Showing papers on "Divisor published in 2018"

Sums of divisor functions in Fq[t] and matrix integrals

Abstract: We study the mean square of sums of the kth divisor function $$d_k(n)$$ over short intervals and arithmetic progressions for the rational function field over a finite field of q elements. In the limit as $$q\rightarrow \infty $$ we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums of $$d_k(n)$$ in terms of a lattice point count. This lattice point count can in turn be calculated in terms of a certain piecewise polynomial function, which we analyse. Our results suggest general conjectures for the corresponding classical problems over the integers, which agree with the few cases where the answer is known.

Adaptive approximation in arithmetic circuits: A low-power unsigned divider design

Abstract: Many approximate arithmetic circuits have been proposed for high-performance and low-power applications. However, most designs are either hardware-efficient with a low accuracy or very accurate with a limited hardware saving, mostly due to the use of a static approximation. In this paper, an adaptive approximation approach is proposed for the design of a divider. In this design, division is computed by using a reduced-width divider and a shifter by adaptively pruning the input bits. Specifically, for a 2n/n division 2k/k bits are selected starting from the most significant ‘1’ in the dividend/divisor. At the same time, redundant least significant bits (LSBs) are truncated or if the number of remaining LSBs is smaller than 2k for the dividend or k for the divisor, ‘0’s are appended to the LSBs of the input. To avoid overflow, a 2(k + 1)/(k + 1) divider is used to compute the division of the 2k-bit dividend and the k-bit divisor, both with the most significant bits being ‘0’. Thus, k < n is a key variable that determines the size of the divider and the accuracy of the approximate design. Finally, an error correction circuit is proposed to recover the error caused by the shifter by using OR gates. The synthesis results in an industrial 28nm CMOS process show that the proposed 16/8 approximate divider using an 8/4 accurate divider is 2.5χ as fast and consumes 34.42% of the power of the accurate 16/8 design. Compared with the other approximate dividers, the proposed design is significantly more accurate at a similar power-delay product. Moreover, simulation results show that the proposed approximate divider outperforms the other designs in two image processing applications.

The Circle Problem of Gauss and the Divisor Problem of Dirichlet—Still Unsolved

Abstract: Let r2(n) denote the number of representations of the positive integer n as a sum of two squares, and let d(n) denote the number of positive divisors of n Gauss and Dirichlet were evidently the fi

Multifractal analysis of the Brjuno function

Abstract: The Brjuno function B is a 1-periodic, nowhere locally bounded function, introduced by Yoccoz because it encapsulates a key information concerning analytic small divisor problems in dimension 1. We show that \(T^p_\alpha \) regularity, introduced by Calderon and Zygmund, is the only one which is relevant in order to unfold the pointwise regularity properties of B; we determine its \(T^p_\alpha \) regularity at every point and show that it is directly related to the irrationality exponent \(\tau (x)\): its p-exponent at x is exactly \(1/\tau (x)\). This new example of multifractal function puts into light a new link between dynamical systems and fractal geometry. Finally we also determine the Holder exponent of a primitive of B.

A boundedness conjecture for minimal log discrepancies on a fixed germ

Abstract: We consider the following conjecture: on a klt germ (X,x), for every finite set I there is a positive integer N with the property that for every R-ideal J on X with exponents in I, there is a divisor E over X that computes the minimal log discrepancy of (X,J) at x and such that its discrepancy k_E is bounded above by N. We show that this implies Shokurov's ACC conjecture for minimal log discrepancies on a fixed klt germ and give some partial results towards the conjecture.

Higher genus relative and orbifold Gromov-Witten invariants

Abstract: Given a smooth projective variety $X$ and a smooth divisor $D\subset X$. We study relative Gromov-Witten invariants of $(X,D)$ and the corresponding orbifold Gromov-Witten invariants of the $r$-th root stack $X_{D,r}$. For sufficiently large $r$, we prove that orbifold Gromov-Witten invariants of $X_{D,r}$ are polynomials in $r$. Moreover, higher genus relative Gromov-Witten invariants of $(X,D)$ are exactly the constant terms of the corresponding higher genus orbifold Gromov-Witten invariants of $X_{D,r}$. We also provide a new proof for the equality between genus zero relative and orbifold Gromov-Witten invariants, originally proved by Abramovich-Cadman-Wise \cite{ACW}. When $r$ is sufficiently large and $X=C$ is a curve, we prove that stationary relative invariants of $C$ are equal to the stationary orbifold invariants in all genera.

