Projective Freeness of Algebras of Real Symmetric Functions

Let \mathbb Dⁿ:={z=(z₁,?, z_n) ? \mathbb Cⁿ:|z_j| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]ⁿ{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cⁿ{\mathbb {C}^n}. Consider the ring

Truncated Complex Moment Problems with a $ \widetilde{zz} $ Relation

We solve the truncated complex moment problem for measures supported on the variety K o \mathcal{K}\equiv { z ? \in C: z [(z)\tilde]\widetilde{z} = A+Bz+C [(z)\tilde]\widetilde{z} +Dz ² ,D 1 \neq 0}. Given a doubly indexed finite sequence of complex numbers g o g⁽²ⁿ⁾:g₀₀,g₀₁,g₁₀,?,g_0,2n,g_1,2n-1,?,g_2n-1,1,g_2n,0 \gamma\equiv\gamma^{(2n)}:\gamma_{00},\gamma_{01},\gamma_{10},\ldots,\gamma_{0,2n},\gamma_{1,2n-1},\ldots,\gamma_{2n-1,1},\gamma_{2n,0} , there exists a positive Borel measure m\mu supported in K \mathcal{K} such that g_ij=ò[`(z)]<sup>i</sup>z<sup>j</sup> dm (0 ￡ 1+j ￡ 2n) \gamma_{ij}=\int\overline{z}^{i}z^{j}\,d\mu\,(0\leq1+j\leq2n) if and only if the moment matrix M(n)( g\gamma ) is positive, recursively generated, with a column dependence relation Z [(Z)\tilde]\widetilde{Z} = A1+BZ +C [(Z)\tilde]\widetilde{Z} +DZ <sup>2</sup>, and card V(g) 3\mathcal{V}(\gamma)\geq rank M(n), where V(g)\mathcal{V}(\gamma) is the variety associated to g \gamma . The last condition may be replaced by the condition that there exists a complex number g<sub>n,n+1</sub> \gamma_{n,n+1} satisfying g<sub>n+1,n</sub> o [`(g)]_n,n+1=Ag_n,n-1+Bg_n,n+Cg_n+1,n-1+Dg_n,n+1 \gamma_{n+1,n}\equiv\overline{\gamma}_{n,n+1}=A\gamma_{n,n-1}+B\gamma_{n,n}+C\gamma_{n+1,n-1}+D\gamma_{n,n+1} . We combine these results with a recent theorem of J. Stochel to solve the full complex moment problem for K \mathcal{K} , and we illustrate the connection between the truncated and full moment problems for other varieties as well, including the variety z ^k = p(z, [(Z)\tilde] \widetilde{Z} ), deg p < k.     

Exotic smooth structures on small 4-manifolds with odd signatures

Let M be (2n-1)\mathbbCP²#2n[`(\mathbbCP)]<sup>2</sup>(2n-1)\mathbb{CP}^{2}\#2n\overline{\mathbb{CP}}{}^{2} for any integer n≥1. We construct an irreducible symplectic 4-manifold homeomorphic to M and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to M. We also construct such exotic smooth structures when M is \mathbbCP<sup>2</sup>#4[`(\mathbbCP)]²\mathbb{CP}{}^{2}\#4\overline {\mathbb{CP}}{}^{2} or 3\mathbbCP²#k[`(\mathbbCP)]²3\mathbb{CP}{}^{2}\#k\overline{\mathbb{CP}}{}^{2} for k=6,8,10.     

Purity for overconvergence

Let X \hookrightarrow[`(X)]{X \hookrightarrow \overline{X}} be an open immersion of smooth varieties over a field of characteristic p > 0 such that the complement is a simple normal crossing divisor and [`(Z)] í Z í [`(X)]{\overline{Z}\subseteq Z \subseteq \overline{X}} closed subschemes of codimension at least 2. In this paper, we prove that the canonical restriction functor between the categories of overconvergent F-isocrystals F-Isoc<sup>f</sup>(X,[`(X)]) ? F-Isoc^f(X\Z,[`(X)]\[`(Z)]){F-{\rm Isoc}^\dagger(X,\overline{X}) \longrightarrow F-{\rm Isoc}^\dagger(X{\setminus}Z, \overline{X}{\setminus}\overline{Z})} is an equivalence of categories. We also give an application of our result to the equivalence of certain categories.     

