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Congruences mod p for a prime p and partial products of the numbers 1,…, p − 1 are obtained. Bibliography: 2 titles.

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.

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

This paper is a survey of results on countable families with natural degree spectra. These results were obtained by a modification of the methodology proposed by Wechner, who first found a family of sets with the spectrum consisting precisely of nonzero Turing degrees. Based on this method, many researchers obtained examples of families with other natural spectra. In addition, in this paper we extend these results and present new examples of natural spectra. In particular, we construct a family of finite sets with the spectrum consisting of exactly non-K-trivial degrees and also we find new sufficient conditions on $$ {\Delta}_2^0 $$ -degree a, which guarantees that the class {x : x ≰ a} is the degree spectrum of some family. Finally, we give a survey of our recent results on the degree spectra of α-families, where α is an arbitrary computable ordinal.

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.

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

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

A dynamical system $$ {\displaystyle \begin{array}{ll}{u}_{tt}-\Delta u-\nabla 1\mathrm{n}\;\rho \cdot \nabla u=0& in\kern0.6em {\mathrm{\mathbb{R}}}_{+}^3\times \left(0,T\right),\\ {}{\left.u\right|}_{t=0}={\left.{u}_t\right|}_{t=0}=0& in\kern0.6em \overline{{\mathrm{\mathbb{R}}}_{+}^3},\\ {}{\left.{u}_z\right|}_{z=0}=f& for\kern0.36em 0\le t\le T,\end{array}} $$ is under consideration, where ρ = ρ(x, y, z) is a smooth positive function; f = f(x, y, t) is a boundary control; u = uf (x, y, z, t) is a solution. With the system one associates a response operator R : f ↦ uf|z = 0. The inverse problem is to recover the function ρ via the response operator. A short representation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided. If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way to use them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems.

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 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.

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).

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, the structure of relative disposition of 3-vertex cutsets in a 3-connected graph is studied. All such cutsets are divided into structural units – complexes of flowers, of cuts, of single cutsets, and trivial complexes. The decomposition of the graph by a complex of each type is described in detail. It is proved that for any two complexes C1 and C2 of a 3-connected graph G there is a unique part of the decomposition of G by C1 that contains C2. The relative disposition of complexes is described with the help of a hypertree T (G) – a hypergraph any cycle of which is a subset of a certain hyperedge. It is also proved that each nonempty part of the decomposition of G by the set of all of its 3-vertex cutsets is either a part of the decomposition of G by one of the complexes or corresponds to a hyperedge of T (G). This paper can be considered as a continuation of studies begun in the joint paper by D. V. Karpov and A. V. Pastor “On the structure of a 3-connected graph,” published in 2011. Bibliography: 10 titles.

In the paper, the elementary net closure problem is considered. An elementary net (net without a diagonal) σ = (σij)i ≠ j of additive subgroups σij of field k is called “closed” if elementary net group E(σ) does not contain new elementary transvections. Elementary net σ = (σij) is called “supplemented” if table (with a diagonal) σ = (σij), 1 ≤ i, j ≤ n, is a (full) net for some additive subgroups σii of field k. The supplemented elementary nets are closed. The necessary and sufficient condition for the supplementarity of elementary net σ = (σij) is the implementation of inclusions σijσjiσij ⊆ σij (for any i ≠ j). The question (Kourovka Notebook, Problem 19.63) is investigated of whether it true that, for closure of elementary net σ = (σij) it suffices to implement inclusions $$\sigma _{{ij}}^{2}{{\sigma }_{{ji}}}$$ ⊆ σji for any i ≠ j (here, ($$\sigma _{{ij}}^{2}$$ denotes the additive subgroup of field k generated by the squares from σij). The elementary nets for which the latter inclusions are satisfied are called “weakly supplemented elementary nets.” The concepts of supplemented and weakly supplemented elementary nets coincide for fields of odd characteristic. Thus, the aforementioned question of the sufficiency of weak supplementarity for the closure of an elementary net is relevant for the fields of characteristics 0 and 2. In this paper, examples of weakly supplemented but not supplemented elementary nets are constructed for the fields of characteristics 0 and 2. An example of a closed elementary net that is not weakly supplemented is constructed.

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 $$\begin{array}{} \displaystyle 8n+12\alpha +76\leq F(G)\leq 8(n-1)-7\alpha + (\alpha+6)^3 ~\mbox{for each}~ G\in {\it\Omega}^{\alpha}_n. \end{array}$$

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.

