Estimation of the solutions of the Sturm-Liouville equation

Exact estimates are presented for the solutions of the problem $\ddot y + \lambda ^2 p(t)y = 0, y(0) = 0, \dot y(0) = 1$ withp(t) satisfying one of the following conditions: $$(i) |p(t)| \leqslant M< \infty ; (ii) 0< \omega _1 \leqslant p(t) \leqslant \omega _2< \infty ; (iii) \mathop {sup}\limits_x \int_x^{x + T} {p(t)dt = P_T /T.} $$ The extremal solutions are found.     

Optimal estimates on rotation number of almost periodic systems

In this paper, we will give some optimal estimates on the rotation number of the linear equation $\ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + p(t)x = 0,$\ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + p(t)x = 0, and that of the asymmetric equation: $\ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + p(t)x_{ + } + q(t)x_{ - } = 0,$\ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + p(t)x_{ + } + q(t)x_{ - } = 0, where p(t) and q(t) are almost periodic functions and x ₊ = max{ x,0} ,  x _- = min{ x,0} .x_{ + } = \max \{ x,0\} ,\;x_{ - } = \min \{ x,0\} . These estimates are obtained by introducing some kind of new norms in the spaces of almost periodic functions.     

The Shuddering Pendulum

This paper treats the rich mathematical structure of the (dimensionless) equation of motion governing the behavior of an elastically restrained simple pendulum subject to a downward force of magnitude f(t) applied to its bob with $\dot{f}(t)>0$\dot{f}(t)>0 for all t>0 and f(t)→∞ as t→∞:

A new goldfish model

A new integrable (indeed, solvable) model of goldfish type is identified, and some of its properties are discussed. A version of its Newtonian equations of motion reads as follows:

Convergence of a Conservative Difference Scheme for the Zakharov Equations in Two Dimensions

1.IntroductionTheZakharovequationsdescribephysicalphenomenainPlas,,,.112].TheglobalexistenceofaweaksolutionfortheZakharovequationswascollsideredbySulemandSulelnin[11].Theexistenceanduniquenessofasmoothsolutioninonedimensionareprovedprovidedthatsmoothinitialdataaredescribed.Forsmallinitialdata,theexistenceofaweaksolutionfortheZa,kharovequationsintwoandthreedimensionsisobtained.NumericalmethodsfortileZakharovequationsinonedimensionwereconsideredin[l],[2],[4],[5]and[10].Aspectralmethodisusedt…     

Boundedness of Solutions for Duffing Equation with Low Regularity in Time

It is shown that all solutions are bounded for Duffing equation x+ x~(2n+1)+2∑i=nPj(t)x~j= 0, provided that for each n + 1 ≤ j ≤ 2 n, P_j ∈ C~y(T~1) with γ 1-1/n and for each j with 0 ≤ j ≤ n, Pj ∈ L(T~1) where T~1= R/Z.     

A Remark on a Problem of Rolf Nevanlinna

Let T ( f ) and N ( r,c ) denote the usual Nevanlinna characteristic and the counting function for the c -points of a meromorphic function f , respectively. Using a result of Miles and Shea ( Quart. J. Math. Oxford , 24 (2), (1973), 377-383) and two simple estimates for trigonometric functions, we show in connection with a 1929 problem of Nevanlinna for meromorphic functions f of finite order 1 < u < X $$ \limsup\limits_{r\rightarrow \infty } { N(r, 0)+N(r, \infty ) \over T(r, \,f)}\ge {2\sqrt 2 \over \pi} {|\sin \pi \lambda | \over D(\lambda )}\ge (0.9)\, {{|\sin \pi \lambda | \over {D(\lambda )}, }} $$ with D ( u ) = q +|sin ~ u | for $ q\le \lambda \le q + \fraca {1}{2} $ and D ( u ) = q + 1 for $ q + {\fraca {1} {2}} \le \lambda \lt q + 1 $ , where $ q = \lfloor \lambda \rfloor $ .     

关于图中存在不含给定k-因子的[a,b]-因子的度条件

设$1\leq a<b, 0\leq k$是整数. 设$G$是一个含有$k$-因子$Q$且阶为$|G|$的图. 设\delta(G)$表示$G$的最小度, 且$\delta(G)\geq a+k$. 如果$Q$连通, 设$\varepsilon=k$, 否则设$\varepsilon=k+1$.证明:当$b\geq a+\varepsilon-1$时, 如果对$G$的任意两个不相邻的点$x$和$y$都有max$\{d_G(x),d_G(y)\}\geq {\rm max}\{{{a|G|} \over {a+b}},{{(|G|+(a-1)(2a+b+\varepsilon-2))} \over {b+1}}\}+k$, 那么$G$有一个$[a, b]$-因子$F$ 使得 $E(F)\cap E(Q)=\emptyset$. 这个度条件是最佳的, 条件$b\geqa+\varepsilon-1$不能去掉. 进一步,得到图存在含给定$k$-因子的$[a, b]$-因子的度条件.     

