Σ-definability of uncountable models of <Emphasis Type="Italic">c</Emphasis>-simple theories

We show that each c-simple theory with an additional discreteness condition has an uncountable model Σ-definable in ℍ$ \mathbb{H} $ \mathbb{H} ($ \mathbb{L} $ \mathbb{L} ), where $ \mathbb{L} $ \mathbb{L} is a dense linear order. From this we establish the same for all c-simple theories of finite signature that are submodel complete.     

A$\mathcal{T}$-algebras and extensions of A$\mathbb{T}$-algebras

Lin and Su classified A$ \mathcal{T} $ \mathcal{T} -algebras of real rank zero. This class includes all A$ \mathbb{T} $ \mathbb{T} -algebras of real rank zero as well as many C*-algebras which are not stably finite. An A$ \mathcal{T} $ \mathcal{T} -algebra often becomes an extension of an A$ \mathbb{T} $ \mathbb{T} -algebra by an AF-algebra. In this paper, we show that there is an essential extension of an A$ \mathbb{T} $ \mathbb{T} -algebra by an AF-algebra which is not an A$ \mathcal{T} $ \mathcal{T} -algebra. We describe a characterization of an extension E of an A$ \mathbb{T} $ \mathbb{T} -algebra by an AF-algebra if E is an A$ \mathcal{T} $ \mathcal{T} -algebra.     

Phase portraits for quadratic homogeneous polynomial vector fields on $$ \mathbb{S}^2 $$

Let X be a homogeneous polynomial vector field of degree 2 on $ \mathbb{S}^2 $ \mathbb{S}^2 . We show that if X has at least a non-hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on $ \mathbb{S}^2 $ \mathbb{S}^2 is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16 ^th Hilbert’s problem restricted to this class of polynomial vector fields to the study of two particular families. Moreover, we present two criteria for studying the nonexistence of periodic orbits for homogeneous polynomial vector fields on $ \mathbb{S}^2 $ \mathbb{S}^2 of degree n.     

On a semilattice of numberings

We study some properties of a $ \mathfrak{c} $ \mathfrak{c} -universal semilattice $ \mathfrak{A} $ \mathfrak{A} with the cardinality of the continuum, i.e., of an upper semilattice of m-degrees. In particular, it is shown that the quotient semilattice of such a semilattice modulo any countable ideal will be also $ \mathfrak{c} $ \mathfrak{c} -universal. In addition, there exists an isomorphism $ \mathfrak{A} $ \mathfrak{A} such that $ {\mathfrak{A} \mathord{\left/ {\vphantom {\mathfrak{A} {\iota \left( \mathfrak{A} \right)}}} \right. \kern-\nulldelimiterspace} {\iota \left( \mathfrak{A} \right)}} $ {\mathfrak{A} \mathord{\left/ {\vphantom {\mathfrak{A} {\iota \left( \mathfrak{A} \right)}}} \right. \kern-\nulldelimiterspace} {\iota \left( \mathfrak{A} \right)}} will be also $ \mathfrak{c} $ \mathfrak{c} -universal. Furthermore, a property of the group of its automorphisms is obtained. To study properties of this semilattice, the technique and methods of admissible sets are used. More exactly, it is shown that the semilattice of mΣ-degrees $ L_{m\Sigma }^{\mathbb{H}\mathbb{F}\left( S \right)} $ L_{m\Sigma }^{\mathbb{H}\mathbb{F}\left( S \right)} on the hereditarily finite superstructure $ \mathbb{H}\mathbb{F} $ \mathbb{H}\mathbb{F} (S) over a countable set S will be a $ \mathfrak{c} $ \mathfrak{c} -universal semilattice with the cardinality of the continuum.     

The Bergman kernel function

In this note, we point out that a large family of n×n matrix valued kernel functions defined on the unit disc $ \mathbb{D} \subseteq \mathbb{C} $ \mathbb{D} \subseteq \mathbb{C} , which were constructed recently in [9], behave like the familiar Bergman kernel function on $ \mathbb{D} $ \mathbb{D} in several different ways. We show that a number of questions involving the multiplication operator on the corresponding Hilbert space of holomorphic functions on $ \mathbb{D} $ \mathbb{D} can be answered using this likeness.     

Complete classification of surfaces with a canonical principal direction in the Euclidean space \mathbb{E} 3

In the present paper we classify all surfaces in $ \mathbb{E} $ \mathbb{E} ³ with a canonical principal direction. Examples of this type of surfaces are constructed. We prove that the only minimal surface with a canonical principal direction in the Euclidean space $ \mathbb{E} $ \mathbb{E} ³ is the catenoid.     

