EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR NONLINEAR BOUNDARY VALUE PROBLEMS OF VOLTERRA－HAMMERSTEIN TYPE INTEGRODIFFERENTIAL EQUATION

ＥＸＩＳＴＥＮＣＥＡＮＤＵＮＩＱＵＥＮＥＳＳＯＦＳＯＬＵＴＩＯＮＳＦＯＲＮＯＮＬＩＮＥＡＲＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＳＯＦＶＯＬＴＥＲＲＡ－ＨＡＭＭＥＲＳＴＥＩＮＴＹＰＥＩＮＴＥＧＲＯＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮ￥ＷａｎｇＧ...     

ON THE BOUNDED AND UNBOUNDED SOLUTIONS OF ONE DIMENSIONAL NONLINEAR REACTION－DIFFUSION PROBLEM

ＯＮＴＨＥＢＯＵＮＤＥＤＡＮＤＵＮＢＯＵＮＤＥＤＳＯＬＵＴＩＯＮＳＯＦＯＮＥＤＩＭＥＮＳＩＯＮＡＬＮＯＮＬＩＮＥＡＲＲＥＡＣＴＩＯＮ－ＤＩＦＦＵＳＩＯＮＰＲＯＢＬＥＭ￥ＧＥＷＥＩＧＡＯＲ．Ｏ．ＷＥＢＥＲＡｂｓｔｒａｃｔ：Ｔｈｅｅｘｉｓｔｅｎｃｅｏｆｂ...     

A NECESSARY AND SUFFICIENT CONDITION OF EXISTENCE OF GLOBAL SOLUTIONS FOR SOME NONLINEAR HYPERBOLIC EQUATIONS

ＡＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＯＮＤＩＴＩＯＮＯＦＥＸＩＳＴＥＮＣＥＯＦＧＬＯＢＡＬＳＯＬＵＴＩＯＮＳＦＯＲＳＯＭＥＮＯＮＬＩＮＥＡＲＨＹＰＥＲＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＺＨＡＮＧＱＵＡＮＤＥ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅ...     

EXACT BOUNDS OF THE MODIFIED LPT ALGORITHMS APPLYING TO PARALLEL MACHINES SCHEDULING WITH NONSIMULTANEOUS MACHINE AVAILABLE TIMES

ＥＸＡＣＴＢＯＵＮＤＳＯＦＴＨＥＭＯＤＩＦＩＥＤＬＰＴＡＬＧＯＲＩＴＨＭＳＡＰＰＬＹＩＮＧＴＯＰＡＲＡＬＬＥＬＭＡＣＨＩＮＥＳＳＣＨＥＤＵＬＩＮＧＷＩＴＨＮＯＮＳＩＭＵＬＴＡＮＥＯＵＳＭＡＣＨＩＮＥＡＶＡＩＬＡＢＬＥＴＩＭＥＳＬＩＮＧＵＯＨＵＩ，Ｈ...     

UNIFORM STABILITY AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF 2-DIMENSIONAL MAGNETOHYDRODYNAMICS EQUATIONS

ＵＮＩＦＯＲＭＳＴＡＢＩＬＩＴＹＡＮＤＡＳＹＭＰＴＯＴＩＣＢＥＨＡＶＩＯＲＯＦＳＯＬＵＴＩＯＮＳＯＦ２－ＤＩＭＥＮＳＩＯＮＡＬＭＡＧＮＥＴＯＨＹＤＲＯＤＹＮＡＭＩＣＳＥＱＵＡＴＩＯＮＳＺＨＡＮＧＬＩＮＧＨＡＩＭａｎｕｓｃｒｉｐｔｒｅｃｅｉｖｅｄＪｕ...     

EXISTENCE AND UNIQUENESS OF GLOBAL SOLUTION FOR CERTAIN NONLINEAR DEGENERATE PARABOLIC SYSTEMS

ＥＸＩＳＴＥＮＣＥＡＮＤＵＮＩＱＵＥＮＥＳＳＯＦＧＬＯＢＡＬＳＯＬＵＴＩＯＮＦＯＲＣＥＲＴＡＩＮＮＯＮＬＩＮＥＡＲＤＥＧＥＮＥＲＡＴＥＰＡＲＡＢＯＬＩＣ　ＳＹＳＴＥＭＳＬＵＧＵＯＦＵ（卢国富）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，Ｐｕｔｉ...     

