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We introduce a graph Ramsey game called Ramsey, Paper, Scissors. This game has two players, Proposer and Decider. Starting from an empty graph on $n$ vertices, on each turn Proposer proposes a potential edge and Decider simultaneously decides (without knowing Proposer's choice) whether to add it to the graph. Proposer cannot propose an edge which would create a triangle in the graph. The game ends when Proposer has no legal moves remaining, and Proposer wins if the final graph has independence number at least $s$. We prove a threshold phenomenon exists for this game by exhibiting randomized strategies for both players that are optimal up to constants. Namely, there exist constants $0B\sqrt{n}\log{n}$. This is a factor of $\Theta(\sqrt{\log{n}})$ larger than the lower bound coming from the off-diagonal Ramsey number $r(3,s)$.

The aim of this paper is to discuss the main result in the paper by D.Y. Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective function has not local extrema with the mentioned constraints for $p\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer. We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995]., 12 pages; in this version we added the forgotten condition $F(x) \ne 0$ for $x\in (a,b)$ on page 3

This short paper is motivated by the paper of Bu et al. [C. Bu, C. Feng, P. Dong, A note on computational formulas for the Drazin inverse of certain block matrices, J. Appl. Math. Comput.(38) (2012) 631-640], where the authors gave additive formula for Drazin inverse for matrices under new conditions, and two representations under some specific conditions. Here is shown that the additive formula is not valid for all matrices which satisfy given conditions. Also, here is proved that the representations which were given in mentioned paper do not extend the results given by Hartwig et al. [R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2 _ 2 block matrix, SIAM J. Matrix. Anal. Appl. (27)(2006) 757-771 ], in fact they are equivalent.

In this short paper, we will show that the space of real valued uniformly continuous functions defined on a metric space (X,d) is a ring if and only if every subset A ? X has one of the following properties: ? A is Bourbaki-bounded, i.e., every uniformly continuous function on X is bounded on A. ? A contains an infinite uniformly isolated subset, i.e., there exist ? > 0 and an infinite subset F ? A such that d(a,x) ? ? for every a ? F, x ? X n \{a}.

This short paper concerns the existence of curves with low gonality on smooth hypersurfaces $$X\subset \mathbb {P}^{n+1}$$ . After reviewing a series of results on this topic, we report on a recent progress we achieved as a product of the Workshop Birational geometry of surfaces, held at University of Rome “Tor Vergata” on January 11th–15th, 2016. In particular, we obtained that if $$X\subset \mathbb {P}^{n+1}$$ is a very general hypersurface of degree $$d\geqslant 2n+2$$ , the least gonality of a curve $$C\subset X$$ passing through a general point of X is $$\mathrm {gon}(C)=d-\left\lfloor \frac{\sqrt{16n+1}-1}{2}\right\rfloor $$ , apart from some exceptions we list.

In this short paper we study the d-Hermite condition about stably free modules for skew $$\textit{PBW}$$ extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and closely related with these questions, we will prove Kronecker’s theorem about the radical of finitely generated ideals for some particular types of skew $$\textit{PBW}$$ extensions.

We study the classical \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {NP}}$$\end{document}NP-hard problems of finding maximum-size subsets from given sets of k terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/MaxNDP is currently not well understood; the best known lower bound is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2^{\varOmega (\sqrt{\log n})}}$$\end{document}2Ω(logn), assuming \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {NP}\not \subseteq \mathsf {DTIME}(n^{\mathcal {O}(\log n)})}$$\end{document}NP⊈DTIME(nO(logn)). This constitutes a significant gap to the best known approximation upper bound of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}(\sqrt{n})}$$\end{document}O(n) due to Chekuri et al. (Theory Comput 2:137–146, 2006), and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (Combinatorica 7(4):365–374, 1987) introduce the technique of randomized rounding for LPs; their technique gives an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}(1)}$$\end{document}O(1)-approximation when edges (or nodes) may be used by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}\left( \log n/\log \log n\right) }$$\end{document}Ologn/loglogn paths. In this paper, we strengthen the fundamental results above. We provide new bounds formulated in terms of the feedback vertex set number r of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following results:For MaxEDP, we give an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}(\sqrt{r} \log ({k}r))}$$\end{document}O(rlog(kr))-approximation algorithm. Up to a logarithmic factor, our result strengthens the best known ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}(\sqrt{n})}$$\end{document}O(n) due to Chekuri et al., as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r\le n}$$\end{document}r≤n.Further, we show how to route \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varOmega ({\text {OPT}}^{*})}$$\end{document}Ω(OPT∗) pairs with congestion bounded by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}(\log (kr)/\log \log (kr))}$$\end{document}O(log(kr)/loglog(kr)), strengthening the bound obtained by the classic approach of Raghavan and Thompson.For MaxNDP, we give an algorithm that gives the optimal answer in time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(k+r)^{\mathcal {O}(r)}\cdot n}$$\end{document}(k+r)O(r)·n. This is a substantial improvement on the run time of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2^kr^{\mathcal {O}(r)}\cdot n}$$\end{document}2krO(r)·n, which can be obtained via an algorithm by Scheffler. We complement these positive results by proving that MaxEDP is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {NP}}$$\end{document}NP-hard even for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r=1}$$\end{document}r=1, and MaxNDP is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {W}[1]}$$\end{document}W[1]-hard when r is the parameter. This shows that neither problem is fixed-parameter tractable in r unless \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {FPT}= \mathsf {W}[1]}$$\end{document}FPT=W[1] and that our approximability results are relevant even for very small constant values of r.

0.1. How it all began. When the second author translated a result on algebraic sur faces into the language of lattice polygons using toric geometry, he got a simple inequality for lattice polygons. This inequality had originally been discovered by Scott [12]. The first author then found a third proof. Subsequently, both authors went through a phase of polygon addiction. Once you get started drawing lattice polygons on graph paper and discovering relations between their numerical invariants, it is not so easy to stop! (The gentle reader has been warned.) Thus, it was just unavoidable that the authors came up with new inequalities: Scott's inequality can be sharpened if one takes into account another invariant, which is de fined by peeling off the skins of the polygons like an onion (see Section 3).

