On the global attractivity of two nonlinear difference equations

The main objective of this paper is to study some qualitative behavior of the solutions of the two difference equations

$ {x_{n + 1}} = {{{a{x_n} - b{x_{n - k}}}} \left/ {{\left( {c{x_n} - d{x_{n - k}}} \right)}} \right.}\quad n = 0,\,1,\,2, \ldots, $

and

$ {x_{n + 1}} = {{{a{x_{n - k}} + b{x_n}}} \left/ {{\left( {c{x_n} - d{x_{n - k}}} \right)}} \right.}\quad n = 0,\,1,\,2, \ldots, $

where the initial conditions x???k, ?, x???1, x0 are arbitrary positive real numbers and the coefficients a, b, c, and d are positive constants, while k is a positive integer number.

     

Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

For x = (x ₁, x ₂, …, x _n ) ∈ (0, 1 ] ⁿ and r ∈ { 1, 2, … , n}, a symmetric function F _n (x, r) is defined by the relation

Singular limits for 2-dimensional elliptic problem involving exponential nonlinearity with nonlinear gradient term

Given a bounded open regular set W ì \mathbbR²{\Omega \subset \mathbb{R}^2} and x₁, x₂, ?, x_m ? W{x_1, x_2, \ldots, x_m \in \Omega}, we give a sufficient condition for the problem

Solution of Initial Value Problems with Monogenic Initial Functions in Banach Spaces with L p -Norm