ON GLOBAL SOLUTION FOR A CLASS OF SYSTEMS OF MULTI－DIMENSIONAL GENERALIZED ZAKHAROV TYPE EQUATION

ＯＮＧＬＯＢＡＬＳＯＬＵＴＩＯＮＦＯＲＡＣＬＡＳＳＯＦＳＹＳＴＥＭＳＯＦＭＵＬＴＩ－ＤＩＭＥＮＳＩＯＮＡＬＧＥＮＥＲＡＬＩＺＥＤＺＡＫＨＡＲＯＶＴＹＰＥＥＱＵＡＴＩＯＮＧＵＯＢＯＬＩＮＧ（郭柏灵）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄａｎｄＣｏｍｐｕ...     

ADMISSIBILITY OF LINEAR ESTIMATE OF REGRESSION COEFFICIENTS IN GROWTH CURVE MODEL UNDER MATRIX LOSS

ＡＤＭＩＳＳＩＢＩＬＩＴＹＯＦＬＩＮＥＡＲＥＳＴＩＭＡＴＥＯＦＲＥＧＲＥＳＳＩＯＮＣＯＥＦＦＩＣＩＥＮＴＳＩＮＧＲＯＷＴＨＣＵＲＶＥＭＯＤＥＬＵＮＤＥＲＭＡＴＲＩＸＬＯＳＳＷＡＮＧＸＵＥＲＥＮ（王学仁）（ＤｅｐａｒｔｍｅｎｔｏｆＳｔａｔｉｓｔｉｃｓ，...     

LARGE－TIME BEHMIOR OF SOLUTIONS FOR THE SYSTEM OF COMPRESSIBLE ADIABATIC FLOW THROUGHPOROUS MEDIA

ＬＡＲＧＥ－ＴＩＭＥＢＥＨＭＩＯＲＯＦＳＯＬＵＴＩＯＮＳＦＯＲＴＨＥＳＹＳＴＥＭＯＦＣＯＭＰＲＥＳＳＩＢＬＥＡＤＩＡＢＡＴＩＣＦＬＯＷＴＨＲＯＵＧＨＰＯＲＯＵＳＭＥＤＩＡ￥ＸＩＡＯＬＩＮＧ（Ｌ．ＨＳＩＡＯ）Ｄ．ＳＥＲＲＥ（ＩｎｓｔｉｔｕｔｅｏｆＭａｔ...     

ON THE DIFFERENTIABILITY OF THE PARITY PROGRESSIVE POPULATION SEMIGROUP

ＯＮＴＨＥＤＩＦＦＥＲＥＮＴＩＡＢＩＬＩＴＹＯＦＴＨＥＰＡＲＩＴＹＰＲＯＧＲＥＳＳＩＶＥＰＯＰＵＬＡＴＩＯＮＳＥＭＩＧＲＯＵＰ￥ＳＨＩＤＥＭＩＮＧＡＮＤＹＡＮＧＬＵＳＨＡＮ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＺｈｅｎｇｚｈｏｕＵｎｉｖｅ...     

Metric theorems related to the Kronecker-Weyl theorem modm

Let (Ω, ?,P) be the infinite product of identical copies of the unit interval probability space. For a Lebesgue measurable subsetI of the unit interval, let \(A(N,I,\omega ) = \# \left\{ {n \leqslant N|\omega _n \varepsilon I} \right\}\) , where ω=(ω₁,ω₂,...). For integersm>1, and 0≤r<m, define $$\varepsilon (k,r,m,I,\omega ) = \left\{ {\begin{array}{*{20}c} {1\,if\,A(k,I,\omega ) \equiv r(\bmod m)} \\ {0\,otherwise} \\ \end{array} } \right.$$ and $$\eta (k,m,I,\omega ) = \left\{ {\begin{array}{*{20}c} {1\,if\,(A(k,I,\omega ),m) \equiv 1} \\ {0\,otherwise.} \\ \end{array} } \right.$$ A theorem ofK. L. Chung yields an iterated logarithm law and a central limit theorem for sums of the variables ε(k) and η(k).     

Difference schemes for one class of variational inequalities and fourth-order mixed boundary-value problem

