CALCULATION OF HAMILTONIAN CONSTRAINT AND MANDELSTAM IDENTITIES OVER EXTENDED KNOT FAMILIES{φi}2^2,{φi}3^4 and {φi}4^6

The actions of the Hamiltonian constraint onto the members of the extended knot families {φi}2^2, {φi}3^4 and {φi}4^6
, and the check of their invariance under the Mandelstam identities are given in the extended loop representation of loop quantum gravity.     

Teichmüller curves in genus three and just likely intersections in $\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}$

We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmüller curves. For the stratum \(\Omega\mathcal{M}_{3}(4)\), consisting of holomorphic one-forms with a single zero, our approach to finiteness uses the Harder-Narasimhan filtration of the Hodge bundle over a Teichmüller curve to obtain new information on the locations of the zeros of eigenforms. By passing to the boundary of moduli space, this gives explicit constraints on the cusps of Teichmüller curves in terms of cross-ratios of six points on \(\mathbf{P}^{1}\).These constraints are akin to those that appear in Zilber and Pink’s conjectures on unlikely intersections in diophantine geometry. However, in our case one is lead naturally to the intersection of a surface with a family of codimension two algebraic subgroups of \(\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}\) (rather than the more standard \(\mathbf{G}_{m}^{n}\)). The ambient algebraic group lies outside the scope of Zilber’s Conjecture but we are nonetheless able to prove a sufficiently strong height bound.For the generic stratum \(\Omega\mathcal{M}_{3}(1,1,1,1)\), we obtain global torsion order bounds through a computer search for subtori of a codimension-two subvariety of \(\mathbf{G}_{m}^{9}\). These torsion bounds together with new bounds for the moduli of horizontal cylinders in terms of torsion orders yields finiteness in this stratum. The intermediate strata are handled with a mix of these techniques.     

Null curves in {\mathbb{C}^3} and Calabi–Yau conjectures

For any open orientable surface M and convex domain ${\Omega\subset \mathbb{C}^3,}$ there exist a Riemann surface N homeomorphic to M and a complete proper null curve F : N → Ω. This result follows from a general existence theorem with many applications. Among them, the followings:

For any convex domain Ω in ${\mathbb{C}^2}$ there exist a Riemann surface N homeomorphic to M and a complete proper holomorphic immersion F : N → Ω. Furthermore, if ${D \subset \mathbb{R}^2}$ is a convex domain and Ω is the solid right cylinder ${\{x \in \mathbb{C}^2 \,|\, \mbox{Re}(x) \in D\},}$ then F can be chosen so that Re(F) : N → D is proper.

There exist a Riemann surface N homeomorphic to M and a complete bounded holomorphic null immersion ${F:N \to {\rm SL}(2, \mathbb{C}).}$

There exists a complete bounded CMC-1 immersion ${X:M \to \mathbb{H}^3.}$

For any convex domain ${\Omega \subset \mathbb{R}^3}$ there exists a complete proper minimal immersion (X _j ) _j=1,2,3 : M → Ω with vanishing flux. Furthermore, if ${D \subset \mathbb{R}^2}$ is a convex domain and ${\Omega=\{(x_j)_{j=1,2,3} \in \mathbb{R}^3 \,|\, (x_1,x_2) \in D\},}$ then X can be chosen so that (X ₁, X ₂) : M → D is proper.

Any of the above surfaces can be chosen with hyperbolic conformal structure.     

