Algebraic surfaces with an infinite set of skew symmetry planes. Mutual arrangement of linear spans of four orbits of symmetry directions

The paper considers the mathematical model development technique to build a vector field of the shape deviations when machining flat surfaces of shell parts on multi-operational machines under conditions of anisotropic rigidity in technological system (TS). The technological system has an anisotropic rigidity, as its elastic strains do not obey the accepted concepts, i.e. the rigidity towards the coordinate axes of the machine is the same, and they occur only towards the external force. The record shows that the diagrams of elastic strains of machine units are substantially different from the circumference. The issues to ensure the specified accuracy require that there should be mathematical models describing kinematic models and physical processes of mechanical machining under conditions of the specific TS. There are such models for external and internal surfaces of rotation [2,3], which are successfully implemented in practice. Flat surfaces (FS) of shell parts (SP) are both assembly and processing datum surfaces. Therefore, on them special stipulations are made regarding deviations of shape and mutual arrangement. The axes of the main bearing holes are coordinated with respect to them. The joints that ensure leak tightness and distributed load on the product part are closed on these surfaces. The paper deals with the analytical construction of the vector field F, which describes with appropriate approximation the real surface obtained as a result of modeling the process of machining flat surfaces (MFS) through face milling under conditions of anisotropic properties.

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The reflection of an electromagnetic plane wave by an infinite set of plates. III

Quarterly of Applied Mathematics ◽

10.1090/qam/38239 ◽

1950 ◽

Vol 8 (3) ◽

pp. 281-291 ◽

Cited By ~ 22

Author(s):

Albert E. Heins

Keyword(s):

Plane Wave ◽

Electromagnetic Plane Wave ◽

Infinite Set ◽

Electromagnetic Plane

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A Note on On-Line Ramsey Numbers for Some Paths

Mathematics ◽

10.3390/math9070735 ◽

2021 ◽

Vol 9 (7) ◽

pp. 735

Author(s):

Tomasz Dzido ◽

Renata Zakrzewska

Keyword(s):

Ramsey Number ◽

Ramsey Numbers ◽

Minimum Number ◽

On Line ◽

Infinite Set ◽

Do So

We consider the important generalisation of Ramsey numbers, namely on-line Ramsey numbers. It is easiest to understand them by considering a game between two players, a Builder and Painter, on an infinite set of vertices. In each round, the Builder joins two non-adjacent vertices with an edge, and the Painter colors the edge red or blue. An on-line Ramsey number r˜(G,H) is the minimum number of rounds it takes the Builder to force the Painter to create a red copy of graph G or a blue copy of graph H, assuming that both the Builder and Painter play perfectly. The Painter’s goal is to resist to do so for as long as possible. In this paper, we consider the case where G is a path P4 and H is a path P10 or P11.

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Density of sets with missing differences and applications

Mathematica Slovaca ◽

10.1515/ms-2021-0006 ◽

2021 ◽

Vol 71 (3) ◽

pp. 595-614

Author(s):

Ram Krishna Pandey ◽

Neha Rai

Keyword(s):

Number Theory ◽

Asymptotic Density ◽

Theory Problem ◽

Distance Graphs ◽

Positive Integers ◽

Infinite Set

Abstract For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M-sets, denoted by μ(M), where an M-set is a set of non-negative integers in which no two elements differ by an element in M. In 1973, Cantor and Gordon find μ(M) for |M| ≤ 2. Partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ(M) for two families namely, M = {a, b,a + b, n(a + b)} and M = {a, b, b − a, n(b − a)}. For both of these families, we find some exact values and some bounds on μ(M). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M. So, as an application, we also study these coloring parameters associated with these families.

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