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We show that a smooth embedded paper Moebius band must have aspect ratio at least $$\begin{aligned}\lambda _1= \frac{2 \sqrt{4-2 \sqrt{3}}+4}{\root 4 \of {3} \sqrt{2}+2 \sqrt{2 \sqrt{3}-3}} =1.69497\ldots \end{aligned}$$ This bound comes more than 3/4 of the way from the old known bound of $$\pi /2=1.5708\ldots $$ to the conjectured bound of $$\sqrt{3} = 1.732\ldots $$

Sendov’s conjecture states that if all the zeroes of a complex polynomial P(z) of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of P(z). In a paper that appeared in 2014, Degot proved that, for each a ∈ (0, 1), there exists an integer N such that for any polynomial P(z) with degree greater than N, if P(a) = 0 and all zeroes lie inside the unit disk, the disk |z − a| ≤ 1 contains a critical point of P(z). Based on this result, we derive an explicit formula N(a) for each a ∈ (0, 1) and, consequently obtain a uniform bound N for all a ∈ [α, β] where 0 < α < β < 1. This (partially) addresses the questions posed in Degot’s paper.

Wu et al. (Nature 566:378–382, 2019) introduced a new indicator measuring disruption ($${DI}_{1}$$DI1). Bornmann et al. (Do disruption index indicators measure what they propose to measure? The comparison of several indicator variants with assessments by peers, 2019. https://arxiv.org/abs/1911.08775) compared variants of the disruption index and pointed to $${DI}_{5}$$DI5 as an interesting variant. The calculation of a field-specific version of $${DI}_{5}$$DI5 (focusing on disruptiveness within the same field) for Scientometrics papers in the current study reveals that the variant is possibly able to identify landmark papers in scientometrics. This result is in contrast to the Scientometrics analysis previously published by Bornmann and Tekles (Scientometrics 120(1):331–336, 2019) based on the original disruption index ($${DI}_{1}$$DI1).

O. Einstein (2008) proved Bollob\'as-type theorems on intersecting families of ordered sets of finite sets and subspaces. Unfortunately, we report that the proof of a theorem on ordered sets of subspaces had a mistake. We prove two weaker variants., Comment: 6 pages. Improved bound for Theorem 4

In their celebrated paper (Furst et al., Math. Syst. Theory 17(1), 13---27 (12)), Furst, Saxe, and Sipser used random restrictions to reveal the weakness of Boolean circuits of bounded depth, establishing that constant-depth and polynomial-size circuits cannot compute the parity function. Such local restrictions have played important roles and have found many applications in complexity analysis and algorithm design over the past three decades. In this article, we give a brief overview of two intriguing applications of local restrictions: the first one is for the Isomorphism Conjecture and the second one is for moderately exponential time algorithms for the Boolean formula satisfiability problem.

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

We give a simple proof and a multivariable generalization of an identity due to E. Alkan concerning a weighted average of the Ramanujan sums. We deduce identities for other weighted averages of the Ramanujan sums with weights concerning logarithms, values of arithmetic functions for gcd's, the Gamma function, the Bernoulli polynomials, and binomial coefficients., Comment: 8 pages

In 1935, Erdos proved that all natural numbers can be written as a sum of a square of a prime and a square-free number. In 1939, Pillai derived an asymptotic formula for the number of such representations. The mathematical review of Pillai’s paper stated that the proof of the above result contained inaccuracies, thus casting a doubt on the correctness of the paper. In this paper, we re-examine Pillai’s paper and show that his argument was essentially correct. Afterwards, we improve the error term in Pillai’s theorem using the Bombieri–Vinogradov theorem.

The archimedean solids Cubus simus (snub cube) and Dodecaedron simum (snub dodecahedron) cannot be constructed by ruler and compass. We explain that for general reasons their vertices can be constructed via paper folding on the faces of a cube, respectively dodecahedron, and we present explicit folding constructions. The construction of the Cubus simus is particularly elegant. We also review and prove the construction rules of the other Archimedean solids., Comment: 13 pages, 12 figures, v2: as published in Geometriae Dedicata

Given a convex body $$K$$K, consider the smallest number $$N$$N so that there is a point $$P\in \partial K$$PźźK such that every circle centred at $$P$$P intersects $$\partial K$$źK in at most $$N$$N points. In 1946 Erdźs conjectured that $$N=2$$N=2 for all $$K$$K, but there are convex bodies for which this is not the case. As far as we know there is no known global upper bound. We show that no convex body has $$N=\infty $$N=ź and that there are convex bodies for which $$N = 6$$N=6.