Reheating and Dark Radiation after Fibre Inflation

Abstract: We study perturbative reheating at the end of fibre inflation where the inflaton is a closed string modulus with a Starobinsky-like potential. We first derive the spectral index $n_s$ and the tensor-to-scalar ratio $r$ as a function of the number of efoldings and the parameter $R$ which controls slow-roll breaking corrections. We then compute the inflaton couplings and decay rates into ultra-light bulk axions and visible sector fields on D7-branes wrapping the inflaton divisor. This leads to a reheating temperature of order $10^{10}$ GeV which requires $52$ efoldings. Ultra-light axions contribute to dark radiation even if $\Delta N_{\rm eff}$ is almost negligible in the generic case where the visible sector D7-stack supports a non-zero gauge flux. If the parameter $R$ is chosen to be small enough, $n_s\simeq 0.965$ is then in perfect agreement with current observations while $r$ turns out to be of order $r\simeq 0.007$. If instead the flux on the inflaton divisor is turned off, $\Delta N_{\rm eff}\lesssim 0.6$ which, when used as a prior for Planck data, requires $n_s\simeq 0.99$. After $R$ is fixed to obtain such a value of $n_s$, primordial gravity waves are larger since $r\simeq 0.01$.

Uniform boundedness in terms of ramification

Abstract: Let $$d\ge 1$$ be fixed. Let F be a number field of degree d, and let E / F be an elliptic curve. Let $$E(F)_{\text {tors}}$$ be the torsion subgroup of E(F). In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a constant B(d), which depends on d but not on the chosen field F or on the curve E / F, such that the size of $$E(F)_{\text {tors}}$$ is bounded by B(d). Moreover, Merel gave a bound (exponential in d) for the largest prime that may be a divisor of the order of $$E(F)_{\text {tors}}$$ . In 1996, Parent proved a bound (also exponential in d) for the largest p-power order of a torsion point that may appear in $$E(F)_{\text {tors}}$$ . It has been conjectured, however, that there is a bound for the size of $$E(F)_{\text {tors}}$$ that is polynomial in d. In this article we show that under certain hypotheses there is a linear bound for the largest p-power order of a torsion point defined over F, which in fact is linear in the maximum ramification index of a prime ideal of the ring of integers F over (p).

The wobbly divisors of the moduli space of rank-$2$ vector bundles

Abstract: Let $X$ be a smooth projective complex curve of genus $g \geq 2$ and let $\M_X(2,\Lambda)$ be the moduli space of semi-stable rank-$2$ vector bundles over $X$ with fixed determinant $\Lambda$. We show that the wobbly locus, i.e., the locus of semi-stable vector bundles admitting a non-zero nilpotent Higgs field is a union of divisors $\Ww_k \subset \M_X(2,\Lambda)$. We show that on one wobbly divisor the set of maximal subbundles is degenerate. We also compute the class of the divisors $\Ww_k$ in the Picard group of $\M_X(2,\Lambda)$.

On the sum of digits of special sequences in finite fields

Abstract: In $$\mathbb {F}_q$$ , Dartyge and Sarkozy introduced the notion of digits and studied some properties of the sum of digits function. We will provide sharp estimates for the number of elements of special sequences of $$\mathbb {F}_q$$ whose sum of digits is prescribed. Such special sequences of particular interest include the set of n-th powers for each $$n\ge 1$$ and the set of elements of order d in $$\mathbb {F}_q^*$$ for each divisor d of $$q-1$$ . We provide an optimal estimate for the number of squares whose sum of digits is prescribed. Our methods combine A. Weil bounds with character sums, Gaussian sums and exponential sums.

