Asymptotics of the Sum of a Sine Series with a Convex Slowly Varying Sequence of Coefficients

We study the asymptotic behavior in a neighborhood of zero of the sum of a sine series g(b,x)=∑k=1∞bksinkx whose coefficients constitute a convex slowly varying sequence b. The main term of the asymptotics of the sum of such a series was obtained by Aljančić, Bojanić, and Tomić. To estimate the deviation of g(b,x) from the main term of its asymptotics bm(x)/x, m(x)=[π/x], Telyakovskiĭ used the piecewise-continuous function σ(b,x)=x∑k=1m(x)−1k2(bk−bk+1). He showed that the difference g(b,x)−bm(x)/x in some neighborhood of zero admits a two-sided estimate in terms of the function σ(b,x) with absolute constants independent of b. Earlier, the author found the sharp values of these constants. In the present paper, the asymptotics of the function g(b,x) on the class of convex slowly varying sequences in the regular case is obtained.

The Holonomy Groupoids of Singularly Foliated Bundles

We define a notion of connection in a fibre bundle that is compatible with a singular foliation of the base. Fibre bundles equipped with such connections are in plentiful supply, arising naturally for any Lie groupoid-equivariant bundle, and simultaneously generalising regularly foliated bundles in the sense of Kamber-Tondeur and singular foliations. We define hierarchies of diffeological holonomy groupoids associated to such bundles, which arise from the parallel transport of jet/germinal conservation laws. We show that the groupoids associated in this manner to trivial singularly foliated bundles are quotients of Androulidakis-Skandalis holonomy groupoids, which coincide with Androulidakis-Skandalis holonomy groupoids in the regular case. Finally we prove functoriality of all our constructions under appropriate morphisms.

Spectral Analysis of One-Dimensional Dirac System with Summable Potential and Sturm- Liouville Operator with Distribution Coefficients

We consider one-dimensional Dirac operatorLP,U with Birkhoff regular boundary conditions and summable potential P(x) on[0, ]. We introduce strongly and weakly regular operators. In both cases, asymptotic formulas for eigenvalues are found. In these formulas, we obtain main asymptotic terms and estimates for the second term. We specify these estimates depending on the functional class of the potential: Lp[0,] with p [1,2] and the Besov space Bp,p'[0,] with p [1,2] and (0,1/p). Additionally, we prove that our estimates are uniform on balls Pp,R Then we get asymptotic formulas for normalized eigenfunctions in the strongly regular case with the same residue estimates in uniform metric on x [0,]. In the weakly regular case, the eigenvalues 2n and 2n+1 are asymptotically close and we obtain similar estimates for two-dimensional Riesz projectors. Next, we prove the Riesz basis property in the space (L2[0,])2 for a system of eigenfunctions and associated functions of an arbitrary strongly regular operatorLP,U. In case of weak regularity, the Riesz basicity of two-dimensional subspaces is proved. In parallel with theLP,U operator, we consider the SturmLiouville operator Lq,U generated by the differential -y'' + q(x)y expressionwith distribution potential q of first-order singularity (i.e., we assume that the primitive u = q(1) belongs to L2[0, ]) and Birkhoff-regular boundary conditions. We reduce to this case -(1y')'+i(y)'+iy'+0y, operators of more general form where '1,,0(-1)L2and 10. For operator Lq,U, we get the same results on the asymptotics of eigenvalues, eigenfunctions, and basicity as for operator LP,U . Then, for the Dirac operator LP,U, we prove that the Riesz basis constant is uniform over the ballsPp,R for p1 or 0. The problem of conditional basicity is naturally generalized to the problem of equiconvergence of spectral decompositions in various metrics. We prove the result on equiconvergence by varying three indices: fL[0,] (decomposable function), PL[0,] (potential), and Sm-Sm00,m in L[0,] (equiconvergence of spectral decompositions in the corresponding norm). In conclusion, we prove theorems on conditional and unconditional basicity of the system of eigenfunctions and associated functions of operator LP,U in the spaces L[0,],2, and in various Besov spaces Bp,q[0,].

Management of people with epilepsy during the COVID-19 pandemic: a national survey among epileptologists in China

Background Compared to the healthy people, people with comorbid medical conditions are more vulnerable in the context of the coronavirus disease 2019 (COVID-19) pandemic, including the people with epilepsy. Besides a consensus recommendation by multi-national epilepsy specialists, the situation of the epilepsy management during the pandemic has seldom been reported. Methods The China Association Against Epilepsy carried out an online nationwide survey among its board members in April 2020. One hundred and thirty board members from 22 provinces, 5 autonomous regions, and 4 municipalities across China responded to the questionnaires. They reported the situation of clinical practice and gave opinions on the management of people with epilepsy between January 13th and March 31st, 2020, a time period concentrated with confirmed COVID-19 cases. Results The proportions of patients consulting through telephone or online (88.4%) and of patients with regular case review (93.9%) were highest in the high-risk area, as reported by the responders. The patients in the high-risk area were more likely to have increased episodes of seizures (17.7%), aggravated psychological disorders (30.2%), and less accessibility to anti-seizure medications (ASMs) (77.2%). Regular ASMs supply (74.6%), medical consultation (69.2%), and psychological aids (29.2%) were urgently needed for people with epilepsy. Conclusions This study demonstrated the most common dilemma faced by people with epilepsy in policy circumstances during the COVID-19 epidemic in China. The opinions raised by Chinese epileptologists may provide reference for epilepsy care in other countries.

