Exceptional Set Research Articles

Double square moments and subconvexity bounds for Rankin-Selberg L-functions of holomorphic cusp forms

Let $f$ and $g$ be holomorphic cusp forms of weights $k_1$and $k_2$ for the congruence subgroups$\Gamma_0(~\mathcal{N}_1~)$ and $\Gamma_0(~\mathcal{N}_2~)$,respectively. In this paper the square moment of theRankin-Selberg $L$-function for $f$ and $g$ in the aspect of both weights in short intervals is bounded, when$k_1^\varepsilon\ll~k_2\ll~k_1^{1-\varepsilon}$.These bounds are the mean Lindelof hypothesis in one caseand subconvexity bounds on average in other cases. Thesesquare moment estimates also imply subconvexity bounds forindividual $L(\frac12+{\rm~i}t,f\times~g)$ for all $g$ when $f$is chosen outside a small exceptional set. In the bestcase scenario the subconvexity bound obtained reaches theWeyl-type bound proved by Lau et al. (2006) inboth the $k_1$ and $k_2$ aspects.

ON THE LONGEST BLOCK FUNCTION IN CONTINUED FRACTIONS

For an irrational number $x\in [0,1)$, let $x=[a_{1}(x),a_{2}(x),\ldots ]$ be its continued fraction expansion with partial quotients $\{a_{n}(x):n\geq 1\}$. Given $\unicode[STIX]{x1D6E9}\in \mathbb{N}$, for $n\geq 1$, the $n$th longest block function of $x$ with respect to $\unicode[STIX]{x1D6E9}$ is defined by $L_{n}(x,\unicode[STIX]{x1D6E9})=\max \{k\geq 1:a_{j+1}(x)=\cdots =a_{j+k}(x)=\unicode[STIX]{x1D6E9}~\text{for some}~j~\text{with}~0\leq j\leq n-k\}$, which represents the length of the longest consecutive sequence whose elements are all $\unicode[STIX]{x1D6E9}$ from the first $n$ partial quotients of $x$. We consider the growth rate of $L_{n}(x,\unicode[STIX]{x1D6E9})$ as $n\rightarrow \infty$ and calculate the Hausdorff dimensions of the level sets and exceptional sets arising from the longest block function.

The Impact of Zinc Oxide Nanoparticles on Male (In)Fertility

Zinc oxide nanoparticles (ZnO NPs) are among nanoscale materials, attracting increasing attention owing to their exceptional set of characteristics, which makes these engineered nanoparticles a great option for improving the quality and effectiveness of diagnosis and treatment. The capacity of ZnO NPs to induce reactive oxygen species (ROS) production, DNA damage, and apoptosis represents a promise for their use in both cancer therapy and microbial treatment. However, their intrinsic toxicity together with their easy entrance and accumulation in organism have raised some concerns regarding the biomedical use of these NPs. Several studies have reported that ZnO NPs might induce cytotoxic effects on the male reproductive system, compromising male fertility. Despite some advances in this area, the knowledge of the effects of ZnO NPs on male fertility is still scarce. Overall, a brief outline of the major ZnO NPs biomedical applications and promises in terms of diagnostic and therapeutic use will also be explored. Further, this review intends to discuss the effect of ZnO NPs exposure on the male reproductive system and speculate their effects on male (in)fertility.

On extending Artin's conjecture to composite moduli in function fields

In 1927, Artin hypothesized that for any given non-zero integer a other than 1, −1, or a perfect square, there exists infinitely many primes p for which a is a primitive root modulo p. In 1967, Hooley proved it under the assumption of the generalized Riemann hypothesis. Since then, there are many analogues and generalization of this conjecture. In this paper, we work on its generalization to composite moduli in the function fields setting. Let A=Fq[t] be the ring of polynomials over the finite field Fq and 0≠a∈A. Let C be the A-Carlitz module. Let a be a fixed element in A. For n∈A, C(A/nA) is a finite A-module. The set of all annihilators of C(A/nA) is an ideal and generated by a monic polynomial, denoted by λ(n). Similarly, The set of all annihilators of the submodule of C(A/nA) generated by a is an ideal and let la(n) be its monic generator. We say that a is a primitive root of n, if λ(n)=la(n). DefineNa(x):=|{n∈A|deg⁡n=x,nis monic,ais a primitiveroot of n}|We prove that for a given non-constant a∈A, a∉E, an exceptional set, there exists an unbounded set V of integers such thatliminfx∈VNa(x)/qx=0 This result is analogous to Li's theorem for Artin's conjecture on composite moduli. It is the first time that this kind of results holds in the setting of the function fields.

