On the structure of large sum-free sets of integers

A set of integers is called sum-free if it contains no triple (x, y, z) of not necessarily distinct elements with x + y = z. In this paper, we provide a structural characterisation of sum-free subsets of {1, 2,..., n} of density at least 2/5 ? c, where c is an absolute positive constant. As an application, we derive a stability version of Hu’s Theorem [Proc. Amer. Math. Soc. 80 (1980), 711–712] about the maximum size of a union of two sum-free sets in {1, 2,..., n}. We then use this result to show that the number of subsets of {1, 2,..., n} which can be partitioned into two sum-free sets is Θ(2^4n/5), confirming a conjecture of Hancock, Staden and Treglown [arXiv:1701.04754].     

Maximum number of sum-free colorings in finite abelian groups

An r-coloring of a subset A of a finite abelian group G is called sum-free if it does not induce a monochromatic Schur triple, i.e., a triple of elements a, b, c ∈ A with a + b = c. We investigate κ_r,G, the maximum number of sum-free r-colorings admitted by subsets of G, and our results show a close relationship between κ_r,G and largest sum-free sets of G.Given a sufficiently large abelian group G of type I, i.e., |G| has a prime divisor q with q ≡ 2 (mod 3). For r = 2, 3 we show that a subset A ? G achieves κ_r,G if and only if A is a largest sum-free set of G. For even order G the result extends to r = 4, 5, where the phenomenon persists only if G has a unique largest sum-free set. On the contrary, if the largest sum-free set in G is not unique, then A attains κ_r,G if and only if it is the union of two largest sum-free sets (in case r = 4) and the union of three (“independent”) largest sum-free sets (in case r = 5).Our approach relies on the so called container method and can be extended to larger r in case G is of even order and contains sufficiently many largest sum-free sets.     

A Classification of Spectrum-determined Circulant Digraphs

Some researchers have proved that ádám’s conjecture is wrong. However, under special conditions, it is right. Let Z_n be a cyclic group of order n and C_n(S) be the circulant digraph of Z_n with respect to S ? Z_n\{0}. In the literature, some people have used a spectral method to solve the isomorphism for the circulants of prime-power order. In this paper, we also use the spectral method to characterize the circulants of order p^aq^bw^c(where p, q and w are all distinct primes), and to make ádám’s conjecture right.     

Phase transition for SIR model with random transition rates on complete graphs

We are concerned with the susceptible-infective-removed (SIR) model with random transition rates on complete graphs C _n with n vertices. We assign independent and identically distributed (i.i.d.) copies of a positive random variable ξ on each vertex as the recovery rates and i.i.d. copies of a positive random variable ρ on each edge as the edge infection weights. We assume that a susceptible vertex is infected by an infective one at rate proportional to the edge weight on the edge connecting these two vertices while an infective vertex becomes removed with rate equals the recovery rate on it, then we show that the model performs the following phase transition when at t = 0 one vertex is infective and others are susceptible. There exists λ _c > 0 such that when λ < λ _c ; the proportion r∞ of vertices which have ever been infective converges to 0 weakly as n → +∞ while when λ > λ _c ; there exist c(λ) > 0 and b(λ) > 0 such that for each n ≥ 1 with probability p ≥ b(λ); the proportion r_∞ ≥ c(λ): Furthermore, we prove that λ _c is the inverse of the production of the mean of ρ and the mean of the inverse of ξ.     

A Generalization of Strassen’s Functional LIL

Let X ₁,X ₂,… be a sequence of i.i.d. mean zero random variables and let S _n denote the sum of the first n random variables. We show that whenever we have with probability one, lim?sup? _n→∞|S _n |/c _n =α ₀<∞ for a regular normalizing sequence {c _n }, the corresponding normalized partial sum process sequence is relatively compact in C[0,1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting.     

Parabolic equation with a singular potential

We study the existence of a nonnegative generalized solution of an initial-boundary value problem for the heat equation with a singular potential in an arbitrary bounded domain Ω ? R ⁿ , n ≥ 3, containing the unit ball. We show that if the condition ∝ _Ω V ^n/2+s |x| ^s dx ≤ c _n is satisfied for some s ≥ 0 and c _n = c _n (n, s, Ω) > 0, then the problem in question has a nonnegative solution.     