A Hierarchical Innovation-Related Crowdsourcing Decision in Fast Fashion Industry

Due to scarcity of designers in fast fashion industry and proliferation of the Internet, small- and medium-sized garment makers have gradually turned to external designers to enhance their innovation efficiency via crowdsourcing initiative. However, few have investigated the issue of fast fashion customized-design matching decision in the crowdsourcing context. Different from previous works, we split crowdsourcing matching decision process into three hierarchical submodels in terms of three key factors, namely, surplus, due date, and goodwill. From a dynamic perspective, we first develop a two-sided matching model where garment makers and online designers select one another by maximizing their total surpluses with an aim to reach robust final pairs and derive the corresponding conditions under which the optimal pairs can be obtained. Then, the extensions of the matching model are conducted by incorporating the critical factors of due date and garment makers’ goodwill, respectively. Followed by that, an improved Gale–Shapley algorithm is devised to solve the crowdsourcing matching problems. The results illustrate when garment makers exceed online designers in number, crowdsourcing design tasks without due-date constraint are more attractive for designers’ participation than those with due-date constraint, and garment makers intend to share the incremental surpluses with designers to maximize the total surpluses. By contrast, when online designers surpass garment makers in number, designers prefer due-date design tasks to those without it. In addition, regardless of whether under the irregular or regular case, the model with goodwill concern always outperforms the two others. Moreover, celebrated garment makers are invited to post design tasks, thus enabling to entice more designers’ engagement in crowdsourcing activities, which in turn facilitating to transfer myopic designers to strategic ones. Finally, sensitivity analysis further verifies the models are stable and robust.

Kernel regression, minimax rates and effective dimensionality: Beyond the regular case

We investigate if kernel regularization methods can achieve minimax convergence rates over a source condition regularity assumption for the target function. These questions have been considered in past literature, but only under specific assumptions about the decay, typically polynomial, of the spectrum of the the kernel mapping covariance operator. In the perspective of distribution-free results, we investigate this issue under much weaker assumption on the eigenvalue decay, allowing for more complex behavior that can reflect different structure of the data at different scales.

Spectral analysis of the spin-boson Hamiltonian with two bosons for arbitrary coupling and bounded dispersion relation

We study the spectrum of the spin-boson Hamiltonian with two bosons for arbitrary coupling [Formula: see text] in the case when the dispersion relation is a bounded function. We derive an explicit description of the essential spectrum which consists of the so-called two- and three-particle branches that can be separated by a gap if the coupling is sufficiently large. It turns out, that depending on the location of the coupling constant and the energy level of the atom (w.r.t. certain constants depending on the maximal and the minimal values of the boson energy) as well as the validity or the violation of the infrared regularity type conditions, the essential spectrum is either a single finite interval or a disjoint union of at most six finite intervals. The corresponding critical values of the coupling constant are determined explicitly and the asymptotic lengths of the possible gaps are given when [Formula: see text] approaches to the respective critical value. Under minimal smoothness and regularity conditions on the boson dispersion relation and the coupling function, we show that discrete eigenvalues can never accumulate at the edges of the two-particle branch. Moreover, we show the absence of the discrete eigenvalue accumulation at the edges of the three-particle branch in the infrared regular case.

Reference analysis of non-regular models and nonparametric Bayes modeling of large data

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] Bayesian analysis is a principled approach, which makes inference about the parameter, by combining the information gained from the data and the prior belief about the parameter. There's no convergence on the choice of priors, and often different motivations for prior lead to different areas of study in Bayesian statistics. This work is motivated by two such choices, namely: reference priors and nonparametric priors. Reference priors arise out of the need to specify priors in a non-subjective manner, i.e. objective manner. Reference priors maximize the amount of information gained from the data about the parameter, in information theoretical sense. The appeal of reference priors lies in the fact that it has nice frequentist properties even for small sample size and often avoids marginalization paradoxes in Bayesian analysis. However, reference prior algorithms are typically available when the posterior is asymptotically normal and Fisher's information matrix is well-defined. In statistical parlance, such models are called regular case or regular model. Recently, Berger et al. (2009) [1] proposed a general expression of reference prior for single continuous parameter model, which is applicable for both regular and non-regular case. Motivated by Berger et al. (2009) [1], we explore reference prior methodology for a general model. Specifically, we derive expression of reference prior for single continuous parameter truncated exponential family and a general expression of conditional reference prior for multi group continuous parameter model. Furthermore, we demonstrate the usefulness of our work by deriving reference priors for models which have no known existing reference priors. We also extend Datta and Ghosh (1996) [2]'s invariance result for reference prior of regular model to general model. Nonparametric priors arise out of the need to specify priors over a large support.

Commercial Law

Commercial Law offers a fresh, modern, and stimulating account of the subject, thereby helping students better understand this important area of law. It provides thorough coverage of all key aspects of the syllabus, including the law of agency, the sale of goods, international trade, and methods of payment, finance, and security. A range of learning features is employed throughout the book to encourage understanding of the law, and to demonstrate how the principles behind it play out in practical domestic and international commercial transactions. Practical, fictional case studies are referred to in example boxes throughout the book, demonstrating the types of legal issues and problems that the law is intended to regulate, and helping students to understand the context and practical application of the law. The book includes: regular case boxes throughout the text to highlight cases of importance, providing a succinct account of the material facts of the case, a clear account of the court’s decision and reasoning, and, where appropriate, commentary on the decision; key legislation boxes to help students understand which statutory provisions are of fundamental importance; and definitions of key terms, which appear in the margins the first time the term is used, thus ensuring that students are not confused by the terminology of the subject.