In the mentioned paper of Hadwiger an upper bound is given for α (A)·N(A), where α (A) is the density of covering ℝn by lattice translates ofA⊂ℝn andN(A) is the number of neighbours ofA. Using “dual” descriptions of α (A) andN(A), in the paper a sharpening of Hadwiger's inequality as well as a lower bound for α (A)·N(A) are proved.

In this short paper, we derive an alternative proof for some known (van den Berg & Gilkey 2015) short-time asymptotics of the heat content in a compact full-dimensional submanifolds S with smooth boundary. This includes formulae like $$\begin{aligned} \int _{S} \exp (t\Delta ) (f \mathbb {1}_{S}) \,\mathrm {d}V= \int _S f \,\mathrm {d}V- \sqrt{\frac{t}{\pi }} \int _{\partial S} f \,\mathrm {d}A+ o(\sqrt{t}),\quad t \rightarrow 0^+, \end{aligned}$$ ∫ S exp ( t Δ ) ( f 1 S ) d V = ∫ S f d V - t π ∫ ∂ S f d A + o ( t ) , t → 0 + , and explicit expressions for similar expansions involving other powers of $$\sqrt{t}$$ t . By the same method, we also obtain short-time asymptotics of $$\int _S \exp (t^m\Delta ^m)(f \mathbb {1}_S)\,\mathrm {d}V$$ ∫ S exp ( t m Δ m ) ( f 1 S ) d V , $$m \in \mathbb N$$ m ∈ N , and more generally for one-parameter families of operators $$t \mapsto k(\sqrt{-t\Delta })$$ t ↦ k ( - t Δ ) defined by an even Schwartz function k.

We characterize one-dimensional Euclidean preference profiles with a small number of alternatives and voters. We show that every single-peaked preference profile withtwovoters is one-dimensional Euclidean, and that every preference profile with up to five alternatives is one-dimensional Euclidean if and only if it is both single-peaked and single-crossing. By the work of Chen et al. (Social Choice and Welfare 48(2):409–432, 2017), we thus obtain that the smallest single-peaked and single-crossing preference profiles that arenotone-dimensional Euclidean consist of three voters and six alternatives.

A model was developed for long term metformin tissue retention based upon temporally inclusive models of serum/plasma concentration (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C $$\end{document}C) having power function tails called the gamma-Pareto type I convolution (GPC) model and was contrasted with biexponential (E2) and noncompartmental (NC) metformin models. GPC models of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C $$\end{document}C have a peripheral venous first arrival of drug-times parameter, early \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C $$\end{document}C peaks and very slow washouts of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C $$\end{document}C. The GPC, E2 and NC models were applied to a total of 148 serum samples drawn from 20 min to 72 h following bolus intravenous metformin in seven healthy mongrel dogs. The GPC model was used to calculate area under the curve (AUC), clearance (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ CL $$\end{document}CL), and functions of time, f(t), for drug mass remaining (M), apparent volume of distribution (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{d}$$\end{document}Vd), as well as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{1/2}\ f(t)$$\end{document}t1/2f(t) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C $$\end{document}C, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M $$\end{document}M and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{d}$$\end{document}Vd. The GPC models of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C $$\end{document}C yielded metformin \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ CL $$\end{document}CL-values that were 84.8% of total renal plasma flow (RPF) as estimated from meta-analysis. The GPC \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ CL $$\end{document}CL-values were significantly less than the corresponding NC and E2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ CL $$\end{document}CL-values of 104.7% and 123.7% of RPF, respectively. The GPC plasma/serum only model predicted 78.9% drug \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M $$\end{document}M average urinary recovery at 72 h; similar to prior human urine drug \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M $$\end{document}M collection results. The GPC model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{1/2}$$\end{document}t1/2 of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M $$\end{document}M, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C $$\end{document}C and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_d$$\end{document}Vd, were asymptotically proportional to elapsed time, with a constant limiting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{1/2}$$\end{document}t1/2 ratio of M/C averaging 7.0 times, a result in keeping with prior simultaneous \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C $$\end{document}C and urine \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M $$\end{document}M collection studies and exhibiting a rate of apparent volume growth of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_d$$\end{document}Vd that achieved limiting constant values. A simulated constant average drug mass multidosing protocol exhibited increased \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_d$$\end{document}Vd and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{1/2}$$\end{document}t1/2 with elapsing time, effects that have been observed experimentally during same-dose multidosing. The GPC heavy-tailed models explained multiple documented phenomena that were unexplained with lighter-tailed models. Electronic supplementary material The online version of this article (10.1007/s10928-019-09666-z) contains supplementary material, which is available to authorized users.