Limit Boundary Value Problems of a class of Nonlinear Differential Equations of the Second order

In this paper, the existence and uniqueness of solution of the limit boundary value problem $\[\ddot x = f(t,x)g(\dot x)\]$(F) $\[a\dot x(0) + bx(0) = c\]$(A) $\[x( + \infty ) = 0\]$(B) is considered, where $\[f(t,x),g(\dot x)\]$ are continuous functions on $\[\{ t \ge 0, - \infty < x,\dot x < + \infty \} \]$ such that the uniqueness of solution together with thier continuous dependence on initial value are ensured, and assume: 1)$\[f(t,0) \equiv 0,f(t,x)/x > 0(x \ne 0);\]$; 2) f(t,x)/x is nondecreasing in x>0 for fixed t and non-increasing in x<0 for fixed t, 3)$\[g(\dot x) > 0\]$, In theorem 1, farther assume: 4) $\[\int\limits_0^{ \pm \infty } {dy/g(y) = \pm \infty } \]$ Condition (A) may be discussed in the following three cases $x(0)=p(p \neq 0)$(A_1) $\[x(0) = q(q \ne 0)\]$(A_2) $\[x(0) = kx(0) + r{\rm{ }}(k > 0,r \ne 0)\]$(A_3) The notation $\[f(t,x) \in {I_\infty }\]$ will refer to the function f(t,x) satisfying $\[\int_0^{ + \infty } {\alpha tf(t,\alpha )dt = + \infty } \]$ for each $\alpha \neq 0$, Theorem. 1. For each $p \neq 0$, the boundary value problem (F), (A_1), (B) has a solution if and only if $f(t,x) \in I_{\infty}$ Theorem 2. For each$q \neq 0$, the boundary value problem (F), (A_2), (B) has a solution if and only if $f(t, x) \in I_{\infty}$. Theorem 3. For each k>0 and $r \neq 0$, the boundary value problem (F), (A_3), (B) has a solution if and only if f(t, x) \in I_{\infty}, Theorem 4. The boundary value problem (F), (A_j), (B) has at most one solution for j=l, 2, 3. .     

New Finding on Factoring Prime Power RSA Modulus $N = p^r q$

This paper proposes three new attacks. In the first attack we consider the class of the public exponents satisfying an equation e X-N Y +(ap~r+ bq~r)Y = Z for suitably small positive integers a, b. Applying continued fractions we show thatY/Xcan be recovered among the convergents of the continued fraction expansion of e/N. Moreover, we show that the number of such exponents is at least N~(2/(r+1)-ε)where ε≥ 0 is arbitrarily small for large N. The second and third attacks works upon k RSA public keys(N_i, e_i) when there exist k relations of the form e_ix-N_iy_i +(ap_i~r + bq_i~r )y_i = z_i or of the form e_ix_i-N_iy +(ap_i~r + bq_i~r )y = z_i and the parameters x, x_i, y, y_i, z_i are suitably small in terms of the prime factors of the moduli. We apply the LLL algorithm, and show that our strategy enables us to simultaneously factor k prime power RSA moduli.     

GLOBAL EXISTENCE OF THE SOLUTIONS TO NONLINEAR HYPERBOLIC EQUATIONS IN EXTERIOR DOMAINS

This paper deals with the following IBV problem of nonlinear hyperbolic equations u_(tt)- sum from i, j=1 to n a_(jj)(u, Du)u_(x_ix_j)=b(u, Du), t>0, x∈Ω, u(O, x) =u~0(x), u_t(O, x) =u~1(v), x∈Ω, u(t, x)=O t>O, x∈()Ω,where Ωis the exterior domain of a compact set in R~n, and |a_(ij)(y)-δ_(ij)|= O(|y|~k), |b(y)|=O(|y|~(k+1)), near y=O. It is proved that under suitable assumptions on the smoothness,compatibility conditions and the shape of Ω, the above problem has a unique global smoothsolution for small initial data, in the case that k=1 add n≥7 or that k=2 and n≥4.Moreover, the solution ham some decay properties as t→ + ∞.     

圈的$L(d_1,d_2,\ldots,d_t)$-数$\lambda(C_n;d_1,d_2,\ldots,d_t)$

An L（d0,d2,...,dt）-labeling of a graph G is a function f from its vertex set V（G） to the set {0,1,..., k} for some positive integer k such that If（x） - f（y）l ≥di, if the distance between vertices x and y in G is equal to i for i = 1,2,...,t. The L（d1,d2,...,dt）-number λ（G;d1,d2,... ,dt） of G is the smallest integer number k such that G has an L（d1,d2,...,dr）- labeling with max{f （x）｜x ∈ V（G）} = k. In this paper, we obtain the exact values for λ（Cn; 2, 2, 1） and λ（Cn; 3, 2, 1）, and present lower and upper bounds for λ（Cn; 2,..., 2, 1,..., 1）     

