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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 $r(3,s)$.

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.

We give common generalizations of the Menon-type identities by Sivaramakrishnan (1969) and Li, Kim, Qiao (2019). Our general identities involve arithmetic functions of several variables, and also contain, as special cases, identities for gcd-sum type functions. We point out a new Menon-type identity concerning the lcm function. We present a simple character free approach for the proof., Comment: 14 pages

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

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)

A record mentioned in the paper by Moree (Abh Math Sem Univ Hamburg 77:109–127, 2007) was recently improved on by Akbary and Scholten. However, the record mentioned was not the then record. The then record, due to Gallot (2004), actually slightly improves on that obtained recently by Akbary and Scholten.

In this note, we slightly generalize Theorem 2 in the paper by M. Tunç and point out that the assumption of Theorem 3 is not sufficient. A misuse of the term 'mean' is also discussed.

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.

This note is related to a nice short paper of M. Ochiai. We prove in a very fast way that the two-parameter family of Heegaard diagrams, constructed by Ochiai, encodes the genuine three-sphere. The result is obtained, up to isotopy, by using a sequence of only three moves in this order: a Whitehead–Zieschang reduction, a band sum and a cancellation of a handle.

We generalize and give a simple proof of a Theorem of Blasco. For a topological space X , C(X) denotes the algebra of real valued continuous functions on X under the operation (αf + gh)(x) := αf(x) + h(x)g(x) for each f, g, h ∈ C(X) and α ∈ R. An algebra A on X means a subalgebra of C(X) which contains the constant functions. By a ∗-algebra A on X we mean an algebra on X and all bounded functions in A separate points from the closed sets. Let A be an algebra on X and π : A −→ R be a (algebra) homomorphism and α be a cardinal number. π is said to be α-evaluating if for each subset B of A with cardinality at most α there exits xB ∈ X such that π(f) = f(xB) for each f ∈ B. If α = |N|(=the cardinal number of the set of the natural numbers N), the α-evaluating homomorphism is called countable evaluating. Let α be a cardinal number. Recall that a topological space X is called αLindelof if for each open cover U has a subcover V such that the cardinality of V is at most α. A lindelof space means a |N|-Lindelof space. The following theorem is given in [1] as one of the main results. Theorem 1. Let X be a completely regular Hausdorff space. Then X is Lindelof if and only if each countable evaluating homomorphism π on a ∗-algebra A on X is point evaluating. Received June 9, 2004; accepted June 30, 2004. Communicated by Bor-Luh Lin. 2000 Mathematics Subject Classification: Primary 54C35, 46E25.

Most of the known continued fraction expansions of special functions are limit periodic. This means that the classical approximants S n (0) are normally not the best ones to use for approximations. In this paper we suggest a number of approximants S n (w n ) which converge faster. The estimation of the improvement and bounds for the error |f - Sn(wn)| (which we still call the truncation error) are mainly obtained by means of Thron's parabola sequence theorem and the oval sequence theorem.

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 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.

The purpose of this note is to present a new proof of a generalized form of Day's fixed point theorem. The proof we give is suggested by the work of T. Mitchell in his paper, Function algebras, means, and fixed points, [2]. The version of Day's theorem which we present here has not appeared explicitly in the literature before, and seems especially well suited for application to questions concerning fixed point properties of topological semigroups. 1. Preliminaries. We adopt the terminology and notation of [2] except where otherwise specified. New terminology will be introduced as needed. Let y be a convex compactum (compact convex set in a real locally convex linear topological space E), and let A( Y) denote the Banach space of all (real) continuous affine functions on Y under the supremum norm. Observe that A(Y) contains every function of the form h=f\Y + r where fe E* and r is real; thus A(Y) separates points of Y.

The circumstances explained in the footnote on p. 61 of the former paper with this title might well have necessitated a re-writing of the whole. Fortunately it appears that only a few short comments are required. For example it may be noted that the first question in §4 can be answered by counting the degrees of freedom in the two configurations. Eight points of a twisted cubic have freedom 20; four pairs of planes drawn at random through four lines of a regulus have freedom 21; therefore the eight planes of the second paragraph of §4 cannot always lead back to the eight points of §2. This is corroborated by the corresponding numbers in [4], which are 31 and 34.

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

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....

Let G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice $$\mathcal {L}(H)$$ L ( H ) embeds in $$\mathcal {L}(G)$$ L ( G ) . It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.