Integral-type operators from a mixed norm space to a bloch-type space on the unit ball

Let $ \mathbb{B} $ \mathbb{B} be the unit ball in ℂ ⁿ and let H($ \mathbb{B} $ \mathbb{B} ) be the space of all holomorphic functions on $ \mathbb{B} $ \mathbb{B} . We introduce the following integral-type operator on H($ \mathbb{B} $ \mathbb{B} ):

Linear maps that strongly preserve regular matrices over the Boolean algebra

The set of all m × n Boolean matrices is denoted by $ \mathbb{M} $ \mathbb{M} _m,n . We call a matrix A ∈ $ \mathbb{M} $ \mathbb{M} _m,n regular if there is a matrix G ∈ $ \mathbb{M} $ \mathbb{M} _n,m such that AGA = A. In this paper, we study the problem of characterizing linear operators on $ \mathbb{M} $ \mathbb{M} _m,n that strongly preserve regular matrices. Consequently, we obtain that if min{m, n} ⩽ 2, then all operators on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserve regular matrices, and if min{m, n} ⩾ 3, then an operator T on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserves regular matrices if and only if there are invertible matrices U and V such that T(X) = UXV for all X ε $ \mathbb{M} $ \mathbb{M} _m,n , or m = n and T(X) = UX ^T V for all X ∈ $ \mathbb{M} $ \mathbb{M} _n .     

Completely nonmeasurable unions

Assume that no cardinal κ < 2 ^ω is quasi-measurable (κ is quasi-measurable if there exists a κ-additive ideal $ \mathbb{I} $ \mathbb{I} of X contains uncountably many pairwise disjoint subfamilies $ \mathbb{I} $ \mathbb{I} -Bernstein unions ∪ $ \mathbb{I} $ \mathbb{I} -Bernstein if A and X \ A meet each Borel $ \mathbb{I} $ \mathbb{I} -positive subset B ⊆ X). This result is a generalization of the Four Poles Theorem (see [1]) and results from [2] and [4].     

A characteristic property of the Euclidean disc

In this paper the following is proved: Let K ⊂ $ \mathbb{E}^2 $ \mathbb{E}^2 be a smooth strictly convex body, and let L ⊂ $ \mathbb{E}^2 $ \mathbb{E}^2 be a line. Assume that for every point x ∈ L/K the two tangent segments from x to K have the same length, and the line joining the two contact points passes through a fixed point in the plane. Then K is an Euclidean disc.     

Spaces with fibered approximation property in dimension <Emphasis Type="Italic">n</Emphasis>

A metric space M is said to have the fibered approximation property in dimension n (briefly, M ∈ FAP(n)) if for any ɛ > 0, m ≥ 0 and any map g: $ \mathbb{I} $ \mathbb{I} ^m × $ \mathbb{I} $ \mathbb{I} ⁿ → M there exists a map g′: $ \mathbb{I} $ \mathbb{I} ^m × $ \mathbb{I} $ \mathbb{I} ⁿ → M such that g′ is ɛ-homotopic to g and dim g′ ({z} × $ \mathbb{I} $ \mathbb{I} ⁿ ) ≤ n for all z ∈ $ \mathbb{I} $ \mathbb{I} ^m . The class of spaces having the FAP(n)-property is investigated in this paper. The main theorems are applied to obtain generalizations of some results due to Uspenskij [11] and Tuncali-Valov [10].     

On additive properties of product sets in an arbitrary finite field

It is proved that for any two subsets A and B of an arbitrary finite field $ \mathbb{F}_q $ \mathbb{F}_q such that |A||B| > q, the identity 10AB = $ \mathbb{F}_q $ \mathbb{F}_q holds. Under the assumption |A||B| ⩾2q, this improves to 8AB = $ \mathbb{F}_q $ \mathbb{F}_q .     

Cardinalities of definable sets in superstructures over models

We introduce the notion of a superstructure over a model. This is a generalization of the notion of the hereditarily finite superstructure ℍ$ \mathbb{F}\mathfrak{M} $ \mathbb{F}\mathfrak{M} over a model $ \mathfrak{M} $ \mathfrak{M} . We consider the question on cardinalities of definable (interpretable) sets in superstructures over λ-homogeneous and λ-saturated models.     