In this short paper, we extend the classical Hoffman-Meeks Halfspace Theorem [Hoffman and Meeks, 'The strong halfspace theorem for minimal surfaces', Invent. Math. 101 (1990) 373-377] to self-shrinkers, that is: Let $P$ be a hyperplane passing through the origin. The only properly immersed self-shrinker $\Sigma $ contained in one of the closed half-space determined by $P$ is $\Sigma = P$. Our proof is geometric and uses a catenoid type hypersurface discovered by Kleene-Moller [Kleene and Moller, 'Self-shrinkers with a rotational symmetry', Trans. Amer. Math. Soc. 366 (2014) 3943-3963]. Also, using a similar geometric idea, we obtain that the only self-shrinker properly immersed in an closed cylinder $ \overline {B^{k+1} (R)} \times {\mathbb R}^{n-k}\subset {\mathbb R}^{n+1}$, for some $k\in \{1, \ldots, n\}$ and radius $R$, $R \leqslant \sqrt {2k}$, is the cylinder ${\mathbb S}^k (\sqrt {2k}) \times {\mathbb R}^{n-k}$. We also extend the above results for $\lambda $-hypersurfaces.

When the reduced twisted $C^*$-algebra $C^*_r(\mathcal{G}, c)$ of a non-principal groupoid $\mathcal{G}$ admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r(\mathcal{G}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid $\mathcal{S}$ of $\mathcal{G}$. In this paper, we study the relationship between the original groupoids $\mathcal{S}, \mathcal{G}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum $\mathfrak{B}$ of the Cartan subalgebra $C^*_r(\mathcal{S}, c)$. We then show that the quotient groupoid $\mathcal{G}/\mathcal{S}$ acts on $\mathfrak{B}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly we show that, if the quotient map $\mathcal{G}\to\mathcal{G}/\mathcal{S}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on $\mathcal{G}/\mathcal{S} \ltimes \mathfrak{B}$., 32 pages

In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of P^3 ramified along a surface of degree 2d with defect has at least d(2d-1) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in P^(2n+2) with defect for d sufficiently large., v2: A proof for the Ciliberto-Di Gennaro conjecture is added (Section 5); Some minor corrections in the other sections. v3: some minor corrections in the abstract v4: The proof for the Ciliberto-Di Gennaro conjecture has been modified; The paper is split into two parts, the complete intersection case will be discussed in a different paper

In this paper, we study a problem posed by Furstenberg on intersections between × 2 , × 3 \times 2, \times 3 invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used to obtain further improvements. For example, we show that if A 2 , A 3 ⊂ [ 0 , 1 ] A_2,A_3\subset [0,1] are closed and × 2 , × 3 \times 2, \times 3 invariant respectively, assuming that dim ⁡ A 2 + dim ⁡ A 3 > 1 \dim A_2+\dim A_3>1 then A 2 ∩ ( u A 3 + v ) A_2\cap (uA_3+v) is sparse (defined in this paper) and has box dimension zero uniformly with respect to the real parameters u , v u,v such that u u and u − 1 u^{-1} are both bounded away from 0 0 .

In our previous paper, viewing $D^b(K(l))$ as a noncommutative curve, where $K(l)$ is the Kronecker quiver with $l$-arrows, we introduced categorical invariants via counting of noncommutative curves. Roughly, these invariants are sets of subcategories in a given category and their quotients. The noncommutative curve-counting invariants are obtained by restricting the subcategories to be equivalent to $D^b(K(l))$. The general definition, however, defines a larger class of invariants and many of them behave properly with respect to fully faithful functors. Here, after recalling the definition, we focus on the examples and extend our studies beyond counting. We enrich our invariants with the following structures: the inclusion of subcategories makes them partially ordered sets and considering semi-orthogonal pairs of subcategories as edges amounts to directed graphs. It turns out that the problem for counting $D^b(A_k)$ in $D^b(A_n)$ has a geometric combinatorial parallel - counting of maps between polygons. Estimating the numbers counting noncommutative curves in $D^b({\mathbb P}^2)$ modulo the group of autoequivalences, we prove finiteness and that the exact determining of these numbers leads to a solution of Markov problem. Via homological mirror symmetry, this gives a new approach to this problem. Regarding the structure of a partially ordered set mentioned above, we initiate intersection theory of noncommutative curves focusing on the case of noncommutative genus zero. The above-mentioned structure of a directed graph (and related simplicial complex) is a categorical analogue of the classical curve complex, introduced by Harvey and Harrer. The paper contains pictures of the graphs in many examples and also presents an approach to Markov conjecture via counting of subgraphs in a graph associated with $D^b({{\mathbb{P}}}^2)$. Some of the results proved here were announced in a previous work.

Let us fix a prime $p$. The Erd\H{o}s-Ginzburg-Ziv problem asks for the minimum integer $s$ such that any collection of $s$ points in the lattice $\mathbb{Z}^n$ contains $p$ points whose centroid is also a lattice point in $\mathbb{Z}^n$. For large $n$, this is essentially equivalent to asking for the maximum size of a subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero. In this paper, we give a new upper bound for this problem for any fixed prime $p\geq 5$ and large $n$. In particular, we prove that any subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero has size at most $C_p\cdot \left(2\sqrt{p}\right)^n$, where $C_p$ is a constant only depending on $p$. For $p$ and $n$ going to infinity, our bound is of the form $p^{(1/2)\cdot (1+o(1))n}$, whereas all previously known upper bounds were of the form $p^{(1-o(1))n}$ (with $p^n$ being a trivial bound). Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method. This method and its consequences were already applied by Naslund as well as by Fox and the author to prove bounds for the problem studied in this paper. However, using some key new ideas, we significantly improve their bounds., Comment: 11 pages

