Embeddability of restricted linear spaces

In this note we examine the problem of embedding into finite projective planes finite linear spaces with p points and q lines satisfying (q ? p)² ? p and q ? 2.     

On the order of an automorphism of a smooth hypersurface

In this paper we give an effective criterion as to when a positive integer q is the order of an automorphism of a smooth hypersurface of dimension n and degree d, for every d ≥ 3, n ≥ 2, (n, d) ≠ (2, 4), and gcd(q, d) = gcd(q, d ? 1) = 1. This allows us to give a complete criterion in the case where q = p is a prime number. In particular, we show the following result: If X is a smooth hypersurface of dimension n and degree d admitting an automorphism of prime order p then p < (d ? 1) ⁿ⁺¹; and if p > (d ? 1) ⁿ then X is isomorphic to the Klein hypersurface, n = 2 or n + 2 is prime, and p = Φ _n+2(1 ? d) where Φ _n+2 is the (n+2)-th cyclotomic polynomial. Finally, we provide some applications to intermediate jacobians of Klein hypersurfaces.     

Small point sets of PG(n, p 3h ) intersecting each line in 1 mod p h points

The main result of this paper is that point sets of PG(n, q), q = p ^3h , p ≥ 7 prime, of size < 3(q ^n-1 + 1)/2 intersecting each line in 1 modulo ${\sqrt[3] q}$ points (these are always small minimal blocking sets with respect to lines) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size < 3(p ^3(n-1) + 1)/2 with respect to lines are always linear.     

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.     

Maximal function and Riesz potential on p-adic linear spaces

For maximal function and Riesz potential on p-adic linear space ? _p ⁿ we give sufficient conditions of its boundedness in generalized Morrey spaces. For radial weights of special kind this condition for Riesz potential is sharp. Also we prove that if Riesz potential I ^α(f) exists at point b, then b is L ^q Lebesgue point for some q.     

Atomic decomposition characterizations of weighted multiparameter Hardy spaces

Let w ?? A _??. In this paper, we introduce weighted-(p, q) atomic Hardy spaces H _w ^p,q (? ⁿ ×? ^m ) for 0 < p ? 1, q >q _w and show that the weighted Hardy space H _w ^p (? ⁿ × ? ^m ) defined via Littlewood-Paley square functions coincides with H _w ^p,q (? ⁿ × ? ^m ) for 0 < p ? 1, q > q _w . As applications, we get a general principle on the H _w ^p (? ⁿ × ? ^m ) to L _w ^p (? ⁿ ×? ^m ) boundedness and a boundedness criterion for two parameter singular integrals on the weighted Hardy spaces.     

On approximation of real numbers by algebraic numbers of bounded degree

Dirichlet proved that for any real irrational number ξ there exist infinitely many rational numbers p/q such that |ξ−p/q|<q⁻². The correct generalization to the case of approximation by algebraic numbers of degree ?n, n>2, is still unknown. Here we prove a result which improves all previous estimates concerning this problem for n>2.     

Boundedness of Schrödinger Type Propagators on Modulation Spaces

It is known that Fourier integral operators arising when solving Schrödinger-type operators are bounded on the modulation spaces ? ^p,q , for 1≤p= q≤∞, provided their symbols belong to the Sjöstrand class M ^∞,1. However, they generally fail to be bounded on ? ^p,q for p≠q. In this paper we study several additional conditions, to be imposed on the phase or on the symbol, which guarantee the boundedness on ? ^p,q for p≠q, and between ? ^p,q →? ^q,p , 1≤q<p≤∞. We also study similar problems for operators acting on Wiener amalgam spaces, recapturing, in particular, some recent results for metaplectic operators. Our arguments make heavily use of the uncertainty principle.     

Asymptotics of the complexity of systems of integer linear forms for additive computations

The computational complexity of integer linear forms is studied. By l ₂(A) we denote the minimal number of the additions and subtractions required for computing the system of p linear forms in q variables x ₁, x ₂, …, x _q that are defined by an integer matrix A of size p × q (repeated use of the results of intermediate computation is permitted). We show that l ₂(A) ? log D(A), where D(A) is the maximum of the absolute values of the minors of A over all minors from order 1 to order min (p, q) (Theorem 1). Moreover, for each sequence of matrices A(n) of size p(n) × q(n) satisfying the condition p + q = o ((log log D(A))^1/2) as n → ∞ the bound l ₂(A) ? log D(A) + o(log D(A)) is valid (Theorem 2). Hence, for all fixed (and even weakly increasing) sizes of matrices that determine a system of integer linear forms, the upper bound on the computational complexity of this system is asymptotically equal to the lower bound.     

