Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Derived categories of cyclic covers and their branch divisors

Abstract: Given a variety Y with a rectangular Lefschetz decomposition of its derived category, we consider a degree n cyclic cover \(X \rightarrow Y\) ramified over a divisor \(Z \subset Y\). We construct semiorthogonal decompositions of \(\mathrm {D^b}(X)\) and \(\mathrm {D^b}(Z)\) with distinguished components \({\mathcal {A}}_X\) and \({\mathcal {A}}_Z\) and prove the equivariant category of \({\mathcal {A}}_X\) (with respect to an action of the nth roots of unity) admits a semiorthogonal decomposition into \(n-1\) copies of \({\mathcal {A}}_Z\). As examples, we consider quartic double solids, Gushel–Mukai varieties, and cyclic cubic hypersurfaces.

On Dirichlet Series for Sums of Squares

Abstract: Hardy and Wright (An Introduction to the Theory of Numbers, 5th edn., Oxford, 1979) recorded elegant closed forms for the generating functions of the divisor functions σ k (n) and σ k 2 (n) in the terms of Riemann Zeta function ζ(s) only. In this paper, we explore other arithmetical functions enjoying this remarkable property. In Theorem 2.1 below, we are able to generalize the above result and prove that if f i and g i are completely multiplicative, then we have $$ \sum\limits_{n = 1}^\infty {\frac{{({f_{1*{g_1}}})\cdot({f_{2*{g_2}}})(n)}}{{{n^s}}}} = \frac{{{L_{{f_1}{f_2}}}(s){L_{{g_1}{g_2}}}(s){L_{{f_1}{g_2}}}(s){L_{{g_1}{f_2}}}(s)}}{{{L_{{f_1}{f_2}{g_1}{g_2}}}(2s)}} $$ where \( {L_f}(s): = \sum olimits_{n = 1}^\infty {f(n){n^{ - s}}} \) is the Dirichlet series corresponding to f. Let r N (n) be the number of solutions of x 1 2 + … + x N 2 = n and r 2, p (n) be the number of solutions of x 2 + Py 2 = n. One of the applications of Theorem 2.1 is to obtain closed forms, in terms of ζ (s) and Dirichlet L-functions, for the generating functions of r N (n), r N 2 (n), r 2, p (n) and r 2, p (n)2 for certain N and P. We also use these generating functions to obtain asymptotic estimates of the average values for each function for which we obtain a Dirichlet series.

Permutations of finite fields for check digit systems

Abstract: Let q be a prime power. For a divisor n of q ? 1 we prove an asymptotic formula for the number of polynomials of the form $$f(X)=\frac{a-b}{n}\left(\sum_{j=1}^{n-1}X^{j(q-1)/n}\right)X+\frac{a+b(n-1)}{n}X\in\mathbb{F}_q[X]$$ such that the five (not necessarily different) polynomials f(X), f(X)±X and f(f(X))±X are all permutation polynomials over $${\mathbb{F}_q}$$ . Such polynomials can be used to define check digit systems that detect the most frequent errors: single errors, adjacent transpositions, jump transpositions, twin errors and jump twin errors.

Multi-bit programmable frequency divider

Abstract: A multi-bit, programmable, modular digital frequency divider divides an input frequency by an m-bit integer divisor to produce an output frequency. The integer divisor re-initializes m-number of flip-flop stages with the divisor input at the end of every output clock. Each divisor bit is gated to a D-input through a respective data multiplexer controlled by a clock output. A run/initialize mode controller receives the input frequency and produces the divided output frequency and controls the timing of the re-initialization.