scholarly journals Novel Parametric Solutions for the Ideal and Non-Ideal Prouhet Tarry Escott Problem

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1775
Author(s):  
Srikanth Raghavendran ◽  
Veena Narayanan
Keyword(s):  
Upper Bound ◽  
Existence Of Solutions ◽  
Parametric Solution ◽  
Parametric Solutions ◽  
Perfect Squares ◽  
The Ideal ◽  
Second Degree

The present study aims to develop novel parametric solutions for the Prouhet Tarry Escott problem of second degree with sizes 3, 4 and 5. During this investigation, new parametric representations for integers as the sum of three, four and five perfect squares in two distinct ways are identified. Moreover, a new proof for the non-existence of solutions of ideal Prouhet Tarry Escott problem with degree 3 and size 2 is derived. The present work also derives a three parametric solution of ideal Prouhet Tarry Escott problem of degree three and size two. The present study also aimed to discuss the Fibonacci-like pattern in the solutions and finally obtained an upper bound for this new pattern.

scholarly journals A Viable Approach to Mitigating Irreproducibility

Stats ◽  
2021 ◽  
Vol 4 (1) ◽  
pp. 205-215
Author(s):  
David Trafimow ◽  
Tonghui Wang ◽  
Cong Wang
Keyword(s):  
Upper Bound ◽  
Recent Article ◽  
Practical Problem ◽  
The Real ◽  
Viable Approach ◽  
Real Universe ◽  
The Ideal

In a recent article, Trafimow suggested the usefulness of imagining an ideal universe where the only difference between original and replication experiments is the operation of randomness. This contrasts with replication in the real universe where systematicity, as well as randomness, creates differences between original and replication experiments. Although Trafimow showed (a) that the probability of replication in the ideal universe places an upper bound on the probability of replication in the real universe, and (b) how to calculate the probability of replication in the ideal universe, the conception is afflicted with an important practical problem. Too many participants are needed to render the approach palatable to most researchers. The present aim is to address this problem. Embracing skewness is an important part of the solution.

Nilpotent Ideals in Alternative Rings

1980 ◽  
Vol 23 (3) ◽  
pp. 299-303 ◽  
Author(s):  
Michael Rich
Keyword(s):  
Upper Bound ◽  
Associative Ring ◽  
Analogous Result ◽  
Alternative Ring ◽  
Locally Nilpotent ◽  
Open Question ◽  
The Ideal

It is well known and immediate that in an associative ring a nilpotent one-sided ideal generates a nilpotent two-sided ideal. The corresponding open question for alternative rings was raised by M. Slater [6, p. 476]. Hitherto the question has been answered only in the case of a trivial one-sided ideal J (i.e., in case J2 = 0) [5]. In this note we solve the question in its entirety by showing that a nilpotent one-sided ideal K of an alternative ring generates a nilpotent two-sided ideal. In the process we find an upper bound for the index of nilpotency of the ideal generated. The main theorem provides another proof of the fact that a semiprime alternative ring contains no nilpotent one-sided ideals. Finally we note the analogous result for locally nilpotent one-sided ideals.

scholarly journals A new approach to the Tarry–Escott problem

2017 ◽  
Vol 13 (02) ◽  
pp. 393-417 ◽  
Author(s):  
Ajai Choudhry
Keyword(s):  
Positive Integer ◽  
New Method ◽  
New Approach ◽  
Parametric Solutions ◽  
Ideal Solutions ◽  
The Ideal

In this paper we describe a new method of obtaining ideal solutions of the well-known Tarry–Escott problem, that is, the problem of finding two distinct sets of integers [Formula: see text] and [Formula: see text] such that [Formula: see text], [Formula: see text], where [Formula: see text] is a given positive integer. When [Formula: see text], only a limited number of parametric/numerical ideal solutions of the Tarry–Escott problem are known. In this paper, by applying the new method mentioned above, we find several new parametric ideal solutions of the problem when [Formula: see text]. The ideal solutions obtained by this new approach are more general and, very frequently, simpler than the ideal solutions obtained by the earlier methods. We also obtain new parametric solutions of certain diophantine systems that are closely related to the Tarry–Escott problem. These solutions are also more general and simpler than the solutions of diophantine systems published earlier.

