Exceptional Set Research Articles

Rectangular Summation of Multiple Fourier Series and Multi-parametric Capacity

We consider the class of multiple Fourier series associated with functions in the Dirichlet space of the polydisc. We prove that every such series is summable with respect to unrestricted rectangular partial sums, everywhere except for a set of zero multi-parametric logarithmic capacity. Conversely, given a compact set in the torus of zero capacity, we construct a Fourier series in the class which diverges on this set, in the sense of Pringsheim. We also prove that the multi-parametric logarithmic capacity characterizes the exceptional sets for the radial variation and radial limits of Dirichlet space functions. As a by-product of the methods of proof, the results also hold in the vector-valued setting.

Office Espionage: A Synthesis of Legal, Ethical, and Professional Standards Considerations Regarding Unauthorized Covert Recording Within the Policing Workplace

Abstract Is it illegal, unethical, or even misconduct for police officers to covertly record conversations with colleagues and managers? Non-consensual recordings of meetings and conversations are becoming increasingly prevalent within policing owing to the ubiquity of smartphones. Whilst there is little in the way of legislative restraint, nevertheless, it is capable of causing serious harm and should not be considered a morally neutral act. It is possible for surreptitious recordings to be ethically justified, but the threshold for such justification is high and requires an exceptional set of circumstances. The article goes on to present a tripartite typology of the behaviour and from this to develop a misconduct framework for assessing the seriousness of the behaviour and making a number of recommendations to reduce the incidence of covert recording.

A randomized comparison of extended-release naltrexone with or without patient navigation vs enhanced treatment-as-usual for incarcerated adults with opioid use disorder

The high prevalence of opioid use among justice-involved adults make jails an exceptional setting to initiate opioid use disorder (OUD) treatment, but optimal strategies for delivering these interventions are still not well understood. The objective of this study was to conduct a randomized controlled trial to assess the effectiveness of extended-release naltrexone (XR-NTX, Vivitrol®; Alkermes Inc) alone or in conjunction with patient navigation (XR-NTX + PN) for jail inmates with OUD. We randomized a sample of 135 sentenced jail inmates with moderate to severe OUD to (1) XR-NTX only; (2) XR-NTX + PN; or (3) enhanced treatment-as-usual (ETAU) with drug education, each initiated prior to release from jail. We scheduled follow-up data assessments at 1, 3, 6, and 12 months post-release. Primary outcomes were opioid use (based on Timeline Followback Interview and Addiction Severity Index) and meeting CIDI DSM-5 criteria for OUD 6 months postrelease. We also measured treatment adherence, HIV risk, and recidivism. XR-NTX participants received a mean of 2.26 of 7 possible injections compared to XR-NTX + PN participants, who received a mean of 2.93 injections (Cohen's d = 0.33, 95% CI: −0.09 to 0.74). Thirty-six percent of patients in XR-NTX + PN attended at least one postrelease PN session. We found no significant differences by study condition six months after release from jail for the primary outcomes of any opioid use (ETAU: 17%, XR-NTX: 16%, XR-NTX + PN: 29%) and past 30-day OUD (ETAU: 8%, XR-NTX: 11%, XR-NTX + PN: 10%). Secondary outcomes of rearrest and HIV risk also were similar across groups, with the exception of lower sex-related HIV risk among those in the XR-NTX condition at 12 months. This study did not show superior outcomes of XR-NTX or XR-NTX + PN with regard to opioid use or recidivism outcomes, relative to ETAU. It did, however, highlight the difficulties with adherence to XR-NTX and PN interventions in OUD patients initiating treatment in jail.

Bridging the gap between visual temporary memory and working memory: The role of stimuli distinctiveness.

Visual working memory (VWM) is traditionally assumed to be immune to proactive interference (PI). However, in a recent study (Endress & Potter, 2014), performance in a visual memory task was superior when all items were unique and hence interference from previous trials was impossible, compared to a standard condition in which a limited set of repeating items was used and stimuli from previous trials could interfere with the current trial. Furthermore, when all the items were unique, the estimated memory capacity far exceeded typical capacity estimates. Consequently, the researchers suggested the existence of a separate memory buffer, the "temporary memory," which has an unbounded capacity for meaningful items. However, before accepting this conclusion, methodological differences between the repeated-unique procedure and typical estimates of VWM should be considered. Here, we tested the extent to which the exceptional set of heterogeneous, complex, meaningful real-world objects contributed to the large PI in the repeated-unique procedure. Thus, the same paradigm was employed with a set of real-world objects and with homogenous sets (e.g., houses, faces) in which the items were meaningful, yet less visually distinct, and participants had to rely on subtle visual details to perform the task. The results revealed a large PI effect for real-world heterogeneous objects, but substantially smaller effects for the homogenous sets. These findings suggest that there is no need to postulate a new memory buffer. Instead, we suggest that VWM capacity and vulnerability to PI are highly influenced by task characteristics, and specifically, by the stimuli distinctiveness. (PsycInfo Database Record (c) 2020 APA, all rights reserved).