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)

Let $f\in\mathbb{Z}[T]$ be any polynomial of degree $d>1$ and $F\in\mathbb{Z}[X_{0},...,X_{n}]$ an irreducible homogeneous polynomial of degree $e>1$ such that the projective hypersurface $V(F)$ is smooth. In this paper we give a bound for \[ N(f,F,B):=|\{\textbf{x}\in\mathbb{Z}^{n+1}:\max_{0\leq i\leq n}|x_{i}|\leq B,\exists t\in\mathbb{Z}\text{ such that }f(t)=F(\textbf{x})\}|, \] To do this, we introduce a generalization of the Heath-Brown and Munshi's power sieve and we extend two results by Deligne and Katz on estimates for additive and multiplicative characters in many variables., Theorem 1 has been improved. The paper has been reorganized to improve the exposition

The word maps $$ \tilde{w}:\kern0.5em {\mathrm{GL}}_m{(D)}^{2k}\to {\mathrm{GL}}_n(D) $$ and $$ \tilde{w}:\kern0.5em {D}^{\ast 2k}\to {D}^{\ast } $$ for a word $$ w=\prod \limits_{i=1}^k\left[{x}_i,{y}_i\right], $$ where D is a division algebra over a field K, are considered. It is proved that if $$ \tilde{w}\left({D}^{\ast 2k}\right)=\left[{D}^{\ast },{D}^{\ast}\right], $$ then $$ \tilde{w}\left({\mathrm{GL}}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right), $$ where En(D) is the subgroup of GLn(D), generated by transvections, and Z(En(D)) is its center. Furthermore if, in addition, n > 2, then $$ \tilde{w}\left({E}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right). $$ The proof of the result is based on an analog of the “Gauss decomposition with prescribed semisimple part” (introduced and studied in two papers of the second author with collaborators) in the case of the group GLn(D), which is also considered in the present paper.

Liouville quantum gravity (LQG) and the Brownian map (TBM) are two distinct models of measure-endowed random surfaces. LQG is defined in terms of a real parameter $$\gamma $$, and it has long been believed that when $$\gamma = \sqrt{8/3}$$, the LQG sphere should be equivalent (in some sense) to TBM. However, the LQG sphere comes equipped with a conformal structure, and TBM comes equipped with a metric space structure, and endowing either one with the other’s structure has been an open problem for some time. This paper is the first in a three-part series that unifies LQG and TBM by endowing each object with the other’s structure and showing that the resulting laws agree. The present work considers a growth process called quantum Loewner evolution (QLE) on a $$\sqrt{8/3}$$-LQG surface $${\mathcal {S}}$$ and defines $$d_{{\mathcal {Q}}}(x,y)$$ to be the amount of time it takes QLE to grow from $$x \in {\mathcal {S}}$$ to $$y \in {\mathcal {S}}$$. We show that $$d_{{\mathcal {Q}}}(x,y)$$ is a.s. determined by the triple $$({\mathcal {S}},x,y)$$ (which is far from clear from the definition of QLE) and that $$d_{{\mathcal {Q}}}$$ a.s. satisfies symmetry (i.e., $$d_{{\mathcal {Q}}}(x,y) = d_{{\mathcal {Q}}}(y,x)$$) for a.a. (x, y) pairs and the triangle inequality for a.a. triples. This implies that $$d_{{\mathcal {Q}}}$$ is a.s. a metric on any countable sequence sampled i.i.d. from the area measure on $${\mathcal {S}}$$. We establish several facts about the law of this metric, which are in agreement with similar facts known for TBM. The subsequent papers will show that this metric a.s. extends uniquely and continuously to the entire $$\sqrt{8/3}$$-LQG surface and that the resulting measure-endowed metric space is TBM.

This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function $${{\,\mathrm{depth}\,}}R/I^{(t)} = \dim R -{{\,\mathrm{pd}\,}}I^{(t)} - 1$$ , where $$I^{(t)}$$ denotes the t-th symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and $${{\,\mathrm{pd}\,}}$$ denotes the projective dimension. It has been an open question whether the function $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is non-increasing if I is a squarefree monomial ideal. We show that $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is almost non-increasing in the sense that $${{\,\mathrm{depth}\,}}R/I^{(s)} \ge {{\,\mathrm{depth}\,}}R/I^{(t)}$$ for all $$s \ge 1$$ and $$t \in E(s)$$ , where $$\begin{aligned} E(s) = \bigcup _{i \ge 1}\{t \in {\mathbb {N}}|\ i(s-1)+1 \le t \le is\} \end{aligned}$$ (which contains all integers $$t \ge (s-1)^2+1$$ ). The range E(s) is the best possible since we can find squarefree monomial ideals I such that $${{\,\mathrm{depth}\,}}R/I^{(s)} < {{\,\mathrm{depth}\,}}R/I^{(t)}$$ for $$t \not \in E(s)$$ , which gives a negative answer to the above question. Another open question asks whether the function $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is always constant for $$t \gg 0$$ . We are able to construct counter-examples to this question by monomial ideals. On the other hand, we show that if I is a monomial ideal such that $$I^{(t)}$$ is integrally closed for $$t \gg 0$$ (e.g. if I is a squarefree monomial ideal), then $${{\,\mathrm{depth}\,}}R/I^{(t)}$$ is constant for $$t \gg 0$$ with $$\begin{aligned} \lim _{t \rightarrow \infty }{{\,\mathrm{depth}\,}}R/I^{(t)} = \dim R - \dim \oplus _{t \ge 0}I^{(t)}/{\mathfrak {m}}I^{(t)}. \end{aligned}$$ Our last result (which is the main contribution of this paper) shows that for any positive numerical function $$\phi (t)$$ which is periodic for $$t \gg 0$$ , there exist a polynomial ring R and a homogeneous ideal I such that $${{\,\mathrm{depth}\,}}R/I^{(t)} = \phi (t)$$ for all $$t \ge 1$$ . As a consequence, for any non-negative numerical function $$\psi (t)$$ which is periodic for $$t \gg 0$$ , there is a homogeneous ideal I and a number c such that $${{\,\mathrm{pd}\,}}I^{(t)} = \psi (t) + c$$ for all $$t \ge 1$$ .