Hyponormal Toeplitz Operators on the Weighted Bergman Space

Consider j = f +[`(g)]\varphi = f + \overline {g}, where f and g are polynomials, and let T_jT_{\varphi} be the Toeplitz operators with the symbol j\varphi. It is known that if T_jT_{\varphi} is hyponormal then |f￠(z)|² 3 |g￠(z)|²|f'(z)|^{2} \geq |g'(z)|^{2} on the unit circle in the complex plane. In this paper, we show that it is also a necessary and sufficient condition under certain assumptions. Furthermore, we present some necessary conditions for the hyponormality of T_jT_{\varphi} on the weighted Bergman space, which generalize the results of I. S. Hwang and J. Lee.     

When is the Uvarov transformation positive definite?

Let L\cal{L} be a positive definite bilinear functional, then the Uvarov transformation of L\cal{L} is given by  U(p,q) = L(p,q) + m p(a)[`(q)](a<sup>-1</sup>) +[`(m)] p([`(a)]<sup>-1</sup>)\,\mathcal{U}(p,q) = \mathcal{L}(p,q) + m\,p(\alpha)\overline{q}(\alpha^{-1}) + \overline{m}\,p(\overline{\alpha}^{-1}) [`(q)]([`(a)])\overline{q}(\overline{\alpha}) where $|\alpha| > 1, m \in \mathbb{C}$|\alpha| > 1, m \in \mathbb{C}. In this paper we analyze conditions on m for U\cal{U} to be positive definite in the linear space of polynomials of degree less than or equal to n. In particular, we show that m has to lie inside a circle in the complex plane defined by α, n and the moments associated with L\cal{L}. We also give an upper bound for the radius of this circle that depends only on α and n. This and other conditions on m are visualized for some examples.     

On properties of entire solutions of linear differential equations with polynomial coefficients

We investigate properties of entire solutions of differential equations of the form

Partitions involving D.H. Lehmer numbers

Let n ≥ 2 be a fixed integer, let q and c be two integers with q > n and (n, q) = (c, q) = 1. For every positive integer a which is coprime with q we denote by [`(a)]<sub>c</sub>{\overline{a}_{c}} the unique integer satisfying 1 ￡ [`(a)]_c ￡ q{1\leq\overline{a}_{c} \leq{q}} and a[`(a)]_c o c(mod q){a\overline{a}_{c} \equiv{c}({\rm mod}\, q)}. Put

Smooth Universal Taylor Series

Let W í \Bbb C\Omega \subseteq {\Bbb C} be a simply connected domain in \Bbb C{\Bbb C} , such that {￥} è[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}] is connected. If g is holomorphic in Ω and every derivative of g extends continuously on [`(W)]\bar{\Omega} , then we write g ∈ A^∞ (Ω). For g ∈ A^∞ (Ω) and z ? [`(W)]\zeta \in \bar{\Omega} we denote S_N (g,z)(z) = ?^N_l=0\fracg^(l) (z)l ! (z-z)^lS_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l . We prove the existence of a function f ∈ A^∞(Ω), such that the following hold:

On Bernstein–Walsh-type lemmas in regions of the complex plane

Let G ì \mathbb C G \subset {\mathbb C} be a finite region bounded by a Jordan curve L: = ?G L: = \partial G , let W: = \textext[`(G)] \Omega : = {\text{ext}}\bar{G} (with respect to [`(\mathbb C)] {\overline {\mathbb C}} ), $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} , and let w = F(z) w = \Phi (z) be a univalent conformal mapping of Ω onto Δ normalized by $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 . By A _p (G); p > 0; we denote a class of functions f analytic in G and satisfying the condition

A fast and simple algorithm for the computation of Legendre coefficients

We present an O(N log N){\mathcal{O}({N\,{\rm log}\,N})} algorithm for the calculation of the first N coefficients in an expansion of an analytic function in Legendre polynomials. In essence, the algorithm consists of an integration of a suitably weighted function along an ellipse, a task which can be accomplished with Fast Fourier Transform, followed by some post-processing. The mathematical underpinning of this algorithm is an old formula that expresses expansion coefficients [^(f)]_m{\hat{f}_m} as infinite linear combinations of derivatives. We evaluate the latter with the Cauchy theorem, thereby expressing each [^(f)]_m{\hat{f}_m} as a scaled integral of f(z)j_m(z)/z^m+1{f(z)\varphi_m(z)/z^{m+1}} along an appropriate contour, where j_m{\varphi_m} is a slowly converging hypergeometric function. Next, we transform j_m{\varphi_m} into another hypergeometric function which converges rapidly. Once we replace the latter function by its truncated Taylor expansion and choose an appropriate elliptic contour, we obtain an expression for the [^(f)]_m{\hat{f}_m}s which is amenable to rapid computation.     

