The maximum and minimum degree of the random <Emphasis Type="Italic">r</Emphasis>-uniform <Emphasis Type="Italic">r</Emphasis>-partite hypergraphs

In this paper we consider the random r-uniform r-partite hypergraph model H(n ₁, n ₂, ···, n _r; n, p) which consists of all the r-uniform r-partite hypergraphs with vertex partition {V ₁, V ₂, ···, V _r} where |V _i| = n _i = n _i(n) (1 ≤ i ≤ r) are positive integer-valued functions on n with n ₁ +n ₂ +···+n _r = n, and each r-subset containing exactly one element in V _i (1 ≤ i ≤ r) is chosen to be a hyperedge of H _p ∈ H (n ₁, n ₂, ···, n _r; n, p) with probability p = p(n), all choices being independent. Let

$${\Delta _{{V_1}}} = {\Delta _{{V_1}}}\left( H \right)$$

and

$${\delta _{{V_1}}} = {\delta _{{V_1}}}\left( H \right)$$

be the maximum and minimum degree of vertices in V ₁ of H, respectively;

$${X_{d,{V_1}}} = {X_{d,{V_1}}}\left( H \right),{Y_{d,{V_1}}} = {Y_{d,{V_1}}}\left( H \right)$$

,

$${Z_{d,{V_1}}} = {Z_{d,{V_1}}}\left( H \right)and{Z_{c,d,{V_1}}} = {Z_{c,d,{V_1}}}\left( H \right)$$

be the number of vertices in V ₁ of H with degree d, at least d, at most d, and between c and d, respectively. In this paper we obtain that in the space H(n ₁, n ₂, ···, n _r; n, p),

$${X_{d,{V_1}}},{Y_{d,{V_1}}},{Z_{d,{V_1}}}and{Z_{c,d,{V_1}}}$$

all have asymptotically Poisson distributions. We also answer the following two questions. What is the range of p that there exists a function D(n) such that in the space H(n ₁, n ₂, ···, n _r; n, p),

$$\mathop {\lim }\limits_{n \to \infty } P\left( {{\Delta _{{V_1}}} = D\left( n \right)} \right) = 1$$

? What is the range of p such that a.e., H _p ∈ H (n ₁, n ₂, ···, n _r; n, p) has a unique vertex in V ₁ with degree

$${\Delta _{{V_1}}}\left( {{H_p}} \right)$$

? Both answers are p = o (log n ₁/N), where

$$N = \mathop \prod \limits_{i = 2}^r {n_i}$$

. The corresponding problems on

$${\delta _{{V_i}}}\left( {{H_p}} \right)$$

also are considered, and we obtained the answers are p ≤ (1 + o(1))(log n ₁/N) and p = o (log n ₁/N), respectively.

     

On the divisor function in short intervals

Let d(n) denote the number of positive divisors of the natural number n. The aim of this paper is to investigate the validity of the asymptotic formula

$\begin{array}{lll}\sum \limits_{x < n \leq x+h(x)}d(n)\sim h(x)\log x\end{array}$

for \({x \to + \infty,}\) assuming a hypothetical estimate on the mean

$\begin{array}{lll} \int \limits_X^{X+Y}(\Delta(x+h(x))-\Delta (x))^2\,{d}x, \end{array}$

which is a weakened form of a conjecture of M. Jutila.

     

Uniqueness of a positive radially symmetric Solution of the Dirichlet problem for certain nonlinear differential equation of the second order in a ball

We consider the Dirichlet problem

$u_\Gamma = 0$

for the nonlinear differential equation

$\Delta u + \left| x \right|^m \left| u \right|^p = 0, x \in S,$

with constant m ≥ 0 and p > 1 in the unit ball S = {x ∈ R ⁿ : |x| < 1}(n ≥ 3) with the boundary Γ. We prove that with p ≤ m+n/n?2 this problem has a unique positive radially symmetric solution.

     

Standard monomials for <Emphasis Type="Italic">q</Emphasis>-uniform families and a conjecture of Babai and Frankl

Let n, k, α be integers, n, α>0, p be a prime and q=p ^α. Consider the complete q-uniform family

$\mathcal{F}\left( {k,q} \right) = \left\{ {K \subseteq \left[ n \right]:\left| K \right| \equiv k(mod q)} \right\}$

We study certain inclusion matrices attached to F(k,q) over the field\(\mathbb{F}_p \). We show that if l≤q?1 and 2l≤n then

$rank_{\mathbb{F}_p } I(\mathcal{F}(k,q),\left( {\begin{array}{*{20}c} {\left[ n \right]} \\ { \leqslant \ell } \\ \end{array} } \right)) \leqslant \left( {\begin{array}{*{20}c} n \\ \ell \\ \end{array} } \right)$

This extends a theorem of Frankl [7] obtained for the case α=1. In the proof we use arguments involving Gröbner bases, standard monomials and reduction. As an application, we solve a problem of Babai and Frankl related to the size of some L-intersecting families modulo q.     

