Conservative fragments of {{S}^{1}_{2}} and {{R}^{1}_{2}}

Conservative subtheories of ${{R}^{1}_{2}}$ and ${{S}^{1}_{2}}$ are presented. For ${{S}^{1}_{2}}$ , a slight tightening of Je?ábek??s result (Math Logic Q 52(6):613?C624, 2006) that ${T^{0}_{2} \preceq_{\forall \Sigma^{b}_{1}}S^{1}_{2}}$ is presented: It is shown that ${T^{0}_{2}}$ can be axiomatised as BASIC together with induction on sharply bounded formulas of one alternation. Within this ${\forall\Sigma^{b}_{1}}$ -theory, we define a ${\forall\Sigma^{b}_{0}}$ -theory, ${T^{-1}_{2}}$ , for the ${\forall\Sigma^{b}_{0}}$ -consequences of ${S^{1}_{2}}$ . We show ${T^{-1}_{2}}$ is weak by showing it cannot ${\Sigma^{b}_{0}}$ -define division by 3. We then consider what would be the analogous ${\forall\hat\Sigma^{b}_{1}}$ -conservative subtheory of ${R^{1}_{2}}$ based on Pollett (Ann Pure Appl Logic 100:189?C245, 1999. It is shown that this theory, ${{T}^{0,\left\{2^{(||\dot{id}||)}\right\}}_{2}}$ , also cannot ${\Sigma^{b}_{0}}$ -define division by 3. On the other hand, we show that ${{S}^{0}_{2}+open_{\{||id||\}}}$ -COMP is a ${\forall\hat\Sigma^{b}_{1}}$ -conservative subtheory of ${R^{1}_{2}}$ . Finally, we give a refinement of Johannsen and Pollett (Logic Colloquium?? 98, 262?C279, 2000) and show that ${\hat{C}^{0}_{2}}$ is ${\forall\hat\Sigma^{b}_{1}}$ -conservative over a theory based on open _cl-comprehension.     

Stability of contact discontinuity for steady Euler system in infinite duct

In this paper, we prove stability of contact discontinuities for full Euler system. We fix a flat duct ${\mathcal{N}_0}$ of infinite length in ${\mathbb{R}^2}$ with width W ₀ and consider two uniform subsonic flow ${{U_l}^{\pm}=(u_l^{\pm}, 0, pl,\rho_l^{\pm})}$ with different horizontal velocity in ${\mathcal{N}_0}$ divided by a flat contact discontinuity ${\Gamma_{cd}}$ . And, we slightly perturb the boundary of ${\mathcal{N}_0}$ so that the width of the perturbed duct converges to ${W_0+\omega}$ for ${|\omega| < \delta}$ at ${x=\infty}$ for some ${\delta >0 }$ . Then, we prove that if the asymptotic state at left far field is given by ${{U_l}^{\pm}}$ , and if the perturbation of boundary of ${\mathcal{N}_0}$ and ${\delta}$ is sufficiently small, then there exists unique asymptotic state ${{U_r}^{\pm}}$ with a flat contact discontinuity ${\Gamma_{cd}^*}$ at right far field( ${x=\infty}$ ) and unique weak solution ${U}$ of the Euler system so that U consists of two subsonic flow with a contact discontinuity in between, and that U converges to ${{U_l}^{\pm}}$ and ${{U_r}^{\pm}}$ at ${x=-\infty}$ and ${x=\infty}$ respectively. For that purpose, we establish piecewise C ¹ estimate across a contact discontinuity of a weak solution to Euler system depending on the perturbation of ${\partial\mathcal{N}_0}$ and ${\delta}$ .     

Positive solutions to m-point boundary value problem of fractional differential equation

In this paper, we study the multiplicity of positive solutions to the following m-point boundary value problem of nonlinear fractional differential equations: Dqu(t) + f(t, u(t)) = 0, 0 < t < 1, u(0) = 0, u(1) =sum (μiDpu(t)|t = ξi ) from i =1 to ∞ m-2, where q ∈R , 1 相似文献   

Fractional harmonic maps into manifolds in odd dimension n > 1

In this paper we consider critical points of the following nonlocal energy $$\begin{array}{ll}{\mathcal{L}}_n(u) = \int_{{I\!\!R}^n}| ({-\Delta})^{n/4} u(x)|^2 dx, \qquad(1)\end{array}$$ where ${u \in \dot{H}^{n/2}({I\!\!R}^n,{\mathcal{N}}), {\mathcal{N}} \subset {I\!\!R}^m}$ is a compact k dimensional smooth manifold without boundary and n > 1 is an odd integer. Such critical points are called n/2-harmonic maps into ${{\mathcal{N}}}$ . We prove that ${(-\Delta) ^{n/4} u\in L^p_{loc}({I\!\!R}^n)}$ for every p ≥  1 and thus ${u \in C^{0,\alpha}_{loc}({I\!\!R}^n)}$ , for every 0 < α < 1. The local Hölder continuity of n/2-harmonic maps is based on regularity results obtained in [4] for nonlocal Schrödinger systems with an antisymmetric potential and on some new 3-terms commutators estimates.     

