Hyperbolic Chevalley groups on ℂ<Superscript>2</Superscript>

Let Γ ? U (1, 1) be the subgroup generated by the complex reflections. Suppose that Γ acts discretely on the domain K = {(z ₁, z ₂) ∈ ?² ||z ₁|² ? |z ₂|² < 0} and that the projective group PΓ acts on the unit disk B = {|z ₁/z ₂| < 1} as a Fuchsian group of signature (n ₁, ..., n _s ), s ? 3, n _i ? 2. For such groups, we prove a Chevalley type theorem, i.e., find a necessary and sufficient condition for the quotient space K/Γ to be isomorphic to ?² ? {0}.     

Inversion and subspaces of a finite field

Consider two F _q -subspaces A and B of a finite field, of the same size, and let A ^?1 denote the set of inverses of the nonzero elements of A. The author proved that A ^?1 can only be contained in A if either A is a subfield, or A is the set of trace zero elements in a quadratic extension of a field. Csajbók refined this to the following quantitative statement: if A ^?1 ? B, then the bound |A ^?1∩B| ≤ 2|B|/q ? 2 holds. He also gave examples showing that his bound is sharp for |B| ≤ q ³. Our main result is a proof of the stronger bound |A ^?1 ∩ B| ≤ |B|/q · (1 + O _d (q ^?1/2)), for |B| = q ^d with d > 3. We also classify all examples with |B| ≤ q ³ which attain equality or near-equality in Csajbók’s bound.     

Economical separability in free groups

Consider the rank n free group F _n with basis X. Bogopol’ski? conjectured in [1, Problem 15.35] that each element w ∈ F _n of length |w| ≥ 2 with respect to X can be separated by a subgroup H ≤ F _n of index at most C log |w| with some constant C. We prove this conjecture for all w outside the commutant of F _n , as well as the separability by a subgroup of index at most |w|/2 + 2 in general.     

A numerical algorithm for solving a nonstationary problem of optimal control

Let {φ _n ^(α,β) (z)} _n=0 ^∞ be a system of Jacobi polynomials orthonormal on the circle |z| = 1 with respect to the weight (1 ? cos τ)^α+1/2(1 + cos τ)^β+1/2 (α, β > ?1), and let \(\psi _n^{\left( {\alpha ,\beta } \right)*} \left( z \right): = z^n \overline {\psi _n^{\left( {\alpha ,\beta } \right)} \left( {{1 \mathord{\left/ {\vphantom {1 {\bar z}}} \right. \kern-\nulldelimiterspace} {\bar z}}} \right)}\)). We establish relations between the polynomial φ _n ^(α,?1/2) (z) and the nth (C, α ? 1/2)-mean of the Maclaurin series for the function (1 ? z)^?α?3/2 and also between the polynomial φ _n ^(α,?1/2)* (z) and the nth (C, α + 1/2)-mean of the Maclaurin series for the function (1 ? z)^?α?1/2. We use these relations to derive an asymptotic formula for φ _n ^(α,?1/2) (z); the formula is uniform inside the disk |z| < 1. It follows that φ _n ^(α,?1/2) (z) ≠ 0 in the disk |z| ≤ ρ for fixed φ ∈ (0, 1) and α > ?1 if n is sufficiently large.     

Tauber-Konstanten für Verschiedene Tauberbedingungen bei den Kreisverfahren der Limitierungstheorie

Till now, we know Tauberian constants for the ‘Kreisverfahren’ with the conditions lim sup |n ^1/2 a _n|<∞ and lim sup |n ¹ a _n|<∞. Now, we obtain constants for the more general condition lim sup |n ^pa_n|<∞ with anyp(=∞<p<+∞). These constants are not always 0 or ∞, even if 1/2<p<1; therefore the Tauberian condition lim sup |n ^pa_n|<∞ is ‘appropriate’ for 1/2≦p≦1.     

On a result by geronimus

In 1935, Ya.L. Geronimus found the best integral approximation on the period [?π,π) of the function sin(n + 1)t ? 2q sin nt, q ∈ ?, by the subspace of trigonometric polynomials of degree at most n ? 1. This result is an integral analog of the known theorem by E.I. Zolotarev (1868). At present, there are several methods of proving this fact. We propose one more variant of the proof. In the case |q| ≥ 1, we apply the (2π/n)-periodization and the fact that the function | sin nt| is orthogonal to the harmonic cos t on the period. In the case |q| < 1, we use the duality relations for Chebyshev’s theorem (1859) on a rational function least deviating from zero on a closed interval with respect to the uniform metric.     

