Page 690 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 690
GENERAL TOPICS
The Number of Solutions to ax + by + cz = n and its Relation to
Quadratic Residues
Damanvir Binner, dbinner@sfu.ca
Simon Fraser University, Canada
We find a very efficient formula for calculating the number of solutions of the equation ax +
by + cz = n in non-negative integer triples (x, y, z), where a, b, c and n are given natural
numbers. This formula involves some summations of floor functions of fractions. To quickly
evaluate these sums, we find a reciprocity relation which generalizes a well-known reciprocity
relation of Gauss, related to the law of quadratic reciprocity. Further, by counting the number
of solutions of the equation px + qy + z = q(p−1) in two different ways, we prove that the above
2
result of Gauss is equivalent to a well-known result of Sylvester related to the Frobenius Coin
Problem. This work has been published in the Journal of Integer Sequences.
A Unified Framework for Constructing Centralized Coded Caching
Schemes
Minquan Cheng, chengqinshi@hotmail.com
Guangxi Normal University, China
Coauthors: Jinyu Wang, Xi Zhong
In caching system, we prefer to design a coded caching scheme with the rate R and the packet
number F as small as possible since the efficiency of transmission in the peak traffic times
increases with the decreasing of R and the realizing complexity increases with the increasing of
F. In this talk, we first introduce a framework for constructing coded caching schemes, which
can represent almost all of the previously known schemes. Based on this framework, we obtain
a new scheme, which generalizes the schemes constructed by Shangguan et al. (IEEE Trans.
Inf. Theory, 64, 5755-5766, 2018) and Yan et al. ( IEEE Commun. Lett., 22, 236-239, 2018)
and has better performance compared with these two schemes since it has advantages on the user
number, the coded gains and the flexible memory size. Then the relationships between a coded
caching scheme and orthogonal array, covering array are derived respectively. Consequently a
tight lower bound on the packet number F is derived since the packet number of the schemes
constructed by Yan et al. (IEEE Trans. Inf. Theory 63, 5821-5833, 2017) gets this lower bound.
Finally based on orthogonal array, we construct a new scheme which has the same user number,
memory size and transmission rate as the scheme constructed by Shangguan et al. (IEEE Trans.
Inf. Theory, 64, 5755-5766, 2018) but has smaller packet number.
The Construction of the Shortest Trajectory on a 2D Surface Using the
Level Lines Information
Olena Iarmosh, elena.iarmosh@gmail.com
Simon Kuznets Kharkiv National University of Economics, Ukraine
Coauthor: Oleh M. Lytvyn
In the modern world, the capabilities to develop extensive transport networks are totally or
almost exhausted in most of the big cities. Therefore, the issues of the optimal patterns of
688
The Number of Solutions to ax + by + cz = n and its Relation to
Quadratic Residues
Damanvir Binner, dbinner@sfu.ca
Simon Fraser University, Canada
We find a very efficient formula for calculating the number of solutions of the equation ax +
by + cz = n in non-negative integer triples (x, y, z), where a, b, c and n are given natural
numbers. This formula involves some summations of floor functions of fractions. To quickly
evaluate these sums, we find a reciprocity relation which generalizes a well-known reciprocity
relation of Gauss, related to the law of quadratic reciprocity. Further, by counting the number
of solutions of the equation px + qy + z = q(p−1) in two different ways, we prove that the above
2
result of Gauss is equivalent to a well-known result of Sylvester related to the Frobenius Coin
Problem. This work has been published in the Journal of Integer Sequences.
A Unified Framework for Constructing Centralized Coded Caching
Schemes
Minquan Cheng, chengqinshi@hotmail.com
Guangxi Normal University, China
Coauthors: Jinyu Wang, Xi Zhong
In caching system, we prefer to design a coded caching scheme with the rate R and the packet
number F as small as possible since the efficiency of transmission in the peak traffic times
increases with the decreasing of R and the realizing complexity increases with the increasing of
F. In this talk, we first introduce a framework for constructing coded caching schemes, which
can represent almost all of the previously known schemes. Based on this framework, we obtain
a new scheme, which generalizes the schemes constructed by Shangguan et al. (IEEE Trans.
Inf. Theory, 64, 5755-5766, 2018) and Yan et al. ( IEEE Commun. Lett., 22, 236-239, 2018)
and has better performance compared with these two schemes since it has advantages on the user
number, the coded gains and the flexible memory size. Then the relationships between a coded
caching scheme and orthogonal array, covering array are derived respectively. Consequently a
tight lower bound on the packet number F is derived since the packet number of the schemes
constructed by Yan et al. (IEEE Trans. Inf. Theory 63, 5821-5833, 2017) gets this lower bound.
Finally based on orthogonal array, we construct a new scheme which has the same user number,
memory size and transmission rate as the scheme constructed by Shangguan et al. (IEEE Trans.
Inf. Theory, 64, 5755-5766, 2018) but has smaller packet number.
The Construction of the Shortest Trajectory on a 2D Surface Using the
Level Lines Information
Olena Iarmosh, elena.iarmosh@gmail.com
Simon Kuznets Kharkiv National University of Economics, Ukraine
Coauthor: Oleh M. Lytvyn
In the modern world, the capabilities to develop extensive transport networks are totally or
almost exhausted in most of the big cities. Therefore, the issues of the optimal patterns of
688
