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

Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

Let C( \mathbbR^m ) C\left( {{\mathbb{R}^m}} \right) be the space of bounded and continuous functions x:\mathbbR^m ? \mathbbR x:{\mathbb{R}^m} \to \mathbb{R} equipped with the norm

Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem

We consider the space A(\mathbbT)A(\mathbb{T}) of all continuous functions f on the circle \mathbbT\mathbb{T} such that the sequence of Fourier coefficients [^(f)] = { [^(f)]( k ), k ? \mathbbZ }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l ¹(ℤ). The norm on A(\mathbbT)A(\mathbb{T}) is defined by || f ||_A(\mathbbT) = || [^(f)] ||_{l¹ (\mathbbZ)}\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if f:\mathbbT ? \mathbbT\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that || e^inf ||_A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that || e^inf ||_A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear.     

Singular oscillatory integrals on {\mathbb{R}^n}

Let ${\mathcal{P}_{d,n}}Let P_d,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRⁿ{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? P_d,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that

supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      5. Analytic cohomology groups in top degrees of Zariski open sets in {\mathbb{P}^n}      Gabriel Chiriacescu  Mihnea Colţoiu  Cezar Joiţa 《Mathematische Zeitschrift》2010,264(3):671-677 We show that if A is a closed analytic subset of \mathbbPn{\mathbb{P}^n} of pure codimension q then Hi(\mathbbPn\ A,F){H^i(\mathbb{P}^n{\setminus} A,{\mathcal F})} are finite dimensional for every coherent algebraic sheaf F{{\mathcal F}} and every i 3 n-[\fracn-1q]{i\geq n-\left[\frac{n-1}{q}\right]} . If n-1 3 2q we show that Hn-2(\mathbbPn\ A,F)=0{n-1\geq 2q\,{\rm we show that}\, H^{n-2}(\mathbb{P}^n{\setminus} A,{\mathcal F})=0} .      6. Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}      Mohamed Benrhouma  Hichem Ounaies 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(5):647-662 In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem {l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      7. Uniform estimates for order statistics and Orlicz functions      Yehoram Gordon  Alexander E. Litvak  Carsten Schütt  Elisabeth Werner 《Positivity》2012,16(1):1-28 We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ 1, … , ξ n and a vector of scalars x = (x 1, … , x n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min1 ￡ i ￡ n |xixi|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max1 ￡ i ￡ n|xixi|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||yx||M{\|y_x\|_M} of the vector y x  = (1/x 1, … , 1/x n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ 1|, G(t) = \mathbb P ({ |x1| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ 1 is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò0t e-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò0se-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.      8. On the regular growth of Dirichlet series absolutely convergent in a half-plane      Yu. V. Stets’  M. M. Sheremeta 《Ukrainian Mathematical Journal》2011,63(5):797-814 For the Dirichlet series F(s) = ?n = 1￥ anexp{ sln } F(s) = \sum\nolimits_{n = 1}^\infty {{a_n}\exp \left\{ {s{\lambda_n}} \right\}} with abscissa of absolute convergence σ a =0, we establish conditions for (λ n ) and (a n ) under which lnM( s, F ) = TR( 1 + o(1) )exp{ rR / | s| } \ln M\left( {\sigma, F} \right) = {T_R}\left( {1 + o(1)} \right)\exp \left\{ {{{{{\varrho_R}}} \left/ {{\left| \sigma \right|}} \right.}} \right\} for σ ↑ 0, where M( s, F ) = sup{ | F( s+ it ) |:t ? \mathbbR } M\left( {\sigma, F} \right) = \sup \left\{ {\left| {F\left( {\sigma + it} \right)} \right|:t \in \mathbb{R}} \right\} and T R and ϱ R are positive constants.      9. The Hermite Property of a Causal Wiener Algebra Used in Control Theory      Amol Sasane 《Complex Analysis and Operator Theory》2010,4(1):97-107 Let \mathbbC+ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra A = { s( ? \mathbbC+ ) ? [^(f)]a (s) + ?k = 0￥ fk e - stk | lfa ? L1 (0,￥),(fk )k 3 0 ? l1, 0 = t0 < t1 < t2 < ? }{{{\mathcal{A}}}} = \left\{ s( \in {{{\mathbb{C}}}}_ + ) \mapsto \hat{f}_a (s) + \sum\limits_{k = 0}^\infty {f_k e^{ - st_k }}\bigg | \bigg.{\begin{array}{l}{f_a \in L^1 (0,\infty ),(f_k )_{k \geq 0} \in \ell^{1}, } \cr {{0 = t_0 < t_1 < t_2 < \ldots}} \end{array}} \right\}      10. On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}      Manass��s de Souza 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(2):199-215 In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbRN{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form -DN u+ V(x)|u|N-2u=\fracf(x,u)|x|a   in  \mathbbRN,    N 3 2,-\Delta_N\,u+ V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^a}\quad{\rm in} \, \mathbb{R}^N,\quad N\geq 2,      11. Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential      Y. Belaud  A. Shishkov 《Journal of Evolution Equations》2010,10(4):857-882 We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ut + Lu + a(x) |u|q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb RN{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò01 s-1 (meas\nolimits {x ? W: |a(x)| ￡ s })q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.      12. Approximations of the exponential of an operator with periodic coefficients      S. E. Pastukhova 《Journal of Mathematical Sciences》2012,181(5):668-700 We consider the operator exponential e −tA , t > 0, where A is a selfadjoint positive definite operator corresponding to the diffusion equation in \mathbbRn {\mathbb{R}^n} with measurable 1-periodic coefficients, and approximate it in the operator norm ||   ·   ||L2( \mathbbRn ) ? L2( \mathbbRn ) {\left\| {\; \cdot \;} \right\|_{{{L^2}\left( {{\mathbb{R}^n}} \right) \to {L^2}\left( {{\mathbb{R}^n}} \right)}}} with order O( t - \fracm2 ) O\left( {{t^{{ - \frac{m}{2}}}}} \right) as t → ∞, where m is an arbitrary natural number. To construct approximations we use the homogenized parabolic equation with constant coefficients, the order of which depends on m and is greater than 2 if m > 2. We also use a collection of 1-periodic functions N α (x), x ? \mathbbRn x \in {\mathbb{R}^n} , with multi-indices α of length | a| \leqslant m \left| \alpha \right| \leqslant m , that are solutions to certain elliptic problems on the periodicity cell. These results are used to homogenize the diffusion equation with ε-periodic coefficients, where ε is a small parameter. In particular, under minimal regularity conditions, we construct approximations of order O(ε m ) in the L 2-norm as ε → 0. Bibliography: 14 titles.      