A string attractor of a string $T[1..|T|]$ is a set of positions $Γ$ of $T$ such that any substring $w$ of $T$ has an occurrence that crosses a position in $Γ$, i.e., there is a position $i$ such that $w = T[i..i+|w|-1]$ and the intersection $[i,i+|w|-1]\cap Γ$ is nonempty. The size of the smallest string attractor of Fibonacci words is known to be $2$. We completely characterize the set of all smallest string attractors of Fibonacci words, and show a recursive formula describing the $2^{n-4} + 2^{\lceil n/2 \rceil - 2}$ distinct position pairs that are the smallest string attractors of the $n$th Fibonacci word for $n \geq 7$. Similarly, the size of the smallest string attractor of period-doubling words is known to be $2$. We also completely characterize the set of all smallest string attractors of period-doubling words, and show a formula describing the two distinct position pairs that are the smallest string attractors of the $n$th period-doubling word for $n\geq 2$. Our results show that strings with the same smallest attractor size can have a drastically different number of distinct smallest attractors.
In this paper, we will give a structure theory for signed graphs with fixed smallest eigenvalue and investigate signed graphs with smallest eigenvalue greater than $-1-\sqrt{2}$. Given a real number $λ\leq -1$, we show that the following hold for each signed graph $(G,σ)$ with smallest eigenvalue at least $λ$ and large minimum valency: $\mathrm{(i)}$ there exist dense induced subgraphs $N_1, \dots, N_r$ in $(G,σ)$ such that each vertex lies in at most $\lfloor -λ\rfloor$ $N_i$'s and almost all edges of $(G,σ)$ lie in at least one of the $N_i$'s; $\mathrm{(ii)}$ if $λ>-1-\sqrt{2}$, then $(G,σ)$ has smallest eigenvalue at least $-2$ and $(G,σ)$ is $1$-integrable.
In this article, we study the smallest distances between the zeros of Gaussian analytic functions over compact Riemann surfaces. Our main result is that, after appropriate rescaling, the point process of the smallest distances converge to a Poisson point process with a universal rate. Furthermore, the locations where these smallest distances occur tend to follow a uniform measure with respect to the volume form. As a consequence, the limiting density of the $k$-th rescaled smallest distance is proportional to $x^{4k-1}e^{-x^4}$ for any $k\geq 1$. Analogous results hold for the classical Gaussian Entire Functions.
Inspired by Andrews' and Bachraoui's work on partitions with repeated smallest part, we extend the concept to overpartitions. We study overpartitions with the restriction that the smallest non-overlined part appears exactly $k$ times and every overlined part is bigger than this part. We prove results expressing the generating functions of these overpartitions (and their subclass where no part has the same parity as the smallest part, among others) as linear combinations of the $q$-Pochhammer symbols with rational functions in $q$ as coefficients.
In a recent paper, Jin, Liu, and Xia \cite{JLX} presented some modulo 4 congruences for $\overline{spt2}(n)$, the number of smallest parts in the overpartitions of $n$ where the smallest part is even and is not overlined. In this paper, we extend the list of such congruences in two directions. First, we prove some new individual congruences for $\overline{spt2}(n)$. Then, we provide a number of infinite families of Ramanujan-like congruences satisfied by $\overline{spt2}(n)$.
A suffixient set is a novel combinatorial object that captures the essential information of repetitive strings in a way that, provided with a random access mechanism, supports various forms of pattern matching. In this paper, we study the size $χ$ of the smallest suffixient set as a repetitiveness measure. First, we study its sensitivity to various string operations. We show that $χ$ cannot increase by more than 2 after appending or prepending a character to the string. As a consequence, we are able to give simple linear-time online algorithms to compute smallest suffixient sets. We also show that, although reversing the string can increase $χ$ by an arbitrary $O(n)$ value, it always holds $χ(T)/χ(T^R)\le 2$. We also prove lower and upper bounds for the additive or multiplicative increase of $χ$ after applying arbitrary edit operations, or rotating the text. In particular, we show that the additive increase can be as large as $Ω(\sqrt{n})$ for all those operations. Secondly, we place $χ$ in between known repetitiveness measures. In particular, we show $χ= O(r)$ (where $r$ is the number of runs in the Burrows-Wheeler Transform of the string), that there are string families where $χ=
Based on Welzl's algorithm for smallest circles and spheres we develop a simple linear time algorithm for finding the smallest circle enclosing a point cloud on a sphere. The algorithm yields correct results as long as the point cloud is contained in a hemisphere, but the hemisphere does not have to be known in advance and the algorithm automatically detects whether the hemisphere assumption is met. For the full-sphere case, that is, if the point cloud is not contained in a hemisphere, we provide hints on how to adapt existing linearithmic time algorithms for spherical Voronoi diagrams to find the smallest enclosing circle.
