Several Bounds for the K-Tower of Hanoi Puzzle

Abstract This paper follows a previous one on the computation of spatial displacements (Ravani and Ge, 1992). The first paper dealt with the problem of computing spatial displacements from a minimum number of simple features of points, lines, planes, and their combinations. The present paper deals with the same problem using a redundant set of the simple geometric features. The problem for redundant information is formulated as a least squares problem which includes all simple features. A Clifford algebra is used to unify the handling of various feature information. An algorithm for determining the best orientation is developed which involves finding the eigenvector associated with the least eigenvalue of a 4 × 4 symmetric matrix. The best translation is found to be a rational cubic function of the best orientation. Special cases are discussed which yield the best orientation in closed form. In addition, simple algorithms are provided for automatic generation of body-fixed coordinate frames from various feature information. The results have applications in robot and world model calibration for off-line programming and computer vision.

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On the Star Puzzle

GANIT Journal of Bangladesh Mathematical Society ◽

10.3329/ganit.v39i0.44158 ◽

2019 ◽

Vol 39 ◽

pp. 1-14

Author(s):

AAK Majumdar

Keyword(s):

Recurrence Relation ◽

Additional Condition ◽

Tower Of Hanoi ◽

Minimum Number

In the star puzzle, there are four pegs, the usual three pegs, S, P and D, and a fourth one at 0. Starting with a tower of n discs on the peg P, the objective is to transfer it to the peg D, in minimum number of moves, under the conditions of the classical Tower of Hanoi problem and the additional condition that all disc movements are either to or from the fourth peg. Denoting by MS(n) the minimum number of moves required to solve this variant, MS(n) satisfies the recurrence relation . This paper studies rigorously and extensively the above recurrence relation, and gives a solution of it. GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 1-14

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On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

The Electronic Journal of Combinatorics ◽

10.37236/6516 ◽

2018 ◽

Vol 25 (3) ◽

Author(s):

Megumi Asada ◽

Ryan Chen ◽

Florian Frick ◽

Frederick Huang ◽

Maxwell Polevy ◽

...

Keyword(s):

Geometric Property ◽

Convex Sets ◽

Convex Hulls ◽

Open Problems ◽

Higher Dimensional ◽

Special Cases ◽

Minimum Number ◽

Point Set ◽

Colored Version ◽

Special Case

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

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Scalable Algorithms for Server Allocation in Infostations

Handbook of Research on Scalable Computing Technologies ◽

10.4018/978-1-60566-661-7.ch027 ◽

2010 ◽

pp. 645-656

Author(s):

Alan A. Bertossi ◽

M. Cristina Pinotti ◽

Phalguni Gupta

Keyword(s):

Bin Packing ◽

Multiprocessor Scheduling ◽

Interval Graph ◽

Scalable Algorithms ◽

Coverage Area ◽

Special Cases ◽

Minimum Number ◽

Server Allocation ◽

On Line ◽

Web Surfing

The server allocation problem arises in isolated infostations, where mobile users going through the coverage area require immediate high-bit rate communications such as web surfing, file transferring, voice messaging, email and fax. Given a set of service requests, each characterized by a temporal interval and a category, an integer k, and an integer hc for each category c, the problem consists in assigning a server to each request in such a way that at most k mutually simultaneous requests are assigned to the same server at the same time, out of which at most hc are of category c, and the minimum number of servers is used. Since this problem is computationally intractable, a scalable 2-approximation online algorithm is exhibited. Generalizations of the problem are considered, which contain bin-packing, multiprocessor scheduling, and interval graph coloring as special cases, and admit scalable on-line algorithms providing constant approximations.

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Dimer coverings on the Tower of Hanoi graph

International Journal of Modern Physics B ◽

10.1142/s0217979219500437 ◽

2019 ◽

Vol 33 (07) ◽

pp. 1950043

Author(s):

Wei-Bang Li ◽

Shu-Chiuan Chang

Keyword(s):

Diatomic Molecules ◽

Upper Bounds ◽

Lower And Upper Bounds ◽

Tower Of Hanoi ◽

N Stage ◽

Dimension 2 ◽

The Difference

We present the number of dimer coverings Nd(n) on the Tower of Hanoi graph THd(n) at n stage with dimension 2 [Formula: see text][Formula: see text]d[Formula: see text][Formula: see text] 5. When the number of vertices v(n) is even, Nd(n) gives the number of close-packed dimers; when the number of vertices is odd, it is impossible to have a close-packed configurations and one of the outmost vertices is allowed to be unoccupied. We define the entropy of absorption of diatomic molecules per vertex as S[Formula: see text][Formula: see text]=[Formula: see text][Formula: see text] Nd(n)/v(n), that can be shown exactly for TH2, while its lower and upper bounds can be derived in terms of the results at a certain n for THd(n) with 3 [Formula: see text][Formula: see text]d[Formula: see text][Formula: see text] 5. We find that the difference between the lower and upper bounds converges rapidly to zero as n increases, such that the value of S[Formula: see text] with d[Formula: see text]=[Formula: see text]3 and 5 can be calculated with at least 100 correct digits.