This paper deals with the initial value problem of the type

On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data

We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations

l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      7. 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\}      8. Global L p estimates for degenerate Ornstein–Uhlenbeck operators      Marco Bramanti  Giovanni Cupini  Ermanno Lanconelli  Enrico Priola 《Mathematische Zeitschrift》2010,266(4):789-816 We consider a class of degenerate Ornstein–Uhlenbeck operators in ${\mathbb{R}^{N}}We consider a class of degenerate Ornstein–Uhlenbeck operators in \mathbbRN{\mathbb{R}^{N}} , of the kind A o ?i, j=1p0aij?xixj2 + ?i, j=1Nbijxi?xj\mathcal{A}\equiv\sum_{i, j=1}^{p_{0}}a_{ij}\partial_{x_{i}x_{j}}^{2} + \sum_{i, j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}      9. Differential Operators Associated to the Cauchy-Fueter Operator in Quaternion Algebra      Thanh Van Nguyen 《Advances in Applied Clifford Algebras》2011,21(3):591-605 This paper deals with the initial value problem of the type ${\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)$ $w(0,x)=\varphi(x)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(2)$ where t is the time, L is a linear first order operator (matrix-type) in Quaternionic Analysis and ${\varphi}This paper deals with the initial value problem of the type \frac?w?t=L(t,x,w,\frac?w?xi)                                 (1){\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)      10. Numerical verification of condition for approximately midconvex functions      P. Spurek  Ja. Tabor 《Aequationes Mathematicae》2012,83(3):223-237 Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR+ ? \mathbbR+{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if f (\fracx + y2)-\fracf(x) + f(y)2 ￡ a(||x - y||)    for  x, y ? V.f \left(\frac{x + y}{2}\right)-\frac{f(x) + f(y)}{2}\leq \alpha(\|x - y\|)\quad {\rm for} \, x, y \in V.      11. The Nevai Condition      Jonathan Breuer  Yoram Last  Barry Simon 《Constructive Approximation》2010,32(2):221-254 We study Nevai’s condition that for orthogonal polynomials on the real line, $K_{n}(x,x_{0})^{2}K_{n}(x_{0},x_{0})^{-1}\,d\rho(x)\to\delta_{x_{0}}We study Nevai’s condition that for orthogonal polynomials on the real line, Kn(x,x0)2Kn(x0,x0)-1 dr(x)?dx0K_{n}(x,x_{0})^{2}K_{n}(x_{0},x_{0})^{-1}\,d\rho(x)\to\delta_{x_{0}} , where K n is the Christoffel–Darboux kernel. We prove that it holds for the Nevai class of a finite gap set uniformly on the spectrum, and we provide an example of a regular measure on [−2,2] where it fails on an interval.      12. On the global well-posedness of a class of Boussinesq�CNavier�CStokes systems      Changxing Miao  Liutang Xue 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(6):707-735 In this paper we consider the following 2D Boussinesq–Navier–Stokes systems lll?t u + u ·?u + ?p = - n|D|a u + qe2       ?t q+u·?q = - k|D|b q               div u = 0{\begin{array}{lll}\partial_t u + u \cdot \nabla u + \nabla p = - \nu |D|^\alpha u + \theta e_2\\ \quad\quad \partial_t \theta+u\cdot\nabla \theta = - \kappa|D|^\beta \theta \\ \quad\quad\quad\quad\quad{\rm div} u = 0\end{array}}      13. Metrizability of the space of {\mathbb{R}} -places of a real function field      Michał Machura  Murray Marshall  Katarzyna Osiak 《Mathematische Zeitschrift》2010,266(1):237-242 For n = 1, the space of ${\mathbb{R}}For n = 1, the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x1,?, xn){\mathbb{R}(x_1,\ldots, x_n)} is homeomorphic to the real projective line. For n ≥ 2, the structure is much more complicated. We prove that the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x, y){\mathbb{R}(x, y)} is not metrizable. We explain how the proof generalizes to show that the space of \mathbbR{\mathbb{R}} -places of any finitely generated formally real field extension of \mathbbR{\mathbb{R}} of transcendence degree ≥ 2 is not metrizable. We also consider the more general question of when the space of \mathbbR{\mathbb{R}} -places of a finitely generated formally real field extension of a real closed field is metrizable.      