Stabilization for the vibrations modeled by the ‘standard linear model’ of viscoelasticity

We study the stabilization of vibrations of a flexible structure modeled by the ‘standard linear model’ of viscoelasticity in a bounded domain in ℝ ⁿ with a smooth boundary. We prove that amplitude of the vibrations remains bounded in the sense of a suitable norm in a space $ \mathbb{X} $ \mathbb{X} , defined explicitly in (22) subject to a restriction on the uncertain disturbing forces on $ \mathbb{X} $ \mathbb{X} . We also estimate the total energy of the system over time interval [0, T] for any T > 0, with a tolerance level of the disturbances. Finally, when the input disturbances are insignificant, uniform exponential stabilization is obtained and an explicit form for the energy decay rate is derived. These results are achieved by a direct method under undamped mixed boundary conditions.     

The Hasse-Witt invariant in some towers of function fields over finite fields

In this article we investigate the p-rank of function fields in several good towers. To do this we first recall and establish some properties of the behaviour of the p-rank under extensions. Then we compute the p-ranks of function fields in several optimal towers over a quadratic field $ \mathbb{F}_{q^2 } $ \mathbb{F}_{q^2 } , as well as for a specific good tower over a cubic field $ \mathbb{F}_{q^3 } $ \mathbb{F}_{q^3 } , which was introduced by Bassa, Garcia and Stichtenoth.     

A construction of maximal linearly independent array

Let M be a subset of r-dimensional vector space Vτ （F2） over a finite field F2, consisting of n nonzero vectors, such that every t vectors of M are linearly independent over F2. Then M is called （n, t）-linearly independent array of length n over Vτ（F2）. The （n, t）-linearly independent array M that has the maximal number of elements is called the maximal （r, t）-linearly independent array, and the maximal number is denoted by M（r, t）. It is an interesting combinatorial structure, which has many applications in cryptography and coding theory. It can be used to construct orthogonal arrays, strong partial balanced designs. It can also be used to design good linear codes, In this paper, we construct a class of maximal （r, t）-linearly independent arrays of length r ＋ 2, and provide some enumerator theorems.     

Trace class Toeplitz operators with unbounded symbols on weighted Bergman spaces

In this note we construct a function φ in L2（Bn,dμ） which is unbounded on any neighborhood of each boundary point of Bn such that Tφ is a trace class operator on weighted Bergman space Lα2（Bn,dμ） for several complex variables.     

Wavelet approximation and Fourier widths of classes of periodic functions of several variables. I

We obtain characterizations (and prove the corresponding equivalence of norms) of function spaces B _pq ^sm ($ \mathbb{I} $ \mathbb{I} ^k ) and L _pq ^sm ($ \mathbb{I} $ \mathbb{I} ^k ) of Nikol’skii-Besov and Lizorkin-Triebel types, respectively, in terms of representations of functions in these spaces by Fourier series with respect to a multiple system $ \mathcal{W}_m^\mathbb{I} $ \mathcal{W}_m^\mathbb{I} of Meyer wavelets and in terms of sequences of the Fourier coefficients with respect to this system. We establish order-sharp estimates for the approximation of functions in B _pq ^sm ($ \mathbb{I} $ \mathbb{I} ) and L _pq ^sm ($ \mathbb{I} $ \mathbb{I} ^k ) by special partial sums of these series in the metric of L _r ($ \mathbb{I} $ \mathbb{I} ^k ) for a number of relations between the parameters s, p, q, r, and m (s = (s ₁, ..., s _n ) ∈ ℝ₊ ⁿ , 1 ≤ p, q, r ≤ ∞, m = (m ₁, ..., m _n ) ∈ ℕ ⁿ , k = m ₁ +... + m _n , and $ \mathbb{I} $ \mathbb{I} = ℝ or $ \mathbb{T} $ \mathbb{T} ). In the periodic case, we study the Fourier widths of these function classes.     

Imaginary powers of a Laguerre differential operator

Imaginary powers associated to the Laguerre differential operator $ L_\alpha = - \Delta + |x|^2 + \sum _{i = 1}^d \frac{1} {{x_i^2 }}(\alpha _i^2 - 1/4) $ L_\alpha = - \Delta + |x|^2 + \sum _{i = 1}^d \frac{1} {{x_i^2 }}(\alpha _i^2 - 1/4) are investigated. It is proved that for every multi-index α = (α₁,...α _d ) such that α _i ≧ −1/2, α _i ∉ (−1/2, 1/2), the imaginary powers $ \mathcal{L}_\alpha ^{ - i\gamma } ,\gamma \in \mathbb{R} $ \mathcal{L}_\alpha ^{ - i\gamma } ,\gamma \in \mathbb{R} , of a self-adjoint extension of L _α, are Calderón-Zygmund operators. Consequently, mapping properties of $ \mathcal{L}_\alpha ^{ - i\gamma } $ \mathcal{L}_\alpha ^{ - i\gamma } follow by the general theory.