Sturm–Liouville problems in weighted spaces in domains with nonsmooth edges. I

We consider (in general noncoercive) mixed problems in a bounded domain D in ? ⁿ for a second-order elliptic partial differential operator A(x, ?). It is assumed that the operator is written in divergent form in D, the boundary operator B(x, ?) is the restriction of a linear combination of the function and its derivatives to ?D and the boundary of D is a Lipschitz surface. We separate a closed set Y ? ?D and control the growth of solutions near Y. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, where the weight is a power of the distance to the singular set Y. Finally, we prove the completeness of the root functions associated with L.The article consists of two parts. The first part published in the present paper, is devoted to exposing the theory of the special weighted Sobolev–Slobodetskii? spaces in Lipschitz domains. We obtain theorems on the properties of these spaces; namely, theorems on the interpolation of these spaces, embedding theorems, and theorems about traces. We also study the properties of the weighted spaces defined by some (in general) noncoercive forms.     

On the Completeness of Products of Harmonic Functions and the Uniqueness of the Solution of the Inverse Acoustic Sounding Problem

It is proved that the family of all pairwise products of regular harmonic functions on D and of the Newtonian potentials of points on the line L ? Rⁿ is complete in L₂(D), where D is a bounded domain in Rⁿ, n ≥ 3, such that \(\bar D\) ∩ L = ?. This result is used in the proof of uniqueness theorems for the inverse acoustic sounding problem in R³.     

Sufficient conditions for the existence of 0’-limitwise monotonic functions for computable <Emphasis Type="Italic">η</Emphasis>-like linear orders

We find new sufficient conditions for the existence of a 0’-limitwise monotonic function defining the order for a computable η-like linear order L, i.e., of a function G such that L ∑ _q∈? G(q). Namely, we define the notions of left local maximal block and right local maximal block and prove that if the sizes of these blocks in a computable η-like linear order L are bounded then there is a 0’-limitwise monotonic function G with L = ∑ _q∈? G(q).     

On recognizability of some finite simple orthogonal groups by spectrum

It is proved that, if G is a finite group that has the same set of element orders as the simple group D _p (q), where p is prime, p ≥ 5 and q ∈ {2, 3, 5}, then the commutator group of G/F(G) is isomorphic to D _p (q), the subgroup F(G) is equal to 1 for q = 5 and to O _q (G) for q ∈ {2, 3}, F(G) ≤ G′, and |G/G′| ≤ 2.     

A Note on MDS Codes, <Emphasis Type="Italic">n</Emphasis>-Arcs and Complete Designs

A generalized incidence matrix of a design over GF(q) is any matrix obtained from the (0, 1)-incidence matrix by replacing ones with nonzero elements from GF(q). The dimension d _q of a design D over GF(q) is defined as the minimum value of the q-rank of a generalized incidence matrix of D. It is proved that the dimension d _q of the complete design on n points having as blocks all w-subsets, is greater that or equal to n ? w + 1, and the equality d _q = n ? w + 1 holds if and only if there exists an [n, n ? w + 1, w] MDS code over GF(q), or equivalently, an n-arc in PG(w ? 2, q).     

A mapping method in inverse Sturm-Liouville problems with singular potentials

In the space L ₂[0, π], the Sturm-Liouville operator L _D(y) = ?y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L ₂[0, π], i.e., q ∈ W ₂ ^?1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L _D is solved in the subspace of odd real functions σ(π/2 ? x) = ?σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.     

On the polyhedral method of solving problems of control strategy synthesis

Based on the method of nonlinear capacity, we study the nonexistence of nonnegative monotonic solutions for the quasilinear elliptic inequality of the form ?Δ_pu ≥ u^q in a half-space in terms of the parameters p and q.     

On the properties of maps connected with inverse Sturm-Liouville problems

Let L _D be the Sturm-Liouville operator generated by the differential expression L _y = ?y″ + q(x)y on the finite interval [0, π] and by the Dirichlet boundary conditions. We assume that the potential q belongs to the Sobolev space W ₂ ^? [0, π] with some ? ≥ ?1. It is well known that one can uniquely recover the potential q from the spectrum and the norming constants of the operator L _D. In this paper, we construct special spaces of sequences ? ₂ ^θ in which the regularized spectral data {s _k } _?∞ ^∞ of the operator L _D are placed. We prove the following main theorem: the map F _q = {s _k } from W ₂ ^? to ? ₂ ^θ is weakly nonlinear (i.e., it is a compact perturbation of a linear map). A similar result is obtained for the operator L _DN generated by the same differential expression and the Dirichlet-Neumann boundary conditions. These results serve as a basis for solving the problem of uniform stability of recovering a potential. Note that this problem has not been considered in the literature. The uniform stability results are formulated here, but their proof will be presented elsewhere.     