13. Global existence and uniqueness for the magnetic Hartree equation      Pei Cao 《Journal of Evolution Equations》2011,11(4):811-825 In this paper, we consider lliut=Hu+\frac1|x|*|u|2u,    (x,t) ? \mathbbRN×\mathbbR.\begin{array}{ll}iu_{t}=Hu+\frac{1}{|x|}*|u|^{2}u,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}.\end{array}      14. On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality      E. A. Sevost’yanov 《Ukrainian Mathematical Journal》2011,63(1):84-97 For open discrete mappings f:D\{ b } ? \mathbbR3 f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR3 D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR3 b \in {\mathbb{R}^3} , we prove the following statement: Let a point y 0 belong to [`(\mathbbR3)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K I (x, f) and outer dilatation K O (x, f) of the mapping f at the point x satisfy certain conditions. Let B f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y 0, the set V ∩ f(B f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRn g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRn U \subset {\mathbb{R}^n} such that A ⊂ U.      15. Blow up of subcritical quantities at the first singular time of the mean curvature flow      Nam Q. Le 《Geometriae Dedicata》2011,151(1):361-371 Consider a family of smooth immersions F(·,t) : Mn? \mathbbRn+1{F(\cdot,t)\,:\,{M^n\to \mathbb{R}^{n+1}}} of closed hypersurfaces in \mathbbRn+1{\mathbb{R}^{n+1}} moving by the mean curvature flow \frac?F(p,t)?t = -H(p,t)·n(p,t){\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}, for t ? [0,T){t\in [0,T)}. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example ò0tòMs\frac|A|n + 2 log (2 + |A|) dmds,{\int_{0}^{t}\int_{M_{s}}\frac{{\vert{\it A}\vert}^{n + 2}}{ log (2 + {\vert{\it A}\vert})}} d\mu ds, blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.      16. Multiple solutions for a superlinear and periodic elliptic system on \mathbbRN{\mathbb{R}^N}      Fukun Zhao  Leiga Zhao  Yanheng Ding 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2011,15(6):495-511 This paper is concerned with the following periodic Hamiltonian elliptic system {l-Du+V(x)u=g(x,v) in  \mathbbRN,-Dv+V(x)v=f(x,u) in  \mathbbRN,u(x)? 0 and v(x)?0 as  |x|?￥,\left \{\begin{array}{l}-\Delta u+V(x)u=g(x,v)\, {\rm in }\,\mathbb{R}^N,\\-\Delta v+V(x)v=f(x,u)\, {\rm in }\, \mathbb{R}^N,\\ u(x)\to 0\, {\rm and}\,v(x)\to0\, {\rm as }\,|x|\to\infty,\end{array}\right.      17. The structure of {\phi}-stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature      Pak Tung Ho 《Mathematische Annalen》2010,348(2):319-332 Suppose (N n , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as ${P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}Suppose (N n , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as P(g)=Rm￥(g)=R(g)-2Dglog f-|?glog f|g2{P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}, where dm=f dvol(g){dm=\phi\,dvol(g)} and R(g) is the scalar curvature of (N n , g). In this paper, under a technical assumption on f{\phi}, we prove that f{\phi}-stable minimal oriented hypersurface in the three-dimensional manifold with nonnegative P-scalar curvature must be conformally equivalent to either the complex plane \mathbbC{\mathbb{C}} or the cylinder \mathbbR×\mathbbS1{\mathbb{R}\times\mathbb{S}^1}.      18. Density of the polynomials in Hardy and Bergman spaces of slit domains      John R. Akeroyd 《Arkiv f?r Matematik》2011,49(1):1-16 It is shown that for any t, 0<t<∞, there is a Jordan arc Γ with endpoints 0 and 1 such that G\{1} í \mathbbD:={z:|z| < 1}\Gamma\setminus\{1\}\subseteq\mathbb{D}:=\{z:|z|<1\} and with the property that the analytic polynomials are dense in the Bergman space \mathbbAt(\mathbbD\G)\mathbb{A}^{t}(\mathbb{D}\setminus\Gamma) . It is also shown that one can go further in the Hardy space setting and find such a Γ that is in fact the graph of a continuous real-valued function on [0,1], where the polynomials are dense in Ht(\mathbbD\G)H^{t}(\mathbb{D}\setminus\Gamma) ; improving upon a result in an earlier paper.      19. Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces      Markus Haase 《Integral Equations and Operator Theory》2006,56(2):197-228 We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( Aa ) ?R( Aa ) )q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as { x  ?  X|t - q\textRea y1 ( tA )x, t - q\textRea y2 ( tA )x ? L*p ( ( 0,￥ );X ) } {\left\{ {x\, \in \,X|t^{{ - \theta {\text{Re}}\alpha }} \psi _{1} {\left( {tA} \right)}x,\,t^{{ - \theta {\text{Re}}\alpha }} \psi _{2} {\left( {tA} \right)}x \in L_{*}^{p} {\left( {{\left( {0,\infty } \right)};X} \right)}} \right\}}      20. On the selfsimilar solutions of a diffusion convection equation      Abdelilah Gmira  Benyouness Bettioui 《NoDEA : Nonlinear Differential Equations and Applications》2002,9(3):277-294 This paper is concerned with the equation¶¶ div(| ?u| p-2?u)+e| ?U| q+bx?U+aU=0, for  x ? \mathbbRN div(| \nabla u| ^{p-2}\nabla u)+\varepsilon \left| \nabla U\right| ^q+\beta x\nabla U+\alpha U=0,{\rm \ for}\;x\in \mathbb{R}^N ¶¶ where $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 and a,b, m \alpha ,\beta, \mu are positive parameters. We study the existence, uniqueness of radial solutions u(r). Also, qualitative behavior of u(r) are presented.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Analytic cohomology groups in top degrees of Zariski open sets in {\mathbb{P}^n}

