Decorated marked surfaces: spherical twists versus braid twists

We are interested in the 3-Calabi-Yau categories \({\mathcal {D}}\) arising from quivers with potential associated to a triangulated marked surface \(\mathbf {S}\) (without punctures). We prove that the spherical twist group \(\mathrm{ST}\) of \({\mathcal {D}}\) is isomorphic to a subgroup (generated by braid twists) of the mapping class group of the decorated marked surface \({\mathbf {S}}_\bigtriangleup \). Here \({\mathbf {S}}_\bigtriangleup \) is the surface obtained from \(\mathbf {S}\) by decorating with a set of points, where the number of points equals the number of triangles in any triangulations of \(\mathbf {S}\). For instance, when \(\mathbf {S}\) is an annulus, the result implies that the corresponding spaces of stability conditions on \({\mathcal {D}}\) are contractible.     

最大度大于等于7的平面图的线性荫度

图的染色问题在组合优化、计算机科学和Hessians矩阵的网络计算等方面具有非常重要的应用。其中图的染色中有一种重要的染色——线性荫度,它是一种非正常的边染色,即在简单无向图中,它的边可以分割成线性森林的最小数量。研究最大度$\bigtriangleup（G）\geq7$的平面图$G$的线性荫度,证明了对于两个固定的整数$i$,$j\in\{5,6,7\}$,如果图$G$中不存在相邻的含弦$i$,$j$-圈,则图$G$的线性荫度为$\lceil\frac\bigtriangleup2\rceil$。     

Stability of the equator map for the Hessian energy

In this paper we show that the equator map is a minimizer of the Hessian energy in for and is unstable for

     

Gaussian curvature in the negative case

In this paper, we reinvestigate an old problem of prescribing Gaussian curvature in the negative case.

In 1974, Kazdan and Warner considered the equation

on any compact two dimensional manifold with . They showed that there exists a number , such that the equation is solvable for every \alpha > \alpha_o$"> and it is not solvable for .

Then one may naturally ask:

Is the equation solvable for ?

In this paper, we answer the question affirmatively. We show that there exists at least one solution for .

     

Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is

where is a bounded open subset of , , is the so-called Laplace operator, , is a Radon measure with bounded variation on , , , and and belong to the Lorentz spaces , , and , respectively. In particular we prove the existence under the assumptions that , belongs to the Lorentz space , , and is small enough.

     

Almost Everywhere Convergence of Fejér Means of Two-dimensional Triangular Walsh–Fourier Series

In 1987 Harris proved (Proc Am Math Soc 101(4):637–643, 1987)—among others—that for each \(1\le p<2\) there exists a two-dimensional function \(f\in L^p\) such that its triangular Walsh–Fourier series diverges almost everywhere. In this paper we investigate the Fejér (or (C, 1)) means of the triangle two variable Walsh–Fourier series of \(L^1\) functions. Namely, we prove the a.e. convergence \(\sigma _n^{\bigtriangleup }f = \frac{1}{n}\sum _{k=0}^{n-1}S_{k, n-k}f\rightarrow f\) (\(n\rightarrow \infty \)) for each integrable two-variable function f.     

Continuous maps with the disjoint support property

Periodica Mathematica Hungarica - It is well known that for each $$n\ge 0$$ , there is a continuous map $$ f :S^{n}\rightarrow \partial \bigtriangleup ^{n+1}$$ with the disjoint support property....     

One Non-Relativistic Particle Coupled to a Photon Field

We investigate the ground state energy of an electron coupled to a photon field. First, we regard the self-energy of a “free” electron, which we describe by the Pauli-Fierz Hamiltonian. We show that, in the case of small values of the coupling constant a \alpha , the leading order term is represented by 2ap^-1(L - ln[1 + L]) 2\alpha\pi^{-1}(\Lambda - \textrm{ln}[1 + \Lambda]) .¶ Next we put the electron in the field of an arbitrary external potential V , such that the corresponding Schrödinger operator p² + V has at least one eigenvalue, and show that by coupling to the radiation field the binding energy increases, at least for small enough values of the coupling constant a \alpha . Moreover, we provide concrete numbers for a \alpha , the ultraviolet cut-off, and the radiative correction for which our procedure works.     

