The gluing formula of the refined analytic torsion for an acyclic Hermitian connection

In the previous study by Huang and Lee (arXiv:1004.1753) we introduced the well-posed boundary conditions ${{\mathcal P}_{-, {\mathcal L}_{0}}}$ and ${{\mathcal P}_{+, {\mathcal L}_{1}}}$ for the odd signature operator to define the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the gluing formula of the refined analytic torsion for an acyclic Hermitian connection with respect to the boundary conditions ${{\mathcal P}_{-, {\mathcal L}_{0}}}$ and ${{\mathcal P}_{+, {\mathcal L}_{1}}}$ . In this case the refined analytic torsion consists of the Ray-Singer analytic torsion, the eta invariant and the values of the zeta functions at zero. We first compare the Ray-Singer analytic torsion and eta invariant subject to the boundary condition ${{\mathcal P}_{-, {\mathcal L}_{0}}}$ or ${{\mathcal P}_{+, {\mathcal L}_{1}}}$ with the Ray-Singer analytic torsion subject to the relative (or absolute) boundary condition and eta invariant subject to the APS boundary condition on a compact manifold with boundary. Using these results together with the well known gluing formula of the Ray-Singer analytic torsion subject to the relative and absolute boundary conditions and eta invariant subject to the APS boundary condition, we obtain the main result.     

The Inviscid Limit for the Navier–Stokes Equations with Slip Condition on Permeable Walls

We consider the Navier–Stokes equations in a 2D-bounded domain with general non-homogeneous Navier slip boundary conditions prescribed on permeable boundaries, and study the vanishing viscosity limit. We prove that solutions of the Navier–Stokes equations converge to solutions of the Euler equations satisfying the same Navier slip boundary condition on the inflow region of the boundary. The convergence is strong in Sobolev’s spaces $W^{1}_{p}, p>2$ , which correspond to the spaces of the data.     

Symmetric Willmore surfaces of revolution satisfying natural boundary conditions

We consider the Willmore-type functional $$\mathcal{W}_{\gamma}(\Gamma):= \int\limits_{\Gamma} H^2 \; dA -\gamma \int\limits_{\Gamma} K \; dA,$$ where H and K denote mean and Gaussian curvature of a surface Γ, and ${\gamma \in [0,1]}$ is a real parameter. Using direct methods of the calculus of variations, we prove existence of surfaces of revolution generated by symmetric graphs which are solutions of the Euler-Lagrange equation corresponding to ${\mathcal{W}_{\gamma}}$ and which satisfy the following boundary conditions: the height at the boundary is prescribed, and the second boundary condition is the natural one when considering critical points where only the position at the boundary is fixed. In the particular case γ = 0 these boundary conditions are arbitrary positive height α and zero mean curvature.     

On the isomorphic mapping of weighted spaces by an elliptic operator with degeneration on the domain boundary

We consider the first boundary value problem for a second-order elliptic equation with degeneration on the entire twice continuously differentiable boundary of a two-dimensional domain Ω; this problem has a generalized solution in the weighted space . We find conditions ensuring that the generalized solution belongs to the narrower space , which permits obtaining estimates for the convergence rate of an approximate finite-element solution to the exact solution in the norms of the spaces and L ₂(Ω).     

On the dependency for asymptotically independent estimates

Suppose ${\widehat{\theta}_1}$ and ${\widehat{\theta}_2}$ are asymptotically independent non-lattice with a joint second order Edgeworth expansion in n ^?1/2. Then the ?? dependency coefficient is $$\alpha \left(\widehat{\theta}_1, \widehat{\theta}_2 \right) = n^{-1/2} C + O \left(n^{-1} \right),$$ where ${C = (4 \pi)^{-1}\exp (-1/2) (\tau^2_1 + \tau^2_2) ^{1/2}}$ for ${\tau_1, \tau_2}$ their joint skewness coefficients.     

Convolution functions of several variables with generalized bounded variation

Let L ¹ be the class of all complex-valued functions, with period 2π in each variable, in the space , where $\mathbb{T} = [0,2\pi )$ is the one-dimensional torus. Here, it is observed that L ¹ * E ? E for E = Lip(p; α ₁, α ₂, ..., α _N ) over , for , for , and for in the sense of Vitali as well as Hardy.     

Continuous embeddings of Besov-Morrey function spaces

We study embeddings of spaces of Besov-Morrey type, M Bp1,q1s1,r1(Rd ) → M Bp2 ,q2s2 ,r2 (R d ), and obtain necessary and sufficient conditions for this. Moreover, we can also characterise the special weighted situation Bp1 ,r1s1 (R d , w) → M Bp2 ,q2s2 ,r2 (Rd ) for a Muckenhoupt A ∞ weight w, with wα(x) = |x|α , α -d1, as a typical example.     

