scholarly journals Number of Directions Determined by a Set in $$\mathbb {F}_{q}^{2}$$ and Growth in $$\mathrm {Aff}(\mathbb {F}_{q})$$

2021 ◽  
Author(s):  
Daniele Dona
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Upper Bound ◽  
Finite Plane ◽  
Structural Theorem ◽  
Prime Fields ◽  
The One

AbstractWe prove that a set A of at most q non-collinear points in the finite plane $$\mathbb {F}_{q}^{2}$$ F q 2 spans more than $${|A|}/\!{\sqrt{q}}$$ | A | / q directions: this is based on a lower bound by Fancsali et al. which we prove again together with a different upper bound than the one given therein. Then, following the procedure used by Rudnev and Shkredov, we prove a new structural theorem about slowly growing sets in $$\mathrm {Aff}(\mathbb {F}_{q})$$ Aff ( F q ) for any finite field $$\mathbb {F}_{q}$$ F q , generalizing the analogous results by Helfgott, Murphy, and Rudnev and Shkredov over prime fields.

BALANCED SETS AND THE VECTOR GAME

2008 ◽  
Vol 04 (03) ◽  
pp. 339-347 ◽  
Author(s):  
ZHIVKO NEDEV ◽  
ANTHONY QUAS
Keyword(s):  
Field Theory ◽  
Lower Bound ◽  
Finite Field ◽  
Upper Bound ◽  
Log P ◽  
Prime Factorization ◽  
Minimum Cardinality ◽  
Player Game

We consider the notion of a balanced set modulo N. A nonempty set S of residues modulo N is balanced if for each x ∈ S, there is a d with 0 < d ≤ N/2 such that x ± d mod N both lie in S. We define α(N) to be the minimum cardinality of a balanced set modulo N. This notion arises in the context of a two-player game that we introduce and has interesting connections to the prime factorization of N. We demonstrate that for p prime, α(p) = Θ( log p), giving an explicit algorithmic upper bound and a lower bound using finite field theory and show that for N composite, α(N) = min p|Nα(p).

scholarly journals The number of olfactory stimuli that humans can discriminate is still unknown

eLife ◽  
2015 ◽  
Vol 4 ◽  
Author(s):  
Richard C Gerkin ◽  
Jason B Castro
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Olfactory Stimuli ◽  
The One ◽  
Reported Data

It was recently proposed (<xref ref-type="bibr" rid="bib2">Bushdid et al., 2014</xref>) that humans can discriminate between at least a trillion olfactory stimuli. Here we show that this claim is the result of a fragile estimation framework capable of producing nearly any result from the reported data, including values tens of orders of magnitude larger or smaller than the one originally reported in (<xref ref-type="bibr" rid="bib2">Bushdid et al., 2014</xref>). Additionally, the formula used to derive this estimate is well-known to provide an upper bound, not a lower bound as reported. That is to say, the actual claim supported by the calculation is in fact that humans can discriminate at most one trillion olfactory stimuli. We conclude that there is no evidence for the original claim.

ESTIMATE OF THE HAUSDORFF MEASURE OF THE SIERPINSKI TRIANGLE

Fractals ◽  
2009 ◽  
Vol 17 (02) ◽  
pp. 137-148
Author(s):  
PÉTER MÓRA
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Upper Bound ◽  
Open Problem ◽  
Dimensional Hausdorff Measure ◽  
Sierpinski Triangle ◽  
The One

It is well-known that the Hausdorff dimension of the Sierpinski triangle Λ is s = log 3/ log 2. However, it is a long standing open problem to compute the s-dimensional Hausdorff measure of Λ denoted by [Formula: see text]. In the literature the best existing estimate is [Formula: see text] In this paper we improve significantly the lower bound. We also give an upper bound which is weaker than the one above but everybody can check it easily. Namely, we prove that [Formula: see text] holds.

scholarly journals Roots of sparse polynomials over a finite field

2016 ◽  
Vol 19 (A) ◽  
pp. 196-204 ◽  
Author(s):  
Zander Kelley
Keyword(s):  
Finite Field ◽  
Upper Bound ◽  
Log P ◽  
Diffie Hellman ◽  
Sparse Polynomials ◽  
Prime Fields