In a paper from 2001 (Journal of the LMS), Diestel and Leader offered a proof that a connected graph has a normal spanning tree if and only if it does not contain a minor from two specific forbidden classes of graphs, all of cardinality $\aleph_1$. Unfortunately, their proof contains a gap, and their result is incorrect. In this paper, we construct a third type of obstruction: an $\aleph_1$-sized graph without a normal spanning tree that contains neither of the two types described by Diestel and Leader as a minor. Further, we show that any list of forbidden minors characterising the graphs with normal spanning trees must contain graphs of arbitrarily large cardinality., Comment: 9 pages. arXiv admin note: text overlap with arXiv:2005.02833

Let $S \subset \Bbb R^n$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\Bbb R^n$. If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^q(X)} \leq C \| f \|_{L^p(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$: there is a number $0 < \alpha \leq n$ such that $|X \cap B_R| \leq c \, R^\alpha$ for all balls $B_R$ in $\Bbb R^n$ of radius $R \geq 1$. On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^q$ against the measure $\chi_X dx$. Our approach consists of replacing the characteristic function $\chi_X$ of $X$ by an appropriate weight function $H$, and studying the resulting estimate in three different regimes: small values of $\alpha$, intermediate values of $\alpha$, and large values of $\alpha$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted H\"{o}lder-type inequality that holds for general non-negative Lebesgue measurable functions on $\Bbb R^n$, and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du-Zhang theorem in the range $0 < \alpha < n/2$., Comment: 31 pages. Minor revision

For a separable finite diffuse measure space $${\mathcal {M}}$$ M and an orthonormal basis $$\{\varphi _n\}$$ { φ n } of $$L^2({\mathcal {M}})$$ L 2 ( M ) consisting of bounded functions $$\varphi _n\in L^\infty ({\mathcal {M}})$$ φ n ∈ L ∞ ( M ) , we find a measurable subset $$E\subset {\mathcal {M}}$$ E ⊂ M of arbitrarily small complement $$|{\mathcal {M}}{\setminus } E| | M \ E | < ϵ , such that every measurable function $$f\in L^1({\mathcal {M}})$$ f ∈ L 1 ( M ) has an approximant $$g\in L^1({\mathcal {M}})$$ g ∈ L 1 ( M ) with $$g=f$$ g = f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of $${\mathcal {M}}=G/H$$ M = G / H being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.

This paper is concerned with the qualitative analysis of solutions to the following class of quasilinear problems $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{\Phi }u=f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$ - Δ Φ u = f ( x , u ) in Ω , u = 0 on ∂ Ω , where $$\Delta _{\Phi }u=\mathrm{div}\,(\varphi (x,|\nabla u|)\nabla u)$$ Δ Φ u = div ( φ ( x , | ∇ u | ) ∇ u ) and $$\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds$$ Φ ( x , t ) = ∫ 0 | t | φ ( x , s ) s d s is a generalized N-function. We assume that $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N is a smooth bounded domain that contains two open regions $$\Omega _N,\Omega _p$$ Ω N , Ω p with $${\overline{\Omega }}_N \cap {\overline{\Omega }}_p=\emptyset $$ Ω ¯ N ∩ Ω ¯ p = ∅ . The features of this paper are that $$-\Delta _{\Phi }u$$ - Δ Φ u behaves like $$-\Delta _N u $$ - Δ N u on $$\Omega _N$$ Ω N and $$-\Delta _p u $$ - Δ p u on $$\Omega _p$$ Ω p , and that the growth of $$f:\Omega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$$ f : Ω × R → R is like that of $$e^{\alpha |t|^{\frac{N}{N-1}}}$$ e α | t | N N - 1 on $$\Omega _N$$ Ω N and as $$|t|^{p^{*}-2}t$$ | t | p ∗ - 2 t on $$\Omega _p$$ Ω p when |t| is large enough. The main result establishes the existence of solutions in a suitable Musielak–Sobolev space in the case of high perturbations with respect to the values of a positive parameter.

A classical theorem of Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In the present paper we extend Koksma's theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every $x>1$ the pair correlations of the fractional parts of $(x^n)_{n=1}^{\infty}$ are asymptotically Poissonian. The proof is based on a martingale approximation method., Version 2: some minor changes. The paper will appear in the Israel Journal of Mathematics

In Sarkozy (2001) the third author of this paper presented two conjectures on the additive decomposability of the sequence of ”smooth” (or ”friable”) numbers. Elsholtz and Harper (2015) proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of S -unit equations).

In this paper, we prove that for the oper stratification of the de Rham moduli space $M_{\mathrm{dR}}(X,r)$, the closed oper stratum is the unique minimal stratum with dimension $r^2(g-1)+g+1$, and the open dense stratum consisting of irreducible flat bundles with stable underlying vector bundles is the unique maximal stratum., Comment: This paper comes from the last section of arXiv:1905.10765v1 as an independent paper. Comments are welcome! To appear in manuscripta mathematica

In this paper we study 1/k geodesics, those closed geodesics that minimize on all subintervals of length $L/k$, where $L$ is the length of the geodesic. We develop new techniques to study the minimizing properties of these curves on doubled polygons, and demonstrate a sequence of doubled polygons whose closed geodesics exhibit unbounded minimizing properties. We also compute the length of the shortest closed geodesic on doubled odd-gons and show that this length approaches 4 times the diameter., Comment: This paper is a result of a SUMRY (REU) project at Yale

