a,b} {(0, a), (0, b), (1,a), (2, b)} is a relation from A to B.
More informationPeriodicity & State Transfer Some Results Some Questions. Periodic Graphs. Chris Godsil. St John s, June 7, Chris Godsil Periodic Graphs
St John s, June 7, 2009 Outline 1 Periodicity & State Transfer 2 Some Results 3 Some Questions Unitary Operators Suppose X is a graph with adjacency matrix A. Definition We define the operator H X (t)
More informationONE CAN HEAR THE EULER CHARACTERISTIC OF A SIMPLICIAL COMPLEX
ONE CAN HEAR THE EULER CHARACTERISTIC OF A SIMPLICIAL COMPLEX OLIVER KNILL y G g(x, y) with g = L 1 but also agrees with Abstract. We prove that that the number p of positive eigenvalues of the connection
More informationORBITAL DIGRAPHS OF INFINITE PRIMITIVE PERMUTATION GROUPS
ORBITAL DIGRAPHS OF INFINITE PRIMITIVE PERMUTATION GROUPS SIMON M. SMITH Abstract. If G is a group acting on a set Ω and α, β Ω, the digraph whose vertex set is Ω and whose arc set is the orbit (α, β)
More informationMP 472 Quantum Information and Computation
MP 472 Quantum Information and Computation http://www.thphys.may.ie/staff/jvala/mp472.htm Outline Open quantum systems The density operator ensemble of quantum states general properties the reduced density
More informationKernel Method: Data Analysis with Positive Definite Kernels
Kernel Method: Data Analysis with Positive Definite Kernels 2. Positive Definite Kernel and Reproducing Kernel Hilbert Space Kenji Fukumizu The Institute of Statistical Mathematics. Graduate University
More informationResearch Statement. MUHAMMAD INAM 1 of 5
MUHAMMAD INAM 1 of 5 Research Statement Preliminaries My primary research interests are in geometric inverse semigroup theory and its connections with other fields of mathematics. A semigroup M is called
More informationNonlinear instability of half-solitons on star graphs
Nonlinear instability of half-solitons on star graphs Adilbek Kairzhan and Dmitry Pelinovsky Department of Mathematics, McMaster University, Canada Workshop Nonlinear Partial Differential Equations on
More informationVery few Moore Graphs
Very few Moore Graphs Anurag Bishnoi June 7, 0 Abstract We prove here a well known result in graph theory, originally proved by Hoffman and Singleton, that any non-trivial Moore graph of diameter is regular
More informationMathematical foundations - linear algebra
Mathematical foundations - linear algebra Andrea Passerini passerini@disi.unitn.it Machine Learning Vector space Definition (over reals) A set X is called a vector space over IR if addition and scalar
More informationarxiv: v1 [quant-ph] 17 Jun 2009
Quantum search algorithms on the hypercube Birgit Hein and Gregor Tanner School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK e-mail: gregor.tanner@nottingham.ac.uk
More informationCombinatorial Hopf algebras in particle physics I
Combinatorial Hopf algebras in particle physics I Erik Panzer Scribed by Iain Crump May 25 1 Combinatorial Hopf Algebras Quantum field theory (QFT) describes the interactions of elementary particles. There
More information2 JAKOBSEN, JENSEN, JNDRUP AND ZHANG Let U q be a quantum group dened by a Cartan matrix of type A r together with the usual quantized Serre relations
QUADRATIC ALGEBRAS OF TYPE AIII; I HANS PLESNER JAKOBSEN, ANDERS JENSEN, SREN JNDRUP, AND HECHUN ZHANG 1;2 Department of Mathematics, Universitetsparken 5 DK{2100 Copenhagen, Denmark E-mail: jakobsen,
More informationA Class of Vertex Transitive Graphs
Volume 119 No. 16 2018, 3137-3144 ISSN: 1314-3395 (on-line version) url: http://www.acadpubl.eu/hub/ http://www.acadpubl.eu/hub/ A Class of Vertex Transitive Graphs 1 N. Murugesan, 2 R. Anitha 1 Assistant
More informationExploring Quantum Chaos with Quantum Computers
Exploring Quantum Chaos with Quantum Computers Part II: Measuring signatures of quantum chaos on a quantum computer David Poulin Institute for Quantum Computing Perimeter Institute for Theoretical Physics
More informationQuantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.
Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.
More informationQuantitative Oppenheim theorem. Eigenvalue spacing for rectangular billiards. Pair correlation for higher degree diagonal forms
QUANTITATIVE DISTRIBUTIONAL ASPECTS OF GENERIC DIAGONAL FORMS Quantitative Oppenheim theorem Eigenvalue spacing for rectangular billiards Pair correlation for higher degree diagonal forms EFFECTIVE ESTIMATES
More informationSolution Set of Homework # 2. Friday, September 09, 2017
Temple University Department of Physics Quantum Mechanics II Physics 57 Fall Semester 17 Z. Meziani Quantum Mechanics Textboo Volume II Solution Set of Homewor # Friday, September 9, 17 Problem # 1 In
More informationOn the automorphism groups of PCC. primitive coherent configurations. Bohdan Kivva, UChicago
On the automorphism groups of primitive coherent configurations Bohdan Kivva (U Chicago) Advisor: László Babai Symmetry vs Regularity, WL2018 Pilsen, 6 July 2018 Plan for this talk Babai s classification
More informationUniversal spectral statistics of Andreev billiards: semiclassical approach
Universal spectral statistics of Andreev billiards: semiclassical approach Sven Gnutzmann, 1, Burkhard Seif, 2, Felix von Oppen, 1, and Martin R. Zirnbauer 2, 1 Institut für Theoretische Physik, Freie
More informationTopological Qubit Design and Leakage
Topological Qubit Design and National University of Ireland, Maynooth September 8, 2011 Topological Qubit Design and T.Q.C. Braid Group Overview Designing topological qubits, the braid group, leakage errors.
More informationBoundary Conditions associated with the Left-Definite Theory for Differential Operators
Boundary Conditions associated with the Left-Definite Theory for Differential Operators Baylor University IWOTA August 15th, 2017 Overview of Left-Definite Theory A general framework for Left-Definite
More informationValerio Cappellini. References
CETER FOR THEORETICAL PHYSICS OF THE POLISH ACADEMY OF SCIECES WARSAW, POLAD RADOM DESITY MATRICES AD THEIR DETERMIATS 4 30 SEPTEMBER 5 TH SFB TR 1 MEETIG OF 006 I PRZEGORZAłY KRAKÓW Valerio Cappellini
More informationCounting the average size of Markov graphs
JOURNAL OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4866, ISSN (o) 2303-4947 www.imvibl.org /JOURNALS / JOURNAL Vol. 7(207), -6 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA
More informationRiesz bases of Floquet modes in semi-infinite periodic waveguides and implications
Riesz bases of Floquet modes in semi-infinite periodic waveguides and implications Thorsten Hohage joint work with Sofiane Soussi Institut für Numerische und Angewandte Mathematik Georg-August-Universität
More informationMS 3011 Exercises. December 11, 2013
MS 3011 Exercises December 11, 2013 The exercises are divided into (A) easy (B) medium and (C) hard. If you are particularly interested I also have some projects at the end which will deepen your understanding
More informationMATHEMATICS. Course Syllabus. Section A: Linear Algebra. Subject Code: MA. Course Structure. Ordinary Differential Equations
MATHEMATICS Subject Code: MA Course Structure Sections/Units Section A Section B Section C Linear Algebra Complex Analysis Real Analysis Topics Section D Section E Section F Section G Section H Section
More informationUnderstanding the log expansions in quantum field theory combinatorially
Understanding the log expansions in quantum field theory combinatorially Karen Yeats CCC and BCCD, February 5, 2016 Augmented generating functions Take a combinatorial class C. Build a generating function
More informationProblem 1. Let f n (x, y), n Z, be the sequence of rational functions in two variables x and y given by the initial conditions.
