Let $G_n$ be a random geometric graph with vertex set $[n]$ based on $n$ i.i.d.\ random vectors $X_1,\ldots,X_n$ drawn from an unknown density $f$ on $\R^d$. An edge $(i,j)$ is present when $\|X_i -X_j\| \le r_n$, for a given threshold $r_n$ possibly depending upon $n$, where $\| \cdot \|$ denotes Euclidean distance. We study the problem of estimating the dimension $d$ of the underlying space when we have access to the adjacency matrix of the graph but do not know $r_n$ or the vectors $X_i$. The main result of the paper is that there exists an estimator of $d$ that converges to $d$ in probability as $n \to \infty$ for all densities with $\int f^5 < \infty$ whenever $n^{3/2} r_n^d \to \infty$ and $r_n = o(1)$. The conditions allow very sparse graphs since when $n^{3/2} r_n^d \to 0$, the graph contains isolated edges only, with high probability. We also show that, without any condition on the density, a consistent estimator of $d$ exists when $n r_n^d \to \infty$ and $r_n = o(1)$.
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Let $G$ be a graph of order $n$. A classical upper bound for the domination number of a graph $G$ having no isolated vertices is $\lfloor\frac{n}{2}\rfloor$. However, for several families of graphs, we have $\gamma(G) \le \lfloor\sqrt{n}\rfloor$ which gives a substantially improved upper bound. In this paper, we give a condition necessary for a graph $G$ to have $\gamma(G) \le \lfloor\sqrt{n}\rfloor$, and some conditions sufficient for a graph $G$ to have $\gamma(G) \le \lfloor\sqrt{n}\rfloor$. We also present a characterization of all connected graphs $G$ of order $n$ with $\gamma(G) = \lfloor\sqrt{n}\rfloor$. Further, we prove that for a graph $G$ not satisfying $rad(G)=diam(G)=rad(\overline{G})=diam(\overline{G})=2$, deciding whether $\gamma(G) \le \lfloor\sqrt{n}\rfloor$ or $\gamma(\overline{G}) \le \lfloor\sqrt{n}\rfloor$ can be done in polynomial time. We conjecture that this decision problem can be solved in polynomial time for any graph $G$.
For a commutative ring $R,$ with non-zero zero divisors $Z^{\ast}(R)$. The zero divisor graph $\Gamma(R)$ is a simple graph with vertex set $Z^{\ast}(R)$, and two distinct vertices $x,y\in V(\Gamma(R))$ are adjacent if and only if $x\cdot y=0.$ In this note, we provide counter examples to the eigenvalues, the energy and the second Zagreb index related to zero divisor graphs of rings obtained in [Johnson and Sankar, J. Appl. Math. Comp. (2023), \cite{johnson}]. We correct the eigenvalues (energy) and the Zagreb index result for the zero divisor graphs of ring $\mathbb{Z}_{p}[x]/\langle x^{4} \rangle.$ We show that for any prime $p$, $\Gamma(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ is non-hyperenergetic and for prime $p\geq 3$, $\Gamma(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ is hypoenergetic. We give a formulae for the topological indices of $\Gamma(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ and show that its Zagreb indices satisfy Hansen and Vuki$\check{c}$cevi\'c conjecture \cite{hansen}.
We quantify the minimax rate for a nonparametric regression model over a convex function class $\mathcal{F}$ with bounded diameter. We obtain a minimax rate of ${\varepsilon^{\ast}}^2\wedge\mathrm{diam}(\mathcal{F})^2$ where \[\varepsilon^{\ast} =\sup\{\varepsilon>0:n\varepsilon^2 \le \log M_{\mathcal{F}}^{\operatorname{loc}}(\varepsilon,c)\},\] where $M_{\mathcal{F}}^{\operatorname{loc}}(\cdot, c)$ is the local metric entropy of $\mathcal{F}$ and our loss function is the squared population $L_2$ distance over our input space $\mathcal{X}$. In contrast to classical works on the topic [cf. Yang and Barron, 1999], our results do not require functions in $\mathcal{F}$ to be uniformly bounded in sup-norm. In addition, we prove that our estimator is adaptive to the true point, and to the best of our knowledge this is the first such estimator in this general setting. This work builds on the Gaussian sequence framework of Neykov [2022] using a similar algorithmic scheme to achieve the minimax rate. Our algorithmic rate also applies with sub-Gaussian noise. We illustrate the utility of this theory with examples including multivariate monotone functions, linear functionals over ellipsoids, and Lipschitz classes.
