Combinatorics of topmost discs of multi-peg Tower of Hanoi problem
|
|
- Anis Gillian Mitchell
- 6 years ago
- Views:
Transcription
1 Combinatorics of tomost discs of multi-eg Tower of Hanoi roblem Sandi Klavžar Deartment of Mathematics, PEF, Unversity of Maribor Koroška cesta 160, 000 Maribor, Slovenia Uroš Milutinović Deartment of Mathematics, PEF, University of Maribor Koroška cesta 160, 000 Maribor, Slovenia Ciril Petr Institute of Information Sciences Prešernova 17, 000 Maribor, Slovenia Abstract Combinatorial roerties of the multi-eg Tower of Hanoi roblem on n discs and egs are studied. To-mas are introduced as mas which reflect tomost discs of regular states. We study these mas from several oints of view. We also count the number of edges in grahs of the multi-eg Tower of Hanoi roblem and in this way obtain some combinatorial identities. 1 Introduction The Tower of Hanoi roblem osed in 1884 [1] is by now very well understood. The classical roblem consists of finding the minimum number of moves necessary to transfer a tower of n discs from one eg to another. Several variants and generalizations of the roblem have been roosed, cf., for instance, [5, 7]. Paers [, 3, 9] nicely survey the toic and/or give the corresonding large bibliograhy. When the classical roblem with three egs is generalized to more egs, the roblem becomes notoriously difficult although Hinz [4] susects that the roblem might be solvable by closer examining the grahs associated to the roblem. Suorted by the Ministry of Science and Technology of Slovenia under the grant J
2 In this aer we introduce a artial descrition of regular states of the roblem by considering only the tomost disc of every eg. Formally this is done via so-called to-mas which assign to every regular state a vector whose comonents are labels of tomost discs. For a fixed number of egs this descrition is olynomial in the number of discs, in contrast to the usual descrition of regular states which is exonential in the number of discs. In the rest of this section we introduce the concets and notations needed later. In the next section we describe several roerties of tomas, for instance we comute sizes of their images and classify their unique reimages. In Section 3 we then aly our considerations to grahs of multi-eg Tower of Hanoi roblem and to obtain certain combinatorial identities related to the Stirling numbers of the second kind. The multi-eg Tower of Hanoi roblem consists of 3 egs numbered 0, 1,... 1 and n 1 discs of different sizes. Discs will be numbered 1,,..., n and we assume that they are ordered by size, disc 1 being the smallest one. Initially all discs lie on eg 0 in small-on-large ordering. The objective is to transfer all the discs to eg 1 in the minimum number of legal moves. A legal move is a transfer of the tomost disc from one eg to another eg such that no disc can be moved onto a smaller one. As usual, a state is regular if no larger disc is laced on a smaller one. A regular state can be uniquely described with an n-tule r ( Z ) n = {0, 1,..., 1} n. More recisely, we set r = (r 1, r,...,r n ), where r d denotes the eg on which the disc d is laced. An examle of a regular state on = 5 egs with n = 6 discs is shown in Figure 1. The corresonding 6-tule r is also given (as well as a 5-tule s to be defined in the next section). Figure 1: A regular state A regular state is erfect if all the discs lie on the same eg. In Figure an examle of a erfect state is shown with all discs on eg 3.
3 Figure : A erfect state We call a regular state a sread state, if all the discs are on different egs, see Figure 3. Clearly, for n > there are no sread states. Figure 3: A sread state A regular state is an almost sread state, if for some k 1, discs 1,...,k are on a common eg, while discs k + 1,...,n are each on a rivate eg, cf. Figure 4. Note that a erfect state is an almost sread state with k = n and a sread state is an almost sread state with k = 1. Let r = (r 1, r,...,r n 1, r n ) be a regular state. Then r is a erfect state if and only if all the comonents r i are equal. The state r is a sread state if and only if r i r j for any i j. Finally, r is an almost sread state if and only if there exists k 1 such that all r i are equal for i k and r i r j r 1 for i, j > k, i j. Finally, as usual, let (n) m = n(n 1) (n m + 1) and let S(n, k) denote the Stirling numbers of the second kind. 3