Extending the Torelli map to toroidal compactifications of Siegel space

It has been known since the 1970s that the Torelli map M  _g →A  _g , associating to a smooth curve its Jacobian, extends to a regular map from the Deligne–Mumford compactification [`(\operatorname M)]<sub>g</sub>\overline {\operatorname {M}}_{g} to the 2nd Voronoi compactification [`(\operatorname A)]_g^vor\overline {\operatorname {A}}_{g}^{\mathrm {vor}}. We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification [`(\operatorname A)]<sub>g</sub><sup>perf</sup>\overline {\operatorname {A}}_{g}^{\mathrm {perf}} is also regular, and moreover [`(\operatorname A)]_g^vor\overline {\operatorname {A}}_{g}^{\mathrm {vor}} and [`(\operatorname A)]<sub>g</sub><sup>perf</sup>\overline {\operatorname {A}}_{g}^{\mathrm {perf}} share a common Zariski open neighborhood of the image of [`(\operatorname M)]_g\overline {\operatorname {M}}_{g}. We also show that the map to the Igusa monoidal transform (central cone compactification) is not regular for g≥9; this disproves a 1973 conjecture of Namikawa.     

Asymptotic Behavior of Variable-Coefficient Toeplitz Determinants

For a continuous function s\sigma defined on [0,1]×\mathbbT[0,1]\times\mathbb{T}, let \ops\op\sigma stand for the (n+1)×(n+1)(n+1)\times(n+1) matrix whose (j,k)(j,k)-entries are equal to \frac1 2pò₀^2p s( \fracjn,e^iq) e^-i(j-k)q  dq,        j,k = 0,1,...,n . \displaystyle \frac{1} {2\pi}\int_0^{2\pi} \sigma \left( \frac{j}{n},e^{i\theta}\right) e^{-i(j-k)\theta} \,d\theta, \qquad j,k =0,1,\dots,n~. These matrices can be thought of as variable-coefficient Toeplitz matrices or as the discrete analogue of pseudodifferential operators. Under the assumption that the function s\sigma possesses a logarithm which is sufficiently smooth on [0,1]×\mathbbT[0,1]\times\mathbb{T}, we prove that the asymptotics of the determinants of \ops\op\sigma are given by det[\ops] ~ G[s]⁽ⁿ⁺¹⁾E[s]     \text as   n?￥ , \det \left[\op\sigma\right] \sim G[\sigma]^{(n+1)}E[\sigma] \quad \text{ as \ } n\to\infty~, where G[s]G[\sigma] and E[s]E[\sigma] are explicitly determined constants. This formula is a generalization of the Szegö Limit Theorem. In comparison with the classical theory of Toeplitz determinants some new features appear.     

Averaging of directional derivatives in vertices of nonobtuse regular triangulations

For a shape-regular triangulation ${\mathcal{T}_h}For a shape-regular triangulation _h{\mathcal{T}_h} without obtuse angles of a bounded polygonal domain W ì ?²{\Omega\subset\Re^2} , let L_h{\mathcal L_h} be the space of continuous functions linear on the triangles from T_h{\mathcal{T}_h} and Π _h the interpolation operator from C([`(W)]){C(\overline\Omega)} to L<sub>h</sub>{\mathcal L_h} . This paper is devoted to the following classical problem: Find a second-order approximation of the derivative ?u/?z(a){\partial u/\partial z(a)} in a direction z of a function u ? C<sup>3</sup>([`(W)]){u\in C^3(\overline\Omega)} in a vertex a in the form of a linear combination of the constant directional derivatives ?P_h(u)/?z{\partial \Pi_h(u)/\partial z} on the triangles surrounding a. An effective procedure for such an approximation is presented, its error is proved to be of the size O(h ²), an operator W_h: L_h?L_h×L_h{\mbox{W}_h: \mathcal L_h\longrightarrow\mathcal L_h\times\mathcal L_h} relating a second-order approximation W _h [Π _h (u)] of ?u{\nabla u} to every u ? C³([`(W)]){u\in C^3(\overline\Omega)} is constructed and shown to be a so-called recovery operator. The accuracy of the presented approximation is compared with the accuracies of the local approximations by other known techniques numerically.     