We present a new and straightforward derivation of a family $\mathcal{F}(h,\tau)$ of exponential splittings of Strang-type for the general linear evolutionary equation with two linear components. One component is assumed to be a time-independent, unbounded operator, while the other is a bounded one with explicit time dependence. The family $\mathcal{F}(h,\tau)$ is characterized by the length of the time-step $h$ and a continuous parameter $\tau$, which defines each member of the family. It is shown that the derivation and error analysis follows from two elementary arguments: the variation of constants formula and specific quadratures for integrals over simplices. For these Strang-type splittings, we prove their convergence which, depending on some commutators of the relevant operators, may be of first or second order. As a result, error bounds appear in terms of commutator bounds. Based on the explicit form of the error terms, we establish the influence of $\tau$ on the accuracy of $\mathcal{F}(h,\tau)$, allowing us to investigate the optimal value of $\tau$. This simple yet powerful approach establishes the connection between exponential integrators and splitting methods. Furthermore, the present approach can be easily applied to the derivation of higher-order splitting methods under similar considerations. Needless to say, the obtained results also apply to Strang-type splittings in the case of time independent-operators. To complement rigorous results, we present numerical experiments with various values of $\tau$ based on the linear Schr\"odinger equation.
Kleene's computability theory based on the S1-S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing's 'machine model' which formalises computing with real numbers. A fundamental distinction in Kleene's framework is between normal and non-normal functionals where the former compute the associated Kleene quantifier $\exists^n$ and the latter do not. Historically, the focus was on normal functionals, but recently new non-normal functionals have been studied based on well-known theorems, the weakest among which seems to be the uncountability of the reals. These new non-normal functionals are fundamentally different from historical examples like Tait's fan functional: the latter is computable from $\exists^2$, while the former are computable in $\exists^3$ but not in weaker oracles. Of course, there is a great divide or abyss separating $\exists^2$ and $\exists^3$ and we identify slight variations of our new non-normal functionals that are again computable in $\exists^2$, i.e. fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, bounded variation, and semi-continuity from real analysis.
Given a graph $G$, the optimization version of the graph burning problem seeks for a sequence of vertices, $(u_1,u_2,...,u_k) \in V(G)^k$, with minimum $k$ and such that every $v \in V(G)$ has distance at most $k-i$ to some vertex $u_i$. The length $k$ of the optimal solution is known as the burning number and is denoted by $b(G)$, an invariant that helps quantify the graph's vulnerability to contagion. This paper explores the advantages and limitations of an $\mathcal{O}(mn + kn^2)$ deterministic greedy heuristic for this problem, where $n$ is the graph's order, $m$ is the graph's size, and $k$ is a guess on $b(G)$. This heuristic is based on the relationship between the graph burning problem and the clustered maximum coverage problem, and despite having limitations on paths and cycles, it found most of the optimal and best-known solutions of benchmark and synthetic graphs with up to 102400 vertices.
Subshifts are colorings of $\mathbb{Z}^d$ defined by families of forbidden patterns. Given a subshift and a finite pattern, its extender set is the set of admissible completions of this pattern. It has been conjectured that the behavior of extender sets, and in particular their growth called extender entropy (arXiv:1711.07515), could provide a way to separate the classes of sofic and effective subshifts. We prove here that both classes have the same possible extender entropies: exactly the $\Pi_3$ real numbers of $[0,+\infty)$. We also consider computational properties of extender entropies for subshifts with some language or dynamical properties: computable language, minimal and some mixing properties.
