Polylogarithmic Research Articles

Dynamic data structures for k-nearest neighbor queries

Our aim is to develop dynamic data structures that support k-nearest neighbors (k-NN) queries for a set of n point sites in the plane in O(f(n)+k) time, where f(n) is some polylogarithmic function of n. The key component is a general query algorithm that allows us to find the k-NN spread over t substructures simultaneously, thus reducing an O(tk) term in the query time to O(k). Combining this technique with the logarithmic method allows us to turn any static k-NN data structure into a data structure supporting both efficient insertions and queries. For the fully dynamic case, this technique allows us to recover the deterministic, worst-case, O(log2⁡n/log⁡log⁡n+k) query time for the Euclidean distance claimed before, while preserving the polylogarithmic update times. We adapt this data structure to also support fully dynamic geodesic k-NN queries among a set of sites in a simple polygon. For this purpose, we design a shallow cutting based, deletion-only k-NN data structure. More generally, we obtain a dynamic planar k-NN data structure for any type of distance functions for which we can build vertical shallow cuttings. We apply all of our methods in the plane for the Euclidean distance, the geodesic distance, and general, constant-complexity, algebraic distance functions.

Linear Forms in Polylogarithms

Let $r, \,m$ be positive integers. Let $x$ be a rational number with $0 \le x <1$. Consider $\Phi_s(x,z) =\displaystyle\sum_{k=0}^{\infty}\frac{z^{k+1}}{{(k+x+1)}^s}$ the $s$-th Lerch function with $s=1, 2, \cdots, r$. When $x=0$, this is a polylogarithmic function. Let $\alpha_1, \cdots, \alpha_m$ be pairwise distinct algebraic numbers of arbitrary degree over the rational number field, with $0<|\alpha_j|<1 \,\,\,(1\leq j \leq m)$. In this article, we show a criterion for the linear independence, over an algebraic number field containing $\mathbb{Q}(\alpha_1, \cdots, \alpha_m)$, of all the $rm+1$ numbers : $\Phi_1(x,\alpha_1)$, $\Phi_2(x,\alpha_1), $ $\cdots , \Phi_r(x,\alpha_1)$, $\Phi_1(x,\alpha_2)$, $\Phi_2(x,\alpha_2), $ $\cdots , \Phi_r(x,\alpha_2), \cdots, \cdots, \Phi_1(x,\alpha_m)$, $\Phi_2(x,\alpha_m)$, $\cdots , \Phi_r(x,\alpha_m)$ and $1$. This is the first result that gives a sufficient condition for the linear independence of values of the Lerch functions at several distinct algebraic points, not necessarily lying in the rational number field nor in quadratic imaginary fields. We give a complete proof with refinements and quantitative statements of the main theorem announced in [10], together with a proof in detail on the non-vanishing Wronskian of Hermite type.

On the new bicomplex generalization of Hurwitz–Lerch zeta function with properties and applications

Abstract In the recent years, various authors introduced different generalizations of the Hurwitz–Lerch zeta function and discussed its various properties. The main aim of our study is to introduce a new bicomplex generalization of the Hurwitz–Lerch zeta function using the new generalized form of the beta function that involves the Appell series and Lauricella functions. The new bicomplex generalization of the Hurwitz–Lerch zeta function reduces to some already known functions like the Hurwitz–Lerch zeta function, Hurwitz zeta function, Riemann zeta function and polylogarithmic function. Its different properties such as recurrence relation, summation formula, differentiation formula, generating relations and integral representations are investigated. All results induced are general in nature and reducible to already known results. As an application of the new bicomplex generalization of the Hurwitz–Lerch zeta function, we have developed a new generalized form of fractional kinetic equation and obtained its solution using the natural transform.