Following a result of Hatori et al. (J Math Anal Appl 326:281–296, 2007), we give here a spectral characterization of an isomorphism from a $$C^\star $$ -algebra onto a Banach algebra. We then use this result to show that a $$C^\star $$ -algebra A is isomorphic to a Banach algebra B if and only if there exists a surjective function $$\phi :A\rightarrow B$$ satisfying (i) $$\sigma \left( \phi (x)\phi (y)\phi (z)\right) =\sigma \left( xyz\right) $$ for all $$x,y,z\in A$$ (where $$\sigma $$ denotes the spectrum), and (ii) $$\phi $$ is continuous at $$\mathbf 1$$ . In particular, if (in addition to (i) and (ii)) $$\phi (\mathbf 1)=\mathbf 1$$ , then $$\phi $$ is an isomorphism. An example shows that (i) cannot be relaxed to products of two elements, as is the case with commutative Banach algebras. The results presented here also elaborate on a paper of Bresar and Spenko (J Math Anal Appl 393:144–150, 2012), and a paper of Bourhim et al. (Arch Math 107:609–621, 2016).

In 1964, A. N. Sharkovskii published a paper in which he introduced an ordering relation on the set of positive integers. His ordering had the property that if a continuous self-map of an interval has a periodic point of some period p, then it also has periodic points of any period larger than p in this ordering. The least number in this ordering is 3. Thus, if a continuous self-map of an interval has a point of period 3, then it has points of any period. In 1975, this result was rediscovered by Lie and Yorke, who published it in their paper “Period three implies chaos.” Their work has led to the international recognition of Sharkovskii’s theorem. Since then, numerous papers on properties of self-maps of an interval have appeared. In 1994, even a conference named “Thirty Years after Sharkovskii’s Theorem: New Perspectives” was held. One of the research directions is estimating the number of periodic trajectories which a map satisfying the conditions of Sharkovskii’s theorem must have. In 1985, Bau-Sen Du published a paper in which he obtained an exact lower bound for the number of periodic trajectories of given period. In the present paper, a new, significantly shorter and more natural, proof of this result is given.

Let E∗ G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to E∗ G-isomorphism, there exists at most one Hughes-free division E∗G-ring. However, the existence of a Hughes-free division E∗ G-ring DE∗G for an arbitrary locally indicable group G is still an open question. Nevertheless, DE∗G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether DE∗G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists DE[G] and it is universal. In Appendix we give a description of DE[G] when G is a RFRS group, This paper is partially supported by the Spanish Ministry of Science and Innovation through the grant MTM2017-82690-P and the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S4). I would like to thank Dawid Kielak and an anonymous referee for useful suggestions and comments

Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if $K \subset \mathbb{R}^{n}$ is any set with equal Hausdorff and packing dimensions, then $$ \dim_{\mathrm{H}} π_{e}(K) = \min\{\dim_{\mathrm{H}} K,1\} $$ for almost every $e \in S^{n - 1}$. Here $π_{e}$ stands for orthogonal projection to $\mathrm{span}(e)$. The primary purpose of this paper is to present proofs for Lutz and Stull's projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatorial-geometric arguments, such as discretised versions of Kaufman's "potential theoretic" method, the pigeonhole principle, and a lemma of Katz and Tao. A secondary purpose is to slightly generalise Lutz and Stull's theorems: the versions in this paper apply to orthogonal projections to $m$-planes in $\mathbb{R}^{n}$, for all $0 < m < n$., 11 pages. v2: Incorporated referee suggestions

