scholarly journals Between 2- and 3-Colorability

10.37236/4673 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Alan Frieze ◽  
Wesley Pegden
Keyword(s):  
Positive Integer ◽  
Random Graphs ◽  
The Other ◽  
Chromatic Numbers ◽  
Other Hand ◽  
Odd Cycles

We consider the question of the existence of homomorphisms between $G_{n,p}$ and odd cycles when $p=c/n$, $1<c\leq 4$. We show that for any positive integer $\ell$, there exists $\epsilon=\epsilon(\ell)$ such that if $c=1+\epsilon$ then w.h.p. $G_{n,p}$ has a homomorphism from $G_{n,p}$ to $C_{2\ell+1}$ so long as its odd-girth is at least $2\ell+1$. On the other hand, we show that if $c=4$ then w.h.p. there is no homomorphism from $G_{n,p}$ to $C_5$. Note that in our range of interest, $\chi(G_{n,p})=3$ w.h.p., implying that there is a homomorphism from $G_{n,p}$ to $C_3$.  These results imply the existence of random graphs with circular chromatic numbers $\chi_c$ satisfying $2<\chi_c(G)<2+\delta$ for arbitrarily small $\delta$, and also that $2.5\leq \chi_c(G_{n,\frac 4 n})<3$ w.h.p.

scholarly journals II. Note on legendre’s coefficients

1878 ◽  
Vol 27 (185-189) ◽  
pp. 381-383
Keyword(s):  
Positive Integer ◽  
Royal Society ◽  
The Other ◽  
Definite Integral ◽  
Other Hand

In the “Proceedings of the Royal Society,” vol. xxvii, pp. 63-71, Professor Adams has given, by an inductive process, the development of the product of any two of Legendre’s Coefficients in a series of the Coefficients; from this is immediately deduced the value of the integral between the limits -1 and +1 of the product of any three of the Coefficients. On the other hand, if we know the value of this definite integral, we can immediately deduce the development of the product of any two of the Coefficients. Thus it may be of interest to give a brief investigation of the value of the definite integral. I follow the notation adopted by Professor Adams. The formula to be established is ∫ 1 -1 P m P n P p dμ = 2/2 s + 1 A ( s - m ) A ( s - n ) A ( s - p ) / A( s ), where 2 s = m + n + P , and the functional symbol A( r ) is thus defined: if r is a positive integer A ( r ) = 1. 3. 5. . (2 r - 1) /1. 2. 3. . r , and in all other cases A ( r ) is to be considered zero, except when r = 0, and then it is to be considered = 1.

MODULES WITH COINDEPENDENT MAXIMAL SUBMODULES

2011 ◽  
Vol 10 (01) ◽  
pp. 73-99 ◽  
Author(s):  
PATRICK F. SMITH
Keyword(s):  
Positive Integer ◽  
Direct Summand ◽  
The Other ◽  
Property A ◽  
Other Hand ◽  
Semisimple Module ◽  
Maximal Submodule

Let R be a ring with identity. A unital left R-module M has the min-property provided the simple submodules of M are independent. On the other hand a left R-module M has the complete max-property provided the maximal submodules of M are completely coindependent, in other words every maximal submodule of M does not contain the intersection of the other maximal submodules of M. A semisimple module X has the min-property if and only if X does not contain distinct isomorphic simple submodules and this occurs if and only if X has the complete max-property. A left R-module M has the max-property if [Formula: see text] for every positive integer n and distinct maximal submodules L, Li (1 ≤ i ≤ n) of M. It is proved that a left R-module M has the complete max-property if and only if M has the max-property and every maximal submodule of M/Rad M is a direct summand, where Rad M denotes the radical of M, and in this case every maximal submodule of M is fully invariant. Various characterizations are given for when a module M has the max-property and when M has the complete max-property.

scholarly journals A Note on Sparse Random Graphs and Cover Graphs

10.37236/1497 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Tom Bohman ◽  
Alan Frieze ◽  
Miklós Ruszinkó ◽  
Lubos Thoma
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
The Other ◽  
Other Hand ◽  
Cover Graph

It is shown in this note that with high probability it is enough to destroy all triangles in order to get a cover graph from a random graph $G_{n,p}$ with $p\le \kappa \log n/n$ for any constant $\kappa < 2/3$. On the other hand, this is not true for somewhat higher densities: If $p\ge \lambda (\log n)^3 / (n\log\log n)$ with $\lambda > 1/8$ then with high probability we need to delete more edges than one from every triangle. Our result has a natural algorithmic interpretation.

scholarly journals On the Quantum Chromatic Number of a Graph

10.37236/999 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Ashley Montanaro ◽  
Michael W. Newman ◽  
Simone Severini ◽  
Andreas Winter
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
First Principles ◽  
Clique Number ◽  
The Other ◽  
Quantum Model ◽  
Other Hand ◽  
Orthogonal Representations

We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince a referee that they have a colouring of the graph.After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is $2$, nor if it is $3$ in a restricted quantum model; on the other hand, we exhibit a graph on $18$ vertices and $44$ edges with chromatic number $5$ and quantum chromatic number $4$.

scholarly journals Maximal sublattices and Frattini sublattices of bounded lattices

1997 ◽  
Vol 63 (1) ◽  
pp. 110-127 ◽  
Author(s):  
M. E. Adams ◽  
Ralph Freese ◽  
J. B. Nation ◽  
Jürg Schmid
Keyword(s):  
Positive Integer ◽  
Finite Lattice ◽  
The Other ◽  
Bounded Lattice ◽  
Other Hand ◽  
Finite Lattices