In this paper we will give a structure theory for regular graphs with fixed smallest eigenvalue. As a consequence of this theory, we show that a $k$-regular graph with smallest eigenvalue $-λ$ has clique number linear in $k$ if $k$ is large with respect to $λ$.
In this article, we introduce the notion of almost consecutive partitions. A partition is almost consecutive if every term is consecutive, with the possible exception of the smallest one. We find formulas relating to the smallest parts of consecutive and almost consecutive partitions. We also find an alternate combinatorial interpretation of the number of almost consecutive partitions of a given integer $n$ and an asymptotic formula for this quantity.
Using group theoretic methods only, we prove the uniqueness of the smallest embedding cover of a profinite group, Problem 36.2.25 of Field Arithmetic, 4th edition.
For a fixed positive integer $k$, let $C(k,n)$ denote the number of two-color partitions of $n$ with odd smallest part and restrictions on even parts, and let $C_k(q)$ be its generating function. We show that $C(1,n)\equiv d(2n-1)\pmod{4}$ and obtain congruences modulo $2$ and $4$ for $C(k,n)$ when $k=2,3$. Using $q$-series methods we derive closed formulas for $C_k(q)$ in terms of eta-quotients and formulate Ramanujan-type congruences for the limiting sequence arising from $\lim_{k\to\infty} C_k(q)$.
In this paper, we give infinitely many examples of (non-isomorphic) connected $k$-regular graphs with smallest eigenvalue in half open interval $[-1-\sqrt2, -2)$ and also infinitely many examples of (non-isomorphic) connected $k$-regular graphs with smallest eigenvalue in half open interval $[α_1, -1-\sqrt2)$ where $α_1$ is the smallest root$(\approx -2.4812)$ of the polynomial $x^3+2x^2-2x-2$. From these results, we determine the largest and second largest limit points of smallest eigenvalues of regular graphs less than -2. Moreover we determine the supremum of the smallest eigenvalue among all connected 3-regular graphs with smallest eigenvalue less than -2 and we give the unique graph with this supremum value as its smallest eigenvalue.
A coloring of a digraph is a partition of its vertex set such that each class induces a digraph with no directed cycles. A digraph is $k$-chromatic if $k$ is the minimum number of classes in such partition, and a digraph is oriented if there is at most one arc between each pair of vertices. Clearly, the smallest $k$-chromatic digraph is the complete digraph on $k$ vertices, but determining the order of the smallest $k$-chromatic oriented graphs is a challenging problem. It is known that the smallest $2$-, $3$- and $4$-chromatic oriented graphs have $3$, $7$ and $11$ vertices, respectively. In 1994, Neumann-Lara conjectured that a smallest $5$-chromatic oriented graph has $17$ vertices. We solve this conjecture and show that the correct order is $19$.
We establish higher dimensional versions of a recent theorem by Chen and Haynes [Int. J. Number Theory 19 (2023), 1405-1413] on the expected value of the smallest denominator of rational points in a randomly shifted interval of small length, and of the closely related 1977 Kruyswijk-Meijer conjecture recently proved by Balazard and Martin [Bull. Sci. Math. 187 (2023), Paper No. 103305]. We express the distribution of smallest denominators in terms of the void statistics of multidimensional Farey fractions and prove convergence of the distribution function and certain finite moments. The latter was previously unknown even in the one-dimensional setting. We furthermore obtain a higher dimensional extension of Kargaev and Zhigljavsky's work on moments of the distance function for the Farey sequence [J. Number Theory 65 (1997), 130-149] as well as new results on pigeonhole statistics.
We provide upper and lower bounds on the smallest eigenvalue of grounded Laplacian matrices (which are matrices obtained by removing certain rows and columns of the Laplacian matrix of a given graph). The gap between the upper and lower bounds depends on the ratio of the smallest and largest components of the eigenvector corresponding to the smallest eigenvalue of the grounded Laplacian. We provide a graph-theoretic bound on this ratio, and subsequently obtain a tight characterization of the smallest eigenvalue for certain classes of graphs. Specifically, for Erdos-Renyi random graphs, we show that when a (sufficiently small) set $S$ of rows and columns is removed from the Laplacian, and the probability $p$ of adding an edge is sufficiently large, the smallest eigenvalue of the grounded Laplacian asymptotically almost surely approaches $|S|p$. We also show that for random $d$-regular graphs with a single row and column removed, the smallest eigenvalue is $Θ(\frac{d}{n})$. Our bounds have applications to the study of the convergence rate in continuous-time and discrete-time consensus dynamics with stubborn or leader nodes.