Regularized sample average approximation for high-dimensional stochastic optimization under low-rankness

This paper concerns a high-dimensional stochastic programming (SP) problem of minimizing a function of expected cost with a matrix argument. To this problem, one of the most widely applied solution paradigms is the sample average approximation (SAA), which uses the average cost over sampled scenarios as a surrogate to approximate the expected cost. Traditional SAA theories require the sample size to grow rapidly when the problem dimensionality increases. Indeed, for a problem of optimizing over a p-by-p matrix, the sample complexity of the SAA is given by \({\widetilde{O}}(1)\cdot \frac{p^2}{\epsilon ^2}\cdot {polylog}(\frac{1}{\epsilon })\) to achieve an \(\epsilon \)-suboptimality gap, for some poly-logarithmic function \({polylog}(\,\cdot \,)\) and some quantity \({\widetilde{O}}(1)\) independent of dimensionality p and sample size n. In contrast, this paper considers a regularized SAA (RSAA) with a low-rankness-inducing penalty. We demonstrate that, when the optimal solution to the SP is of low rank, the sample complexity of RSAA is \({\widetilde{O}}(1)\cdot \frac{p}{\epsilon ^3}\cdot {polylog}(p,\,\frac{1}{\epsilon })\), which is almost linear in p and thus indicates a substantially lower dependence on dimensionality. Therefore, RSAA can be more advantageous than SAA especially for larger scale and higher dimensional problems. Due to the close correspondence between stochastic programming and statistical learning, our results also indicate that high-dimensional low-rank matrix recovery is possible generally beyond a linear model, even if the common assumption of restricted strong convexity is completely absent.

Various Series Related to the Polylogarithmic Function

We derive some series related to the polylogarithmic function, and we also give a new proof to the existing series. Our approach is based on using the summation and integral representation methods. We obtain various interesting series as a consequence.

Harmonični niz sa polilogaritamskim funkcijama

Introduction/purpose: Some sums of the polylogarithmic function associated with harmonic numbers are established. Methods: The approach is based on using the summation methods. Results: This paper generalizes the results of the zeta function series associated with the harmonic numbers. Conclusions: Various interesting series as the consequence of the generalization are obtained.

Linear independence criteria for generalized polylogarithms with distinct shifts

For a given rational number $x$ and an integer $s\geq 1$, consider a generalized polylogarithmic function, often called the Lerch function, defined by $$\Phi _{s}(x,z)= \sum _{k=0}^{\infty }\frac{z^{k+1}}{(k+x+1)^s}.$$ We prove the linear independence

Measurement of the two-photon excitation cross-section of the 6p′[3/2]2 and 6p′[1/2]0 levels of Xe I at the wavelengths 224.3 and 222.6 nm

The two-photon excitation cross-section is a key parameter for the two-photon absorption laser induced fluorescence (TALIF) method, which is commonly used to measure atomic densities in gaseous media, especially for plasma diagnostics. The method consists in recording the fluorescence signal that follows the resonant absorption of two photons of UV light. Calibration often relies on comparing the signal recorded in the studied sample with the fluorescence produced, at a similar wavelength, in a noble gas vapor, the density of which can be easily known. The ratio of the involved cross-sections however plays an essential role for the accuracy of such measurements. Yet the two-photon excitation cross-section of atomic xenon, which is often used as the reference for oxygen density measurements, was measured only once, at the wavelengths of interest. The aim of the present study has been to consolidate the experimental value of that key parameter. The cross-section is found equal to and for the 6p′[3/2]2 and 6p′[1/2]0 levels, respectively. For the 6p′[3/2]2 level this is more than twice smaller than previously admitted. Even though the necessarily large relative uncertainty of a non-linear cross-section attaches a relatively large uncertainty to this factor of one half, the result suggests that atomic densities already measured by Xe-calibrated TALIF may have to be revised to significantly lower values. The experiments performed also provide an opportunity to revisit the validity of the approximations used for quantitative TALIF measurements and the collisional broadening and pressure shift of the two-photon 6p′[1/2]0 line. A new formula has been used to describe the two-photon absorption of a Gaussian beam in a long gas cell, which makes the decrease of the beam intensity a simple analytic expression even in strong absorption regime, based on a polylogarithmic function of the absorption rate variable.