Inequalities for the beta function

We prove the following inequalities involving Euler’s beta function. (i) Let α and β be real numbers. The inequalities $\left( {\frac{{y^{z - x} }} {{x^{z - y} z^{y - x} }}} \right)^\alpha \leqslant \frac{{B(x,x)^{z - y} B(z,z)^{y - x} }} {{B(y,y)^{z - x} }} \leqslant \left( {\frac{{y^{z - x} }} {{x^{z - y} z^{y - x} }}} \right)^\beta $ hold for all x, y, z with 0 < x ≤ y ≤ z if and only if α ≤ 1/2 and β ≥ 1. (ii) Let a and b be non-negative real numbers. For all positive real numbers x and y we have $\delta (a,b) \leqslant \frac{{x^a B(x + b,y) + y^a B(x,y + b)}} {{(x + y)^a B(x,y)}} \leqslant \Delta (a,b) $ with the best possible bounds $\delta (a,b) = \min \{ 2^{ - a} ,2^{1 - a - b} \} and\Delta (a,b) = \max \{ 1,2^{1 - a - b} \} . $ .     

Further extension of the Heinz-Kato-Furuta inequality

Let be a bounded operator on a Hilbert space and positive definite operators. Kato has shown that if and for all , then where are operator monotone functions defined on such that . Furuta has shown that Let be any continuous operator monotone functions, and set for We will show that is well defined and Moreover, we will extend this result for unbounded closed operators densely defined on

     

On Global Asymptotic Stability of Second Order Nonlinear Differential Systems

This paper investigates the global asymptotic stability of the autonomous planar systems $ \dot {x} = p_2(y)q_2(x)y $ , $ \dot {y} = p_3(y)q_3(x)x + p_3(y)q_4(x)y $ and $ \dot {x} = f_1(x) + h_2(x)y $ , $ \dot {y} = f_3(x) + h_4(x)y $ , under the assumption that all functions involved in the equations are continuous and that the origin is a unique equilibrium. We present necessary and sufficient conditions for the origin to be globally asymptotically stable.     

关于一类中心循环的有限P-群的自同构群

确定了一类中心循环的有限p-群G的自同构群.设G=X_3(p~m)~(*n)*Z_(p~(m+r)),其中m≥1,n≥1和r≥0,并且X_3(p~m)=x,y|x~(p~m)=y~(p~m)=1,[x,y]~(p~m)=1,[x,[x,y]]=[y,[x,y]]=1.Aut_nG表示Aut G中平凡地作用在N上的元素形成的正规子群,其中G'≤N≤ζG,|N|=p~(m+s),0≤s≤r,则(i)如果p是一个奇素数,那么AutG/Aut_nG≌Z_(p~((m+s-1)(p-1))),Aut_nG/InnG≌Sp(2n,Z_(p~m))×Z_(p~(r-s)).(ii)如果p=2,那么AutG/Aut_nG≌H,其中H=1(当m+s=1时)或者Z_(2~(m+s-2))×Z_2(当m+s≥2时).进一步地,Aut_nG/InnG≌K×L,其中K=Sp(2n,Z_(2~m))(当r0时)或者O(2n,Z_(2~m))(当r=0时),L=Z_(2~(r-1))×Z_2(当m=1,s=0,r≥1时)或者Z_(2~(r-s)).     

单纯正交表OA$_\lambda(3, 5, v)$的存在性

An orthogonal array of strength t,degree k,order v and index λ,denoted by OAλ(t,k,v),is a λvt× k array on a v symbol set such that each λvt× t subarray contains each t-tuple exactly λ times.An OAλ(t,k,v) is called simple and denoted by SOAλ(t,k,v)if it contains no repeated rows.In this paper,it is proved that the necessary conditions for the existence of an SOAλ(3,5,v) with λ≥ 2 are also sufficient with possible exceptions where v = 6 and λ∈ {3,7,11,13,15,17,19,21,23,25,29,33}.     

关于一个典型函数Fourier级数的Gibbs现象

In this paper,we point out that the Fourier series of a classical function∑∞k=1 sin kx/k has the Gibbs phenomenon in the neighborhood of zero.Furthermore,we estimate the upper bound of its partial sum and get:supn≥1‖n∑k=1sin kx/k‖∫x0sin x/xdx=1.85194,which is better than that in[1].     

Regularity of Positive Solutions for an Integral System on Heisenberg Group

In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, \begin{equation} \left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.\end{equation} for $x\in \mathbb{H}^n$, where $0<\alpha 1$ satisfying $\frac{1}{p+1} $+ $\frac{1}{q+1} + \frac{1}{r+1} = \frac{Q+α+β}{Q}.$ We show that positive solution triples $(u,v,w)\in L^{p+1}(\mathbb{H}^n)\times L^{q+1}(\mathbb{H}^n)\times L^{r+1}(\mathbb{H}^n)$ are bounded and they converge to zero when $|x|→∞.$     

Towers of Borel functions

We give mathematical reformulations of the cardinals and in terms of families of Borel functions. As an application we show that is invariant under the addition of a single Cohen real.