Weighted polyharmonic equation with Navier boundary conditions in a half space

We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space:?????(-?)~mu(x)=u~p(x)/|x|~s,in R_+~n,u(x)=-?u(x)=…=(-?)~(m-1)u(x)=0,on ?R_+~n,(0.1)where m is any positive integer satisfying 02mn.We first prove that the positive solutions of(0.1)are super polyharmonic,i.e.,(-?)~iu0,i=0,1,...,m-1.(0.2) For α=2m,applying this important property,we establish the equivalence between (0.1) and the integral equation u(x)=c_n∫R_+~n(1/|x-y|~(n-α)-1/|x~*-y|~(n-α))u~p(y)/|y|~sdy,(0.3) where x~*=(x1,...,x_(n-1),-x_n) is the reflection of the point x about the plane R~(n-1).Then,we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of(0.3),in whichαcan be any real number between 0 and n.By some Pohozaev type identities in integral forms,we prove a Liouville type theorem—the non-existence of positive solutions for(0.1).     

A discrete version of Koldobsky’s slicing inequality

Let #K be a number of integer lattice points contained in a set K. In this paper we prove that for each d ∈ N there exists a constant C(d) depending on d only, such that for any origin-symmetric convex body K ? R ^d containing d linearly independent lattice points

$$\# K \leqslant C\left( d \right)\max \left( {\# \left( {K \cap H} \right)} \right)vo{l_d}{\left( K \right)^{\frac{{d - m}}{d}}},$$

where the maximum is taken over all m-dimensional subspaces of R ^d . We also prove that C(d) can be chosen asymptotically of order O(1) ^d d ^d?m . In particular, we have order O(1) ^d for hyperplane slices. Additionally, we show that if K is an unconditional convex body then C(d) can be chosen asymptotically of order O(d) ^d?m .

     

Ergodic theorems with arithmetical weights

We prove that the divisor function d(n) counting the number of divisors of the integer n is a good weighting function for the pointwise ergodic theorem. For any measurable dynamical system (X, A, ν, τ) and any f ∈ L ^p (ν), p > 1, the limit

$$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{\Sigma _{k = 1}^nd\left( k \right)}}\sum\limits_{k = 1}^n {d\left( k \right)f\left( {{\tau ^k}x} \right)} $$

exists ν-almost everywhere. The proof is based on Bourgain’s method, namely the circle method based on the shift model. Using more elementary ideas we also obtain similar results for other arithmetical functions, like the θ(n) function counting the number of squarefree divisors of n and the generalized Euler totient function J _s (n) = Σ _d|n d ^s μ(n/d), s > 0.

     

Modified boundary element method for problems on oscillations of flat membranes

In the space L ² of real-valued measurable 2π-periodic functions that are square summable on the period [0, 2π], the Jackson-Stechkin inequality

$$E_n (f) \leqslant \mathcal{K}_n (\delta ,\omega )\omega (\delta ,f), f \in L^2 $$

, is considered, where E _n (f) is the value of the best approximation of the function f by trigonometric polynomials of order at most n and ω(δ, f) is the modulus of continuity of the function f in L ² of order 1 or 2. The value

$$\mathcal{K}_n (\delta ,\omega ) = \sup \left\{ {\frac{{E_n (f)}}{{\omega (\delta ,f)}}:f \in L^2 } \right\}$$

is found at the points δ = 2π/m (where m ∈ ?) for m ≥ 3n ² + 2 and ω = ω ₁ as well as for m ≥ 11n ⁴/3 ? 1 and ω = ω ₂.

     

On the well-posedness for stochastic fourth-order Schrödinger equations

The influence of the random perturbations on the fourth-order nonlinear Schrödinger equations,

$iu_t + \Delta ^2 u + \varepsilon \Delta u + \lambda |u|^{p - 1} u = \dot \xi ,(t,x) \in \mathbb{R}^ + \times \mathbb{R}^n ,n \geqslant 1,\varepsilon \in \{ - 1,0, + 1\} ,$

, is investigated in this paper. The local well-posedness in the energy space H ²(? ⁿ ) are proved for \(p > \tfrac{{n + 4}}{{n + 2}}\), and p ≤ 2^# ? 1 if n ≥ 5. Global existence is also derived for either defocusing or focusing L ²-subcritical nonlinearities.

