scholarly journals An Extremal Graph Problem with a Transcendental Solution

2018 ◽  
Vol 28 (2) ◽  
pp. 303-324 ◽  
Author(s):  
DHRUV MUBAYI ◽  
CAROLINE TERRY
Keyword(s):  
Number Theory ◽  
Related Problem ◽  
Extremal Graph ◽  
Natural Question ◽  
Extremal Combinatorics ◽  
Easy Consequence ◽  
Vertex Set ◽  
Schanuel's Conjecture ◽  
Transcendental Solution ◽  
Shed Light

We prove that the number of multigraphs with vertex set {1, . . .,n} such that every four vertices span at most nine edges isan2+o(n2)whereais transcendental (assuming Schanuel's conjecture from number theory). This is an easy consequence of the solution to a related problem about maximizing the product of the edge multiplicities in certain multigraphs, and appears to be the first explicit (somewhat natural) question in extremal graph theory whose solution is transcendental. These results may shed light on a question of Razborov, who asked whether there are conjectures or theorems in extremal combinatorics which cannot be proved by a certain class of finite methods that include Cauchy–Schwarz arguments.Our proof involves a novel application of Zykov symmetrization applied to multigraphs, a rather technical progressive induction, and a straightforward use of hypergraph containers.

scholarly journals The Extremal Cacti on Multiplicative Degree-Kirchhoff Index

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 83
Author(s):  
Fangguo He ◽  
Zhongxun Zhu
Keyword(s):  
Effective Resistance ◽  
Extremal Graph ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Degree Of Vertex ◽  
Vertex Set

For a graph G, the resistance distance r G ( x , y ) is defined to be the effective resistance between vertices x and y, the multiplicative degree-Kirchhoff index R ∗ ( G ) = ∑ { x , y } ⊂ V ( G ) d G ( x ) d G ( y ) r G ( x , y ) , where d G ( x ) is the degree of vertex x, and V ( G ) denotes the vertex set of G. L. Feng et al. obtained the element in C a c t ( n ; t ) with first-minimum multiplicative degree-Kirchhoff index. In this paper, we first give some transformations on R ∗ ( G ) , and then, by these transformations, the second-minimum multiplicative degree-Kirchhoff index and the corresponding extremal graph are determined, respectively.

scholarly journals DISCRETE METRIC SPACES: STRUCTURE, ENUMERATION, AND 0-1 LAWS

2019 ◽  
Vol 84 (4) ◽  
pp. 1293-1325 ◽  
Author(s):  
DHRUV MUBAYI ◽  
CAROLINE TERRY
Keyword(s):  
Metric Spaces ◽  
Extremal Combinatorics ◽  
Binary Relations ◽  
Easy Consequence ◽  
First Order ◽  
Colored Graphs ◽  
Long Line ◽  
Image Position ◽  
Almost All ◽  
Continuous Setting

AbstractFix an integer $r \ge 3$. We consider metric spaces on n points such that the distance between any two points lies in $\left\{ {1, \ldots ,r} \right\}$. Our main result describes their approximate structure for large n. As a consequence, we show that the number of these metric spaces is $\left\lceil {{{r + 1} \over 2}} \right\rceil ^{\left( {\matrix{ n \cr 2 \cr } } \right) + o\left( {n^2 } \right)} .$Related results in the continuous setting have recently been proved by Kozma, Meyerovitch, Peled, and Samotij [34]. When r is even, our structural characterization is more precise and implies that almost all such metric spaces have all distances at least $r/2$. As an easy consequence, when r is even, we improve the error term above from $o\left( {n^2 } \right)$ to $o\left( 1 \right)$, and also show a labeled first-order 0-1 law in the language ${\cal L}_r $, consisting of r binary relations, one for each element of $[r]$ . In particular, we show the almost sure theory T is the theory of the Fraïssé limit of the class of all finite simple complete edge-colored graphs with edge colors in $\left\{ {r/2, \ldots ,r} \right\}$.Our work can be viewed as an extension of a long line of research in extremal combinatorics to the colored setting, as well as an addition to the collection of known structures that admit logical 0-1 laws.

A theorem on hyperhypersimple sets

1963 ◽  
Vol 28 (4) ◽  
pp. 273-278 ◽  
Author(s):  
Donald A. Martin
Keyword(s):  
Main Concern ◽  
Natural Question ◽  
Easy Consequence ◽  
Recursively Enumerable ◽  
Simple Set

Let be the class of recursively enumerable (r.e.) sets with infinite complements. A set M ϵ is maximal if every superset of M which is in is only finitely different from M. In [1] Friedberg shows that maximal sets exist, and it is an easy consequence of this fact that every non-simple set in has a maximal superset. The natural question which arises is whether or not this is also true for every simple set (Ullian [2]). In the present paper this question is answered negatively. However, the main concern of this paper is with demonstrating, and developing a few consequences of, what might be called the “density” of hyperhypersimple sets.