A fast and secure public-key image encryption scheme based on Mordell elliptic curves

Encryption schemes based on elliptic curves (ECs) have shown high resistance against modern cryptanalysis. However, these schemes involve heavy computation for the generation of points over ECs, and therefore they are not suitable for communication systems such as sensor networks and Internet of Things. Thus, we propose a fast and secure public-key image encryption scheme. In this scheme, the sender and receiver agree on a public EC. Then the scheme performs pixel-masking and pixel-scrambling procedures using random numbers and a dynamic substitution box over ECs that are isomorphic to the public EC. For a fixed public EC, we prove that our scheme has a computational complexity that is a polylogarithmic function of the size of the plain text. In other words, the computational complexity of our scheme is independent of the point generation over ECs. We tested the security of our scheme against modern attacks by encrypting all standard images in the USC-SIPI image database. Compared to some state-of-the-art cryptosystems, our scheme is fast and has high security.

Definite Integral of Arctangent and Polylogarithmic Functions Expressed as a Series

We present a method using contour integration to evaluate the definite integral of arctangent reciprocal logarithmic integrals in terms of infinite sums. In a similar manner, we evaluate the definite integral involving the polylogarithmic function L i k ( y ) in terms of special functions. In various cases, these generalizations give the value of known mathematical constants such as Catalan’s constant G, Aprey’s constant ζ ( 3 ) , the Glaisher–Kinkelin constant A, l o g ( 2 ) , and π .

Low Growth Equational Complexity

AbstractThe equational complexity function $\beta \nu \,:\,{\open N} \to {\open N}$ of an equational class of algebras bounds the size of equation required to determine the membership of n-element algebras in . Known examples of finitely generated varieties with unbounded equational complexity have growth in Ω(nc), usually for c ≥ (1/2). We show that much slower growth is possible, exhibiting $O(\log_{2}^{3}(n))$ growth among varieties of semilattice-ordered inverse semigroups and additive idempotent semirings. We also examine a quasivariety analogue of equational complexity, and show that a finite group has polylogarithmic quasi-equational complexity function, bounded if and only if all Sylow subgroups are abelian.

Some applications of degenerate poly-Bernoulli numbers and polynomials

Abstract In this paper, we consider degenerate poly-Bernoulli numbers and polynomials associated with a polylogarithmic function and a p-adic invariant integral on ℤ p {\mathbb{Z}_{p}} . By using umbral calculus, we derive some identities of those numbers and polynomials.

Minimum vertex cover in generalized random graphs with power law degree distribution

In this paper we study the approximability of the minimum vertex cover problem in power law graphs. In particular, we investigate the behavior of a standard 2-approximation algorithm together with a simple pre-processing step when the input is a random sample from a generalized random graph model with power law degree distribution. More precisely, if the probability of a vertex of degree i to be present in the graph is ci−β, where β>2 and c is a normalizing constant, the expected approximation ratio is 1+ζ(β)−1Liβ(e−ρ(β)), where ζ(β) is the Riemann Zeta function of β, Li(β) is the polylogarithmic special function of β and ρ(β)=Liβ−2(1e)ζ(β−1).

An efficient distributed algorithm for constructing small dominating sets

The dominating set problem asks for a small subset D of nodes in a graph such that every node is either in D or adjacent to a node in D. This problem arises in a number of distributed network applications, where it is important to locate a small number of centers in the network such that every node is nearby at least one center. Finding a dominating set of minimum size is NP-complete, and the best known approximation is logarithmic in the maximum degree of the graph and is provided by the same simple greedy approach that gives the well-known logarithmic approximation result for the closely related set cover problem.We describe and analyze new randomized distributed algorithms for the dominating set problem that run in polylogarithmic time, independent of the diameter of the network, and that return a dominating set of size within a logarithmic factor from optimal, with high probability. In particular, our best algorithm runs in O(log n log Δ) rounds with high probability, where n is the number of nodes, Δ is one plus the maximum degree of any node, and each round involves a constant number of message exchanges among any two neighbors; the size of the dominating set obtained is within O (log Δ) of the optimal in expectation and within O(log n) of the optimal with high probability. We also describe generalizations to the weighted case and the case of multiple covering requirements.

