scholarly journals Genus Ranges of 4-Regular Rigid Vertex Graphs

10.37236/3825 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Dorothy Buck ◽  
Egor Dolzhenko ◽  
Natasa Jonoska ◽  
Masahico Saito ◽  
Karin Valencia
Keyword(s):  
Cyclic Order ◽  
Consecutive Integers ◽  
Orientable Surfaces

A rigid vertex of a graph is one that has a prescribed cyclic order of its incident edges. We study orientable genus ranges of 4-regular rigid vertex graphs. The (orientable) genus range is a set of genera values over all orientable surfaces into which a graph is embedded cellularly, and the embeddings of rigid vertex graphs are required to preserve the prescribed cyclic order of incident edges at every vertex. The genus ranges of 4-regular rigid vertex graphs are sets of consecutive integers, and we address two questions: which intervals of integers appear as genus ranges of such graphs, and what types of graphs realize a given genus range. For graphs with $2n$ vertices ($n>1$), we prove that all intervals $[a, b]$ for all $a<b \leq n$, and singletons $[h, h]$ for some $h\leq n$, are realized as genus ranges. For graphs with $2n-1$ vertices ($n\geq 1$), we prove that all intervals $[a, b]$ for all $a<b \leq n$ except $[0,n]$, and $[h,h]$ for some $h\leq n$, are realized as genus ranges. We also provide constructions of graphs that realize these ranges.

Algorithm of the Cyclic-Order Graph Program (Implementation and Usage)

2019 ◽  
Vol 7 (3) ◽  
Author(s):  
Štefan Berežný ◽  
Ján Buša Jr.
Keyword(s):  
Program Implementation ◽  
Cyclic Order

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

scholarly journals Invariants of Stable Maps between Closed Orientable Surfaces

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 215
Author(s):  
Catarina Mendes de Jesus S. ◽  
Pantaleón D. Romero
Keyword(s):  
Point Of View ◽  
Stable Maps ◽  
Smooth Maps ◽  
Orientable Surfaces

In this paper, we will consider the problem of constructing stable maps between two closed orientable surfaces M and N with a given branch set of curves immersed on N. We will study, from a global point of view, the behavior of its families in different isotopies classes on the space of smooth maps. The main goal is to obtain different relationships between invariants. We will provide a new proof of Quine’s Theorem.

On triplets with descending largest prime factors

2001 ◽  
Vol 38 (1-4) ◽  
pp. 45-50 ◽  
Author(s):  
A. Balog
Keyword(s):  
Prime Factor ◽  
Prime Factors ◽  
Consecutive Integers

For an integer n≯1 letP(n) be the largest prime factor of n. We prove that there are infinitely many triplets of consecutive integers with descending largest prime factors, that is P(n - 1) ≯P(n)≯P(n+1) occurs for infinitely many integers n.

Finite fields and the 1-chromatic number of orientable surfaces

2009 ◽  
pp. n/a-n/a
Author(s):  
Vladimir P. Korzhik
Keyword(s):  
Chromatic Number ◽  
Finite Fields ◽  
Orientable Surfaces

Graphs of stable maps between closed orientable surfaces

2016 ◽  
Vol 36 (3) ◽  
pp. 1185-1194 ◽  
Author(s):  
C. Mendes de Jesus
Keyword(s):  
Stable Maps ◽  
Orientable Surfaces

A Polynomial Invariant of Graphs On Orientable Surfaces

2001 ◽  
Vol 83 (3) ◽  
pp. 513-531 ◽  
Author(s):  
Béla Bollobás ◽  
Oliver Riordan
Keyword(s):  
Polynomial Invariant ◽  
Orientable Surfaces

An Algorithm to Slice 3-D Shapes for Reconstruction in Prototyping Systems

1991 ◽  
Author(s):  
K. L. Chalasani ◽  
B. Grogan ◽  
A. Bagchi ◽  
C. C. Jara-Almonte ◽  
A. A. Ogale ◽  
...  
Keyword(s):  
Data Structure ◽  
Rapid Prototyping ◽  
Surface Finish ◽  
Cyclic Order ◽  
Contour Tracing ◽  
Building Process ◽  
Dense Grid ◽  
A Chain ◽  
Definition Of

Abstract Rapid Prototyping (RP) processes reduce the time consumed in the manufacture of a prototype by producing parts directly from a CAD representation, without tooling. The StereoLithography Apparatus (SLA), and most other recent RP processes build a 3-D object from 2.5-D layers. Slicing is the process of defining layers to be built by the system. In this paper a framework is proposed for the development of algorithms for the representation and definition of layers for use in the SLA, with a view to determine if the slicing algorithms will affect surface finish in any significant manner. Currently, it is not possible to automatically vary slice thicknesses within the same object, using the existent algorithm. Also, it would be useful to use a dense grid for hatching or skin filling any given layer, or to change the hatch-pattern if desired. In addition, simulation of the layered building process would be helpful, so that the user can prespecify parameters that need to be varied during the process. The proposed framework incorporates these and other features. Two approaches for determining contours on each slice are suggested and their implementation is discussed. In the first, the layers are defined by the intersections of a plane with the surfaces defining the object. The plane is moved up from the base of the object as it is being built in increments. All intersections found are stored in a data structure, and sorted in head to tail fashion to define a contour for all closed areas on a layer. The second approach uses a scanline-type search to look for an intersection that will trigger a contour-tracing procedure. The contour-tracer is invoked whenever an unused edge is found in the search. This saves storage and sorting times, because the contour is determined as a chain of edges, in cyclic order. It is envisaged that results of this work on the SLA can be applied to other RP processes entailing layered building.

Stable bi-maps from closed orientable surfaces to R x R

2021 ◽  
Author(s):  
Catarina Mendes de Jesus ◽  
Erica Boizan Batista ◽  
João Carlos Ferreira Costa
Keyword(s):  
Orientable Surfaces

Note on Pairs of Consecutive Integers the Sum of whose Squares is an Integral Square

1902 ◽  
Vol 23 ◽  
pp. 264-267
Author(s):  
Thomas Muir
Keyword(s):  
Consecutive Integers

The solution of the problem of finding such pairs of integers is not a thing of yesterday, as may be seen by consulting Hutton's translation of Ozanam's Recreations, i. pp. 46–8 (1814).

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