In this paper the substantiation of the partial scheme of the averaging method for impulsive differential inclusions with fuzzy right-hand side in terms of R - solutions on the finite interval is considered.Consider the impulsive differential inclusion with the fuzzy right-hand side $$\dot x \in \varepsilon F(t,x) ,\ t \not= t_i,\ x(0)\in X_0,\quad\Delta x \mid _{t=t_i} \in \varepsilon I_i (x),\qquad\qquad\qquad\qquad\qquad\qquad\qquad (1)$$ where $t\in \mathbb{R}_+ $ is time, $x \in \mathbb{R}^n $ is a phase variable, $\varepsilon > 0 $ is a small parameter,$ F \colon \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{E}^n,$ $I_i \colon \mathbb{R}^n \to \mathbb{E}^n $ are fuzzy mappings, moments $t_i$ are enumerated in the increasing order.Associate with inclusion (1) the following partial averaged differential inclusion $$\dot\xi \in \varepsilon \widetilde F (t, \xi ),\ t \not= s_j ,\ \xi (0) \in X_0,\quad \Delta \xi \vert _{t=s_j} \in \varepsilon K_j (\xi ),\qquad\qquad\qquad\qquad\qquad\qquad\quad (2),$$ where the fuzzy mappings $ \widetilde F \colon \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{E}^n ; \quad K_j \colon \mathbb{R} \to \mathbb{E}^n $ satisfy the condition $$\lim _{T \to \infty } \frac 1T D \Big( \int\limits_t^{t+T} F(t,x) dt + \sum_{t \leq t_i < t+T} I_i (x),\int\limits_t^{t+T} \widetilde F(t,x)dt +\sum_{t \leq s_j < t+T} K_j (x) \Big) = 0,\quad\quad (3)$$ moments $s_j$ are enumerated in the increasing order. In the paper is proved the following main theorem:{\sl Let in the domain $ Q = \lbrace t \geq 0 , x \in G\subset \mathbb{R}^n \rbrace $ the following conditions fulfill:$1)$ fuzzy mappings $ F (t,x), \widetilde F(t,x), I_i(x),K_j(x) $are continuous, uniformly bounded with constant $M$, concave in $x,$ satisfy Lipschitz condition in $x$ with constant $ \lambda ;$$2)$ uniformly with respect to $t, x$ limit (3) exists and $\frac 1T i(t,t+T) \leq d < \infty ,\ \frac 1T j(t,t+T) \leq d < \infty,$where $i(t,t+T)$ and $j(t,t+T)$ are the quantities of impulse moments $t_i$ and $s_j$ on the interval$ [ t, t+T ] $;$3)$ {\rm R}-solutions of inclusion (2) for all $ X_0 \subset G^{\prime} \subset G $for $ t \in [0,L^{\ast} \varepsilon ^{-1} ] $ belong to the domain $G$ with a $ \rho $- neighborhood.Then for any $\eta > 0 $ and $L \in (0,L^{\ast}]$ there exists $\varepsilon _0 (\eta,L) \in (0,\sigma ] $ such that for all $\varepsilon \in (0, \varepsilon _0 ]$ and $t \in [0,L \varepsilon ^{-1}] $ the inequality holds:$D(R(t, \varepsilon ), \widetilde R (t, \varepsilon)) < \eta,$ where $R(t, \varepsilon), \widetilde R(t, \varepsilon ) $ are the {\rm R-} solutions of inclusions (1) and (2), $R(0, \varepsilon ) = \widetilde R (0, \varepsilon).$

In this paper we study some cube packing problems. In particular we are interested in compact subsets of $\mathbb{R}^n,n\geq 2$, which contain boundaries of cubes with all side lengths in $(0,1)$. We show here that such sets must have lower box dimension at least $n-0.5$ and we will also provide sharp examples. We also show here that such sets must be large in general in a precise sense which is also introduced in this paper., 9 pages. arXiv admin note: text overlap with arXiv:1711.06533

The original definition of cluster algebras by Fomin and Zelevinsky has been categorified and generalised in several ways over the course of the past 20 years, giving rise to cluster theory. This study lead to Iyama and Yang's generalised cluster categories $\mathcal{T}/\mathcal{T}^{fd}$ coming from $n$-Calabi-Yau triples $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$. In this paper, we use some classic tools of homological algebra to give a deeper understanding of such categories $\mathcal{T}/\mathcal{T}^{fd}$. Let $k$ be a field, $n\geq 3$ an integer and $\mathcal{T}$ a $k$-linear triangulated category with a triangulated subcategory $\mathcal{T}^{fd}$ and a subcategory $\mathcal{M}=\text{add}(M)$ such that $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$ is an $n$-Calabi-Yau triple. In this paper, we prove some properties of the triangulated categories $\mathcal{T}$ and $\mathcal{T}/\mathcal{T}^{fd}$. Our first result gives a relation between the Hom-spaces in these categories, using limits and colimits. Our second result is a Gap Theorem in $\mathcal{T}$, showing when the truncation triangles split. Moreover, we apply our two theorems to present an alternative proof to a result by Guo, originally stated in a more specific setup of dg $k$-algebras $A$ and subcategories of the derived category of dg $A$-modules. This proves that $\mathcal{T}/\mathcal{T}^{fd}$ is Hom-finite and $(n-1)$-Calabi-Yau, its object $M$ is $(n-1)$-cluster tilting and the endomorphism algebras of $M$ over $\mathcal{T}$ and over $\mathcal{T}/\mathcal{T}^{fd}$ are isomorphic. Note that these properties make $\mathcal{T}/\mathcal{T}^{fd}$ a generalisation of the cluster category., Comment: 17 pages. Final accepted version to appear in the Pacific Journal of Mathematics

This paper is concerned with voting processes on graphs where each vertex holds one of two different opinions. In particular, we study the \emph{Best-of-two} and the \emph{Best-of-three}. Here at each synchronous and discrete time step, each vertex updates its opinion to match the majority among the opinions of two random neighbors and itself (the Best-of-two) or the opinions of three random neighbors (the Best-of-three). Previous studies have explored these processes on complete graphs and expander graphs, but we understand significantly less about their properties on graphs with more complicated structures. In this paper, we study the Best-of-two and the Best-of-three on the stochastic block model $G(2n,p,q)$, which is a random graph consisting of two distinct Erdős-Renyi graphs $G(n,p)$ joined by random edges with density $q\leq p$. We obtain two main results. First, if $p=\omega(\log n/n)$ and $r=q/p$ is a constant, we show that there is a phase transition in $r$ with threshold $r^*$ (specifically, $r^*=\sqrt{5}-2$ for the Best-of-two, and $r^*=1/7$ for the Best-of-three). If $r>r^*$, the process reaches consensus within $O(\log \log n+\log n/\log (np))$ steps for any initial opinion configuration with a bias of $\Omega(n)$. By contrast, if $r r^*$, we show that, for any initial opinion configuration, the process reaches consensus within $O(\log n)$ steps. To the best of our knowledge, this is the first result concerning multiple-choice voting for arbitrary initial opinion configurations on non-complete graphs.