Properly coloured Hamiltonian cycles in edge-coloured complete graphs

Let K ^c _n be an edge-coloured complete graph on n vertices. Let Δ_mon(K^c _n) denote the largest number of edges of the same colour incident with a vertex of K^c _n. A properly coloured cycleis a cycle such that no two adjacent edges have the same colour. In 1976, BollobÁs and Erd?s[6] conjectured that every K^c _n with Δ_mon(K^c _n)<?n/2?contains a properly coloured Hamiltonian cycle. In this paper, we show that for any ε>0, there exists an integer n₀ such that every K^c _n with Δ_mon(K^c _n)<(1/2–ε)n and n≥n₀ contains a properly coloured Hamiltonian cycle. This improves a result of Alon and Gutin [1]. Hence, the conjecture of BollobÁs and Erd?s is true asymptotically.     

Phragmén-Lindelöf Theorems and p-harmonic Measures for Sets Near Low-dimensional Hyperplanes

We prove estimates of a p-harmonic measure, p∈(n?m,∞], for sets in Rⁿ which are close to an m-dimensional hyperplane Λ?Rⁿ, m∈[0,n?1]. Using these estimates, we derive results of Phragmén-Lindelöf type in unbounded domains Ω?Rⁿ?Λ for p-subharmonic functions. Moreover, we give local and global growth estimates for p-harmonic functions, vanishing on sets in Rⁿ, which are close to an m-dimensional hyperplane.     

Certain pairs of irreducible characters of the groups <Emphasis Type="Italic">S</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The investigation of the pairs of irreducible characters of the symmetric group S _n that have the same set of roots in one of the sets A _n and S _n ? A _n is continued. All such pairs of irreducible characters of the group S _n are found in the case when the least of the main diagonal’s lengths of the Young diagrams corresponding to these characters does not exceed 2. Some arguments are obtained for the conjecture that alternating groups A _n have no pairs of semiproportional irreducible characters.     

A characterization of a certain real hypersurface of type (A<Subscript>2</Subscript>) in a complex projective space

In the class of real hypersurfaces M ^2n?1 isometrically immersed into a nonflat complex space form \(\widetilde {{M_n}}\left( c \right)\) of constant holomorphic sectional curvature c (≠ 0) which is either a complex projective space ?P ⁿ (c) or a complex hyperbolic space ?H ⁿ (c) according as c > 0 or c < 0, there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds. In this paper, inspired by a simple characterization of all ruled real hypersurfaces in \(\widetilde {{M_n}}\left( c \right)\), we consider a certain real hypersurface of type (A₂) in ?P ⁿ (c) and give a geometric characterization of this Hopf manifold.     

On the maximum rank of a real binary form

We show that a real binary form f of degree n has n distinct real roots if and only if for any \({(\alpha,\beta)\in\mathbb{R}^2{\setminus}\{0\}}\) all the forms αf _x + βf _y have n ? 1 distinct real roots. This answers to a question of Comon and Ottaviani (On the typical rank of real binary forms, available at arXiv:math/0909.4865, 2009), and allows to complete their argument to show that f has symmetric rank n if and only if it has n distinct real roots.     

Solution of the Cameron-Erdös problem for groups of prime order

A subset A of a group G is sum-free if a + b does not belong to A for any a, b ∈ A. Asymptotics of the number of sum-free sets in groups of prime order are proved.     

On a Diophantine Inequality with Prime Numbers of a Special Type

We consider the Diophantine inequality |p ₁ ^c + p ₂ ^c + p ₃ ^c ? N| < (logN)^?E, where 1 < c < 15/14, N is a sufficiently large real number and E > 0 is an arbitrarily large constant. We prove that the above inequality has a solution in primes p₁, p₂, p₃ such that each of the numbers p₁ + 2, p₂ + 2 and p₃ + 2 has at most [369/(180 ? 168c)] prime factors, counted with multiplicity.     

The real symmetric matrices of odd order with a P-set of maximum size

Suppose that A is a real symmetric matrix of order n. Denote by m_A(0) the nullity of A. For a nonempty subset α of {1, 2,..., n}, let A(α) be the principal submatrix of A obtained from A by deleting the rows and columns indexed by α. When m_A(α)(0) = m_A(0)+|α|, we call α a P-set of A. It is known that every P-set of A contains at most ?n/2? elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As a first step to the full characterization of these graphs of odd order, we establish some conditions for such graphs G under which there is a real symmetric matrix A whose graph is G and contains a P-set of size (n ? 1)/2.     

