On a conjecture of Butler and Graham

Motivated by a hat guessing problem proposed by Iwasawa, Butler and Graham made the following conjecture on the existence of a certain way of marking the coordinate lines in [k] ⁿ : there exists a way to mark one point on each coordinate line in [k] ⁿ , so that every point in [k] ⁿ is marked exactly a or b times as long as the parameters (a, b, n, k) satisfies that there are nonnegative integers s and t such that s + t = k ⁿ and as + bt = nk ^n?1. In this paper we prove this conjecture for any prime number k. Moreover, we prove the conjecture for the case when a = 0 for general k.     

On multiplicative functions which are additive on sums of primes

Let ${f : \mathbb{N} \to \mathbb{C}}$ be a multiplicative function satisfying f(p ₀) ≠ 0 for at least one prime number p ₀, and let k ≥ 2 be an integer. We show that if the function f satisfies f(p ₁ + p ₂ + . . . + p _k ) = f(p ₁) + f(p ₂) + . . . + f(p _k ) for any prime numbers p ₁, p ₂, . . . ,p _k then f must be the identity f(n) = n for each ${n \in \mathbb{N}}$ . This result for k = 2 was established earlier by Spiro, whereas the case k = 3 was recently proved by Fang. In the proof of this result for k ≥ 6 we use a recent result of Tao asserting that every odd number greater than 1 is the sum of at most five primes.     

On the minimum length of ternary linear codes

From the geometrical point of view, we prove that [g ₃(6, d) + 1, 6, d]₃ codes exist for d = 118–123, 283–297 and that [g ₃(6, d), 6, d]₃ codes for d = 100, 341, 342 and [g ₃(6, d) + 1, 6, d]₃ codes for d = 130, 131, 132 do not exist, where ${g_3(k,\,d)=\sum_{i=0}^{k-1}\left\lceil d/3^i \right\rceil}$ . These determine the exact value of n ₃(6, d) for d = 100, 118–123, 130, 131, 132, 283–297, 341, 342, where n _q (k, d) is the minimum length n for which an [n, k, d] _q code exists.     

On the central limit theorem for f (n k x)

By a classical observation in analysis, lacunary subsequences of the trigonometric system behave like independent random variables: they satisfy the central limit theorem, the law of the iterated logarithm and several related probability limit theorems. For subsequences of the system ( f (nx)) _n≥1 with 2π-periodic ${f\in L^2}$ this phenomenon is generally not valid and the asymptotic behavior of ( f (n _k x)) _k≥1 is determined by a complicated interplay between the analytic properties of f (e.g., the behavior of its Fourier coefficients) and the number theoretic properties of n _k . By the classical theory, the central limit theorem holds for f (n _k x) if n _k  = 2 ^k , or if n _k+1/n _k → α with a transcendental α, but it fails e.g., for n _k  = 2 ^k  ? 1. The purpose of our paper is to give a necessary and sufficient condition for f (n _k x) to satisfy the central limit theorem. We will also study the critical CLT behavior of f (n _k x), i.e., the question what happens when the arithmetic condition of the central limit theorem is weakened “infinitesimally”.     

Fast Growing Solutions of Linear Differential Equations in the Unit Disc

For $n \in \mathbb{N}$ , the n-order of an analytic function f in the unit disc D is defined by $$\sigma _{{{M,n}}} (f) = {\mathop {\lim \sup }\limits_{r \to 1^{ - } } }\frac{{\log ^{ + }_{{n + 1}} M(r,f)}} {{ - \log (1 - r)}},$$ where log⁺ x  =  max{log x, 0}, log ⁺ ₁ x  =  log ⁺ x, log ⁺ _n+1 x  =  log ⁺ log ⁺ _n x, and M(r, f) is the maximum modulus of f on the circle of radius r centered at the origin. It is shown, for example, that the solutions f of the complex linear differential equation $$f^{{(k)}} + a_{{k - 1}} (z)f^{{(k - 1)}} + \cdots + a_{1} (z)f^{\prime} + a_{0} (z)f = 0,\quad \quad \quad (\dag)$$ where the coefficients are analytic in D, satisfy σ _M,n+1(f)  ≤  α if and only if σ _M,n (a _j )  ≤  α for all j  =  0, ..., k ? 1. Moreover, if q ∈{0, ..., k ? 1} is the largest index for which $\sigma _{M,n} ( a_{q}) = {\mathop {\max }\limits_{0 \leq j \leq k - 1} }{\left\{ {\sigma _{{M,n}} {\left( {a_{j} } \right)}} \right\}}$ , then there are at least k ? q linearly independent solutions f of ( $\dag$ ) such that σ _M,n+1(f) = σ _M,n (a _q ). Some refinements of these results in terms of the n-type of an analytic function in D are also given.     