A difference scheme is constructed for the solution of the variational equation $$\begin{gathered} a\left( {u, v} \right)---u \geqslant \left( {f, v---u} \right)\forall v \varepsilon K,K \{ vv \varepsilon W_2^2 \left( \Omega \right) \cap \mathop {W_2^1 \left( \Omega \right)}\limits^0 ,\frac{{\partial v}}{{\partial u}} \geqslant 0 a.e. on \Gamma \} ; \hfill \\ \Omega = \{ x = (x_1 ,x_2 ):0 \leqslant x_\alpha< l_\alpha ,\alpha = 1, 2\} \Gamma = \bar \Omega - \Omega ,a(u, v) = \hfill \\ = \int\limits_\Omega {\Delta u\Delta } vdx \equiv (\Delta u,\Delta v, \hfill \\ \end{gathered} $$ The following bound is obtained for this scheme: $$\left\| {y - u} \right\|_{W_2 \left( \omega \right)}^2 = 0(h^{(2k - 5)/4} )u \in W_2^k \left( \Omega \right),\left\| {y - u} \right\|_{W_2^2 \left( \omega \right)} = 0(h^{\min (k - 2;1,5)/2} ),u \in W_\infty ^k \left( \Omega \right) \cap W_2^3 \left( \Omega \right)$$ The following bounds are obtained for the mixed boundary-value problem: $$\begin{gathered} \left\| {y - u} \right\|_{W_2^2 \left( \omega \right)} = 0\left( {h^{\min \left( {k - 2;1,5} \right)} } \right),u \in W_\infty ^k \left( \Omega \right),\left\| {y - u} \right\|_{W_2^2 \left( \omega \right)} = 0\left( {h^{k - 2,5} } \right), \hfill \\ u \in W_2^k \left( \Omega \right),k \in \left[ {3,4} \right] \hfill \\ \end{gathered} $$ .     

On the normalizing multiplier of the generalized Jackson kernel

We consider the question of evaluating the normalizing multiplier $$\gamma _{n,k} = \frac{1}{\pi }\int_{ - \pi }^\pi {\left( {\frac{{sin\tfrac{{nt}}{2}}}{{sin\tfrac{t}{2}}}} \right)^{2k} dt} $$ for the generalized Jackson kernel J _n,k (t). We obtain the explicit formula $$\gamma _{n,k} = 2\sum\limits_{p = 0}^{\left[ {k - \tfrac{k}{n}} \right]} {( - 1)\left( {\begin{array}{*{20}c} {2k} \\ p \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {k(n + 1) - np - 1} \\ {k(n - 1) - np} \\ \end{array} } \right)} $$ and the representation $$\gamma _{n,k} = \sqrt {\frac{{24}}{\pi }} \cdot \frac{{(n - 1)^{2k - 1} }}{{\sqrt {2k - 1} }}\left[ {1\frac{1}{8} \cdot \frac{1}{{2k - 1}} + \omega (n,k)} \right],$$ , where $$\left| {\omega (n,k)} \right| < \frac{4}{{(2k - 1)\sqrt {ln(2k - 1)} }} + \sqrt {12\pi } \cdot \frac{{k^{\tfrac{3}{2}} }}{{n - 1}}\left( {1 + \frac{1}{{n - 1}}} \right)^{2k - 2} .$$ .     

Optimal methods for the approximate calculation of functional on classesW r L ∞

Пусть T_n(f)={L₁(f), ..., L_n(f)} — набор линейных функционал ов, заданных на простран стве \(C_{(r - 1)} (\parallel f\parallel _{C_{(r - 1)} } = \mathop {\max }\limits_{0 \leqq i \leqq r - 1} \parallel f^{(i)} \parallel _C );A_{n,r}\) — множество всех так их наборов функцио налов; С_{2n, 2} — множество всех н аборов из 2n функциона лов вида $$T_{2n} (f) = \{ f(x_1 ), \ldots ,f(x_n ),f'(x_1 ), \ldots ,f'(x_n )\}$$ и s: Е_n→Е₁. Доказано, что е слиW _∞ ^r множество всех 2π-периодических функ цийfεW∞0, 2π_r, то приr=1,2,3,... ирε(1, ∞) и $$\begin{gathered} \mathop {\inf }\limits_{T_{2n} \in A_{2n,r} } \parallel \mathop {\inf }\limits_s \mathop {\sup }\limits_{f \in W_\infty ^r } |f( \cdot ) - s(T_{2n} ,f, \cdot )|\parallel _p = \parallel \varphi _{n,r} \parallel _p \hfill \\ \mathop {\inf }\limits_{T_{2n} \in C_{2n,2} } \parallel \mathop {\inf }\limits_s \mathop {\sup }\limits_{f \in W_\infty ^r } |f( \cdot ) - s(T_{2n} ,f, \cdot )|\parallel _p = \parallel \parallel \varphi _{n,r} \parallel _\infty - \varphi _{n,r} \parallel _p , \hfill \\ \end{gathered}$$ где ?_n,r —r-й периодичес кий интеграл, в средне м равный нулю на периоде, от фун кции ?_n, 0t=sign sinnt. При этом указан ы оптимальные методы приближенного вычис ления.     

