Log N-log Research Articles

Probing the cosmological principle using the slope of log N-log S relationship for quasars

We study the dipole signal in the slope x of the log N–log S relationship for quasars using the CatWISE2020 catalog of infrared sources. Here N is the number of sources with flux density greater than S. The slope is extracted by using a maximized log-likelihood method as well as Bayesian analysis. We obtain the value x = 1.579 ± 0.001 for a quasar sample of 1355352 sources. We extract the dipole signal in this parameter by employing χ 2 minimization, assuming a sky model of x up to the quadrupole term. We find that the dipole amplitude |D| is 0.005 ± 0.002 and dipole direction (l, b) in Galactic coordinate system equal to (201.50° ± 27.87°, -29.37° ± 19.86°). The direction of dipole anisotropy is found to be very close to the hemispherical power asymmetry (l, b)=(221°,-27°) in the Cosmic Microwave Background (CMB). The dipole signal is also extracted using Bayesian analysis and found to be in good agreement with that obtained using χ 2 minimization. We also obtain a signal of quadrupole anisotropy which is found to be correlated with the ecliptic poles and can be attributed to ecliptic bias.

The SRG/eROSITA All-Sky Survey

Clusters of galaxies can be used as powerful probes to study astrophysical processes on large scales, test theories of the growth of structure, and constrain cosmological models. The driving science goal of the SRG/eROSITA All-Sky Survey is to assemble a large sample of X-ray clusters with a well-defined selection function to determine the evolution of the mass function and, hence, the cosmological parameters. We present here a catalog of 12 247 optically confirmed galaxy groups and clusters detected in the 0.2–2.3 keV as extended X-ray sources in a 13 116 deg2 region in the western Galactic half of the sky, which eROSITA surveyed in its first six months of operation. The clusters in the sample span the redshift range 0.003 < z < 1.32. The majority (68%) of these clusters, 8361 sources, represent new discoveries without known counterparts in the literature. The mass range of the sample covers three orders of magnitude from 5 × 1012 Msun to 2 × 1015Msun. We construct a sample for cosmology with a higher purity level (~95%) than the primary sample, comprising 5259 securely detected and confirmed clusters in the 12791 deg2 common footprint of eRASS1 and the DESI Legacy Survey DR10. We characterize the X-ray properties of each cluster, including their flux, luminosity and temperature, the total mass, gas mass, gas mass fraction, and mass proxy YX. These are determined within two apertures, 300 kpc, and the overdensity radius R500, and are calculated by applying a forward modeling approach with a rigorous X-ray background treatment, K-factor, and the Galactic absorption corrections. Population studies utilizing log N-log S, the number of clusters detected above a given flux limit, and the luminosity function show overall agreement with the previous X-ray surveys after accounting for the survey completeness and purity through the selection function. The first eROSITA All-Sky Survey provides an unprecedented sample of galaxy groups and clusters selected in the X-ray band. The eRASS1 cluster catalog demonstrates the excellent performance of eROSITA for extended source detection, consistent with the pre-launch expectations for the final all-sky survey, eRASS:8.

Lower bounds for piercing and coloring boxes

Given a family B of axis-parallel boxes in Rd, let τ denote its piercing number, and ν its independence number. It is an old question whether τ/ν can be arbitrarily large for given d≥2. Here, for every ν, we construct a family of axis-parallel boxes achievingτ≥Ωd(ν)⋅(log⁡νlog⁡log⁡ν)d−2. This not only answers the previous question for every d≥3 positively, but also matches the best known upper bound up to double-logarithmic factors.Our main construction has further implications about the Ramsey and coloring properties of configurations of boxes as well. We show the existence of a family of n boxes in Rd, whose intersection graph has clique and independence number Od(n1/2)⋅(log⁡nlog⁡log⁡n)−(d−2)/2. This is the first improvement over the trivial upper bound Od(n1/2), and matches the best known lower bound up to double-logarithmic factors. Finally, for every ω satisfying log⁡nlog⁡log⁡n≪ω≪n1−ε, we construct an intersection graph of n boxes with clique number at most ω, and chromatic number Ωd,ε(ω)⋅(log⁡nlog⁡log⁡n)d−2. This matches the best known upper bound up to a factor of Od((log⁡ω)(log⁡log⁡n)d−2).

