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We show that for any Hilbert space of distributions on $\textbf{R}^d$ which is translation and modulation invariant, is equal to $L^2(\textbf{R}^d)$, with the same norm apart from a multiplicative constant., Comment: 6 pages

We investigate the error between any discrete solution of the implicit marker-and-cell (MAC) numerical scheme for compressible Navier–Stokes equations in the low Mach number regime and an exact strong solution of the incompressible Navier–Stokes equations. The main tool is the relative energy method suggested on the continuous level in Feireisl et al. (2012, Relative entropies, suitable weak solutions, and weak–strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech., 14, 717–730). Our approach highlights the fact that numerical and mathematical analyses are not two separate fields of mathematics. The result is achieved essentially by exploiting in detail the synergy of analytical and numerical methods. We get an unconditional error estimate in terms of explicitly determined positive powers of the space–time discretization parameters and Mach number in the case of well-prepared initial data and in terms of the boundedness of the error if the initial data are ill prepared. The multiplicative constant in the error estimate depends on a suitable norm of the strong solution but it is independent of the numerical solution itself (and of course, on the discretization parameters and the Mach number). This is the first proof that the MAC scheme is unconditionally and uniformly asymptotically stable in the low Mach number regime.

We suggest a new method for coloring generalized Kneser graphs based on hypergraphs with high discrepancy and a small number of edges. The main result provides a proper coloring of K ( n , n ∕ 2 − t , s ) in ( 4 + o ( 1 ) ) ( s + t ) 2 colors, which is produced by Hadamard matrices. Also, we show that for colorings by independent set of a natural type, this result is the best possible up to a multiplicative constant. Our method extends to Kneser hypergraphs as well.

The hitting and mixing times are two often-studied quantities associated with Markov chains. Yuval Peres, Perla Sousi and Roberto Oliveira showed that the mixing times and “worst-case” hitting times of reversible Markov chains on finite state spaces are “equivalent”—that is, equal up to some universal multiplicative constant. We have extended this strong connection between mixing and hitting times to Markov chains satisfying the strong Feller property in an earlier work. In the present paper, we further extend the results to include Metropolis–Hastings chains, the popular Gibbs sampler (from statistics), and Glauber dynamics (from statistical physics), which make “one-dimensional” updates and thus do not satisfy the strong Feller property. We also apply this result to obtain decomposition bounds for such Markov chains. Our main tools come from nonstandard analysis.

We prove that for k ≪ 4√n regular resolution requires length nΩ(k) to establish that an Erdős–Rényi graph with appropriately chosen edge density does not contain a k-clique. This lower bound is optimal up to the multiplicative constant in the exponent and also implies unconditional nΩ(k) lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs. The first, second, and fourth authors were supported by the European Research Council under the European Union’s Horizon 2020 Research and Innovation Programme/ERC grant agreement no. 648276 AUTAR. The third and fifth authors were supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ ERC grant agreement no. 279611 as well as by Swedish Research Council grants 621-2012-5645 and 2016-00782, and the second author did part of this work while at KTH Royal Institute of Technology supported by the same grants. The last author was supported by the Russian Foundation for Basic Research.

Given a linear equation $\mathcal{L}$, a set $A \subseteq [n]$ is $\mathcal{L}$-free if $A$ does not contain any `non-trivial' solutions to $\mathcal{L}$. In this paper we consider the following three general questions: (i) What is the size of the largest $\mathcal{L}$-free subset of $[n]$? (ii) How many $\mathcal{L}$-free subsets of $[n]$ are there? (iii) How many maximal $\mathcal{L}$-free subsets of $[n]$ are there? We completely resolve (i) in the case when $\mathcal{L}$ is the equation $px+qy=z$ for fixed $p,q\in \mathbb N$ where $p\geq 2$. Further, up to a multiplicative constant, we answer (ii) for a wide class of such equations $\mathcal{L}$, thereby refining a special case of a result of Green. We also give various bounds on the number of maximal $\mathcal{L}$-free subsets of $[n]$ for three-variable homogeneous linear equations $\mathcal{L}$. For this, we make use of container and removal lemmas of Green., Comment: 20 pages, final version. To appear in European Journal of Combinatorics