An Energy-Efficient, Yet Highly-Accurate, Approximate Non-Iterative Divider

Abstract: In1 this paper, we present a highly accurate and energy efficient non-iterative divider, which uses multiplication as its main building block In this structure, the division operation is performed by first reforming both dividend and divisor inputs, and then multiplying the rounded value of the scaled dividend by the reciprocal of the rounded value of the scaled divisor Precisely, the interval representing the fractional value of the scaled divisor is partitioned into non-overlapping sub-intervals, and the reciprocal of the scaled divisor is then approximated with a linear function in each of these sub-intervals The efficacy of the proposed divider structure is assessed by comparing its design parameters and accuracy with state-of-the-art, non-iterative approximate dividers as well as exact dividers in 45nm digital CMOS technology Circuit simulation results show that the mean absolute relative error of the proposed structure for doing 1 32-bit division is less than 02%, while the proposed structure has significantly lower energy consumption than the exact divider Finally, the effectiveness of the proposed divider in one image processing application is reported and discussed

A Parallel Probabilistic Approach to Factorize a Semiprime

Abstract: In accordance with the distributive traits of semiprimes’ divisors, the article proposes an approach that can find out the small divisor of a semiprime by parallel computing. The approach incorporates a deterministic search with a probabilistic search, requires less memory and can be implemented on ordinary multicore computers. Experiments show that certain semiprimes of 27 to 46 decimal-bits can be validly factorized with the approach on personal computer in expected time.

Multi-point codes over Kummer extensions

Abstract: This paper is concerned with the construction of algebraic geometric codes defined from Kummer extensions. It plays a significant role in the study of such codes to describe bases for the Riemann–Roch spaces associated with totally ramified places. Along this line, we present an explicit characterization of Weierstrass semigroups and pure gaps. Additionally, we determine the floor of a certain type of divisor introduced by Maharaj, Matthews and Pirsic. Finally, we apply these results to find multi-point codes with excellent parameters. As one of the examples, a presented code with parameters $$[254,228,\geqslant 16]$$ over $$ {\mathbb {F}}_{64} $$ yields a new record.

Log BPS numbers of log Calabi-Yau surfaces

Abstract: Let $(S,E)$ be a log Calabi-Yau surface pair with $E$ a smooth divisor. We define new conjecturally integer-valued counts of $\mathbb{A}^1$-curves in $(S,E)$. These log BPS numbers are derived from genus 0 log Gromov-Witten invariants of maximal tangency along $E$ via a formula analogous to the multiple cover formula for disk counts. A conjectural relationship to genus 0 local BPS numbers is described and verified for del Pezzo surfaces and curve classes of arithmetic genus up to 2. We state a number of conjectures and provide computational evidence.

Triple correlations of Fourier coefficients of cusp forms

Abstract: We treat an unbalanced shifted convolution sum of Fourier coefficients of cusp forms. As a consequence, we obtain an upper bound for correlation of three Hecke eigenvalues of holomorphic cusp forms \(\sum _{H\le h\le 2H}W\left( \frac{h}{H}\right) \sum _{X\le n\le 2X}\lambda _{1}(n-h)\lambda _{2}(n)\lambda _{3}(n+h)\), which is nontrivial provided that \(H\ge X^{2/3+\varepsilon }\). The result can be viewed as a cuspidal analogue of a recent result of Blomer on triple correlations of divisor functions.

Correction to: Explicit upper bound for the average number of divisors of irreducible quadratic polynomials.

Abstract: Consider the divisor sum $$\sum _{n\le N}\tau (n^2+2bn+c)$$ for integers b and c. We improve the explicit upper bound of this average divisor sum in certain cases, and as an application, we give an improvement in the maximal possible number of $$D(-1)$$ -quadruples. The new tool is a numerically explicit Polya–Vinogradov inequality, which has not been formulated explicitly before but is essentially due to Frolenkov–Soundararajan.

Some multidimensional integrals in number theory and connections with the Painlev\'e V equation

Abstract: We study piecewise polynomial functions $\gamma_k(c)$ that appear in the asymptotics of averages of the divisor sum in short intervals. Specifically, we express these polynomials as the inverse Fourier transform of a Hankel determinant that satisfies a Painlev\'e V equation. We prove that $\gamma_k(c)$ is very smooth at its transition points, and also determine the asymptotics of $\gamma_k(c)$ in a large neighbourhood of $k=c/2$. Finally, we consider the coefficients that appear in the asymptotics of elliptic Aliquot cycles.