3. In construction 1.24., definition (9'), ff~(O,ab-1)should replaced by ff~(O,ba-1).~ However, (9') does not suffice be to make Q(m) into a right topological semigroup. It has been overlooked that for b +0, and i = 2m + k + I n ~(O,O)+ ~i(O,O).~ In $~$~.l(O,a).K(O,bna-1) converges to order to get a right topological semigroup, we have to = ~(O,O) and o' = ~(0,0). The resulting semiidentify o group is also semitopological, but its underlying topological space is not a surface. The statements about the ideals of Q(m) should be altered accordingly. The statement about Q(O) is correct.

LetA be aC*-algebra on the separable Hilbert space ℋ, and let ℛ be the von Neumann generated byA. LetG be a topological group anda→φ(a) a representation ofG into the group of *-automorphisms ofA. Suppose that each φ(a) extends to a *-automorphism of ℛ, and suppose thata→〈φ(a)(T)x, y〉 is continuous for eachT inA andx, y and ℋ. Then, for a large class of groupsG, one has automatically thata→〈φ(a)(T)x,y〉 is continuous for allT in ℛ andx, y in ℋ.

DR. WATSON has shown in his letter to NATURE of March 31 (p. 512) how the general methods of Maxwell and Boltzmann may be applied to the particular problem discussed by Prof. Burnside. He has also pointed out an error in Burnside's reasoning—namely, the non-introduction of the factor u − U + c?, whereby Burnside's conclusions at variance with the Maxwell-Boltzmann law of partition of energy are vitiated.

AN easy solution of the paper-band puzzle described by Prof. C. V. Boys in NATURE of June 9, p. 774, is obtained as follows: Hold the hand with thumb up and palm towards you; place the paper band over the index finger, letting the ends hang down. Observe which way the original four half-twists were applied. Treat the nearest of these to the index finger on the palm side of the hand as if it were that of an ordinary single half-twist band; which complete, by looping up one-half of the band over the finger (the other twists being pushed out of the way into the remaining half). Then apply the surfaces one upon another at the finger; and turn the other half of the band inside out so as to get rid of two of the twists. It will be found to fit exactly upon the first half, as required.

The right hand side of Equation (3.10) of this paper should read g(x, Uo)NAlb(x) Uo rather than g(x, Uo)+NAlb(X)Uo. This error in sign is sufficient to invalidate partially the results claimed. With the corrected sign the right hand side of (3.10) is always positive if (i) N = 0 or if (ii) g(x, u )>NAlb (x ) u for u > 0 for some positive integer N > 0. In the first case our analysis demonstrates existence of a positive solution under hypotheses H-1 to H-5 for all 2 in 0 NAt b(x) u for u > 0 for some integer N > 0 ; we then conclude that a positive solution exists for all 2 in the internal 0__ O. The author wishes to point out that for the problem (1.1), (1.2) of nonlinear reactor dynamics the existence of a solution for all 2 > 0 follows from the work of NORMAN LEVJNSON (Dirichlet Problem for A u=f(P, u), J. Math. Mech. 12, 567-575 (1963)). We have not yet been able to establish that a positive solution must exist for 2>A~. An unpublished theorem of H. B. KELLER shows that positive solutions are unique for 2 > 0. Thus, on the basis of the corrected results we know that there exists a unique positive solution for all 2 in 0 A~ still remains open, though we conjecture that it must be valid.

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.

Let $$p$$ p and $$q$$ q be two odd primes with $$p=Mf+1$$ p = Mf + 1 and $$M$$ M is even. A new construction of $$M$$ M -ary sequences of period $$pq$$ pq with low periodic autocorrelation is presented in this paper based on interleaving the $$M$$ M -ary power residue sequence of period $$p$$ p according to the quadratic residue with respect to $$q$$ q . This construction can generate the well-known twin-prime sequence and generalized cyclotomy sequence of order two if $$M=2$$ M = 2 . For $$M=4$$ M = 4 , a new class of quaternary sequences of period $$pq$$ pq with maximal nontrivial autocorrelation value being either $$\sqrt{5}$$ 5 or $$3$$ 3 is obtained. This achieves the best known results for such kind of quaternary sequences.