We consider a flip dynamics for directed (1+d)-dimensional lattice paths with length L. The model can be interpreted as a higher dimensional version of the simple exclusion process, the latter corresponding to the case d=1. We prove that the mixing time of the associated Markov chain scales like L^2\log L up to a d-dependent multiplicative constant. The key step in the proof of the upper bound is to show that the system satisfies a logarithmic Sobolev inequality on the diffusive scale L^2 for every fixed d, which we achieve by a suitable induction over the dimension together with an estimate for adjacent transpositions. The lower bound is obtained with a version of Wilson's argument for the one-dimensional case. Keywords: exclusion process, adjacent transpositions, logarithmic Sobolev inequality, mixing time

We establish sharp asymptotic estimates for the diameter of primitive zonotopes when their dimension is fixed. We also prove that, for infinitely many integers $k$, the largest possible diameter of a lattice zonotope contained in the hypercube $[0,k]^d$ is uniquely achieved by a primitive zonotope. As a consequence, we obtain that this largest diameter grows like $k^{d/(d+1)}$ up to an explicit multiplicative constant, when $d$ is fixed and $k$ goes to infinity, providing a new lower bound on the largest possible diameter of a lattice polytope contained in $[0,k]^d$., 12 pages

We consider a model for complex networks that was recently proposed as a model for complex networks by Krioukov et al. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has been previously shown that this model has various properties associated with complex networks, including a power-law degree distribution and a strictly positive clustering coefficient. The model is specified using three parameters : the number of nodes $N$, which we think of as going to infinity, and $\alpha, \nu > 0$ which we think of as constant. Roughly speaking $\alpha$ controls the power law exponent of the degree sequence and $\nu$ the average degree. Earlier work of Kiwi and Mitsche has shown that when $\alpha < 1$ (which corresponds to the exponent of the power law degree sequence being $< 3$) then the diameter of the largest component is a.a.s.~polylogarithmic in $N$. Friedrich and Krohmer have shown it is a.a.s.~$\Omega(\log N)$ and they improved the exponent of the polynomial in $\log N$ in the upper bound. Here we show the maximum diameter over all components is a.a.s.~$O(\log N)$ thus giving a bound that is tight up to a multiplicative constant., Comment: very minor corrections since the last version

Given a linear equation L , a set A ⊆ [ n ] is L -free if A does not contain any ‘non-trivial’ solutions to L . In this paper we consider the following three general questions: (i) What is the size of the largest L -free subset of [n]? (ii) How many L -free subsets of [n] are there? (iii) How many maximal L -free subsets of [n] are there? We completely resolve (i) in the case when L is the equation p x + q y = z for fixed p , q ∈ N where p ≥ 2 . Further, up to a multiplicative constant, we answer (ii) for a wide class of such equations L , thereby refining a special case of a result of Green [15]. We also give various bounds on the number of maximal L -free subsets of [n] for three-variable homogeneous linear equations L . For this, we make use of container and removal lemmas of Green [15].

In analogy to the corresponding measures of pseudorandomness for quaternary sequences introduced by Mauduit and Sarkozy (for m-ary sequences) we introduce the well-distribution measure and correlation measure of order k for sequences over \(\mathbb F_4\). Using any fixed bijection from \(\mathbb F_4\) to the set of complex fourth roots of unity, we analyze the relation of these pseudorandomness measures for sequences over \(\mathbb F_4\) and for the corresponding quaternary sequences. More precisely, we show that they differ only by a multiplicative constant (depending only on k). We also apply the results for deriving new quaternary pseudorandom sequences from pseudorandom sequences over \(\mathbb F_4\) and vice versa.

In this note we confirm a conjecture raised by Benjamini et al. (SIAM J Discrete Math 28(2):767---785, 2014) on the acquaintance time of graphs, proving that for all graphs G with n vertices it holds that $$\mathcal {AC}(G) = O(n^{3/2})$$AC(G)=O(n3/2). This is done by proving that for all graphs G with n vertices and maximum degree $$\varDelta $$Δ it holds that $$\mathcal {AC}(G) \le 20 \varDelta n$$AC(G)≤20Δn. Combining this with the bound $$\mathcal {AC}(G) \le O(n^2/\varDelta )$$AC(G)≤O(n2/Δ) from Benjamini et al. (SIAM J Discrete Math 28(2):767---785, 2014) gives the uniform upper bound of $$O(n^{3/2})$$O(n3/2) for all n-vertex graphs. This bound is tight up to a multiplicative constant. We also prove that for the n-vertex path $$P_n$$Pn it holds that $$\mathcal {AC}(P_n)=n-2$$AC(Pn)=n-2. In addition we show that the barbell graph $$B_n$$Bn consisting of two cliques of sizes $${\lceil n/2\rceil }$$źn/2ź and $${\lfloor n/2\rfloor }$$źn/2ź connected by a single edge also has $$\mathcal {AC}(B_n) = n-2$$AC(Bn)=n-2. This shows that it is possible to add $$\varOmega (n^2$$Ω(n2) edges a graph without changing its $$\mathcal {AC}$$AC value.