4 Figure 4: An almost sread state To-mas Information about tomost discs suffices to know all the ossible legal moves from a given regular state. From this reason we introduce maings T n, : ( Z ) n ( Z n+1 ) as T n, (r 1, r,...,r n ) = (s 0, s 1,..., s 1 ), where { 0; rk i for 1 k n, s i = min {k ; r k = i}; otherwise. 1 k n Hence, the comonent s i of s is the tomost disc on eg i, if eg i is nonemty; otherwise s i = 0. We will briefly refer to these maings as to-mas. As usual, R(T n, ) denotes the image of T n, and R(T n, ) its size. Observe that in general a to-ma need not be surjective. For instance, the 4-tules (, 3, 4, 5) and (, 1, 4, 4) are not in the image of some to-ma. In the first case the smallest disc is not resent and in the second case the disc 4 is suosed to be simultaneously on two discs. We begin with the lemma which is useful for comuting R(T n, ). Lemma.1 Let s = (s 0, s 1,..., s 1 ) ( Z n+1 ). Then s R(T n, ) if and only if the following two conditions are fulfilled: (i) i {0,..., 1} : s i = 1, (ii) i, j {0,..., 1}: s i = s j (i = j s i = 0). Proof. The conditions are clearly necessary. Indeed, the smallest disc must be one of the to discs and no disc can be tomost simultaneously on two egs. Assume now that s = (s 0, s 1,..., s 1 ) fulfills the two conditions. We need to show that there exists r = (r 1, r,...,r n ) ( Z ) n with T n, (r) = s. 4
5 We define the comonents r i as follows. For any i with s i 0 we set r si = i. By condition (ii) such i is uniquely determined. For all the other discs i (i.e. for all those which do not aear as the tomost discs) we set r i = r 1. (Note that by condition (i) the comonent r 1 has already been defined.) It is now easy to verify that r ( Z ) n reresents a regular state and that T n, (r) = s. Proosition. For any n 1 and 3 we have 1 ( ) 1 R(T n, ) = (n 1) k 1. k k=0 Proof. Let s R(T n, ). Then, by condition (i) of Lemma.1 we have s i = 1 for some i. There are ossibilities for that. After fixing this i, there may be k emty egs for any k with 0 k 1. For a fixed k there are ( ) 1 k ossible selections of k emty egs. The remaining 1 k egs are nonemty, and by the condition (ii) of Lemma.1, which asserts that the tomost discs must be different, we infer that there are ( )( ) ( ) n 1 (n 1) 1) (n 1) ( 1 k 1) = (n 1) k 1 ossibilities to select the 1 k to discs. Let be fixed. Then, as R(T n, ) ( Z n+1 ), the size of R(T n, ) is a olynomial function in n. This observation can be made more recise as follows: Corollary.3 Let 3. Then R(T n, ) = n 1 + O(n ). Proof. Result follows by noting that the leading term from Proosition. is obtained for k = 0. Recall that the number of regular states is n, i.e. the number of regular states is exonential in the number of discs. Therefore Corollary.3 seems to be interesting from the algorithmic oint of view, because the exonential number of regular states is a source of difficulties in studying the grahs of the Tower of Hanoi roblem. Let s R(T n, ). In order to determine T n, 1 (s) we introduce the following function. For a ositive integer d and a -tule s = (s 0, s 1,...,s 1 ) let h(d, s) = {i {0,..., 1} ; 0 < s i < d}. Now we have: 5
6 Lemma.4 Let s R(T n, ). Then T 1 n, (s) = 1; h(n + 1, s) = n, n h(i, s); otherwise. i=1 i s 0,...,s 1 Proof. Suose that h(n + 1, s) = n. Then every disc in (any element of) T 1 n, (s) is tomost. It follows that the reimage of s is unique, i.e. T 1 n, (s) = 1. Assume now that at least one disc in T 1 n, (s), say i, is not tomost. Then h(i, s) is nonzero and reresents the number of egs on which we can lace the disc i in a regular state corresonding to s. Therefore T 1 n, (s) is at least i h(i, s), where i runs over all discs that are not tomost. On the other hand, a disc i can only be laced on a eg j if i < s j. If there are several discs that can be laced on the same eg, then they must be on this eg sorted by their sizes, i.e. there is only one ossibility to do that. Therefore, T 1 n, (s) is at most i h(i, s). The case when the reimage of s is unique is characterized in the next theorem. Theorem.5 Let T n, (r) = s. Then T n, 1 (s) = {r} if and only if r is an almost sread state. Proof. Suose first that r is a erfect state, a sread state, or an almost sread state. Then, using Lemma.4, it is easy to verify that T 1 n, (s) = 1. Conversely, let T 1 n, (s) = 1. By Lemma.4 we then either have h(n + 1, s) = n or i h(i, s) = 1. In the first case r is a sread state and thus an almost sread state. Assume now that i h(i, s) = 1. It follows that the index set of this roduct is nonemty and for any such i we have h(i, s) = 1. Let i be such number with h(i, s) = 1. Clearly, i > 1. Moreover, Lemma.1, imlies that there exists a eg j such that s j = 1. Therefore, i < s j and as h(i, s) = 1 disc i must lie on eg s j. Hence, all the discs i with h(i, s) = 1 must be on the same disc (i.e. on the disc s j ). Now, if the number of such discs i is n 1, then r is a erfect state which is an almost sread state. Otherwise, for any other disc k we have k = s t for some t. In other words, all the other discs are tomost. They clearly lie on egs different from s j, and we can conclude that we have an almost sread state. 6
