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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).

Very recently, I. Altun, M. Aslantas and H. Sahin [1] introduced the notion of $$p$$ -proximal contractions and surveyed the existence of best proximity points for such class of non-self mappings in metric spaces. In this note we show that this existence result is a straightforward consequence of the same conclusion in fixed point theory.

We correct some mistakes in the paper “A mass formula for negacyclic codes of length 2k and some good negacyclic codes over $\mathbb {Z}_{4}+u\mathbb {Z}_{4}$ ” (Bandi et al. Cryptogr. Commun. 9, 241–272, 2017).

If $$z\in {\mathbb {C}}$$ and $$1\le n$$ is a natural number then $$\begin{aligned} \sum _{d_1 d_2 =n} (1-z^{p_1})\cdots (1-z^{p_m}) z^{q_1 e_{1}+\cdots +q_i e_{i} }=1, \end{aligned}$$ where $$d_1=p_1^{r_1}\dots p_m^{r_m }$$ , $$d_2=q_1^{e_1}\dots q_i^{e_i }$$ are the prime decompositions of $$d_1, d_2$$ . This is one of the identities involving arithmetic functions that we prove using ideas from the paper of Dressler and van de Lune [3].

The main result of this paper states that if $${\mathcal {F}}$$ is a subgroup-closed saturated formation of full characteristic, then the $${\mathcal {F}}$$ -residual of a K- $${\mathcal {F}}$$ -subnormal subgroup S of a finite group G is a large subgroup of G provided that the $${\mathcal {F}}$$ -hypercentre of every subgroup X of G containing S is contained in the $${\mathcal {F}}$$ -residual of X. This extends a recent result of Aivazidis, Safonova and Skiba.

Very recently Bordelles, Dai, Heyman, Pan and Shparlinski studied asymptotic behaviour of the quantity $$\begin{aligned} S_f(x) := \sum _{n\leqslant x} f\left( \left[ \frac{x}{n}\right] \right) , \end{aligned}$$and established some asymptotic formulas for $$S_f(x)$$ under three different types of assumptions on f. In this short note we improve some of their results.

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 this short paper, we look into a conclusion drawn by Alzalg (J Optim Theory Appl 169:32---49, 2016). We think the conclusion drawn in the paper is incorrect by pointing out three things. First, we provide a counterexample that the proposed inner product does not satisfy bilinearity. Secondly, we offer an argument why a pth-order cone cannot be self-dual under any reasonable inner product structure on $$\mathbb {R}^n$$Rn. Thirdly, even under the assumption that all elements operator commute, the inner product becomes an official inner product and the arbitrary-order cone can be shown as a symmetric cone, we think this condition is still unreasonable and very stringent so that the result can only be applied to very few cases.

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.

Let a(n) be the Fourier coefficient of a holomorphic cusp form on some discrete subgroup of \(SL_2({\mathbb R})\). This note is to refine a recent result of Hofmann and Kohnen on the non-positive (resp. non-negative) product of \(a(n)a(n+r)\) for a fixed positive integer r.

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

Directed graphs with an odd number of vertices n, where each vertex has both (n − 1)/2 incoming and outgoing edges, have a rich structure. We were lead to their study by both the Borromean rings and the game rockpaper-scissors. An interesting interplay between groups, graphs, topological links, and matrices reveals the structure of these objects, and for larger values of n, extensive computation produces some surprises. Perhaps most surprising is how few of the larger graphs have any symmetry and those with symmetry possess very little. In the final section, we dramatically sped up the computation by first computing a “profile” for each graph.

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

Problem $$1$$ of (Bulman-Fleming et al. Commun Algebra 34:1291–1317, 2006) asks: what conditions on a pomonoid $$S$$ and a convex, proper right ideal $$K$$ , guarantee that $$S/K$$ has property $$(P_w)$$ ? This question was first answered in Proposition 3.2 of (Qiao and Li Semigroup Forum 77:306–315, 2008) and later in Lemma $$5$$ of (Kilp Semigroup Forum 86:592–602, 2013). In this note we first show, by counterexample, that the result of Qiao and Li is incorrect, and then demonstrate a condition equivalent to that of Kilp.

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.

The paper [3] contains an upper bound to the weighted density of a packing of circles on the unit sphere with radii from a given finite set. This bound is attained by many packings and has applications to problems of solidity. In the present note it is shown that a certain condition imposed on the set of admissible radii can be removed by modifying the original proof of the theorem.

The authors of [KMM] consider a system of two functional inequalities for a function $$f : {\mathbb{R}} \rightarrow {\mathbb{R}}$$ , and they show that, if certain arithmetical conditions and inequalities for certain parameters are fulfilled, f has to be a polynomial provided that f is continuous at some point x0. This result is derived here under the weaker condition that for some x0 the limit $${\rm lim}_{x \rightarrow x_0} f(x)$$ exists. Moreover, another system of inequalities is given leading to the same result on the nature of f. The methods used also give natural explanations for the fact that Bernoulli numbers play an important role in this context.