Estimates of sums \({R_{nk}}\left( x \right) = \sum\limits_{m = n}^\infty {{P_{mk}}\left( x \right)} \) are established. Here, Pn0(x)= Pn(x), \({R_{nk}}\left( x \right) = \int\limits_.^x {{P_{n,k - 1}}\left( y \right)dy} \), Pn is the Legendre polynomial with standard normalization Pn(1) = 1. With k = 1 in the main interval [–1, 1] the sum decreases with increasing n as n–1, and in the half-open interval [–1, 1), as n–3/2. With k > 1 the point x = 1 does not need to be excluded. The sum decreases as n-k–1/2. Moreover, a small increase in the multiplicative constant permits to obtain the estimate \(|{R_{nk}}\left( {\cos \theta } \right)| < \frac{{C{{\sin }^{k - 3/2}}\theta }}{{{n^{k + 1/2}}}}\), where C depends weakly on k (but not on n, θ). In passing, a Mehler–Dirichlet-type integral for Rnk(cos θ) is deduced.

We compute the multiplicative constant in the large gap asymptotics of the Meijer-G point process. This point process generalizes the Bessel point process and appears at the hard edge of Cauchy--Laguerre multi-matrix models and of certain product random matrix ensembles., Comment: 33 pages, 6 figures

We consider the Goldstein-Taylor model, which is a 2-velocity BGK model, and construct the “optimal” Lyapunov functional to quantify the convergence to the unique normalized steady state. The Lyapunov functional is optimal in the sense that it yields decay estimates in \(L^2\)-norm with the sharp exponential decay rate and minimal multiplicative constant. The modal decomposition of the Goldstein-Taylor model leads to the study of a family of 2-dimensional ODE systems. Therefore we discuss the characterization of “optimal” Lyapunov functionals for linear ODE systems with positive stable diagonalizable matrices. We give a complete answer for optimal decay rates of 2-dimensional ODE systems, and a partial answer for higher dimensional ODE systems.

La corrección de los efectos atmosféricos en imágenes de satélite es una tarea crucial para el entendimiento de los patrones espacio-temporales de las reflectancias e índices de vegetación en el análisis de la dinámica de la vegetación. En el presente trabajo se analizó el problema de inversión de modelos de transferencia de radiación de la atmósfera mediante el uso de los patrones invariantes de las líneas del suelo y de la vegetación densa, bajo un esquema de restricciones múltiples o def inición de relaciones a cumplir. Las inversiones de radiación se invirtieron al minimizar una función objetivo o de mérito asociada a las restricciones disponibles y a las relaciones funcionales de compactación (análisis multivariado) de simulaciones de radiación. Las líneas del suelo resultaron mejores con las dos restricciones disponibles, que para el uso de una sola restricción. En el caso general de usar todas las restricciones disponibles para las líneas del suelo y la vegetación densa, se lograron precisiones de clasif icación correcta del binomio atmósfera-aerosol mayores al 80%, sin conocimiento previo de esta información. En conclusión, cuando se usan las restricciones de las líneas del suelo y de la vegetación densa, las estimaciones mejoran para las constantes multiplicativas y las soluciones óptimas de los esquemas de inversión se ubican dentro de los mejores cinco valores que minimizan las funciones objetivo propuestas.

The dimension of a poset P, denoted $$\dim (P)$$dim(P), is the least positive integer d for which P is the intersection of d linear extensions of P. The maximum dimension of a poset P with $$|P|\le 2n+1$$|P|≤2n+1 is n, provided $$n\ge 2$$n?2, and this inequality is tight when P contains the standard example $$S_n$$Sn. However, there are posets with large dimension that do not contain the standard example $$S_2$$S2. Moreover, for each fixed $$d\ge 2$$d?2, if P is a poset with $$|P|\le 2n+1$$|P|≤2n+1 and P does not contain the standard example $$S_d$$Sd, then $$\dim (P)=o(n)$$dim(P)=o(n). Also, for large n, there is a poset P with $$|P|=2n$$|P|=2n and $$\dim (P)\ge (1-o(1))n$$dim(P)?(1-o(1))n such that the largest d so that P contains the standard example $$S_d$$Sd is o(n). In this paper, we will show that for every integer $$c\ge 1$$c?1, there is an integer $$f(c)=O(c^2)$$f(c)=O(c2) so that for large enough n, if P is a poset with $$|P|\le 2n+1$$|P|≤2n+1 and $$\dim (P)\ge n-c$$dim(P)?n-c, then P contains a standard example $$S_d$$Sd with $$d\ge n-f(c)$$d?n-f(c). From below, we show that $$f(c)={\varOmega }(c^{4/3})$$f(c)=Ω(c4/3). On the other hand, we also prove an analogous result for fractional dimension, and in this setting f(c) is linear in c. Here the result is best possible up to the value of the multiplicative constant.

