Unique Integer Research Articles

An experiment on Lowest Unique Integer Games

We experimentally study Lowest Unique Integer Games (LUIGs). In a LUIG, N (>= 3) players submit a positive integer up to M and the player choosing the smallest number not chosen by anyone else wins. LUIGs are simplified versions of real systems such as lottery games and Lowest/Highest Unique Bid Auctions that have been attracting attention from scholars, yet experimental studies are still scarce. Here, we consider four LUIGs with N={3,4} and M={3,4}. We find that (a) choices made by a majority of subjects over 50 rounds of a LUIG were not significantly different from that in the symmetric mixed-strategy Nash equilibrium (MSE) of the LUIG; however, (b) those subjects who behaved significantly differently from what the MSE predicts won the game more frequently than those who behaved similarly to what the MSE predicts.

Rational Polynomials That Take Integer Values at the Fibonacci Numbers

An integer-valued polynomial on a subset S of ℤ is a polynomial f (x) ∊ ℚ [x] with the property f (S) ⊆ ℤ. This article describes the ring of such polynomials in the special case that S is the Fibonacci numbers. An algorithm is described for finding a regular basis, i.e., an ordered sequence of polynomials, the nth one of degree n, with which any such polynomial can be expressed as a unique integer linear combination.

The number of large prime factors of integers and normal numbers

In a series of papers, we constructed large families of normal numbers using the concatenation of the values of the largest prime factor P(n), as n runs through particular sequences of positive integers. A similar approach using the smallest prime factor function also allowed for the construction of normal numbers. Letting ω(n) stand for the number of distinct prime factors of the positive integer n, we then showed that the concatenation of the successive values of |ω(n)-⌊loglogn⌋| in a fixed base q≥2, as n runs through the integers n≥3, yields a normal number. Here we prove the following. Let q≥2 be a fixed integer. Given an integer n≥n 0 =max(q,3), let N be the unique positive integer satisfying q N ≤n<q N+1 and let h(n,q) stand for the residue modulo q of the number of distinct prime factors of n located in the interval [logN,N]. Setting x N :=e N , we then create a normal number in base q using the concatenation of the numbers h(n,q), as n runs through the integers ≥x n 0 .

Radient: Scalable, memory efficient name lookup algorithm for named data networking

Named Data Networking (NDN) aims to discard the existing host-centric networking paradigm just to replace it with a more practical Content-Centric Networking (CCN) paradigm. CCN allows users to fetch and distribute contents directly using their names. NDN router stores all incoming content requests (׳/׳-delimited string components) in the Pending Interest Table (PIT) until they are satisfied. Multiple requests for the same content are merged in a single PIT entry and when the requested content is available, it is forwarded simultaneously to all the requesters. Although NDN has several benefits over the existing IP-based network, replacing IP addresses with names increases memory consumption and lookup cost. One possible way to restrict memory usage is to use name encoding, i.e., to encode identical components of a name with a unique integer. In this paper, we proposed a novel memory efficient name encoding scheme (called, Radient) for PIT and evaluated it extensively. Our results show that the Radient scheme can reduce memory consumption by 35.45% compared to the ENPT for 29 million names.

Waiter–Client and Client–Waiter planarity, colorability and minor games

For a finite set X, a family of sets F⊆2X and a positive integer q, we consider two types of two player, perfect information games with no chance moves. In each round of the (1:q) Waiter–Client game (X,F), the first player, called Waiter, offers the second player, called Client, q+1 elements of the board X which have not been offered previously. Client then chooses one of these elements which he claims and the remaining q elements are claimed by Waiter. Waiter wins this game if by the time every element of X has been claimed by some player, Client has claimed all elements of some A∈F; otherwise Client is the winner. Client–Waiter games are defined analogously, the main difference being that Client wins the game if he manages to claim all elements of some A∈F and Waiter wins otherwise. In this paper we study the Waiter–Client and Client–Waiter versions of the non-planarity, Kt-minor and non-k-colorability games. For each such game, we give a fairly precise estimate of the unique integer q at which the outcome of the game changes from Client’s win to Waiter’s win. We also discuss the relation between our results, random graphs, and the corresponding Maker–Breaker and Avoider–Enforcer games.