7 3 To-mas and combinatorial identities We have seen in the revious section that the image of a to-ma is of olynomial size in the number of discs. Therefore it is natural to ask which information can be deduced from the image of T n, itself. Here we show how to comute the number of legal moves from a regular state r using only information of T n, (r). Proosition 3.1 Let T n, (r) = s. Then the number of legal moves from r is ( h(n + 1, s) 1 ) 1 h(n + 1, s). Proof. Observe first that the number of nonemty egs is h(n + 1, s), and so the number of emty egs is h(n+1, s). Any tomost disc of a eg can be moved to any emty eg. Therefore, there are h(n+1, s)( h(n+1, s)) legal moves of this kind. It remains to consider the legal moves in which a tomost disc is moved to a nonemty eg. Recall that the h(n + 1, s) tomost discs are different. Thus, the smallest one can be moved to any other of h(n+1, s) 1 nonemty egs, the second smallest can be moved to h(n + 1, s), and so on. Therefore, there are h(n+1,s) 1 i=1 i = 1 (h(n + 1, s) 1)h(n + 1, s) different moves from a nonemty onto a nonemty eg. Summing the above exressions we get the result. The number of tomost discs that can be moved to a eg i is equal to h(s i, s). Moreover, if s j = 0, i.e. if eg j is emty, then by definition we have h(s j, s) = 0. Therefore, the number of legal moves in which a tomost disc is moved to an nonemty eg is equal to i h(s i, s) and so we also have (cf. the above roof) 1 h(s i, s) = 1 (h(n + 1, s) 1)h(n + 1, s). i=0 By Theorem 3.1 we thus only need to know the number of nonemty egs h(n + 1, s) in order to comute the number of legal moves. In order to obtain some additional results, we shall consider grahs of the multi-eg tower of Hanoi roblem. They are defined as follows. The grah G n = (V n, En ) of n discs and egs has regular states as vertices, two vertices being adjacent if one state is obtained from the other by a legal move. 7
8 Theorem 3. For any n 1 and 3 we have: (k( 1 ) 1 ) n 1 k S(n, k)() k = ( 1) i+1 ( ) n i 1. k=1 Proof. We will rove the theorem by counting the number of edges in G n in two ways. Note first that the number of regular states where exactly k egs are nonemty is equal to S(n, k)() k. Therefore, using Proosition 3.1, we have E n = 1 k=1 i=0 (k( 1 ) 1 k ) S(n, k)() k. Consider now the set of states in which the largest disc is fixed on some eg. We infer that the corresonding vertices induce a subgrah of G n isomorhic to G n 1. There are such subgrahs and they form a artition of Vn. Two vertices belonging to two such subgrahs are adjacent if and only if they differ exactly in osition of the largest disc. Since all the remaining discs excet the largest one lie on discs (i.e. on the egs that are not involved in the move of the largest disc), there are V n 1 edges connecting two such subgrahs. It follows that the number of edges of G n can be exressed recursively as E n = En 1 + ( ) V n 1. Thus, E n is equal to: ( ) E n 1 + V n 1 = ( ( ) ) ( ) E n + V n + V n 1 = ( ( ( ) ) ( ) ) ( ) E n 3 + V n 3 + V n + V n 1 = ( ( ( ( ) ) ( ) ) ) ( )... E 1 + V 1 + V V n 1 = ( ) ( ) ( ) ( ) ((...( + ( ) 1 ) + ( ) ) +...) + ( ) n 1 = }{{} n 1 n 1 i=0 ( ) i ( ) (n 1) i. } {{ } n 1 8
9 Combining the above exressions the result follows. Using standard methods the right-hand side exression of Theorem 3. can be summed u. In this way we obtain: Corollary 3.3 For any n 1 and 3 we have: k=1 (k( 1 ) 1 k ) S(n, k)() k = ( ) [ n ( ) n ]. In the roof of Theorem 3. we observed that the number of regular states where exactly k egs are nonemty is S(n, k)() k. On the other hand the number of all regular states is n. Thus in assing we get the following well-known identity, cf. [6, 8]: Corollary 3.4 n = S(n, k)() k = k=1 n S(n, k)() k. In this formula the choice of the uer bound for summation makes no difference, since S(n, k) = 0, if n < k, and () k = 0, if < k. k=1 References [1] N. Claus (= E. Lucas), La Tour d Hanoi, Jeu de calcul, Science et Nature 1(8) (1884) [] A.M. Hinz, The Tower of Hanoi, Enseign. Math. 35 (1989) [3] A.M. Hinz, Shortest aths between regular states of the Tower of Hanoi, Inform. Sci. 63 (199) [4] A.M. Hinz, The Tower of Hanoi, Proceedings of the International Congress in Algebra and Combinatorics 1997, Hong Kong (China), to aear. [5] C.S. Klein and S. Minsker, The suer towers of Hanoi roblem: large rings on small rings, Discrete Math. 114 (1993) [6] J.H. van Lint and R.M. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge,
10 [7] D.G. Pool, The towers and triangles of Professor Claus (or, Pascal knows Hanoi), Math. Magazine 67 (1994) [8] J. Riordan, Combinatorial Identities, John Wiley & Sons, New York, [9] P.K. Stockmeyer, The Tower of Hanoi: A historical survey and bibliograhy, manuscrit, Setember
Simple explicit formulas for the Frame-Stewart numbers
Simple explicit formulas for the Frame-Stewart numbers Sandi Klavžar Department of Mathematics, PEF, University of Maribor Koroška cesta 160, 2000 Maribor, Slovenia sandi.klavzar@uni-lj.si Uroš Milutinović
More informationHanoi Graphs and Some Classical Numbers
Hanoi Graphs and Some Classical Numbers Sandi Klavžar Uroš Milutinović Ciril Petr Abstract The Hanoi graphs Hp n model the p-pegs n-discs Tower of Hanoi problem(s). It was previously known that Stirling
More informationElementary Analysis in Q p
Elementary Analysis in Q Hannah Hutter, May Szedlák, Phili Wirth November 17, 2011 This reort follows very closely the book of Svetlana Katok 1. 1 Sequences and Series In this section we will see some
More informationDOMINATION IN DEGREE SPLITTING GRAPHS S , S t. is a set of vertices having at least two vertices and having the same degree and T = V S i