On the permutability of Backlund transformations

Let(?)=B_ηu:2(q-(?))+(⊿((?)-2q))+(2q_x+(?)_x))η=0,2(r-(?)+(⊿(2(?)-r)+(r_x+2(?)_x))η=0,u=(q,r)~Tbe the Backlund transformation (BT) of the hierarchy of AKNS equations,where η is a parameterand Δ=integral from -∞ to x (qr-(?))dx′.It is shown in this paper the infinitesimal BT B_(η+ε)B_η~(-1) admits thefollowing expansionB_(η+ε)B_η~(-1)u=u+εsum from n=0 to ∞ β_n(JL~(n+1)u)η~n,β_n=1+(-1)~n2~(-n-1),where L is the recurrence operator of the hierarchy and ε is an infinitesimal parameter.Thisexpansion implies the equivalence between the permutabiliy of BTs and the involution in pairs ofconserved densities.     

A reciprocity formula for quadratic forms

LetQ(x) denote a quadratic form over the rational integers in four variables (x=(x₁,...,x₄)). ThenQ is representable as a symmetric matrix. Assume this matrix to be non-singular modp(p≠2 prime); then the “inverse” quadratic formQ ^?1 modp can be defined. Letf:?⁴→? be defined such that the Fourier transformf exists and the sum $$\sum\limits_{x \in \mathbb{Z}^4 } {f(c x), c \in \mathbb{R}, c \ne 0} $$ is convergent. Furthermore, letm=p ₁...p _k be the product ofk distinct primes withm>1, 2×m; let $$\varepsilon = \prod\limits_{i = 1}^k {\left( {\frac{{\det Q}}{{p_i }}} \right)} \ne 0$$ for the Legendre symbol $$\left( {\frac{ \cdot }{p}} \right)$$ ; define $$B_i (Q,x) = \left\{ {\begin{array}{*{20}c} {1 for Q(x) \equiv 0\bmod p_i } \\ , \\ {0 for Q(x)\not \equiv 0\bmod p_i } \\ \end{array} } \right.$$ and forr∈?,r>0, $$F(Q,f,r) = \sum\limits_{x \in \mathbb{Z}^4 } {\left( {\prod\limits_{i = 1}^k {\left( {B_i (Q,x) - \frac{1}{{p_i }}} \right)} } \right)f(r^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} x)} $$ Then we have $$F(Q,f,m) = \varepsilon F(Q^{ - 1} ,\hat f,m)$$      

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

Solvability for Some Higher Order Multi-Point Boundary Value Problems at Resonance

In this paper, by using the Mawhin’s continuation theorem, we obtain an existence theorem for some higher order multi-point boundary value problems at resonance in the following form: $$\begin{array}{lll}x^{(n)}(t) = f(t,x(t),x'(t),\ldots,x^{(n-1)}(t))+e(t),\ t\in(0,1),\\x^{(i)}(0) = 0, i=0,1,\ldots,n-1,\ i\neq p, \\x^{(k)}(1) = \sum\limits_{j=1}^{m-2}{\beta_j}x^{(k)}(\eta_j),\end{array}$$ where ${f:[0,1]\times \mathbb{R}^n \to \mathbb{R}=(-\infty,+\infty)}$ is a continuous function, ${e(t)\in L^1[0,1], p, k\in\{0,1,\ldots,n-1\}}$ are fixed, m ≥ 3 for p ≤ k (m ≥ 4 for p > k), ${\beta_j \in \mathbb{R}, j=1,2,\ldots,m-2, 0 < \eta_1 < \eta_2 < \cdots < \eta_{m-2} <1 }$ . We give an example to demonstrate our results.     

Strong convergence in nonparametric estimation of regression functions

The nonparametric regression problem has the objective of estimating conditional expectation. Consider the model $$Y = R(X) + Z$$ , where the random variableZ has mean zero and is independent ofX. The regression functionR(x) is the conditional expectation ofY givenX = x. For an estimator of the form $$R_n (x) = \sum\limits_{i = 1}^n {Y_i K{{\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} \mathord{\left/ {\vphantom {{\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} {\sum\limits_{i = 1}^n {K\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} }}} \right. \kern-\nulldelimiterspace} {\sum\limits_{i = 1}^n {K\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} }}} $$ , we obtain the rate of strong uniform convergence $$\mathop {\sup }\limits_{x\varepsilon C} \left| {R_n (x) - R(x)} \right|\mathop {w \cdot p \cdot 1}\limits_ = o({{n^{{1 \mathord{\left/ {\vphantom {1 {(2 + d)}}} \right. \kern-\nulldelimiterspace} {(2 + d)}}} } \mathord{\left/ {\vphantom {{n^{{1 \mathord{\left/ {\vphantom {1 {(2 + d)}}} \right. \kern-\nulldelimiterspace} {(2 + d)}}} } {\beta _n \log n}}} \right. \kern-\nulldelimiterspace} {\beta _n \log n}}),\beta _n \to \infty $$ . HereX is ad-dimensional variable andC is a suitable subset ofR ^d .