Prime Cubes Research Articles

Exceptional sets for sums of five and six almost equal prime cubes

We investigate the exceptional sets of natural number n which can be represented as sums of five and six cubes of almost equal primes, i.e. $${n=p_1^3+\cdots+p_s^3}$$ (s=5,6). It is established that almost all natural numbers n subject to certain congruence conditions have the above representation with $${|p_j-(n/s)^{\frac{1}{3}}|\leq n^{\theta_s/3+\varepsilon}}$$ ( $${1\leq j\leq s}$$ ), where $${\theta_5=8/9+\varepsilon}$$ and $${\theta_6=5/6+\varepsilon}$$ .

Tetravalent half-arc-transitive graphs of order a product of three primes

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Let n be a product of three primes. The problem on the classification of the tetravalent half-arc-transitive graphs of order n has been considered by Xu (1992), Feng et al. (2007) and Wang and Feng (2010), and it was solved for the cases where n is a prime cube or twice a product of two primes. In this paper, we solve this problem for the remaining cases. In particular, there exist some families of these graphs which have a solvable automorphism group but are not metacirculants.

Exceptional set for sums of almost equal prime cubes

We prove that almost all integers N satisfying some necessary congruence conditions can be written as sum of the form N = p 13 +…+ p 1 s with | p i( N/s )1/3| ≤ N 1/3- θ s , where θ s = 7/261 - 2 e , 5/159 - e , 11/333 - e , 19/561 - e for s = 5, 6, 7, 8, respectively.

Sums of nine almost equal prime cubes

We prove that each sufficiently large odd integer N can be written as sum of the form N = p 1 3 + p 2 3 + ⋯ + p 9 3 with |p j − (N/9)1/3| ⩽ N (1/3)−θ , where p j , j = 1, 2, …, 9, are primes and θ = (1/51) − ɛ.

Characterization of p-schemes of prime cube order

Let p be a prime, let S be a non-commutative p-scheme of order p 3 , and assume that S is not thin. We prove that each closed subset of order p 2 of S is an elementary abelian p-group. We also construct non-commutative schurian p-schemes of order p 3 .

Sums of almost equal prime cubes

We prove that almost all integers N satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 5; 6; 7; 8, i.e., N = p 1 3 + ... + p 3 with |p i − (N/j)1/3| ≦ $$ N^{1/3 - \delta _j + \varepsilon } $$ (1 ≦ i ≦ j), for δ j = 1/45; 1/30; 1/25; 2/45, respectively.

Sums of two relatively prime cubes

Sums of two relatively prime cubes

Crossing Graphs as Joins of Graphs and Cartesian Products of Median Graphs

For a partial cube $G$ its crossing graph $G^\#$ is the graph whose vertices are the T-classes of $G$, two classes being adjacent if they cross on some cycle in $G$. The following problem posed in [S. Klavzcar and H. M. Mulder, SIAM J. Discrete Math., 15 (2002), pp. 235-251, Problem 7.1] is considered: What can be said about the partial cube $G$ if $G^\#$ is the join $A\oplus B$ of graphs $A$ and $B$ with at least one edge? It is proved that for arbitrary graphs $A$ and $B$, where at least one of them contains an edge, there exists a Cartesian prime partial cube $G$ such that $G^\# = A\oplus B$. On the other hand, if $G$ is a median graph, then $G^\# = A\oplus B$ if and only if $G=H\,\square\, K$, where $H^\# = A$ and $K^\# = B$. Along the way some new facts about partial cubes are obtained; for instance, a bipartite graph of radius 2 is a partial cube if and only if it is $K_{2,3}$-free.

Normality of Tetravalent Cayley Graphs of Odd Prime–cube Order and Its Application

Let p be an odd prime. In this paper we prove that all tetravalent connected Cayley graphs of order p3 are normal. As an application, a classification of tetravalent symmetric graphs of odd prime–cube order is given.