Stratified spaces and synthetic Ricci curvature bounds

Abstract: We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K $\in$ R on the regular set, the cone angle along the stratum of codimension two is smaller than or equal to 2$\pi$ and its dimension is at most equal to N. This gives a new wide class of geometric examples of metric measure spaces satisfying the RCD(K, N) curvature-dimension condition, including for instance spherical suspensions, orbifolds, K{\"a}hler-Einstein manifolds with a divisor, Einstein manifolds with conical singularities along a curve. We also obtain new analytic and geometric results on stratied spaces, such as Bishop-Gromov volume inequality, Laplacian comparison, L{\'e}vy-Gromov isoperimetric inequality.

Explicit upper bound for the average number of divisors of irreducible quadratic polynomials.

Abstract: Consider the divisor sum $$\sum _{n\le N}\tau (n^2+2bn+c)$$ for integers b and c. We extract an asymptotic formula for the average divisor sum in a convenient form, and provide an explicit upper bound for this sum with the correct main term. As an application we give an improvement of the maximal possible number of $$D(-1)$$ -quadruples.

Combinatorial identities and Titchmarsh's divisor problem for multiplicative functions

Abstract: Given a multiplicative function $f$ which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum $\sum_{|h|

Black holes and higher depth mock modular forms

Abstract: By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi-Yau threefold, we derive modular properties of the generating function of BPS degeneracies of D4-D2-D0 black holes in type IIA string theory compactified on the same space. Mathematically, these BPS degeneracies are the generalized Donaldson-Thomas invariants counting coherent sheaves with support on a divisor $\cal D$, at the large volume attractor point. For $\cal D$ irreducible, this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping M5-brane on $\cal D$ and is therefore known to be modular. Instead, when $\cal D$ is the sum of $n$ irreducible divisors ${\cal D}_i$, we show that the generating function acquires a modular anomaly. We characterize this anomaly for arbitrary $n$ by providing an explicit expression for a non-holomorphic modular completion in terms of generalized error functions. As a result, the generating function turns out to be a (mixed) mock modular form of depth $n-1$.

Influence of Divisor-ratio to Distribution of Semiprime's Divisor

Abstract: For a semiprime that consists in two distinct odd prime divisors, this article makes an investigation on the distribution of the small divisor by analyzing the divisor-ratio that is calculated by the big divisor divided by the small one. It proves that, the small divisor must be a divisor of an odd integer lying in an interval that is uniquely determined by the divisor-ratio, and the length of the interval decreases exponentially with the increment of the ratio. Accordingly, a big divisor-ratio means the small divisor can be found in a small interval whereas a small divisor-ratio means it has to find the small divisor in a large interval. The proved theorems and corollaries can provide certain theoretical supports for finding out the small divisor of the semiprime.

Explicit rationality of some cubic fourfolds

Abstract: Recent results of Hassett, Kuznetsov and others pointed out countably many divisors $C_d$ in the open subset of $\mathbb{P}^{55}=\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^5}(3)))$ parametrizing all cubic 4-folds and lead to the conjecture that the cubics corresponding to these divisors should be precisely the rational ones. Rationality has been proved by Fano for the first divisor $C_{14}$ and in [arXiv:1707.00999] for the divisors $C_{26}$ and $C_{38}$. In this note we describe explicit birational maps from a general cubic fourfold in $C_{14}$, in $C_{26}$ and in $C_{38}$ to $\mathbb{P}^4$, providing concrete geometric realizations of the more abstract constructions in [arXiv:1707.00999].

$S$-parts of values of univariate polynomials,\\ binary forms and decomposable forms at integral points

Abstract: Let $S$ be a finite set of primes. The $S$-part $[m]_S$ of a non-zero integer $m$ is the largest positive divisor of $m$ that is composed of primes from $S$. In 2013, Gross and Vincent proved that if $f(X)$ is a polynomial with integer coefficients and with at least two roots in the complex numbers, then for every integer $x$ at which $f(x)$ is non-zero, we have (*) $[f(x)]_S\leq c\cdot |f(x)|^d$, where $c$ and $d$ are effectively computable and $d 1/n$, provided we do not require effectivity of $c$. Further, we show that such an inequality does not hold anymore with $d=1/n$ and sufficiently small $c$. In addition we prove a density result, giving for every $\epsilon>0$ an asymptotic estimate with the right order of magnitude for the number of integers $x$ with absolute value at most $B$ such that $f(x)$ has $S$-part at least $|f(x)|^{\epsilon}$. The result of Gross and Vincent, as well as the other results mentioned above, are generalized to values of binary forms and decomposable forms at integral points. Our main tools are Baker type estimates for linear forms in complex and $p$-adic logarithms, the $p$-adic Subspace Theorem of Schmidt and Schlickewei, and a recent general lattice point counting result of Barroero and Widmer.