For a $t$-nomial $f(x)=\sum _{i=1}^{t}c_{i}x^{a_{i}}\in \mathbb{F}_{q}[x]$, we show that the number of distinct, nonzero roots of $f$ is bounded above by $2(q-1)^{1-\unicode[STIX]{x1D700}}C^{\unicode[STIX]{x1D700}}$, where $\unicode[STIX]{x1D700}=1/(t-1)$ and $C$ is the size of the largest coset in $\mathbb{F}_{q}^{\ast }$ on which $f$ vanishes completely. Additionally, we describe a number-theoretic parameter depending only on $q$ and the exponents $a_{i}$ which provides a general and easily computable upper bound for $C$. We thus obtain a strict improvement over an earlier bound of Canetti et al. which is related to the uniformity of the Diffie–Hellman distribution. Finally, we conjecture that $t$-nomials over prime fields have only $O(t\log p)$ roots in $\mathbb{F}_{p}^{\ast }$ when $C=1$.

scholarly journals Sur les solutions friables de l'équation a+b=c

2013 ◽  
Vol 154 (3) ◽  
pp. 439-463 ◽  
Author(s):  
SARY DRAPPEAU
Keyword(s):  
Lower Bound ◽  
Asymptotic Behaviour ◽  
Recent Work ◽  
Upper Bound ◽  
Smooth Function ◽  
Riemann Hypothesis ◽  
Smooth Solutions ◽  
Compactly Supported ◽  
The One

AbstractIn a recent paper [5], Lagarias and Soundararajan study the y-smooth solutions to the equation a+b=c. Conditionally under the Generalised Riemann Hypothesis, they obtain an estimate for the number of those solutions weighted by a compactly supported smooth function, as well as a lower bound for the number of bounded unweighted solutions. In this paper, we prove a more precise conditional estimate for the number of weighted solutions that is valid when y is relatively large with respect to x, so as to connect our estimate with the one obtained by La Bretèche and Granville in a recent work [2]. We also prove, conditionally under the Generalised Riemann Hypothesis, the conjectured upper bound for the number of bounded unweighted solutions, thus obtaining its exact asymptotic behaviour.

ON THE MINIMAL NUMBER OF SMALL ELEMENTS GENERATING FINITE PRIME FIELDS

2017 ◽  
Vol 96 (2) ◽  
pp. 177-184
Author(s):  
MARC MUNSCH
Keyword(s):  
Finite Field ◽  
General Result ◽  
Upper Bound ◽  
Prime Fields ◽  
Almost All ◽  
Small Elements

In this note, we give an upper bound for the number of elements from the interval $[1,p^{1/4\sqrt{e}+\unicode[STIX]{x1D716}}]$ necessary to generate the finite field $\mathbb{F}_{p}^{\ast }$ with $p$ an odd prime. The general result depends on the distribution of the divisors of $p-1$ and can be used to deduce results which hold for almost all primes.

scholarly journals The Degree/Diameter Problem in Maximal Planar Bipartite graphs

10.37236/4468 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Cristina Dalfó ◽  
Clemens Huemer ◽  
Julián Salas
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Upper Bounds ◽  
The One

The $(\Delta,D)$ (degree/diameter) problem consists of finding the largest possible number of vertices $n$ among all the graphs with maximum degree $\Delta$ and diameter $D$. We consider the $(\Delta,D)$ problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the $(\Delta,2)$ problem, the number of vertices is $n=\Delta+2$; and for the $(\Delta,3)$ problem, $n= 3\Delta-1$ if $\Delta$ is odd and $n= 3\Delta-2$ if $\Delta$ is even. Then, we prove that, for the general case of the $(\Delta,D)$ problem, an upper bound on $n$ is approximately $3(2D+1)(\Delta-2)^{\lfloor D/2\rfloor}$, and another one is $C(\Delta-2)^{\lfloor D/2\rfloor}$ if $\Delta\geq D$ and $C$ is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on $n$ for maximal planar bipartite graphs, which is approximately $(\Delta-2)^{k}$ if $D=2k$, and $3(\Delta-3)^k$ if $D=2k+1$, for $\Delta$ and $D$ sufficiently large in both cases.