Two problems of the theory of quadratic maps

Given quadratic forms q ₁, …, q _k , two questions are studied: Under what conditions does the set of common zeros of these quadratic forms consist of the only point x = 0? When is the maximum of these quadratic forms nonnegative or positive for any x ≠ 0? Criteria for each of these conditions to hold are obtained. These criteria are stated in terms of matrices determining the quadratic forms under consideration.     

Bessel Property of the System of Root Functions of a Second-Order Singular Operator on an Interval

For the system of root functions of an operator defined by the differential operation ?u″ + p(x)u′ + q(x)u, x ∈ G = (0, 1), with complex-valued singular coefficients, sufficient conditions for the Bessel property in the space L₂(G) are obtained and a theorem on the unconditional basis property is proved. It is assumed that the functions p(x) and q(x) locally belong to the spaces L₂ and W₂^?1, respectively, and may have singularities at the endpoints of G such that q(x) = q_R(x) +q′S(x) and the functions q_S(x), p(x), q ₂ ^S (x)w(x), p²(x)w(x), and q_R(x)w(x) are integrable on the whole interval G, where w(x) = x(1 ? x).     

Blow-up of <Emphasis Type="Italic">p</Emphasis>-Laplacian evolution equations with variable source power

We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power

$${s_j} = {\beta _j} + \overline {{\beta _{n - j}}}p$$

where Ω is either a bounded domain or the whole space ? ^N , q(x) is a positive and continuous function defined in Ω with 0 < q _? = inf q(x) ? q(x) ? sup q(x) = q+ < ∞. It is demonstrated that the equation with variable source power has much richer dynamics with interesting phenomena which depends on the interplay of q(x) and the structure of spatial domain Ω, compared with the case of constant source power. For the case that Ω is a bounded domain, the exponent p ? 1 plays a crucial role. If q+ > p ? 1, there exist blow-up solutions, while if q ₊ < p ? 1, all the solutions are global. If q _? > p ? 1, there exist global solutions, while for given q _? < p ? 1 < q ₊, there exist some function q(x) and Ω such that all nontrivial solutions will blow up, which is called the Fujita phenomenon. For the case Ω = ? ^N , the Fujita phenomenon occurs if 1 < q _? ? q ₊ ? p ? 1 + p/N, while if q _? > p ? 1 + p/N, there exist global solutions.

     

Estimation of the depth of reversible circuits consisting of NOT,CNOT and 2-CNOT gates

The paper discusses the asymptotic depth of a reversible circuits consisting of NOT, CNOT and 2-CNOT gates. The reversible circuit depth function D(n, q) is introduced for a circuit implementing a mapping f: Z₂ⁿ → Z₂ⁿ as a function of n and the number q of additional inputs. It is proved that for the case of implementation of a permutation from A(Z₂ⁿ) with a reversible circuit having no additional inputs the depth is bounded as D(n, 0) ? 2ⁿ/(3log₂n). It is also proved that for the case of transformation f: Z₂ⁿ → Z₂ⁿ with a reversible circuit having q₀ ~ 2ⁿ additional inputs the depth is bounded as D(n,q₀) ? 3n.     

The Einstein-Kähler metric on the third Cartan-Hartogs domain

In this paper we discuss the Einstein-Kahler metric on the third Cartan-Hartogs domain Y111（n, q; K）. Firstly we get the complete Einstein Kahler metric with explicit form on Y111（n, q; K） in the case of K=q/2 ＋ 1/q-1. Secondly we obtain the holomorphic sectional curvature under this metric and get the sharp estimate for this holomorphic curvature. Finally we prove that the complete Einstein-Kahler metric is equivalent to the Bergman metric on Y111（n, q; K） in case of K=q/2＋1/q-1.     