An upper bound for least solutions of the exponential Diophantine equation D1x2 - D2y2 = λkz

Let D1, D2, D, k, λ be fixed integers such that D1 ≥ 1, D2 ≥ 1, gcd (D1, D2) = 1, D = D1D2 is not a square, ∣k∣ &gt; 1, gcd (D, k) = 1 and λ = 1 or 4 according as 2 ∤ k or not. In this paper, we prove that every solution class S(l) of the equation D1x2-D2y2 = λkz, gcd (x, y) = 1, z &gt; 0, has a unique positive integer solution [Formula: see text] satisfying [Formula: see text] and [Formula: see text], where z runs over all integer solutions (x,y,z) of S(l),(u1,v1) is the fundamental solution of Pell's equation u2 - Dv2 = 1. This result corrects and improves some previous results given by M. H. Le.

A note on p-adic valuations of Schenker sums

A prime number p is called a Schenker prime if there exists such n ∈ N+ that p n and p | an, where an = ∑n j=0 n! j! n j is a socalled Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning p-adic valuations of an in case when p is a Schenker prime. In particular, they asked whether for each k ∈ N+ there exists a unique positive integer nk < 5 such that v5(am·5k+nk) ≥ k for each nonnegative integer m. We prove that for every k ∈ N+ the inequality v5(an) ≥ k has exactly one solution modulo 5. This confirms the first conjecture stated by the mentioned authors. Moreover, we show that if 37 n then v37(an) ≤ 1, which means that the second conjecture stated by the mentioned authors is not true.

ON THE DIOPHANTINE EQUATION $483^x + 485^y = z^2$

In this paper, we show that the Diophantine equation 483 x +485 y = z 2 has a unique non-negative integer solution where x, y and z are non-negative

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

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.

ON THE DIOPHANTINE EQUATION $5^x + 7^y = z^2$

In this paper, we prove that the Diophantine equation 5 x +63 y = z 2 has a unique non-negative integer solution where x, y and z are non-negative integers. The solution (x,y,z) is (0,1,8).

A Novel Approach for Secure Routing Through BGP Using Symmetric Key

The Border Gateway Protocol (BGP) is the path vector routing protocol that connects different autonomous systems.. These ASes have unique integer numbers which assign by IANA organization. The traditional BGP protocol is not sufficient to provide security and authentication for AS path and verification of AS number ownership as well as network IP prefix. The BGP remains vulnerable to various types of misconfiguration by users and attacks. Many secure BGP algorithms have been proposed but complexity of algorithm and attack on that models still remain open problem. In this paper, we propose an efficient model for SBGP; initially establish trust relationship between BGP peers. In this process BGP use TCP protocol for reliable communication. The BGP routers will attempt to create secure BGP session by exchanging BGP Open messages. During this Open messages master BGP router generate private key with help of cyclic shifting of ASCII of password called cyclic shift algorithm. Then hash of this private key send towards neighbour. Instead of key exchange, we use hashing algorithm, we generate hash of only key through SHA-1. This hash code for private key sent with Open messages during session establishment. When this Open messages receive by neighbor BGP routers, first it generate key using same password with same algorithm & generate hash code for same and then compare both hash code. If it matches then establish secure session with master BGP router & accept the Autonomous system number which is sent by master router during Open Messages. In this manner both BGP speakers make trust relationship between each other & then exchange route UPDATE within secure channel. If hash code at receiver end does not match then simply receiving BGP router discard Open messages and does not make connection with unauthorized AS number. If malicious router wants to inject false route or false ip prefix then it does not create secure session by lack of secure private key. So malicious router does not participate in above BGP routing process.