Algebraic Methods for the Construction of Algebraic-Difference Equations With Desired Behavior

Abstract: For a given system of algebraic and difference equations, written as an Auto-Regressive (AR) representation $A(\sigma)\beta(k)=0$, where $\sigma $ denotes the shift forward operator and $A\left( \sigma \right) $ a regular polynomial matrix, the forward-backward behavior of this system can be constructed by using the finite and infinite elementary divisor structure of $A\left( \sigma \right) $. This work studies the inverse problem: Given a specific forward-backward behavior, find a family of regular or non-regular polynomial matrices $A\left( \sigma \right) $, such that the constructed system $A\left( \sigma \right) \beta \left( k\right) =0$ has exactly the prescribed behavior. It is proved that this problem can be reduced either to a linear system of equations problem or to an interpolation problem and an algorithm is proposed for constructing a system satisfying a given forward and/or backward behavior.

Two applications of the spectrum of numbers

Abstract: Let the base $${\beta}$$ be a complex number, $${|\beta| > 1}$$ , and let $${A \subset \mathbb{C}}$$ be a finite alphabet of digits. The A-spectrum of $${\beta}$$ is the set $${{S}_{A}(\beta) = {\{\Sigma^{n}_{k=0} {a}_{k}\beta^{k} | n\in \mathbb{N}, {a}_{k} \in A\}}}$$ . We show that the spectrum $${{S}_{A}(\beta)}$$ has an accumulation point if and only if 0 has a particular $${(\beta, A)}$$ -representation, said to be rigid. The first application is restricted to the case that $${\beta > 1}$$ and the alphabet is A = {−M, . . . , M}, $${{M \geq}}$$ 1 integer. We show that the set $${{Z}_{\beta, M}}$$ of infinite $${(\beta, A)}$$ -representations of 0 is recognizable by a finite Buchi automaton if and only if the spectrum $${{S}_{A}(\beta)}$$ has no accumulation point. Using a result of Akiyama–Komornik and Feng, this implies that $${{Z}_{\beta, M}}$$ is recognizable by a finite Buchi automaton for any positive integer $${M \geq\lceil {\beta\rceil-1}}$$ if and only if $${{\beta}}$$ is a Pisot number. This improves the previous bound $${M \geq \lceil \beta\rceil}$$ . For the second application the base and the digits are complex. We consider the on-line algorithm for division of Trivedi and Ercegovac generalized to a complex numeration system. In on-line arithmetic the operands and results are processed in a digit serial manner, starting with the most significant digit. The divisor must be far from 0, which means that no prefix of the $${(\beta,A)}$$ -representation of the divisor can be small. The numeration system $${(\beta,A)}$$ is said to allow preprocessing if there exists a finite list of transformations on the divisor which achieve this task. We show that $${(\beta,A)}$$ allows preprocessing if and only if the spectrum $${{S}_{A}(\beta)}$$ has no accumulation point.

Untrodden pathways in the theory of the restricted partition function $p(n, N)$

Abstract: We obtain a finite analogue of a recent generalization of an identity in Ramanujan's Notebooks. Differentiating it with respect to one of the parameters leads to a result whose limiting case gives a finite analogue of Andrews' famous identity for $\textup{spt}(n)$. The latter motivates us to extend the theory of the restricted partition function $p(n, N)$, namely, the number of partitions of $n$ with largest parts less than or equal to $N$, by obtaining the finite analogues of rank and crank for vector partitions as well as of the rank and crank moments. As an application of the identity for our finite analogue of the spt-function, namely $\textup{spt}(n, N)$, we prove an inequality between the finite second rank and crank moments. The other results obtained include finite analogues of a recent identity of Garvan, an identity relating $d(n, N)$ and lpt$(n, N)$, namely the finite analogues of the divisor and largest parts functions respectively, and a finite analogue of the Beck-Chern theorem. We also conjecture an inequality between the finite analogues of $k^{\textup{th}}$ rank and crank moments.