A new tool of the classification of DNA sequences is introduced. The method is based on 2D-dynamic graphs and their descriptors. Using the descriptors created by centers of masses, moments of inertia, angles between the x axis and the principal axis of inertia of the 2D-dynamic graphs one can obtain classification diagrams in which similar sequences are clustered in separated areas.

In this short paper, we will prove the following Theorem: Let \(\mathcal {F}\) be a codimension-one foliation of a complete and connected Riemannian manifold \(M^{n+1}\) with constant sectional curvature \(c {\leq} 0\). Suppose that \(\mathcal {F} \) is transversely orientable, i.e., there exists an unit vector field \(N \epsilon {\mathfrak{X}}(M)\) such that N is normal to the leaves of \(\mathcal {F} \). Suppose that N is geodesic. Then, if the foliation \(\mathcal {F} \) has an umbilical leaf, every leaf of foliation \(\mathcal {F} \) must be umbilical.

In this short paper, we first introduce the concept ofZ-mappings under which the image of aZ-continuous poset (respectivelyZ-algebraic poset,Z-inductive poset) is still aZ-continuous poset (respectivelyZ-algebraic poset,Z-inductive poset). We then give a classification theorem ofZ-continuous posets which generalizes an earlier work of R.-E. Hoffmann in [3].

IN the recent article by Prof. Robert B. Grieves and Dibakar Bhattacharyya1, Fig. 2 shows a relation between two variables with the statement “… The residual DSS concentrations are related in Fig. 2 to the ratio of DSS to trivalent iron in the feed. As xi/zi was increased, xr remained relatively constant and then rose sharply above a feed ratio of approximately 1.0.”. In view of the close attention given to statistical aspects of data analysis in other parts of the article, the statement about the data of Fig. 2 appeared somewhat questionable. Accordingly, I read off the numbers from the graph as best I could with a piece of superimposed graph paper and replotted the data on log–log co-ordinates. The data suggested a relation of the form xr=√(2xi/zi), so I prepared a graph as shown in Fig. 1. Within the variability of the data, it does not appear to me that the data can be considered approximately constant below xi/zi of 1, but rather that a regular increase is observed over the full range of the values. Certainly the variability of the data does not warrant a definite conclusion on the relationship.

A NEW volume of Proceedings of the Bologna Mathematical Congress begins with a short paper by Max Bruckner, on the old problem of how many different polyhedra are possible of n sides—with the limiting condition that all the corners shall be trihedral. In my book on “Growth and Form” I dealt with the kindred problem of the possible number of arrangements of a plane assemblage of cells, their partition-walls all meeting three-by-three, as is actually the case in a system of soap-bubbles or of living cells. The problem of the polyhedra is just as interesting to the biologist; for any natural clump of cells, such as a totally segmented egg or ‘morula’, may be looked on as a polyhedron and may be studied accordingly.

A combinatorial polytope $P$ is said to be projectively unique if it has a single realization up to projective transformations. Projective uniqueness is a geometrically compelling property but is difficult to verify. In this paper, we merge two approaches to projective uniqueness in the literature. One is primarily geometric and is due to McMullen, who showed that certain natural operations on polytopes preserve projective uniqueness. The other is more algebraic and is due to Gouveia, Macchia, Thomas, and Wiebe. They use certain ideals associated to a polytope to verify a property called graphicality that implies projective uniqueness. In this paper, we show that that McMullen's operations preserve not only projective uniquness but also graphicality. As an application, we show that large families of order polytopes are graphic and thus projectively unique.

The Weighted Tree Augmentation problem (WTAP) is a fundamental problem in network design. In this paper, we consider this problem in the online setting. We are given an n-vertex spanning tree T and an additional set L of edges (called links) with costs. Then, terminal pairs arrive one-by-one and our task is to maintain a low-cost subset of links F such that every terminal pair that has arrived so far is 2-edge-connected in T cup F. This online problem was first studied by Gupta, Krishnaswamy and Ravi (SICOMP 2012) who used it as a subroutine for the online survivable network design problem. They gave a deterministic O(log^2 n)-competitive algorithm and showed an Omega(log n) lower bound on the competitive ratio of randomized algorithms. The case when T is a path is also interesting: it is exactly the online interval set cover problem, which also captures as a special case the parking permit problem studied by Meyerson (FOCS 2005). The contribution of this paper is to give tight results for online weighted tree and path augmentation problems. The main result of this work is a deterministic O(log n)-competitive algorithm for online WTAP, which is tight up to constant factors.