14. On a question of Erdös and Graham      I. E. Shparlinski 《Archiv der Mathematik》2002,78(6):445-448 We prove that for any $ \varepsilon > 0 $ \varepsilon > 0 there is k (e) k (\varepsilon) such that for any prime p and any integer c there exist k \leqq k(e) k \leqq k(\varepsilon) pairwise distinct integers xi with 1 \leqq xi \leqq pe, i = 1, ?, k 1 \leqq x_{i} \leqq p^{\varepsilon}, i = 1, \ldots, k , and such that¶¶?i=1k [1/(xi)] o c    (mod p). \sum\limits_{i=1}^k {{1}\over{x_i}} \equiv c\quad (\mathrm{mod}\, p). ¶¶ This gives a positive answer to a question of Erdös and Graham.      15. Notes on the nuclearity of random subalgebras of O2{\mathcal {O}_2}      Bernhard Burgstaller 《Monatshefte für Mathematik》2009,265(4):1-11 There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x 1, x 2, ... we have limn ? ￥\mathbb P(C*(x1,?,xn) is nuclear)=0{\rm {lim}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra O2{\mathcal {O}_2} such that for an independent sequence x 1, x 2, ... of μ-distributed random elements x i we have lim infn ? ￥\mathbb P(C*(x1,?,xn) is nuclear)=0{\rm {lim\, inf}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1] d ) is d.      16. Epimorphisms, dominions and permutative semigroups      Noor Mohammad Khan  Aftab Hussain Shah 《Semigroup Forum》2010,80(1):181-190 We show that any right semicommutative semigroup satisfying the minimum condition on principal right ideals is saturated. Further we show that if a supersaturated semigroup S has a globally idempotent ideal U which satisfies a nontrivial permutation identity $x_{1}x_{2}\ldots x_{n}=x_{i_{1}}x_{i_{2}}\ldots x_{i_{n}}We show that any right semicommutative semigroup satisfying the minimum condition on principal right ideals is saturated. Further we show that if a supersaturated semigroup S has a globally idempotent ideal U which satisfies a nontrivial permutation identity x1x2?xn=xi1xi2?xinx_{1}x_{2}\ldots x_{n}=x_{i_{1}}x_{i_{2}}\ldots x_{i_{n}} such that i 1=1 and i n ≠n, then U is also supersaturated.      17. An inverse spectral theorem for Kre?n strings with a negative eigenvalue      Harald Woracek 《Monatshefte für Mathematik》2012,33(3):105-149 A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ￥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L2(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D\mathfrakmDx{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as f￠(x) + z ò0L f(y) d\mathfrakm(y) = 0,    x ? \mathbb R, f￠(0-) = 0.f^{\prime}(x) + z \int_0^L f(y)\,d\mathfrak{m}(y) = 0,\quad x \in\mathbb R,\ f^{\prime}(0-) = 0.      18. Multiple solutions to a nonlinear Schrödinger equation with Aharonov–Bohm magnetic potential      Mónica Clapp  Andrzej Szulkin 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):229-248 We consider the magnetic nonlinear Schrödinger equations $\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}$ in ${\Omega=\mathcal{O}\times \mathbb{R}}We consider the magnetic nonlinear Schr?dinger equations ll(-i?+ sA)2 u + u   =  |u|p-2 u,     p ? (2, 6),         (-i?+ sA) 2u   =  |u|4 u,\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}      19. Projective Freeness of Algebras of Real Symmetric Functions      Amol Sasane 《Complex Analysis and Operator Theory》2012,6(2):465-475 Let \mathbb Dn:={z=(z1,?, zn) ? \mathbb Cn:|zj| < 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 Cr([`(\mathbbD)]n;\mathbb C) = {f:[`(\mathbbD)]n? \mathbb C:f   is   continuous   and  f(z)=[`(f([`(z)]))]   (z ? [`(\mathbbD)]n)}C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C}) =\left\{f: \overline{\mathbb{D}}^n\rightarrow \mathbb {C}:f \,\, {\rm is \,\, continuous \,\, and}\,\, f(z)=\overline{f(\overline{z})} \;(z\in \overline{\mathbb{D}}^n)\right\}      20. Hermitian-Symmetric Inequalities in Hilbert Space      Stephan Ramon Garcia 《Complex Analysis and Operator Theory》2009,3(4):835-846 We establish several conditions which are equivalent to |[Bx,x]| ￡ áAx, x ?,    "x ? H|[Bx,x]| \leq \langle Ax, x \rangle,\quad \forall x \in {{\mathcal{H}}}      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

