Arithmetic functions monotonic at consecutive arguments

2014 ◽  
Vol 51 (2) ◽  
pp. 155-164
Author(s):  
Jean-Marie Koninck ◽  
Florian Luca
Keyword(s):  
Large Class ◽  
The Other ◽  
Arithmetic Functions ◽  
Arbitrary Integer ◽  
Other Hand ◽  
Positive Integers

For a large class of arithmetic functions f, it is possible to show that, given an arbitrary integer κ ≤ 2, the string of inequalities f(n + 1) < f(n + 2) < … < f(n + κ) holds for in-finitely many positive integers n. For other arithmetic functions f, such a property fails to hold even for κ = 3. We examine arithmetic functions from both classes. In particular, we show that there are only finitely many values of n satisfying σ2(n − 1) < σ2 < σ2(n + 1), where σ2(n) = ∑d|nd2. On the other hand, we prove that for the function f(n) := ∑p|np2, we do have f(n − 1) < f(n) < f(n + 1) in finitely often.

scholarly journals On the Turán Properties of Infinite Graphs

10.37236/771 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Andrzej Dudek ◽  
Vojtěch Rödl
Keyword(s):  
Infinite Graph ◽  
Infinite Graphs ◽  
The Other ◽  
Other Hand ◽  
Turan's Theorem ◽  
Vertex Set ◽  
Turán’S Theorem ◽  
Positive Integers

Let $G^{(\infty)}$ be an infinite graph with the vertex set corresponding to the set of positive integers ${\Bbb N}$. Denote by $G^{(l)}$ a subgraph of $G^{(\infty)}$ which is spanned by the vertices $\{1,\dots,l\}$. As a possible extension of Turán's theorem to infinite graphs, in this paper we will examine how large $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$ can be for an infinite graph $G^{(\infty)}$, which does not contain an increasing path $I_k$ with $k+1$ vertices. We will show that for sufficiently large $k$ there are $I_k$–free infinite graphs with ${1\over 4}+{1\over 200} < \liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$. This disproves a conjecture of J. Czipszer, P. Erdős and A. Hajnal. On the other hand, we will show that $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}\le{1\over 3}$ for any $k$ and such $G^{(\infty)}$.

scholarly journals Equations with powers of singular moduli

2019 ◽  
Vol 15 (03) ◽  
pp. 445-468 ◽  
Author(s):  
Antonin Riffaut
Keyword(s):  
Rational Number ◽  
The Other ◽  
Other Hand ◽  
Singular Moduli ◽  
Cm Points ◽  
The One ◽  
Straight Lines ◽  
Positive Integers

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli [Formula: see text] such that the numbers [Formula: see text], [Formula: see text] and [Formula: see text] are linearly dependent over [Formula: see text] for some positive integers [Formula: see text], must be of degree at most [Formula: see text]. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in [Formula: see text] defined over [Formula: see text]. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to a hyperbola [Formula: see text], where [Formula: see text].

scholarly journals Explicit isogenies in quadratic time in any characteristic

2016 ◽  
Vol 19 (A) ◽  
pp. 267-282 ◽  
Author(s):  
Luca De Feo ◽  
Cyril Hugounenq ◽  
Jérôme Plût ◽  
Éric Schost
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Large Class ◽  
The Other ◽  
Base Field ◽  
Other Hand ◽  
Previous Algorithm ◽  
The Cost ◽  
Interesting Alternative ◽  
Refined Version

Consider two ordinary elliptic curves$E,E^{\prime }$defined over a finite field$\mathbb{F}_{q}$, and suppose that there exists an isogeny$\unicode[STIX]{x1D713}$between$E$and$E^{\prime }$. We propose an algorithm that determines$\unicode[STIX]{x1D713}$from the knowledge of$E$,$E^{\prime }$and of its degree$r$, by using the structure of the$\ell$-torsion of the curves (where $\ell$ is a prime different from the characteristic $p$of the base field). Our approach is inspired by a previous algorithm due to Couveignes, which involved computations using the$p$-torsion on the curves. The most refined version of that algorithm, due to De Feo, has a complexity of $\tilde{O} (r^{2})p^{O(1)}$base field operations. On the other hand, the cost of our algorithm is$\tilde{O} (r^{2})\log (q)^{O(1)}$, for a large class of inputs; this makes it an interesting alternative for the medium- and large-characteristic cases.