The s-Riesz transform of an s-dimensional measure in ℝ2 is unbounded for 1 < s < 2

In this paper, we prove that for s ∈ (1, 2) there exists no totally lower irregular finite positive Borel measure µ in ?² with such that ${\left\| {R\mu } \right\|_{{L^\infty }({m_2})}} < + \infty $ , where Rµ = µ*x/|x| ^s+1 and m ₂ is the Lebesgue measure in ?². Combined with known results of Prat and Vihtilä, this shows that for any s ∈ (0, 1) ∪ (1, 2) and any finite positive Borel measure in ?² with , we have ${\left\| {R\mu } \right\|_{{L^\infty }({m_2})}} = \infty $ .     

Boundary value problem for the Burgers system

We consider the boundary value problem $\begin{gathered} div(\rho V) = 0, \rho |\Gamma _1 = \rho 0, \hfill \\ \rho (V,\nabla V) = v\Delta V, V|\Gamma = V^0 \hfill \\ \end{gathered} $ for a vector functionV=(V ₁,V ₂) and a scalar function ρ>-0 in a rectangular domain Ω ? ?² with boundary Γ. Here $\Gamma _1 = \{ x \in \Gamma :(V^0 ,n)< 0\} , V_1^0 |_\Gamma > 0, v = const > 0.$ We prove that this problem is solvable in Hölder classes.     

Filippov Lemma for certain second order differential inclusions

In the paper we give an analogue of the Filippov Lemma for the second order differential inclusions with the initial conditions y(0) = 0, y??(0) = 0, where the matrix A ?? ? ^d×d and multifunction is Lipschitz continuous in y with a t-independent constant l. The main result is the following: Assume that F is measurable in t and integrably bounded. Let y ₀ ?? W ^2,1 be an arbitrary function fulfilling the above initial conditions and such that where p ₀ ?? L ¹[0, 1]. Then there exists a solution y ?? W ^2,1 to the above differential inclusions such that a.e. in [0, 1], .     

An inverse problem of the three-dimensional wave equation for a general annular vibrating membrane with piecewise smooth boundary conditions

This paper deals with the very interesting problem about the influence of piecewise smooth boundary conditions on the distribution of the eigenvalues of the negative Laplacian inR ³. The asymptotic expansion of the trace of the wave operator $\widehat\mu (t) = \sum\limits_{\upsilon = 1}^\infty {\exp \left( { - it\mu _\upsilon ^{1/2} } \right)} $ for small ?t? and $i = \sqrt { - 1} $ , where $\{ \mu _\nu \} _{\nu = 1}^\infty $ are the eigenvalues of the negative Laplacian $ - \nabla ^2 = - \sum\limits_{k = 1}^3 {\left( {\frac{\partial }{{\partial x^k }}} \right)} ^2 $ in the (x ¹,x ²,x ³), is studied for an annular vibrating membrane Ω inR ³ together with its smooth inner boundary surfaceS ₁ and its smooth outer boundary surfaceS ₂. In the present paper, a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth componentsS ^* _i(i=1, …,m) ofS ₁ and on the piecewise smooth componentsS ^* _i(i=m+1, …,n) ofS ₂ such that $S_1 = \bigcup\limits_{i = 1}^m {S_i^* } $ and $S_2 = \bigcup\limits_{i = m + 1}^n {S_i^* } $ are considered. The basic problem is to extract information on the geometry of the annular vibrating membrane ω from complete knowledge of its eigenvalues by analyzing the asymptotic expansions of the spectral function $\widehat\mu (t)$ for small ?t?.     

Lower large deviations for the maximal flow through a domain of {\mathbb{R}^d} in first passage percolation

We consider the standard first passage percolation model in the rescaled graph ${\mathbb{Z}^d/n}$ for d??? 2, and a domain ?? of boundary ?? in ${\mathbb{R}^d}$ . Let ??¹ and ??² be two disjoint open subsets of ??, representing the parts of ?? through which some water can enter and escape from ??. We investigate the asymptotic behaviour of the flow ${\phi_n}$ through a discrete version ?? _n of ?? between the corresponding discrete sets ${\Gamma^{1}_{n}}$ and ${\Gamma^{2}_{n}}$ . We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the lower large deviations of ${\phi_n/ n^{d-1}}$ below a certain constant are of surface order.     