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

Global L p estimates for degenerate Ornstein–Uhlenbeck operators

We consider a class of degenerate Ornstein–Uhlenbeck operators in ${\mathbb{R}^{N}}We consider a class of degenerate Ornstein–Uhlenbeck operators in \mathbbR^N{\mathbb{R}^{N}} , of the kind

A o ?i, j=1p0aij?xixj2 + ?i, j=1Nbijxi?xj\mathcal{A}\equiv\sum_{i, j=1}^{p_{0}}a_{ij}\partial_{x_{i}x_{j}}^{2} + \sum_{i, j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}      9. Differential Operators Associated to the Cauchy-Fueter Operator in Quaternion Algebra      Thanh Van Nguyen 《Advances in Applied Clifford Algebras》2011,21(3):591-605 This paper deals with the initial value problem of the type ${\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)$ $w(0,x)=\varphi(x)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(2)$ where t is the time, L is a linear first order operator (matrix-type) in Quaternionic Analysis and ${\varphi}This paper deals with the initial value problem of the type \frac?w?t=L(t,x,w,\frac?w?xi)                                 (1){\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)      10. Numerical verification of condition for approximately midconvex functions      P. Spurek  Ja. Tabor 《Aequationes Mathematicae》2012,83(3):223-237 Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR+ ? \mathbbR+{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if f (\fracx + y2)-\fracf(x) + f(y)2 ￡ a(||x - y||)    for  x, y ? V.f \left(\frac{x + y}{2}\right)-\frac{f(x) + f(y)}{2}\leq \alpha(\|x - y\|)\quad {\rm for} \, x, y \in V.      11. The Nevai Condition      Jonathan Breuer  Yoram Last  Barry Simon 《Constructive Approximation》2010,32(2):221-254 We study Nevai’s condition that for orthogonal polynomials on the real line, $K_{n}(x,x_{0})^{2}K_{n}(x_{0},x_{0})^{-1}\,d\rho(x)\to\delta_{x_{0}}We study Nevai’s condition that for orthogonal polynomials on the real line, Kn(x,x0)2Kn(x0,x0)-1 dr(x)?dx0K_{n}(x,x_{0})^{2}K_{n}(x_{0},x_{0})^{-1}\,d\rho(x)\to\delta_{x_{0}} , where K n is the Christoffel–Darboux kernel. We prove that it holds for the Nevai class of a finite gap set uniformly on the spectrum, and we provide an example of a regular measure on [−2,2] where it fails on an interval.      12. On the global well-posedness of a class of Boussinesq�CNavier�CStokes systems      Changxing Miao  Liutang Xue 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(6):707-735 In this paper we consider the following 2D Boussinesq–Navier–Stokes systems lll?t u + u ·?u + ?p = - n|D|a u + qe2       ?t q+u·?q = - k|D|b q               div u = 0{\begin{array}{lll}\partial_t u + u \cdot \nabla u + \nabla p = - \nu |D|^\alpha u + \theta e_2\\ \quad\quad \partial_t \theta+u\cdot\nabla \theta = - \kappa|D|^\beta \theta \\ \quad\quad\quad\quad\quad{\rm div} u = 0\end{array}}      13. Metrizability of the space of {\mathbb{R}} -places of a real function field      Michał Machura  Murray Marshall  Katarzyna Osiak 《Mathematische Zeitschrift》2010,266(1):237-242 For n = 1, the space of ${\mathbb{R}}For n = 1, the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x1,?, xn){\mathbb{R}(x_1,\ldots, x_n)} is homeomorphic to the real projective line. For n ≥ 2, the structure is much more complicated. We prove that the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x, y){\mathbb{R}(x, y)} is not metrizable. We explain how the proof generalizes to show that the space of \mathbbR{\mathbb{R}} -places of any finitely generated formally real field extension of \mathbbR{\mathbb{R}} of transcendence degree ≥ 2 is not metrizable. We also consider the more general question of when the space of \mathbbR{\mathbb{R}} -places of a finitely generated formally real field extension of a real closed field is metrizable.      