Situational Irony

2017 ◽  
Author(s):  
G. R. F. Ferrari
Keyword(s):  
Large Class ◽  
The Other ◽  
Other Hand

When a situation feels like a story, we call it ironic. It seems audience-directed, though we know it is not. On the other hand, a large class of situations that strike people as ironic are simple incongruities, with no apparent connection to how stories work. This chapter therefore proposes a model that applies equally to ironies of simple incongruity and story-like ironies of peripety. It is the play of the mundane against the surprising (and not surprise alone) that makes a situation ironic. This characteristic of situational irony bears comparison with how, in stories, surprises that a plot holds in store must emerge from a plausible nexus of events, rather than springing from nowhere. It turns out, however, that ironies of simple incongruity manifest the same pattern: what makes their incongruity exquisite is that it shows up against a mundane background. Situational ironies ‘speak’ to their audiences, then, in the manner of a three-quarters-on intimation.

On generalised Fibonaccian and Lucasian numbers

2006 ◽  
Vol 90 (518) ◽  
pp. 215-222 ◽  
Author(s):  
Peter Hilton ◽  
Jean Pedersen
Keyword(s):  
Residue Class ◽  
The Other ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Residue Classes ◽  
Other Hand ◽  
Positive Integers ◽  
Striking Fact

In [1, Chapter 3, Section 2], we collected together results we had previously obtained relating to the question of which positive integers m were Lucasian, that is, factors of some Lucas number L n. We pointed out that the behaviors of odd and even numbers m were quite different. Thus, for example, 2 and 4 are both Lucasian but 8 is not; for the sequence of residue classes mod 8 of the Lucas numbers, n ⩾ 0, reads and thus does not contain the residue class 0*. On the other hand, it is a striking fact that, if the odd number s is Lucasian, then so are all of its positive powers.

LOWER TYPE DIMENSIONS OF SOME MORAN SETS

Fractals ◽  
2019 ◽  
Vol 27 (05) ◽  
pp. 1950082
Author(s):  
JIAOJIAO YANG
Keyword(s):  
Large Class ◽  
The Other ◽  
Moran Set ◽  
Lower Type ◽  
Fractal Sets ◽  
Lipschitz Mappings ◽  
Other Hand ◽  
Lower Spectrum

In this paper, we discuss the lower type dimensions for some Moran sets. On one hand, for Moran set [Formula: see text] with [Formula: see text], we prove that [Formula: see text], where the supremum is taken over all quasi-Lipschitz mappings [Formula: see text]. On the other hand, we obtain the lower spectrum formula for homogeneous Moran sets. In the proof a lower spectrum formula for a large class of fractal sets is established.

scholarly journals On the size of Diophantine m-tuples

2002 ◽  
Vol 132 (1) ◽  
pp. 23-33 ◽  
Author(s):  
ANDREJ DUJELLA
Keyword(s):  
The Other ◽  
Famous Conjecture ◽  
Nonzero Integer ◽  
Other Hand ◽  
Parametric Families ◽  
Diophantine Quintuples ◽  
Positive Integers

