Random embedding of $${\ell_p^n}$$ into $${\ell_r^N}$$

For any 0 < p < 2 and any natural numbers N > n, we give an explicit definition of a random operator \({S : \ell_p^n \to \mathbb{R}^N}\) such that for every 0 < r < p < 2 with r ≤ 1, the operator \({S_r = S : \ell_p^n \to \ell_r^N}\) satisfies with overwhelming probability that \({\|S_r\| \, \|(S_r)_{| {\rm Im}\, S}^{-1}\| \le C(p,r)^{n/(N-n)}}\), where C(p, r) > 0 is a real number depending only on p and r. One of the main tools that we develop is a new type of multidimensional Esseen inequality for studying small ball probabilities.     

2-Arc-transitive cyclic covers of $$K_{n,n}$$

In this paper, a complete classification is achieved of all the regular covers of the complete bipartite graphs \(K_{n,n}\) with cyclic covering transformation group, whose fibre-preserving automorphism group acts 2-arc-transitively. All these covers consist of one threefold covers of \(K_{6,6}\), one twofold cover of \(K_{12, 12}\) and one infinite family X(r, p) of p-fold covers of \(K_{p^r,p^r}\) with p a prime and r an integer such that \(p^r\ge 3\). This infinite family X(r, p) can be derived by a very simple and nice voltage assignment f as follows: \(X(r, p)=K_{p^r, p^r}\times _f \mathbb {Z}_p\), where \(K_{p^r, p^r}\) is a complete bipartite graph with the bipartition \(V=\{ \alpha \bigm |\alpha \in V(r,p)\}\cup \{ \alpha '\bigm |\alpha \in V(r,p)\}\) for the r-dimensional vector space V(r, p) over the field of order p and \(f_{\alpha ,\beta '}=\sum _{i=1}^ra_ib_i,\,\, \mathrm{for\,\,all}\,\,\alpha =(a_i)_r, \beta =(b_i)_r\in V(r,p)\).     

Structural adaptive deconvolution under $${\mathbb{L}_p}$$-losses

In this paper, we address the problem of estimating a multidimensional density f by using indirect observations from the statistical model Y = X + ε. Here, ε is a measurement error independent of the random vector X of interest and having a known density with respect to Lebesgue measure. Our aim is to obtain optimal accuracy of estimation under \({\mathbb{L}_p}\)-losses when the error ε has a characteristic function with a polynomial decay. To achieve this goal, we first construct a kernel estimator of f which is fully data driven. Then, we derive for it an oracle inequality under very mild assumptions on the characteristic function of the error ε. As a consequence, we getminimax adaptive upper bounds over a large scale of anisotropic Nikolskii classes and we prove that our estimator is asymptotically rate optimal when p ∈ [2,+∞]. Furthermore, our estimation procedure adapts automatically to the possible independence structure of f and this allows us to improve significantly the accuracy of estimation.     

Set-theoretic complete intersection monomial curves in $${\mathbb{P}^n}$$

In this paper, we give a sufficient numerical criterion for a monomial curve in a projective space to be a set-theoretic complete intersection. Our main result generalizes a similar statement proven by Keum for monomial curves in three-dimensional projective space. We also prove that there are infinitely many set-theoretic complete intersection monomial curves in the projective n?space for any suitably chosen n ? 1 integers. In particular, for any positive integers p, q, where gcd(p, q) = 1, the monomial curve defined by p, q, r is a set-theoretic complete intersection for every \({r \geq pq( q - 1)}\).     

Under-recurrence in the Khintchine recurrence theorem

The Khintchine recurrence theorem asserts that in a measure preserving system, for every set A and ε > 0, we have μ(A ∩ T^?nA) ≥ μ(A)² ? ε for infinitely many n ∈ N. We show that there are systems having underrecurrent sets A, in the sense that the inequality μ(A ∩ T^?nA) < μ(A)² holds for every n ∈ N. In particular, all ergodic systems of positive entropy have under-recurrent sets. On the other hand, answering a question of V. Bergelson, we show that not all mixing systems have under-recurrent sets. We also study variants of these problems where the previous strict inequality is reversed, and deduce that under-recurrence is a much more rare phenomenon than over-recurrence. Finally, we study related problems pertaining to multiple recurrence and derive some interesting combinatorial consequences.     

