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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 $r(3,s)$.

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.

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.

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

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.

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.

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 \(nd(n)\).

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).

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 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 < ~

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.

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.

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 [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

Within the class of offspring distributions with given meanm>1 and probability of no offspringp o, the probabilityq of ultimate extinction in a Galton-Watson branching process starting from one individual satisfiesp 0

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.

IN this paper Haldane points out (p. 142) a difference between his results for the mean and variance of x2 in a 2 x n-fold contingency table when the expectation p is fixed and the results obtained by me in my paper (Annals of Eugenics, Vol. VII, part III, p. 211). The difference is that I have (n 1) throughout where Haldane has n. Haldane writes, "my own results would appear to be slightly more accurate than Cochran's", which might, I think, give the impression that both Haldane's results and mine are only approximations. In fact, both results are mathematically exact, the difference between them being one of definition of x2. My paper is almost entirely concerned with the distribution of x2 when the expectation p is not known. In the results which I gave for the distribution of x2 when p is known, I retained the term S (x -x-)2 in the numerator of x2 instead of S (x np)2, to facilitate comparison between this and my other results. Thus my x2 has (n 1) degrees of freedom, whereas Haldane's x2 has n degrees of freedom. Unfortunately I did not emphasize this point in the passage concerned, and as it may have appeared misleading to others besides Haldane, I welcome this opportunity of drawing attention to it. Haldane's x2 is, of course, the one which is normally appropriate in testing the departure from independence when the expectation is known.

F(k) ={xkj x F}; F,=F; Fk= (Fk_l,F). Let (u, v, 0)=u, (u, v, 1)=(u, v), (u, v, n)=((u, v, n-1), v). Then Sanov [3] proved that (1. 1) (u, v, apa _1) p-a E F(p#)Fapa+?, /3 a = 1, 2, where p is a prime. In this paper, (1.1) is proved for the cases a= 1, 2; 3 arbitrary. A slight generalization of these results is also proved. Sanov's proof involved an investigation of ideals in a Lie Ring. In this paper, Hall's Collection Process will be used. The method also yields other formulas, e.g. (1.2) (u, v, p2 1)P' (E F2p2_pF(p) (1.3) (u, v, pa+l 1)p~l C F2pa+l paF(pl), a = 1 2, and can be used to produce numerous formulas of a similar nature. The author hopes that some of these formulas and/or the method may be of use in solving other group-theoretic problems. The author was unable to use the method to prove (1.1) for a =3. Note that for a=f = 1, (1. 1) becomes (1.4) (U, v, p 1) E F(p)Fv+1. (1.4) plays an important role in the theory of the Restricted Burnside Problem.

C. Hobby and C. R. B. Wright [2] have just published the following Theorem A. However, their proof seems to contain an error.2 The notation of [2] is used except that G. is not reserved for the nth term of the lower central series of G: +/(G) denotes the Frattini subgroup of G; (G, H) means the group generated by the commutators g-lh-1gh where gEG, hEH; (A1, A2, * * *, A.+,) is defined inductively as ((A1, A2, . , A.), A.+,); HCG means that H is properly included in G.

I present the development of a modular origami design based upon the Pajarita, a figure from the traditional Spanish paper-folding art. The work is a modular cube decorated with Pajaritas with the ...

‡ Abstract. In this short paper we study the game of cops and robbers, which is played on the vertices of some fixed graph G. Cops and a robber are allowed to move along the edges of G, and the goal of cops is to capture the robber. The cop number cðGÞ of G is the minimum number of cops required to win the game. Meyniel conjectured a long time ago that Oð ffiffiffi p Þ cops are enough for any connected G on n vertices. Improving several previous results, we prove that the cop number of an n-vertex graph is at most n2 −ð1þoð1ÞÞ ffiffiffiffiffiffiffiffi log n p .A similar result independently and slightly before us was also obtained by Lu and Peng.