Finding an Optimal Set of Cutter Radii for 2D Pocket Machining

2004 ◽  
Author(s):  
Irena Nadjakova ◽  
Sara McMains
Keyword(s):  
Upper Bound ◽  
Approximation Ratio ◽  
Pocket Machining ◽  
Ratio Set ◽  
Tool Set ◽  
The Given ◽  
Optimal Set ◽  
Predetermined Number ◽  
Points Of Discontinuity ◽  
The Ideal

We describe an approach to finding an optimal (within a requested approximation ratio) set of cutter radii for machining a given 2-dimensional pocket. We do not assume that there is a pre-determined set of cutter radii to choose from or a predetermined number of cutters to use. Given an initial set of cutters to choose from, we derive an upper bound on the approximation ratio of what is achievable choosing from this set compared to the ideal set. We then use this bound to subdivide the intervals between the given radii until the requested approximation ratio is achieved. We also look at the machinable area as a function of the tool radius. We show that this area is continuous everywhere, except at a certain set of radii determinable by constructing the Voronoi diagram of the pocket. This lets us avoid subdividing the intervals around the points of discontinuity, improving both running time and the size of the output tool set.

scholarly journals Improved Upper Bound to Product Rate Variation Problem

2020 ◽  
Vol 37 (1-2) ◽  
pp. 47-54
Author(s):  
Shree Ram Khadka
Keyword(s):  
Upper Bound ◽  
Feasible Solution ◽  
Exact Algorithms ◽  
Maximum Deviation ◽  
Rate Variation ◽  
Sequencing Problem ◽  
Variation Problem ◽  
Total Product ◽  
Cumulative Production ◽  
The Ideal

The sequencing problem which minimizes the deviation between the actual (integral) and the ideal (rational) cumulative production of a variety of models of a common base product is called the product rate variation problem. If the objective is to minimize the maximum deviation, the problem is bottleneck product rate variation problem and the problem with the objective of minimizing all the deviations is the total product rate variation problem. The problem has been widely studied with several pseudo-polynomial time exact algorithms and heurism-tics. The lower bound of a feasible solution to the problem has been investigated to be tight. However, the upper bound of a feasible solution had been established in the literature which could further be improved. In this paper, we propose the improved upper bound for BPRVP and TPRVP.

scholarly journals On The Binomial Edge Ideal of a Pair of Graphs

10.37236/2987 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Sara Saeedi Madani ◽  
Dariush Kiani
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Betti Numbers ◽  
Edge Ideal ◽  
Closed Graph ◽  
Linear Relations ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Generic Matrix ◽  
The Ideal

We characterize all pairs of graphs $(G_1,G_2)$, for which the binomial edge ideal $J_{G_1,G_2}$ has linear relations. We show that $J_{G_1,G_2}$ has a linear resolution if and only if $G_1$ and $G_2$ are complete and one of them is just an edge. We also compute some of the graded Betti numbers of the binomial edge ideal of a pair of graphs with respect to some graphical terms. In particular, we show that for every pair of graphs $(G_1,G_2)$ with girth (i.e. the length of a shortest cycle in the graph) greater than 3, $\beta_{i,i+2}(J_{G_1,G_2})=0$, for all $i$. Moreover, we give a lower bound for the Castelnuovo-Mumford regularity of any binomial edge ideal $J_{G_1,G_2}$ and hence the ideal of adjacent $2$-minors of a generic matrix. We also obtain an upper bound for the regularity of $J_{G_1,G_2}$, if $G_1$ is complete and $G_2$ is a closed graph.

EXISTENCE RESULTS FOR THE STATIC CONTACT PROBLEM WITH COULOMB FRICTION

1998 ◽  
Vol 08 (03) ◽  
pp. 445-468 ◽  
Author(s):  
C. ECK ◽  
J. JARUSEK
Keyword(s):  
Contact Problem ◽  
Coefficient Of Friction ◽  
Upper Bound ◽  
Penalty Method ◽  
Coulomb Friction ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Static Contact ◽  
The Coefficient Of Friction ◽  
Smoothing Procedure

We prove the existence of solutions to the static contact problem with Coulomb friction, provided that the coefficient of friction is small enough. The proof employs the penalty method and a certain smoothing procedure for the friction functional. Using optimal trace estimates for the solutions of the Lamé equations, we calculate an upper bound for the admissible coefficient of friction which is greater than the corresponding bounds proposed by Necas, Jarusek and Haslinger (1980) and by Jarusek (1983).