In the present paper, we study fundamental groups of curves in positive characteristic. Let $$X^{\bullet }$$ be a pointed stable curve of type $$(g_{X}, n_{X})$$ over an algebraically closed field of characteristic $$p>0$$, $$\Gamma _{X^{\bullet }}$$ the dual semi-graph of $$X^{\bullet }$$, and $$\Pi _{X^{\bullet }}$$ the admissible fundamental group of $$X^{\bullet }$$. In the present paper, we study a kind of group-theoretical invariant $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ associated to the isomorphism class of $$\Pi _{X^{\bullet }}$$ called the limit of p-averages of $$\Pi _{X^{\bullet }}$$, which plays a central role in the theory of anabelian geometry of curves over algebraically closed fields of positive characteristic. Without any assumptions concerning $$\Gamma _{X^{\bullet }}$$, we give a lower bound and a upper bound of $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$. In particular, we prove an explicit formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ under a certain assumption concerning $$\Gamma _{X^{\bullet }}$$ which generalizes a formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ obtained by Tamagawa. Moreover, if $$X^{\bullet }$$ is a component-generic pointed stable curve, we prove an explicit formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ without any assumptions concerning $$\Gamma _{X^{\bullet }}$$, which can be regarded as an averaged analogue of the results of Nakajima, Zhang, and Ozman–Pries concerning p-rank of abelian etale coverings of projective generic curves for admissible coverings of component-generic pointed stable curves.

We begin with the reference measure P0 induced by simple, symmetric nearest neighbor continuous time random walk on Zd starting at 0 with jump rate 2d and then define, for β ⩾ 0, t > 0, the Gibbs probability measure Pβ,t by specifying its density with respect to P0 as $${{d{P_{\beta, t}}} \over {d{P^0}}} = {Z_{\beta, t}}{(0)^{-1}}{{\rm{e}}^{\beta \int_0^t {{\delta _0}({x_s})ds}}},$$ (0.1) where $${Z_{\beta, t}}(0) \equiv {E^0}{\rm{[}}{{\rm{e}}^{\beta \;\int_0^t {{\delta _0}({x_s})ds}}}].$$. This Gibbs probability measure provides a simple model for a homopolymer with an attractive potential at the origin. In a previous paper (Cranston and Molchanov, 2007), we showed that for dimensions d ⩾ 3 there is a phase transition in the behavior of these paths from the diffusive behavior for β below a critical parameter to the positive recurrent behavior for β above this critical value. The critical value was determined by means of the spectral properties of the operator Δ + βδ0, where Δ is the discrete Laplacian on Zd. This corresponds to a transition from a diffusive or stretched-out phase to a globular phase for the polymer. In this paper we give a description of the polymer at the critical value where the phase transition takes place. The behavior at the critical parameter is dimension-dependent.

Prime end boundaries $\partial_P\Omega$ of domains $\Omega$ are studied in the setting of complete doubling metric measure spaces supporting a $p$-Poincar\'e inequality. Notions of rectifiably (in)accessible- and (in)finitely far away prime ends are introduced and employed in classification of prime ends. We show that, for a given domain, the prime end capacity of the collection of all rectifiably inaccessible prime ends together will all non-singleton prime ends is zero. We show the resolutivity of continouous functions on $\partial_P\Omega$ which are Lipschitz continuous with respect to the Mazurkiewicz metric when restricted to the collection $\partial_{SP}\Omega$ of all accessible prime ends. Furthermore, bounded perturbations of such functions in $\partial_P\Omega\setminus\partial_{SP}\Omega$ yield the same Perron solution. In the final part of the paper, we demonstrate the (resolutive) Kellogg property with respect to the prime end boundary of bounded domains in the metric space. Notions given in this paper are illustrated by a number of examples., Comment: 23 pages, 3 figures

In this paper, the compositional structure of a finite group G is investigated, which has the Sylow 2-subgroup that is permutable with some non p-nilpotent biprimary subgroups, which contain the Sylow р-subgroup of G for all odd simple divisors of the р order of the group G, and such biprimary subgroups are taken one by one for each odd р, and mark the set SB(G). In this work, the existence of the subset SB(G)* in SB(G) is proved, which consists of р-closed subgroups. The main result of this paper is as follows: if the Sylow 2-subgroup of the group G is permutable with all subgroups SB(G)*, then G may have simple non-abelian compositional factors only of L2(7) type, if p > 3, and additionally of L2(3f) type, f = 3a, a ≥ 1, if p = 3.

In the short note of 1927, Urysohn constructed the metric space R that is nowhere locally separable. There is no publication with indications that R is a (noncomplete) ℝ-tree that has valency c at each point. The author in 1989, as well as Polterovich and Shnirelman in 1997, constructed ℝ-trees isometric to R unaware of the paper by Urysohn. In this paper the author considers various constructions of the ℝ-tree R and of the minimal complete ℝ-tree of valency c including R, as well as the characterizations of ℝ-trees, their properties, and connections with ultrametric spaces.