In this paper, we develop a fast and accurate pseudospectral method to approximate numerically the half Laplacian $(-\Delta)^{1/2}$ of a function on $\mathbb{R}$, which is equivalent to the Hilbert transform of the derivative of the function. The main ideas are as follows. Given a twice continuously differentiable bounded function $u\in\mathcal C_b^2(\mathbb{R})$, we apply the change of variable $x=L\cot(s)$, with $L>0$ and $s\in[0,\pi]$, which maps $\mathbb{R}$ into $[0,\pi]$, and denote $(-\Delta)_s^{1/2}u(x(s)) \equiv (-\Delta)^{1/2}u(x)$. Therefore, by performing a Fourier series expansion of $u(x(s))$, the problem is reduced to computing $(-\Delta)_s^{1/2}e^{iks} \equiv (-\Delta)^{1/2}[(x + i)^k/(1+x^2)^{k/2}]$. On a previous work, we considered the case with $k$ even for the more general power $\alpha/2$, with $\alpha\in(0,2)$, so here we focus on the case with $k$ odd. More precisely, we express $(-\Delta)_s^{1/2}e^{iks}$ for $k$ odd in terms of the Gaussian hypergeometric function ${}_2F_1$, and also as a well-conditioned finite sum. Then, we use a fast convolution result, that enable us to compute very efficiently $\sum_{l = 0}^Ma_l(-\Delta)_s^{1/2}e^{i(2l+1)s}$, for extremely large values of $M$. This enables us to approximate $(-\Delta)_s^{1/2}u(x(s))$ in a fast and accurate way, especially when $u(x(s))$ is not periodic of period $\pi$. As an application, we simulate a fractional Fisher's equation having front solutions whose speed grows exponentially.
We describe a new dependent-rounding algorithmic framework for bipartite graphs. Given a fractional assignment $y$ of values to edges of graph $G = (U \cup V, E)$, the algorithms return an integral solution $Y$ such that each right-node $v \in V$ has at most one neighboring edge $f$ with $Y_f = 1$, and where the variables $Y_e$ also satisfy broad nonpositive-correlation properties. In particular, for any edges $e_1, e_2$ sharing a left-node $u \in U$, the variables $Y_{e_1}, Y_{e_2}$ have strong negative-correlation properties, i.e. the expectation of $Y_{e_1} Y_{e_2}$ is significantly below $y_{e_1} y_{e_2}$. This algorithm is based on generating negatively-correlated Exponential random variables and using them in a contention-resolution scheme inspired by an algorithm Im & Shadloo (2020). Our algorithm gives stronger and much more flexible negative correlation properties. Dependent rounding schemes with negative correlation properties have been used for approximation algorithms for job-scheduling on unrelated machines to minimize weighted completion times (Bansal, Srinivasan, & Svensson (2021), Im & Shadloo (2020), Im & Li (2023)). Using our new dependent-rounding algorithm, among other improvements, we obtain a $1.398$-approximation for this problem. This significantly improves over the prior $1.45$-approximation ratio of Im & Li (2023).
The advent of large-scale inference has spurred reexamination of conventional statistical thinking. In a Gaussian model for $n$ many $z$-scores with at most $k < \frac{n}{2}$ nonnulls, Efron suggests estimating the location and scale parameters of the null distribution. Placing no assumptions on the nonnull effects, the statistical task can be viewed as a robust estimation problem. However, the best known robust estimators fail to be consistent in the regime $k \asymp n$ which is especially relevant in large-scale inference. The failure of estimators which are minimax rate-optimal with respect to other formulations of robustness (e.g. Huber's contamination model) might suggest the impossibility of consistent estimation in this regime and, consequently, a major weakness of Efron's suggestion. A sound evaluation of Efron's model thus requires a complete understanding of consistency. We sharply characterize the regime of $k$ for which consistent estimation is possible and further establish the minimax estimation rates. It is shown consistent estimation of the location parameter is possible if and only if $\frac{n}{2} - k = \omega(\sqrt{n})$, and consistent estimation of the scale parameter is possible in the entire regime $k < \frac{n}{2}$. Faster rates than those in Huber's contamination model are achievable by exploiting the Gaussian character of the data. The minimax upper bound is obtained by considering estimators based on the empirical characteristic function. The minimax lower bound involves constructing two marginal distributions whose characteristic functions match on a wide interval containing zero. The construction notably differs from those in the literature by sharply capturing a scaling of $n-2k$ in the minimax estimation rate of the location.
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