Heterogeneity in Lowest Unique Integer Game

2014 ◽  
Author(s):  
Takashi Yamada ◽  
Nobuyuki Hanaki
Keyword(s):  
Unique Integer

scholarly journals Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Wensheng Wang ◽  
Anwei Zhu
Keyword(s):  
Random Walk ◽  
Large Deviation ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Law ◽  
Iterated Logarithm ◽  
Cluster Set ◽  
Random Scenery ◽  
Unique Integer

Let X={Xi,i≥1} be a sequence of real valued random variables, S0=0 and Sk=∑i=1kXi  (k≥1). Let σ={σ(x),x∈Z} be a sequence of real valued random variables which are independent of X’s. Denote by Kn=∑k=0nσ(⌊Sk⌋)  (n≥0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋≤a<⌊a⌋+1. It is assumed that σ’s belong to the domain of attraction of a stable law with index 0<β<2. In this paper, by employing conditional argument, we investigate large deviation inequalities, some sufficient conditions for Chover-type laws of the iterated logarithm and the cluster set for random walk in random scenery Kn. The obtained results supplement to some corresponding results in the literature.

On the zeros of linear combinations of derivatives of the Riemann zeta function, II

2018 ◽  
Vol 14 (02) ◽  
pp. 371-382
Author(s):  
K. Paolina Koutsaki ◽  
Albert Tamazyan ◽  
Alexandru Zaharescu
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
Linear Combinations ◽  
The Real ◽  
The Riemann Zeta Function ◽  
Unique Integer ◽  
Riemann Zeta ◽  
Derivatives Of

The relevant number to the Dirichlet series [Formula: see text], is defined to be the unique integer [Formula: see text] with [Formula: see text], which maximizes the quantity [Formula: see text]. In this paper, we classify the set of all relevant numbers to the Dirichlet [Formula: see text]-functions. The zeros of linear combinations of [Formula: see text] and its derivatives are also studied. We give an asymptotic formula for the supremum of the real parts of zeros of such combinations. We also compute the degree of the largest derivative needed for such a combination to vanish at a certain point.

scholarly journals Self-complementing permutations of k-uniform hypergraphs

2009 ◽  
Vol Vol. 11 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Artur Szymański ◽  
Adam Pawel Wojda
Keyword(s):  
Positive Integer ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Unique Integer ◽  
International Audience ◽  
Sigma E

Graphs and Algorithms International audience A k-uniform hypergraph H = ( V; E) is said to be self-complementary whenever it is isomorphic with its complement (H) over bar = ( V; ((V)(k)) - E). Every permutation sigma of the set V such that sigma(e) is an edge of (H) over bar if and only if e is an element of E is called self-complementing. 2-self-comlementary hypergraphs are exactly self complementary graphs introduced independently by Ringel ( 1963) and Sachs ( 1962). <br> For any positive integer n we denote by lambda(n) the unique integer such that n = 2(lambda(n)) c, where c is odd. <br> In the paper we prove that a permutation sigma of [1, n] with orbits O-1,..., O-m O m is a self-complementing permutation of a k-uniform hypergraph of order n if and only if there is an integer l >= 0 such that k = a2(l) + s, a is odd, 0 <= s <= 2(l) and the following two conditions hold: <br> (i)n = b2(l+1) + r,r is an element of {0,..., 2(l) - 1 + s}, and <br> (ii) Sigma(i:lambda(vertical bar Oi vertical bar)<= l) vertical bar O-i vertical bar <= r. <br> For k = 2 this result is the very well known characterization of self-complementing permutation of graphs given by Ringel and Sachs.

scholarly journals Tridiagonal pairs of type III with height one

2019 ◽  
Vol 35 ◽  
pp. 555-582 ◽  
Author(s):  
Xue Li ◽  
Bo Hou ◽  
Suogang Gao
Keyword(s):  
Vector Space ◽  
Closed Field ◽  
Positive Dimension ◽  
Algebraically Closed Field ◽  
Type Iii ◽  
Tridiagonal Pairs ◽  
Unique Integer ◽  
Tridiagonal Pair