Topological spaces compact with respect to a set of filters

If is a family of filters over some set I, a topological space X is sequencewise -compact if for every I-indexed sequence of elements of X there is such that the sequence has an F-limit point. Countable compactness, sequential compactness, initial κ-compactness, [λ; µ]-compactness, the Menger and Rothberger properties can all be expressed in terms of sequencewise -compactness for appropriate choices of . We show that sequencewise -compactness is preserved under taking products if and only if there is a filter such that sequencewise -compactness is equivalent to F-compactness. If this is the case, and there exists a sequencewise -compact T ₁ topological space with more than one point, then F is necessarily an ultrafilter. The particular case of sequential compactness is analyzed in detail.     

On the asymptotic form of convex hulls of Gaussian random fields

We consider a centered Gaussian random field X = {X _t : t ∈ T} with values in a Banach space $\mathbb{B}$ defined on a parametric set T equal to ? ^m or ? ^m . It is supposed that the distribution of X _t is independent of t. We consider the asymptotic behavior of closed convex hulls W _n = conv{X _t : t ∈ T _n}, where (T _n ) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (b _n ) _n≥1 with probability 1, (in the sense of Hausdorff distance), where the limit set is the concentration ellipsoid of . The asymptotic behavior of the mathematical expectations Ef(W _n ), where f is some function, is also studied.     

Double Hilbert transform on

We introduce a space , where is the testing function space whose functions are infinitely differentiable and have bounded support, and is the space the double Hilbert transform acting on the testing function space. We prove that the double Hilbert transform is a homeomorphism from onto itself.     

Wavelet transform and radon transform on the quaternion Heisenberg group

Let Q be the quaternion Heisenberg group,and let P be the affine automorphism group of Q.We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on L2(Q).A class of radial wavelets is constructed.The inverse wavelet transform is simplified by using radial wavelets.Then we investigate the Radon transform on Q.A Semyanistyi–Lizorkin space is introduced,on which the Radon transform is a bijection.We deal with the Radon transform on Q both by the Euclidean Fourier transform and the group Fourier transform.These two treatments are essentially equivalent.We also give an inversion formula by using wavelets,which does not require the smoothness of functions if the wavelet is smooth.In addition,we obtain an inversion formula of the Radon transform associated with the sub-Laplacian on Q.     

The Pólya–Chebotarev problem and inverse polynomial images

Consider the problem, usually called the Pólya–Chebotarev problem, of finding a continuum in the complex plane including some given points such that the logarithmic capacity of this continuum is minimal. We prove that each connected inverse image ${\mathcal {T}}_{n}^{-1} ([-1,1])$ of a polynomial  ${\mathcal {T}}_{n}$ is always the solution of a certain Pólya–Chebotarev problem. By solving a nonlinear system of equations for the zeros of ${\mathcal {T}}_{n}^{2}-1$ , we are able to construct polynomials ${\mathcal {T}}_{n}$ with a connected inverse image.     

Regularity of minimizers of non-isotropic integrals of the calculus of variations

The regularity of the minimizers of a special type of non-isotropic variational minimization problem is studied. The particularity of the potential of energy is that it has different growth rate with respect to different parts of the derivatives of the function. In particular, the model treated in this paper can be described as $$\Phi (Du) = |\partial _1 u|^2 + |\partial _2 u|^2 + |\partial _3 u|^2 + |\partial _3 u - |^p .$$ By using a result of P.Marcellini (cf. [4]) and perturbation method, it is proved that the minimizer of the Dirichlet boundary value problem is a function of W _loc ^{1, ∞} .     

Harmonic mappings and conformal minimal immersions of Riemann surfaces into {\mathbb {R}^{\rm N}}

We prove that for any open Riemann surface ${\mathcal{N}}$ , natural number N ≥ 3, non-constant harmonic map ${h:\mathcal{N} \to \mathbb{R}}$ ^N?2 and holomorphic 2-form ${\mathfrak{H}}$ on ${\mathcal{N}}$ , there exists a weakly complete harmonic map ${X=(X_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}$ with Hopf differential ${\mathfrak{H}}$ and ${(X_j)_{j=3,\ldots,{\sc N}}=h.}$ In particular, there exists a complete conformal minimal immersion ${Y=(Y_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}$ such that ${(Y_j)_{j=3,\ldots,{\sc N}}=h}$ . As some consequences of these results (1) there exist complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss map is non-degenerate and fails to intersect N hyperplanes of ${\mathbb{CP}^{{\sc N}-1}}$ in general position. (2) There exist complete non-proper embedded minimal surfaces in ${\mathbb{R}^{\sc N},}$ ${\forall\,{\sc N} >3 .}$