Identifying invariant ergodic subsets and barriers to mixing by cutting and shuffling: Study in a birotated hemisphere

Mixing by cutting and shuffling can be mathematically described by the dynamics of piecewise isometries (PWIs), higher dimensional analogs of one-dimensional interval exchange transformations. In a two-dimensional domain under a PWI, the exceptional set, E[over ¯], which is created by the accumulation of cutting lines (the union of all iterates of cutting lines and all points that pass arbitrarily close to a cutting line), defines where mixing is possible but not guaranteed. There is structure within E[over ¯] that directly influences the mixing potential of the PWI. Here we provide computational and analytical formalisms for examining this structure by way of measuring the density and connectivity of ɛ-fattened cutting lines that form an approximation of E[over ¯]. For the example of a PWI on a hemispherical shell studied here, this approach reveals the subtle mixing behaviors and barriers to mixing formed by invariant ergodic subsets (confined orbits) within the fractal structure of the exceptional set. Some PWIs on the shell have provably nonergodic exceptional sets, which prevent mixing, while others have potentially ergodic exceptional sets where mixing is possible since ergodic exceptional sets have uniform cutting line density. For these latter exceptional sets, we show the connectivity of orbits in the PWI map through direct examination of orbit position and shape and through a two-dimensional return plot to explain the necessity of orbit connectivity for mixing.

Exceptional set in Waring–Goldbach problem: Two squares, two cubes and two sixth powers

Let R(n) denote the number of representations of an even integer n as the sum of two squares, two cubes and two sixth powers of primes, and by $$\mathcal {E}(N)$$ we denote the number of even integers $$n \leqslant N$$ such that the expected asymptotic formula for R(n) fails to hold. In this paper, it is proved that $$\mathcal {E}(N) \ll N^{\frac{127}{288} + \varepsilon }$$ for any $$\varepsilon >0$$ .

Matching for a family of infinite measure continued fraction transformations

As a natural counterpart to Nakada's $ \alpha $-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter in this family and that the exceptional set has Hausdorff dimension 1. Due to this matching property, we can construct a planar version of the natural extension for a large part of the parameter space. We use this to obtain an explicit expression for the density of the unique infinite $ \sigma $-finite absolutely continuous invariant measure and to compute the Krengel entropy, return sequence and wandering rate of the corresponding maps.

A lemma on the difference quotients

Using a new Borel type growth lemma, we extend the difference analogue of the lemma on the logarithmic derivative due to Halburd and Korhonen to the case of meromorphic functions $f(z)$ such that $\log T(r,f)\leq r/(\log r)^{2+\nu}$, $\nu>0$, for all sufficiently large $r$. The method by Halburd and Korhonen implies an estimate for the lemma on difference quotients, where the exceptional set is of finite logarithmic measure. We show the necessity of this set by proving that it must be of infinite linear measure for meromorphic functions whose deficiency is dependent on the choice of the origin. In addition, we show that there is an infinite sequence of $r$ in the set for which $m(r,f(z+c)/f(z))$ is not small compared to $T(r,f)$ for entire functions constructed by Miles. We also give a discrete version of Borel type growth lemma and use it to extend Halburd's result on first order discrete equations of Malmuist type.

On exceptional sets of transcendental functions with integer coefficients: solution of a problem of Mahler

On exceptional sets of transcendental functions with integer coefficients: solution of a problem of Mahler

On the Exceptional Set of the Sum of a Prime Number and a Fixed Degree of a Prime Number

On the Exceptional Set of the Sum of a Prime Number and a Fixed Degree of a Prime Number