MORE ON THE DIOPHANTINE EQUATION 2^x + 3^y = z^2

In this paper, we show that (1,0,2) is a unique non-negative integer solution (x,y,z) for the Diophantine equation 3 x + 85 y = z 2 where x,y and z are non-negative integers. This result implies that (1,0,0,2) is a solution (x,u,v,z) for the Diophantine equation 3 x + 5 u 17 v = z 2 .

Minimal Decoupling Indices for Linear Systems

We show that for any right invertible linear system, there is a unique minimum list integers that represent the least expensive increases of infinite structure to produce for decoupling by non regular static state feedback, without changing the essential orders. In this case, this original list of minimal decoupling indices allows to establish necessary and sufficient conditions when the couplings between R* and the rest of the system do not affect the part that must change the structure at infinity, which is, to our knowledge, a second original contribution.

An Effective Approach to Offline Arabic Handwriting Recognition

Segmentation is the most challenging part of the Arabic handwriting recognition, due to the unique characteristics of Arabic writing that allows the same shape to denote different characters. In this paper, an off-line Arabic handwriting recognition system is proposed. The processing details are presented in three main stages. Firstly, the image is skeletonized to one pixel thin. Secondly, transfer each diagonally connected foreground pixel to the closest horizontal or vertical line. Finally, these orthogonal lines are coded as vectors of unique integer numbers; each vector represents one letter of the word. In order to evaluate the proposed techniques, the system has been tested on the IFN/ENIT database, and the experimental results show that our method is superior to those methods currently available.

THE SYMMETRIC CROSSCAP NUMBER OF THE GROUPS OF SMALL-ORDER

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 &gt; 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) &lt; 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.

On-line Certification for All: The PINVOX Algorithm

A protoype algorithm: PINVOX (â??Personal&#x0D; Identification Number by Voice") for on-line certification is&#x0D; introduced to guarantee that scholars have followed, i.e.,&#x0D; listened and watched, a complete recorded lecture with the&#x0D; option of earning a certificate or diploma of completion&#x0D; after remotely attending courses. It is based on the injection&#x0D; of unique, randomly selected and pre-recorded integer&#x0D; numbers (or single letters or words) within the audio trace&#x0D; of a video stream at places where silence is automatically&#x0D; detected. The certificate of completion or â??virtual&#x0D; attendanceâ? is generated on-the-fly after the successful&#x0D; identification of the embedded PINVOX code by a video&#x0D; viewer student.

Note on “A unique integer mathematical model for scheduling deteriorating jobs with rate-modifying activities on a single machine”

In this note, we study scheduling problems with simultaneous considerations of deteriorating jobs and rate-modifying activities on a single machine. The job-dependent deterioration effect and the job-independent deterioration effect models are examined with the objective of minimizing the makespan. We propose polynomial time solutions for all the studied problems.

On minimum sum representations for weighted voting games

A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the “yea” voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. In Freixas and Molinero (Ann. Oper. Res. 166:243–260, 2009) the authors have classified all weighted voting games without a unique minimum sum representation for up to 8 voters. Here we exhaustively classify all weighted voting games consisting of 9 voters which do not admit a unique minimum sum integer weight representation.

MINIMUM NUMBER OF FOX COLORS FOR SMALL PRIMES

This article concerns exact results on the minimum number of colors of a Fox coloring over the integers modulo r, of a link with non-null determinant.Specifically, we prove that whenever the least prime divisor of the determinant of such a link and the modulus r is 2, 3, 5, or 7, then the minimum number of colors is 2, 3, 4, or 4 (respectively) and conversely.We are thus led to conjecture that for each prime p there exists a unique positive integer, mp, with the following property. For any link L of non-null determinant and any modulus r such that p is the least prime divisor of the determinant of L and the modulus r, the minimum number of colors of L modulo r is mp.

On Minimum Sum Representations for Weighted Voting Games

A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the yea voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. In Freixas and Molinero (2009) the authors have classified all weighted voting games without a unique minimum sum representation for up to 8 voters.Here we exhaustively classify all weighted voting games consisting of 9 voters, which do not admit a unique minimum sum integer weight representation.