Journal of Analysis and Comutation, Vol 8, No 1, (January-June 2012) : 1-8 ISSN : 0973-2861 J A C Serials Publications DOMINATION IN DEGREE SPLITTING GRAPHS B BASAVANAGOUD 1*, PRASHANT V PATIL 2 AND SUNILKUMAR
More informationOn the Toppling of a Sand Pile
Discrete Mathematics and Theoretical Comuter Science Proceedings AA (DM-CCG), 2001, 275 286 On the Toling of a Sand Pile Jean-Christohe Novelli 1 and Dominique Rossin 2 1 CNRS, LIFL, Bâtiment M3, Université
More informationARTICLE IN PRESS. Hanoi graphs and some classical numbers. Received 11 January 2005; received in revised form 20 April 2005
PROD. TYPE: COM PP: -8 (col.fig.: nil) EXMATH2002 DTD VER:.0. ED: Devanandh PAGN: Vijay -- SCAN: SGeetha Expo. Math. ( ) www.elsevier.de/exmath Hanoi graphs and some classical numbers Sandi Klavžar a,,
More informationON FREIMAN S 2.4-THEOREM
ON FREIMAN S 2.4-THEOREM ØYSTEIN J. RØDSETH Abstract. Gregory Freiman s celebrated 2.4-Theorem says that if A is a set of residue classes modulo a rime satisfying 2A 2.4 A 3 and A < /35, then A is contained
More informationMATH 2710: NOTES FOR ANALYSIS
MATH 270: NOTES FOR ANALYSIS The main ideas we will learn from analysis center around the idea of a limit. Limits occurs in several settings. We will start with finite limits of sequences, then cover infinite
More informationGRACEFUL NUMBERS. KIRAN R. BHUTANI and ALEXANDER B. LEVIN. Received 14 May 2001
IJMMS 29:8 2002 495 499 PII S06720200765 htt://immshindawicom Hindawi Publishing Cor GRACEFUL NUMBERS KIRAN R BHUTANI and ALEXANDER B LEVIN Received 4 May 200 We construct a labeled grah Dn that reflects
More informationThe Euler Phi Function
The Euler Phi Function 7-3-2006 An arithmetic function takes ositive integers as inuts and roduces real or comlex numbers as oututs. If f is an arithmetic function, the divisor sum Dfn) is the sum of the
More informationp-adic Properties of Lengyel s Numbers
1 3 47 6 3 11 Journal of Integer Sequences, Vol. 17 (014), Article 14.7.3 -adic Proerties of Lengyel s Numbers D. Barsky 7 rue La Condamine 75017 Paris France barsky.daniel@orange.fr J.-P. Bézivin 1, Allée
More informationARTICLE IN PRESS Discrete Mathematics ( )
Discrete Mathematics Contents lists available at ScienceDirect Discrete Mathematics journal homeage: wwwelseviercom/locate/disc Maximizing the number of sanning trees in K n -comlements of asteroidal grahs
More informationRANDOM WALKS AND PERCOLATION: AN ANALYSIS OF CURRENT RESEARCH ON MODELING NATURAL PROCESSES
RANDOM WALKS AND PERCOLATION: AN ANALYSIS OF CURRENT RESEARCH ON MODELING NATURAL PROCESSES AARON ZWIEBACH Abstract. In this aer we will analyze research that has been recently done in the field of discrete
More information18.312: Algebraic Combinatorics Lionel Levine. Lecture 12
8.3: Algebraic Combinatorics Lionel Levine Lecture date: March 7, Lecture Notes by: Lou Odette This lecture: A continuation of the last lecture: comutation of µ Πn, the Möbius function over the incidence
More informationLilian Markenzon 1, Nair Maria Maia de Abreu 2* and Luciana Lee 3
Pesquisa Oeracional (2013) 33(1): 123-132 2013 Brazilian Oerations Research Society Printed version ISSN 0101-7438 / Online version ISSN 1678-5142 www.scielo.br/oe SOME RESULTS ABOUT THE CONNECTIVITY OF
More information#A37 INTEGERS 15 (2015) NOTE ON A RESULT OF CHUNG ON WEIL TYPE SUMS
#A37 INTEGERS 15 (2015) NOTE ON A RESULT OF CHUNG ON WEIL TYPE SUMS Norbert Hegyvári ELTE TTK, Eötvös University, Institute of Mathematics, Budaest, Hungary hegyvari@elte.hu François Hennecart Université
More informationMath 4400/6400 Homework #8 solutions. 1. Let P be an odd integer (not necessarily prime). Show that modulo 2,
MATH 4400 roblems. Math 4400/6400 Homework # solutions 1. Let P be an odd integer not necessarily rime. Show that modulo, { P 1 0 if P 1, 7 mod, 1 if P 3, mod. Proof. Suose that P 1 mod. Then we can write
More informationBOUNDS FOR THE SIZE OF SETS WITH THE PROPERTY D(n) Andrej Dujella University of Zagreb, Croatia
GLASNIK MATMATIČKI Vol. 39(59(2004, 199 205 BOUNDS FOR TH SIZ OF STS WITH TH PROPRTY D(n Andrej Dujella University of Zagreb, Croatia Abstract. Let n be a nonzero integer and a 1 < a 2 < < a m ositive
More information#A47 INTEGERS 15 (2015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS
#A47 INTEGERS 15 (015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS Mihai Ciu Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit No. 5,
More informationThe Graph Accessibility Problem and the Universality of the Collision CRCW Conflict Resolution Rule
The Grah Accessibility Problem and the Universality of the Collision CRCW Conflict Resolution Rule STEFAN D. BRUDA Deartment of Comuter Science Bisho s University Lennoxville, Quebec J1M 1Z7 CANADA bruda@cs.ubishos.ca
More informationStrong Matching of Points with Geometric Shapes
Strong Matching of Points with Geometric Shaes Ahmad Biniaz Anil Maheshwari Michiel Smid School of Comuter Science, Carleton University, Ottawa, Canada December 9, 05 In memory of Ferran Hurtado. Abstract
