On character sums over flat numbers

Let $q\geqslant2$ be an integer, $\chi$ be any non-principal character mod $q$, and $H=H(q)\leqslant q.$ In this paper the authors prove some estimates for character sums of the form $$\mathcal{W}(\chi,H;q)=\sum_{n\in\mathscr{F}(H)}\chi(n),$$ where \mathscr{F}(H)=\left\{n\in\mathbb{Z}|(n,q)=1,1\leqslant n,\bar{n}\leqslant q, |n-\bar{n}|\leqslant H\}, $\bar{n}$ is defined by $n\bar{n}\equiv1\pmod q.$Comment: 9 pages, with a complete proof of Theorem 3, Section 5. Accepted by J. Number Theor

Dynamic Assortment Optimization with Changing Contextual Information

In this paper, we study the dynamic assortment optimization problem under a finite selling season of length $T$. At each time period, the seller offers an arriving customer an assortment of substitutable products under a cardinality constraint, and the customer makes the purchase among offered products according to a discrete choice model. Most existing work associates each product with a real-valued fixed mean utility and assumes a multinomial logit choice (MNL) model. In many practical applications, feature/contexutal information of products is readily available. In this paper, we incorporate the feature information by assuming a linear relationship between the mean utility and the feature. In addition, we allow the feature information of products to change over time so that the underlying choice model can also be non-stationary. To solve the dynamic assortment optimization under this changing contextual MNL model, we need to simultaneously learn the underlying unknown coefficient and makes the decision on the assortment. To this end, we develop an upper confidence bound (UCB) based policy and establish the regret bound on the order of $\widetilde O(d\sqrt{T})$, where $d$ is the dimension of the feature and $\widetilde O$ suppresses logarithmic dependence. We further established the lower bound $\Omega(d\sqrt{T}/K)$ where $K$ is the cardinality constraint of an offered assortment, which is usually small. When $K$ is a constant, our policy is optimal up to logarithmic factors. In the exploitation phase of the UCB algorithm, we need to solve a combinatorial optimization for assortment optimization based on the learned information. We further develop an approximation algorithm and an efficient greedy heuristic. The effectiveness of the proposed policy is further demonstrated by our numerical studies.Comment: 4 pages, 4 figures. Minor revision and polishing of presentatio

Izravan način rješavanja Burgersove jednadžbe

By utilizing the transformation given in our article Fizika A 14 (2005) 233 and introducing an intermediate function f(y), a direct approach is presented to solve the Burgers equation which does not require introducing the trial function. Based on it, abundant types of explicit exact solutions of the Burgers equation, including the solitary wave solutions, the singular traveling wave solutions, the triangle function periodic wave solutions, the rational solution etc., are successfully derived.Primjenom pretvorbe objavljene u našem radu Fizika A 14 (2005) 233 i uvodeći međufunkciju f(y), razvili smo izravan način rješavanja Burgersove jednadžbe bez uvođenja probne funkcije. Na toj smo osnovi izveli niz eksplicitnih egzaktnih rješenja Burgersove jednadžbe, uključujući solitonska valna rješenja, singularna rješenja za putujuće valove, trokutna periodička valna rješenja i racionalna rješenja