On power deformations of univalent functions

For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f′(0) ? 1 = 0 and f (z) ≠ 0, 0 < |z| < 1, we consider the power deformation f _c (z) = z(f (z)/z) ^c for a complex number c. We determine those values c for which the operator \({f \mapsto f_c}\) maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), |z| < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.     

On multiple zeros of a partial theta function

We consider the partial theta function θ(q, x) := ∑_j=0^∞q^j(j+1)/2x^j, where x ∈ ? is a variable and q ∈ ?, 0 < |q| < 1, is a parameter. We show that, for any fixed q, if ζ is a multiple zero of the function θ(q, · ), then |ζ| ≤ 8¹¹.     

<Emphasis Type="Italic">L</Emphasis>-Approximation of a linear combination of the poisson kernel and its conjugate kernel by trigonometric polynomials

A linear combination Π _q,α = cos(απ/2)P + sin(απ/2)Q of the Poisson kernel P(t) = 1/2 + q cos t + q ² cos 2t + ... and its conjugate kernel Q(t) = q sin t + q ² sin 2t + ... is considered for α ∈ ? and |q| < 1. A new explicit formula is found for the value E _n?1(Π _q,α ) of the best approximation in the space L = L _2π of the function Π _q,α by the subspace of trigonometric polynomials of order at most n ? 1. More exactly, it is proved that \(E_{n - 1} \left( {\prod _{q,\alpha } } \right) = \left. {\frac{{\left| q \right|^n \left( {1 - q^2 } \right)}}{{1 - q^{4n} }}} \right\|\left. {\frac{{\cos \left( {nt - {{\alpha \pi } \mathord{\left/ {\vphantom {{\alpha \pi } 2}} \right. \kern-\nulldelimiterspace} 2}} \right) - q^{2n} \cos \left( {nt + {{\alpha \pi } \mathord{\left/ {\vphantom {{\alpha \pi } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}{{1 - q^2 - 2q \cos t}}} \right\|_L\). In addition, the value E _n?1(Π _q,α ) is represented as a rapidly convergent series.     

Turán-Type Results for Complete <Emphasis Type="Italic">h</Emphasis>-Partite Graphs in Comparability and Incomparability Graphs

We consider an h-partite version of Dilworth’s theorem with multiple partial orders. Let P be a finite set, and let <₁,...,< _r be partial orders on P. Let G(P, <₁,...,< _r ) be the graph whose vertices are the elements of P, and x, y ∈ P are joined by an edge if x< _i y or y< _i x holds for some 1 ≤ i ≤ r. We show that if the edge density of G(P, <₁, ... , < _r ) is strictly larger than 1 ? 1/(2h ? 2) ^r , then P contains h disjoint sets A ₁, ... , A _h such that A ₁ < _j ... < _j A _h holds for some 1 ≤ j ≤ r, and |A ₁| = ... = |A _h | = Ω(|P|). Also, we show that if the complement of G(P, <) has edge density strictly larger than 1 ? 1/(3h ? 3), then P contains h disjoint sets A ₁, ... , A _h such that the elements of A _i are incomparable with the elements of A _j for 1 ≤ i < j ≤ h, and |A ₁| = ... = |A _h | = |P|^1?o(1). Finally, we prove that if the edge density of the complement of G(P, <₁, <₂) is α, then there are disjoint sets A, B ? P such that any element of A is incomparable with any element of B in both <₁ and <₂, and |A| = |B| > n ^1?γ(α), where γ(α) → 0 as α → 1. We provide a few applications of these results in combinatorial geometry, as well.     

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

Let G be a finite group. The main result of this paper is as follows: If G is a finite group, such that Γ(G) = Γ(²G₂(q)), where q = 3²ⁿ⁺¹ for some n ≥ 1, then G has a (unique) nonabelian composition factor isomorphic to ² G ₂(q). We infer that if G is a finite group satisfying |G| = |² G ₂(q)| and Γ(G) = Γ (² G ₂(q)) then G ? = ² G ₂(q). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications of this result are also considered to the problem of recognition by element orders of finite groups.     

Equalities on the number of conjugacy classes of the Sylow p-subgroups of GL(n,q)

We denote by Gn the group of the upper unitriangular matrices over F_q, the finite field with q = p^t elements, and r(G_n) the number of conjugacy classes of G_n. In this paper, we obtain the value of r(G_n) modulo (q² -1)(q -1). We prove the following equalities     

Coefficient estimates of negative powers and inverse coefficients for certain starlike functions

For ?1≤B<A≤1, let \(\mathcal {S}^{*}(A,B)\) denote the class of normalized analytic functions \(f(z)= z+{\sum }_{n=2}^{\infty }a_{n} z^{n}\) in |z|<1 which satisfy the subordination relation z f ^′(z)/f(z)?(1 + A z)/(1 + B z) and Σ^?(A,B) be the corresponding class of meromorphic functions in |z|>1. For \(f\in \mathcal {S}^{*}(A,B)\) and λ>0, we shall estimate the absolute value of the Taylor coefficients a _n (?λ,f) of the analytic function (f(z)/z)^?λ . Using this we shall determine the coefficient estimate for inverses of functions in the classes \(\mathcal {S}^{*}(A,B)\) and Σ^?(A,B).     