Consider an oriented curve $\Gamma$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding in the Euclidean space $\boldsymbol R^3$. This can be expressed as the image of an "origami map" $\Phi:D\to \boldsymbol R^3$ such that $\Gamma$ is the singular set of $\Phi$, the word "origami" coming from the Japanese term for paper folding. We call the singular set image $C:=\Phi(\Gamma)$ the crease of $\Phi$ and the singular set $\Gamma$ the crease pattern of $\Phi$. We are interested in the number of origami maps whose creases and crease patterns are $C$ and $\Gamma$, respectively. Two such possibilities have been known. In the authors' previous work, two other new possibilities and an explicit example with four such non-congruent distinct curved foldings were established. In this paper, we determine the possibility of the number $N$ of congruence classes of curved foldings with the same crease and crease pattern. As a consequence, if $C$ is a non-closed simple arc, then $N=4$ if and only if both $\Gamma$ and $C$ do not admit any symmetries. On the other hand, when $C$ is a closed curve, there are infinitely many distinct possibilities for curved foldings with the same crease and crease pattern, in general., Comment: 29 pages, 7 figures

In this paper we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group $R$ has a nontrivial proper normal subgroup $N$ with the property that the automorphism group of the digraph fixes each $N$-coset setwise, and the cases where it does not. In this paper, we deal with undirected graphs in the case where the regular group has such a nontrivial proper normal subgroup., Comment: 27 pages

The partition crank is a statistic on partitions introduced by Freeman Dyson to explain Ramanujan's congruences. In this paper, we prove that the crank is asymptotically equidistributed modulo Q, for any odd number Q. To prove this, we obtain effective bounds on the error term from Zapata Rolon's asymptotic estimate for the crank function. We then use those bounds to prove the surjectivity and strict log-subadditivity of the crank function., 14 pages 1) Some clarifications and corrections have been made. 2) This paper will appear in Ramanujan Journal

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

Let $$\alpha = (1+\sqrt{5})/2$$ and define the lower and upper Wythoff sequences by $$a_i = \lfloor i \alpha \rfloor $$ , $$b_i = \lfloor i \alpha ^2 \rfloor $$ for $$i \ge 1$$ . In a recent interesting paper, Kawsumarng et al. proved a number of results about numbers representable as sums of the form $$a_i + a_j$$ , $$b_i + b_j$$ , $$a_i + b_j$$ , and so forth. In this paper I show how to derive all of their results, using one simple idea and existing free software called Walnut. The key idea is that for each of their sumsets, there is a relatively small automaton accepting the Fibonacci representation of the numbers represented. I also show how the automaton approach can easily prove other results.

This paper deals with a class of singular problems involving the fractional $$p(x,\cdot )$$ -Laplace operator of the form $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}_{p(x,\cdot )}u(x)= \frac{\lambda }{u^{\gamma (x)}}+u^{q(x)-1} &{} \hbox {in }\Omega , \\ u>0, \;\;\text {in}\;\; \Omega &{} \hbox {} \\ u=0 \;\;\text {on}\;\;{\mathbb {R}}^N\setminus \Omega , &{} \hbox {} \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a smooth bounded domain in $${\mathbb {R}}^N$$ ( $$N\ge 3$$ ), $$00$$ small enough. To our best knowledge, this paper is one of the first attempts in the study of singular problems involving fractional $$p(x,\cdot )$$ -Laplace operators.

It is known that the generating function f of a sequence of Toeplitz matrices {Tn(f)}n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f) are real for all n, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol $\mathfrak {f}$ f , appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of Tn(f). This eigenvalue symbol $\mathfrak {f}$ f is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function $\mathfrak {f}$ f . The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor $\mathfrak {f}$ f is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of $\mathfrak {f}$ f . Future research directions are outlined at the end of the paper.