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The Tower of Hanoi Problem with Evildoer Discs

Journal of Bangladesh Academy of Sciences ◽

10.3329/jbas.v43i2.45742 ◽

2020 ◽

Vol 43 (2) ◽

pp. 205-209

Author(s):

AAK Majumdar

Keyword(s):

Explicit Form ◽

Optimality Equation ◽

Tower Of Hanoi ◽

New Variant ◽

Minimum Number ◽

Academy Of Sciences

This paper deals with a variant of the classical Tower of Hanoi problem with n ( ≥ 1) discs, of which r discs are evildoers, each of which can be placed directly on top of a smaller disc any number of times. Denoting by E(n, r) the minimum number of moves required to solve the new variant, is given a scheme find the optimality equation satisfied by E(n, r). An explicit form of E(n, r) is then obtained. Journal of Bangladesh Academy of Sciences, Vol. 43, No. 2, 205-209, 2019

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Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

Combinatorics Probability Computing ◽

10.1017/s0963548312000156 ◽

2012 ◽

Vol 21 (4) ◽

pp. 611-622 ◽

Cited By ~ 2

Author(s):

A. KOSTOCHKA ◽

M. KUMBHAT ◽

T. ŁUCZAK

Keyword(s):

Chromatic Number ◽

Upper Bound ◽

Upper Bounds ◽

Uniform Hypergraph ◽

Lower And Upper Bounds ◽

Uniform Hypergraphs ◽

Minimum Number

A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by χCF(). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph with m edges, χCF() is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

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Properties of the Solution Set for Generalized Equilibrium Problems with Lower and Upper Bounds in FC-Metric Spaces

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.671-674.1557 ◽

2013 ◽

Vol 671-674 ◽

pp. 1557-1560

Author(s):

Kai Ting Wen

Keyword(s):

Metric Spaces ◽

Equilibrium Problems ◽

Optimization Problems ◽

Upper Bounds ◽

Solution Set ◽

Lower And Upper Bounds ◽

Paper Properties ◽

Special Cases ◽

Mathematical Problems ◽

Generalized Equilibrium

The equilibrium problem includes many fundamental mathematical problems, e.g., optimization problems, saddle point problems, fixed point problems, economics problems, comple- mentarity problems, variational inequality problems, mechanics, engineering, and others as special cases. In this paper, properties of the solution set for generalized equilibrium problems with lower and upper bounds in FC-metric spaces are studied. In noncompact setting, we obtain that the solution set for generalized equilibrium problems with lower and upper bounds is nonempty and compact. Our results improve and generalize some recent results in the reference therein.

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Dicycle Cover of Hamiltonian Oriented Graphs

Journal of Discrete Mathematics ◽

10.1155/2016/7942192 ◽

2016 ◽

Vol 2016 ◽

pp. 1-5

Author(s):

Khalid A. Alsatami ◽

Hong-Jian Lai ◽

Xindong Zhang

Keyword(s):

Bipartite Graphs ◽

Upper Bounds ◽

Oriented Graphs ◽

Minimum Number ◽

Complete Bipartite Graphs ◽

Number Of Classes

A dicycle cover of a digraph D is a family F of dicycles of D such that each arc of D lies in at least one dicycle in F. We investigate the problem of determining the upper bounds for the minimum number of dicycles which cover all arcs in a strong digraph. Best possible upper bounds of dicycle covers are obtained in a number of classes of digraphs including strong tournaments, Hamiltonian oriented graphs, Hamiltonian oriented complete bipartite graphs, and families of possibly non-Hamiltonian digraphs obtained from these digraphs via a sequence of 2-sum operations.

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On restricted domination in graphs

Mathematica Slovaca ◽

10.2478/s12175-007-0035-2 ◽

2007 ◽

Vol 57 (5) ◽

Cited By ~ 1

Author(s):

Vladimir Samodivkin

Keyword(s):

Dominating Set ◽

Domination Number ◽

Upper Bounds ◽

Minimum Number ◽

Domination In Graphs ◽

Restricted Domination Number

AbstractThe k-restricted domination number of a graph G is the minimum number d k such that for any subset U of k vertices of G, there is a dominating set in G including U and having at most d k vertices. Some new upper bounds in terms of order and degrees for this number are found.

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