14. On a question of Erdös and Graham      I. E. Shparlinski 《Archiv der Mathematik》2002,78(6):445-448 We prove that for any $ \varepsilon > 0 $ \varepsilon > 0 there is k (e) k (\varepsilon) such that for any prime p and any integer c there exist k \leqq k(e) k \leqq k(\varepsilon) pairwise distinct integers xi with 1 \leqq xi \leqq pe, i = 1, ?, k 1 \leqq x_{i} \leqq p^{\varepsilon}, i = 1, \ldots, k , and such that¶¶?i=1k [1/(xi)] o c    (mod p). \sum\limits_{i=1}^k {{1}\over{x_i}} \equiv c\quad (\mathrm{mod}\, p). ¶¶ This gives a positive answer to a question of Erdös and Graham.      15. Notes on the nuclearity of random subalgebras of O2{\mathcal {O}_2}      Bernhard Burgstaller 《Monatshefte für Mathematik》2009,265(4):1-11 There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x 1, x 2, ... we have limn ? ￥\mathbb P(C*(x1,?,xn) is nuclear)=0{\rm {lim}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra O2{\mathcal {O}_2} such that for an independent sequence x 1, x 2, ... of μ-distributed random elements x i we have lim infn ? ￥\mathbb P(C*(x1,?,xn) is nuclear)=0{\rm {lim\, inf}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1] d ) is d.      16. Epimorphisms, dominions and permutative semigroups      Noor Mohammad Khan  Aftab Hussain Shah 《Semigroup Forum》2010,80(1):181-190 We show that any right semicommutative semigroup satisfying the minimum condition on principal right ideals is saturated. Further we show that if a supersaturated semigroup S has a globally idempotent ideal U which satisfies a nontrivial permutation identity $x_{1}x_{2}\ldots x_{n}=x_{i_{1}}x_{i_{2}}\ldots x_{i_{n}}We show that any right semicommutative semigroup satisfying the minimum condition on principal right ideals is saturated. Further we show that if a supersaturated semigroup S has a globally idempotent ideal U which satisfies a nontrivial permutation identity x1x2?xn=xi1xi2?xinx_{1}x_{2}\ldots x_{n}=x_{i_{1}}x_{i_{2}}\ldots x_{i_{n}} such that i 1=1 and i n ≠n, then U is also supersaturated.      17. An inverse spectral theorem for Kre?n strings with a negative eigenvalue      Harald Woracek 《Monatshefte für Mathematik》2012,33(3):105-149 A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ￥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L2(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D\mathfrakmDx{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as f￠(x) + z ò0L f(y) d\mathfrakm(y) = 0,    x ? \mathbb R, f￠(0-) = 0.f^{\prime}(x) + z \int_0^L f(y)\,d\mathfrak{m}(y) = 0,\quad x \in\mathbb R,\ f^{\prime}(0-) = 0.      18. Multiple solutions to a nonlinear Schrödinger equation with Aharonov–Bohm magnetic potential      Mónica Clapp  Andrzej Szulkin 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):229-248 We consider the magnetic nonlinear Schrödinger equations $\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}$ in ${\Omega=\mathcal{O}\times \mathbb{R}}We consider the magnetic nonlinear Schr?dinger equations ll(-i?+ sA)2 u + u   =  |u|p-2 u,     p ? (2, 6),         (-i?+ sA) 2u   =  |u|4 u,\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}      19. Projective Freeness of Algebras of Real Symmetric Functions      Amol Sasane 《Complex Analysis and Operator Theory》2012,6(2):465-475 Let \mathbb Dn:={z=(z1,?, zn) ? \mathbb Cn:|zj| < 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 Cr([`(\mathbbD)]n;\mathbb C) = {f:[`(\mathbbD)]n? \mathbb C:f   is   continuous   and  f(z)=[`(f([`(z)]))]   (z ? [`(\mathbbD)]n)}C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C}) =\left\{f: \overline{\mathbb{D}}^n\rightarrow \mathbb {C}:f \,\, {\rm is \,\, continuous \,\, and}\,\, f(z)=\overline{f(\overline{z})} \;(z\in \overline{\mathbb{D}}^n)\right\}      20. Hermitian-Symmetric Inequalities in Hilbert Space      Stephan Ramon Garcia 《Complex Analysis and Operator Theory》2009,3(4):835-846 We establish several conditions which are equivalent to |[Bx,x]| ￡ áAx, x ?,    "x ? H|[Bx,x]| \leq \langle Ax, x \rangle,\quad \forall x \in {{\mathcal{H}}}      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