Precise Asymptotic Formulas for Semilinear Eigenvalue Problems

. We consider the nonlinear Sturm-Liouville problem¶¶-u"(t) = | u(t) | ^p-1u(t) - lu(t), t ? I :=(0,1), u(0) = u(1) = 0 -u'(t) = \mid u(t)\mid^{p-1}u(t) - \lambda u(t), t \in I :=(0,1), u(0) = u(1) = 0 ,¶¶ where p > 1 and l ? R \lambda \in {\bf R} is an eigenvalue parameter. To investigate the global L²-bifurcation phenomena, we establish asymptotic formulas for the n-th bifurcation branch l = l_n (a) \lambda = \lambda_n (\alpha) with precise remainder term, where a \alpha is the L² norm of the eigenfunction associated with l \lambda .     

Centralizers and Central Idempotents of Semigroup Rings

. Let A be a nonempty subset of an associative ring R . Call the subring CR(A)={r] R\mid ra=ar \quadfor all\quad a] A} of R the centralizer of A in R . Let S be a semigroup. Then the subsemigroup S'= {s] S\mid sa=sb \quador\quad as=bs \quadimplies\quad a=b \quadfor all a,b] S} of S is called the C -subsemigroup. In this paper, the centralizer CR[S](R[M]) for the semigroup ring R[S] will be described, where M is any nonempty subset of S' . An non-zero idempotent e is called the central idempotent of R[S] if e lies in the center of R[S] . Assume that S\backslash S' is a commutative ideal of S and Annl(R)=0 . Then we show that the supporting subsemigroup of any central idempotent of R[S] must be finite.     

Invariant characters and coprime actions on finite nilpotent groups

Abstract. Let S be a subgroup of SLn(R), where R is a commutative ring with identity and n \geqq 3n \geqq 3. The order of S, o(S), is the R-ideal generated by x_ij, x_ii - x_jj (i 1 j)x_{ij},\ x_{ii} - x_{jj}\ (i \neq j), where (x_ij) ? S(x_{ij}) \in S. Let En(R) be the subgroup of SLn(R) generated by the elementary matrices. The level of S, l(S), is the largest R-ideal \frak q\frak {q} with the property that S contains all the \frak q\frak {q}-elementary matrices and all conjugates of these by elements of En(R). It is clear that l(S) \leqq o(S)l(S) \leqq o(S). Vaserstein has proved that, for all R and for all n \geqq 3n \geqq 3, the subgroup S is normalized by En(R) if and only if l(S) = o(S)     

L p Norms of Eigenfunctions in the Completely Integrable Case

The eigenfunctions e^iál,x? e^{i\langle\lambda,x\rangle} of the Laplacian on a flat torus have uniformly bounded L^p norms. In this article, we prove that for every other quantum integrable Laplacian, the L^p norms of the joint eigenfunctions blow up at least at the rate || j_k || L^p 3 C(e)l_k^{[(p-2)/(4p)]-e} \| \varphi_k \| L^{p} \geq C(\epsilon)\lambda_{k}^{{p-2\over4p}-\epsilon} when p > 2. This gives a quantitative refinement of our recent result [TZ1] that some sequence of eigenfunctions must blow up in L^p unless (M,g) is flat. The better result in this paper is based on mass estimates of eigenfunctions near singular leaves of the Liouville foliation.     

Nonlinear time-harmonic Maxwell equations in a bounded domain: Lack of compactness

We survey recent results on ground and bound state solutions E:?→R~3 of the problem {▽(▽×E)+}λE=|E|~(P-2)E in Ω,v×E=0 on Ω on a bounded Lipschitz domain ??R~3,where?×denotes the curl operator in R~3.The equation describes the propagation of the time-harmonic electric field R{E(χ)e~(iwt)}in a nonlinear isotropic material ? withλ=-μεω~2≤0,where μ andεstand for the permeability and the linear part of the permittivity of the material.The nonlinear term|E|~(P-2)E with 2p≤2*=6 comes from the nonlinear polarization and the boundary conditions are those for?surrounded by a perfect conductor.The problem has a variational structure;however the energy functional associated with the problem is strongly indefinite and does not satisfy the Palais-Smale condition.We show the underlying difficulties of the problem and enlist some open questions.     