The Erdös-Kac theorem for additive arithmetical semigroups

We show that the Erdös-Kac theorem for additive arithmetical semigroups can be proved under the condition that the counting function of elements has the asymptotics G(n) = q ⁿ (A + O(1/(lnn)^k) as n → ∞ with A > 0, q > 1, and arbitrary k ∈ ? and that P(n) = O(q ⁿ /n) for the number of prime elements of degree n. This improves a result of Zhang.     

Ck-regularity for the $$\bar \partial $$ -equation with a support condition

Let D be a C ^d q-convex intersection, d ≥ 2, 0 ≤ q ≤ n ? 1, in a complex manifold X of complex dimension n, n ≥ 2, and let E be a holomorphic vector bundle of rank N over X. In this paper, C ^k -estimates, k = 2, 3,...,∞, for solutions to the \(\bar \partial \)-equation with small loss of smoothness are obtained for E-valued (0, s)-forms on D when n ? q ≤ s ≤ n. In addition, we solve the \(\bar \partial \)-equation with a support condition in C ^k -spaces. More precisely, we prove that for a \(\bar \partial \)-closed form f in C _0,q ^k (X D,E), 1 ≤ q ≤ n ? 2, n ≥ 3, with compact support and for ε with 0 < ε < 1 there exists a form u in C _0,q?1 ^k?ε (X D,E) with compact support such that \(\bar \partial u = f\) in \(X\backslash \bar D\). Applications are given for a separation theorem of Andreotti-Vesentini type in C ^k -setting and for the solvability of the \(\bar \partial \)-equation for currents.     

Lattices of topologies of unary algebras of the variety $$ \mathcal{A}_{1,1} $$

We consider the variety of unary algebras 〈A, f, g〉 defined by the identities f(g(x)) = g(f(x)) = x. We describe algebras of this variety, whose topology lattices are modular, distributive, linearly ordered, complemented, or pseudocomplemented.     

Homogenization of hyperbolic equations

We consider a self-adjoint matrix elliptic operator A _{ε, ε} > 0, on L ₂(R ^d ;C ⁿ ) given by the differential expression b(D)*g(x/ε)b(D). The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice; b(D) is an (m × n)-matrix first order differential operator such that m ≥ n and the symbol b(ξ) has maximal rank. We study the operator cosine cos(τA _ε ^1/2 ), where τ ∈ R. It is shown that, as ε → 0, the operator cos(τA _ε ^1/2 ) converges to cos(τ(A ⁰)^1/2) in the norm of operators acting from the Sobolev space H ^s (R ^d ;C ⁿ ) (with a suitable s) to L ₂(R ^d ;C ⁿ ). Here A ₀ is the effective operator with constant coefficients. Sharp-order error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation ? _τ ² u _ε (x, τ) = ?A _ε u _ε (x, τ).     

A new family of differentially 4-uniform permutations over $$\mathbb{F}_{2^{2k} }$$ for odd k

We study the differential uniformity of a class of permutations over \(\mathbb{F}_{2^n } \) with n even. These permutations are different from the inverse function as the values x^?1 are modified to be (γx)^? on some cosets of a fixed subgroup 〈γ〉 of \(\mathbb{F}_{2^n }^* \). We obtain some sufficient conditions for this kind of permutations to be differentially 4-uniform, which enable us to construct a new family of differentially 4-uniform permutations that contains many new Carlet-Charpin-Zinoviev equivalent (CCZ-equivalent) classes as checked by Magma for small numbers n. Moreover, all of the newly constructed functions are proved to possess optimal algebraic degree and relatively high nonlinearity.     