The Polynomial Carleson operator

We prove affirmatively the one-dimensional case of a conjecture of Stein regarding the $L^p$-boundedness of the Polynomial Carleson operator for $1 \lt p\lt \infty$. Our proof relies on two new ideas: (i) we develop a framework for higher-order wave-packet analysis that is consistent with the time-frequency analysis of the (generalized) Carleson operator, and (ii) we introduce a local analysis adapted to the concepts of mass and counting function, which yields a new tile discretization of the time-frequency plane that has the major consequence of eliminating the exceptional sets from the analysis of the Carleson operator. As a further consequence, we are able to deliver the full $L^p$-boundedness range and prove directly---without interpolation techniques---strong $L^2$ bound for the (generalized) Carleson operator, answering a question raised by C. Fefferman.

New Exceptional Sets and Convergence of the Square Partial Sums of Walsh-Fourier Series

For double Walsh-Fourier series and with f ∈ L2([0, 1) × [0, 1)) we prove two almost orthogonality results relative to the linearized maximal square partial sums operator SN(x,y)f (x, y). Assumptions are N(x, y) non-decreasing as a function of x and of y and, roughly speaking, partial derivatives with approximately constant ratio $${{N_y^\prime \left({x,y} \right)} \over {N_x^\prime \left({x,y} \right)}} \cong {2^{{n_0}}}$$ for all x and y, where n0 is any fixed non-negative integer. Estimates, independent of N(x, y) and n0, are then extended to Lr, 1 10.

Dynamics of conformal foliations

Abstract The purpose of this article is to review the author's results on the existence and structure of minimal sets and attractors of conformal foliations. Results on strong transversal equivalence of conformal foliations are also presented. Connections with works of other authors are indicated. Examples of conformal foliations with exceptional, exotic and regular minimal sets which are attractors are constructed.

Exceptional Algebraic Sets for Infinite Discrete Groups of $$PSL(3,\mathbb {C})$$

In this note we show that the exceptional algebraic set for an infinite discrete group in $$PSL(3,{{\mathbb {C}}})$$ should be a finite union of: complex lines, copies of the Veronese curve or copies of the cubic $$xy^2-z^3$$ .

Orientation at singularities of harmonic functions

We find a simple expression in complex terms for homogeneous harmonic polynomials, which we use to express the Laurent series of a harmonic function around an isolated singularity. Also, we show a residue theorem and study the orientation at isolated singularities through the use of complex dilatation, focusing on those points where orientation is not preserved nor reversed, making essential the concept of exceptional set and extending it to isolated singularities.

Source-count Distribution of Gamma-Ray Blazars

Abstract With 10 years of operation and an exceptional data set, the Fermi-Large Area Telescope allows us to unveil the detailed composition of the extragalactic γ-ray sky above 100 MeV. In this paper, we derive the intrinsic source-count distribution (logN–logS) of extragalactic sources (i.e., blazars) at via the efficiency correction method. With this approach, we are able to measure the distribution down to a photon flux of and to an energy flux of ∼10−12 . In both cases, the logN–logS becomes flatter at low fluxes. Moreover, we show that this logN–logS is representative of the blazar population (assuming the majority of unassociated sources are blazars) and allows us to constrain its evolution quite effectively. Among recently proposed evolutionary models, we find that the Pure Density Evolution model best describes the evolutionary properties of the blazar population and that their integrated emission accounts for of the total extragalactic γ-ray background.

Predation of loggerhead sea turtle eggs across Georgia’s barrier islands

Nesting beach management is a vital element of the population recovery efforts for the vulnerable loggerhead sea turtle (Caretta caretta) across the globe. In the southeast United Sates, turtle nests are threatened by numerous anthropogenic and natural threats, including predation of eggs by native and non-native predators. We analyzed loggerhead nest predation and other egg; loss using an exceptional 10-year data set (2009–2018) that covered nesting beaches on 12 islands on the Georgia coast. Our objectives were to 1) determine which predators cause the greatest loss of loggerhead sea turtle eggs, 2) evaluate whether non-native species have a higher; rate of predation than native species, and 3) compare predation rates to other major sources of egg loss across these islands. Our results show that under current strategies for nest management: 1) non-native feral hogs and native raccoons have the greatest impact as predators on sea turtle eggs; 2) non-native predators have caused significantly more egg loss across Georgia’s coast than native species, but the impact varies greatly by species; and 3) losses to predation are similar in magnitude to post-management losses from tides and storms over the last decade. We recommend the continued use of multiple management techniques, including nest screening and; targeted predator management, but caution that predator controls should be focused on those demonstrated to cause significant losses in order to prioritize conservation funding.

On large values of Weyl sums

A special case of the Menshov–Rademacher theorem implies for almost all polynomials x1Z+…+xdZd∈R[Z] of degree d for the Weyl sums satisfy the upper bound|∑n=1Nexp⁡(2πi(x1n+…+xdnd))|⩽N1/2+o(1),N→∞. Here we investigate the exceptional sets of coefficients (x1,…,xd) with large values of Weyl sums for infinitely many N, and show that in terms of the Baire categories and Hausdorff dimension they are quite massive, in particular of positive Hausdorff dimension in any fixed cube inside of [0,1]d. We also use a different technique to give similar results for sums with just one monomial xnd. We apply these results to show that the set of poorly distributed modulo one polynomials is rather massive as well.