Linear cycles of consecutive lengths

A well-known result of Verstraëte [23] shows that for each integer k≥2 every graph G with average degree at least 8k contains cycles of k consecutive even lengths, the shortest of which is of length at most twice the radius of G. We establish two extensions of Verstraëte's result for linear cycles in linear r-uniform hypergraphs.We show that for any fixed integers r≥3 and k≥2, there exist constants c1=c1(r) and c2=c2(r), such that every n-vertex linear r-uniform hypergraph G with average degree d(G)≥c1k contains linear cycles of k consecutive even lengths, the shortest of which is of length at most 2⌈log⁡nlog⁡(d(G)/k)−c2⌉. In particular, as an immediate corollary, we retrieve the current best known upper bound on the linear Turán number of C2kr with improved coefficients.Furthermore, we show that for any fixed integers r≥3 and k≥2, there exist constants c3=c3(r) and c4=c4(r) such that every n-vertex linear r-uniform hypergraph with average degree d(G)≥c3k, contains linear cycles of k consecutive lengths, the shortest of which has length at most 6⌈log⁡nlog⁡(d(G)/k)−c4⌉+6. In both cases for given average degree d, the length of the shortest cycles cannot be improved up to the constant factors c2,c4.

Building squares with optimal state complexity in restricted active self-assembly

Tile Automata is a recently defined model of self-assembly that borrows many concepts from cellular automata to create active self-assembling systems where changes may be occurring within an assembly without requiring attachment. This model has been shown to be powerful even with limited assembly size, but many fundamental questions have yet to be explored. Here, we study the state complexity of assembling n×n squares in seeded Tile Automata systems where growth starts from a seed and tiles attach one at a time, similar to the abstract Tile Assembly Model. We provide optimal bounds for three classes of seeded Tile Automata systems (all without detachment), which vary in the amount of complexity allowed in the transition rules. We show that, in general, seeded Tile Automata systems require Θ(log14⁡n) states. For single-transition systems, where only one state may change in a transition rule, we show a bound of Θ(log13⁡n), and for deterministic systems, where each pair of states may only have one associated transition rule, a bound of Θ((log⁡nlog⁡log⁡n)12). Along the way, we provide optimal bounds for the subroutines of building binary strings and building O(log⁡n)×n rectangles.

Multi-robot motion planning for unit discs with revolving areas

We study the problem of motion planning for a collection of n labeled unit disc robots in a polygonal environment. We assume that the robots have revolving areas around their start and final positions: that each start and each final is contained in a radius 2 disc lying in the free space, not necessarily concentric with the start or final position, which is free from other start or final positions. This assumption allows a weakly-monotone motion plan, in which robots move according to an ordering as follows: during the turn of a robot R in the ordering, it moves fully from its start to final position, while other robots do not leave their revolving areas. As R passes through a revolving area, a robot R′ that is inside this area may move within the revolving area to avoid a collision. Notwithstanding the existence of a motion plan, we show that minimizing the total traveled distance in this setting, specifically even when the motion plan is restricted to be weakly-monotone, is APX-hard, ruling out any polynomial-time (1+ε)-approximation algorithm.On the positive side, we present the first constant-factor approximation algorithm for computing a feasible weakly-monotone motion plan. The total distance traveled by the robots is within an O(1) factor of that of the optimal motion plan, which need not be weakly monotone. Our algorithm extends to an online setting in which the polygonal environment is fixed but the initial and final positions of robots are specified in an online manner. Finally, we observe that the overhead in the overall cost that we add while editing the paths to avoid robot-robot collision can vary significantly depending on the ordering we chose. Finding the best ordering in this respect is known to be NP-hard, and we provide a polynomial time O(log⁡nlog⁡log⁡n)-approximation algorithm for this problem.

A note for approximating the submodular cover problem over integer lattice with low adaptive and query complexities

This paper studies the Diminishing Return Submodular Cover (DRSC) problem as follows: given a diminishing return submodular function f:Z+E↦R+ over the ground set E of size n, a positive integer B as the maximum value of a coordinate of a vector in Z+E and a threshold α>0, the problem asks to find the vector with the smallest size x so that f(x)≥α. This problem has been attracted by much research recently due to its importance in machine learning and combinatorial optimization. However, because of high query and adaptive complexities, the existing algorithms might still not be efficient enough when working with large input data. This paper proposes a randomized and bicriteria approximation algorithm that has O(log⁡n) adaptive complexity and O((n+log⁡nlog⁡B)log⁡(nB)) query complexity. Our algorithm significantly reduces both adaptive and query complexities compared to some state-of-the-art algorithms by factors of Ω(log⁡(m)log2⁡(mnlog⁡B)(1+log⁡(log⁡B)/log⁡(n))), Ω(min⁡{n/log⁡(n),log⁡(B)}) respectively, where m=f(B⋅1).

