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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.

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.

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

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.

Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We show that if there exists a continuous linear unbounded operator between l-K\"{o}the spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding K\"{o}the matrices when every continuous linear operator between l-K\"{o}the spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-K\"{o}the spaces, under a splitting condition, causes the existence of a common basic subspace.

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.

This expository paper presents known results on distribution functions g(x) of the sequence of blocks where xn is an increasing sequence of positive integers. Also presents results of the set G(Xn) of all distribution functions g(x). Specially: - continuity of g(x); - connectivity of G(Xn); - singleton of G(Xn); - one-step g(x); - uniform distribution of Xn, n = 1, 2, . . . ; - lower and upper bounds of g(x); - applications to bounds of ; - many examples, e.g., , where pn is the nth prime, is uniformly distributed. The present results have been published by 25 papers of several authors between 2001-2013.

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.

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.

Let G = (V,E) be a graph and k be a positive integer. A k-dominating set of G is a subset S ⊆ V such that each vertex in V \S has atleast k neighbors in S. A Roman k-dominating function on G is a functionf : V → {0,1,2} such that every vertex v with f(v) = 0 is adjacent toat least k vertices v 1 ,v 2 ,...,v k with f(v i ) = 2 for i = 1,2,...,k. In thepaper titled “Roman k-domination in graphs” (J. Korean Math. Soc. 46(2009), no. 6, 1309–1318) K. Kammerling and L. Volkmann showed thatfor any graph G with n vertices, γ kR (G) + γ kR (G) ≥ min {2n,4k + 1},and the equality holds if and only if n ≤ 2k or k ≥ 2 and n = 2k + 1or k = 1 and G or G has a vertex of degree n − 1 and its complementhas a vertex of degree n − 2. In this paper we find a counterexampleof Kammerling and Volkmann’s result and then give a correction to theresult. 1. IntroductionLet G = (V,E) be agraphwith vertexset V = V (G) andedge set E = E(G).A k-dominatingsetof G is a subset S ⊆ V such that every vertex in V \S has atleast k neighbors in S. The k-domination numberγ

In Variations on a theme in paper folding, Burkard Polster [6] discusses an itera tive construction due to Hilton and Pedersen [3] for approximating any acute rational angle (rational multiple of n radians) to any desired accuracy by folding a strip of paper. Hilton and Pedersen, motivated by the construction of regular polygons and star-polygons, focus on the construction of rational angles. The construction employs a periodic sequence composed of two types of folding moves, guided by number theoretic properties of the angle's fractional representation (in fact, analysis of the construction yields new results in number theory?see [4], [1], and [2]). This note generalizes this construction to any acute angle, rational or irrational. For irrational angles, the number-theoretic prescription must obviously be replaced by something new; we show how to derive the appropriate aperiodic sequence of folding moves, via a connection between the folding construction and a simple dynamical system. Hilton and Pedersen's construction recursively generates a sequence of angles a0, otx, ... by a pair of folding operations on a strip of paper which we term Folds and Switchfolds. The angles ak appear between an edge of the strip and a crease created by folding. The Fold operation generates ak+x by bisecting ak (Figure 1)

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.

A new property of a complete intersection in a graded Noetherian ring is proved; this result, in conjunction with the Jacobian criterion, is then applied to the description of the loci of singularities in a family of affine varieties introduced in a well-known paper on forms of many variables.

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

Fine and Wilf's theorem has recently been extended to words having three periods. Following the method of the authors we extend it to an arbitrary number of periods and deduce from that a characterization of generalized Arnoux-Rauzy sequences or episturmian infinite words.

Let c e IN. For d e ℤ with gcd (d, c) = 1 let δ(d, c) be defined by d · δ(d, c) 1 mod c, 1 ⩽ δ(d, c) ⩽ c. Let s, t e IN with 1 ⩽ s ⩽ c, 1 ⩽ t ⩽ c. The main result is that for arbitrary fixed e > 0, but uniformly over c, s and t Equivalently, the points (d, δ(c,d)) are approximately uniformly distributed in [0, c] × [0, c]: The two-dimensional discrepancy of {(d,δ (c, d)): 1 ⩽ d ⩽ c and (c, d)= 1} in [0,c] × [0,c] is Oe(ce-1/2).