We show that if A is a closed analytic subset of \mathbbPⁿ{\mathbb{P}^n} of pure codimension q then Hⁱ(\mathbbPⁿ\ A,F){H^i(\mathbb{P}^n{\setminus} A,{\mathcal F})} are finite dimensional for every coherent algebraic sheaf F{{\mathcal F}} and every i 3 n-[\fracn-1q]{i\geq n-\left[\frac{n-1}{q}\right]} . If n-1 3 2q we show that H^n-2(\mathbbPⁿ\ A,F)=0{n-1\geq 2q\,{\rm we show that}\, H^{n-2}(\mathbb{P}^n{\setminus} A,{\mathcal F})=0} .     

Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}

In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem

{l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      7. Uniform estimates for order statistics and Orlicz functions      Yehoram Gordon  Alexander E. Litvak  Carsten Schütt  Elisabeth Werner 《Positivity》2012,16(1):1-28 We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ 1, … , ξ n and a vector of scalars x = (x 1, … , x n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min1 ￡ i ￡ n |xixi|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max1 ￡ i ￡ n|xixi|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||yx||M{\|y_x\|_M} of the vector y x  = (1/x 1, … , 1/x n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ 1|, G(t) = \mathbb P ({ |x1| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ 1 is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò0t e-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò0se-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.      8. On the regular growth of Dirichlet series absolutely convergent in a half-plane      Yu. V. Stets’  M. M. Sheremeta 《Ukrainian Mathematical Journal》2011,63(5):797-814 For the Dirichlet series F(s) = ?n = 1￥ anexp{ sln } F(s) = \sum\nolimits_{n = 1}^\infty {{a_n}\exp \left\{ {s{\lambda_n}} \right\}} with abscissa of absolute convergence σ a =0, we establish conditions for (λ n ) and (a n ) under which lnM( s, F ) = TR( 1 + o(1) )exp{ rR