The Approximate Rectangle of Influence Drawability Problem

Let ε 1,ε 2 be two non negative numbers. An approximate rectangle of influence drawing (also called (ε 1,ε 2)-RID) is a proximity drawing of a graph such that: (i) if u,v are adjacent vertices then their rectangle of influence “scaled down” by the factor \(\frac{1}{1+\varepsilon_{1}}\) does not contain other vertices of the drawing; (ii) if u,v are not adjacent, then their rectangle of influence “blown-up” by the factor 1+ε 2 contains some vertices of the drawing other than u and v. Firstly, we prove that all planar graphs have an (ε 1,ε 2)-RID for any ε 1>0 and ε 2>0, while there exist planar graphs which do not admit an (ε 1,0)-RID and planar graphs which do not admit a (0,ε 2)-RID. Then, we investigate (0,ε 2)-RIDs; we prove that both the outerplanar graphs and a suitably defined family of graphs without separating 3-cycles admit this type of drawing. Finally, we study polynomial area approximate rectangle of influence drawings and prove that (0,ε 2)-RIDs of proper track planar graphs (a superclass of the outerplanar graphs) can be computed in polynomial area for any ε 2>2. As for values of ε 2 such that ε 2≤2, we describe a drawing algorithm that computes (0,ε 2)-RIDs of binary trees in area \(O(n^{c + f(\varepsilon_{2})})\), where c is a positive constant, f(ε 2) is a poly-logarithmic function that tends to infinity as ε 2 tends to zero, and n is the number of vertices of the input tree.

Higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials

In this paper, we consider higher-order Frobenius-Euler polynomials, associated with poly-Bernoulli polynomials, which are derived from polylogarithmic function. These polynomials are called higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials. The purpose of this paper is to give various identities of those polynomials arising from umbral calculus.

Differential Subordination Results for Certain Integrodifferential Operator and Its Applications

We introduce an integrodifferential operator Js,b(f) which plays an important role in the Geometric Function Theory. Some theorems in differential subordination for Js,b(f) are used. Applications in Analytic Number Theory are also obtained which give new results for Hurwitz‐Lerch Zeta function and Polylogarithmic function.

Coloring [formula omitted]-free intersection graphs of geometric objects in the plane

The intersection graph of a collection C of sets is the graph on the vertex set C, in which C1,C2∈C are joined by an edge if and only if C1∩C2≠0̸. Erdős conjectured that the chromatic number of triangle-free intersection graphs of n segments in the plane is bounded from above by a constant. Here we show that it is bounded by a polylogarithmic function of n, which is the first nontrivial bound for this problem. More generally, we prove that for any t and k, the chromatic number of every Kk-free intersection graph of n curves in the plane, every pair of which have at most t points in common, is at most (ctlognlogk)clogk, where c is an absolute constant and ct only depends on t. We establish analogous results for intersection graphs of convex sets, x-monotone curves, semialgebraic sets of constant description complexity, and sets that can be obtained as the union of a bounded number of sets homeomorphic to a disk.Using a mix of results on partially ordered sets and planar separators, for large k we improve the best known upper bound on the number of edges of a k-quasi-planar topological graph with n vertices, that is, a graph drawn in the plane with curvilinear edges, no k of which are pairwise crossing. As another application, we show that for every ε>0 and for every positive integer t, there exist δ>0 and a positive integer n0 such that every topological graph with n≥n0 vertices, at least n1+ε edges, and no pair of edges intersecting in more than t points, has at least nδ pairwise intersecting edges.

A promiseBQP-complete string rewriting problem

We consider the following combinatorial problem. We are given three strings s, t, and t'of length L over some fixed finite alphabet and an integer m that is polylogarithmic inL. We have a symmetric...

Note edge-colourings of K n,n with no long two-coloured cycles

Consider the set of all proper edge-colourings of a graph G with n colours. Among all such colourings, the minimum length of a longest two-coloured cycle is denoted L(n, G). The problem of understanding L(n, G) was posed by Haggkvist in 1978 and, specifically, L(n, K n,n ) has received recent attention. Here we construct, for each prime power q ≥ 8, an edge-colouring of K n,n with n colours having all two-coloured cycles of length ≤ 2q 2, for integers n in a set of density 1 − 3/(q − 1). One consequence is that L(n, K n,n ) is bounded above by a polylogarithmic function of n, whereas the best known general upper bound was previously 2n − 4.