We present a theorem in the present paper on an approximation to functions of a Holder class on a countable union of segments lying on a positive ray by entire functions of order 1/2 bounded on this ray. Problems related to the approximation of entire functions on subsets of the semiaxis by using entire functions of order 1/2 are closely related to problems of approximating functions on subsets of the whole axis using entire functions of exponential type but have their own specifics. We consider segments In in this paper with lengths of order n such that the distance between In and In + 1 is also of order n. Cases of the whole semiaxis or the union of finitely many segments and a ray were considered in previous papers. As for the problem of approximating functions of the Holder class on the union of a countable set of segments on the whole axis, it turns out that the approximation rate at neighborhoods of the segment endpoints as the type of the functions increases is higher than that in a neighborhood of their midpoints.

We continue in this paper to study one of the models of a discrete analog of the Renyi problem, also known as the parking problem. Suppose that n and i are integers satisfying n ≥ 0 and 0 ≤ i ≤ n – 1. We place an open interval (i, i + 1) in the segment [0, n] with i being a random variable taking values 0, 1, 2, …, n – 1 with equal probability for all n ≥ 2. If n < 2, then we say that the interval does not fit. After placing the first interval, two free segments [0, i] and [i + 1, n] are formed and independently filled with intervals of unit length according to the same rule, and so on. At the end of the process of filling the segment [0, n] with intervals of unit length, the distance between any two adjacent unit intervals does not exceed one. Suppose now that Xn is the number of unit intervals placed. In our earlier work published in 2018, we studied the asymptotic behavior of the first moments of random variable Xn. In contrast to the classical case, the exact expressions for the expectation, variance, and third central moment were obtained. The asymptotic behavior of all central moments of random variable Xn is investigated in this paper and the asymptotic normality for Xn is proved.

We have, in this paper, constructed a new non-geometric embedding of some braid group into the mapping class group of a surface which is induced by the 3-fold branched covering over a disk with some branch points. There is a lift $$\tilde{\beta }_i$$ of the half-Dehn twist $$\beta _i$$ on the disk with some marked points to some surface via the 3-fold covering. We show how this lift $$\tilde{\beta }_i$$ acts on the fundamental group of the surface, and also show that $$\tilde{\beta }_i$$ equals the product of two (inverse) Dehn twists. Two adjacent lifts satisfy the braid relation, hence such lifts induce a homomorphism $$\phi : B_k \rightarrow \Gamma _{g,b}$$ . In this paper we give a concrete description of this homomorphism and show that it is injective by the Birman–Hilden theory. Furthermore, we show that the map on the level of classifying spaces of groups is compatible with the action of little 2-cube operad so that it induces a trivial homomorphism on the stable homology.

This paper is concerned with a parabolic–parabolic Keller–Segel system $$\begin{aligned} \left\{ \begin{array}{ll} u_t = \nabla \cdot ( \nabla u - u \nabla v ) &{} \quad \text{ in } \, \Omega \times (0,T), \\ v_t = \Delta v - \alpha v + u &{} \quad \text{ in } \, \Omega \times (0,T) \end{array} \right. \end{aligned}$$with a constant $$ \alpha \ge 0 $$ and nonnegative initial data in a smoothly bounded domain $$ \Omega \subset \mathbb {R}^2 $$ under the Neumann boundary condition or in $$ \Omega = \mathbb {R}^2 $$. It was introduced as a model of aggregation of bacteria, which is mathematically translated as finite-time blowup. A solution (u, v) is said to blow up at $$ t = T 0 $$, and type II otherwise. It was shown in Mizoguchi (J Funct Anal 271:3323–3347, 2016) that each blowup is type II in radial case. In this paper, we obtain the conclusion in general case.

In this paper, we mainly study a family of unbounded non-hyperbolic domains in $$\mathbb {C}^{n+m}$$, called Fock–Bargmann–Hartogs domains $$D_{n,m}(\mu )$$ ($$\mu >0$$) which are defined as a Hartogs type domains with the fiber over each $$z\in \mathbb {C}^{n}$$ being a ball of radius $$e^{-\frac{\mu }{2} {\Vert z\Vert }^{2}}$$. The purpose of this paper is twofold. Firstly, we obtain necessary and sufficient conditions for Rawnsley’s $$\varepsilon $$-function $$\varepsilon _{(\alpha ,g)}(\widetilde{w})$$ of $$\big (D_{n,m}(\mu ), g(\mu ;\nu )\big )$$ to be a polynomial in $$\Vert \widetilde{w}\Vert ^2$$, where $$g(\mu ;\nu )$$ is a Kahler metric associated with the Kahler potential $$\nu \mu {\Vert z\Vert }^{2} -\ln (e^{-\mu {\Vert z\Vert }^{2}}-\Vert w\Vert ^2)$$. Secondly, using above results, we study the Berezin quantization on $$D_{n,m}(\mu )$$ with the metric $$\beta g(\mu ;\nu )$$$$(\beta >0)$$.