This work has been motivated by recent papers that quantify the density of values of generic quadratic forms and other polynomials at integer points, in particular ones that use Rogers' second moment estimates. In this paper we establish such results in a very general framework. Given any subhomogeneous function (a notion to be defined) $f: \mathbb{R}^n \to \mathbb{R}$, we derive a necessary and sufficient condition on the approximating function $\psi$ for guaranteeing that a generic element $f\circ g$ in the $G$-orbit of $f$ is $\psi$-approximable; that is, $|f\circ g(\mathbf{v})| \le \psi(\|\mathbf{v}\|)$ for infinitely many $\mathbf{v} \in \mathbb{Z}^n$. We also deduce a sufficient condition in the case of uniform approximation. Here, $G$ can be any closed subgroup of $\rm{ASL}_n(\mathbb{R})$ satisfying certain axioms that allow for the use of Rogers-type estimates., Comment: 26 pages; misprints corrected, concluding remarks added

This paper is the 2nd in a series of planned papers that provide 1st bijective proofs of alternating sign matrix (ASM) results. Based on the main result from the 1st paper, we construct a bijective proof of the enumeration formula for ASMs and of the fact that ASMs are equinumerous with descending plane partitions. We are also able to refine these bijections by including the position of the unique $1$ in the top row of the matrix. Our constructions rely on signed sets and related notions. The starting point for these constructions were known “computational” proofs, but the combinatorial point of view led to several drastic modifications. We also provide computer code where all of our constructions have been implemented.

PurposeThe purpose of this current paper is to deal with the study of non-constant entire solutions of some non-linear complex differential equations in connection to Brück conjecture, by using the theory of complex differential equation. The results generalize the results due to Pramanik et al.Design/methodology/approach39B32, 30D35.FindingsIn the current paper, we mainly study the Brück conjecture and the various works that confirm this conjecture. In our study we find that the conjecture can be generalized for differential monomials under some additional conditions and it generalizes some works related to the conjecture. Also we can take the complex number a in the conjecture to be a small function. More precisely, we obtain a result which can be restate in the following way: Let f be a non-constant entire function such that σ2(f)<∞, σ2(f) is not a positive integer and δ(0, f)>0. Let M[f] be a differential monomial of f of degree γM and α(z), β(z)∈S(f) be such that max{σ(α), σ(β)} <σ(f). If M[f]+β and fγM−α share the value 0 CM, then M[f]+βfγM−α=c,where c≠0 is a constant.Originality/valueThis is an original work of the authors.

Aspace X is said to have the absolutely strongly star -𝒤-Hurewicz (ASS𝒤H) property if for each sequence (𝒰 n : n ∈ 𝕅)of opencovers of X and each dense subset Y of X, there is a sequence (Fn : n ∈ 𝕅) of finite subsets of Y such that for each x ∈ X, {n ∈ 𝕅 : x ∉ St(Fn , 𝒰 n )}∈ 𝒤, where 𝒤 is the proper admissible ideal of 𝕅. In this paper, we investigate the relationship between the ASS𝒤H property and other related properties and study the topological properties of the ASS𝒤H property. This paper generalizes several results of Song [25] to the larger class of spaces having the ASS𝒤H properties.

In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data $$(\varSigma ,\gamma ,H)$$ . We prove that given a metric $$\gamma $$ on $${{\mathbf {S}}}^{n-1}$$ ( $$3\le n\le 7$$ ), $$({{\mathbf {S}}}^{n-1},\gamma ,H)$$ admits no fill-in of NNSC metrics provided the prescribed mean curvature H is large enough (Theorem 4). Moreover, we prove that if $$\gamma $$ is a positive scalar curvature (PSC) metric isotopic to the standard metric on $${{\mathbf {S}}}^{n-1}$$ , then the much weaker condition that the total mean curvature $$\int _{{{\mathbf {S}}}^{n-1}}H\,{{\mathrm {d}}}\mu _\gamma $$ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (Four lectures on scalar curvature, 2019, see P. 23). In the second part of this paper, we investigate the $$\theta $$ -invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.

In the present paper, we consider the nonparametric regression model with random design based on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbf{X}_\mathrm{t},\mathbf{Y}_\mathrm{t})_{\mathrm{t}\ge 0}$$\end{document}(Xt,Yt)t≥0 a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{d}\times \mathbb {R}^{q}$$\end{document}Rd×Rq-valued strictly stationary and ergodic continuous time process, where the regression function is given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(\mathbf{x},\psi ) = \mathbb {E}(\psi (\mathbf{Y}) \mid \mathbf{X} = \mathbf{x}))$$\end{document}m(x,ψ)=E(ψ(Y)∣X=x)), for a measurable function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi : \mathbb {R}^{q} \rightarrow \mathbb {R}$$\end{document}ψ:Rq→R. We focus on the estimation of the location \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Theta }}$$\end{document}Θ (mode) of a unique maximum of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(\cdot , \psi )$$\end{document}m(·,ψ) by the location \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widehat{{\varvec{\Theta }}}_\mathrm{T}$$\end{document}Θ^T of a maximum of the Nadaraya–Watson kernel estimator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{m}_\mathrm{T}(\cdot , \psi )$$\end{document}m^T(·,ψ) for the curve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(\cdot , \psi )$$\end{document}m(·,ψ). Within this context, we obtain the consistency with rate and the asymptotic normality results for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widehat{{\varvec{\Theta }}}_\mathrm{T}$$\end{document}Θ^T under mild local smoothness assumptions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(\cdot , \psi )$$\end{document}m(·,ψ) and the design density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\cdot )$$\end{document}f(·) of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{X}$$\end{document}X. Beyond ergodicity, any other assumption is imposed on the data. This paper extends the scope of some previous results established under the mixing condition. The usefulness of our results will be illustrated in the construction of confidence regions.