On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms

Abstract In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the Ramanujan Δ-function, we show that, for any ϵ > 0 \epsilon>0 , there exist infinitely many natural numbers 𝑛 such that τ ⁢ ( p n ) \tau(p^{n}) has at least 2 ( 1 - ϵ ) ⁢ log ⁡ n log ⁡ log ⁡ n 2^{(1-\epsilon)\frac{\log n}{\log\log n}} distinct prime factors for almost all primes 𝑝. This improves and refines the existing bounds. We also study lower bounds for absolute norms of radicals of non-zero Fourier coefficients of modular forms alluded to above.

On parallel time in population protocols

The parallel time of a population protocol is defined as the average number of required interactions in which an agent in the protocol participates, i.e., the quotient between the total number of interactions required by the protocol and the total number n of agents, or just roughly the number of required rounds, where a round stands for a sequence of n consecutive interactions. This naming triggers an intuition that at least the expected number of parallel steps sufficient to implement a round is O(1). In a single parallel step only mutually independent interactions can be involved. We show that when the transition function of a population protocol is treated as a black box then the expected maximum number of parallel steps necessary to implement a round is Ω(log⁡nlog⁡log⁡n). We also provide a combinatorial argument for a matching upper bound on the expected number of parallel steps under additional assumptions. Further, we extend these bounds by showing that the situation changes dramatically for sequences of m=Ω(nlog⁡n) interactions. Then, the expected number of parallel steps required to implement such sequences is Θ(mn) under the aforementioned additional assumptions. Thus, it asymptotically coincides with the notion of parallel time, i.e., O(mn), for sequences of interactions produced by protocols solving any non-trivial problems requiring Ω(nlog⁡n) interactions.

On the gaps between consecutive primes

Abstract Let p n p_{n} denote the 𝑛-th prime. We prove that, for any m ≥ 1 m\geq 1 , there exist infinitely many 𝑛 such that p n - p n - m ≤ C m p_{n}-p_{n-m}\leq C_{m} for some large constant C m > 0 C_{m}>0 , and p n + 1 - p n ≥ c m ⁢ log ⁡ n ⁢ log ⁡ log ⁡ n ⁢ log ⁡ log ⁡ log ⁡ log ⁡ n log ⁡ log ⁡ log ⁡ n p_{n+1}-p_{n}\geq\frac{c_{m}\log n\log\log n\log\log\log\log n}{\log\log\log n} for some small constant c m > 0 c_{m}>0 . Furthermore, for any fixed positive integer 𝑙, there are many positive integers 𝑘 with ( k , l ) = 1 (k,l)=1 such that p ′ ⁢ ( k , l ) ≥ c ⁢ k ⋅ log ⁡ k ⁢ log ⁡ log ⁡ k ⁢ log ⁡ log ⁡ log ⁡ log ⁡ k log ⁡ log ⁡ log ⁡ k , p^{\prime}(k,l)\geq ck\cdot\frac{\log k\log\log k\log\log\log\log k}{\log\log\log k}, where p ′ ⁢ ( k , l ) p^{\prime}(k,l) denotes the least prime of the form k ⁢ n + l kn+l with n ≥ 1 n\geq 1 , which improves the previous result of Prachar.

Finding binary words with a given number of subsequences

We relate binary words with a given number of subsequences to continued fractions of rational numbers with a given denominator. We deduce that there are binary strings of length O(log⁡nlog⁡log⁡n) with exactly n subsequences; this can be improved to O(log⁡n) under assumption of Zaremba's conjecture.

XXL-HSC: An updated catalogue of high-redshift (z ≥ 3.5) X-ray AGN in the XMM-XXL northern field