For any symplectic manifold $${(M,\omega )}$$ , the set of Hamiltonian diffeomorphisms $${{\text {Ham}}^c(M,\omega )}$$ forms a group and $${{\text {Ham}}^c(M,\omega )}$$ contains an important subset $${{\text {Aut}}(M,\omega )}$$ which consists of time one flows of autonomous(time-independent) Hamiltonian vector fields on M. One might expect that $${{\text {Aut}}(M,\omega )}$$ is a very small subset of $${{\text {Ham}}^c(M,\omega )}$$ . In this paper, we estimate the size of the subset $${{\text {Aut}}(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric which was introduced by Hofer. Polterovich and Shelukhin proved that the complement $${{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}$$ is a dense subset of $${{\text {Ham}}^c(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric if $${(M,\omega )}$$ is a closed symplectically aspherical manifold where Conley conjecture is established (Polterovich and Schelukhin in Sel Math 22(1):227–296, 2016). In this paper, we generalize above theorem to general closed symplectic manifolds and general conv! ex symplectic manifolds. So, we prove that the set of all non-autonomous Hamiltonian diffeomorphisms $${{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}$$ is a dense subset of $${{\text {Ham}}^c(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric if $${(M,\omega )}$$ is a closed or convex symplectic manifold without relying on the solution of Conley conjecture.

In 1964, A.N. Sharkovskii published an article in which he introduced a special ordering on the set of positive integers. This ordering has the property that if p ◃ q and a mapping of a closed bounded interval into itself has a point of period p, then it has a point of period q. The least number with respect to this ordering is 3. Thus, if a mapping has a point of period 3, then it has points of any periods. In 1975, the latter result was rediscovered by Li and Yorke, who published the paper “Period Three Implies Chaos.” In the present paper, an exact lower bound for the number of trajectories of a given period for a mapping of a closed bounded interval into itself having a point of period 3 is given. The key point of the reasoning consisted in solution of a combinatorial problem the answer to which is expressed in terms of Lucas numbers. As a consequence, an explicit formula for the number of necklaces of a special type is obtained. We also consider a piecewise linear unimodular mapping of [0, 1] into itself for which it is possible to find points of an arbitrary given period.

In 2005, using a famous lemma of Atkin and Swinnerton-Dyer (Some properties of partitions, Proc. Lond. Math. Soc. (3)4(1954), 84–106), Yesilyurt (Four identities related to third order mock theta functions in Ramanujan’s lost notebook, Adv. Math. 190(2005), 278–299) proved four identities for third order mock theta functions found on pages 2 and 17 in Ramanujan’s lost notebook. The primary purpose of this paper is to offer new proofs in the spirit of what Ramanujan might have given in the hope that a better understanding of the identities might be gained. Third order mock theta functions are intimately connected with ranks of partitions. We prove new dissections for two rank generating functions, which are keys to our proof of the fourth, and the most difficult, of Ramanujan’s identities. In the last section of this paper, we establish new relations for ranks arising from our dissections of rank generating functions.

Let rk(n) denote the number of lattice points on a k-dimensional sphere of radius $$ \sqrt{n} $$ . The generating function $$ {\zeta}_k(s)=\sum \limits_{n=1}^{\infty }{r}_k(n){n}^{-s},\kern0.5em k\ge 2, $$ is Epstein’s zeta function. The paper considers the Riesz mean of the type $$ {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{1}{\Gamma \left(\rho +1\right)}\sum \limits_{n\le x}{\left(x-n\right)}^{\rho }{r}_3(n), $$ where ρ > 0; the error term Δρ(x; ζ3) is defined by $$ {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{\uppi^{3/2}{x}^{\rho +3/2}}{\Gamma \left(\rho +5/2\right)}+\frac{x^{\rho }}{\Gamma \left(\rho +1\right)}{\zeta}_3(0)+{\Delta}_{\rho}\left(x;{\zeta}_3\right). $$ K. Chandrasekharan and R. Narasimhan (1962, MR25#3911) proved that $$ {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\Big({x}^{1/2+\rho /2\Big)}& \left(\rho >1\right),\\ {}{\Omega}_{\pm}\left({x}^{1/2+\rho /2}\right)& \left(\rho \ge 0\right).\end{array}} $$ In the present paper, it is proved that $$ {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\left(x\log x\right)& \left(\rho =1\right),\\ {}O\left({x}^{2/3+\rho /3+\varepsilon}\right)& \left(1/2