Integrability of moduli and regularity of denjoy counterexamples

We study the regularity of exceptional actions of groups by $C^{1,\alpha}$ diffeomorphisms on the circle, i.e. ones which admit exceptional minimal sets, and whose elements have first derivatives that are continuous with concave modulus of continuity $\alpha$. Let $G$ be a finitely generated group admitting a $C^{1,\alpha}$ action $\rho$ with a free orbit on the circle, and such that the logarithms of derivatives of group elements are uniformly bounded at some point of the circle. We prove that if $G$ has spherical growth bounded by $c n^{d-1}$ and if the function $1/\alpha^d$ is integrable near zero, then under some mild technical assumptions on $\alpha$, there is a sequence of exceptional $C^{1,\alpha}$ actions of $G$ which converge to $\rho$ in the $C^1$ topology. As a consequence for a single diffeomorphism, we obtain that if the function $1/\alpha$ is integrable near zero, then there exists a $C^{1,\alpha}$ exceptional diffeomorphism of the circle. This corollary accounts for all previously known moduli of continuity for derivatives of exceptional diffeomorphisms. We also obtain a partial converse to our main result. For finitely generated free abelian groups, the existence of an exceptional action, together with some natural hypotheses on the derivatives of group elements, puts integrability restrictions on the modulus $\alpha$. These results are related to a long-standing question of D. McDuff concerning the length spectrum of exceptional $C^1$ diffeomorphisms of the circle.

Exceptional sets for the definition of quasiregularity

We establish some quasiregularity conditions in terms of local boundedness of the inferior linear dilatation outside some exceptional sets E.

An improved bound on the packing dimension of Furstenberg sets in the plane

Let 0 \leq s \leq 1 . A set K \subset \mathbb{R}^{2} is a Furstenberg s -set if for every unit vector e \in S^{1} , some line L_{e} parallel to e satisfies \dim_H [K \cap L_{e}] \geq s. The Furstenberg set problem, introduced by T. Wolff in 1999, asks for the best lower bound for the dimension of Furstenberg s -sets. Wolff proved that \dim_H K \geq \max\{s + 1/2,2s\} and conjectured that \dim_H K \geq (1 + 3s)/2 . The only known improvement to Wolff's bound is due to Bourgain, who proved in 2003 that \dim_H K \geq 1 + \epsilon for Furstenberg 1/2 -sets K , where \epsilon > 0 is an absolute constant. In the present paper, I prove a similar \epsilon -improvement for all 1/2 < s < 1 , but only for packing dimension: \dim_p K \geq 2s + \epsilon for all Furstenberg s -sets K \subset \mathbb{R}^{2} , where \epsilon > 0 only depends on s . The proof rests on a new incidence theorem for finite collections of planar points and tubes of width \delta > 0 . As another corollary of this theorem, I obtain a small improvement for Kaufman's estimate from 1968 on the dimension of exceptional sets of orthogonal projections. Namely, I prove that if K \subset \mathbb{R}^{2} is a linearly measurable set with positive length, and 1/2 < s < 1 , then \dim_H \{e \in S^{1} : \dim_p \pi_{e}(K) \leq s\} \leq s - \epsilon for some \epsilon > 0 depending only on s . Here \pi_{e} is the orthogonal projection onto the line spanned by e .

On exceptional sets of biquadratic Waring-Goldbach problem

TextIn this paper, we investigate exceptional sets in the biquadratic Waring-Goldbach problem. For example, we prove that all but O(N60/61) integers subject to the necessary local conditions can be represented as the sum of seven biquadrates of primes. Furthermore, the method in this paper can be easily used for exceptional sets for sums of fifth and higher of powers of primes. VideoFor a video summary of this paper, please visit https://youtu.be/PGa8P-Px8bk.

Exceptional Set of Waring-Goldbach Problem with Unequal Powers of Primes

In this paper, it is proved that with at most $O(N^{17/42+\varepsilon})$ exceptions, all even positive integer $n$, $n \in [N/2,N]$, can be represented in the form $p_{1}^{2} + p_{2}^{2} + p_{3}^{3} + p_{4}^{3} + p_{5}^{4} + p_{6}^{4}$, where $p_{1}$, $p_{2}$, $p_{3}$, $p_{4}$, $p_{5}$, $p_{6}$ are prime numbers. This improves a recent result $O(N^{13/16+\varepsilon})$ due to Zhang and Li [13].