Let K denote an algebraically closed field with characteristic 0. Let V denote a vector space over K with finite positive dimension, and let A, A∗ denote a tridiagonal pair on V  of diameter d.  Let V0, . . . , Vd  denote a standard ordering of  the eigenspaces of A on V , and let θ0, . . . , θd denote the corresponding eigenvalues of A. It is assumed that d ≥ 3.  Let ρi  denote the dimension of Vi. The sequence ρ0, ρ1, . . . , ρd is called the shape of the tridiagonal pair. It is known that ρ0 = 1 and there  exists  a  unique  integer  h (0 ≤ h ≤ d/2)  such  that  ρi−1 < ρi  for  1 ≤ i ≤ h,  ρi−1 = ρi  for  h < i ≤ d − h,  and  ρi−1 > ρi for d − h < i ≤ d. The integer h is known as the height of the tridiagonal pair. In this paper, it is showed that the shape of a tridiagonal pair of type III with height one is either 1, 2, 2, . . ., 2, 1 or 1, 3, 3, 1.  In each case, an interesting basis is found for V and the actions of A, A∗ on this basis are described.

SCHEDULING WITH POSITION-BASED DETERIORATING JOBS AND MULTIPLE DETERIORATING RATE-MODIFYING ACTIVITIES

2014 ◽  
Vol 31 (01) ◽  
pp. 1450009 ◽  
Author(s):  
GUIYI WEI ◽  
YONG QIU ◽  
MIN JI
Keyword(s):  
Polynomial Time ◽  
Single Machine ◽  
Completion Time ◽  
Heuristic Algorithms ◽  
Deteriorating Jobs ◽  
Single Machine Scheduling ◽  
Integer Program ◽  
Total Completion Time ◽  
Unique Integer ◽  
Optimal Polynomial

In a recent paper, Ozturkoglu and Bulfin (Ozturkoglu, Y. and RL Bulfin (2011). A unique integer mathematical model for scheduling deteriorating jobs with rate-modifying activities on a single machine. The International Journal of Advanced Manufacturing Technology, 57, 753–762.) formulate a unique integer program to solve the single-machine scheduling for the objectives of minimizing makespan and total completion time. They also propose efficient heuristic algorithms for solving large size problems. However their heuristics are not optimal and so the NP-hardness of the considered problem is still open. In this note, we show that a more general problem can be optimally solved in polynomial time. We also provide optimal polynomial-time solution algorithm for the parallel-machine case to minimize total completion time.

THE SYMMETRIC CROSSCAP NUMBER OF THE GROUPS OF SMALL-ORDER

2012 ◽  
Vol 12 (02) ◽  
pp. 1250164 ◽  
Author(s):  
J. J. ETAYO ◽  
E. MARTÍNEZ
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
The Other ◽  
Klein Surfaces ◽  
Exceptional Groups ◽  
Minimal Genus ◽  
Other Hand ◽  
Unique Integer ◽  
Symmetric Crosscap Number

Every finite group G acts as an automorphism group of some non-orientable Klein surfaces without boundary. The minimal genus of these surfaces is called the symmetric crosscap number and denoted by [Formula: see text]. It is known that 3 cannot be the symmetric crosscap number of a group. Conversely, it is also known that all integers that do not belong to nine classes modulo 144 are the symmetric crosscap number of some group. Here we obtain infinitely many groups whose symmetric crosscap number belong to each one of six of these classes. This result supports the conjecture that 3 is the unique integer which is not the symmetric crosscap number of a group. On the other hand, there are infinitely many groups with symmetric crosscap number 1 or 2. For g > 2 the number of groups G with [Formula: see text] is finite. The value of [Formula: see text] is known when G belongs to certain families of groups. In particular, if o(G) < 32, [Formula: see text] is known for all except thirteen groups. In this work we obtain it for these groups by means of a one-by-one analysis. Finally we obtain the least genus greater than two for those exceptional groups whose symmetric crosscap number is 1 or 2.

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