Precise inequalities for norms of functions,third partial,second mixed,or directional derivatives

For functions f which are bounded throughout the plane R₂ together with the partial derivatives f(^3,0) f(_0,3), inequalities $$\left\| {f^{(1,1)} } \right\| \leqslant \sqrt[3]{3}\left\| f \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left\| {f^{(3,0)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left\| {f^{(0,3)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} ,\left\| {f_e^{(2)} } \right\| \leqslant \sqrt[3]{3}\left\| f \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left( {\left\| {f^{(3,0)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left| {e_1 } \right| + \left\| {f^{(0,3)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left| {e_2 } \right|} \right)^2 ,$$ are established, where ∥?∥denotes the upper bound on R² of the absolute values of the corresponding function, andf _fe ⁽²⁾ is the second derivative in the direction of the unit vector e=(e₁, e₂). Functions are exhibited for which these inequalities become equalities.     

THE SECTIONS OF ODD UNIVALENT FUNCTIONS

Let $\[{S_k}\]$ be the class of functions $\[f(z) = z + \sum\limits_{m = 1}^\infty {b_{mk + 1}^{(k)}{z^{mk + 1}}} \]$ which are regular and univalent in $\[\left| z \right| < 1\]$ and denote $\[S_n^{(k)}(z) = z + \sum\limits_{m = 1}^\infty {b_{mk + 1}^{(k)}{z^{mk + 1}}} \]$. The authors prove that the functions $\[S_n^{(2)}(z)\]$ are starlike in $\[\left| z \right| < \frac{1}{{\sqrt 3 }}\]$.     

具有p-Laplace算子的一类高阶奇异边值问题解的存在性

本文研究下面问题的正解其中Φp(s)=|s|p-2s,p>1．f在点x(i)=0,i=0,．．．,n-2可能是奇异的．证明建立在Leray-Schauder拓扑度和Vitali收敛定理的基础上．     

3$\times$3阶上三角型算子矩阵的可能谱

Let H1, H2 and H3 be infinite dimensional separable complex Hilbert spaces. We denote by M（D,V,F） a 3×3 upper triangular operator matrix acting on Hi ＋H2＋ H3 of theform M（D,E,F）=（A D F 0 B F 0 0 C）.For given A ∈ B（H1）, B ∈ B（H2） and C ∈ B（H3）, the sets ∪D,E,F^σp（M（D,E,F））,∪D,E,F ^σr（M（D,E,F））,∪D,E,F ^σc（M（D,E,F）） and ∪D,E,F σ（M（D,E,F）） are characterized, where D ∈ B（H2,H1）, E ∈B（H3, H1）, F ∈ B（H3,H2） and σ（·）, σp（·）, σr（·）, σc（·） denote the spectrum, the point spectrum, the residual spectrum and the continuous spectrum, respectively.     

一类n-阶m-点奇异边值问题的正解

研究n-阶m-点奇异边值问题其中h(t)允许在t=0,t=1处奇异,f(t,v_0,v_1,…,v_(n-2))允许在v_i=0(i=0,1,…,n-2)处奇异.利用锥拉伸与压缩不动点定理得到了上述奇异边值问题正解的存在性.     

STRONG UNIFORM CONSISTENCY FOR DENSITY ESTIMATOR FROM RANDOMLY CENSORED DATA

Let X_1,…,X_n be a sequence of independent identically distributed random variableswith distribution function F and density function f.The X_are censored on the right byY_i,where the Y_i are i.i.d.r.v.s with distribution function G and also independent of theX_i.One only observesLet S=1-F be survival function and S be the Kaplan-Meier estimator,i.e.,where Z_are the order statistics of Z_i and δ_((i))are the corresponping censoring indicatorfunctions.Define the density estimator of X_i by where =1-and h_n(>0)↓0.     

DISTRIBUTION OF THE（0，∞）ACCUMULATIVE LINES OF MEROMORPHIC FUNCTIONS

Suppose that f(z)is a meromorphic function of order λ（0&lt;λ&lt;+∞）and of lower order μ in the plane.Let ρ be a positive number such that μ≤ρ≤λ.(1)If f^(l)(z)(0≤l&lt;+∞）has p(1≤p&lt;+∞）finite nonzero deficient valnes αi(i=1,…，p)with deficiencies δ（αi,f^(l)),then f(z)has a (0,∞）accumulative line of order ≥ρin any angular domain whose vertex is at the origin and whose magnitude is larger than max(π/ρ，2π-4/ρ ∑i=1^p arcsin √δ（αi,f^(l))/2).(2)If f(z) has only p(0&lt;p&lt;+∞）（0,∞）,accumulative lines of order≥ρ：arg z=θk(0≤θ1&lt;θ2&lt;…&lt;θp&lt;2π，θp+1=θ1+2π）,then λ≤π/ω，where ω=min I≤k≤p(θk+1-θk),provided that f^(l)(z)(0≤l&lt;+∞）has a finite nonzero deficient value.     