In this short paper we establish a (non-trivial) lower bound on the degree two entry h 2 of a Gorenstein h-vector of any given socle degree e and any codimension r. In particular, when e = 4, that is, for Gorenstein h-vectors of the form h = (1, r, h 2 , r, 1), our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say f(r), that h 2 may assume. In fact, we show that limr→∞ f(r) r 2/3 =6 2/3 . In general, we wonder whether our lower bound is sharp for all integers e > 4 and r > 2.

Let G be a finite nonabelian group, P ∈Sylp(G), and bcl(G) the largest length of conjugacy classes of G. In this short paper, we prove that in general and |P/Op(G)| < bcl(G) in the case where P is abelian.

A Hurwitz group is any non-trivial finite group that can be (2,3,7)-generated; that is, generated by elements $x$ and $y$ satisfying the relations $x^2 = y^3 = (xy)^7 = 1$ . In this short paper a complete answer is given to a 1965 question by John Leech, showing that the centre of a Hurwitz group can be any given finite abelian group. The proof is based on a recent theorem of Lucchini, Tamburini and Wilson, which states that the special linear group ${\rm SL}_n(q)$ is a Hurwitz group for every integer $n \geqslant 287$ and every prime-power $q$ .

Suppose each of the squares of an n x n chessboard is equipped with an indicator light and a button. If the button of a square is pressed, the light of that square will change from off to on and vice versa; the same happens to the lights of all the edge-adjacent squares. Initially all lights are off. Now, consider the following question: is it possible to press a sequence of buttons in such a way that in the end all lights are on? We will refer to this problem as the All-Ones Problem. A moment 's reflection will show that pressing a button twice has the same effect as not pressing it at all. Thus a solution to our problem can be described by a subset of all squares (namely a set of squares whose buttons when pressed in an arbitrary order will render all lights on) rather than a sequence. In fact a set X of squares is a solution to the All-Ones Problem if and only if for every square s the number of squares in X adjacent to or equal to s is odd. Consequently, we will call such a set an odd-parity cover. Trial and error in conjunction with a pad of graph paper will readily produce solutions for n ~ 4. A little more experimentation shows that an odd-parity cover shou ld one exist--is difficult to construct even for n = 5 o r 6 . The brute-force approach to the problem, namely exhaustive search over all subsets of {1 . . . . n} x {1 . . . . n}, presents 2 n2 candidates, and the search becomes infeasible for moderate values of n even with the help of a computer. A less brute-force method would be to try to solve the system terpreted as a matrix over GF(2)) and 1 is the vector with all components equal to 1. This method, which involves n 2 equations, again becomes unwieldy for small values of n. For a similar approach to a game related to the All-Ones Problem, see [3]. In any case, Figure 1 shows odd-parity covers for n = 4, 5, 8. Several questions come to mind. For which n does a solution to the All-Ones Problem exist? More generally, how many odd-parity covers are there for an n x n board? What happens if the adjacency condition is changed--say , to an octal array (where a cell in the center has eight neighbors)? Can one replace an n x n rectangular grid by some other arrangement of sites and still obtain a solution? To answer some of these questions, we first rephrase the problem in terms of cellular automata.

A well known diameter search problem for finite groups with respect to its systems of generators is considered. The problem can be formulated as follows: find the diameter of a group over its system of generators. The diameter of a group over a specific system of generators is the diameter of the corresponding Cayley graph. It is considered alternating groups with classic irreducible system of generators consisting of cycles with length three of the form $(1,2,k)$. The main part of the paper concentrates on analysis how even permutations decompose with respect to this system of generators. The rules for moving generators from permutation's decomposition from left to right and from right to left are introduced. These rules give rise for transformations of decompositions, that do not increase their lengths. They are applied for removing fixed points of a permutation, that were included in its decomposition. Based on this rule the stability of system of generators is proved. The strict growing property of the system of generators is also proved, as the corollary of transformation rules and the stability property. It is considered homogeneous theory, that was introduced in the previous author's paper. For the series of alternating groups with systems of generators mentioned above it is shown that this series is uniform and homogeneous. It makes possible to apply the homogeneous down search algorithm to compute the diameter. This algorithm is applied and exact values of diameters for alternating groups of degree up to 43 are computed.