On the Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with K-Quasiconformal Boundary

Let G be a finite domain in the complex plane with K-quasicon formal boundary, z ₀ be an arbitrary fixed point in G and p>0. Let j_p ( z ): = ò_x0 ^x [ f( z) ]^2/8 dz\varphi _p \left( z \right): = \int_{x_0 }^x {\left[ {\phi \left( \zeta \right)} \right]^{2/8} } d\zeta , and let \iint_c | j_p ( z ) - P_x¹ (z) |^p d0_x \iint\limits_c {\left| {\varphi _p \left( z \right) - P_x^1 (z)} \right|^p d0_x } in the class \mathop ?_n \mathop \prod \limits_n of all polynomials of degree [`(G)]\bar G in case of $p > 2 - \frac{{K^2 + 1}}{{2K^4 }}$p > 2 - \frac{{K^2 + 1}}{{2K^4 }} .     

Characterization of field homomorphisms and derivations by functional equations

Summary. Let K and [`(K)] \overline K be fields containing \Bbb Q {\Bbb Q} . We characterize pairs of additive functions f,g: K ?[`(K)] f,g: K \to \overline K satisfying a functional equation¶¶ g(x^ln) = f(x^l)ⁿ     \textrespectively        g(x^ln) = Ax^ln + x^ln-lf(x^l) g(x^{ln}) = f(x^l)^n \quad \text{respectively} \qquad g(x^{ln}) = Ax^{ln} + x^{ln-l}f(x^l) ,¶where n ? \Bbb Z \{0,1} n \in {\Bbb Z} \setminus \{0,1\} , l ? \Bbb N l\in {\Bbb N} and A ? K A \in K .     

On the sets of branch points of mappings more general than quasiregular

It is shown that if a point x ₀ ∊ ℝ ⁿ , n ≥ 3, is an essential isolated singularity of an open discrete Q-mapping f : D → [`(\mathbb R<sup>n</sup>)] \overline {\mathbb {R}^n} , B <sub>f</sub> is the set of branch points of f in D; and a point z <sub>0</sub> ∊ [`(\mathbb Rⁿ)] \overline {\mathbb {R}^n} is an asymptotic limit of f at the point x ₀; then, for any neighborhood U containing the point x ₀; the point z ₀ ∊ [`(f( B<sub>f</sub> ?U ))] \overline {f\left( {B_f \cap U} \right)} provided that the function Q has either a finite mean oscillation at the point x <sub>0</sub> or a logarithmic singularity whose order does not exceed n − 1: Moreover, for n ≥ 2; under the indicated conditions imposed on the function Q; every point of the set [`(\mathbb Rⁿ)] \overline {\mathbb {R}^n} \ f(D) is an asymptotic limit of f at the point x ₀. For n ≥ 3, the following relation is true: [`(\mathbbR<sup>n</sup> )] \f( D ) ì [`(f B_f )] \overline {\mathbb{R}^n } \backslash f\left( D \right) \subset \overline {f\,B_f } . In addition, if ￥ ? f( D ) \infty \notin f\left( D \right) , then the set f B _f is infinite and x₀ ? [`(B_f )] x_0 \in \overline {B_f } .     

Bound states to critical quasilinear Schr?dinger equations

In this paper, we consider the critical quasilinear Schr?dinger equations of the form

The composition operators on weighted Bloch space

We will characterize the boundedness and compactness of the composition operators on weighted Bloch space B _log = { f ? H(D): sup_{z ? D} (1-| z|²) ( log\frac21-| z|² )| f￠(z)| B_{ \log }= \{ f \in H(D): \sup_{z \in D } (1-\left| z\right|^2) \left( \log \frac{2}{1-\left| z\right|^2} \right)\left| f'(z)\right| < +￥} +\infty \} , where H(D) be the class of all analytic functions on D.     

Irregular Wavelet Frames and Gabor Frames

Given g { l^\fracn2 g( l_j x - kb ) }_jezjezⁿ ,where  l_j \left\{ {\lambda ^{\frac{n}{2}} g\left( {\lambda _j x - kb} \right)} \right\}_{j\varepsilon zj\varepsilon z^n } ,where\;\lambda _j > 0 and b > 0. Sufficient conditions for the wavelet system to constitute a frame for L ²(R ⁿ ) are given. For a class of functions g{ e^{zrib( j,x )} g( x - l_k ) }_{jezⁿ ,kez}\left\{ {e^{zrib\left( {j,x} \right)} g\left( {x - \lambda _k } \right)} \right\}_{j\varepsilon z^n ,k\varepsilon z} to be a frame.