scholarly journals Mining Low-Variance Biclusters to Discover Coregulation Modules in Sequencing Datasets

2012 ◽  
Vol 20 (1) ◽  
pp. 15-27 ◽  
Author(s):  
Zhen Hu ◽  
Raj Bhatnagar
Keyword(s):  
Transcription Factors ◽  
High Throughput ◽  
Upper Bound ◽  
High Throughput Sequencing ◽  
Subspace Clustering ◽  
Mean Value ◽  
Variance Bounds ◽  
Clustering Techniques ◽  
Gene Sets ◽  
The Ideal

High-throughput sequencing (CHIP-Seq) data exhibit binding events with possible binding locations and their strengths, followed by interpretation of the locations of peaks. Recent methods tend to summarize all CHIP-Seq peaks detected within a limited up and down region of each gene into one real-valued score in order to quantify the probability of regulation in a region. Applying subspace clustering techniques on these scores can help discover important knowledge such as the potential co-regulation or co-factor mechanisms. The ideal biclusters generated would contain subsets of genes and transcription factors (TF) such that the cell-values in biclusters are distributed around a mean value with very low variance. Such biclusters would indicate TF sets regulating gene sets with very similar probability values. However, most existing biclustering algorithms neither enforce low variance as the desired property of a bicluster, nor use variance as a guiding metric while searching for the desirable biclusters. In this paper we present an algorithm that searches a space of all overlapping biclusters organized in a lattice, and uses an upper bound on variance values of biclusters as the guiding metric. We show the algorithm to be an efficient and effective method for discovering the possibly overlapping biclusters under pre-defined variance bounds. We present in this paper our algorithm, its results with synthetic, CHIP-Seq and motif datasets, and compare them with the results obtained by other algorithms to demonstrate the power and effectiveness of our algorithm.

scholarly journals Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2779
Author(s):  
Petr Karlovsky
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Natural Numbers ◽  
Parametric Solutions ◽  
Special Cases ◽  
Positive Integers

Diophantine equations ∏i=1nxi=F∑i=1nxi with xi,F∈ℤ+ associate the products and sums of n natural numbers. Only special cases have been studied so far. Here, we provide new parametric solutions depending on F and the divisors of F or F2. One of these solutions shows that the equation of any degree with any F is solvable. For n = 2, exactly two solutions exist if and only if F is a prime. These solutions are (2F,2F) and (F + 1, F(F + 1)). We generalize an upper bound for the sum of solution terms from n = 3 established by Crilly and Fletcher in 2015 to any n to be F+1F+n−1 and determine a lower bound to be nnFn−1. Confining the solutions to n-tuples consisting of distinct terms, equations of the 4th degree with any F are solvable but equations of the 5th to 9th degree are not. An upper bound for the sum of terms of distinct-term solutions is conjectured to be F+1F+n−2n−1!/2+1/n−2!. The conjecture is supported by computation, which also indicates that the upper bound equals the largest sum of solution terms if and only if F=n+k−2n−2!−1, k∈ℤ+. Computation provides further insights into the relationships between F and the sum of terms of distinct-term solutions.

scholarly journals SYMMETRIC HOMOGENEOUS DIOPHANTINE EQUATIONS OF ODD DEGREE

2013 ◽  
Vol 09 (04) ◽  
pp. 867-876 ◽  
Author(s):  
M. A. REYNYA
Keyword(s):  
Diophantine Equations ◽  
Parametric Solution ◽  
Elementary Approach ◽  
Symmetric Form ◽  
Model Case ◽  
Parametric Solutions ◽  
Odd Degree

An elementary approach for finding non-trivial parametric solutions to homogeneous symmetric Diophantine equations of odd degree in sufficiently many variables is presented. The method is based on studying a model case of quintic symmetric Diophantine equations in six variables. We prove that every symmetric form of odd degree n ≥ 5 in 6 ⋅ 2n - 5 variables has a rational parametric solution depending on 2n-8 parameters. We also present a solution to a problem of Waring type: if F(x1,…, xN) is a symmetric form of odd degree n ≥ 5 in N = 6 ⋅ 2n-4 variables, then for any q ∈ ℚ the equation F(xi) = q has a rational parametric solution depending on 2n - 6 parameters.

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