Existence of solutions for the critical elliptic system with inverse square potentials

Let Ω ? 0 be an open bounded domain in R ^N (N ≥ 3) and $2^* (s) = \tfrac{{2(N - s)}} {{N - 2}}$ , 0 < s < 2. We consider the following elliptic system of two equations in H ₀ ¹ (Ω) × H ₀ ¹ (Ω): $$- \Delta u - t\frac{u} {{\left| x \right|^2 }} = \frac{{2\alpha }} {{\alpha + \beta }}\frac{{\left| u \right|^{\alpha - 2} u\left| v \right|^\beta }} {{\left| x \right|^s }} + \lambda u, - \Delta v - t\frac{v} {{\left| x \right|^2 }} = \frac{{2\beta }} {{\alpha + \beta }}\frac{{\left| u \right|^\alpha \left| v \right|^{\beta - 2} v}} {{\left| x \right|^s }} + \mu v,$$ where λ, µ > 0 and α, β > 1 satisfy α + β = 2*(s). Using the Moser iteration, we prove the asymptotic behavior of solutions at the origin. In addition, by exploiting the Mountain-Pass theorem, we establish the existence of solutions.     

Interior W 1,p Regularity and Hölder Continuity of Weak Solutions to a Class of Divergence Kolmogorov Equations with Discontinuous Coefficients

We consider a class of Kolmogorov equation $$Lu={\sum^{p_0}_{i,j=1}{\partial_{x_i}}(a_{ij}(z){\partial_{x_j}}u)}+{\sum^{N}_{i,j=1}b_{ij}x_{i}{\partial_{x_j}}u-{\partial_t}u}={\sum^{p_0}_{j=1}{\partial_{x_j}}F_{j}(z)}$$ in a bounded open domain ${\Omega \subset \mathbb{R}^{N+1}}$ , where the coefficients matrix (a _ij (z)) is symmetric uniformly positive definite on ${\mathbb{R}^{p_0} (1 \leq p_0 < N)}$ . We obtain interior W ^1,p (1 < p < ∞) regularity and Hölder continuity of weak solutions to the equation under the assumption that coefficients a _ij (z) belong to the ${VMO_L\cap L^\infty}$ and ${({b_{ij}})_{N \times N}}$ is a constant matrix such that the frozen operator ${L_{z_0}}$ is hypoelliptic.     

Asymptotic properties for the loglog laws under positive association

Let {X _n : n ?? 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $S_n = \sum\limits_{k = 1}^n {X_k }$ , $Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$ , n ?? 1. Suppose that $0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$ . In this paper, we prove that if E|X ₁|^2+?? < for some ?? ?? (0, 1], and $\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$ for some ?? > 1, then for any b > ?1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x ₊ = max{x, 0}, N is a standard normal random variable, and ??(·) is a Gamma function.     

A generalized Cole-Hopf transformation for a two-dimensional burgers equation with a variable coefficient

The direct method is applied to the two dimensional Burgers equation with a variable coefficient (u _t + uu _x ? u _xx ) _x + s(t)u _yy = 0 is transformed into the Riccati equation $H' - \tfrac{1} {2}H^2 + \left( {\tfrac{\rho } {2} - 1} \right)H = 0$ via the ansatz $u\left( {x,y,t} \right) = \tfrac{1} {{\sqrt t }}H(\rho ) + \tfrac{y} {{2\sqrt t }}\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y$ , provided that s(t) = t ^?3/2. Further, a generalized Cole-Hopf transformations $u\left( {x,y,t} \right) = \tfrac{y} {{2\sqrt t }} - \tfrac{2} {{\sqrt t }}\tfrac{{U_\rho (\rho ,r)}} {{U(\rho ,r)}}$ , $\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y$ , r(t) = log t is derived to linearize (u _t + uu _x ? u _xx ) _x + t ^?3/2 u _yy to the parabolic equation $U_r = U_{\rho \rho } + \left( {\tfrac{\rho } {2} - 1} \right)U_\rho$ .     