On the primitive divisors of the recurrent sequence $$u_{n+1}=(4\rm{cos}^2(2\pi/7)-1)\it{u}_{n}-u_{n-\rm{1}}$$ with applications to group theory

Consider the sequence of algebraic integers u_n given by the starting values u₀ = 0, u₁ = 1 and the recurrence \(u_{n+1}=(4\rm{cos}^2(2\pi/7)-1)\it{u}_{n}-u_{n-\rm{1}}\). We prove that for any n ? {1, 2, 3, 5, 8, 12, 18, 28, 30} the n-th term of the sequence has a primitive divisor in \(\mathbb{Z}[2\rm{cos}(2\pi/7)]\). As a consequence we deduce that for any suffciently large n there exists a prime power q such that the group PSL₂(q) can be generated by a pair x, y with \(x^2=y^3=(xy)^7=1\) and the order of the commutator [x, y] is exactly n. The latter result answers in affrmative a question of Holt and Plesken.     

On solutions to two-dimensional systems realizing the transition from an unstable equilibrium to a stable cycle

For a two-dimensional dynamical system at ?∞ < t < +∞, a process describing the transition from an arbitrary neighborhood of an unstable equilibrium to a stable limit cycle is studied. The system is reduced to the Poincaré normal form. An approximate solution is constructed as a polynomial of degree 2N containing only even degrees of the small parameter ε. The functional classes to which the coefficients of this polynomial belong are described. The function space containing the exact solution differing from the approximate one by O(ε^2N+1) is determined.     

On homogenization for non-self-adjoint locally periodic elliptic operators

In this note we consider the homogenization problem for a matrix locally periodic elliptic operator on R ^d of the form A ^ε = ?divA(x, x/ε)?. The function A is assumed to be Hölder continuous with exponent s ∈ [0, 1] in the “slow” variable and bounded in the “fast” variable. We construct approximations for (A ^ε ? μ)^?1, including one with a corrector, and for (?Δ) ^s/2(A ^ε ? μ)^?1 in the operator norm on L ₂(R ^d ) ⁿ . For s ≠ 0, we also give estimates of the rates of approximation.     

Output stabilization for a class of uncertain systems

The system

$$\frac{{dx}}{{dt}} = A\left( \cdot \right)x + B\left( \cdot \right)u,{\kern 1pt} \frac{{dy}}{{dt}} = A\left( \cdot \right)y + B\left( \cdot \right)u + D\left( {C*y - v} \right)$$

where v = C*x is an output, u = S*y is a control, A(·) ∈ R ^{n × n} , B(·) ∈ R ^{n × (n–p)}, C ∈ R ^{n × (n–p)}, and D ∈ R ^{n × (n–p)}, is considered. The elements α_ij(·) and β_ij(·) of the matrices A(·) and B(·) are arbitrary functionals satisfying the conditions

$$\mathop {\sup }\limits_{\left( \cdot \right)} |{\alpha _{ij}}\left( \cdot \right)| < \infty \left( {i,j \in 1,n} \right),\mathop {\sup }\limits_{\left( \cdot \right)} |{\beta _{ij}}\left( \cdot \right)| < \infty \left( {i \in 1,n,j \in 1,n - p} \right).$$

It is assumed that A(·) ∈ Z ₁ ∪ Z ₃ and A*(·) ∈ Z ₁ ∪ Z ₃, where Z ₁ is the class of matrices in which the first p elements of the kth superdiagonal are sign-definite and the elements above them are sufficiently small. The class Z ₃ differs from Z _t1 in that the elements between this superdiagonal and the (k + 1)th row are sufficiently small. If k > p, then the elements of the p × p square in the upper left corner of the matrix are sufficiently small as well. By using special quadratic Lyapunov functions, a matrix D for which y(t)–x(t) → 0 exponentially as t → ∞ is first found, and then a matrix S for which the vectors x(t) and y(t) have the same property is constructed.

