Finite or Infinite Number of Solutions of Polynomial Congruences in Two Positive Integer Variables

2014 ◽  
pp. 11-26
Author(s):  
Thomas Bier
Keyword(s):  
Positive Integer ◽  
Infinite Number ◽  
Number Of Solutions ◽  
Integer Variables ◽  
Polynomial Congruences

On the kth root partition function

2021 ◽  
pp. 1-15
Author(s):  
Ya-Li Li ◽  
Jie Wu
Keyword(s):  
Partition Function ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Real Number ◽  
Number Of Solutions ◽  
Asymptotic Development ◽  
Number Formula

For any positive integer [Formula: see text], let [Formula: see text] be the number of solutions of the equation [Formula: see text] with integers [Formula: see text], where [Formula: see text] is the integral part of real number [Formula: see text]. Recently, Luca and Ralaivaosaona gave an asymptotic formula for [Formula: see text]. In this paper, we give an asymptotic development of [Formula: see text] for all [Formula: see text]. Moreover, we prove that the number of such partitions is even (respectively, odd) infinitely often.

Free-surface flows with two stagnation points

1996 ◽  
Vol 324 ◽  
pp. 393-406 ◽  
Author(s):  
J.-M. Vanden-Broeck ◽  
F. Dias
Keyword(s):  
Free Surface ◽  
Infinite Number ◽  
Wave Breaking ◽  
Free Surface Flows ◽  
Number Of Solutions ◽  
Surface Flows ◽  
Stagnation Points ◽  
Surface Profiles ◽  
Countably Infinite

Symmetric suction flows are computed. The flows are free-surface flows with two stagnation points. The configuration is related to the modelling of wave breaking at the bow of a ship. It is shown that there is a countably infinite number of solutions and that the free-surface profiles are characterized by waves.

scholarly journals ON SOLUTIONS TO SOME POLYNOMIAL CONGRUENCES IN SMALL BOXES

2013 ◽  
Vol 89 (2) ◽  
pp. 300-307
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Exponential Sums ◽  
Polynomial Systems ◽  
Side Length ◽  
Character Sum ◽  
Number Of Solutions ◽  
Polynomial Congruences

AbstractWe use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solutions in boxes with the side length below ${p}^{1/ 2} $, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.

scholarly journals A note on the lower bound of representation functions

2021 ◽  
pp. 1-8
Author(s):  
Xing-Wang Jiang ◽  
Csaba Sándor ◽  
Quan-Hui Yang
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Number Of Solutions

For a set [Formula: see text] of nonnegative integers, let [Formula: see text] denote the number of solutions to [Formula: see text] with [Formula: see text], [Formula: see text]. Let [Formula: see text] be the Thue–Morse sequence and [Formula: see text]. Let [Formula: see text] and [Formula: see text] be a positive integer such that [Formula: see text] for all [Formula: see text]. Previously, the first author proved that if [Formula: see text] and [Formula: see text], then [Formula: see text] for all [Formula: see text]. In this paper, we prove that the above lower bound is nearly best possible. We also get some other results.

A Proposed Method to Group the Solutions From Dimensional Synthesis: Planar Triads for Six Precision Position

1992 ◽  
Author(s):  
Xiancheng Lu ◽  
Chuen-Sen Lin
Keyword(s):  
Infinite Number ◽  
Numerical Solutions ◽  
Angular Displacement ◽  
Dimensional Synthesis ◽  
Number Of Solutions ◽  
Jacobian Matrices ◽  
Design Task ◽  
Numerical Examples ◽  
Computer Automation ◽  
Constraint Sets

Abstract In this paper, a method has been proposed to group into six sets the infinite number of solutions from dimensional synthesis of planar triads for six precision positions. The proposed method reveals the relationships between the different configurations of the compatibility linkage and the sets of numerical solutions from dimensional synthesis. By checking the determinant signs and the contunities of values of the sub-Jacobian matrices and their derivatives with respect to the independent angular displacement for all constraint sets in the compatibility linkage, it enables the computer to identify and group the synthesized solutions. Numerical examples have been given to verify the applicability of this method. Six sets of the partial triad Burmester curves have been plotted based on grouped solutions. Suitable solutions can be easily found from the partial triad Burmester curves and utilized for the prescribed design task. This method provides a useful tool to group the dimensional synthesis solutions and enhances the computer automation in the design of linkage mechanisms.

On the values of representation functions III

2019 ◽  
Vol 16 (03) ◽  
pp. 511-522
Author(s):  
Xing-Wang Jiang
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Binary Representation ◽  
Number Of Solutions

For a set [Formula: see text] of nonnegative integers, let [Formula: see text] denote the number of solutions to [Formula: see text] with [Formula: see text]. Let [Formula: see text] be the set of all nonnegative integers [Formula: see text] with an even number of ones in the binary representation of [Formula: see text] and let [Formula: see text]. Let [Formula: see text] and [Formula: see text] be a positive integer such that [Formula: see text] for all [Formula: see text]. In 2011, Chen proved that, if [Formula: see text] and [Formula: see text], then [Formula: see text] for all [Formula: see text]. In this paper, we improve the lower bound by proving that [Formula: see text] for all [Formula: see text].

On the quaternary form x2 + xy + 7y2 + z2 + zt + 7t2

2016 ◽  
Vol 12 (07) ◽  
pp. 1791-1800 ◽  
Author(s):  
Dongxi Ye
Keyword(s):  
Positive Integer ◽  
Number Of Solutions ◽  
Quaternary Form

For any positive integer [Formula: see text], we state and prove a formula for the number of solutions in integers of [Formula: see text]

scholarly journals Problems of Menotactic Orientation According to the Polarized Light of the Sky

1975 ◽  
Vol 30 (1-2) ◽  
pp. 88-90 ◽  
Author(s):  
Kuno Kirschfeld ◽  
M. Lindauer ◽  
H. Martin
Keyword(s):  
Infinite Number ◽  
Polarized Light ◽  
The Sun ◽  
Number Of Solutions ◽  
Special Cases ◽  
Linearly Polarized ◽  
Vector Direction

Abstract It is shown that the knowledge of the E-vector direction of the linearly polarized light at any point of the sky alone is insufficient for the determination of the position of the sun. If the E-vector direction of a second point is not known the knowledge of at least one other parameter is necessary. This parameter might be the height of the sun over the horizon. With the knowledge of the height the infinite number of solutions for the sun’s position becomes reduced to two, or in special cases to one. These cases are derived.

Design of Helical Springs for Minimum Weight, Volume, and Length

1959 ◽  
Vol 81 (1) ◽  
pp. 37-40 ◽  
Author(s):  
R. T. Hinkle ◽  
I. E. Morse
Keyword(s):  
Infinite Number ◽  
Minimum Weight ◽  
Allowable Stress ◽  
Number Of Solutions ◽  
Helical Springs ◽  
Additional Requirement ◽  
Outside Diameter ◽  
Selection Of

In the design of helical springs where the load, deflection, allowable stress, and material are specified, there are an infinite number of solutions. In this paper, equations and graphs are presented for the selection of a spring index that will result in a spring of minimum weight, volume, or length. If, an addition to these requirements, the inside or outside diameter of the spring is fixed, there is only one solution. Equations and graphs are included for the selection of the spring index which will satisfy this additional requirement.

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