Proof of three conjectures on congruences

This paper proves three conjectures on congruences involving central binomial coefficients or Lucas sequences.Let p be an odd prime and let a be a positive integer.It is shown that if p=1(mod 4)or a1then where(—)denotes the Jacobi symbol.This confirms a conjecture of the second author.A conjecture of Tauraso is also confirmed by showing that where the Lucas numbers Lo,L_1,L_2,...are defined by L_0=2,L_1=1 and L_n+1=L_n+L_n-l(n=1,2,3,...).The third theorem states that if p=5 then F_p~a-(p~a/5)mod p~3 can be determined in the following way:which appeared as a conjecture in a paper of Sun and Tauraso in 2010.     

Sets Computing the Symmetric Tensor Rank

Let ${\nu_{d} : \mathbb{P}^{r} \rightarrow \mathbb{P}^{N}, N := \left( \begin{array}{ll} r + d \\ \,\,\,\,\,\, r \end{array} \right)- 1,}$ denote the degree d Veronese embedding of ${\mathbb{P}^{r}}$ . For any ${P\, \in \, \mathbb{P}^{N}}$ , the symmetric tensor rank sr(P) is the minimal cardinality of a set ${\mathcal{S} \subset \nu_{d}(\mathbb{P}^{r})}$ spanning P. Let ${\mathcal{S}(P)}$ be the set of all ${A \subset \mathbb{P}^{r}}$ such that ${\nu_{d}(A)}$ computes sr(P). Here we classify all ${P \,\in\, \mathbb{P}^{n}}$ such that sr(P) <  3d/2 and sr(P) is computed by at least two subsets of ${\nu_{d}(\mathbb{P}^{r})}$ . For such tensors ${P\, \in\, \mathbb{P}^{N}}$ , we prove that ${\mathcal{S}(P)}$ has no isolated points.     

Isolated boundary singularities of semilinear elliptic equations

Given a smooth domain ${\Omega\subset\mathbb{R}^N}$ such that ${0 \in \partial\Omega}$ and given a nonnegative smooth function ?? on ???, we study the behavior near 0 of positive solutions of ???u?=?u ^q in ?? such that u =? ?? on ???\{0}. We prove that if ${\frac{N+1}{N-1} < q < \frac{N+2}{N-2}}$ , then ${u(x)\leq C |x|^{-\frac{2}{q-1}}}$ and we compute the limit of ${|x|^{\frac{2}{q-1}} u(x)}$ as x ?? 0. We also investigate the case ${q= \frac{N+1}{N-1}}$ . The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.     

Instability of Standing Wave for the Klein–Gordon–Hartree Equation

The instability property of the standing wave uω(t, x) = eiωtφ(x) for the Klein–Gordon– Hartree equation     

Profile of the least energy solution of a singular perturbed Neumann problem with mixed powers

We consider the problem ${\varepsilon^{2}\Delta u - u^q + u^p = 0\,{\rm in}\,\Omega,\,u > 0\,{\rm in}\,\Omega,\,\frac{\partial u}{\partial \nu} = 0\,{\rm on}\,\partial\Omega }$ where Ω is a smooth bounded domain in ${\mathbb{R}^N}$ , ${1 < q < p < {N+2\over N-2}}$ if N ≥ 2 and ${\varepsilon}$ is a small positive parameter. We determine the location and shape of the least energy solution when ${\varepsilon \rightarrow 0.}$      

Sharp constants in the doubly weighted Hardy-Littlewood-Sobolev inequality

It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows,Z Rn Z Rn f(x)g(y)|x||x.y||y|dxdy6 B(p,q,,,,n)kfkLp(Rn)kgkLq(Rn).The main purpose of this paper is to give the sharp constants B(p,q,,,,n)for the above inequality for three cases:(i)p=1 and q=1;(ii)p=1 and 1q 6∞,or 1p 6∞and q=1;(iii)1p,q∞and 1p+1q=1.In addition,the explicit bounds can be obtained for the case 1p,q∞and 1p+1q1.     

Composition Operators on Spaces of Fractional Cauchy Transforms

For ?? > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let ?? be an analytic self-map of the unit disc D. The map ?? induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$ . Various conditions on ?? are given, sufficient to imply that C _?? is bounded on ${\mathcal{F}_{\alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$ . Relations are found between the behavior of C _?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded.     