In this paper, we study noncommutative surface singularities arising from orders. The singularities we study are mild in the sense that they have finite representation type or, equivalently, are log terminal in the sense of the Mori minimal model program for orders (Chan and Ingalls in Invent Math 161(2):427–452, 2005). These were classified independently by Artin (in terms of ramification data) and Reiten–Van den Bergh (in terms of their AR-quivers). The first main goal of this paper is to connect these two classifications, by going through the finite subgroups $$G \subset {{{\,\mathrm{GL}\,}}_2}$$ , explicitly computing $$H^2(G,k^*)$$ , and then matching these up with Artin’s list of ramification data and Reiten–Van den Bergh’s AR-quivers. This provides a semi-independent proof of their classifications and extends the study of canonical orders in Chan et al. (Proc Lond Math Soc (3) 98(1):83–115, 2009) to the case of log terminal orders. A secondary goal of this paper is to study noncommutative analogues of plane curves which arise as follows. Let $$B = k_{\zeta } \llbracket x,y \rrbracket $$ be the skew power series ring where $$\zeta $$ is a root of unity, or more generally a terminal order over a complete local ring. We consider rings of the form $$A = B/(f)$$ where $$f \in Z(B)$$ which we interpret to be the ring of functions on a noncommutative plane curve. We classify those noncommutative plane curves which are of finite representation type and compute their AR-quivers.

In this paper, we introduce and investigate two new subclasses of analytic and bi-univalent functions defined in the open unit disc. We use the Faber polynomial expansions to find upper bounds for the $n$th$~ n\geq 3 $ Taylor-Maclaurin coefficients $\left\vert a_{n}\right\vert $ of functions belong to these new subclasses with $a_{k}=0$ for $2\leq k\leq n-1$, also we find non-sharp estimates on the first two coefficients $\left\vert a_{2}\right\vert $ and $\left\vert a_{3}\right\vert $. The results, which are presented in this paper, would generalize those in related earlier works of several authors.

The classical Morse–Sard theorem claims that for a mapping v : R n → R m + 1 of class C k the measure of critical values v ( Z v , m ) is zero under condition k ≥ n − m . Here the critical set, or m-critical set is defined as Z v , m = { x ∈ R n : rank ∇ v ( x ) ≤ m } . Further Dubovitskiĭ in 1957 and independently Federer and Dubovitskiĭ in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the C k category. Here we formulate and prove a bridge theorem that includes all the above results as particular cases: namely, if a function v : R n → R d belongs to the Holder class C k , α , 0 ≤ α ≤ 1 , then for every q > m the identity H μ ( Z v , m ∩ v − 1 ( y ) ) = 0 holds for H q -almost all y ∈ R d , where μ = n − m − ( k + α ) ( q − m ) . Intuitively, the sense of this bridge theorem is very close to Heisenberg's uncertainty principle in theoretical physics: the more precise is the information we receive on measure of the image of the critical set, the less precisely the preimages are described, and vice versa. The result is new even for the classical C k -case (when α = 0 ); similar result is established for the Sobolev classes of mappings W p k ( R n , R d ) with minimal integrability assumptions p = max ⁡ ( 1 , n / k ) , i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some N-properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces. The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015). We also crucially use very deep Y. Yomdin's entropy estimates of near critical values for polynomials (based on algebraic geometry tools).

A hypergraph $$H=(V,E)$$ has property $$B_k$$ if there exists a 2-coloring of the set $$V$$ such that each edge contains at least $$k$$ vertices of each color. We let $$m_{k,g}(n)$$ and $$m_{k,b}(n)$$ , respectively, denote the least number of edges of an $$n$$ -homogeneous hypergraph without property $$B_k$$ which contains either no cycles of length at least $$g$$ or no two edges intersecting in more than $$b$$ vertices. In the paper, upper bounds for these quantities are given. As a consequence, we obtain results for $$m^{*}_k(n)$$ , i.e., for the least number of edges of an $$n$$ -homogeneous simple hypergraph without property $$B_k$$ . Let $$\Delta(H)$$ be the maximal degree of vertices of a hypergraph $$H$$ . By $$\Delta_k(n,g)$$ we denote the minimal degree $$\Delta$$ such that there exists an $$n$$ -homogeneous hypergraph $$H$$ with maximal degree $$\Delta$$ and girth at least $$g$$ but without property $$B_k$$ . In the paper, an upper bound for $$\Delta_k(n,g)$$ is obtained.

In this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [Proximal normal structure and nonexpansive mappings, Studia Math. 171 (2005), 283–293] immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper [Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.

All groups in the family of Baumslag-Solitar groups (i.e., groups of the form G(m,n) = 〈a, b; a−1bma = bn〉, where m and n are nonzero integers) for which the residual nilpotence condition holds if and only if the residual p-finiteness condition holds for some prime number p are described. It has turned out, in particular, that the group G(pr, −pr), where p is an odd prime and r ≥ 1, is residually nilpotent, but it is residually q-finite for no prime q. Thus, an answer to the existence problem for noncyclic one-relator groups possessing such a property (formulated by McCarron in his 1996 paper) is obtained. A simple proof of the statement that an arbitrary residually nilpotent noncyclic one-relator group which has elements of finite order is residual p-finite for some prime p, which was announced in the same paper of McCarron, is also given.