Our interest here is to find the invader in a two species, diffusive and competitive Lotka–Volterra system in the particular case of travelling wave solutions. We investigate the role of diffusion in homogeneous domains. We might expect a priori two different cases: strong interspecific competition and weak interspecific competition. In this paper, we study the first one and obtain a clear conclusion: the invading species is, up to a fixed multiplicative constant, the more diffusive one.

The $n$th order fractional Brownian motion was introduced by Perrin et al. It is the (upto a multiplicative constant) unique self-similar Gaussian process with Hurst index $H \in (n-1,n)$, having $n$th order stationary increments. We provide a transfer principle for the $n$th order fractional Brownian motion, i.e., we construct a Brownian motion from the $n$the order fractional Brownian motion and then represent the $n$the order fractional Brownian motion by using the Brownian motion in a non-anticipative way so that the filtrations of the $n$the order fractional Brownian motion and the associated Brownian motion coincide. By using this transfer principle, we provide the prediction formula for the $n$the order fractional Brownian motion and also a representation formula for all the Gaussian processes that are equivalent in law to the $n$th order fractional Brownian motion., 20 pages, 20 figures

Here a new notion of fractional length of a smooth curve, which depends on a parameter $\sigma$, is introduced that is analogous to the fractional perimeter functional of sets that has been studied in recent years. It is shown that in an appropriate limit the fractional length converges to the traditional notion of length up to a multiplicative constant. Since a curve that connects two points of minimal length must have zero curvature, the Euler--Lagrange equation associated with the fractional length is used to motivate a nonlocal notion of curvature for a curve. This is analogous to how the fractional perimeter has been used to define a nonlocal mean curvature., Comment: 24 pages, 3 figures

We define two versions of the extremal length of elements of the fundamental group of the twice punctured complex plane and give upper and lower bounds for two versions of this invariant. The bounds differ by a multiplicative constant. The main motivation comes from 3-braid invariants and their application.

A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in an n-vertex graph not containing C2k, the cycle of length 2k, is O(n1+1/k). Simonovits established a corresponding supersaturation result for C2k’s, showing that there exist positive constants C,c depending only on k such that every n-vertex graph G with e(G)⩾ Cn1+1/k contains at least c(e(G)/v(G))2k copies of C2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant).In this paper we extend Simonovits' result to a supersaturation result of r-uniform linear cycles of even length in r-uniform linear hypergraphs. Our proof is self-contained and includes the r = 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.

Let p be a prime and let A = (a 1,...,a l ) be a sequence of nonzero elements in $\mathbb{F}_p $ . In this paper, we study the set of all 0---1 solutions to the equation $$a_1 x_1 + \cdots + a_\ell x_\ell = 0$$ We prove that whenever l?p, this set actually characterizes A up to a nonzero multiplicative constant, which is no longer true for l

We study the quasiadditivity property (a version of superadditivity with a multiplicative constant) of variational capacity in metric spaces with respect to Whitney type covers. We characterize this property in terms of a Mazya type capacity condition, and also explore the close relation between quasiadditivity and Hardy's inequality., Comment: 15 pages

In this note we asymptotically determine the maximum number of hyperedges possible in an r -uniform, connected n -vertex hypergraph without a Berge path of length k , as n and k tend to infinity. We show that, unlike in the graph case, the multiplicative constant is smaller with the assumption of connectivity.

We consider the problem of detecting an anomaly in the form of a path of correlations hidden in white noise. We provide a minimax lower bound and a test that, under mild assumptions, is able to achieve the lower bound up to a multiplicative constant., arXiv admin note: text overlap with arXiv:1504.06984

A fooling-set matrix is a square matrix with nonzero diagonal, but at least one in every pair of diagonally opposite entries is 0. Dietzfelbinger et al. ’96 proved that the rank of such a matrix is at least \(\sqrt{n}\), for a matrix of order n. It is known that the bound is tight (up to a multiplicative constant).