More information3 Properties of Dedekind domains
18.785 Number theory I Fall 2016 Lecture #3 09/15/2016 3 Proerties of Dedekind domains In the revious lecture we defined a Dedekind domain as a noetherian domain A that satisfies either of the following
More informationON THE LEAST SIGNIFICANT p ADIC DIGITS OF CERTAIN LUCAS NUMBERS
#A13 INTEGERS 14 (014) ON THE LEAST SIGNIFICANT ADIC DIGITS OF CERTAIN LUCAS NUMBERS Tamás Lengyel Deartment of Mathematics, Occidental College, Los Angeles, California lengyel@oxy.edu Received: 6/13/13,
More informationRepresenting Integers as the Sum of Two Squares in the Ring Z n
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 17 (2014), Article 14.7.4 Reresenting Integers as the Sum of Two Squares in the Ring Z n Joshua Harrington, Lenny Jones, and Alicia Lamarche Deartment
More information1 1 c (a) 1 (b) 1 Figure 1: (a) First ath followed by salesman in the stris method. (b) Alternative ath. 4. D = distance travelled closing the loo. Th
18.415/6.854 Advanced Algorithms ovember 7, 1996 Euclidean TSP (art I) Lecturer: Michel X. Goemans MIT These notes are based on scribe notes by Marios Paaefthymiou and Mike Klugerman. 1 Euclidean TSP Consider
More informationApproximating min-max k-clustering
Aroximating min-max k-clustering Asaf Levin July 24, 2007 Abstract We consider the roblems of set artitioning into k clusters with minimum total cost and minimum of the maximum cost of a cluster. The cost
More informationCHAPTER 2: SMOOTH MAPS. 1. Introduction In this chapter we introduce smooth maps between manifolds, and some important
CHAPTER 2: SMOOTH MAPS DAVID GLICKENSTEIN 1. Introduction In this chater we introduce smooth mas between manifolds, and some imortant concets. De nition 1. A function f : M! R k is a smooth function if
More informationAnalysis of some entrance probabilities for killed birth-death processes
Analysis of some entrance robabilities for killed birth-death rocesses Master s Thesis O.J.G. van der Velde Suervisor: Dr. F.M. Sieksma July 5, 207 Mathematical Institute, Leiden University Contents Introduction
More informationGENERICITY OF INFINITE-ORDER ELEMENTS IN HYPERBOLIC GROUPS
GENERICITY OF INFINITE-ORDER ELEMENTS IN HYPERBOLIC GROUPS PALLAVI DANI 1. Introduction Let Γ be a finitely generated grou and let S be a finite set of generators for Γ. This determines a word metric on
More informationMATH 361: NUMBER THEORY EIGHTH LECTURE
MATH 361: NUMBER THEORY EIGHTH LECTURE 1. Quadratic Recirocity: Introduction Quadratic recirocity is the first result of modern number theory. Lagrange conjectured it in the late 1700 s, but it was first
More informationarxiv:math/ v2 [math.nt] 21 Oct 2004
SUMS OF THE FORM 1/x k 1 + +1/x k n MODULO A PRIME arxiv:math/0403360v2 [math.nt] 21 Oct 2004 Ernie Croot 1 Deartment of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 ecroot@math.gatech.edu
More informationA construction of bent functions from plateaued functions
A construction of bent functions from lateaued functions Ayça Çeşmelioğlu, Wilfried Meidl Sabancı University, MDBF, Orhanlı, 34956 Tuzla, İstanbul, Turkey. Abstract In this resentation, a technique for
More informationLinear mod one transformations and the distribution of fractional parts {ξ(p/q) n }
ACTA ARITHMETICA 114.4 (2004) Linear mod one transformations and the distribution of fractional arts {ξ(/q) n } by Yann Bugeaud (Strasbourg) 1. Introduction. It is well known (see e.g. [7, Chater 1, Corollary
More informationMatching Transversal Edge Domination in Graphs
Available at htt://vamuedu/aam Al Al Math ISSN: 19-9466 Vol 11, Issue (December 016), 919-99 Alications and Alied Mathematics: An International Journal (AAM) Matching Transversal Edge Domination in Grahs
More informationAn Inverse Problem for Two Spectra of Complex Finite Jacobi Matrices
Coyright 202 Tech Science Press CMES, vol.86, no.4,.30-39, 202 An Inverse Problem for Two Sectra of Comlex Finite Jacobi Matrices Gusein Sh. Guseinov Abstract: This aer deals with the inverse sectral roblem
More information216 S. Chandrasearan and I.C.F. Isen Our results dier from those of Sun [14] in two asects: we assume that comuted eigenvalues or singular values are
Numer. Math. 68: 215{223 (1994) Numerische Mathemati c Sringer-Verlag 1994 Electronic Edition Bacward errors for eigenvalue and singular value decomositions? S. Chandrasearan??, I.C.F. Isen??? Deartment
More informationMATH 6210: SOLUTIONS TO PROBLEM SET #3
MATH 6210: SOLUTIONS TO PROBLEM SET #3 Rudin, Chater 4, Problem #3. The sace L (T) is searable since the trigonometric olynomials with comlex coefficients whose real and imaginary arts are rational form
More informationQUADRATIC RECIPROCITY
QUADRATIC RECIPROCITY JORDAN SCHETTLER Abstract. The goals of this roject are to have the reader(s) gain an areciation for the usefulness of Legendre symbols and ultimately recreate Eisenstein s slick
More informationGaps in Semigroups. Université Pierre et Marie Curie, Paris 6, Equipe Combinatoire - Case 189, 4 Place Jussieu Paris Cedex 05, France.