/

| s| } \ln M\left( {\sigma, F} \right) = {T_R}\left( {1 + o(1)} \right)\exp \left\{ {{{{{\varrho_R}}} \left/ {{\left| \sigma \right|}} \right.}} \right\} for σ ↑ 0, where M( s, F ) = sup{ | F( s+ it ) |:t ? \mathbbR } M\left( {\sigma, F} \right) = \sup \left\{ {\left| {F\left( {\sigma + it} \right)} \right|:t \in \mathbb{R}} \right\} and T R and ϱ R are positive constants.      9. The Hermite Property of a Causal Wiener Algebra Used in Control Theory      Amol Sasane 《Complex Analysis and Operator Theory》2010,4(1):97-107 Let \mathbbC+ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra A = { s( ? \mathbbC+ ) ? [^(f)]a (s) + ?k = 0￥ fk e - stk | lfa ? L1 (0,￥),(fk )k 3 0 ? l1, 0 = t0 < t1 < t2 < ? }{{{\mathcal{A}}}} = \left\{ s( \in {{{\mathbb{C}}}}_ + ) \mapsto \hat{f}_a (s) + \sum\limits_{k = 0}^\infty {f_k e^{ - st_k }}\bigg | \bigg.{\begin{array}{l}{f_a \in L^1 (0,\infty ),(f_k )_{k \geq 0} \in \ell^{1}, } \cr {{0 = t_0 < t_1 < t_2 < \ldots}} \end{array}} \right\}      10. On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}      Manass��s de Souza 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(2):199-215 In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbRN{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form -DN u+ V(x)|u|N-2u=\fracf(x,u)|x|a   in  \mathbbRN,    N 3 2,-\Delta_N\,u+ V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^a}\quad{\rm in} \, \mathbb{R}^N,\quad N\geq 2,      11. Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential      Y. Belaud  A. Shishkov 《Journal of Evolution Equations》2010,10(4):857-882 We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ut + Lu + a(x) |u|q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb RN{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò01 s-1 (meas\nolimits {x ? W: |a(x)| ￡ s })q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.      12. Approximations of the exponential of an operator with periodic coefficients      S. E. Pastukhova 《Journal of Mathematical Sciences》2012,181(5):668-700 We consider the operator exponential e −tA , t > 0, where A is a selfadjoint positive definite operator corresponding to the diffusion equation in \mathbbRn {\mathbb{R}^n} with measurable 1-periodic coefficients, and approximate it in the operator norm ||   ·   ||L2( \mathbbRn ) ? L2( \mathbbRn ) {\left\| {\; \cdot \;} \right\|_{{{L^2}\left( {{\mathbb{R}^n}} \right) \to {L^2}\left( {{\mathbb{R}^n}} \right)}}} with order O( t - \fracm2 ) O\left( {{t^{{ - \frac{m}{2}}}}} \right) as t → ∞, where m is an arbitrary natural number. To construct approximations we use the homogenized parabolic equation with constant coefficients, the order of which depends on m and is greater than 2 if m > 2. We also use a collection of 1-periodic functions N α (x), x ? \mathbbRn x \in {\mathbb{R}^n} , with multi-indices α of length | a| \leqslant m \left| \alpha \right| \leqslant m , that are solutions to certain elliptic problems on the periodicity cell. These results are used to homogenize the diffusion equation with ε-periodic coefficients, where ε is a small parameter. In particular, under minimal regularity conditions, we construct approximations of order O(ε m ) in the L 2-norm as ε → 0. Bibliography: 14 titles.      13. Global existence and uniqueness for the magnetic Hartree equation      Pei Cao 《Journal of Evolution Equations》2011,11(4):811-825 In this paper, we consider lliut=Hu+\frac1|x|*|u|2u,    (x,t) ? \mathbbRN×\mathbbR.\begin{array}{ll}iu_{t}=Hu+\frac{1}{|x|}*|u|^{2}u,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}.\end{array}      14. On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality      E. A. Sevost’yanov 《Ukrainian Mathematical Journal》2011,63(1):84-97 For open discrete mappings f:D\{ b } ? \mathbbR3 f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR3 D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR3 b \in {\mathbb{R}^3} , we prove the following statement: Let a point y 0 belong to [`(\mathbbR3)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K I (x, f) and outer dilatation K O (x, f) of the mapping f at the point x satisfy certain conditions. Let B f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y 0, the set V ∩ f(B f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRn g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRn U \subset {\mathbb{R}^n} such that A ⊂ U.      15. Blow up of subcritical quantities at the first singular time of the mean curvature flow      Nam Q. Le 《Geometriae Dedicata》2011,151(1):361-371 Consider a family of smooth immersions F(·,t) : Mn? \mathbbRn+1{F(\cdot,t)\,:\,{M^n\to \mathbb{R}^{n+1}}} of closed hypersurfaces in \mathbbRn+1{\mathbb{R}^{n+1}} moving by the mean curvature flow \frac?F(p,t)?t = -H(p,t)·n(p,t){\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}, for t ? [0,T){t\in [0,T)}. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example ò0tòMs\frac|A|n + 2 log (2 + |A|) dmds,{\int_{0}^{t}\int_{M_{s}}\frac{{\vert{\it A}\vert}^{n + 2}}{ log (2 + {\vert{\it A}\vert})}} d\mu ds, blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.      16. Multiple solutions for a superlinear and periodic elliptic system on \mathbbRN{\mathbb{R}^N}      Fukun Zhao  Leiga Zhao  Yanheng Ding 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2011,15(6):495-511 This paper is concerned with the following periodic Hamiltonian elliptic system {l-Du+V(x)u=g(x,v) in  \mathbbRN,-Dv+V(x)v=f(x,u) in  \mathbbRN,u(x)? 0 and v(x)?0 as  |x|?￥,\left \{\begin{array}{l}-\Delta u+V(x)u=g(x,v)\, {\rm in }\,\mathbb{R}^N,\\-\Delta v+V(x)v=f(x,u)\, {\rm in }\, \mathbb{R}^N,\\ u(x)\to 0\, {\rm and}\,v(x)\to0\, {\rm as }\,|x|\to\infty,\end{array}\right.      17. The structure of {\phi}-stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature      Pak Tung Ho 《Mathematische Annalen》2010,348(2):319-332 Suppose (N n , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as ${P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}Suppose (N n , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as P(g)=Rm￥(g)=R(g)-2Dglog f-|?glog f|g2{P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}, where dm=f dvol(g){dm=\phi\,dvol(g)} and R(g) is the scalar curvature of (N n , g). In this paper, under a technical assumption on f{\phi}, we prove that f{\phi}-stable minimal oriented hypersurface in the three-dimensional manifold with nonnegative P-scalar curvature must be conformally equivalent to either the complex plane \mathbbC{\mathbb{C}} or the cylinder \mathbbR×\mathbbS1{\mathbb{R}\times\mathbb{S}^1}.      18. Density of the polynomials in Hardy and Bergman spaces of slit domains      John R. Akeroyd 《Arkiv f?r Matematik》2011,49(1):1-16 It is shown that for any t, 0<t<∞, there is a Jordan arc Γ with endpoints 0 and 1 such that G\{1} í \mathbbD:={z:|z| < 1}\Gamma\setminus\{1\}\subseteq\mathbb{D}:=\{z:|z|<1\} and with the property that the analytic polynomials are dense in the Bergman space \mathbbAt(\mathbbD\G)\mathbb{A}^{t}(\mathbb{D}\setminus\Gamma) . It is also shown that one can go further in the Hardy space setting and find such a Γ that is in fact the graph of a continuous real-valued function on [0,1], where the polynomials are dense in Ht(\mathbbD\G)H^{t}(\mathbb{D}\setminus\Gamma) ; improving upon a result in an earlier paper.      19. Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces      Markus Haase 《Integral Equations and Operator Theory》2006,56(2):197-228 We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( Aa ) ?R( Aa ) )q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as { x  ?  X|t - q\textRea y1 ( tA )x, t - q\textRea y2 ( tA )x ? L*p ( ( 0,￥ );X ) } {\left\{ {x\, \in \,X|t^{{ - \theta {\text{Re}}\alpha }} \psi _{1} {\left( {tA} \right)}x,\,t^{{ - \theta {\text{Re}}\alpha }} \psi _{2} {\left( {tA} \right)}x \in L_{*}^{p} {\left( {{\left( {0,\infty } \right)};X} \right)}} \right\}}      20. On the selfsimilar solutions of a diffusion convection equation      Abdelilah Gmira  Benyouness Bettioui 《NoDEA : Nonlinear Differential Equations and Applications》2002,9(3):277-294 This paper is concerned with the equation¶¶ div(| ?u| p-2?u)+e| ?U| q+bx?U+aU=0, for  x ? \mathbbRN div(| \nabla u| ^{p-2}\nabla u)+\varepsilon \left| \nabla U\right| ^q+\beta x\nabla U+\alpha U=0,{\rm \ for}\;x\in \mathbb{R}^N ¶¶ where $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 and a,b, m \alpha ,\beta, \mu are positive parameters. We study the existence, uniqueness of radial solutions u(r). Also, qualitative behavior of u(r) are presented.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Uniform estimates for order statistics and Orlicz functions