n阶非线性常微分方程组边值问题的正解

运用不动点指数理论,研究以下$n$阶非线性常微分方程组边值问题正解的存在性和多重正解的存在性\[\left\{\ay\begin{array}{l}-u^{(n)}=f_1(x,u,v),\q-v^{(n)}=f_2(x,u,v),\\[2mm]u^{(i)}(0)=u^{(p)}(1)= v^{(i)}(0)=v^{(p)}(1)=0.\end{array}\right. \] 这里$n\geq 2$, $i = 0,1,2,\cdots,n-2$, $p \in \{1,2,\cdots,n-1\}$, $f_i\in C([0,1]\times\mathbb R^+\times\mathbb R^+,\mathbb R^+)~(i=1,2)$. 用凹函数刻画非线性项$f_1,f_2$的耦合行为, 因而非线性项 $f_i(i=1,2)$ 既可以都是超线性的, 也可以都是次线性的,还可以是混合非线性的(即其中一个是超线性的, 另一个是次线性的).     

Lower Bounds for the Perron Root of a Sum of Nonnegative Matrices

Let $A^{(l)} (l = 1, \ldots ,k)$ be $n \times n$ nonnegative matrices with right and left Perron vectors $u^{(l)} $ and $v^{(l)} $ , respectively, and let $D^{(l)} $ and $E^{(l)} (l = 1, \ldots ,k)$ be positive-definite diagonal matrices of the same order. Extending known results, under the assumption that $$u^{(1)} \circ v^{(1)} = \ldots = u^{(k)} \circ v^{(k)} \ne 0$$ (where `` $ \circ $ '' denotes the componentwise, i.e., the Hadamard product of vectors) but without requiring that the matrices $A^{(l)} $ be irreducible, for the Perron root of the sum $\sum\nolimits_{l = 1}^k {D^{(l)} A^{(l)} E^{(l)} } $ we derive a lower bound of the form $$\rho \left( {\sum\limits_{l = 1}^k {D^{(l)} A^{(l)} E^{(l)} } } \right) \geqslant \sum\limits_{l = 1}^k {\beta _{l\rho } (A^{(l)} ),{\text{ }}\beta _l >0.} $$ Also we prove that, for arbitrary irreducible nonnegative matrices $A^{{\text{ (}}l{\text{)}}} (l = 1, \ldots ,k),$ , $$\rho \left( {\sum\limits_{l = 1}^k {A^{(l)} } } \right) \geqslant \sum\limits_{l = 1}^k {\alpha _{l\rho } (A^{(l)} ),} $$ where the coefficients ∝₁>0 are specified using an arbitrarily chosen normalized positive vector. The cases of equality in both estimates are analyzed, and some other related results are established. Bibliography: 8 titles.     

Real hypersurfaces of a complex space form

In this paper, we show that an n-dimensional connected non-compact Ricci soliton isometrically immersed in the flat complex space form ${(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle )}$ , with potential vector field of the Ricci soliton is the characteristic vector field of the real hypersurface is an Einstein manifold. We classify connected Hopf hypersurfaces in the flat complex space form ${(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle )}$ and also obtain a characterization for the Hopf hypersurfaces in ${(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle ) }$ .     

Optimal recovery methods for the integral on classes of differentiable functions

Найдены методы восст ановления интеграла по информации $$I\left( f \right) = \left\{ {f^{(j)} \left( {x_i } \right)\left( {j = 0, ..., \gamma _i - 1; i = 1, ..., n; 1 \leqq \gamma _i \leqq r; \gamma _i + ... + \gamma _n \leqq N} \right.} \right\},$$ оптимальные на класс ахW _p ^r ,r=1,2,...; 1≦p≦∞. Это позволило, в частност и, получить наилучшие для классаW _p ^r квадратурные форму лы вида $$\mathop \smallint \limits_0^1 f\left( x \right)dx = \mathop \Sigma \limits_{i = 1}^n \mathop \Sigma \limits_{j = 1}^{\gamma _i - 1} a_{ij} f^{(j)} \left( {x_i } \right) + \mathop \Sigma \limits_{j = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} b_j f^{(2j - 1)} \left( 0 \right) + \mathop \Sigma \limits_{k = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} c_k f^{(2k - 1)} \left( 1 \right) + R\left( f \right)$$ И $$\mathop \smallint \limits_0^1 f\left( x \right)dx = af\left( 0 \right) + \mathop \Sigma \limits_{i = 1}^n \mathop \Sigma \limits_{j = 0}^{\gamma _i - 1} a_{ij} f^{(j)} \left( {x_i } \right) + bf\left( 1 \right) + \mathop \Sigma \limits_{j = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} b_j f^{(2j - 1)} \left( 0 \right) + \mathop \Sigma \limits_{k = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} c_k f^{(2k - 1)} \left( 1 \right) + R\left( f \right).$$      

Schrödinger equation and oscillatory Hilbert transforms of second degree

Let $h(t,x): = p.v. \sum\limits_{n \in Z\backslash \left| 0 \right|} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} = \mathop {\lim }\limits_{N \to \infty } \sum\limits_{0< \left| n \right| \leqslant N} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} $ ( $(i = \sqrt { - 1;} t,x$ -real variables). It is proved that in the rectangle $D: = \left\{ {(t,x):0< t< 1,\left| x \right| \leqslant \frac{1}{2}} \right\}$ , the function h satisfies the followingfunctional inequality: $\left| {h(t,x)} \right| \leqslant \sqrt t \left| {h\left( {\frac{1}{t},\frac{x}{t}} \right)} \right| + c,$ where c is an absolute positive constant. Iterations of this relation provide another, more elementary, proof of the known global boundedness result $\left\| {h; L^\infty (E^2 )} \right\| : = ess sup \left| {h(t,x)} \right|< \infty .$ The above functional inequality is derived from a general duality relation, of theta-function type, for solutions of the Cauchy initial value problem for Schrödinger equation of a free particle. Variation and complexity of solutions of Schrödinger equation are discussed.     