Formal development of ordinal number theory

1955 ◽  
Vol 20 (2) ◽  
pp. 95-104
Author(s):  
Steven Orey
Keyword(s):  
Number Theory ◽  
Ordinal Number ◽  
Ordered Sets ◽  
Stringent Condition ◽  
Easy Consequence ◽  
Basic Properties ◽  
Ordinal Numbers ◽  
Order Types ◽  
Proper Subclass ◽  
Definition Of

In this paper we shall develop a theory of ordinal numbers for the system ML, [6]. Since NF, [2], is a sub-system of ML one could let the ordinal arithmetic developed in [9] serve also as the ordinal arithmetic of ML. However, it was shown in [9] that the ordinal numbers of [9], NO, do not have all the usual properties of ordinal numbers and that theorems contradicting basic results of “intuitive ordinal arithmetic” can be proved.In particular it will be a theorem in our development of ordinal numbers that, for any ordinal number α, the set of all smaller ordinal numbers ordered by ≤ has ordinal number α; this result does not hold for the ordinals of [9] (see [9], XII.3.15). It will also be an easy consequence of our definition of ordinal number that proofs by induction over the ordinal numbers are permitted for arbitrary statements of ML; proofs by induction over NO can be carried through only for stratified statements with no unrestricted class variables.The class we shall take as the class of ordinal numbers, to be designated by ‘ORN’, will turn out to be a proper subclass of NO. This is because in ML there are two natural ways of defining the concept of well ordering. Sets which are well ordered in the sense of [9] we shall call weakly well ordered; sets which satisfy a certain more stringent condition will be called strongly well ordered. NO is the set of order types of weakly well ordered sets, while ORN is the class of order types of strongly well ordered sets. Basic properties of weakly and strongly well ordered sets are developed in Section 2.

scholarly journals Short Proofs of Some Extremal Results

2013 ◽  
Vol 23 (1) ◽  
pp. 8-28 ◽  
Author(s):  
DAVID CONLON ◽  
JACOB FOX ◽  
BENNY SUDAKOV
Keyword(s):  
Graph Theory ◽  
Ramsey Theory ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Extremal Combinatorics ◽  
Additive Combinatorics ◽  
Open Problems

We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.

Elucidating cyclic AMP signaling in subcellular domains with optogenetic tools and fluorescent biosensors

2019 ◽  
Vol 47 (6) ◽  
pp. 1733-1747 ◽  
Author(s):  
Christina Klausen ◽  
Fabian Kaiser ◽  
Birthe Stüven ◽  
Jan N. Hansen ◽  
Dagmar Wachten
Keyword(s):  
Cyclic Amp ◽  
Adenosine Monophosphate ◽  
Adenylyl Cyclases ◽  
Temporal Precision ◽  
Fluorescent Biosensors ◽  
Subcellular Domains ◽  
Spatio Temporal ◽  
Hydrolysis Of ◽  
Shed Light ◽  
Cyclic Amp Signaling

The second messenger 3′,5′-cyclic nucleoside adenosine monophosphate (cAMP) plays a key role in signal transduction across prokaryotes and eukaryotes. Cyclic AMP signaling is compartmentalized into microdomains to fulfil specific functions. To define the function of cAMP within these microdomains, signaling needs to be analyzed with spatio-temporal precision. To this end, optogenetic approaches and genetically encoded fluorescent biosensors are particularly well suited. Synthesis and hydrolysis of cAMP can be directly manipulated by photoactivated adenylyl cyclases (PACs) and light-regulated phosphodiesterases (PDEs), respectively. In addition, many biosensors have been designed to spatially and temporarily resolve cAMP dynamics in the cell. This review provides an overview about optogenetic tools and biosensors to shed light on the subcellular organization of cAMP signaling.

Sentence-Based Experience Logging in New Hearing Aid Users

2020 ◽  
Vol 29 (3S) ◽  
pp. 631-637
Author(s):  
Katja Lund ◽  
Rodrigo Ordoñez ◽  
Jens Bo Nielsen ◽  
Dorte Hammershøi
Keyword(s):  
Hearing Aid ◽  
Patient Experiences ◽  
Online Tool ◽  
Rehabilitation Process ◽  
Hearing Rehabilitation ◽  
Clinical Observations ◽  
Hearing Aid Use ◽  
Wide Range ◽  
Insight Into ◽  
Shed Light

Purpose The aim of this study was to develop a tool to gain insight into the daily experiences of new hearing aid users and to shed light on aspects of aided performance that may not be unveiled through standard questionnaires. Method The tool is developed based on clinical observations, patient experiences, expert involvement, and existing validated hearing rehabilitation questionnaires. Results An online tool for collecting data related to hearing aid use was developed. The tool is based on 453 prefabricated sentences representing experiences within 13 categories related to hearing aid use. Conclusions The tool has the potential to reflect a wide range of individual experiences with hearing aid use, including auditory and nonauditory aspects. These experiences may hold important knowledge for both the patient and the professional in the hearing rehabilitation process.

Pearls Shed Light on Tx for Dark-Skinned Children

2006 ◽  
Vol 37 (4) ◽  
pp. 42
Author(s):  
HEIDI SPLETE
Keyword(s):  
Shed Light
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