Slice Fueter-regular functions, originally called slice Dirac-regular functions, are generalized holomorphic functions defined over the octonion algebra $\mathbb{O}$, recently introduced by M. Jin, G. Ren and I. Sabadini. A function $f:\Omega_D\subset\mathbb{O}\to\mathbb{O}$ is called (quaternionic) slice Fueter-regular if, given any quaternionic subalgebra $\mathbb{H}_\mathbb{I}$ of $\mathbb{O}$ generated by a pair $\mathbb{I}=(I,J)$ of orthogonal imaginary units $I$ and $J$ ($\mathbb{H}_\mathbb{I}$ is a `quaternionic slice' of $\mathbb{O}$), the restriction of $f$ to $\Omega_D\cap\mathbb{H}_\mathbb{I}$ belongs to the kernel of the corresponding Cauchy-Riemann-Fueter operator $\frac{\partial}{\partial x_0}+I\frac{\partial}{\partial x_1}+J\frac{\partial}{\partial x_2}+(IJ)\frac{\partial}{\partial x_3}$. The goal of this paper is to show that slice Fueter-regular functions are standard (complex) slice functions, whose stem functions satisfy a Vekua system having exactly the same form of the one characterizing axially monogenic functions of degree zero. The mentioned standard sliceness of slice Fueter-regular functions is able to reveal their `holomorphic nature': slice Fueter-regular functions have Cauchy integral formulas, Taylor and Laurent series expansions, and a version of Maximum Modulus Principle, and each of these properties is global in the sense that it is true on genuine $8$-dimesional domains of $\mathbb{O}$. Slice Fueter-regular functions are real analytic. Furthermore, we introduce the global concepts of spherical Dirac operator $\Gamma$ and of slice Fueter operator $\bar{\vartheta}_F$ over octonions, which allow to characterize slice Fueter-regular functions as the $\mathscr{C}^2$-functions in the kernel of $\bar{\vartheta}_F$ satisfying a second order differential system associated with $\Gamma$. The paper contains eight open problems., Comment: 33 pages

The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics over $${\mathbb {C}}$$ C and $$\mathbb {R}$$ R in 1906–1907. The analogous problem for finite fields $$\mathbb {F}_q$$ F q with q odd was solved by Dickson in 1908. In 1914, Wilson attempted to classify nets (two-dimensional systems) of conics over finite fields of odd characteristic, but his classification was incomplete and contained some inaccuracies. In a recent article, we completed Wilson’s classification (for q odd) of nets of rank one, namely those containing a repeated line. The aim of the present paper is to introduce and calculate certain combinatorial invariants of these nets, which we expect will be of use in various applications. Our approach is geometric in the sense that we view a net of rank one as a plane in $$\mathrm {PG}(5,q)$$ PG ( 5 , q ) , q odd, that meets the quadric Veronesean in at least one point; two such nets are then equivalent if and only if the corresponding planes belong to the same orbit under the induced action of $$\mathrm {PGL}(3,q)$$ PGL ( 3 , q ) viewed as a subgroup of $$\mathrm {PGL}(6,q)$$ PGL ( 6 , q ) . Since q is odd, the orbits of lines in $$\mathrm {PG}(5,q)$$ PG ( 5 , q ) under this action correspond to the aforementioned pencils of conics in $$\mathrm {PG}(2,q)$$ PG ( 2 , q ) . The main contribution of this paper is to determine the line-orbit distribution of a plane $$\pi $$ π corresponding to a net of rank one, namely, the number of lines in $$\pi $$ π belonging to each line orbit. It turns out that this list of invariants completely determines the orbit of $$\pi $$ π , and we will use this fact in forthcoming work to develop an efficient algorithm for calculating the orbit of a given net of rank one. As a more immediate application, we also determine the stabilisers of nets of rank one in $$\mathrm {PGL}(3,q)$$ PGL ( 3 , q ) , and hence the orbit sizes.

Xiao, Zeng, Li and Helleseth proposed new generalized cyclotomic binary sequences $$s^{\infty }$$ of period $$p^m$$ and showed that these sequences are almost balanced and have very large linear complexity if p is a non-Wieferich prime and $$m=2$$ . Wu, Xu, Chen and Ke determined the values of the k-error linear complexity for $$m=2$$ in terms of the theory of Fermat quotients and the results indicated that sequences $$s^{\infty }$$ have good stability. Edemskiy, Li, Zeng and Helleseth studied the linear complexity of $$s^{\infty }$$ for general integers $$m\ge 2$$ . Furthermore, Ouyang and Xie constructed new $$2p^{m}$$ -periodic binary sequences $${\widehat{s}}^{\infty }$$ and $${\widetilde{s}}^{\infty }$$ and proved that the sequences $${\widehat{s}}^{\infty }$$ and $${\widetilde{s}}^{\infty }$$ are of high linear complexity when $$m\ge 2$$ . In this paper we shall show that despite a high linear complexity the sequences $$s^{\infty }$$ , $${\widehat{s}}^{\infty }$$ and $${\widetilde{s}}^{\infty }$$ have some undesirable features which may not suggest them for cryptography. The properties of multiplicative character sums modulo $$p^m$$ play an important role in the proof of this paper.