The main objective of this paper is to construct a new sequence of operators involving Apostol-Genocchi polynomials based on certain parameters. We investigate the rate of convergence of the operators given in this paper using second-order modulus of continuity and Voronovskaja type approximation theorem. Moreover, we find weighted approximation result of the given operators. Finally, we derive the Kantorovich variant of the given operators and discussed the approximation results.

In a recent paper, Sokół et al. [Applications of k-Fibonacci numbers for the starlike analytic functions, Hacet. J. Math. Stat. 44(1) (2015), 121{127] obtained an upper bound for the Fekete-Szegö functionalϕλwhenλ 2R of functions belong to the classSLkconnected withk-Fibonacci numbers. The main purpose of this paper is to obtain sharp bounds forϕλbothλ 2R andλ 2C.

The paper is devoted to the study of the metric properties of regular and semiregular polyhedra in Euclidean spaces. In the first part, we prove that every regular polytope of dimension greater or equal than 4, and different from 120-cell in $$\mathbb {E}^4 $$ is such that the set of its vertices is a Clifford–Wolf homogeneous finite metric space. The second part of the paper is devoted to the study of special properties of Archimedean solids. In particular, for each Archimedean solid, its description is given as the convex hull of the orbit of a suitable point of a regular tetrahedron, cube or dodecahedron under the action of the corresponding isometry group.

In the paper, we consider the limit behavior of partial-sum random field (r.f.) $$ \left.{S}_n\left({t}_1,{t}_2;\right)X\left(b\left(\mathbf{n}\right)\right)\right)={\sum}_{k=1}^{\left[{n}_1{t}_1\right]}{\sum}_{l=1}^{\left[{n}_2{t}_2\right]}{X}_{k,l}\left(b\left(\mathbf{n}\right)\right), $$ where $$ \left\{{X}_{k,l}\left(b\left(\mathbf{n}\right)\right)={\sum}_{i=0}^{\infty }{\sum}_{j=0}^{\infty }{c}_{i,j}{\upxi}_{k-i,l-j}\left(b\left(\mathbf{n}\right)\right),k,l\in \mathrm{\mathbb{Z}}\right\},n\ge 1, $$ is a family (indexed by n = (n1, n2), ni ≥ 1) of linear r.f.s with filter ci,j = aibj and innovations ξk,l(b(n)) having heavy-tailed tapered distributions with tapering parameter b(n) growing to infinity as n → ∞. In [V. Paulauskas, Limit theorems for linear random fields with tapered innovations. I: The Gaussian case, Lith. Math. J., 61(2):261–273, 2021], we considered the so-called hard tapering as b(n) grows relatively slowly and the limit r.f.s for appropriately normalized Sn(t1, t2;X(b(n))) are Gaussian. In this paper, we consider the case of soft tapering where b(n) grows more rapidly in comparison with the case of hard tapering and stable limit r.f.s.We consider cases where the sequences {ai} and {bj} are long-range, short-range, and negatively dependent.

In this paper, we study a problem posed by Furstenberg on intersections between × 2 , × 3 \times 2, \times 3 invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used to obtain further improvements. For example, we show that if A 2 , A 3 ⊂ [ 0 , 1 ] A_2,A_3\subset [0,1] are closed and × 2 , × 3 \times 2, \times 3 invariant respectively, assuming that dim ⁡ A 2 + dim ⁡ A 3 > 1 \dim A_2+\dim A_3>1 then A 2 ∩ ( u A 3 + v ) A_2\cap (uA_3+v) is sparse (defined in this paper) and has box dimension zero uniformly with respect to the real parameters u , v u,v such that u u and u − 1 u^{-1} are both bounded away from 0 0 .

