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.

Amalgamation in totally non-negative Grassmannians and real regular KP divisors on $M$-curves

Abstract: In this paper we use the fact that every Postnikov planar bicolored (plabic) trivalent graph representing a given irreducible positroid cell $S$ in the totally non-negative Grassmannian $Gr^{TNN}(k,n)$ is dual to a rationally degenerate $M$-curve $\Gamma$, to provide parametrizations of $S$ in terms of real regular KP divisors in the ovals of $\Gamma$ in agreement with the characterization of real regular finite-gap solutions of the Kadomtsev-Petviashvili (KP) II equation found by Dubrovin and Natanzon [22]. Our construction is based on the connection established by the authors [3,5] between real regular finite-gap KP solutions [22] and real regular multi-line KP solitons which are known to be parametrized by points in $Gr^{TNN}(k,n)$ [16,41]. In [3,5] we studied such connection for Le-graphs with a fixed orientation and were not able to prove the invariance of the KP divisor with respect to the many geometric gauge freedoms on the network. Here we both extend the previous construction to any trivalent plabic graph representing the given positroid cell to which the soliton data belong to and we prove the invariance of the divisor on the choice of gauges using the space of totally non-negative relations studied in [7]. Such systems of relations were proposed by Lam [50] in connection with the computation of scattering amplitudes on on-shell diagrams $N=4$ SYM [10] and govern the totally non-negative amalgamation of the little positive Grassmannians, $Gr^{TP}(1,3)$ and $Gr^{TP}(2,3)$, into any given positroid cell $S\subset Gr^{TNN}(k,n)$. In our setting they rule the reality and regularity properties of the KP divisor. Finally, we explain the transformation of both the curve and the divisor under Postnikov moves and reductions and apply our construction to some examples.

On certain integrals involving the Dirichlet divisor problem

Abstract: We prove that $$ \int_1^X\Delta(x)\Delta_3(x)dx \ll X^{13/9}\log^{10/3}X, \qquad \int_1^X\Delta(x)\Delta_4(x)\d x \ll_\varepsilon X^{25/16+\varepsilon}, $$ where $\Delta_k(x)$ is the error term in the asymptotic formula for the summatory function of $d_k(n)$, generated by $\zeta^k(s)$ ($\Delta_2(x) \equiv \Delta(x)$). These bounds are sharper than the ones which follow by the Cauchy-Schwarz inequality and mean square results for $\Delta_k(x)$. We also obtain the analogues of the above bounds when $\Delta(x)$ is replaced by $E(x)$, the error term in the mean square formula for $|\zeta(1/2+it)|$.

A Hardware Algorithm for Integer Division Using the SD2 Representation

Abstract: A hardware algorithm for integer division is proposed. It is based on the radix-2 non-restoring division algorithm. Fast computation is achieved by the use of the radix-2 signed-digit (SD2) representation. The algorithm does not require normalization of the divisor, and hence, does not require an area-consuming leading-one (or zero) detection nor shifts of variable-amount. Combinational (unfolded) implementation of the algorithm yields a regularly structured array divider, and sequential implementation yields compact dividers.

Partially scattered linearized polynomials and rank metric codes

Abstract: A linearized polynomial $f(x)\in\mathbb F_{q^n}[x]$ is called scattered if for any $y,z\in\mathbb F_{q^n}$, the condition $zf(y)-yf(z)=0$ implies that $y$ and $z$ are $\mathbb F_{q}$-linearly dependent. In this paper two generalizations of the notion of a scattered linearized polynomial are defined and investigated. Let $t$ be a nontrivial positive divisor of $n$. By weakening the property defining a scattered linearized polynomial, L-$q^t$-partially scattered and R-$q^t$-partially scattered linearized polynomials are introduced in such a way that the scattered linearized polynomials are precisely those which are both L-$q^t$- and R-$q^t$-partially scattered. They determine linear sets and maximum rank distance codes whose properties are described in this paper.

Recursive filters with uniform power distribution

Abstract: Recursive linear digital filters which operate without roundoff errors, known as cyclotomic filters, have been shown to be useful as tone generators and detectors applicable to pseudo-random noise generation. Touch Tone®, f.s.k. and broadband frequency detection. These filters are modelled by periodic linear recursions with integer feedback coefficients. In certain applications, such as tone detection, it is desirable to have considerable differentiation between the detected and the rejected frequency amplitudes. Other applications such as pseudorandom noise generation require a more uniform power distribution throughout the spectrum. It is shown that the number N(m) of distinct nonzero spectral amplitude levels for such a filter of period m is exactly N(m)=?(m0)/2 when in m0>2, where m0 is the largest square-free divisor of m and ?(m0) is the number of positive integers less than and sharing no prime factor with in m0; N(m)=1 when m0?2. In particular, a uniform power distribution is effected over the entire spectrum of resonating frequencies (i.e. N(m)=1) if and only if m=2a3b for some non-negative integers a and b.