Hausdorff dimension of the large values of Weyl sums

The authors have recently obtained a lower bound of the Hausdorff dimension for the sets of vectors (x1,…,xd)∈[0,1)d with large Weyl sums, namely of vectors for which|∑n=1Nexp⁡(2πi(x1n+…+xdnd))|⩾Nα for infinitely many integers N⩾1. Here we obtain an upper bound for the Hausdorff dimension of these exceptional sets.

On the Nature of γ-th Arithmetic Zeta Functions

Symmetry and elementary symmetric functions are main components of the proof of the celebrated Hermite–Lindemann theorem (about the transcendence of e α , for algebraic values of α ) which settled the ancient Greek problem of squaring the circle. In this paper, we are interested in similar results, but for powers such as e γ log n . This kind of problem can be posed in the context of arithmetic functions. More precisely, we study the arithmetic nature of the so-called γ-th arithmetic zeta function ζ γ ( n ) : = n γ ( = e γ log n ), for a positive integer n and a complex number γ . Moreover, we raise a conjecture about the exceptional set of ζ γ , in the case in which γ is transcendental, and we connect it to the famous Schanuel’s conjecture.

SLOW GROWTH RATE OF THE DIGITS IN ENGEL EXPANSIONS

We are concerned with the Hausdorff dimension of the set [Formula: see text] where [Formula: see text] is the digit of the Engel expansion of [Formula: see text] and [Formula: see text] is a function such that [Formula: see text] as [Formula: see text]. The Hausdorff dimension of [Formula: see text] is studied by Lü and Liu [Hausdorff dimensions of some exceptional sets in Engel expansions, J. Number Theory 185 (2018) 490–498] under the condition that [Formula: see text] grows to infinity. The aim of this paper is to determine the Hausdorff dimension of [Formula: see text] when [Formula: see text] slowly increases to infinity, such as in logarithmic functions and power functions with small exponents. We also provide a detailed analysis of the gaps between consecutive digits. This includes the central limit theorem and law of the iterated logarithm for [Formula: see text] and the Hausdorff dimension of the set [Formula: see text] where [Formula: see text] with the convention [Formula: see text].

Unlocking Platform Technology to Combat Health Pandemics

Effective use of data from digital platforms and related technological ecosystems could be key to mitigating the spread of the COVID-19 pandemic. Data from smartphones, GPS, and wearable fitness trackers can combine with sophisticated algorithms to trace networks of contact with COVID-19 patients. This practice has already been used to successfully slow the spread of the pandemic in Korea and Taiwan warrants immediate, broader consideration. Regarding consumer concerns about data privacy, given the unusual and dire circumstances, government authorities need a set of consent exceptions that allows non-health data to be harnessed for the public health. A swift, thoughtful collaboration between the technology sector and the government could result in regulatory policy changes that have proven potential to save lives.

Hausdorff dimension of an exceptional set in the theory of continued fractions**The research of A Bakhtawar is supported by La Trobe University postgraduate research award. The research of M Hussain is supported by La Trobe University startup grant and Australian Research Council Discovery Project (DP200100994).

In this article we calculate the Hausdorff dimension of the setwhere is any function with limn→∞ Φ(n) = ∞. This in turn contributes to the metrical theory of continued fractions as well as gives insights about the set of Dirichlet non-improvable numbers.

Maximal run-length function for real numbers in beta-dynamical system**This work is supported by NSFC Nos. 11225101, 11601358, 11831007.

Let β > 1 and x ∈ [0, 1) be two real numbers. For any y ∈ [0, 1), the maximal run-length function rx(y, n) (with respect to x) is defined to be the maximal length of the prefix of x’s β-expansion which appears in the first n digits of y’s. In this paper, we study the metric properties of the maximal run-length function and apply them to the hitting time, which generalises many known results. In the meantime, the fractal dimensions of the related exceptional sets are also determined.

Reduced spectral synthesis and compact operator synthesis

We introduce and study the notion of reduced spectral synthesis, which unifies the concepts of spectral synthesis and uniqueness in locally compact groups. We exhibit a number of examples and prove that every non-discrete locally compact group with an open abelian subgroup has a subset that fails reduced spectral synthesis. We introduce compact operator synthesis as an operator algebraic counterpart of this notion and link it to other exceptional sets in operator algebra theory, studied previously. We show that a closed subset E of a second countable locally compact group G satisfies reduced local spectral synthesis if and only if the subset E⁎={(s,t):ts−1∈E} of G×G satisfies compact operator synthesis. We apply our results to questions about the equivalence of linear operator equations with normal commuting coefficients on Schatten p-classes.

Exceptional Set for Sums of Symmetric Mixed Powers of Primes

The main purpose of this paper is to use the Hardy–Littlewood method to study the solvability of mixed powers of primes. To be specific, we consider the even integers represented as the sum of one prime, one square of prime, one cube of prime, and one biquadrate of prime. However, this representation can not be realized for all even integers. In this paper, we establish the exceptional set of this kind of representation and give an upper bound estimate.