Convergence of symmetric Markov chains on {\mathbb{Z}^d}

For each n let ${Y^{(n)}_t}$ be a continuous time symmetric Markov chain with state space ${n^{-1} \mathbb{Z}^d}$ . Conditions in terms of the conductances are given for the convergence of the ${Y^{(n)}_t}$ to a symmetric Markov process Y _t on ${\mathbb{R}^d}$ . We have weak convergence of $\{{Y^{(n)}_t: t \leq t_0\}}$ for every t ₀ and every starting point. The limit process Y has a continuous part and may also have jumps.     

ON HYPONORMAL WEIGHTED SHIFT

It is shown in this paper that the necessary condition that a hyponormal weightedunilateral shift of norm one be unitarily equivalent to a Toeplitz operator is that theassociated weights{a_n}_0~- must stisfy 1-丨a_n丨~2=(1-丨a_0丨~2)~(n 1) _n≥0.As a consequence weobtain that the answer to Abrahamse's Problem 3 is that the Bergman shift is not unitarilyequivalent to a Toeplitz operator.     

ON THE ALGEBRAIC INDEPENDENCE OF CERTAIN POWER SERIES OF ALGEBRAIC NUMBERS

Letf_v(z)=∑a_(v,,k)z~(λ_(v,k))(v=1,…,s)be s power series with algebraic coefficients a_(v,k),convergence radii R_v>0 and sufficientlyrapidly increasing integers λ_(v,k).It is shown that under certain conditions depending only ona_(v,k) and λ_(v,k),(i)f_1(θ_1),…,f_s(θ_s)are algebraically independent for arbitrary algebraicnumbers θ_1,…,θ_s with θ<丨θ_v丨相似文献   

Existence Theory for an Arbitrary Order Fractional Differential Equation with Deviating Argument

In this paper, we are concerned with the existence criteria for positive solutions of the following nonlinear arbitrary order fractional differential equations with deviating argument

A general LIL for trimmed sums of random fields in banach spaces

Given a field of independent identically distributed (i.i.d.) random variables $ \left\{ {X_{\bar n} ;\bar n \in \aleph ^d } \right\} $ indexed by d-tuples of positive integers and taking values in a separable Banach space B, let $ X_{\bar n}^{(r)} = X_{\bar m} $ is the r-th maximum of $ \left\{ {\left\| {X_{\bar k} } \right\|;\bar k \leqq \bar n} \right\} $ and let $ ^{(r)} S_{\bar n} = S_{\bar n} - \left( {X_{\bar n}^{(1)} + \cdots + X_{\bar n}^{(r)} } \right) $ be the trimmed sums, where $ S_{\bar n} = \sum\nolimits_{\bar k \leqq \bar n} {X_{\bar k} } $ . This paper aims to obtain a general law of the iterated logarithm (LIL) for the trimmed sums which improves previous works.     

An Inverse Spectral Problem for a Nonsymmetric Differential Operator: Uniqueness and Reconstruction Formula

We consider an eigenvalue problem for a system on [0, 1]: $$\left\{ {\begin{array}{*{20}l} {\left[ {\left( {\begin{array}{*{20}c} 0 & 1 \\ 1 & 0 \\ \end{array} } \right)\frac{{\text{d}}} {{{\text{d}}x}} + \left( {\begin{array}{*{20}c} {p_{11} (x)} & {p_{12} (x)} \\ {p_{21} (x)} & {p_{22} (x)} \\ \end{array} } \right)} \right]\left( {\begin{array}{*{20}c} {\varphi ^{(1)} (x)} \\ {\varphi ^{(2)} (x)} \\ \end{array} } \right) = \lambda \left( {\begin{array}{*{20}c} {\varphi ^{(1)} (x)} \\ {\varphi ^{(1)} (x)} \\ \end{array} } \right)} \\ {\varphi ^{(2)} (0)\cosh \mu - \varphi ^{(1)} (0)\sinh \mu = \varphi ^{(2)} (1)\cosh \nu + \varphi ^{(1)} (1)\sinh \nu = 0} \\ \end{array} } \right.$$ with constants $$\mu ,\nu \in \mathbb{C}.$$ Under the assumption that p₂₁, p₂₂ are known, we prove a uniqueness theorem and provide a reconstruction formula for p₁₁ and p₁₂ from the spectral characteristics consisting of one spectrum and the associated norming constants.