1.|a b|>|a| |b|

有众所周知}a bl(la! !b{,在这里我们却}a bl》ial }b}.诸肴如下证明:*为一且二_八亘万_压二画三二一’一’I口 bl可(a 吞)1心(a b)(a b) ︼一人U. 一t,一,一.办 一.︸︺叭一办一 一‘U一之一al一一叭U李,一b 叼a︸ 一a_旦口十石 。 b不),雌 !b}}a b}篇-_I,口。、乓乏‘石不十石落j, (l) (2)得{a} }b}}a b} a-下-r十a十O:}a bl》}a! b。 b}bl...‘l,‘ ‘1.|a b|>|a| |b|@曹存富$宁波市北仓区教研室~~…     

复指数函数系在$L_{\alpha}^{p}$空间中的完备性

设函数 $\alpha(t)$在$\bf R$上非负连续 和 $1\le{p}<+{\infty}$, 则 $L_{\alpha}^p=\{f: \int_{-{\infty}}^{\infty}|f(t)e^{-\alpha(t)}|^p\mathrm{d}t<{\infty}\}$ 是Banach空间. 本文中我们得到了一个复指数函数系在$L_{\alpha}^{p}$ 空间中稠密的充分必要条件.     

Positive Solutions of a Nonlinear Elliptic Equation in {x∈R2:|x|>1}

We prove the existence of infinitely many solutions for the nonlinear elliptic equation u+f(.,u)=0 in D with u>0 in D and u=0 on D, where D={xR ²:|x|>1} and f is a measurable function dominated by a regular function q such that q(x,Log|x|) is in a some Kato class.     

Метод многопараметрической интерполяции и теоремы вложения пространств БесоваB_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)

In this paper we determine the method of multi-parameter interpolation and the scales of Lebesgue spaces $B_{\vec p} \left[ {0,2\pi } \right)$ and Besov spaces $B_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)$ , which are generalizations of the Lorentz spacesL _pq [0, 2π) and Besov spacesB _pq ^α [0, 2π). We also prove imbedding theorems.     

On a Right Inverse of a Polynomial of the Laplace in the Weighted Hilbert Space $L^2 (\mathbb{R}^n,e^{−|x|^2} )$

Let $P(∆)$ be a polynomial of the Laplace operator $$∆ = \sum\limits^n_{j=1}\frac{∂^2}{∂x^2_j} \ \ on \ \ \mathbb{R}^n.$$ We prove the existence of a bounded right inverse of the differential operator $P(∆)$ in the weighted Hilbert space with the Gaussian measure, i.e., $L^2(\mathbb{R}^n ,e^{−|x|^2}).$     

$\mathbb{Z}^{k}$-作用一维子系统的Lipschitz跟踪性

本文主要研究了$\mathbb{Z}^{k}$-作用一维子系统的跟踪性质. 文中运用两种等价的方式引入了$\mathbb{Z}^{k}$-作用一维子系统的伪轨以及跟踪性的概念. 对于一个闭黎曼流形上的光滑$\mathbb{Z}^{k}$-作用$T$, 我们通过诱导的非自治动力系统提出了Anosov方向的概念. 借助Bowen几何的方法, 我们证明了$T$沿着任意Anosov方向具有Lipschitz跟踪性.     

复指数系统在$L_{\alpha}^{p}$空间的不完备性

A necessary and sufficient condition is obtained for the incompleteness of complex exponential system in the weighted Banach space Lαp = {f：∫＋∞∞ ｜f（t）e-α（t）｜pdt ＋∞},where 1 ≤ p ＋∞ and α（t） is a weight on R.     

Rigidité conforme des hémisphères {{\rm S}^4_+} et {{\rm S}^6_+}

Let (M, g) be a four or six dimensional compact Riemannian manifold which is locally conformally flat and assume that its boundary is totally umbilical. In this note, we prove that if the Euler characteristic of M is equal to 1 and if its Yamabe invariant is positive, then (M, g) is conformally isometric to the standard hemisphere. As an application and using a result of Hang and Wang (Comm Anal Geom 14(1):91–106, 2006), we prove a rigidity result for these hemispheres regarding the Min-Oo conjecture.     