AbstractWe investigate the number and size of the maximal sublattices of a finite lattice. For any positive integer k, there is a finite lattice L with more that ]L]k sublattices. On the other hand, there are arbitrary large finite lattices which contain a maximal sublattice with only 14 elements. It is shown that every bounded lattice is isomorphic to the Frattini sublattice (the intersection of all maximal sublattices) of a finite bounded lattice.

scholarly journals Dense Graphs With a Large Triangle Cover Have a Large Triangle Packing

2012 ◽  
Vol 21 (6) ◽  
pp. 952-962 ◽  
Author(s):  
RAPHAEL YUSTER
Keyword(s):  
Absolute Constant ◽  
The Other ◽  
Asymptotically Optimal ◽  
Edge Disjoint ◽  
Other Hand ◽  
Dense Graphs ◽  
Large Triangle ◽  
Odd Cycles ◽  
Asymptotic Validity

It is well known that a graph with m edges can be made triangle-free by removing (slightly less than) m/2 edges. On the other hand, there are many classes of graphs which are hard to make triangle-free, in the sense that it is necessary to remove roughly m/2 edges in order to eliminate all triangles.We prove that dense graphs that are hard to make triangle-free have a large packing of pairwise edge-disjoint triangles. In particular, they have more than m(1/4+cβ) pairwise edge-disjoint triangles where β is the density of the graph and c ≥ is an absolute constant. This improves upon a previous m(1/4−o(1)) bound which follows from the asymptotic validity of Tuza's conjecture for dense graphs. We conjecture that such graphs have an asymptotically optimal triangle packing of size m(1/3−o(1)).We extend our result from triangles to larger cliques and odd cycles.

scholarly journals Approximaitly Primary Submodules

2019 ◽  
Vol 24 (5) ◽  
pp. 105
Author(s):  
Ali Sh. Ajeel ◽  
Haibat K. Mohammadali
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
The Other ◽  
Left Module ◽  
Basic Properties ◽  
Other Hand ◽  
Prime Submodules ◽  
Primary Submodules ◽  
Primary Submodule

The study deals with the notion of an approximaitly primary submodules of unitary left -module  over a commutative ring  with identity as a generalization of a primary submodules and approximaitly prime submodules, where a proper submodule  of an -module  is called an approximaitly primary submodule of , if whenever , for , , implies that either  or  for some positive integer  of . Several characterizations, basic properties of this concept are given. On the other hand the relationships of this concept with some classes of modules are studied. Furthermore, the behavior of approximaitly primary submodule under -homomorphism are discussed   http://dx.doi.org/10.25130/tjps.24.2019.098

scholarly journals On the Nonexistence of $k$-Reptile Simplices in $\mathbb R^3$ and $\mathbb R^4$

10.37236/6113 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Jan Kynčl ◽  
Zuzana Patáková
Keyword(s):  
Positive Integer ◽  
The Other ◽  
Dimensional Simplex ◽  
Other Hand

A $d$-dimensional simplex $S$ is called a $k$-reptile (or a $k$-reptile simplex) if it can be tiled by $k$ simplices with disjoint interiors that are all mutually congruent and similar to $S$. For $d=2$, triangular $k$-reptiles exist for all $k$ of the form $a^2, 3a^2$ or $a^2 + b^2$ and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only $k$-reptile simplices that are known for $d \ge 3$, have $k = m^d$, where $m$ is a positive integer. We substantially simplify the proof by Matoušek and the second author that for $d=3$, $k$-reptile tetrahedra can exist only for $k=m^3$. We then prove a weaker analogue of this result for $d=4$ by showing that four-dimensional $k$-reptile simplices can exist only for $k=m^2$.

A study of cometary eruptions through OH radio-lines

1999 ◽  
Vol 173 ◽  
pp. 249-254
Author(s):  
A.M. Silva ◽  
R.D. Miró
Keyword(s):  
Short Duration ◽  
Growth Curves ◽  
The Other ◽  
Initial Mass ◽  
Radio Lines ◽  
Other Hand ◽  
Sudden Rise

AbstractWe have developed a model for theH2OandOHevolution in a comet outburst, assuming that together with the gas, a distribution of icy grains is ejected. With an initial mass of icy grains of 108kg released, theH2OandOHproductions are increased up to a factor two, and the growth curves change drastically in the first two days. The model is applied to eruptions detected in theOHradio monitorings and fits well with the slow variations in the flux. On the other hand, several events of short duration appear, consisting of a sudden rise ofOHflux, followed by a sudden decay on the second day. These apparent short bursts are frequently found as precursors of a more durable eruption. We suggest that both of them are part of a unique eruption, and that the sudden decay is due to collisions that de-excite theOHmaser, when it reaches the Cometopause region located at 1.35 × 105kmfrom the nucleus.

Closing the GAP—A 5Å Scanning Microscope

1969 ◽  
Vol 27 ◽  
pp. 6-7
Author(s):  
A. V. Crewe
Keyword(s):  
Normal Form ◽  
The Other ◽  
Resolving Power ◽  
Scanning Microscope ◽  
Conventional Microscope ◽  
Other Hand ◽  
Closing The Gap ◽  
Thermal Sources ◽  
Better Than ◽  
Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.

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