On the 2-adic valuation of the cardinality of elliptic curves over finite extensions of $$\mathbb {F}_{\varvec{q}} $$

In this paper we study the difference between the 2-adic valuations of the cardinalities \( \# E( \mathbb {F}_{q^k} ) \) and \( \# E( \mathbb {F}_q ) \) of an elliptic curve E over \( \mathbb {F}_q \). We also deduce information about the structure of the 2-Sylow subgroup \( E[ 2^\infty ]( \mathbb {F}_{q^k} ) \) from the exponents of \( E[ 2^\infty ]( \mathbb {F}_q ) \).     

L-Regularity of Markov Sets and of m-Perfect Sets in the Complex Plane

Let E be a compact subset of C. We prove that if E satisfies the following local Markov property: for each polynomial P,

Classes of Idempotents in Hilbert Space

An idempotent operator E in a Hilbert space \({\mathcal {H}}\) \((E^2=1)\) is written as a \(2\times 2\) matrix in terms of the orthogonal decomposition

$$\begin{aligned} {\mathcal {H}}=R(E)\oplus R(E)^\perp \end{aligned}$$

(R(E) is the range of E) as

$$\begin{aligned} E=\left( \begin{array}{l@{\quad }l} 1_{R(E)} &{} E_{1,2} \\ 0 &{} 0 \end{array} \right) . \end{aligned}$$

We study the sets of idempotents that one obtains when \(E_{1,2}:R(E)^\perp \rightarrow R(E)\) is a special type of operator: compact, Fredholm and injective with dense range, among others.

     

On the Topology of Vacuum Spacetimes

We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n + 1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold Sⁿ \Sigma^n to an asymptotically Euclidean solution of the constraints on \mathbbRⁿ \mathbb{R}^n . For any Sⁿ \Sigma^n which does not admit a metric of positive scalar curvature, this provides for the existence of asymptotically flat vacuum spacetimes with no maximal slices. Our main theorem is a special case of a more general gluing construction for nondegenerate solutions of the vacuum constraint equations which have some restrictions on the mean curvature, but for which the mean curvature is not necessarily constant. This generalizes the construction [16], which is restricted to constant mean curvature data.     

Gamma-convergence of nonlocal perimeter functionals

Given ${\Omega\subset\mathbb{R}^{n}}$ open, connected and with Lipschitz boundary, and ${s\in (0, 1)}$ , we consider the functional $$\mathcal{J}_s(E,\Omega)\,=\, \int_{E\cap \Omega}\int_{E^c\cap\Omega}\frac{dxdy}{|x-y|^{n+s}}+\int_{E\cap \Omega}\int_{E^c\cap \Omega^c}\frac{dxdy}{|x-y|^{n+s}}\,+ \int_{E\cap \Omega^c}\int_{E^c\cap \Omega}\frac{dxdy}{|x-y|^{n+s}},$$ where ${E\subset\mathbb{R}^{n}}$ is an arbitrary measurable set. We prove that the functionals ${(1-s)\mathcal{J}_s(\cdot, \Omega)}$ are equi-coercive in ${L^1_{\rm loc}(\Omega)}$ as ${s\uparrow 1}$ and that $$\Gamma-\lim_{s\uparrow 1}(1-s)\mathcal{J}_s(E,\Omega)=\omega_{n-1}P(E,\Omega),\quad \text{for every }E\subset\mathbb{R}^{n}\,{\rm measurable}$$ where P(E, ??) denotes the perimeter of E in ?? in the sense of De Giorgi. We also prove that as ${s\uparrow 1}$ limit points of local minimizers of ${(1-s)\mathcal{J}_s(\cdot,\Omega)}$ are local minimizers of P(·, ??).     

Existence and Stability Results for Renormalized Solutions to Noncoercive Nonlinear Elliptic Equations with Measure Data

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is