Gas in Semigrous J.L. Ramírez Alfonsín Université Pierre et Marie Curie, Paris 6, Equie Combinatoire - Case 189, 4 Place Jussieu Paris 755 Cedex 05, France. Abstract In this aer we investigate the behaviour
More information19th Bay Area Mathematical Olympiad. Problems and Solutions. February 28, 2017
th Bay Area Mathematical Olymiad February, 07 Problems and Solutions BAMO- and BAMO- are each 5-question essay-roof exams, for middle- and high-school students, resectively. The roblems in each exam are
More informationA Note on the Positive Nonoscillatory Solutions of the Difference Equation
Int. Journal of Math. Analysis, Vol. 4, 1, no. 36, 1787-1798 A Note on the Positive Nonoscillatory Solutions of the Difference Equation x n+1 = α c ix n i + x n k c ix n i ) Vu Van Khuong 1 and Mai Nam
More informationEconometrica Supplementary Material
Econometrica Sulementary Material SUPPLEMENT TO WEAKLY BELIEF-FREE EQUILIBRIA IN REPEATED GAMES WITH PRIVATE MONITORING (Econometrica, Vol. 79, No. 3, May 2011, 877 892) BY KANDORI,MICHIHIRO IN THIS SUPPLEMENT,
More informationAveraging sums of powers of integers and Faulhaber polynomials
Annales Mathematicae et Informaticae 42 (20. 0 htt://ami.ektf.hu Averaging sums of owers of integers and Faulhaber olynomials José Luis Cereceda a a Distrito Telefónica Madrid Sain jl.cereceda@movistar.es
More informationVarious Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various Families of R n Norms and Some Open Problems
Int. J. Oen Problems Comt. Math., Vol. 3, No. 2, June 2010 ISSN 1998-6262; Coyright c ICSRS Publication, 2010 www.i-csrs.org Various Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various
More informationCOMMUNICATION BETWEEN SHAREHOLDERS 1
COMMUNICATION BTWN SHARHOLDRS 1 A B. O A : A D Lemma B.1. U to µ Z r 2 σ2 Z + σ2 X 2r ω 2 an additive constant that does not deend on a or θ, the agents ayoffs can be written as: 2r rθa ω2 + θ µ Y rcov
More informationCHAPTER 3: TANGENT SPACE
CHAPTER 3: TANGENT SPACE DAVID GLICKENSTEIN 1. Tangent sace We shall de ne the tangent sace in several ways. We rst try gluing them together. We know vectors in a Euclidean sace require a baseoint x 2
More information#A8 INTEGERS 12 (2012) PARTITION OF AN INTEGER INTO DISTINCT BOUNDED PARTS, IDENTITIES AND BOUNDS
#A8 INTEGERS 1 (01) PARTITION OF AN INTEGER INTO DISTINCT BOUNDED PARTS, IDENTITIES AND BOUNDS Mohammadreza Bidar 1 Deartment of Mathematics, Sharif University of Technology, Tehran, Iran mrebidar@gmailcom
More informationMatching points with geometric objects: Combinatorial results
Matching oints with geometric objects: Combinatorial results Bernardo Ábrego1, Esther Arkin, Silvia Fernández 1, Ferran Hurtado 3, Mikio Kano 4, Joseh S. B. Mitchell, and Jorge Urrutia 1 Deartment of Mathematics,
More informationarxiv: v2 [math.ac] 5 Jan 2018
Random Monomial Ideals Jesús A. De Loera, Sonja Petrović, Lily Silverstein, Desina Stasi, Dane Wilburne arxiv:70.070v [math.ac] Jan 8 Abstract: Insired by the study of random grahs and simlicial comlexes,
More informationwhere x i is the ith coordinate of x R N. 1. Show that the following upper bound holds for the growth function of H:
Mehryar Mohri Foundations of Machine Learning Courant Institute of Mathematical Sciences Homework assignment 2 October 25, 2017 Due: November 08, 2017 A. Growth function Growth function of stum functions.
More informationImprovement on the Decay of Crossing Numbers
Grahs and Combinatorics 2013) 29:365 371 DOI 10.1007/s00373-012-1137-3 ORIGINAL PAPER Imrovement on the Decay of Crossing Numbers Jakub Černý Jan Kynčl Géza Tóth Received: 24 Aril 2007 / Revised: 1 November
More informationA Note on Guaranteed Sparse Recovery via l 1 -Minimization
A Note on Guaranteed Sarse Recovery via l -Minimization Simon Foucart, Université Pierre et Marie Curie Abstract It is roved that every s-sarse vector x C N can be recovered from the measurement vector
More informationA CLASS OF ALGEBRAIC-EXPONENTIAL CONGRUENCES MODULO p. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXI, (2002),. 3 7 3 A CLASS OF ALGEBRAIC-EXPONENTIAL CONGRUENCES MODULO C. COBELI, M. VÂJÂITU and A. ZAHARESCU Abstract. Let be a rime number, J a set of consecutive integers,
More informationA structure theorem for product sets in extra special groups
A structure theorem for roduct sets in extra secial grous Thang Pham Michael Tait Le Anh Vinh Robert Won arxiv:1704.07849v1 [math.nt] 25 Ar 2017 Abstract HegyváriandHennecartshowedthatifB isasufficientlylargebrickofaheisenberg
More informationLenny Jones Department of Mathematics, Shippensburg University, Shippensburg, Pennsylvania Daniel White
#A10 INTEGERS 1A (01): John Selfrige Memorial Issue SIERPIŃSKI NUMBERS IN IMAGINARY QUADRATIC FIELDS Lenny Jones Deartment of Mathematics, Shiensburg University, Shiensburg, Pennsylvania lkjone@shi.eu
More informationThe Fekete Szegő theorem with splitting conditions: Part I
ACTA ARITHMETICA XCIII.2 (2000) The Fekete Szegő theorem with slitting conditions: Part I by Robert Rumely (Athens, GA) A classical theorem of Fekete and Szegő [4] says that if E is a comact set in the
More informationCharacteristics of Fibonacci-type Sequences
Characteristics of Fibonacci-tye Sequences Yarden Blausa May 018 Abstract This aer resents an exloration of the Fibonacci sequence, as well as multi-nacci sequences and the Lucas sequence. We comare and
More informationSOME SUMS OVER IRREDUCIBLE POLYNOMIALS
SOME SUMS OVER IRREDUCIBLE POLYNOMIALS DAVID E SPEYER Abstract We rove a number of conjectures due to Dinesh Thakur concerning sums of the form P hp ) where the sum is over monic irreducible olynomials
More informationTRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES
Khayyam J. Math. DOI:10.22034/kjm.2019.84207 TRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES ISMAEL GARCÍA-BAYONA Communicated by A.M. Peralta Abstract. In this aer, we define two new Schur and