Text and Data Mining: Articles 3 and 4 of the Directive 2019/790/Eu

Our society is in the midst of an explosion of data: ‘there was 5 exabytes of information created between the dawn of civilization through 2003, but that much information is now created every 2 days, and the pace is increasing’. In 2014, there were 2.4 billion internet users. That number grew to 3.8 billion in 2017 and new data is created by the quintillions of bytes every day. Together with mobile devices, the Internet of Things (IoT) contributes to this huge data production. In the big data era, orientating within this magma of online data has become an extremely complex but crucial task, also leading to complex issues in terms of regulation of this new environment. The European Union seems at first to have acknowledged the potential of monitoring data, putting in place measures to unlock TDM potentialities. On 14 September 2016, the European Commission published a Proposal for a Directive on copyright in the Digital Single Market, which was approved into Directive 2019/790/EU on 17 April 2019 (“DSM Directive”). Inter alia, this copyright reform would like to improve access to protected works across borders within the Digital Single Market (DSM) to boost research and innovation. To this end, the DSM Directive includes a set of new mandatory exceptions and limitations. In particular, the reform introduces two specific limitations for TDM. In this chapter, the introduction of mandatory TDM limitations in European law will be assessed against the international and European framework of copyright exceptions and limitations by considering the rationales for such an exception and the positive and negative impacts of the reform. Giving the importance of TDM activities for the economic development in the EU and its innovative environment, the question arises if the reform lives up to the expectations. Although, following our previous suggestions, the scope of the limitation has been broadened, the final text of the reform still limits the full exploitation of the potential of data for research and innovation, for start-ups as well as more generally for the broader access to works and the information they contain.

Slim exceptional set for sums of two squares, two cubes, and two biquadrates of primes

We prove that, with at most $$O(N^{\frac{17}{192}+\varepsilon})$$ exceptions, all even positive integers up to N are expressible in the form p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 4 + p 6 4 , where p1, p2, …, p6 are prime numbers. This gives large improvement of a recent result $$O(N^{\frac{13}{16}+\varepsilon})$$ due to M. Zhang and J. J. Li.

Representation of Analytic Functions by Series of Exponential Monomials in Convex Domains and Its Applications

In this paper lower bounds for entire functions of exponential type and regular growth, zero sets of which have zero condensation indices, are obtained. In this case, the exceptional set consists of pairwise disjoint disks centered at zeroes. Sufficient conditions for radii of these circles are indicated. We also obtain a result on representation of analytic functions in the closure of a bounded convex domain (as well as analytic functions in domain and continuous up to the boundary) by series of exponential monomials. This result extends the classical result of A.F. Leont’ev to the case of multiple zero set of entire function. The obtained result is applied to the problem on distribution of singular points of a sum of series of exponential monomials at the boundary of its convergence domain.

Existence of closed geodesics through a regular point on translation surfaces

We show that on any translation surface, if a regular point is contained in some simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in $${\mathbb {RP}}^1$$. Moreover, the set of points that are not contained in any simple closed geodesic is finite. We also construct explicit examples showing that such points exist on some translation surfaces. For a surface in any hyperelliptic component, we show that this finite exceptional set is actually empty. The proofs of our results use Apisa’s classifications of periodic points and of $$\text {GL}(2,{\mathbb {R}})$$ orbit closures in hyperelliptic components, as well as a recent result of Eskin–Filip–Wright.

The Well Efficiency Criteria Revisited—Development of a General Well Efficiency Criteria (GWEC) Based on Rorabaugh’s Model

The Strategic Water Storage & Recovery (SWSR) Project in Liwa, Abu Dhabi is a leading and unique hydrogeology project in the world because of both its financial and scientific dimensions. The objective of the project is to store desalinized water in the local Liwa aquifer, to be able to supply water to Abu Dhabi in case of emergency. A total of 315 recovery wells have been drilled in pursuance of the scope of the SWSR project. Out of the total 315 wells, 25 wells met construction problems and were removed from the study. The remaining 290 wells have been analyzed using step drawdown tests (SDTs) and the model of Rorabaugh. This provided a large and unique database regarding the parameters of this model: linear aquifer-loss coefficient (B), non-linear well-loss coefficient (C) and the exponent p. Analysis of this exceptional data set revealed noteworthy and novel findings: (1) the range of the exponent p values is found to be very extensive, varying from 0.35 to 6.01. For comparison, the highest values of p given in the literature very seldom exceed 4; (2) p behaves like a lognormal variable; (3) parameters C and p are closely correlated. A semi-logarithmic diagram displays a linear relation of p vs. log C, with a determination coefficient R² = 0.83; (4) A graphical and tabulated procedure, termed General Well Efficiency Criteria, is proposed to assess well efficiency. Given the very wide range of p values implied in this procedure, it becomes possible thus to assess the efficiency of any well analyzed with an SDT. This study finally raises questions about Jacob’s model validity, which assumes that p is constant and equal to 2.