Orlicz-Hardy spaces and their duals

We establish the theory of Orlicz-Hardy spaces generated by a wide class of functions.The class will be wider than the class of all the N-functions.In particular,we consider the non-smooth atomic decomposition.The relation between Orlicz-Hardy spaces and their duals is also studied.As an application,duality of Hardy spaces with variable exponents is revisited.This work is different from earlier works about Orlicz-Hardy spaces H(Rn)in that the class of admissible functions is largely widened.We can deal with,for example,(r)≡(rp1(log(e+1/r))q1,0r 6 1,rp2(log(e+r))q2,r1,with p1,p2∈(0,∞)and q1,q2∈(.∞,∞),where we shall establish the boundedness of the Riesz transforms on H(Rn).In particular,is neither convex nor concave when 0p11p2∞,0p21p1∞or p1=p2=1 and q1,q20.If(r)≡r(log(e+r))q,then H(Rn)=H(logH)q(Rn).We shall also establish the boundedness of the fractional integral operators I of order∈(0,∞).For example,I is shown to be bounded from H(logH)1./n(Rn)to Ln/(n.)(log L)(Rn)for 0n.     

Nikol’skii’s inequality for different metrics and properties of the sequence of norms of the Fourier sums of a function in the Lorentz space

Let (X, Y) be a pair of normed spaces such that X ? Y ? L ₁[0, 1] ⁿ and {e _k } _k be an expanding sequence of finite sets in ? ⁿ with respect to a scalar or vector parameter k, k ∈ ? or k ∈ ? ⁿ . The properties of the sequence of norms $\{ \left\| {S_{e_k } (f)} \right\|x\} _k $ of the Fourier sums of a fixed function f ∈ Y are studied. As the spaces X and Y, the Lebesgue spaces L _p [0, 1], the Lorentz spaces L _p,q [0, 1], L _p,q [0, 1] ⁿ , and the anisotropic Lorentz spaces L _p,q*[0, 1] ⁿ are considered. In the one-dimensional case, the sequence {e _k } _k consists of segments, and in the multidimensional case, it is a sequence of hyperbolic crosses or parallelepipeds in ? ⁿ . For trigonometric polynomials with the spectrum given by step hyperbolic crosses and parallelepipeds, various types of inequalities for different metrics in the Lorentz spaces L _p,q [0, 1] ⁿ and L _p,q*[0, 1] ⁿ are obtained.     

Estimates for wave and Klein-Gordon equations on modulation spaces

We prove that the fundamental semi-group eit(m 2I+|Δ|)1/2(m = 0) of the Klein-Gordon equation is bounded on the modulation space M ps,q(Rn) for all 0 < p,q ∞ and s ∈ R.Similarly,we prove that the wave semi-group eit|Δ|1/2 is bounded on the Hardy type modulation spaces μsp,q(Rn) for all 0 < p,q ∞,and s ∈ R.All the bounds have an asymptotic factor tn|1/p 1/2| as t goes to the infinity.These results extend some known results for the case of p 1.Also,some applications for the Cauchy problems related to the semi-group eit(m2I+|Δ|)1/2 are obtained.Finally we discuss the optimum of the factor tn|1/p 1/2| and raise some unsolved problems.     

Asymptotic enumeration of labeled multigraphs by vertices,edges, and degree parities

We show that the number of labeled (n, q)-multigraphs with some specified p vertices of odd degree is asymptotically independent of p and is in fact asymptotically 2^1?n times the number of (n, q)-multigraphs. We determine the asymptotic number of (n, q)-multigraphs.     

On deep holes of standard Reed-Solomon codes

Determining deep holes is an important open problem in decoding Reed-Solomon codes. It is well known that the received word is trivially a deep hole if the degree of its Lagrange interpolation polynomial equals the dimension of the Reed-Solomon code. For the standard Reed-Solomon codes [p-1, k]p with p a prime, Cheng and Murray conjectured in 2007 that there is no other deep holes except the trivial ones. In this paper, we show that this conjecture is not true. In fact, we find a new class of deep holes for standard Reed-Solomon codes [q-1, k]q with q a power of the prime p. Let q≥4 and 2≤k≤q-2. We show that the received word u is a deep hole if its Lagrange interpolation polynomial is the sum of monomial of degree q-2 and a polynomial of degree at most k-1. So there are at least 2(q-1)qk deep holes if k q-3.     