Spectral properties of the complex airy operator on the half-line

We prove a theorem on the completeness of the system of root functions of the Schrödinger operator L = ?d ²/dx ² + p(x) on the half-line R₊ with a potential p for which L appears to be maximal sectorial. An application of this theorem to the complex Airy operator L _c = ?d ²/dx ² + cx, c = const, implies the completeness of the system of eigenfunctions of L _c for the case in which |arg c| < 2π/3.We use subtler methods to prove a theorem stating that the system of eigenfunctions of this special operator remains complete under the condition that |arg c| < 5π/6.     

On the Convergence of Solutions of Variational Problems with Implicit Pointwise Constraints in Variable Domains

Results on the convergence of minimizers and minimum values of integral and more general functionals J_s: W^1,p(Ω_s) → ? on the sets U_s(h_s) = {v ∈ W^1,p(Ω_s): h_s(v) ≤ 0 a.e. in Ω_s}, where p > 1, {Ω_s} is a sequence of domains contained in a bounded domain Ω of ?ⁿ (n > 2), and {h_s} is a sequence of functions on ?, are announced.     

Seifert manifolds and (1, 1)-knots

The aim of this paper is to investigate the relations between Seifert manifolds and (1, 1)-knots. In particular, we prove that each orientable Seifert manifold with invariants

$\{ Oo,0| - 1;\underbrace {(p,q),...,(p,q)}_{n times},(l,l - 1)\} $

has the fundamental group cyclically presented by G _n ((x ₁ ^q ...x _n ^{q l} x _n ^?p ) and, moreover, it is the n-fold strongly-cyclic covering of the lens space L(|nlq ? p|, q) which is branched over the (1, 1)-knot K(q, q(nl ? 2), p ? 2q, p ? q) if p ≥ 2q and over the (1, 1)-knot K(p? q, 2q ? p, q(nl ? 2), p ? q) if p< 2q.

     

Hausdorff dimension of the maximal run-length in dyadic expansion

For any x ∈ [0, 1), let x = [? ₁, ? ₂, …,] be its dyadic expansion. Call r _n (x):= max{j ? 1: ? _i+1 = … = ? _i+j = 1, 0 ? i ? n ? j} the n-th maximal run-length function of x. P.Erdös and A.Rényi showed that \(\mathop {\lim }\limits_{n \to \infty } \) r _n (x)/log₂ n = 1 almost surely. This paper is concentrated on the points violating the above law. The size of sets of points, whose runlength function assumes on other possible asymptotic behaviors than log₂ n, is quantified by their Hausdorff dimension.     

Ramanujan-type congruences modulo powers of 5 and 7

Let b _? (n) denote the number of ?-regular partitions of n. In 2012, using the theory of modular forms, Furcy and Penniston presented several infinite families of congruences modulo 3 for some values of ?. In particular, they showed that for α, n ≥ 0, b ₂₅ (3^2α+3 n+2 · 3^2α+2-1) ≡ 0 (mod 3). Most recently, congruences modulo powers of 5 for c5(n) was proved by Wang, where c _N (n) counts the number of bipartitions (λ₁,λ₂) of n such that each part of λ₂ is divisible by N. In this paper, we prove some interesting Ramanujan-type congruences modulo powers of 5 for b25(n), B25(n), c25(n) and modulo powers of 7 for c49(n). For example, we prove that for j ≥ 1, \({c_{25}}\left( {{5^{2j}}n + \frac{{11 \cdot {5^{2j}} + 13}}{{12}}} \right) \equiv 0\) (mod 5 ^j+1), \({c_{49}}\left( {{7^{2j}}n + \frac{{11 \cdot {7^{_{2j}}} + 25}}{{12}}} \right) \equiv 0\) (mod 7 ^j+1) and b ₂₅ (3^2α+3 · n+2 · 3^2α+2-1) ≡ 0 (mod 3 · 5^2j-1).     

Sharpened forms of the generalized Schwarz inequality on the boundary

In this paper, a boundary version of the Schwarz inequality is investigated. We obtain more general results at the boundary. If we know the second coefficient in the expansion of the function f(z) = 1 + c_pz^p + c_{p + 1}z^{p + 1}…, then we obtain new inequalities of the Schwarz inequality at boundary by taking into account c_{p + 1} and zeros of the function f(z) ? 1. The sharpness of these inequalities is also proved.