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

This paper investigates the construction of $${\text {MDS}}$$ matrices with generalized Feistel structures ( $${\text {GFS}}$$ ). The approach developed by this paper consists in deriving $${\text {MDS}}$$ matrices from the product of several sparser matrices. This can be seen as a generalization to several matrices of the recursive construction which derives $${\text {MDS}}$$ matrices as the powers of a single companion matrix. In other words, the idea of this paper is to explore a space of matrices with a $${\text {GFS}}$$ structure, and then to search for instantiations of the binary linear functions so that the resulting matrix is both $${\text {MDS}}$$ and efficient to implement with respect to the number of $${\text {XOR}}$$ gates and the depth of the circuit. In this direction we first give some theoretical results on the iteration of $${\text {GFS}}$$ . We then using $${\text {GFS}}$$ with minimal diffusion rounds, propose some types of sparse matrices that are called extended primitive $${\text {GFS}}$$ ( $${\text {EGFS}}$$ ) matrices. Next, by applying binary linear functions to several round of $${\text {EGFS}}$$ matrices, we introduce lightweight $$4\times 4$$ , $$6\times 6$$ and $$8\times 8$$ $${\text {MDS}}$$ matrices that are implemented with 67, 156 and 260 $${\text {XOR}}$$ over 8-bit input, respectively. The results match the best known lightweight $$4\times 4$$ $${\text {MDS}}$$ matrix and improve the best known $$6\times 6$$ and $$8\times 8$$ $${\text {MDS}}$$ matrices. Moreover, we propose $$8\times 8$$ Near- $${\text {MDS}}$$ matrices such that the implementation cost of the proposed matrices are 108 and 204 $${\text {XOR}}$$ over 4-bit and 8-bit inputs, respectively. On the whole, the construction presented in this paper is relatively general and can be applied for other matrix dimensions and finite fields as well.

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

The concept of antimagic labelings of a graph is to produce distinct vertex sums by labeling edges through consecutive numbers starting from one. A long-standing conjecture is that every connected graph, except a single edge, is antimagic. Some graphs are known to be antimagic, but little has been known about sparse graphs, not even trees. This paper studies a weak version called $k$-shifted-antimagic labelings which allow the consecutive numbers starting from $k+1$, instead of starting from 1, where $k$ can be any integer. This paper establishes connections among various concepts proposed in the literature of antimagic labelings and extends previous results in three aspects: $\bullet$ Some classes of graphs, including trees and graphs whose vertices are of odd degrees, which have not been verified to be antimagic are shown to be $k$-shifted-antimagic for sufficiently large $k$. $\bullet$ Some graphs are proved $k$-shifted-antimagic for all $k$, while some are proved not for some particular $k$. $\bullet$ Disconnected graphs are also considered.

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

Given a tournament $T$, a module of $T$ is a subset $M$ of $V(T)$ such that for $x, y\in M$ and $v\in V(T)\setminus M$, $(v,x)\in A(T)$ if and only if $(v,y)\in A(T)$. The trivial modules of $T$ are $\emptyset$, $\{u\}$ $(u\in V(T))$ and $V(T)$. The tournament $T$ is indecomposable if all its modules are trivial; otherwise it is decomposable. Let $T$ be a tournament with at least five vertices. In a previous paper, the authors proved that the smallest number $\delta(T)$ of arcs that must be reversed to make $T$ indecomposable satisfies $\delta(T) \leq \left\lceil \frac{v(T)+1}{4} \right\rceil$, and this bound is sharp, where $v(T) = |V(T)|$ is the order of $T$. In this paper, we prove that if the tournament $T$ is not transitive of even order, then $T$ can be made indecomposable by reversing the arcs of a subtournament of $T$. We denote by $\delta'(T)$ the smallest size of such a subtournament. We also prove that $\delta(T) = \left\lceil \frac{\delta'(T)}{2} \right\rceil$., Comment: 15 pages