X-rays offer a reliable method to identify active galactic nuclei (AGNs). However, in the high-redshift Universe, X-ray AGNs are poorly sampled due to their relatively low space density and the small areas covered by X-ray surveys. In addition to wide-area X-ray surveys, it is important to have deep optical data in order to locate the optical counterparts and determine their redshifts. In this work, we built a high-redshift (z ≥ 3.5) X-ray-selected AGN sample in the XMM-XXL northern field using the most updated [0.5–2 keV] catalogue along with a plethora of new spectroscopic and multi-wavelength catalogues, including the deep optical Subaru Hyper Suprime-Cam (HSC) data, reaching magnitude limits i ∼ 26 mag. We selected all the spectroscopically confirmed AGN and complement this sample with high-redshift candidates that are HSC g- and r-band dropouts. To confirm the dropouts, we derived their photometric redshifts using spectral energy distribution techniques. We obtained a sample of 54 high-z sources (28 with spec-z), the largest in this field so far (almost three times larger than in previous studies), and we estimated the possible contamination and completeness. We calculated the number counts (log N-log S) in different redshift bins and compared our results with previous studies and models. We provide the strongest high-redshift AGN constraints yet at bright fluxes (f0.5 − 2 keV > 10−15 erg s−1 cm−2). The samples of z ≥ 3.5, z ≥ 4, and z ≥ 5 are in agreement with an exponential decline model similar to that witnessed at optical wavelengths. Our work emphasises the importance of using wide-area X-ray surveys with deep optical data to uncover high-redshift AGNs.

Online facility location with mobile facilities

We examine the Online Facility Location problem in an extended version. Fotakis showed a lower bound of Ω(log⁡nlog⁡log⁡n) for the original Online Facility Location problem, where n is the number of clients. This bound holds even on the real line and for randomized algorithms against oblivious adversaries.We propose randomized online algorithms in the following setting: We consider the Euclidean space of arbitrary dimension and allow the facilities to either move arbitrarily or to move at most a constant distance m in each time step. The costs for moving a facility from a to b is D⋅d(a,b) where D≥1 is a constant. Clients are assigned to facilities instantly and irreversibly. The cost for serving the clients is either calculated as soon as a client arrives or at the end of the computation. The algorithms for these two cost models achieve the same competitiveness on the line, which, in contrast to the original Online Facility Location problem, is independent of n. In the case of arbitrary movement, it only depends on D. In the case of a limited movement distance m, it additionally depends on m and the opening costs of the facilities. We show that our results are asymptotically tight on the real line. For the Euclidean space of higher dimensions, we give an algorithm for the model where costs for clients are evaluated immediately. The competitive ratio of our algorithm depends on the above parameters and additionally on the number of optimal facilities.

On the approximability of robust network design

Given the dynamic nature of traffic, we investigate the variant of robust network design where we have to determine the capacity to reserve on each link so that each demand vector belonging to a polyhedral set can be routed. The objective is either to minimize congestion or a linear cost. Routing is assumed to be fractional and dynamic (i.e., dependent on the current traffic vector). We first prove that the robust network design problem with minimum congestion cannot be approximated within any constant factor. Then, using the ETH conjecture, we get a Ω(log⁡nlog⁡log⁡n) lower bound for the approximability of this problem. This implies that the well-known O(log⁡n) approximation ratio established by Räcke in 2008 is tight. Using Lagrange relaxation, we obtain a new proof of the O(log⁡n) approximation. An important consequence of the Lagrange-based reduction and our inapproximability results is that the robust network design problem with linear reservation cost cannot be approximated within any constant ratio. This answers a long-standing open question of Chekuri (2007). We also give another proof of the result of Goyal et al. (2009) stating that the optimal linear cost under static routing can be Ω(log⁡n) more expensive than the cost obtained under dynamic routing. Finally, we show that even if only two given paths are allowed for each commodity, the robust network design problem with minimum congestion or linear cost is hard to approximate within some constant.

Slightly improved lower bounds for homogeneous formulas of bounded depth and bounded individual degree

Kayal, Saha and Tavenas (Theory of Computing, 2018), showed that any bounded-depth homogeneous formula of bounded individual degree (bounded by r) that computes an explicit polynomial over n variables must have size exp⁡(Ω(1r(n4)1/Δ)) for all depths Δ≤O(log⁡nlog⁡r+log⁡log⁡n). In this article we show an improved size lower bound of exp⁡(Ω(Δr(nr2)1/Δ)) for all depths Δ≤O(log⁡nlog⁡r), and for the same explicit polynomial. In comparison to Kayal, Saha and Tavenas (Theory of Computing, 2018) (1) our results give superpolynomial lower bounds in a wider regime of depth Δ, and (2) for all Δ∈[ω(1),o(log⁡nlog⁡r)] our lower bound is asymptotically better.This improvement is due to a finer product decomposition of general homogeneous formulas of bounded-depth. This follows from an adaptation of a new product decomposition for bounded-depth multilinear formulas shown by Chillara, Limaye and Srinivasan (SIAM Journal of Computing, 2019).