We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ ₁, … , ξ _n and a vector of scalars x = (x ₁, … , x _n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min_{1 ￡ i ￡ n} |x_ix_i|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max_{1 ￡ i ￡ n}|x_ix_i|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||y_x||_M{\|y_x\|_M} of the vector y _x  = (1/x ₁, … , 1/x _n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ ₁|, G(t) = \mathbb P ({ |x₁| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ ₁ is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò₀^t e^-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò₀^se^-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.     

On the regular growth of Dirichlet series absolutely convergent in a half-plane

For the Dirichlet series F(s) = ?_{n = 1}^￥ a_nexp{ sl_n } F(s) = \sum\nolimits_{n = 1}^\infty {{a_n}\exp \left\{ {s{\lambda_n}} \right\}} with abscissa of absolute convergence σ _a =0, we establish conditions for (λ _n ) and (a _n ) under which lnM( s, F ) = T_R( 1 + o(1) )exp{ r_R

The Hermite Property of a Causal Wiener Algebra Used in Control Theory

Let \mathbbC₊ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra

On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}

In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbR^N{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form

Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation u_t + Lu + a(x) |u|^q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb R^N{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò₀¹ s^-1 (meas\nolimits {x ? W: |a(x)| ￡ s })^q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.     

Approximations of the exponential of an operator with periodic coefficients

We consider the operator exponential e ^−tA , t > 0, where A is a selfadjoint positive definite operator corresponding to the diffusion equation in \mathbbRⁿ {\mathbb{R}^n} with measurable 1-periodic coefficients, and approximate it in the operator norm ||   ·   ||_{L²( \mathbbRⁿ ) ? L²( \mathbbRⁿ )} {\left\| {\; \cdot \;} \right\|_{{{L^2}\left( {{\mathbb{R}^n}} \right) \to {L^2}\left( {{\mathbb{R}^n}} \right)}}} with order O( t ^{- \fracm2} ) O\left( {{t^{{ - \frac{m}{2}}}}} \right) as t → ∞, where m is an arbitrary natural number. To construct approximations we use the homogenized parabolic equation with constant coefficients, the order of which depends on m and is greater than 2 if m > 2. We also use a collection of 1-periodic functions N _α (x), x ? \mathbbRⁿ x \in {\mathbb{R}^n} , with multi-indices α of length | a| \leqslant m \left| \alpha \right| \leqslant m , that are solutions to certain elliptic problems on the periodicity cell. These results are used to homogenize the diffusion equation with ε-periodic coefficients, where ε is a small parameter. In particular, under minimal regularity conditions, we construct approximations of order O(ε ^m ) in the L ²-norm as ε → 0. Bibliography: 14 titles.     