Removability of isolated singularities for anisotropic elliptic equations with gradient absorption

We study the well-posedness of the third-order degenerate differential equation \(\left( {{P_3}} \right):\alpha {\left( {Mu} \right)^{\prime \prime \prime }}\left( t \right) + {\left( {Mu} \right)^{\prime \prime }}\left( t \right) = \beta Au\left( t \right) + f\left( t \right)\), (t ∈ [0, 2p]) with periodic boundary conditions \(Mu\left( 0 \right) = Mu\left( {2\pi } \right),\;Mu'\left( 0 \right) = Mu'\left( {2\pi } \right),\;Mu''\left( 0 \right) = Mu''\left( {2\pi } \right)\), in periodic Lebesgue–Bochner spaces L^p(T,X), periodic Besov spaces B_p,q^s(T,X) and periodic Triebel–Lizorkin spaces F_p,q^s(T,X), where A, B and M are closed linear operators on a Banach space X satisfying D(A) \( \cap \)D(B) ? D(M) and α, β, γ ∈ R. Using known operator-valued Fourier multiplier theorems, we completely characterize the well-posedness of (P₃) in the above three function spaces.     

Invariants of finite groups generated by generalized transvections in the modular case

We investigate the invariant rings of two classes of finite groups G ≤ GL(n, F _q) which are generated by a number of generalized transvections with an invariant subspace H over a finite field F _q in the modular case. We name these groups generalized transvection groups. One class is concerned with a given invariant subspace which involves roots of unity. Constructing quotient groups and tensors, we deduce the invariant rings and study their Cohen-Macaulay and Gorenstein properties. The other is concerned with different invariant subspaces which have the same dimension. We provide a explicit classification of these groups and calculate their invariant rings.     

Some notes on embedding for anisotropic Sobolev spaces

In this paper, we prove new embedding theorems for generalized anisotropic Sobolev spaces, \(W_{{\Lambda ^{p,q}}(w)}^{{r_1}, \cdots ,{r_n}}\) and \(W_X^{{r_1}, \cdots ,{r_n}}\), where Λ ^p,q (w) is the weighted Lorentz space and X is a rearrangement invariant space in ? ⁿ . The main methods used in the paper are based on some estimates of nonincreasing rearrangements and the applications of B _p weights.     

Decomposition theorems for<Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> spaces

We study the Möbius invariant spacesQ _p andQ _{p, 0} of analytic functions. These scales of spaces include BMOA=Q₁, VMOA=Q_{1, 0} and the Dirichlet space=Q₀. Using the Bergman metric, we establish decomposition theorems for these spaces. We obtain also a fractional derivative characterization for bothQ _p andQ _{p, 0} .     

Information of orderα and typeβ

Information I_α ^β (Q/P) of orderα and typeβ is introduced and it is shown that for every fixedβ, this information is a monotonic increasing function ofα. It is also shown that information of orderα and type 1 is non-negative when\(\sum\limits_{k = 1}^N { q_k } \geqslant \sum\limits_{k = 1}^N { p_k } \), where (q ₁,q ₂ …,q _N) and (p ₁,p ₂, …,p _N) are generalised probability distributions for Q and P respectively.     

On Extraction of Smooth Solutions of a Class of Almost Hypoelliptic Equations with Constant Power

A linear differential operator P(x, D) = P(x₁,... x _n , D₁,..., D _n ) = ∑_αγ_α(x)D^α with coefficients γ_α(x) defined in E ⁿ is called formally almost hypoelliptic in E ⁿ if all the derivatives D^ν_ξP(x, ξ) can be estimated by P(x, ξ), and the operator P(x, D) has uniformly constant power in Eⁿ. In the present paper, we prove that if P(x, D) is a formally almost hypoelliptic operator, then all solutions of equation P(x, D)u = 0, which together with some of their derivatives are square integrable with a specified exponential weight, are infinitely differentiable functions.     

Second order linear <Emphasis Type="Italic">q</Emphasis>-difference equations: Nonoscillation and asymptotics

The paper can be understood as a completion of the q-Karamata theory along with a related discussion on the asymptotic behavior of solutions to the linear q-difference equations. The q-Karamata theory was recently introduced as the theory of regularly varying like functions on the lattice \({q^{{{\Bbb N}_0}}}: = \left\{ {{q^k}:k \in {{\Bbb N}_0}} \right\}\) with q > 1. In addition to recalling the existing concepts of q-regular variation and q-rapid variation we introduce q-regularly bounded functions and prove many related properties. The q-Karamata theory is then applied to describe (in an exhaustive way) the asymptotic behavior as t → ∞ of solutions to the q-difference equation D _q ² y(t) + p(t)y(qt) = 0, where \(p:q^{\mathbb{N}_0 } \to \mathbb{R}\). We also present the existing and some new criteria of Kneser type which are related to our subject. A comparison of our results with their continuous counterparts is made. It reveals interesting differences between the continuous case and the q-case and validates the fact that q-calculus is a natural setting for the Karamata like theory and provides a powerful tool in qualitative theory of dynamic equations.