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

In a little-known paper Hurwitz gave an infinite series representation of the class number for positive definite binary quadratic forms. In this paper we give a similar formula in the indefinite case. We also give a simple proof of Hurwitz's formula and indicate some extensions., Journal of the European Mathematical Society, 23 (12), ISSN:1435-9855, ISSN:1435-9863

Let $\mathcal X$ be a Banach space and let T be a bounded linear operator on $\mathcal {X}$ . We denote by S(T) the set of all complex $\lambda \in \mathcal {C}$ such that T does not have the single-valued extension property. In this paper it is shown that if MC is a 2 × 2 upper triangular operator matrix acting on the Banach space $\mathcal {X} \oplus \mathcal {Y}$ , then the passage from σLD(A) ∪ σLD(B) to σLD(MC) is accomplished by removing certain open subsets of σd(A) ∩ σLD(B) from the former, that is, there is the equality σLD(A) ∪ σLD(B) = σLD(MC) ∪ℵ, where ℵ is the union of certain of the holes in σLD(MC) which happen to be subsets of σd(A) ∩ σLD(B). Generalized Weyl’s theorem and generalized Browder’s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how generalized Weyl’ theorem, generalized Browder’s theorem, generalized a-Weyl’s theorem and generalized a-Browder’s theorem survive for 2 × 2 upper triangular operator matrices on the Banach space.

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

In a paper from 2001 (Journal of the LMS), Diestel and Leader offered a proof that a connected graph has a normal spanning tree if and only if it does not contain a minor from two specific forbidden classes of graphs, all of cardinality $\aleph_1$. Unfortunately, their proof contains a gap, and their result is incorrect. In this paper, we construct a third type of obstruction: an $\aleph_1$-sized graph without a normal spanning tree that contains neither of the two types described by Diestel and Leader as a minor. Further, we show that any list of forbidden minors characterising the graphs with normal spanning trees must contain graphs of arbitrarily large cardinality., Comment: 9 pages. arXiv admin note: text overlap with arXiv:2005.02833

This paper continues the study of P. S. Alexandroff’s problem: When can a Hausdorff space $$X$$ be one-to-one continuously mapped onto a compact Hausdorff space? For a cardinal number $$\tau$$ , the classes of $$a_\tau$$ -spaces and strict $$a_\tau$$ -spaces are defined. A compact space $$X$$ is called an $$a_\tau$$ -space if, for any $$C\in[X]^{\le\tau}$$ , there exists a one-to-one continuous mapping of $$X\setminus C$$ onto a compact space. A compact space $$X$$ is called a strict $$a_\tau$$ -space if, for any $$C\in[X]^{\le\tau}$$ , there exits a one-to-one continuous mapping of $$X\setminus C$$ onto a compact space $$Y$$ , and this mapping can be continuously extended to the whole space $$X$$ . In this paper, we study properties of the classes of $$a_\tau$$ - and strict $$a_\tau$$ -spaces by using Raukhvarger’s method of special continuous paritions.

Let $S \subset \Bbb R^n$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\Bbb R^n$. If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^q(X)} \leq C \| f \|_{L^p(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$: there is a number $0 < \alpha \leq n$ such that $|X \cap B_R| \leq c \, R^\alpha$ for all balls $B_R$ in $\Bbb R^n$ of radius $R \geq 1$. On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^q$ against the measure $\chi_X dx$. Our approach consists of replacing the characteristic function $\chi_X$ of $X$ by an appropriate weight function $H$, and studying the resulting estimate in three different regimes: small values of $\alpha$, intermediate values of $\alpha$, and large values of $\alpha$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted H\"{o}lder-type inequality that holds for general non-negative Lebesgue measurable functions on $\Bbb R^n$, and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du-Zhang theorem in the range $0 < \alpha < n/2$., Comment: 31 pages. Minor revision

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

In Sarkozy (2001) the third author of this paper presented two conjectures on the additive decomposability of the sequence of ”smooth” (or ”friable”) numbers. Elsholtz and Harper (2015) proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of S -unit equations).