It is proved that the set of those natural numbers which cannot be written as n-Ω(n) is of positive lower density. Here Ω(n) is the number of the prime power divisors of n. This is a refinement of a theorem of F. Luca.

Dr. Lerchl expresses his concern about the low variability of results from the comet assay reported in our publication (Pilger et al. 2004). He argues that the findings are suspicious for statistical as well as technical reasons. In addition, he suspects that the blinding of the experiments could have been circumvented. These allegations are mainly derived from controversies in the context of experiments with highfrequency electromagnetic fields (Lerchl and Wilhelm 2010; Rudiger and Adlkofer 2010; Rudiger 2009a, b; Lerchl 2009). However, it should be pointed out that, apart from the objections raised by Lerchl, the results of Rudiger’s group on genotoxic effects of extremely low-frequency electromagnetic fields (ELF-EMF) were unexpected in a completely different sense. In contrast to expectations, intermittent exposure to ELF-EMF was found to induce DNA strand breaks, whereas no changes in DNA fragmentation have been observed upon treatment with continuous ELF-EMF (Ivancsits et al. 2002). Recently, this phenomenon which seems to lack biological plausibility was corroborated by Focke et al. (2010) using the same cells and the same exposure system. One of the main reasons for Lerchl’s claim that the data of the comet tail factors must have been fabricated is rooted in the argument that one E-cell classification more or less would result in a much greater variation than the reported standard deviation of 0.04. Lerchl argues that the resulting range assuming a tail factor of 4.08 would be 3.985–4.175. However, this argument is fundamentally flawed. There cannot be a miraculous addition or loss of one E-cell but this cell must be balanced by the reduction or addition of a cell exhibiting another tail type. The most likely case is that the addition of one E-cell is balanced by the reduction of one D-cell and vice versa. By simple arithmetic, this results in an effect on the tail factor of 0.03 and not in the more than threefold higher figure of 0.095 given by Lerchl. It has to be borne in mind that the five tail types A to E have, under the assumption of only random variation, a multinomial distribution with a variance of each cell type count proportional to p(1 p), where p is the probability for observing a cell of that type. Under control conditions, the fraction of E-cells is very low (about 0.0015). Hence, the average number of E-cells among 1,000 cells is 1.5, and the theoretical standard deviation is about 1.22. Single counts for the data reported in this paper (Pilger et al. 2004) could not be retrieved, and only the tail factors were kept on file, but examination of records from fibroblast controls (n = 50) yielded a standard deviation of 1.25 counts for E-cells. This value is quite close to expectation. The statement by Lerchl ‘‘The effect of one E-cell more or less in one of the 5 replicates on the standard deviation of the mean value is also significant since the standard deviation would change by 0.04’’ is not correct, and it is incomprehensible how Lerchl arrived at this statement. The standard deviation—under the conditions stated by Lerchl—will change according to the following formula

It is shown that an approach earlier used by the author for classification of closed simply connected 6-manifolds (reduction to the problem of calculating certain bordism groups) can also be applied for easily obtaining the results by Barden (1965) on classification of closed simply connected 5-manifolds. Bibliography: 11 titles.

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

We give the asymptotics of the sum \(\sum\nolimits_{x \leqslant n \leqslant x + h} {f(n)}\), where \(h \gg x^{7/12 + \varepsilon }\), for the multiplicative functions \(f(n) = z^{\omega (n)} , z^{\omega (n)} |\mu (n)|, 1/d_k (n)\), and \(nd(n)\).

The size Ramsey number ˆ(F1 ,F 2) is the smallest number of edges that an (F1 ,F 2)-arrowing graph can have. Let Si,n be obtained from the star K1,n by subdividing one edge by new i − 1 vertices. Bielak [Periodica Math. Hung. 18

We prove the existence of an open and dense subset of mapsfeDiff ω ∞ (S2) which have positive topological entropy. It follows that these maps have infinitely many hyperbolic periodic points and an exponential growth rate of hyperbolic periodic points. The proof is an application of Pixton's theorem

P. Lavrenko (t~) Institute of Macromolecular Compounds Russian Academy of Sciences 199004 St. Petersburg, Russia Here, ML is the shift factor (the ratio of M to the contour length L of the macromolecule) and ~b0 is the ~b value at T = 0. q~0 is equal to 2.870 x 1023 tool 1 in the limit of non-draining random coil [4]. To apply Eq. (4) to data on draining chain it is necessary to know the 9 value which is very sensitive to the draining effect. Namely, the 9 value decreases strongly with decrease of both the number of segments L / A in the chain and the reduced diameter d/A (Fig. 1). To choose the 9 value (with the known a one), Dondos and Staikos [1] have proposed (and this is the basis of the method) the relation between the Flory parameter 4~ and the exponent a

An existence theorem is proved for closed convex surfaces whose principal radii of curvature regarded as functions of the unit normal vectorn satisfy the equation $$R_1 R_2 + \Phi (R_1 + R_2 ,R_1 ,R_2 ,n) + c \cdot n = \varphi (n),$$ wheree is an unknown constant vector being sought along with the surface. This equation has been studied by Miranda, who proved a variety of existence and uniqueness theorems about its solutions. The results obtained lead to a certain simplification in the hypotheses imposed on the functions Φ andϕ in these theorems.