Given a graph $G$ and a nondecreasing sequence $S=(s_1,\ldots,s_k)$ of positive integers, the mapping $c:V(G)\longrightarrow \{1,\ldots,k\}$ is called an $S$-packing coloring of $G$ if for any two distinct vertices $x$ and $y$ in $c^{-1}(i)$, the distance between $x$ and $y$ is greater than $s_i$. The smallest integer $k$ such that there exists a $(1,2,\ldots,k)$-packing coloring of a graph $G$ is called the packing chromatic number of $G$, denoted $\chi_{\rho}(G)$. The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier papers; recently it was established that the invariant is unbounded in the class of all subcubic graphs. In this paper, we prove that the packing chromatic number of any 2-connected bipartite subcubic outerplanar graph is bounded by $7$. Furthermore, we prove that every subcubic triangle-free outerplanar graph has a $(1,2,2,2)$-packing coloring, and that there exists a subcubic outerplanar graph with a triangle that does not admit a $(1,2,2,2)$-packing coloring. In addition, there exists a subcubic triangle-free outerplanar graph that does not admit a $(1,2,2,3)$-packing coloring. A similar dichotomy is shown for bipartite outerplanar graphs: every such graph admits an $S$-packing coloring for $S=(1,3,\ldots,3)$, where $3$ appears $\Delta$ times ($\Delta$ being the maximum degree of vertices), and this property does not hold if one of the integers $3$ is replaced by $4$ in the sequence $S$., Comment: 24 pages

Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if $$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2)156 (2002), 79–101] showed that for every irrational number, the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on $\unicode[STIX]{x1D703}$: the explicit map provided by Parry’s method is an isomorphism between the map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ This condition can be relaxed again into $$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}| where $[0;a_{1},a_{2},\ldots ]$ is the continued fraction expansion and $(p_{n}/q_{n})_{n\geq 0}$ the sequence of convergents of $\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$. Whether Parry’s map is an isomorphism for every $\unicode[STIX]{x1D703}$ or not is still an open question, although we expect a positive answer.

In this paper we consider a link $$B_{n,\rho }$$Bn,ρ between Baskakov type operators $$B_{n,\infty }$$Bn,∞ and genuine Baskakov–Durrmeyer type operators $$ B_{n,1}$$Bn,1 depending on a positive real parameter $$\rho $$ρ. The topic of the present paper is the pointwise limit relation $$\left( B_{n,\rho }f\right) \left( x\right) \rightarrow \left( B_{n,\infty }f\right) \left( x\right) $$Bn,ρfx→Bn,∞fx as $$\rho \rightarrow \infty $$ρ→∞ for $$x\ge 0.$$x≥0. As a main result we derive uniform convergence on each compact subinterval of the positive real axis for all continuous functions f of polynomial growth.

The F-index F(G) of a graph G is obtained by the sum of cubes of the degrees of all the vertices in G. It is defined in the same paper of 1972 where the first and second Zagreb indices are introduced to study the structure-dependency of total π-electron energy. Recently, Furtula and Gutman [J. Math. Chem. 53 (2015), no. 4, 1184–1190] reinvestigated F-index and proved its various properties. A connected graph with order n and size m, such that m = n + 2, is called a tricyclic graph. In this paper, we characterize the extremal graphs and prove the ordering among the different subfamilies of graphs with respect to F-index in $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$, where $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$ is a complete class of tricyclic graphs with three, four, six and seven cycles, such that each graph has α ≥ 1 pendant vertices and n ≥ 16 + α order. Mainly, we prove the bounds (lower and upper) of F(G), i.e $$\begin{array}{} \displaystyle 8n+12\alpha +76\leq F(G)\leq 8(n-1)-7\alpha + (\alpha+6)^3 ~\mbox{for each}~ G\in {\it\Omega}^{\alpha}_n. \end{array}$$

The theory of mixed multiplicities of filtrations by $m$-primary ideals in a ring is introduced in a recent paper by Cutkosky, Sarkar and Srinivasan. In this paper, we consider the positivity of mixed multiplicities of filtrations. We show that the mixed multiplicities of filtrations must be nonnegative real numbers and give examples to show that they could be zero or even irrational. When $R$ is analytically irreducible, and $\mathcal I(1),\ldots,\mathcal I(r)$ are filtrations of $R$ by $m_R$-primary ideals, we show that all of the mixed multiplicities $e_R(\mathcal I(1)^{[d_1]},\ldots,\mathcal I(r)^{[d_r]};R)$ are positive if and only if the ordinary multiplicities $e_R(\mathcal I(i);R)$ for $1\le i\le r$ are positive. We extend this to modules and prove a simple characterization of when the mixed multiplicities are positive or zero on a finitely generated module., 15 pages

In 1940, M. G. Krein obtained necessary and sufficient conditions for the extension of a continuous function f defined in an interval (-a, a), a > 0, to a positive definite function on the whole number axis R. In addition, Krein showed that the function 1 - |x|, |x| < a, can be extended to a positive definite one on R if and only if 0 < a ≤ 2, and this function has a unique extension only in the case a = 2. The present paper deals with the problem of uniqueness of the extension of the function 1 - |x|, |x| ≤ a, a G (0,1), for a class of positive definite functions on R whose support is contained in the closed interval [-1,1] (the class T). It is proved that if a ∈ [1/2,1] and Re ϕ(x) = 1 - |x|, |x| ≤ a, for some ϕ ∈ T, then ϕ(x) = (1 - |x|) +, x G R. In addition, for any a G (0,1/2), there exists a function ϕ ∈ T such that ϕ(x) = 1 - |x|, |x| ≤ a, but ϕ(x) ≠ (1 - |x|)+. Also the paper deals with extremal problems for positive definite functions and nonnegative trigonometric polynomials indirectly related to the extension problem under consideration.

