On the Distribution of the Number of Solutions of Random Systems of Equations Which are Known to be Consistent

1996 ◽  
Vol 40 (2) ◽  
pp. 376-383 ◽  
Author(s):  
V. A. Kopytsev
Keyword(s):  
Random Systems ◽  
Systems Of Equations ◽  
Number Of Solutions

scholarly journals How many tuples of group elements have a given property? With an appendix by Dmitrii V. Trushin

2014 ◽  
Vol 24 (04) ◽  
pp. 413-428 ◽  
Author(s):  
Anton A. Klyachko ◽  
Anna A. Mkrtchyan
Keyword(s):  
Systems Of Equations ◽  
Number Of Solutions ◽  
First Order ◽  
Group Language ◽  
The Matrix

Generalizing Solomon's theorem, Gordon and Rodriguez-Villegas have proven recently that, in any group, the number of solutions to a system of coefficient-free equations is divisible by the order of this group whenever the rank of the matrix composed of the exponent sums of ith unknown in jth equation is less than the number of unknowns. We generalize this result in two directions: first, we consider equations with coefficients, and second, we consider not only systems of equations but also any first-order formulae in the group language (with constants). Our theorem implies some amusing facts; for example, the number of group elements whose squares lie in a given subgroup is divisible by the order of this subgroup.

scholarly journals Some New Symmetric Equilateral Embeddings of Platonic and Archimedean Polyhedra

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 382
Author(s):  
Kris Coolsaet ◽  
Stan Schein
Keyword(s):  
Computer Algebra ◽  
Degrees Of Freedom ◽  
State Of The Art ◽  
Edge Length ◽  
Additional Constraint ◽  
Icosahedral Symmetry ◽  
Systems Of Equations ◽  
Number Of Solutions ◽  
Tetrahedral Symmetry ◽  
Reasonable Time

The icosahedron and the dodecahedron have the same graph structures as their algebraic conjugates, the great dodecahedron and the great stellated dodecahedron. All four polyhedra are equilateral and have planar faces—thus “EP”—and display icosahedral symmetry. However, the latter two (star polyhedra) are non-convex and “pathological” because of intersecting faces. Approaching the problem analytically, we sought alternate EP-embeddings for Platonic and Archimedean solids. We prove that the number of equations—E edge length equations (enforcing equilaterality) and 2 E − 3 F face (torsion) equations (enforcing planarity)—and of variables ( 3 V − 6 ) are equal. Therefore, solutions of the equations up to equivalence generally leave no degrees of freedom. As a result, in general there is a finite (but very large) number of solutions. Unfortunately, even with state-of-the-art computer algebra, the resulting systems of equations are generally too complicated to completely solve within reasonable time. We therefore added an additional constraint, symmetry, specifically requiring solutions to display (at least) tetrahedral symmetry. We found 77 non-classical embeddings, seven without intersecting faces—two, four and one, respectively, for the (graphs of the) dodecahedron, the icosidodecahedron and the rhombicosidodecahedron.

ASYMPTOTIC ESTIMATES OF THE NUMBER OF SOLUTIONS OF SYSTEMS OF EQUATIONS WITH DETERMINABLE PARTIAL BOOLEAN FUNCTIONS

2019 ◽  
Vol 53 (2 (249)) ◽  
pp. 127-131
Author(s):  
Ed.V. Yeghiazaryan
Keyword(s):  
Boolean Functions ◽  
Asymptotic Estimate ◽  
Typical Case ◽  
Asymptotic Estimates ◽  
Systems Of Equations ◽  
Number Of Solutions

In this paper we investigate a class of equation systems with determinable partial (not everywhere defined) Boolean functions. We found the asymptotic estimate of the number of solutions of equation systems in the “typical” case (for the whole range of changes in the number of equations).

SINGULAR SPECTRAIN ONE-DIMENSIONAL RANDOM SYSTEMS

1983 ◽  
Vol 44 (C3) ◽  
pp. C3-1555-C3-1556
Author(s):  
T. A.L. Ziman
Keyword(s):  
Random Systems ◽  
One Dimensional

Researching the factors influencing the satisfaction of master students at VNU School of Interdisciplinary Studies

2019 ◽  
Vol 35 (3) ◽  
Author(s):  
Do Huy Thuong ◽  
Nguyen Thi Phuong Hong
Keyword(s):  
Service Industry ◽  
Interdisciplinary Studies ◽  
Number Of Solutions ◽  
Factors Affecting ◽  
Training Institutions ◽  
Market Needs ◽  
Market Strategies ◽  
Students Satisfaction ◽  
Training Service

Improving the quality in order to keep up with the trend in the world is the vital task of training institutions today. Training institutions need to grasp market needs and satisfy the requirements of customers - learners. Nadiri, H., Kandampully, J & Hussain, K. (2009) argue that the managers in education need to apply market strategies that are being used by manufacturing and business enterprises and need to be aware that the role of training institutions is a service industry which is responsible for satisfying learner needs (Elliott & Shin, 2002). Currently, there have been many researches on students’ satisfaction. However, each research has its own objectives and is conducted on different scales. This study is implemented to provide information about the factors affecting master students’ satisfaction with the training service at VNU School of Interdisciplinary Studies (VNU SIS). Through it, the research offers a number of solutions to improving the satisfaction level of the master students at VNU SIS in the coming time.

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