Rings of Constants of the Form k[f]

Let k[X] be the algebra of polynomials in n variables over a field k of characteristic zero, and let f ? k[X]? k. We present a construction of a derivation d of k[X] whose ring of constants is equal to the integral closure of k[f] in k[X]. A similar construction for fields of rational functions is also given.     

A note: Every homogeneous difference equation of degree one admits a reduction in order

Every difference equation x _n+1 = f _n (x _n ,x _n ? 1,…,x _n ? k ) of order k+1 with each mapping f _n being homogeneous of degree 1 on a group G is shown to be equivalent to a system consisting of an equation of order k and a linear equation of order 1.     

Hausdorff dimension of quasi-cirles of polygonal mappings and its applications

We show that the Hausdorff dimension of quasi-circles of polygonal mappings is one.Furthermore,we apply this result to the theory of extremal quasiconformal mappings.Let [μ] be a point in the universal Teichmller space such that the Hausdorff dimension of fμ(Δ) is bigger than one.We show that for every kn∈(0,1) and polygonal differentials ψn,n=1,2,...,the sequence {[kn ψn/|ψn|]} cannot converge to [μ] under the Teichmer metric.     

Powers of Elements and Monomial Ideals #

All monomial ideals I ? k[X ₀,…, X _d ] are classified which satisfy the following condition: If f ∈ I with f ⁿ  ∈ I ⁿ⁺¹ for some n, then f ∈ (X ₀,…, X _d ) I.     

Two conjectures of Grünbaum on arrangements of curves

We prove the following two conjectures of Grünbaum on arrangements of curves in the Euclidean plane: (a) There is no arrangement of n curves such that 4n - 4<f₂(A)<5n - 12. (b) There is no digon-free arrangement of n curves (n?36) such that 4n - 4<f₂(A)<5n - 7. (f₂(A) denotes the number of faces of the arrangement A.) Generalizing (a). we obtain: (c) For each k there is an integer n₀ (depending on k) such that no arrangement of n curves (n ? n₀) satisfies: kn-2k+4<f₂(A)<(k+1)n-k(k-1).     

On the Stability of a k-Cubic Functional Equation in Intuitionistic Fuzzy n-Normed Spaces

In this paper, we investigate some stability results concerning the k-cubic functional equation f(kx + y) + f(kx?y) = kf(x + y) + kf(x?y) + 2k(k²?1)f(x) in the intuitionistic fuzzy n-normed spaces.     

Further gossip problems

n people have distinct bits of information. They can communicate via k-party conference calls. How many such calls are needed to inform everyone of everyone else's information? Let f(n,k) be this minimum number. Then we give a simple proof that f(n,k)= [(n?k)(k?1)]+[nk] for 1?n?k², f(n,k)=2[(n?k)(k?1)] for n>k².In the 2-party case we consider the case in which certain of the calls may permit information flow in only one direction. We show that any 2n-4 call scheme that conveys everone's information to all must contain a 4-cycle, each of whose calls is “two way”, along with some other results. The method follows that of Bumby who first proved the 4-cycle conjecture.     

On the number of combinations without unit separation

Let f(n, k) denote the number of ways of selecting k objects from n objects arrayed in a line with no two selected having unit separation (i.e., having exactly one object between them). Then, if $\text{n ? 2(k ? 1), f(n,k)=}\text{∑}\text{i=0}\text{κ}\text{(}\text{n?k+I?2i}\text{k?2i}\text{)}$ (where $\text{κ = [}\text{k}\text{2}\text{]}$). If n < 2(k ? 1), then f(n, k) = 0. In addition, f(n, k) satisfies the recurrence relation f(n, k) = f(n ? 1, k) + f(n ? 3, k ? 1) + f(n ? 4, k ? 2). If the objects are arrayed in a circle, and the corresponding number is denoted by g(n, k), then for n > 3, g(n, k) = f(n ? 2, k) + 2f(n ? 5, k ? 1) + 3f(n ? 6, k ? 2). In particular, if n ? 2k + 1 then $\text{(n,k)=(}\text{n?k}\text{k}\text{)+(}\text{n?k?1}\text{k?1}\text{)}$.     