Differential Operators Associated to the Cauchy-Fueter Operator in Quaternion Algebra

This paper deals with the initial value problem of the type ${\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)$ $w(0,x)=\varphi(x)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(2)$ where t is the time, L is a linear first order operator (matrix-type) in Quaternionic Analysis and ${\varphi}This paper deals with the initial value problem of the type

\frac?w?t=L(t,x,w,\frac?w?xi)                                 (1){\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)      10. Numerical verification of condition for approximately midconvex functions      P. Spurek  Ja. Tabor 《Aequationes Mathematicae》2012,83(3):223-237 Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR+ ? \mathbbR+{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if f (\fracx + y2)-\fracf(x) + f(y)2 ￡ a(||x - y||)    for  x, y ? V.f \left(\frac{x + y}{2}\right)-\frac{f(x) + f(y)}{2}\leq \alpha(\|x - y\|)\quad {\rm for} \, x, y \in V.      11. The Nevai Condition      Jonathan Breuer  Yoram Last  Barry Simon 《Constructive Approximation》2010,32(2):221-254 We study Nevai’s condition that for orthogonal polynomials on the real line, $K_{n}(x,x_{0})^{2}K_{n}(x_{0},x_{0})^{-1}\,d\rho(x)\to\delta_{x_{0}}We study Nevai’s condition that for orthogonal polynomials on the real line, Kn(x,x0)2Kn(x0,x0)-1 dr(x)?dx0K_{n}(x,x_{0})^{2}K_{n}(x_{0},x_{0})^{-1}\,d\rho(x)\to\delta_{x_{0}} , where K n is the Christoffel–Darboux kernel. We prove that it holds for the Nevai class of a finite gap set uniformly on the spectrum, and we provide an example of a regular measure on [−2,2] where it fails on an interval.      12. On the global well-posedness of a class of Boussinesq�CNavier�CStokes systems      Changxing Miao  Liutang Xue 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(6):707-735 In this paper we consider the following 2D Boussinesq–Navier–Stokes systems lll?t u + u ·?u + ?p = - n|D|a u + qe2       ?t q+u·?q = - k|D|b q               div u = 0{\begin{array}{lll}\partial_t u + u \cdot \nabla u + \nabla p = - \nu |D|^\alpha u + \theta e_2\\ \quad\quad \partial_t \theta+u\cdot\nabla \theta = - \kappa|D|^\beta \theta \\ \quad\quad\quad\quad\quad{\rm div} u = 0\end{array}}      13. Metrizability of the space of {\mathbb{R}} -places of a real function field      Michał Machura  Murray Marshall  Katarzyna Osiak 《Mathematische Zeitschrift》2010,266(1):237-242 For n = 1, the space of ${\mathbb{R}}For n = 1, the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x1,?, xn){\mathbb{R}(x_1,\ldots, x_n)} is homeomorphic to the real projective line. For n ≥ 2, the structure is much more complicated. We prove that the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x, y){\mathbb{R}(x, y)} is not metrizable. We explain how the proof generalizes to show that the space of \mathbbR{\mathbb{R}} -places of any finitely generated formally real field extension of \mathbbR{\mathbb{R}} of transcendence degree ≥ 2 is not metrizable. We also consider the more general question of when the space of \mathbbR{\mathbb{R}} -places of a finitely generated formally real field extension of a real closed field is metrizable.      14. On a question of Erdös and Graham      I. E. Shparlinski 《Archiv der Mathematik》2002,78(6):445-448 We prove that for any $ \varepsilon > 0 $ \varepsilon > 0 there is k (e) k (\varepsilon) such that for any prime p and any integer c there exist k \leqq k(e) k \leqq k(\varepsilon) pairwise distinct integers xi with 1 \leqq xi \leqq pe, i = 1, ?, k 1 \leqq x_{i} \leqq p^{\varepsilon}, i = 1, \ldots, k , and such that¶¶?i=1k [1/(xi)] o c    (mod p). \sum\limits_{i=1}^k {{1}\over{x_i}} \equiv c\quad (\mathrm{mod}\, p). ¶¶ This gives a positive answer to a question of Erdös and Graham.      15. Notes on the nuclearity of random subalgebras of O2{\mathcal {O}_2}      Bernhard Burgstaller 《Monatshefte für Mathematik》2009,265(4):1-11 There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x 1, x 2, ... we have limn ? ￥\mathbb P(C*(x1,?,xn) is nuclear)=0{\rm {lim}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra O2{\mathcal {O}_2} such that for an independent sequence x 1, x 2, ... of μ-distributed random elements x i we have lim infn ? ￥\mathbb P(C*(x1,?,xn) is nuclear)=0{\rm {lim\, inf}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1] d ) is d.      16. Epimorphisms, dominions and permutative semigroups      Noor Mohammad Khan  Aftab Hussain Shah 《Semigroup Forum》2010,80(1):181-190 We show that any right semicommutative semigroup satisfying the minimum condition on principal right ideals is saturated. Further we show that if a supersaturated semigroup S has a globally idempotent ideal U which satisfies a nontrivial permutation identity $x_{1}x_{2}\ldots x_{n}=x_{i_{1}}x_{i_{2}}\ldots x_{i_{n}}We show that any right semicommutative semigroup satisfying the minimum condition on principal right ideals is saturated. Further we show that if a supersaturated semigroup S has a globally idempotent ideal U which satisfies a nontrivial permutation identity x1x2?xn=xi1xi2?xinx_{1}x_{2}\ldots x_{n}=x_{i_{1}}x_{i_{2}}\ldots x_{i_{n}} such that i 1=1 and i n ≠n, then U is also supersaturated.      17. An inverse spectral theorem for Kre?n strings with a negative eigenvalue      Harald Woracek 《Monatshefte für Mathematik》2012,33(3):105-149 A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ￥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L2(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D\mathfrakmDx{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as f￠(x) + z ò0L f(y) d\mathfrakm(y) = 0,    x ? \mathbb R, f￠(0-) = 0.f^{\prime}(x) + z \int_0^L f(y)\,d\mathfrak{m}(y) = 0,\quad x \in\mathbb R,\ f^{\prime}(0-) = 0.      18. Multiple solutions to a nonlinear Schrödinger equation with Aharonov–Bohm magnetic potential      Mónica Clapp  Andrzej Szulkin 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):229-248 We consider the magnetic nonlinear Schrödinger equations $\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}$ in ${\Omega=\mathcal{O}\times \mathbb{R}}We consider the magnetic nonlinear Schr?dinger equations ll(-i?+ sA)2 u + u   =  |u|p-2 u,     p ? (2, 6),         (-i?+ sA) 2u   =  |u|4 u,\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}      19. Projective Freeness of Algebras of Real Symmetric Functions      Amol Sasane 《Complex Analysis and Operator Theory》2012,6(2):465-475 Let \mathbb Dn:={z=(z1,?, zn) ? \mathbb Cn:|zj| < 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 Cr([`(\mathbbD)]n;\mathbb C) = {f:[`(\mathbbD)]n? \mathbb C:f   is   continuous   and  f(z)=[`(f([`(z)]))]   (z ? [`(\mathbbD)]n)}C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C}) =\left\{f: \overline{\mathbb{D}}^n\rightarrow \mathbb {C}:f \,\, {\rm is \,\, continuous \,\, and}\,\, f(z)=\overline{f(\overline{z})} \;(z\in \overline{\mathbb{D}}^n)\right\}      20. Hermitian-Symmetric Inequalities in Hilbert Space      Stephan Ramon Garcia 《Complex Analysis and Operator Theory》2009,3(4):835-846 We establish several conditions which are equivalent to |[Bx,x]| ￡ áAx, x ?,    "x ? H|[Bx,x]| \leq \langle Ax, x \rangle,\quad \forall x \in {{\mathcal{H}}}      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