OBDD minimization based on two-level representation of Boolean functions

In this paper, we analyze the basic properties of some Boolean function classes and propose a low complexity OBDD variable ordering algorithm, which is exact (optimum) to some classes of functions and very effective to general two-level form functions. We show that the class of series-parallel functions, which can be expressed by a factored form where each variable appears exactly once, can yield exact OBDD variable orderings in polynomial time. We also study the thin Boolean functions whose corresponding OBDDs can be represented by the form of thin OBDDs in which the number of nonterminal nodes is equal to the number of input variables. We show,that a thin Boolean function always has an essential prime cube cover and the class of series-parallel functions is a proper subset of thin Boolean functions. We propose a heuristic viewing OBDDs as evaluation machines with function cube covers as their inputs and apply a queuing principle in the algorithm design. Our heuristic, the augmented Dynamic Shortest Cube First algorithm, is proven to be optimum for the series-parallel functions and also be very effective for general two-level form functions. Experimental results on a large number of two-level form benchmark circuits show that the algorithm yields an OBDD total size reduction of over 51 percent with only 7 percent CPU time compared to the well-known network-based Fan-in Heuristic implemented in the SIS package. Comparing to the known exact results, ours is only 49 percent larger in size while only uses 0.001 percent CPU time.

Factoring elementary groups of prime cube order into subsets

Let p p be a prime and let G G be the 3 3 -fold direct product of the cyclic group of order p p . Rédei conjectured if G G is the direct product of subsets A A and B B , each of which contains the identity element of G G , then either A A or B B does not generate all of G G . The paper verifies Rédei’s conjecture for p ≤ 11 p\leq 11 .

On sums and differences of two relative prime cubes

On sums and differences of two relative prime cubes

A new graph approach to minimizing processor fragmentation in hypercube multiprocessors

The authors propose a new approach for subcube and noncubic processor allocations for hypercube multiprocessors. The main idea is to represent available processors in the system by means of a prime cube graph (PC-graph). The PC-graph maintains the inter-relationships between free subcubes and hence reduces both internal and external processor fragmentations. Their simulation results show that the PC-graph approach outperforms the existing allocation strategies by 25% to 50% under certain load conditions. >

Bit decision table for hypercube decomposition

A new algorithm for subcube assignment and decomposition is introduced to solve the fragmentation problem in hyper-cube (n-cube) processor allocation. The sharing density vector introduced in the prime cube graph strategy [1] is used as a vehicle to develop the new algorithm. A bit decision table is devised to provide the key parameters for determining the decomposition. It is proven that the result of the algorithm is at least 22% better than the sharing density vector approach.

An algorithm for multiple output minimization

A computer-aided design procedure for the minimization of multiple-output Boolean functions as encountered in the synthesis of VLSI logic circuits is presented. A fast technique for the determination of essential prime cubes without generating all the prime cubes is among the salient features of the algorithm. A new class of selective prime cubes called valid selective prime cubes is described. This class of prime cubes has proved to be a very powerful tool inasmuch as it guides the algorithm to the minimal set of selective prime cubes while encountering either an independent chain or an interconnected chain of cyclic prime cubes. In many cases, this avoids branching, which is computationally an expensive operation. The algorithm does not generate either the complement or all the prime cubes of the functions. Therefore, it is well suited to minimizing functions with a large complement size and/or a very high number of prime cubes. The algorithm has been implemented in Pascal and evaluated using a large number of programmable logic arrays (PLAs) including those of the Berkeley PLA test set. Results of comparison with ESPRESSO II and McBOOLE indicate that the program produces absolute minimal solutions in most of the cases and near-minimal solutions in a few others. >

Average Values of Quantities Appearing in Multiple Output Boolean Minimization

In connection with the problem of two-level minimization of systems of Boolean functions, formulas are obtained for the following statistical quantities: average number of k cubes, prime k cubes, and essential k cubes of a system of Boolean functions. The parameters appearing in the formulas are the number of variables, the number of functions of the system, and the number of ``one'' vertices of each function. Numerical evaluations are given. Increasing by one the number of variables n of a system of m functions roughly results in multiplying the average numbers of cubes and prime cubes by a factor of about 2.2 to 2.3. The ratio of the average numbers of essential cubes and prime cubes rapidly decreases increasing n or m, so that the minimization algorithms, which obtain the essential cubes before the prime cubes, seem statistically unsuitable to solve ``large'' minimization problems. The average occupation memory occurring in Quine, McCluskey, and Bartee algorithms is also evaluated. Its rate of increase with the number of variables is about 2.5.