     

Global continuum of positive solutions for discrete <Emphasis Type="Italic">p</Emphasis>-Laplacian eigenvalue problems

We discuss the discrete p-Laplacian eigenvalue problem,

$$\left\{ \begin{gathered} \Delta (\phi _p (\Delta u(k - 1))) + \lambda a(k)g(u(k)) = 0,k \in \{ 1,2,...,T\} , \hfill \\ u(0) = u(T + 1) = 0, \hfill \\ \end{gathered} \right.$$

where T > 1 is a given positive integer and φ _p (x):= |x| ^p?2 x, p > 1. First, the existence of an unbounded continuum C of positive solutions emanating from (λ, u) = (0, 0) is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any λ > 0 and all solutions are ordered. Thus the continuum C is a monotone continuous curve globally defined for all λ > 0.

     

Simple proofs and extensions of a result of L. D. Pustylnikov on the nonautonomous Siegel theorem

We present simple proofs of a result of L.D. Pustylnikov extending to nonautonomous dynamics the Siegel theorem of linearization of analytic mappings. We show that if a sequence f _n of analytic mappings of C ^d has a common fixed point f _n (0) = 0, and the maps f _n converge to a linear mapping A∞ so fast that

$$\sum\limits_n {{{\left\| {{f_m} - {A_\infty }} \right\|}_{L\infty \left( B \right)}} < \infty } $$

$${A_\infty } = diag\left( {{e^{2\pi i{\omega _1}}},...,{e^{2\pi i{\omega _d}}}} \right)\omega = \left( {{\omega _1},...,{\omega _q}} \right) \in {\mathbb{R}^d},$$

then f _n is nonautonomously conjugate to the linearization. That is, there exists a sequence h _n of analytic mappings fixing the origin satisfying

$${h_{n + 1}} \circ {f_n} = {A_\infty }{h_n}.$$

The key point of the result is that the functions hn are defined in a large domain and they are bounded. We show that

$${\sum\nolimits_n {\left\| {{h_n} - Id} \right\|} _{L\infty (B)}} < \infty .$$

We also provide results when f _n converges to a nonlinearizable mapping f∞ or to a nonelliptic linear mapping. In the case that the mappings f _n preserve a geometric structure (e. g., symplectic, volume, contact, Poisson, etc.), we show that the hn can be chosen so that they preserve the same geometric structure as the f _n . We present five elementary proofs based on different methods and compare them. Notably, we consider the results in the light of scattering theory. We hope that including different methods can serve as an introduction to methods to study conjugacy equations.

     

Cocentralizing and vanishing derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2 and extended centroid C and let f(x1,..., x _n ) be a multilinear polynomial over C not central-valued on R, while δ is a nonzero derivation of R. Suppose that d and g are derivations of R such that

$\delta (d(f(r_1 , \ldots ,r_n ))f(r_1 , \ldots ,r_n ) - f(r_1 , \ldots ,r_n )g(f(r_1 , \ldots ,r_n ))) = 0$

for all r1,..., r _n ∈ R. Then d and g are both inner derivations on R and one of the following holds: (1) d = g = 0; (2) d = ?g and f(x ₁,..., x _n )² is central-valued on R.

     

(<Emphasis Type="Italic">m</Emphasis>,<Emphasis Type="Italic">r</Emphasis>)-central Riordan arrays and their applications

For integers m > r ≥ 0, Brietzke (2008) defined the (m, r)-central coefficients of an infinite lower triangular matrix G = (d, h) = (d_n,k)_n,k∈N as d_{mn+r,(m?1)n+r}, with n = 0, 1, 2,..., and the (m, r)-central coefficient triangle of G as

$${G^{\left( {m,r} \right)}} = {\left( {{d_{mn + r,\left( {m - 1} \right)n + k + r}}} \right)_{n,k \in \mathbb{N}}}.$$

It is known that the (m, r)-central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array G = (d, h) with h(0) = 0 and d(0), h′(0) ≠ 0, we obtain the generating function of its (m, r)-central coefficients and give an explicit representation for the (m, r)-central Riordan array G^(m,r) in terms of the Riordan array G. Meanwhile, the algebraic structures of the (m, r)-central Riordan arrays are also investigated, such as their decompositions, their inverses, and their recessive expressions in terms of m and r. As applications, we determine the (m, r)-central Riordan arrays of the Pascal matrix and other Riordan arrays, from which numerous identities are constructed by a uniform approach.