Global existence and uniqueness for the magnetic Hartree equation

In this paper, we consider

On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality

For open discrete mappings f:D\{ b } ? \mathbbR³ f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR³ D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR³ b \in {\mathbb{R}^3} , we prove the following statement: Let a point y ₀ belong to [`(\mathbbR³)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K _I (x, f) and outer dilatation K _O (x, f) of the mapping f at the point x satisfy certain conditions. Let B _f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y ₀, the set V ∩ f(B _f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRⁿ g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRⁿ U \subset {\mathbb{R}^n} such that A ⊂ U.     

Blow up of subcritical quantities at the first singular time of the mean curvature flow

Consider a family of smooth immersions F(·,t) : Mⁿ? \mathbbRⁿ⁺¹{F(\cdot,t)\,:\,{M^n\to \mathbb{R}^{n+1}}} of closed hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} moving by the mean curvature flow \frac?F(p,t)?t = -H(p,t)·n(p,t){\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}, for t ? [0,T){t\in [0,T)}. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example ò₀^tò_Ms\frac|A|^{n + 2} log (2 + |A|) dmds,{\int_{0}^{t}\int_{M_{s}}\frac{{\vert{\it A}\vert}^{n + 2}}{ log (2 + {\vert{\it A}\vert})}} d\mu ds, blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.     

Multiple solutions for a superlinear and periodic elliptic system on \mathbbRN{\mathbb{R}^N}

This paper is concerned with the following periodic Hamiltonian elliptic system

The structure of {\phi}-stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature

Suppose (N ⁿ , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as ${P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}Suppose (N ⁿ , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as P(g)=R^m_￥(g)=R(g)-2D_glog f-|?_glog f|_g²{P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}, where dm=f dvol(g){dm=\phi\,dvol(g)} and R(g) is the scalar curvature of (N ⁿ , g). In this paper, under a technical assumption on f{\phi}, we prove that f{\phi}-stable minimal oriented hypersurface in the three-dimensional manifold with nonnegative P-scalar curvature must be conformally equivalent to either the complex plane \mathbbC{\mathbb{C}} or the cylinder \mathbbR×\mathbbS¹{\mathbb{R}\times\mathbb{S}^1}.     

Density of the polynomials in Hardy and Bergman spaces of slit domains

It is shown that for any t, 0<t<∞, there is a Jordan arc Γ with endpoints 0 and 1 such that G\{1} í \mathbbD:={z:|z| < 1}\Gamma\setminus\{1\}\subseteq\mathbb{D}:=\{z:|z|<1\} and with the property that the analytic polynomials are dense in the Bergman space \mathbbA^t(\mathbbD\G)\mathbb{A}^{t}(\mathbb{D}\setminus\Gamma) . It is also shown that one can go further in the Hardy space setting and find such a Γ that is in fact the graph of a continuous real-valued function on [0,1], where the polynomials are dense in H^t(\mathbbD\G)H^{t}(\mathbb{D}\setminus\Gamma) ; improving upon a result in an earlier paper.     

Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces

We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( A^a ) ?R( A^a ) )_q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as

On the selfsimilar solutions of a diffusion convection equation

This paper is concerned with the equation¶¶ div(| ?u| ^p-2?u)+e| ?U| ^q+bx?U+aU=0, for  x ? \mathbbR^N div(| \nabla u| ^{p-2}\nabla u)+\varepsilon \left| \nabla U\right| ^q+\beta x\nabla U+\alpha U=0,{\rm \ for}\;x\in \mathbb{R}^N ¶¶ where $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 and a,b, m \alpha ,\beta, \mu are positive parameters. We study the existence, uniqueness of radial solutions u(r). Also, qualitative behavior of u(r) are presented.