Numerical verification of condition for approximately midconvex functions

Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR₊ ? \mathbbR₊{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if

The Nevai Condition

We study Nevai’s condition that for orthogonal polynomials on the real line, $K_{n}(x,x_{0})^{2}K_{n}(x_{0},x_{0})^{-1}\,d\rho(x)\to\delta_{x_{0}}We study Nevai’s condition that for orthogonal polynomials on the real line, K_n(x,x₀)²K_n(x₀,x₀)^-1 dr(x)?d_x0K_{n}(x,x_{0})^{2}K_{n}(x_{0},x_{0})^{-1}\,d\rho(x)\to\delta_{x_{0}} , where K _n is the Christoffel–Darboux kernel. We prove that it holds for the Nevai class of a finite gap set uniformly on the spectrum, and we provide an example of a regular measure on [−2,2] where it fails on an interval.     

On the global well-posedness of a class of Boussinesq�CNavier�CStokes systems

In this paper we consider the following 2D Boussinesq–Navier–Stokes systems

Metrizability of the space of {\mathbb{R}} -places of a real function field

For n = 1, the space of ${\mathbb{R}}For n = 1, the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x₁,?, x_n){\mathbb{R}(x_1,\ldots, x_n)} is homeomorphic to the real projective line. For n ≥ 2, the structure is much more complicated. We prove that the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x, y){\mathbb{R}(x, y)} is not metrizable. We explain how the proof generalizes to show that the space of \mathbbR{\mathbb{R}} -places of any finitely generated formally real field extension of \mathbbR{\mathbb{R}} of transcendence degree ≥ 2 is not metrizable. We also consider the more general question of when the space of \mathbbR{\mathbb{R}} -places of a finitely generated formally real field extension of a real closed field is metrizable.     