Boundary Values of Harmonic Functions in Spaces of Triebel–Lizorkin Type

Triebel (J Approx Theory 35:275–297, 1982; 52:162–203, 1988) investigated the boundary values of the harmonic functions in spaces of the Triebel–Lizorkin type ${\mathcal F^{\alpha,q}_{p}}$ on ${\mathbb{R}^{n+1}_+}$ by finding an characterization of the homogeneous Triebel–Lizorkin space ${{\bf \dot{F}}^{\alpha,q}_p}$ via its harmonic extension, where ${0 < p < \infty, 0 < q \leq \infty}$ , and ${\alpha < {\rm min}\{-n/p, -n/q\}}$ . In this article, we extend Triebel’s result to α < 0 and ${0 < p, q \leq \infty}$ by using a discrete version of reproducing formula and discretizing the norms in both ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf{\dot{F}}}^{\alpha,q}_p}$ . Furthermore, for α < 0 and ${1 < p,q \leq \infty}$ , the mapping from harmonic functions in ${\mathcal{F}^{\alpha,q}_{p}}$ to their boundary values forms a topological isomorphism between ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf \dot{F}}^{\alpha,q}_p}$ .     

The reverse H?lder inequality for power means

For a function ?? non-negative on the interval [0, 1], the power mean of order ??????0 is defined by the equality $ \mathcal{M}_{\alpha \varphi} (t) = {\left( {\frac{1}{t}\int_0^t {{\varphi^\alpha }(u)du} } \right)^{1/\alpha }},\,0 < t \leqslant 1 $ . We consider the class $ {\widetilde{{RH}}^{\alpha, \beta }}(B) $ of functions ?? satisfying the reverse H?lder inequality $$ {\mathcal{M}_\beta }_\varphi \leqslant B \cdot {\mathcal{M}_\alpha }_\varphi $$ at some ???<???,??·??????0,???>?1. The sharp estimates for the summability exponents of the compositions of power means are established. As a result, we determine the properties of self-improvement of the summability exponents of functions from $ {\widetilde{{RH}}^{\alpha, \beta }}(B) $ .     

On All Fractional （a,b, k）-Critical Graphs

Let a,b,k,r be nonnegative integers with 1≤a≤b and r≥2.LetG be a graph of order n with n(a+b)(r(a+b)-2)+ak/a.In this paper,we first show a characterization for all fractional(a,b,k)-critical graphs.Then using the result,we prove that G is all fractional(a,b,k)-critical if δ(G)≥(r-1)b2/a+k and |NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b for any independent subset {x1,x2,...,xr} in G.Furthermore,it is shown that the lower bound on the condition|NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b is best possible in some sense,and it is an extension of Lu's previous result.     

Multibump bound states for quasilinear Schrödinger systems with critical frequency

In this paper, we are concerned with the multibump solutions for the following quasilinear Schrödinger system in ${\mathbb{R}^N}$ : $$\left\{\begin{array}{ll}-\Delta{u} + \lambda{a(x)u} - \frac{1}{2}(\Delta|u|^2)u = \frac{2\alpha}{\alpha + \beta}|u|^{\alpha-2}|\upsilon|^\beta u, \\-\Delta{\upsilon} + \lambda{b(x)\upsilon} - \frac{1}{2}(\Delta|\upsilon|^2)\upsilon = \frac{2\beta}{\alpha + \beta}|u|^\alpha|\upsilon|^{\beta-2} \upsilon, \\u(x) \rightarrow 0, \upsilon(x) \rightarrow 0 \quad as|x| \rightarrow \infty,\end{array}\right.$$ where λ > 0 is a parameter, α, β > 2 satisfying α + β < 2 · 2*, here ${2^{*} = \frac{2N}{N-2}}$ is the critical Sobolev exponent for ${N \geq 3}$ and a(x), b(x) are nonnegative potentials. Using variational methods, we prove that if the zero sets of a(x) and b(x) have several common isolated connected components ${\Omega_{1}, . . . ,\Omega_{k}}$ such that the interior of ${\Omega_{i} (i = 1, 2, . . . , k)}$ is not empty and ${\partial\Omega_{i} (i = 1, 2, . . . , k)}$ is smooth, then for λ sufficiently large, the system admits, for any nonempty subset ${J \subset \{1, 2, . . . , k\}}$ , a solution which is trapped in a neighborhood of ${\cup_{j\epsilon{J}} \Omega_{j}}$ .