18.217 Problem Set (due Monday, December 03, 2018) Solve as many problems as you want. Turn in your favorite solutions. You can also solve and turn any other claims that were given in class without proofs,
More informationDichotomy Theorems for Counting Problems. Jin-Yi Cai University of Wisconsin, Madison. Xi Chen Columbia University. Pinyan Lu Microsoft Research Asia
Dichotomy Theorems for Counting Problems Jin-Yi Cai University of Wisconsin, Madison Xi Chen Columbia University Pinyan Lu Microsoft Research Asia 1 Counting Problems Valiant defined the class #P, and
More informationMean-Field Theory: HF and BCS
Mean-Field Theory: HF and BCS Erik Koch Institute for Advanced Simulation, Jülich N (x) = p N! ' (x ) ' 2 (x ) ' N (x ) ' (x 2 ) ' 2 (x 2 ) ' N (x 2 )........ ' (x N ) ' 2 (x N ) ' N (x N ) Slater determinant
More informationSome Remarks on the Discrete Uncertainty Principle
Highly Composite: Papers in Number Theory, RMS-Lecture Notes Series No. 23, 2016, pp. 77 85. Some Remarks on the Discrete Uncertainty Principle M. Ram Murty Department of Mathematics, Queen s University,
More informationBefore you begin read these instructions carefully.
MATHEMATICAL TRIPOS Part IB Thursday, 9 June, 2011 9:00 am to 12:00 pm PAPER 3 Before you begin read these instructions carefully. Each question in Section II carries twice the number of marks of each
More informationIntroduction to Quantum Information Hermann Kampermann
Introduction to Quantum Information Hermann Kampermann Heinrich-Heine-Universität Düsseldorf Theoretische Physik III Summer school Bleubeuren July 014 Contents 1 Quantum Mechanics...........................
More informationMarkov Chains, Stochastic Processes, and Matrix Decompositions
Markov Chains, Stochastic Processes, and Matrix Decompositions 5 May 2014 Outline 1 Markov Chains Outline 1 Markov Chains 2 Introduction Perron-Frobenius Matrix Decompositions and Markov Chains Spectral
More informationPermutation groups/1. 1 Automorphism groups, permutation groups, abstract
Permutation groups Whatever you have to do with a structure-endowed entity Σ try to determine its group of automorphisms... You can expect to gain a deep insight into the constitution of Σ in this way.
More informationLecture 1: Graphs, Adjacency Matrices, Graph Laplacian
Lecture 1: Graphs, Adjacency Matrices, Graph Laplacian Radu Balan January 31, 2017 G = (V, E) An undirected graph G is given by two pieces of information: a set of vertices V and a set of edges E, G =
More informationarxiv: v2 [math-ph] 13 Jan 2018
NODAL STATISTICS ON QUANTUM GRAPHS LIOR ALON, RAM BAND, AND GREGORY BERKOLAIKO arxiv:1709.10413v [math-ph] 13 Jan 018 Abstract. It has been suggested that the distribution of the suitably normalized number
More informationRoot systems and optimal block designs
Root systems and optimal block designs Peter J. Cameron School of Mathematical Sciences Queen Mary, University of London Mile End Road London E1 4NS, UK p.j.cameron@qmul.ac.uk Abstract Motivated by a question
More informationThe Hierarchical Product of Graphs
The Hierarchical Product of Graphs Lali Barrière Francesc Comellas Cristina Dalfó Miquel Àngel Fiol Universitat Politècnica de Catalunya - DMA4 April 8, 2008 Outline 1 Introduction 2 Graphs and matrices
More informationFrom Bernstein approximation to Zauner s conjecture
From Bernstein approximation to Zauner s conjecture Shayne Waldron Mathematics Department, University of Auckland December 5, 2017 Shayne Waldron (University of Auckland) Workshop on Spline Approximation
More informationEigenvalue PDFs. Peter Forrester, M&S, University of Melbourne
Outline Eigenvalue PDFs Peter Forrester, M&S, University of Melbourne Hermitian matrices with real, complex or real quaternion elements Circular ensembles and classical groups Products of random matrices
More informationSymmetries and Dynamics of Discrete Systems
Symmetries and Dynamics of Discrete Systems Talk at CASC 2007, Bonn, Germany Vladimir Kornyak Laboratory of Information Technologies Joint Institute for Nuclear Research 19 September 2007 V. V. Kornyak
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationFluctuation statistics for quantum star graphs
Contemporary Mathematics Fluctuation statistics for quantum star graphs J.P. Keating Abstract. Star graphs are examples of quantum graphs in which the spectral and eigenfunction statistics can be determined
More informationAn Isometric Dynamics for a Causal Set Approach to Discrete Quantum Gravity
University of Denver Digital Commons @ DU Mathematics Preprint Series Department of Mathematics 214 An Isometric Dynamics for a Causal Set Approach to Discrete Quantum Gravity S. Gudder Follow this and
More informationA = A U. U [n] P(A U ). n 1. 2 k(n k). k. k=1
Lecture I jacques@ucsd.edu Notation: Throughout, P denotes probability and E denotes expectation. Denote (X) (r) = X(X 1)... (X r + 1) and let G n,p denote the Erdős-Rényi model of random graphs. 10 Random
More informationWhat is a particle? Keith Fratus. July 17, 2012 UCSB
What is a particle? Keith Fratus UCSB July 17, 2012 Quantum Fields The universe as we know it is fundamentally described by a theory of fields which interact with each other quantum mechanically These
More informationSCHRÖDINGER OPERATORS ON GRAPHS AND GEOMETRY III. GENERAL VERTEX CONDITIONS AND COUNTEREXAMPLES.