As early as the 16th century, Simon Jacob, a German reckoning master, noticed that the worst case in computing the greatest common divisor of two numbers by the Euclidean algorithm occurs if these numbers are equimultiples of two consecutive members of the Fibonacci sequence.

In the second paragraph on page 30 [ 1 ], we wrote ‘Making a critical point precritical decreases the number of points about which we have to worry’. Indeed, we stopped worrying about these points so much, that we forgot to mention them in several statements. This includes Lemma 3.13, Theorem 3.14 and the Main Theorem. Thus, the possibility of critical points being precritical (not only sinking or recurrent) has to be added to those statements. In particular, the Main Theorem should read as follows.

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.

In this note we point out that a simple proof of the lower bound of the sets (b, c), and so also of Ξ(b, c), defined in the previous paper [1] can be obtained as a simple application of a general method. By Example 4.6 from [2], if [0, 1] = E0⊃E1⊃ … are sets each of which is a finite union of disjoint closed intervals such that each interval of Ek−1, contains at least mk intervals of Ek which are separated by gaps of lengths at least ek, and if mk≥2 and ek≥ek+1>0, then the dimension of the intersection of Ek is at least

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.

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

iF(y)=( where G c R* is a bounded domain, y: G -» R , y'(x) = (dyydx), and F: M -> R is continuous. Here M denotes the linear space of real N X k matrices. We suppose throughout that K > 2, N > 2. In [7] F is called T-conυex if there exists a convex function /, defined on R, r = (t) ~ 1, such that F(p) = f(τ(p)) for all/? e M, where τ(p) denotes the minors of p of all orders j , 1 y uniformly on G with supx χGG\yj(x) ~ yj(x)\ < C < oo for ally. (Equivalently, if G has sufficiently regular boundary then IF is lsc if and only if IF is sequentially weak* lower semicontinuous on the Sobolev space W(G; R).) A consequence of [7? Theorem 3.6] is that IF lsc implies F polyconvex; that this conclusion is false was pointed out implicitly by Morrey [4, p. 26]. Morrey's remark is based on an example due to Terpstra [8] of a quadratic form

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.

R. A. Brualdi [1] presents a construction yielding matrices whose Birkhoff representation consists of the maximum number of permutation matrices and having 0(2n2) line sum. In this note a counterexample to such a construction is given. Furthermore, a new construction is presented, yielding matrices with lower line sums.

By 350 B.C. the Greeks knew Euclidean constructions for the regular polygons of 4, 8, 16,... sides and for those of 3 and 5 sides-the equilateral triangle and the regular pentagon. From these it was easy to construct regular polygons of 2c x 3, 2c x 5, 2c x 3 x 5 sides, where c is any positive integer, and the Greeks in effect showed how. They got no farther. Young Gauss proved that if N is of the form 2c or 2c times a product of different Fermat primes F,,, then there is a Eucidean construction for a regular polygon of N sides. This form of N is both necessary and sufficient for the possibility of a Eucidean construction....

The object of this note is to point out and correct an error in the paper [21 of Fox, purporting to prove Fenchel's conjecture that a finitely generated. infinite Fuchsian group has a torsion-free normal subgroup of finite index. Fox divided the proof of the main result in his paper into four cases and an error occurs in the proofs of Cases (III) and (IV). A direct proof of Case (III) was given. While a direct proof of Case (IV) can be given, the author shows that it also follows indirectly from the result in the paper [1] of Burns and Solitar. Given any three integers a > 1, b > 1 and c > 1, there exist permutations A of order a and B of order b, such that AB has order c. Fox divided the proof of this lemma into the four cases: (I) ab 0 (mod 2) and max{a, b} ? c < a + b, (II) ab 0 (mod 2) and c : a + b, (III) a b c 1 (mod 2), (IV) a b 1 and c 0 (mod 2), and subcases. In the subcase m(a + b 2) + 2 ? c < (m + l)(b + 1) 1 of Case (III) where m is the largest integer not exceeding (c l)/(b + 1), and it is assumed that a ? b < c, the number of distinct symbols in the permutations [2, p. 64] nl~~~W B ( Wv-sI .. WI2 I7 (vWb B = (vl * *t*v W_ ***W P2) * (V W2 .. * ]Pi+]1)'

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