In a joint work with S. Bianchini [8] (see also [6, 7]), we proved a quadratic interaction estimate for the system of conservation laws $$ \left\{\begin{array}{l}{u}_t+f{(u)}_x=0,\\ {}u\left(t=0\right)={u}_0(x),\end{array}\right. $$ where u : [0, ∞) × ℝ → ℝn, f : ℝn → ℝn is strictly hyperbolic, and Tot.Var.(u0) ≪ 1. For a wavefront solution in which only two wavefronts at a time interact, such an estimate can be written in the form $$ \sum \limits_{t_j\;\mathrm{interaction}\ \mathrm{time}}\frac{\left|\sigma \left({\alpha}_j\right)-\sigma \left({\alpha}_j^{\prime}\right)\right|\left|{\alpha}_j\right|\left|{\alpha}_j^{\prime}\right|}{\left|{\alpha}_j\right|+\left|{\alpha}_j^{\prime}\right|}\le C(f)\mathrm{Tot}.\mathrm{Var}.{\left({u}_0\right)}^2, $$ where αj and $$ {\alpha}_j^{\prime } $$ are the wavefronts interacting at the interaction time tj, σ(·) is the speed, |·| denotes the strength, and C(f) is a constant depending only on f (see [8, Theorem 1.1] or Theorem 3.1 in the present paper for a more general form). The aim of this paper is to provide the reader with a proof for such a quadratic estimate in a simplified setting, in which: • all the main ideas of the construction are presented; • all the technicalities of the proof in the general setting [8] are avoided.

Decomposition spaces are simplicial $\infty$-groupoids subject to a certain exactness condition, needed to induce a coalgebra structure on the space of arrows. Conservative ULF functors (CULF) between decomposition spaces induce coalgebra homomorphisms. Suitable added finiteness conditions define the notion of M\"obius decomposition space, a far-reaching generalisation of the notion of M\"obius category of Leroux. In this paper, we show that the Lawvere-Menni Hopf algebra of M\"obius intervals, which contains the universal M\"obius function (but is not induced by a M\"obius category), can be realised as the homotopy cardinality of a M\"obius decomposition space $U$ of all M\"obius intervals, and that in a certain sense $U$ is universal for M\"obius decomposition spaces and CULF functors., Comment: 35 pages. This paper is one of six papers that formerly constituted the long manuscript arXiv:1404.3202. v3: minor expository improvements. Final version to appear in Adv. Math

In this paper, we first give several properties with respect to subgroups of self-similar groups in the sense of iterate function system (IFS). We then prove that some subgroups of p-adic numbers p are strong self-similar in the sense of IFS, 1306F169, The author thanks editor Stephen Glasby and the anonymous reviewers for their constructive suggestions. This paper is supported by the Anadolu University Research Fund under Contract 1306F169.

This paper is concerned with a Delsarte-type extremal problem. Denote by$${\mathcal {P}}(G)$$P(G)the set of positive definite continuous functions on a locally compact abelian groupG. We consider the function class, which was originally introduced by Gorbachev,$$\begin{aligned}&{\mathcal {G}}(W, Q)_G = \left\{ f \in {\mathcal {P}}(G) \cap L^1(G)~:\right. \\&\qquad \qquad \qquad \qquad \qquad \left. f(0) = 1, ~ {\text {supp}}{f_+} \subseteq W,~ {\text {supp}}{\widehat{f}} \subseteq Q \right\} \end{aligned}$$G(W,Q)G=f∈P(G)∩L1(G):f(0)=1,suppf+⊆W,suppf^⊆Qwhere$$W\subseteq G$$W⊆Gis closed and of finite Haar measure and$$Q\subseteq {\widehat{G}}$$Q⊆G^is compact. We also consider the related Delsarte-type problem of finding the extremal quantity$$\begin{aligned} {\mathcal {D}}(W,Q)_G = \sup \left\{ \int _{G} f(g) \mathrm{d}\lambda _G(g) ~ : ~ f \in {\mathcal {G}}(W,Q)_G\right\} . \end{aligned}$$D(W,Q)G=sup∫Gf(g)dλG(g):f∈G(W,Q)G.The main objective of the current paper is to prove the existence of an extremal function for the Delsarte-type extremal problem$${\mathcal {D}}(W,Q)_G$$D(W,Q)G. The existence of the extremal function has recently been established by Berdysheva and Révész in the most immediate case where$$G={\mathbb {R}}^d$$G=Rd. So, the novelty here is that we consider the problem in the general setting of locally compact abelian groups. In this way, our result provides a far reaching generalization of the former work of Berdysheva and Révész.