We study the problem of finding the smallest graph that does not occur as an induced subgraph of a given graph. This missing induced subgraph has at most logarithmic size and can be found by a brute-force search, in an $n$-vertex graph, in time $n^{O(\log n)}$. We show that under the Exponential Time Hypothesis this quasipolynomial time bound is optimal. We also consider variations of the problem in which either the missing subgraph or the given graph comes from a restricted graph family; for instance, we prove that the smallest missing planar induced subgraph of a given planar graph can be found in polynomial time.
Grammar-based compression is a loss-less data compression scheme that represents a given string $w$ by a context-free grammar that generates only $w$. While computing the smallest grammar which generates a given string $w$ is NP-hard in general, a number of polynomial-time grammar-based compressors which work well in practice have been proposed. RePair, proposed by Larsson and Moffat in 1999, is a grammar-based compressor which recursively replaces all possible occurrences of a most frequently occurring bigrams in the string. Since there can be multiple choices of the most frequent bigrams to replace, different implementations of RePair can result in different grammars. In this paper, we show that the smallest grammars generating the Fibonacci words $F_k$ can be completely characterized by RePair, where $F_k$ denotes the $k$-th Fibonacci word. Namely, all grammars for $F_k$ generated by any implementation of RePair are the smallest grammars for $F_k$, and no other grammars can be the smallest for $F_k$. To the best of our knowledge, Fibonacci words are the first non-trivial infinite family of strings for which RePair is optimal.
In 2017, Andrews, Dixit, Schultz and Yee introduced the function $\overline{\textrm{spt}}_ω(n)$, which denotes the number of smallest parts in the overpartitions of $n$ in which the smallest part is always overlined and all odd parts are less than twice the smallest part. Recently, Baruah and Begum established several internal congruences and congruences modulo small powers of $5$ for $\overline{\textrm{spt}}_ω(n)$. Moreover, they conjectured a family of internal congruences modulo any powers of $5$ and two families of congruences modulo any even powers of $5$. In this paper, we confirm three families of congruences due to Baruah and Begum.
In this paper we study entanglement of the reduced density matrix of a bipartite quantum system in a random pure state. It transpires that this involves the computation of the smallest eigenvalue distribution of the fixed trace Laguerre ensemble of $N\times N$ random matrices. We showed that for finite $N$ the smallest eigenvalue distribution may be expressed in terms of Jack polynomials. Furthermore, based on the exact results, we found, a limiting distribution, when the smallest eigenvalue is suitably scaled with $N$ followed by a large $N$ limit. Our results turn out to be the same as the smallest eigenvalue distribution of the classical Laguerre ensembles without the fixed trace constraint. This suggests in a broad sense, the global constraint does not influence local correlations, at least, in the large $N$ limit. Consequently, we have solved an open problem: The determination of the smallest eigenvalue distribution of the reduced density matrix---obtained by tracing out the environmental degrees of freedom---for a bipartite quantum system of unequal dimensions.
We introduce a new stress dilatometer with exactly the same size and weight as the world's smallest miniature capacitance dilatometer (height x width x depth = 15 mmx14 mmx15 mm, weight: 12 g). To develop this new device, only a single part of the most recently developed mini-dilatometer, the so-called 'body', needs to be replaced. Therefore, the new mini dilatometer with an interchangeable body can be used for high-resolution measurements of thermal expansion and magnetostriction with and without large stress. We also report two novel applications of both mini-dilatometer cell types. Our new setup was installed for the first time in a cryogen-free system (PPMS DynaCool). The first new setup allows the rotation of both dilatometers in situ at any angle between -90 deg > μ > +90 deg in the temperature range from 320 K to 1.8 K. We also installed our mini-cells in a dilution refrigerator insert of a PPMS DynaCool, in which dilatometric measurements are now possible in the temperature range from 4 K to 0.06 K. Because of the limited sample space, such measurements could not be performed so far. For both new applications, we can resolve the impressive length changes to 0.01 A.