The degenerate crossing number\({\text{ cr}^{*}}(G)\) of a graph \(G\) is the minimum number of crossing points of edges in any drawing of \(G\) as a simple topological graph in the plane. This notion was introduced by Pach and Toth who showed that for a graph \(G\) with \(n\) vertices and \(e \ge 4n\) edges \({\text{ cr}^{*}}(G)=\Omega \big (e^4 / n^4\big )\). In this paper we completely resolve the main open question about degenerate crossing numbers and show that \({\text{ cr}^{*}}(G)=\Omega \big (e^3 / n^2 \big )\), provided that \(e \ge 4n\). This bound is best possible (apart for the multiplicative constant) as it matches the tight lower bound for the standard crossing number of a graph.

We consider weak thresholding greedy algorithms with respect to Markushevich bases in general Banach spaces. We find sufficient conditions for the equivalence of boundedness and convergence of the approximants. We also show that if there is a weak thresholding algorithm for the system which gives the best n-term approximation up to a multiplicative constant, then the system is already “greedy”. Similar results are proved for “almost greedy” and “semi-greedy” systems.

We evaluate the determinants of Hankel matrices, whose elements are a linear combination of three successive shifted Catalan numbers. This is done by finding a Jacobi linear functional, such that their moments are, up to a multiplicative constant, the Catalan numbers. The values of such determinants are then expressed in terms of Jacobi polynomials.

For random walk $S_n$ with independent identically distributed steps $X_i$ with $MX\ge 0$, satisfying the right-hand Cramer condition, asymptotical behavior of the Shepp statistics in the case of it large deviations is studied: $\bar\rho_{m,n}:=\max_{k\le m}\max_{i\le n} (S_{k+i}-S_k)\ge\theta n$, $n,m\rightarrow\infty$. For $P(\bar\rho_{n,m}\ge \theta n)$ an explicit asymptotics is obtained, and the multiplicative constant is represented in the form of probability. For $\bar\rho_{m,n}$ and $\tau_n(\theta):=\min\{m:\max_{k\le n}(S_{k+m}-S_{m})\ge \theta n\}$ limit theorems are obtained, and for segments of random walk trajectory functional limit theorems are given.

We discuss some relations between autocorrelations (ACFs) and partial autocorrelations (PACFs) of weakly stationary processes. First, we construct an extension of a process ARIMA(0,d,0) for d ∈ (−∞, 0), which enjoys non-summable partial autocorrelations and autocorrelations decaying as rapidly as ρn ≅ n−1+2d. Such a situation is impossible if the absolute sum of autocorrelations is sufficiently small. We show that then the PACF is less than the ACF up to a multiplicative constant. Our second result complements a similar result of Baxter (1962).

It is an intriguing question to see what kind of information on the structure of an oriented graph $D$ one can obtain if $D$ does not contain a fixed oriented graph $H$ as a subgraph. The related question in the unoriented case has been an active area of research, and is relatively well-understood in the theory of quasi-random graphs and extremal combinatorics. In this paper, we consider the simplest cases of such a general question for oriented graphs, and provide some results on the global behavior of the orientation of $D$. For the case that $H$ is an oriented four-cycle we prove: in every $H$-free oriented graph $D$, there is a pair $A,B\ssq V(D)$ such that $e(A,B)\ge e(D)^{2}/32|D|^{2}$ and $e(B,A)\le e(A,B)/2$. We give a random construction which shows that this bound on $e(A,B)$ is best possible (up to the constant). In addition, we prove a similar result for the case $H$ is an oriented six-cycle, and a more precise result in the case $D$ is dense and $H$ is arbitrary. We also consider the related extremal question in which no condition is put on the oriented graph $D$, and provide an answer that is best possible up to a multiplicative constant. Finally, we raise a number of related questions and conjectures., Comment: 35 pages

We study the problem how to obtain a small drawing of a 3-polytope with Euclidean distance between any two points at least 1. The problem can be reduced to a one-dimensional problem, since it is sufficient to guarantee distinct integer x-coordinates. We develop an algorithm that yields an embedding with the desired property such that the polytope is contained in a 2(n−2)×1 ×1 box. The constructed embedding can be scaled to a grid embedding whose x-coordinates are contained in [0,2(n−2)]. Furthermore, the point set of the embedding has a small spread, which differs from the best possible spread only by a multiplicative constant.