ISSN: 1401-5617 SCHRÖDINGER OPERATORS ON GRAPHS AND GEOMETRY III. GENERAL VERTEX CONDITIONS AND COUNTEREXAMPLES. Pavel Kurasov, Rune Suhr Research Reports in Mathematics Number 1, 2018 Department of Mathematics
More informationPermutation Groups. John Bamberg, Michael Giudici and Cheryl Praeger. Centre for the Mathematics of Symmetry and Computation
Notation Basics of permutation group theory Arc-transitive graphs Primitivity Normal subgroups of primitive groups Permutation Groups John Bamberg, Michael Giudici and Cheryl Praeger Centre for the Mathematics
More informationMULTI DE BRUIJN SEQUENCES
MULTI DE BRUIJN SEQUENCES GLENN TESLER * arxiv:1708.03654v1 [math.co] 11 Aug 2017 Abstract. We generalize the notion of a de Bruijn sequence to a multi de Bruijn sequence : a cyclic or linear sequence
More informationFermionic coherent states in infinite dimensions
Fermionic coherent states in infinite dimensions Robert Oeckl Centro de Ciencias Matemáticas Universidad Nacional Autónoma de México Morelia, Mexico Coherent States and their Applications CIRM, Marseille,
More informationREPRESENTATION THEORY WEEK 7
REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable
More informationAnother algorithm for nonnegative matrices
Linear Algebra and its Applications 365 (2003) 3 12 www.elsevier.com/locate/laa Another algorithm for nonnegative matrices Manfred J. Bauch University of Bayreuth, Institute of Mathematics, D-95440 Bayreuth,
More informationUniversality. Why? (Bohigas, Giannoni, Schmit 84; see also Casati, Vals-Gris, Guarneri; Berry, Tabor)
Universality Many quantum properties of chaotic systems are universal and agree with predictions from random matrix theory, in particular the statistics of energy levels. (Bohigas, Giannoni, Schmit 84;
More informationEigenvalues and edge-connectivity of regular graphs
Eigenvalues and edge-connectivity of regular graphs Sebastian M. Cioabă University of Delaware Department of Mathematical Sciences Newark DE 19716, USA cioaba@math.udel.edu August 3, 009 Abstract In this
More informationUPPER BOUNDS FOR EIGENVALUES OF THE DISCRETE AND CONTINUOUS LAPLACE OPERATORS
UPPER BOUNDS FOR EIGENVALUES OF THE DISCRETE AND CONTINUOUS LAPLACE OPERATORS F. R. K. Chung University of Pennsylvania, Philadelphia, Pennsylvania 904 A. Grigor yan Imperial College, London, SW7 B UK
More informationLectures 6 7 : Marchenko-Pastur Law
Fall 2009 MATH 833 Random Matrices B. Valkó Lectures 6 7 : Marchenko-Pastur Law Notes prepared by: A. Ganguly We will now turn our attention to rectangular matrices. Let X = (X 1, X 2,..., X n ) R p n
More information