Bilinear Relative Equilibria of Identical Point Vortices

A new class of bilinear relative equilibria of identical point vortices in which the vortices are constrained to be on two perpendicular lines, conveniently taken to be the x- and y-axes of a Cartesian coordinate system, is introduced and studied. In the general problem we have m vortices on the y-axis and n on the x-axis. We define generating polynomials q(z) and p(z), respectively, for each set of vortices. A second-order, linear ODE for p(z) given q(z) is derived. Several results relating the general solution of the ODE to relative equilibrium configurations are established. Our strongest result, obtained using Sturm’s comparison theorem, is that if p(z) satisfies the ODE for a given q(z) with its imaginary zeros symmetric relative to the x-axis, then it must have at least n?m+2 simple, real zeros. For m=2 this provides a complete characterization of all zeros, and we study this case in some detail. In particular, we show that, given q(z)=z ²+η ², where η is real, there is a unique p(z) of degree n, and a unique value of η ²=A _n , such that the zeros of q(z) and p(z) form a relative equilibrium of n+2 point vortices. We show that $A_{n} \approx\frac{2}{3}n + \frac{1}{2}$ , as n→∞, where the coefficient of n is determined analytically, the next-order term numerically. The paper includes extensive numerical documentation on this family of relative equilibria.     

Some New Maximal Sets of Mutually Orthogonal Latin Squares

Two ways of constructing maximal sets of mutually orthogonal Latin squares are presented. The first construction uses maximal partial spreads in PG(3, 4) \ PG(3, 2) with r lines, where r ∈ {6, 7}, to construct transversal-free translation nets of order 16 and degree r + 3 and hence maximal sets of r + 1 mutually orthogonal Latin squares of order 16. Thus sets of t MAXMOLS(16) are obtained for two previously open cases, namely for t = 7 and t = 8. The second one uses the (non)existence of spreads and ovoids of hyperbolic quadrics Q ⁺ (2m + 1, q), and yields infinite classes of q ^{2n ? 1} ? 1 MAXMOLS(q ²ⁿ ), for n ≥ 2 and q a power of two, and for n = 2 and q a power of three.     

Existence and scattering theory for Boussinesq type equations with singular data

We study the initial value problem for the generalized Boussinesq equation and prove existence of local and global solutions with singular initial data in weak-Lp spaces. Our class of initial data for global existence is larger than that of Cho and Ozawa (2007) [7]. Long time behavior results are obtained and a scattering theory is proved in that framework. With more structure, we show Sobolev H¹ and Lorentz-type L(p,q) regularity properties for the obtained solutions. The approach employed is unified for all dimensions n?1.     

Application of wavelet bases in linear and nonlinear approximation to functions from Besov spaces

Linear and nonlinear approximations to functions from Besov spaces B _{p, q} ^σ ([0, 1]), σ > 0, 1 ≤ p, q ≤ ∞ in a wavelet basis are considered. It is shown that an optimal linear approximation by a D-dimensional subspace of basis wavelet functions has an error of order D ^{-min(σ, σ + 1/2 ? 1/p)} for all 1 ≤ p ≤ ∞ and σ > max(1/p ? 1/2, 0). An original scheme is proposed for optimal nonlinear approximation. It is shown how a D-dimensional subspace of basis wavelet functions is to be chosen depending on the approximated function so that the error is on the order of D ^?σ for all 1 ≤ p ≤ ∞ and σ > max(1/p ? 1/2, 0). The nonlinear approximation scheme proposed does not require any a priori information on the approximated function.     

Weyl product algebras and modulation spaces

We discuss algebraic properties of the Weyl product acting on modulation spaces. For a certain class of weight functions ω we prove that is an algebra under the Weyl product if p∈[1,∞] and 1?q?min(p,p^′). For the remaining cases p∈[1,∞] and min(p,p^′)<q?∞ we show that the unweighted spaces Mp,q are not algebras under the Weyl product.     

More coverings by rook domains

The set V_kⁿ of all n-tuples (x₁, x₂,…, x_n) with x_i?, $\text{Z}$_k is considered. The problem treated in this paper is determining σ(n, k), the minimum size of a set W ? V_kⁿ such that for each x in V_kⁿ, there is an element in W that differs from x in at most one coordinate. By using a new constructive method, it is shown that σ(n, p) ? (p ? t + 1)p^n?r, where p is a prime and $\text{n = 1 +}\text{t(p}{}^{\mathrm{r?1}}\text{? 1)}\text{(p ? 1)}$ for some integers t and r. The same method also gives σ(7, 3) ? 216. Another construction gives the inequality σ(n, kt) ? σ(n, k)t^n?1 which implies that σ(q + 1, qt) = q^q?1t^q when q is a prime power. By proving another inequality σ(np + 1, p) ? σ(n, p)p^n(p?1), σ(10, 3) ? 5 · 3⁶ and σ(16, 5) ? 13 · 5¹² are obtained.