On a question of Erdös and Graham

We prove that for any $ \varepsilon > 0 $ \varepsilon > 0 there is k (e) k (\varepsilon) such that for any prime p and any integer c there exist k \leqq k(e) k \leqq k(\varepsilon) pairwise distinct integers x_i with 1 \leqq x_i \leqq p^e, i = 1, ?, k 1 \leqq x_{i} \leqq p^{\varepsilon}, i = 1, \ldots, k , and such that¶¶?_i=1^k [1/(x_i)] o c    (mod p). \sum\limits_{i=1}^k {{1}\over{x_i}} \equiv c\quad (\mathrm{mod}\, p). ¶¶ This gives a positive answer to a question of Erdös and Graham.     

Notes on the nuclearity of random subalgebras of O2{\mathcal {O}_2}

There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x ₁, x ₂, ... we have lim_{n ? ￥}\mathbb P(C^*(x₁,?,x_n) is nuclear)=0{\rm {lim}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra O₂{\mathcal {O}_2} such that for an independent sequence x ₁, x ₂, ... of μ-distributed random elements x _i we have lim inf_{n ? ￥}\mathbb P(C^*(x₁,?,x_n) is nuclear)=0{\rm {lim\, inf}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1] ^d ) is d.     

Epimorphisms, dominions and permutative semigroups

We show that any right semicommutative semigroup satisfying the minimum condition on principal right ideals is saturated. Further we show that if a supersaturated semigroup S has a globally idempotent ideal U which satisfies a nontrivial permutation identity $x_{1}x_{2}\ldots x_{n}=x_{i_{1}}x_{i_{2}}\ldots x_{i_{n}}We show that any right semicommutative semigroup satisfying the minimum condition on principal right ideals is saturated. Further we show that if a supersaturated semigroup S has a globally idempotent ideal U which satisfies a nontrivial permutation identity x₁x₂?x_n=x_i1x_i2?x_inx_{1}x_{2}\ldots x_{n}=x_{i_{1}}x_{i_{2}}\ldots x_{i_{n}} such that i ₁=1 and i _n ≠n, then U is also supersaturated.     

An inverse spectral theorem for Kre?n strings with a negative eigenvalue

A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ￥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L²(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D_\mathfrakmD_x{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as

Multiple solutions to a nonlinear Schrödinger equation with Aharonov–Bohm magnetic potential

We consider the magnetic nonlinear Schrödinger equations $\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}$ in ${\Omega=\mathcal{O}\times \mathbb{R}}We consider the magnetic nonlinear Schr?dinger equations

ll(-i?+ sA)2 u + u   =  |u|p-2 u,     p ? (2, 6),         (-i?+ sA) 2u   =  |u|4 u,\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}      19. Projective Freeness of Algebras of Real Symmetric Functions      Amol Sasane 《Complex Analysis and Operator Theory》2012,6(2):465-475 Let \mathbb Dn:={z=(z1,?, zn) ? \mathbb Cn:|zj| < 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 Cr([`(\mathbbD)]n;\mathbb C) = {f:[`(\mathbbD)]n? \mathbb C:f   is   continuous   and  f(z)=[`(f([`(z)]))]   (z ? [`(\mathbbD)]n)}C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C}) =\left\{f: \overline{\mathbb{D}}^n\rightarrow \mathbb {C}:f \,\, {\rm is \,\, continuous \,\, and}\,\, f(z)=\overline{f(\overline{z})} \;(z\in \overline{\mathbb{D}}^n)\right\}      20. Hermitian-Symmetric Inequalities in Hilbert Space      Stephan Ramon Garcia 《Complex Analysis and Operator Theory》2009,3(4):835-846 We establish several conditions which are equivalent to |[Bx,x]| ￡ áAx, x ?,    "x ? H|[Bx,x]| \leq \langle Ax, x \rangle,\quad \forall x \in {{\mathcal{H}}}      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

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

Hermitian-Symmetric Inequalities in Hilbert Space

We establish several conditions which are equivalent to