Let M0 be the set of measures μ on the real line such that open sets are μ*-immeasurable. While attempting to find out whether a set μ*-measurable for all μ in Mo is mapped into a similar set by a continuous function of bounded variation, Maurice Sion develops a theory for what he calls variational measure (4). As an application of the theory, he gets conditions on a function f and a set of measures M in order that f map a set, which is μ*-measurable for all μ ∈ M, into a set of the same kind. In particular he proves for his class M2 (def. 2.5), the following theorem (4, § 8.11).

We take a measurable set E on the positive η-axis and denote by μ(r) the linear measure of the part of E in the interval 0 < η < r. The lower density of E at η = 0 is defined byTheorem by Kawakami [1] asserts that if λ is positive, if a function f(ζ) = f(ξ + iη) is bounded analytic in ξ > 0 and continuous at E, and if f(ζ) → A as ζ → 0 along E, then f(ζ) → A as ζ → 0 in ∣η∣ ≦ kξ for any k > 0.

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)'

(1973). An Addendum to the Paper “A Characterization of the n × n Matrices Over a Finite Field”. The American Mathematical Monthly: Vol. 80, No. 9, pp. 1041-1043.

In [4], Phelps showed that in certain function algebras the unit ball is the closed convex hull of its extreme points. The algebra, C(X), of complex valued continuous functions on a compact Hausdorff space, will always have this property. The class of logmodular algebras which have a Gleason part which is total also was shown to have the property. In this paper we give an elementary proof of the first result (a proof which is, in theory at least, constructive). The simplest nontrivial example of a logmodular algebra with a total part is the disk algebra (i.e. the functions continuous on the closed disk and analytic in the interior). For this algebra we show that the extreme points (in fact the exposed points) of the unit ball form a dense subset of the boundary of the unit ball. Let U be the unit ball of C(X). It is well known that q in U is an extreme point of U if I q(x) I = 1 for all x in X. Now if f in U never vanishes, then f is in the closed convex hull of the extreme points; in factf is between two uniquely determined extreme points. We need only observe that for each x in X,f(x) is halfway between two uniquely determined extreme points of the disk and that these points vary continuously with x. Now it is not necessarily the case that each function in U can be approximated by a nonvanishing function. For a counterexample we need only look at h(z) = z in the algebra of continuous functions on the disk. (If If(z)-zj 1/e. Letfi, i = 1, *, N be the retract maps obtained from these rotations. Then I 1/N Efi(z)-z

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

In a recent paperf entitled A note on the linear fractional group, Todd obtained an abstract definition for the group LF(2, 2) in terms of n+2 generators. Apparently he gave no consideration to the question of the independence of the defining relations, for they can be considerably simplified. First, in view of the condition RSi = Si+iR (which is the same as i£S0^~* = Si), the three generators U, R and S0 are sufficient to generate the entire group. If we give a definition in teims of these three generators alone, the relations

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

This note deals with the correction of a small flaw in the proof of a theorem (and a lemma) in [2]. Also a simpler proof of the theorem in question is given. In the proof of Theorem 2.3 in [2], the statement "that distinct L's cannot belong to a common pure subgroup HL follows immediately from the fact that no two L's generate with G [p] the same subgroup of G" is false as it stands. One has to exercise a little more care in the selection of HL. For example, since the union [xa, Xa]aEA of all the L's (in Lemma 2.1 in [2]) is still linearly independent, there exists according to [1] a pure subgroup H of G supported by S such that HDL for each L. Hence one could choose HL=H for each L. However, all goes well if HL is chosen such that HLn [Xa, XTg aEA =L, as can easily be done. Similar remarks hold for Lemma 3.8 in [2]. A short alternate proof of Theorem 2.3 in [2] is given. In what follows let c be the cardinality of the continuum and Q be the first ordinal whose cardinality is c. The notation and terminology will be the same as that in [2].

In [2], Tosiro Tsuzzuku gave a proof of the following:THEOREM. Let G be a doubly transitive permutation group of degree n, let K be any commutative ring with unit element and let p be the natural representation of G by n × n permutation matrices with elements 0, 1 in K. Then ρ is decomposable as a matrix representation over K if and only ifn is an invertible element of K.For G the symmetric group this result follows from Theorems (2.1) and (4.12) of [1]. The proof given by Tsuzuku is unsatisfactory, although it is perfectly valid when K is a field. The purpose of this note is to give a correct proof of the general case.

M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.