More information(Workshop on Harmonic Analysis on symmetric spaces I.S.I. Bangalore : 9th July 2004) B.Sury
Is e π 163 odd or even? (Worksho on Harmonic Analysis on symmetric saces I.S.I. Bangalore : 9th July 004) B.Sury e π 163 = 653741640768743.999999999999.... The object of this talk is to exlain this amazing
More informationAlmost All Palindromes Are Composite
Almost All Palindromes Are Comosite William D Banks Det of Mathematics, University of Missouri Columbia, MO 65211, USA bbanks@mathmissouriedu Derrick N Hart Det of Mathematics, University of Missouri Columbia,
More informationVerifying Two Conjectures on Generalized Elite Primes
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 12 (2009), Article 09.4.7 Verifying Two Conjectures on Generalized Elite Primes Xiaoqin Li 1 Mathematics Deartment Anhui Normal University Wuhu 241000,
More informationSums of independent random variables
3 Sums of indeendent random variables This lecture collects a number of estimates for sums of indeendent random variables with values in a Banach sace E. We concentrate on sums of the form N γ nx n, where
More informationOn the irreducibility of a polynomial associated with the Strong Factorial Conjecture
On the irreducibility of a olynomial associated with the Strong Factorial Conecture Michael Filaseta Mathematics Deartment University of South Carolina Columbia, SC 29208 USA E-mail: filaseta@math.sc.edu
More informationIntrinsic Approximation on Cantor-like Sets, a Problem of Mahler
Intrinsic Aroximation on Cantor-like Sets, a Problem of Mahler Ryan Broderick, Lior Fishman, Asaf Reich and Barak Weiss July 200 Abstract In 984, Kurt Mahler osed the following fundamental question: How
More informationBest approximation by linear combinations of characteristic functions of half-spaces
Best aroximation by linear combinations of characteristic functions of half-saces Paul C. Kainen Deartment of Mathematics Georgetown University Washington, D.C. 20057-1233, USA Věra Kůrková Institute of
More informationChromaticity of Turán Graphs with At Most Three Edges Deleted
Iranian Journal of Mathematical Sciences and Informatics Vol. 9, No. 2 (2014), 45-64 Chromaticity of Turán Grahs with At Most Three Edges Deleted Gee-Choon Lau a, Yee-hock Peng b, Saeid Alikhani c a Faculty
More informationMarch 4, :21 WSPC/INSTRUCTION FILE FLSpaper2011
International Journal of Number Theory c World Scientific Publishing Comany SOLVING n(n + d) (n + (k 1)d ) = by 2 WITH P (b) Ck M. Filaseta Deartment of Mathematics, University of South Carolina, Columbia,
More informationFactorability in the ring Z[ 5]
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Dissertations, Theses, and Student Research Paers in Mathematics Mathematics, Deartment of 4-2004 Factorability in the ring
More informationMinimum-cost matching in a random graph with random costs
Minimum-cost matching in a random grah with random costs Alan Frieze Tony Johansson Deartment of Mathematical Sciences Carnegie Mellon University Pittsburgh PA 523 U.S.A. November 8, 205 Abstract Let G
More informationPretest (Optional) Use as an additional pacing tool to guide instruction. August 21
Trimester 1 Pretest (Otional) Use as an additional acing tool to guide instruction. August 21 Beyond the Basic Facts In Trimester 1, Grade 8 focus on multilication. Daily Unit 1: Rational vs. Irrational
More informationAdvanced Calculus I. Part A, for both Section 200 and Section 501
Sring 2 Instructions Please write your solutions on your own aer. These roblems should be treated as essay questions. A roblem that says give an examle requires a suorting exlanation. In all roblems, you
More informationOhno-type relation for finite multiple zeta values
Ohno-tye relation for finite multile zeta values Kojiro Oyama Abstract arxiv:1506.00833v3 [math.nt] 23 Se 2017 Ohno s relation is a well-known relation among multile zeta values. In this aer, we rove Ohno-tye
More informationStone Duality for Skew Boolean Algebras with Intersections
Stone Duality for Skew Boolean Algebras with Intersections Andrej Bauer Faculty of Mathematics and Physics University of Ljubljana Andrej.Bauer@andrej.com Karin Cvetko-Vah Faculty of Mathematics and Physics
More informationClassification of rings with unit graphs having domination number less than four
REND. SEM. MAT. UNIV. PADOVA, Vol. 133 (2015) DOI 10.4171/RSMUP/133-9 Classification of rings with unit grahs having domination number less than four S. KIANI (*) - H. R. MAIMANI (**) - M. R. POURNAKI
More informationCERIAS Tech Report The period of the Bell numbers modulo a prime by Peter Montgomery, Sangil Nahm, Samuel Wagstaff Jr Center for Education
CERIAS Tech Reort 2010-01 The eriod of the Bell numbers modulo a rime by Peter Montgomery, Sangil Nahm, Samuel Wagstaff Jr Center for Education and Research Information Assurance and Security Purdue University,
More informationTHE LEAST PRIME QUADRATIC NONRESIDUE IN A PRESCRIBED RESIDUE CLASS MOD 4
THE LEAST PRIME QUADRATIC NONRESIDUE IN A PRESCRIBED RESIDUE CLASS MOD 4 PAUL POLLACK Abstract For all rimes 5, there is a rime quadratic nonresidue q < with q 3 (mod 4 For all rimes 3, there is a rime
More informationSolvability and Number of Roots of Bi-Quadratic Equations over p adic Fields
Malaysian Journal of Mathematical Sciences 10(S February: 15-35 (016 Secial Issue: The 3 rd International Conference on Mathematical Alications in Engineering 014 (ICMAE 14 MALAYSIAN JOURNAL OF MATHEMATICAL
More informationRECIPROCITY LAWS JEREMY BOOHER
RECIPROCITY LAWS JEREMY BOOHER 1 Introduction The law of uadratic recirocity gives a beautiful descrition of which rimes are suares modulo Secial cases of this law going back to Fermat, and Euler and Legendre
More informationOn the normality of p-ary bent functions