In this paper, an algorithm to enumerate non-hyperelliptic superspecial curves of genus $4$ over finite fields of characteristic $p \geq 5$ is constructed. As an application, the algorithm is used to enumerate non-hyperelliptic superspecial curves of genus $4$ over prime fields of characteristic $p \leq 11$. Thanks to the fact that the number of $\mathbb{F}_{p^a}$-isomorphism classes of superspecial curves over $\mathbb{F}_{p^a}$ of a fixed genus depends only on the parity of $a$, this paper contributes to the odd-degree case for genus $4$, whereas our previous work (Kudo-Harashita: Superspecial curves of genus $4$ in small characteristic) contributes to the even-degree case. This paper is the full-version of our conference paper (Kudo-Harashita: Enumerating Superspecial Curves of Genus $4$ over Prime Fields) presented at WCC2017.

This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem $$\begin{aligned} -\Delta \phi _{\sigma _j}=0,\quad \hbox { on }\,\,\Omega ,\quad \partial _\nu \phi _{\sigma _j}=\sigma _j \phi _{\sigma _j}\quad \hbox { on }\,\,\partial \Omega \end{aligned}$$-Δϕσj=0,onΩ,∂νϕσj=σjϕσjon∂Ωin two-dimensional domains $$\Omega $$Ω. In particular, this paper presents a dense family $$\mathcal {A}$$A of simply-connected two-dimensional domains with analytic boundaries such that, for each $$\Omega \in \mathcal {A}$$Ω∈A, the nodal set of the eigenfunction $$\phi _{\sigma _j}$$ϕσj “is not dense at scale $$\sigma _j^{-1}$$σj-1”. This result addresses a question put forth under “Open Problem 10” in Girouard and Polterovich (J Spectr Theory 7(2):321–359, 2017). In fact, the results in the present paper establish that, for domains $$\Omega \in \mathcal {A}$$Ω∈A, the nodal sets of the eigenfunctions $$\phi _{\sigma _j}$$ϕσj associated with the eigenvalue $$\sigma _j$$σj have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each $$\Omega \in \mathcal {A}$$Ω∈A there is a value $$r_1>0$$r1>0 such that for each j there is $$x_j\in \Omega $$xj∈Ω such that $$\phi _{\sigma _j}$$ϕσj does not vanish on the ball of radius $$r_1$$r1 around $$x_j$$xj.

In this paper, we study large $m$ asymptotics of the $l^1$ minimal $m$-partition problem for Dirichlet eigenvalue. For any smooth domain $\Omega\in \mathbb{R}^n$ such that $|\Omega|=1$, we prove that the limit $\lim\limits_{m\rightarrow\infty}l_m^1(\Omega)=c_0$ exists, and the constant $c_0$ is independent of the shape of $\Omega$. Here $l_m^1(\Omega)$ denotes the minimal value of the normalized sum of the first Laplacian eigenvalues for any $m$-partition of $\Omega$., Comment: This paper has been accepted for publication in SCIENCE CHINA Mathematics

Masking of quantum information means that information is hidden from a subsystem and spread over a composite system. Modi et al. proved in [Phys. Rev. Lett. 120, 230501 (2018)] that this is true for some restricted sets of nonorthogonal quantum states and it is not possible for arbitrary quantum states. In this paper, we discuss the problem of masking quantum information encoded in pure and mixed states, respectively. Based on an established necessary and sufficient condition for a set of pure states to be masked by an operator, we find that there exists a set of four states that can not be masked, which implies that to mask unknown pure states is impossible. We construct a masker $S^\sharp$ and obtain its maximal maskable set, leading to an affirmative answer to a conjecture proposed in Modi's paper mentioned above. We also prove that an orthogonal (resp. linearly independent) subset of pure states can be masked by an isometry (resp. injection). Generalizing the case of pure states, we introduce the maskability of a set of mixed states and prove that a commuting subset of mixed states can be masked by an isometry $S^{\diamond}$ while it is impossible to mask all of mixed states by any operator. We also find the maximal maskable sets of mixed states of the isometries ${S^{\sharp}}$ and ${S^{\diamond}}$, respectively.

We use hypergeometric sheaves on $G_m/F_q$, which are particular sorts of rigid local systems, to construct explicit local systems whose arithmetic and geometric monodromy groups are the finite general linear groups $GL_n(q)$ for any $n \ge 2$ and and any prime power $q$, so long as $q > 3$ when $n=2$. This paper continues a program of finding simple (in the sense of simple to remember) families of exponential sums whose monodromy groups are certain finite groups of Lie type, cf. [Gr], [KT1], [KT2], [KT3] for (certain) finite symplectic and unitary groups, or certain sporadic groups, cf. [KRL], [KRLT1], [KRLT2], [KRLT3]. The novelty of this paper is obtaining $GL_n(q)$ in this hypergeometric way. A pullback construction then yields local systems on $A^1/F_q$ whose geometric monodromy groups are $SL_n(q)$. These turn out to recover a construction of Abhyankar., 26 pages