     

On the reflexivity of $$\mathcal {P}_{w}(^{n}E;F)$$

In this paper we prove that if E and F are reflexive Banach spaces and G is a closed linear subspace of the space \(\mathcal {L}_{K}(E;F)\) of all compact linear operators from E into F, then G is either reflexive or non-isomorphic to a dual space. This result generalizes (Israel J Math 21:38-49, 1975, Theorem 2) and gives the solution to a problem posed by Feder (Ill J Math 24:196-205, 1980, Problem 1). We also prove that if E and F are reflexive Banach spaces, then the space \(\mathcal {P}_{w}(^{n}E;F)\) of all n-homogeneous polynomials from E into F which are weakly continuous on bounded sets is either reflexive or non-isomorphic to a dual space.     

Most small $${\varvec{}}$$-grous have an automorhism of order 2

Let f(p, n) be the number of pairwise nonisomorphic p-groups of order \(p^n\), and let g(p, n) be the number of groups of order \(p^n\) whose automorphism group is a p-group. We prove that the limit, as p grows to infinity, of the ratio g(p, n) / f(p, n) equals 1/3 for \(n=6,7\).     

Asymptotic formula for the convolution of a generalized divisor function

An asymptotic formula is obtained for the sum of terms σ _it (n)σ_-it (N - n) (t is real) over 0 < n < N with a remainder estimated by O _ε((1+|t|)^1+ε N ^3/4+ε) for any ε > 0. As a consequence, Porter’s result on a power scale for the average number of steps in the Euclidean algorithm is improved.     

Equiconvergence of expansions in multiple Fourier series and in fourier integrals with “lacunary sequences of partial sums”

We investigate the equiconvergence on T^N = [?π, π)^N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L_p(T^N) and g ∈ L_p(R^N), p > 1, N ≥ 3, g(x) = f(x) on T^N, in the case where the “partial sums” of these expansions, i.e., S_n(x; f) and J_α(x; g), respectively, have “numbers” n ∈ Z^N and α ∈ R^N (n_j = [α_j], j = 1,..., N, [t] is the integral part of t ∈ R¹) containing N ? 1 components which are elements of “lacunary sequences.”     

Homogenization of Schrödinger-type equations

We consider a self-adjoint elliptic operator A_ε, ε> 0, on L₂(R^d; Cⁿ) given by the differential expression b(D)*g(x/ε)b(D). Here \(b(D) = \sum\nolimits_{j = 1}^d {b_j D_j }\) is a first-order matrix differential operator such that the symbol b(ξ) has maximal rank. The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice. We study the operator exponential \({e^{ - i\tau {A_\varepsilon }}}\), where τ ∈ R. It is shown that, as ε → 0, the operator \({e^{ - i\tau {A_\varepsilon }}}\) converges to \({e^{ - i\tau {A^0}}}\) in the norm of operators acting from the Sobolev space H^s(R^d;Cⁿ) (with suitable s) to L₂(R^d;Cⁿ). Here A⁰ is the effective operator with constant coefficients. Order-sharp error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation i?_τu_ε(x, τ) = A_εu_ε(x, τ).     

Orthogonal Polynomials Associated with Equilibrium Measures on $\mathbb {R}$

Let K be a non-polar compact subset of \(\mathbb {R}\) and μ _K denote the equilibrium measure of K. Furthermore, let P _n (?;μ _K ) be the n-th monic orthogonal polynomial for μ _K . It is shown that \(\|P_{n}\left (\cdot ; \mu _{K}\right )\|_{L^{2}(\mu _{K})}\), the Hilbert norm of P _n (?;μ _K ) in L ²(μ _K ), is bounded below by Cap(K) ⁿ for each \(n\in \mathbb {N}\). A sufficient condition is given for\(\left (\|P_{n}\left (\cdot ;\mu _{K}\right )\|_{L^{2}(\mu _{K})}/\text {Cap}(K)^{n}\right )_{n=1}^{\infty }\) to be unbounded. More detailed results are presented for sets which are union of finitely many intervals.