This paper studies the asymptotic behavior of the exact constants of the Nikolskii inequalities for the space $$\Pi _n^d$$ of spherical polynomials of degree at most n on the unit sphere $$\mathbb{S}{^d} \subset {^{d + 1}}$$ as n → ∞. It is shown that for 0 < p < ∞, $$\mathop {\lim }\limits_{x \to \infty } \sup \left\{ {\frac{{{{\left\| P \right\|}_{{L^\infty }({\mathbb{S}^d})}}}}{{{n^{\tfrac{d}{p}}}{{\left\| P \right\|}_{{L^p}({\mathbb{S}^d})}}}}:P \in \Pi _n^d} \right\} = \sup \left\{ {\frac{{{{\left\| f \right\|}_{{L^\infty }({\mathbb{R}^d})}}}}{{{{\left\| f \right\|}_{{L^p}({\mathbb{R}^d})}}}}:f \in \varepsilon _p^d} \right\},$$ where $$\varepsilon _p^d$$ denotes the space of all entire functions of spherical exponential type at most 1 whose restrictions to ℝd belong to the space Lp(ℝd), and it is agreed that 0/0 = 0. It is also proved that for 0 < p < q < ∞, $$\liminf _{n \rightarrow \infty} \sup \left\{\frac{\|P\|_{L^{q}\left(\mathbb{S}^{d}\right)}}{n^{d(1 / p-1 / q)}\|P\|_{L^{p}\left(\mathbb{S}^{d}\right)}}: P \in \Pi_{n}^{d}\right\} \geq \sup \left\{\frac{\|f\|_{L^{q}\left(\mathbb{R}^{d}\right)}}{\|f\|_{L^{p}\left(\mathbb{R}^{d}\right)}}: f \in \mathcal{E}_{p}^{d}\right\}.$$ These results extend the recent results of Levin and Lubinsky for trigonometric polynomials on the unit circle. The paper also determines the exact value of the Nikolskii constant for nonnegative functions with p = 1 and q = ∞: $$\lim _{n \rightarrow \infty} \sup _{0 \leq P \in \Pi_{n}^{d}} \frac{\|P\|_{L^{\infty}\left(\mathbb{S}^{d}\right)}}{\|P\|_{L^{1}\left(\mathbb{S}^{d}\right)}}=\sup _{0 \leq f \in \mathcal{E}_{1}^{d}} \frac{\|f\|_{L^{\infty}\left(\mathbb{R}^{d}\right)}}{\|f\|_{L^{1} \mathbb{R}^{d}}}=\frac{1}{4^{d} \pi^{d / 2} \Gamma(d / 2+1)}.$$

A graph Γ is called a bi-circulant if it admits a cyclic group as a group of automorphisms acting semiregularly on the vertices of Γ with two orbits. The characterization of tetravalent edgetransitive bi-circulants was given in several recent papers. In this paper, a classification is given of connected tetravalent vertex-transitive bi-circulants of order twice an odd integer.

In this paper, we consider weak, directional and strong matrix majorizations. Namely, for square matrices A and B of the same size we say that A is weakly majorized by B if there is a row stochastic matrix X such that A = XB. Further, A is strongly majorized by B if there is a doubly stochastic matrix X such that A = XB. Finally, A is directionally majorized by B if Ax is majorized by Bx for any vector x where the usual vector majorization is used. We introduce the notion of majorization of matrix tuples which is defined as a natural generalization of matrix majorizations: for a chosen type of majorization we say that one tuple of matrices is majorized by another tuple of the same size if every matrix of the “smaller” tuple is majorized by a matrix in the same position in the “bigger” tuple. We say that a linear operator preserves majorization if it maps ordered pairs to ordered pairs and the image of the smaller element does not exceed the image of the bigger one. This paper contains a full characterization of linear operators that preserve weak, strong or directional majorization of tuples of matrices and linear operators that map tuples that are ordered with respect to strong majorization to tuples that are ordered with respect to directional majorization. We have shown that every such operator preserves respective majorization of each component. For all types of majorization we provide counterexamples that demonstrate that the inverse statement does not hold, that is if majorization of each component is preserved, majorization of tuples may not.

Bidimensionality Theory was introduced by [E.D. Demaine, F.V. Fomin, M.Hajiaghayi, and D.M. Thilikos. Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs, J. ACM, 52 (2005), pp.866--893] as a tool to obtain sub-exponential time parameterized algorithms on H-minor-free graphs. In [E.D. Demaine and M.Hajiaghayi, Bidimensionality: new connections between FPT algorithms and PTASs, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2005, pp.590--601] this theory was extended in order to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this work, we establish a third meta-algorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In particular, we prove that every minor (respectively contraction) bidimensional problem that satisfies a separation property and is expressible in Countable Monadic Second Order Logic (CMSO), admits a linear kernel for classes of graphs that exclude a fixed graph (respectively an apex graph) H as a minor. Our results imply that a multitude of bidimensional problems g graph classes. For most of these problems no polynomial kernels on H-minor-free graphs were known prior to our work., Comment: An an earlier version of this paper appeared in SODA 2010. That paper contained preliminary versions of some of the results of this paper

If $G$ is a group, a virtual retract of $G$ is a subgroup which is a retract of a finite index subgroup. Most of the paper focuses on two group properties: property (LR), that all finitely generated subgroups are virtual retracts, and property (VRC), that all cyclic subgroups are virtual retracts. We study the permanence of these properties under commensurability, amalgams over retracts, graph products and wreath products. In particular, we show that (VRC) is stable under passing to finite index overgroups, while (LR) is not. The question whether all finitely generated virtually free groups satisfy (LR) motivates the remaining part of the paper, studying virtual free factors of such groups. We give a simple criterion characterizing when a finitely generated subgroup of a virtually free group is a free factor of a finite index subgroup. We apply this criterion to settle a conjecture of Brunner and Burns., 30 pages, 1 figure. v3: added Lemma 5.8 and made minor corrections following referee's comments. This version of the paper has been accepted for publication

In our previous paper we have elaborated a certain signed count of real lines on real projective n-dimensional hypersurfaces of degree 2n-1. Contrary to the honest "cardinal" count, it is independent of the choice of a hypersurface, and by this reason provides a strong lower bound on the honest count. In this count the contribution of a line is its local input to the Euler number of a certain auxiliary vector bundle. The aim of this paper is to present other, in a sense more geometric, interpretations of this local input. One of them results from a generalization of Segre species of real lines on cubic surfaces and another from a generalization of Welschinger weights of real lines on quintic threefolds., Comment: 20 pages, typos are corrected (most essential, in Proposition 4.3.3)