Noname manuscrit No. (will be inserted by the editor) On the normality of -ary bent functions Ayça Çeşmelioğlu Wilfried Meidl Alexander Pott Received: date / Acceted: date Abstract In this work, the normality
More informationA note on variational representation for singular values of matrix
Alied Mathematics and Comutation 43 (2003) 559 563 www.elsevier.com/locate/amc A note on variational reresentation for singular values of matrix Zhi-Hao Cao *, Li-Hong Feng Deartment of Mathematics and
More information6 Binary Quadratic forms
6 Binary Quadratic forms 6.1 Fermat-Euler Theorem A binary quadratic form is an exression of the form f(x,y) = ax 2 +bxy +cy 2 where a,b,c Z. Reresentation of an integer by a binary quadratic form has
More informationQUADRATIC RECIPROCITY
QUADRATIC RECIPROCITY JORDAN SCHETTLER Abstract. The goals of this roject are to have the reader(s) gain an areciation for the usefulness of Legendre symbols and ultimately recreate Eisenstein s slick
More informationA CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS. 1. Abstract
A CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS CASEY BRUCK 1. Abstract The goal of this aer is to rovide a concise way for undergraduate mathematics students to learn about how rime numbers behave
More informationAdditive results for the generalized Drazin inverse in a Banach algebra
Additive results for the generalized Drazin inverse in a Banach algebra Dragana S. Cvetković-Ilić Dragan S. Djordjević and Yimin Wei* Abstract In this aer we investigate additive roerties of the generalized
More information#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS
#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS Ramy F. Taki ElDin Physics and Engineering Mathematics Deartment, Faculty of Engineering, Ain Shams University, Cairo, Egyt
More informationFactorizations Of Functions In H p (T n ) Takahiko Nakazi
Factorizations Of Functions In H (T n ) By Takahiko Nakazi * This research was artially suorted by Grant-in-Aid for Scientific Research, Ministry of Education of Jaan 2000 Mathematics Subject Classification
More informationQUADRATIC RECIPROCITY
QUADRATIC RECIPROCIT POOJA PATEL Abstract. This aer is an self-contained exosition of the law of uadratic recirocity. We will give two roofs of the Chinese remainder theorem and a roof of uadratic recirocity.
More informationSmall Zeros of Quadratic Forms Mod P m
International Mathematical Forum, Vol. 8, 2013, no. 8, 357-367 Small Zeros of Quadratic Forms Mod P m Ali H. Hakami Deartment of Mathematics, Faculty of Science, Jazan University P.O. Box 277, Jazan, Postal
More informationarxiv: v2 [math.co] 6 Jul 2017
COUNTING WORDS SATISFYING THE RHYTHMIC ODDITY PROPERTY FRANCK JEDRZEJEWSKI arxiv:1607.07175v2 [math.co] 6 Jul 2017 Abstract. This aer describes an enumeration of all words having a combinatoric roerty
More informationFor q 0; 1; : : : ; `? 1, we have m 0; 1; : : : ; q? 1. The set fh j(x) : j 0; 1; ; : : : ; `? 1g forms a basis for the tness functions dened on the i
Comuting with Haar Functions Sami Khuri Deartment of Mathematics and Comuter Science San Jose State University One Washington Square San Jose, CA 9519-0103, USA khuri@juiter.sjsu.edu Fax: (40)94-500 Keywords:
More informationNetwork Configuration Control Via Connectivity Graph Processes
Network Configuration Control Via Connectivity Grah Processes Abubakr Muhammad Deartment of Electrical and Systems Engineering University of Pennsylvania Philadelhia, PA 90 abubakr@seas.uenn.edu Magnus
More informationErnie Croot 1. Department of Mathematics, Georgia Institute of Technology, Atlanta, GA Abstract
SUMS OF THE FORM 1/x k 1 + + 1/x k n MODULO A PRIME Ernie Croot 1 Deartment of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 ecroot@math.gatech.edu Abstract Using a sum-roduct result
More informationUnit Groups of Semisimple Group Algebras of Abelian p-groups over a Field*
JOURNAL OF ALGEBRA 88, 580589 997 ARTICLE NO. JA96686 Unit Grous of Semisimle Grou Algebras of Abelian -Grous over a Field* Nako A. Nachev and Todor Zh. Mollov Deartment of Algebra, Plodi Uniersity, 4000
More informationTowards understanding the Lorenz curve using the Uniform distribution. Chris J. Stephens. Newcastle City Council, Newcastle upon Tyne, UK
Towards understanding the Lorenz curve using the Uniform distribution Chris J. Stehens Newcastle City Council, Newcastle uon Tyne, UK (For the Gini-Lorenz Conference, University of Siena, Italy, May 2005)
More informationThe Arm Prime Factors Decomposition
The Arm Prime Factors Decomosition Arm Boris Nima arm.boris@gmail.com Abstract We introduce the Arm rime factors decomosition which is the equivalent of the Taylor formula for decomosition of integers
More informationThe inverse Goldbach problem
1 The inverse Goldbach roblem by Christian Elsholtz Submission Setember 7, 2000 (this version includes galley corrections). Aeared in Mathematika 2001. Abstract We imrove the uer and lower bounds of the
More informationTHE ERDÖS - MORDELL THEOREM IN THE EXTERIOR DOMAIN
INTERNATIONAL JOURNAL OF GEOMETRY Vol. 5 (2016), No. 1, 31-38 THE ERDÖS - MORDELL THEOREM IN THE EXTERIOR DOMAIN PETER WALKER Abstract. We show that in the Erd½os-Mordell theorem, the art of the region
More informationA review of the foundations of perfectoid spaces
A review of the foundations of erfectoid saces (Notes for some talks in the Fargues Fontaine curve study grou at Harvard, Oct./Nov. 2017) Matthew Morrow Abstract We give a reasonably detailed overview
More informationA Social Welfare Optimal Sequential Allocation Procedure
A Social Welfare Otimal Sequential Allocation Procedure Thomas Kalinowsi Universität Rostoc, Germany Nina Narodytsa and Toby Walsh NICTA and UNSW, Australia May 2, 201 Abstract We consider a simle sequential
More information