Invariable generation of permutation and linear groups

A subset {x1,x2,…,xd} of a group G invariably generates G if {x1g1,x2g2,…,xdgd} generates G for every d-tuple (g1,g2…,gd)∈Gd. We prove that a finite completely reducible linear group of dimension n can be invariably generated by ⌊3n2⌋ elements. We also prove tighter bounds when the field in question has order 2 or 3. Finally, we prove that a transitive [respectively primitive] permutation group of degree n≥2 [resp. n≥3] can be invariably generated by O(nlog⁡n) [resp. O(log⁡nlog⁡log⁡n)] elements.

Mixing times of random walks on dynamic configuration models

The mixing time of a random walk, with or without backtracking, on a random graph generated according to the configuration model on n n vertices, is known to be of order logn log⁡n . In this paper, we investigate what happens when the random graph becomes dynamic, namely, at each unit of time a fraction α n αn of the edges is randomly rewired. Under mild conditions on the degree sequence, guaranteeing that the graph is locally tree-like, we show that for every e∈(0,1) e∈(0,1) the e e -mixing time of random walk without backtracking grows like 2log(1/e)/log(1/(1−α n )) − − − − − − − − − − − − − − − − − − − − − √ 2log⁡(1/e)/log⁡(1/(1−αn)) as n→∞ n→∞ , provided that lim n→∞ α n (logn) 2 =∞ limn→∞αn(log⁡n)2=∞ . The latter condition corresponds to a regime of fast enough graph dynamics. Our proof is based on a randomised stopping time argument, in combination with coupling techniques and combinatorial estimates. The stopping time of interest is the first time that the walk moves along an edge that was rewired before, which turns out to be close to a strong stationary time.

The nearest colored node in a tree

We start a systematic study of data structures for the nearest colored node problem on trees. Given a tree with colored nodes and weighted edges, we want to answer queries (v,c) asking for the nearest node to node v that has color c. This is a natural generalization of the well-known nearest marked ancestor problem. We give an O(n)-space O(log⁡log⁡n)-query solution and show that this is optimal. We also consider the dynamic case where updates can change a node's color and show that in O(n) space we can support both updates and queries in O(log⁡n) time. We complement this by showing that O(polylogn) update time implies Ω(log⁡nlog⁡log⁡n) query time. Finally, we consider the case where updates can change the edges of the tree (link-cut operations). There is a known (top-tree based) solution that requires update time that is roughly linear in the number of colors. We show that this solution is probably optimal by showing that a strictly sublinear update time implies a strictly subcubic time algorithm for the classical all pairs shortest paths problem on a general graph. We also consider versions where the tree is rooted, and the query asks for the nearest ancestor/descendant of node v that has color c, and present efficient data structures for both variants in the static and the dynamic setting.

On finding the Adams consensus tree

This article presents a fast algorithm for finding the Adams consensus tree of a set of conflicting phylogenetic trees with identical leaf labels. Its worst-case running time is O(knlog⁡n), where k is the number of input trees and n is the size of the leaf label set; in comparison, the original algorithm of Adams has a worst-case running time of O(kn2). To achieve subquadratic running time, the centroid path decomposition technique is applied in a novel way that traverses the input trees by following a centroid path in each of them in unison. For k=2, an even faster algorithm running in O(n⋅log⁡nlog⁡log⁡n) time is provided, which relies on an extension of the wavelet tree-based technique of Bose et al. for orthogonal range counting on a grid. Our extended wavelet tree data structure also supports truncated range maximum/minimum queries efficiently.

Lattice-based group signature with verifier-local revocation

Among several post quantum primitives proposed in the past few decades, lattice-based cryptography is considered as the most promising one, due to its underlying rich combinatorial structure, and the worst-case to average-case reductions. The first lattice-based group signature scheme with verifier-local revocation (VLR) is treated as the first quantum-resistant scheme supported member revocation, and was put forward by Langlois et al. This VLR group signature (VLR-GS) has group public key size of O(nm log N log q), and a signature size of O(tm logN log q log β). Nguyen et al. constructed a simple efficient group signature from lattice, with significant advantages in bit-size of both the group public key and the signature. Based on their work, we present a VLR-GS scheme with group public key size of O(nm log q) and signature size of O(tm log q). Our group signature has notable advantages: support of membership revocation, and short in both the public key size and the signature size.