     

An upper bound for the number of uniformly packed binary codes

Under study are the binary codes uniformly packed (in the wide sense) of length n with minimum distance d and covering radius ρ. It is shown that every code of this kind is uniquely determined by the set of its codewords of weights ?n/2? ? ρ, …, ?n/2? + ρ. For odd d, the number of distinct codes is at most

$2^{2^{n - \tfrac{3}{2}\log n + o(log n)} } $

.

     

Fixed points of meromorphic functions and of their differences,divided differences and shifts

Let f(z) be a finite order meromorphic function and let c∈C\{0} be a constant.If f(z)has a Borel exceptional value a∈C,it is proved that max{τ(f(z)),τ(△_cf(z))}=max{τ(f(z)),τ(f(z+c))}=max{τ(△_cf(z)),τ(f(z+c))}=σ(f(z)).If f(z) has a Borel exceptional value b∈(C\{0})∪{∞},it is proved that max{τ(f(z)),τ(△cf(z)/f(z))}=max{τ(△cf(z)/f(z)),τ(f(z+c))}=σ(f(z)) unless f(z) takes a special form.Here τ(g(z)) denotes the exponent of convergence of fixed points of the meromorphic function g(z),and σ(g(z)) denotes the order of growth of g(z).     

Homoclinic solutions for a class of second order discrete Hamiltonian systems

Consider the second order discrete Hamiltonian systems Δ2u(n-1)-L(n)u(n) + ▽W (n, u(n)) = f(n),where n ∈ Z, u ∈ RN and W : Z × RN → R and f : Z → RN are not necessarily periodic in n. Under some comparatively general assumptions on L, W and f , we establish results on the existence of homoclinic orbits. The obtained results successfully generalize those for the scalar case.     

Dynamic control for non-linear systems with delay

We consider some class of non-linear systems of the form

$\dot x = A( \cdot )x + \sum\limits_{i = 1}^l {A_i ( \cdot )x(t - \tau _i (t)) + b( \cdot )u} ,$

where A(·) ∈ ? ^{n × n} , A _i (·) ∈ ? ^{n × n} , b(·) ∈ ? ⁿ , whose coefficients are arbitrary uniformly bounded functionals.

A special type of the Lyapunov-Krasovskii functional is used to synthesize dynamic control described by the equation

$\dot u = \rho ( \cdot )u + (m( \cdot ),x),$

where ρ(·) ∈ ?¹, m(·) ∈ ? ⁿ , which makes the system globally asymptotically stable. Also, the situation is considered where the control u enters into the system not directly but through a pulse element performing an amplitude-frequency modulation.

     

On the phase transitions of (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-SAT

Call a sequence of k Boolean variables or their negations a k-tuple. For a set V of n Boolean variables, let T _k (V) denote the set of all 2 ^k n ^k possible k-tuples on V. Randomly generate a set C of k-tuples by including every k-tuple in T _k (V) independently with probability p, and let Q be a given set of q “bad” tuple assignments. An instance I = (C,Q) is called satisfiable if there exists an assignment that does not set any of the k-tuples in C to a bad tuple assignment in Q. Suppose that θ, q > 0 are fixed and ε = ε(n) > 0 be such that εlnn/lnlnn→∞. Let k ≥ (1 + θ) log₂ n and let \({p_0} = \frac{{\ln 2}}{{q{n^{k - 1}}}}\). We prove that

$$\mathop {\lim }\limits_{n \to \infty } P\left[ {I is satisfiable} \right] = \left\{ {\begin{array}{*{20}c} {1,} & {p \leqslant (1 - \varepsilon )p_0 ,} \\ {0,} & {p \geqslant (1 + \varepsilon )p_0 .} \\ \end{array} } \right.$$

     

The hole probability for Gaussian entire functions

Consider the random entire function

$f(z) = \sum\limits_{n = 0}^\infty {{\phi _n}{a_n}{z^n}} $

, where the ? _n are independent standard complex Gaussian coefficients, and the a _n are positive constants, which satisfy

$\mathop {\lim }\limits_{x \to \infty } {{\log {a_n}} \over n} = - \infty $

.

We study the probability P _H (r) that f has no zeroes in the disk{|z| < r} (hole probability). Assuming that the sequence a _n is logarithmically concave, we prove that

$\log {P_H}(r) = - S(r) + o(S(r))$

, where

$S(r) = 2 \cdot \sum\limits_{n:{a_n}{r^n} \ge 1} {\log ({a_n}{r^n})} $

, and r tends to ∞ outside a (deterministic) exceptional set of finite logarithmic measure.