用不等式|a|-|b|≤|a b|≤|a| |b|解题

（事实上，也成立，证明略）在高中《代数》下册不等式一章给出，教材主要研究了它在绝对值不等式证明中的运用．而在其它方面的运用几乎没有涉及．有关资料一般也未作探讨，为了强化《考试说明》中提及的这一重要基础，下面就其运用举例说明．1证明不等式例1用数学归纳法证明：（1985年全国高考上海试题）证明（1）n=1时，不等式显然成立．（2）假设成立，那么n＝k十1时，＜Silk。OSSI＋ICOSk。i。l＜ksinxl＋Dsinx【＝（k＋1）Isinx【．不等式成立．综合（1）、（2）有lsinn6I。nlsin8I．NZ已知a、b、c是实数，函数（1996年全国（3…     

关于{P2, Ci| i≥3}-覆盖图

设 Ｇ是一个图,若对于 Ｇ的任意一边 Ｇ都有｛Ｐ_２,Ｃｉ｜ｉ－＞３｝－因子含有这条边,则称Ｇ是｛Ｐ_２,Ｃｉ｜ｉ－＞３｝－覆盖图．本文给出连通非二分图Ｇ是｛Ｐ２,Ｃｉ｜ｉ－＞３｝－覆盖图的充要条件为任给Ｓ■Ｖ（Ｇ）,Ｖ（Ｇ）≠Ｓ≠■有ｉ（Ｇ－Ｓ）＿＞｜Ｓ｜－１成立．     

( m,n)-Equidista nt Sets i n $\ mathbb{R}^{k},\ mathbb{S}^{k}$ , a nd ℙk

We call a metric space X (m,n)-equidistant if, when A⊆X has exactly m points, there are exactly n points in X each of which is equidistant from (the points of) A. We prove that, for k≥2, the Euclidean space ℝ ^k contains an (m,1)-equidistant set if and only if k≥m. Although the sphere is (3,2)-equidistant, and ℝ⁴ contain no (4,2)-equidistant sets. We discuss related results about projective spaces, and state a conjecture about analogous to the Double Midset Conjecture.     

双圆盘加权Hardy空间上$T_{z_1^{k_1}z_2^{k_2}+\bar{z}_1^{l_1}\bar{z}_2^{l_2}}$的约化子空间

本文中, 我们主要刻画了Toeplitz算子$T=M_{z^k}+M^*_{z^l}$的约化子空间, 其中 $k_i, l_i$ ($i=1,2$) 均是正整数, $k=(k_1,k_2), l=(l_1,l_2)$ 且 $k\neq l$, $M_{z^k}$, $M_{z^l}$ 是双圆盘加权Hardy空间$\mathcal{H}_\omega^2(\mathbb{D}^2)$上的乘法算子. 对权系数 $\omega$ 适当限制, 我们证明了由 $z^m$ 生成的 $T$ 的约化子空间均是极小的. 特别地, Bergman 空间和加权 Dirichlet 空间 $\mathcal{D}_\delta(\mathbb{D}^2)(\delta>0)$ 均是满足该限制条件的加权Hardy空间. 作为应用, 我们刻画了 $\mathcal{D}_\delta(\mathbb{D}^2)(\delta>0)$ 上 Toeplitz 算子 $T_{z^k+\bar{z}^l}$ 的约化子空间, 该结论是对双圆盘Bergman 空间上相关结论的推广.     

Morse and semistable stratifications of Kähler spacesby ${\Bbb C}^{\ast}$-actions

We prove that the Morse decomposition in the sense of Kirwan and semistable decomposition in the sense of GIT of a -K?hler manifold coincide if the moment map is proper and if the fixed points set has a finite number of connected components. For general K?hler space with holomorphic action of a complex reductive group G, if every component of the moment map is proper, the two decompositions also coincide if each semistable piece is Zariski open in its topological closure and the moment map square is minimal degenerate Morse function in the sense of Kirwan. Author’s address: Department of Mathematics, Tsinghua University, 100084 Beijing, P.R. China     

由H\"{o}rmander向量场构成的抛物方程的 $W_{\ast}^{1,p}$ 正则性

设 $X_{1},\cdots ,X_{q}\ (q相似文献