Let n be a nonzero integer. A set of m positive integers {a1, a2, …, am} is said to have the property D(n) if aiaj+n is a perfect square for all 1 [les ] i [les ] j [les ] m. Such a set is called a Diophantine m-tuple (with the property D(n)), or Pn-set of size m.Diophantus found the quadruple {1, 33, 68, 105} with the property D(256). The first Diophantine quadruple with the property D(1), the set {1, 3, 8, 120}, was found by Fermat (see [8, 9]). Baker and Davenport [3] proved that this Fermat’s set cannot be extended to the Diophantine quintuple, and a famous conjecture is that there does not exist a Diophantine quintuple with the property D(1). The theorem of Baker and Davenport has been recently generalized to several parametric families of quadruples [12, 14, 16], but the conjecture is still unproved.On the other hand, there are examples of Diophantine quintuples and sextuples like {1, 33, 105, 320, 18240} with the property D(256) [11] and {99, 315, 9920, 32768, 44460, 19534284} with the property D(2985984) [19]].

scholarly journals On the Exponent of the NK0-Groups of Virtually Infinite Cyclic Groups

2002 ◽  
Vol 45 (2) ◽  
pp. 180-195 ◽  
Author(s):  
Francis X. Connolly ◽  
Stratos Prassidis
Keyword(s):  
Large Class ◽  
The Other ◽  
Main Difficulty ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Other Hand

AbstractIt is known that the K-theory of a large class of groups can be computed from the K-theory of their virtually infinite cyclic subgroups. On the other hand, Nil-groups appear to be the obstacle in calculations involving the K-theory of the latter. The main difficulty in the calculation of Nil-groups is that they are infinitely generated when they do not vanish. We develop methods for computing the exponent of NK0-groups that appear in the calculation of the K0-groups of virtually infinite cyclic groups.

scholarly journals Branes, quivers and wave-functions

2021 ◽  
Vol 10 (2) ◽  
Author(s):  
Taro Kimura ◽  
Milosz Panfil ◽  
Yuji Sugimoto ◽  
Piotr Sułkowski
Keyword(s):  
Large Class ◽  
Wave Functions ◽  
The Other ◽  
Partition Functions ◽  
Other Hand ◽  
Generating Series

We consider a large class of branes in toric strip geometries, both non-periodic and periodic ones. For a fixed background geometry we show that partition functions for such branes can be reinterpreted, on one hand, as quiver generating series, and on the other hand as wave-functions in various polarizations. We determine operations on quivers, as well as SL(2,\mathbb{Z})SL(2,ℤ) transformations, which correspond to changing positions of these branes. Our results prove integrality of BPS multiplicities associated to this class of branes, reveal how they transform under changes of polarization, and imply all other properties of brane amplitudes that follow from the relation to quivers.

scholarly journals Discrepancy of Matrices of Zeros and Ones

10.37236/1447 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
R. A. Brualdi ◽  
J. Shen
Keyword(s):  
Explicit Formula ◽  
The Other ◽  
Column Vector ◽  
Other Hand ◽  
Minimum Discrepancy ◽  
Positive Integers ◽  
Maximum Discrepancy ◽  
Sum Vector

Let $m$ and $n$ be positive integers, and let $R=(r_1,\ldots, r_m)$ and $ S=(s_1,\ldots, s_n)$ be non-negative integral vectors. Let ${\cal A} (R,S)$ be the set of all $m \times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$, and let $\bar A$ be the $m \times n$ $(0,1)$-matrix where for each $i$, $1\le i \le m$, row $i$ consists of $r_i$ $1$'s followed by $n-r_i$ $0$'s. If $S$ is monotone, the discrepancy $d(A)$ of $A$ is the number of positions in which $\bar A$ has a $1$ and $A$ has a $0$. It equals the number of $1$'s in $\bar A$ which have to be shifted in rows to obtain $A$. In this paper, we study the minimum and maximum $d(A)$ among all matrices $A \in {\cal A} (R,S)$. We completely solve the minimum discrepancy problem by giving an explicit formula in terms of $R$ and $S$ for it. On the other hand, the problem of finding an explicit formula for the maximum discrepancy turns out to be very difficult. Instead, we find an algorithm to compute the maximum discrepancy.

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