Some Zygmund type inequalities for the <Emphasis Type="Italic">s</Emphasis>th derivative of polynomials

For a polynomial P(z) of degree n having no zeros in |z| < 1, it was recently proved in [9] that

$$\left| {{z^s}{P^{\left( s \right)}}\left( z \right) + \beta \frac{{n\left( {n - 1} \right)...\left( {n - s + 1} \right)}}{{{2^s}}}P\left( z \right)} \right| \leqslant \frac{{n\left( {n - 1} \right)...\left( {n - s + 1} \right)}}{2}\left( {\left| {1 + \frac{\beta }{{{2^s}}}} \right| + \left| {\frac{\beta }{{{2^s}}}} \right|} \right)\mathop {\max }\limits_{\left| z \right| = 1} \left| {P\left( z \right)} \right|$$

for every β ∈ C with |β| ≤ 1, 1 ≤ s ≤ n and |z| = 1. In this paper, we obtain the L _p mean extension of the above and other related results for the sth derivative of polynomials.

     

Intrinsic Ultracontractivity for Schrödinger Operators Based on Fractional Laplacians

We study the Feynman-Kac semigroup generated by the Schrödinger operator based on the fractional Laplacian ??(???Δ)^α/2???q in R ^d , for q?≥?0, α?∈?(0,2). We obtain sharp estimates of the first eigenfunction φ ₁ of the Schrödinger operator and conditions equivalent to intrinsic ultracontractivity of the Feynman-Kac semigroup. For potentials q such that lim_{|x| →?∞?} q(x)?=?∞ and comparable on unit balls we obtain that φ ₁(x) is comparable to (|x|?+?1)^???d???α (q(x)?+?1)^???1 and intrinsic ultracontractivity holds iff lim_{|x| →?∞?} q(x)/log|x|?=?∞. Proofs are based on uniform estimates of q-harmonic functions.     

CR deformations for rational homogeneous surface singularities

In this paper, we will present a CR-construction of the versal deformations of the singularitiesV _n ? ?²/? _n ,n ∈ {2,3,4,?} defined by the immersions of ?² into ? ⁿ⁺¹ X ⁿ : (z, w) → (z ⁿ ,z ^n?1 w, ?,zw ^n?1 ,w ⁿ )     

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).     

The weighted Hardy spaces associated to self-adjoint operators and their duality on product spaces

Let L be a non-negative self-adjoint operator acting on L²(R ⁿ ) satisfying a pointwise Gaussian estimate for its heat kernel. Let w be an A _r weight on R ⁿ × R ⁿ , 1 < r < ∞. In this article we obtain a weighted atomic decomposition for the weighted Hardy space H _L,w ^p (R ⁿ ×R ⁿ ), 0 < p ≤ 1 associated to L. Based on the atomic decomposition, we show the dual relationship between H _L,w ¹ (R ⁿ × R ⁿ ) and BMO_L,w(R ⁿ × R ⁿ ).     

Heat Kernels of Non-symmetric Jump Processes: Beyond the Stable Case

Let J be the Lévy density of a symmetric Lévy process in \(\mathbb {R}^{d}\) with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator

$$\mathcal{L}^{\kappa}f(x):= \lim_{{\varepsilon} \downarrow 0} {\int}_{\{z \in \mathbb{R}^{d}: |z|>{\varepsilon}\}} (f(x+z)-f(x))\kappa(x,z)J(z)\, dz\, , $$

where κ(x, z) is a Borel function on \(\mathbb {R}^{d}\times \mathbb {R}^{d}\) satisfying 0 < κ ₀ ≤ κ(x, z) ≤ κ ₁, κ(x, z) = κ(x,?z) and |κ(x, z) ? κ(y, z)|≤ κ ₂|x ? y| ^β for some β ∈ (0, 1]. We construct the heat kernel p ^κ (t, x, y) of \(\mathcal {L}^{\kappa }\), establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel p ^κ .

     

Bohr's inequalities for Hilbert space operators

The classical Bohr's inequality states that

²|z+w|?p²|z|+q²|w|