The Hazen probability paper presently being used for plotting discharge probability curves makes it possible to represent the probability curve as a straight line only in the case of a normal distribution of the series. In the case of a skew distribution of the series the probability curve has a bend. In such cases a special gamma-distribution probability paper should be used for skew distributions. The use of such a graph paper permits representing the probability curve as a straight line for any values of the coefficients of variation and skewness.

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 a paper which appeared a few years ago the authors investigated the exponential representation of functions analytic in the upper half-plane with positive imaginary part there [1]. We refer to that paper in the sequel as A-D. One of the principal results of A-D, there called Theorem A, can be extended to a considerably more general result, the proof of which is perhaps simpler than that given in A-D. We give the extended version of Theorem A here. We will use the notations and results of A-D without further explanations. Before we present the extension and its proof we would like to add some information that by oversight was omitted from the list of fundamental properties of the functions in the class P given in Section 1 of A-D. In such a comprehensive review one should mention that the classical theorem on representation of a positive harmonic function in a circle by a Poisson-Stieltjes integral is due to G. Herglotz [2]. The following results of L. H. Loomis [3] were not given: XVII. For all ~ for which /t[~] = O, the limits lira Im[~b(~ + iq)] ~l~ O and lira hOg(A) exist and are f ini te simultaneously and are equal. Their ~ 0 common value is the symmetric derivative of #(A) at 2 = ~ multiplied by ~r. XVIII. I f for two values of 0 in the interval 0 < 0 < ~

Let A be a partition of the segment [0, 1] into a countable number of disjoint subsets of positive measure, let t∈L1(0,1), let Nt be the smallest rearrangement-invariant order ideal vector lattice in L1(0,1), containing t. In the paper one investigates the properties of the image E(Nt¦A) of the averaging operator with respect to A. In particular, one elucidates under what conditions there exists a function g, g∈L1(0,1), such that E(Nt¦A)⊂Ng. One formulates a generalization of the known Hardy-Littlewood inequality, namely Theorem E(t∣A)≺QE(t*∣A*), where ≺ is the Hardy-Littlewood preorder, t* and A* are the decreasing rearrangements of the function ¦t¦ and (in a special sense) of the partition A, while Q is an absolute constant, 1⩽Q⩽25. One formulates the problem of the smallest value of Q.

One gives an application of the results of the paper mentioned in the title to the problem of the range of the system of initial coefficients on the class Vk,λ(k⩾2, 0⩽λ

An error in the proof, and in the statement of a generalization, of the result that submodular setfunctions can be minimized over the subsets with odd cardinality is corrected.

A simple proof is pointed out for the asymptotic exponential decay of then-step survival probability of a random walk on a finite lattice with traps in the limit asn → ∞. Some bounds are mentioned, which are valid for finiten and for symmetric random walks.

Recently, P. D. Finch and R. Groblicki determined all bivariate probability densities with specified margins. We point out that their result follows immediately from the complete solution to the problem of determining all n-dimensional cumulative probability distribution functions with specified one-dimensional margins, which was solved by one of us in 1959.

It is proved (without the assumption of the continuum hypothesis) that under the condition there exist on the Marcinkiewicz space M(ψ) abnormal functionals which are not localizable.

For every separable Banach spaceX there is a Banach spaceY with a separable dual such thatY ⊕X* ≈Y**. There is also a separable spaceZ so thatZ**/JZ is isomorphic toX.

Dans cet article nous attirons l'attention sur un resultat du aGaunt et que voici: Si pour resoudre l'equation differetielle $$\dot x = f\left( {t,x} \right), x\left( 0 \right) = x_0 $$ on utilise la methode des differences $$\mu \delta \xi _i = h f\left( {t,\xi } \right)_i , \xi _0 = x_0 , \xi _1 $$ la valeur de ξ1 doit etre calculee d'une facon determinee, distincte dex(h). L'erreur varie alors uniformement au sens decrit ci-dessus et on peut appliquer la metamethode deRichardson. Nous donnons un enonce precis et une demonstration elegante que nous etendons ensuite au cas d'une equation differentielle de second ordre. Nous indiquons d'autres generalisations possibles et signalons une methode pratique de calcul pour ξ1.

In sections 2 and 3 two methods for proving the non existence of certain universal Banach spaces, are presented. In section 4 it is proved that every infinite-dimensional conjugate Banach space has a two-dimensional subspace whose unit cell is not a polygon.

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.