A discrete chain of degrees of index sets

1972 ◽  
Vol 37 (1) ◽  
pp. 139-149 ◽  
Author(s):  
Louise Hay
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Local Information ◽  
Truth Table ◽  
Partial Ordering ◽  
Index Set ◽  
Recursively Enumerable ◽  
Index Sets ◽  
The One ◽  
Standard Enumeration

Let {Wi} be a standard enumeration of all recursively enumerable (r.e.) sets, and for any class A of r.e. sets, let θA denote the index set of A = {n ∣ Wn ∈ A}. (Clearly, .) In [1], the index sets of nonempty finite classes of finite sets were classified under one-one reducibility into an increasing sequence {Ym}, 0 ≤ m < ∞. In this paper we examine further properties of this sequence within the partial ordering of one-one degrees of index sets. The main results are as follows: (1) For each m, Ym < Ym + 1 and < Ym + 1; (2) Ym is incomparable to ; (3) Ym + 1 and ; are immediate successors (among index sets) of Ym and m; (4) the pair (Ym + 1, ) is a “least upper bound” for the pair (Ym, ) in the sense that any successor of both Ym and is ≥ Ym + 1or; (5) the pair (Ym, ) is a “greatest lower bound” for the pair (Ym + 1, ) in the sense that any predecessor of both Ym + 1 and is ≤ Ym or . Since and all Ym are in the bounded truth-table degree of K, this yields some local information about the one-one degrees of index sets which are “at the bottom” in the one-one ordering of index sets.

scholarly journals On Ensuring Multilayer Wirability by Stretching Layouts

VLSI Design ◽  
1998 ◽  
Vol 7 (4) ◽  
pp. 365-383
Author(s):  
Teofilo F. Gonzalez ◽  
Si-Qing Zheng
Keyword(s):  
Dynamic Programming ◽  
Lower Bound ◽  
Upper Bound ◽  
Area Increase ◽  
Different Types ◽  
Np Complete ◽  
Layout Area ◽  
The One

Every knock-knee layout is four-layer wirable. However, there are knock-knee layouts that cannot be wired in less than four layers. While it is easy to determine whether a knock-knee layout is one-layer wirable or two-layer wirable, the problem of determining three-layer wirability of knock-knee layouts is NP-complete. A knock-knee layout may be stretched vertically (horizontally) by introducing empty rows (columns) so that it can be wired in fewer than four layers. In this paper we discuss two different types of stretching schemes. It is known that under these two stretching schemes, any knock-knee layout is three-layer wirable by stretching it up to (4/3) of the knock-knee layout area (upper bound). We show that there are knock-knee layouts that when stretched and wired in three layers under scheme I (II) require at least 1.2 (1.07563) of the original layout area. Our lower bound for the area increase factor can be used to guide the search for effective stretching-based dynamic programming three-layer wiring algorithms similar to the one presented in [8].

Success probabilities for second guessers

1980 ◽  
Vol 17 (04) ◽  
pp. 1133-1137 ◽  
Author(s):  
A. O. Pittenger
Keyword(s):  
Lower Bound ◽  
Dimensional Space ◽  
Success Probability ◽  
First Person ◽  
True Location ◽  
Rotationally Symmetric ◽  
The One ◽  
Sharp Lower Bound

Two people independently and with the same distribution guess the location of an unseen object in n-dimensional space, and the one whose guess is closer to the unseen object is declared the winner. The first person announces his guess, but the second modifies his unspoken idea by moving his guess in the direction of the first guess and as close to it as possible. It is shown that if the distribution of guesses is rotationally symmetric about the true location of the unseen object, ¾ is the sharp lower bound for the success probability of the second guesser. If the distribution is fixed and the dimension increases, then for a certain class of distributions, the success probability approaches 1.

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