Small point sets of PG(n, q 3) intersecting each k-subspace in 1 mod q points

The main result of this paper is that point sets of PG(n, q ³), q = p ^h , p ≥ 7 prime, of size less than 3(q ^3(n?k) + 1)/2 intersecting each k-space in 1 modulo q points (these are always small minimal blocking sets with respect to k-spaces) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size less than 3(p ^3(n?k) + 1)/2 with respect to k-spaces are linear. We also give a classification of small linear blocking sets of PG(n, q ³) which meet every (n ? 2)-space in 1 modulo q points.     

On sets of type (q + 1, n)2 in finite three-dimensional projective spaces

It is proved that a k–set of type (q + 1, n)₂ in PG(3, q) either is a plane or it has size k ≥ (q + 1)² and a characterization of some sets of size (q + 1)² is given.     

On set systems with a threshold property

For n,k and t such that 1<t<k<n, a set F of subsets of [n] has the (k,t)-threshold property if every k-subset of [n] contains at least t sets from F and every (k-1)-subset of [n] contains less than t sets from F. The minimal number of sets in a set system with this property is denoted by m(n,k,t). In this paper we determine m(n,4,3)exactly for n sufficiently large, and we show that m(n,k,2) is asymptotically equal to the generalized Turán number T_k-1(n,k,2).     

Commutators of Riesz transforms of magnetic Schrödinger operators

Let \({A=-(\nabla-i{\vec a})\cdot (\nabla-i{\vec a}) +V}\) be a magnetic Schrödinger operator acting on \({L^2({\mathbb R}^n)}\), n ≥  1, where \({{\vec a}=(a_1, \ldots, a_n)\in L^2_{\rm loc}({\mathbb R}^n, {\mathbb R}^n)}\) and \({0\leq V\in L^1_{\rm loc}({\mathbb R}^n)}\). In this paper, we show that when a function \({b\in {\rm BMO}({\mathbb R}^n)}\), the commutators [b, T _k ]f = T _k (b f) ? b T _k f, k = 1, . . . , n, are bounded on \({L^p({\mathbb R}^n)}\) for all 1 < p < 2, where the operators T _k are Riesz transforms (?/?x _k  ? i a _k )A ^?1/2 associated with A.     

Solving Non-Homogeneous Nested Recursions Using Trees

The solutions to certain nested recursions, such as Conolly’s C(n) = C(n?C(n?1)) + C(n?1?C(n?2)), with initial conditions C(1) = 1, C(2) = 2, have a well-established combinatorial interpretation in terms of counting leaves in an infinite binary tree. This tree-based interpretation, and its generalization to a similar k-term nested recursion, only apply to homogeneous recursions and only solve each recursion for one set of initial conditions determined by the tree. In this paper, we extend the tree-based interpretation to solve a non-homogeneous version of the k-term recursion that includes a constant term. To do so we introduce a tree-grafting methodology that inserts copies of a finite tree into the infinite k-ary tree associated with the solution of the corresponding homogeneous k-term recursion. This technique also solves the given non-homogeneous recursion with various sets of initial conditions.     

Permanent formulae from the Veronesean

The two formulae for the permanent of a d × d matrix given by Ryser (1963) and Glynn (2010) fit into a similar pattern that allows generalization because both are related to polarization identities for symmetric tensors, and to the classical theorem of P. Serret in algebraic geometry. The difference between any two formulae of this type corresponds to a set of dependent points on the “Veronese variety” (or “Veronesean”) v _d ([d ? 1]), where v _d ([n]) is the image of the Veronese map v _d acting on [n], the n-dimensional projective space over a suitable field. To understand this we construct dependent sets on the Veronesean and show how to construct small independent sets of size nd + 2 on v _d ([n]). For d = 2 such sets of 2n + 2 points in [n] have been called “associated” and we observe that they correspond to self-dual codes of length 2n + 2.     

PATHS AND CYCLES EMBEDDING ON FAULTY ENHANCED HYPERCUBE NETWORKS

Let Qn,k(n≥3,1≤k≤n-1) be an n-dimensional enhanced hypercube which is an attractive variant of the hypercube and can be obtained by adding some complementary edges,fv and fe be the numbers of faulty vertices and faulty edges,respectively.In this paper,we give three main results.First,a fault-free path P [u,v] of length at least 2n-2fv-1(respectively,2n-2fv-2) can be embedded on Qn,k with fv+fe≤n-1 when d Qn,k(u,v) is odd(respectively,d Qn,k(u,v) is even).Secondly,an Qn,k is(n-2) edgefault-free hyper Hamiltonian-laceable when n(≥3) and k have the same parity.Lastly,a fault-free cycle of length at least 2n-2fv can be embedded on Qn,k with fe≤n-1 and fv+fe≤2n-4.