Mapping prefer-opposite to prefer-one de Bruijn sequences

We present a mapping of the binary prefer-opposite de Bruijn sequence of order n onto the binary prefer-one de Bruijn sequence of order \(n-1\). The mapping is based on the differentiation operator \(D(\langle {b_1,\ldots ,b_l}\rangle ) = \langle b_2-b_1, b_3-b_2,\ldots , b_{l}-b_{l-1} \rangle \) where bit subtraction is modulo two. We show that if we take the prefer-opposite sequence \(\langle {b_1,b_2,\ldots ,b_{2^n}}\rangle \), apply D to get the sequence \(\langle {\hat{b}_1, \ldots , \hat{b}_{2^n-1}}\rangle \) and drop all the bits \(\hat{b}_i\) such that \(\langle {\hat{b}_i,\ldots ,\hat{b}_{i+n-1}}\rangle \) is a substring of \(\langle {\hat{b}_1,\ldots ,\hat{b}_{i+n-2}}\rangle \), we get the prefer-one de Bruijn sequence of order \(n-1\).     

Uniqueness of self-shrinkers to the degree-one curvature flow with a tangent cone at infinity

Given a smooth, symmetric and homogeneous of degree one function \(f\left( \lambda _{1},\ldots ,\lambda _{n}\right) \) satisfying \(\partial _{i}f>0\quad \forall \,i=1,\ldots , n\), and a properly embedded smooth cone \({\mathcal {C}}\) in \({\mathbb {R}}^{n+1}\), we show that under suitable conditions on f, there is at most one f self-shrinker (i.e. a hypersurface \(\Sigma \) in \({\mathbb {R}}^{n+1}\) satisfying \(f\left( \kappa _{1},\ldots ,\kappa _{n}\right) +\frac{1}{2}X\cdot N=0\), where \(\kappa _{1},\ldots ,\kappa _{n}\) are principal curvatures of \(\Sigma \)) that is asymptotic to the given cone \({\mathcal {C}}\) at infinity.     

Spherical metrics with conical singularities on 2-spheres

Suppose that \(\theta _1,\theta _2,\ldots ,\theta _n\) are positive numbers and \(n\ge 3\). We want to know whether there exists a spherical metric on \(\mathbb {S}^2\) with n conical singularities of angles \(2\pi \theta _1,2\pi \theta _2,\ldots ,2\pi \theta _n\). A sufficient condition was obtained by Mondello and Panov (Int Math Res Not 2016(16):4937–4995, 2016). We show that their condition is also necessary when we assume that \(\theta _1,\theta _2,\ldots ,\theta _n \not \in \mathbb {N}\).     

Perpetual Integrals for Lévy Processes

Given a Lévy process \(\xi \), we find necessary and sufficient conditions for almost sure finiteness of the perpetual integral \(\int _0^\infty f(\xi _s)\hbox {d}s\), where \(f\) is a positive locally integrable function. If \(\mu =\mathbb {E}[\xi _1]\in (0,\infty )\) and \(\xi \) has local times we prove the 0–1 law

$$\begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )\in \{0,1\} \end{aligned}$$

with the exact characterization

$$\begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )=0\qquad \Longleftrightarrow \qquad \int ^\infty f(x)\,\hbox {d}x=\infty . \end{aligned}$$

The proof uses spatially stationary Lévy processes, local time calculations, Jeulin’s lemma and the Hewitt–Savage 0–1 law.

     

The convolution algebra

For L a complete lattice L and \(\mathfrak {X}=(X,(R_i)_I)\) a relational structure, we introduce the convolution algebra \(L^{\mathfrak {X}}\). This algebra consists of the lattice \(L^X\) equipped with an additional \(n_i\)-ary operation \(f_i\) for each \(n_i+1\)-ary relation \(R_i\) of \(\mathfrak {X}\). For \(\alpha _1,\ldots ,\alpha _{n_i}\in L^X\) and \(x\in X\) we set \(f_i(\alpha _1,\ldots ,\alpha _{n_i})(x)=\bigvee \{\alpha _1(x_1)\wedge \cdots \wedge \alpha _{n_i}(x_{n_i}):(x_1,\ldots ,x_{n_i},x)\in R_i\}\). For the 2-element lattice 2, \(2^\mathfrak {X}\) is the reduct of the familiar complex algebra \(\mathfrak {X}^+\) obtained by removing Boolean complementation from the signature. It is shown that this construction is bifunctorial and behaves well with respect to one-one and onto maps and with respect to products. When L is the reduct of a complete Heyting algebra, the operations of \(L^\mathfrak {X}\) are completely additive in each coordinate and \(L^\mathfrak {X}\) is in the variety generated by \(2^\mathfrak {X}\). Extensions to the construction are made to allow for completely multiplicative operations defined through meets instead of joins, as well as modifications to allow for convolutions of relational structures with partial orderings. Several examples are given.     

Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces?

We choose some special unit vectors \({\mathbf {n}}_1,\ldots ,{\mathbf {n}}_5\) in \({\mathbb {R}}^3\) and denote by \({\mathscr {L}}\subset {\mathbb {R}}^5\) the set of all points \((L_1,\ldots ,L_5)\in {\mathbb {R}}^5\) with the following property: there exists a compact convex polytope \(P\subset {\mathbb {R}}^3\) such that the vectors \({\mathbf {n}}_1,\ldots ,{\mathbf {n}}_5\) (and no other vector) are unit outward normals to the faces of P and the perimeter of the face with the outward normal \({\mathbf {n}}_k\) is equal to \(L_k\) for all \(k=1,\ldots ,5\). Our main result reads that \({\mathscr {L}}\) is not a locally-analytic set, i.e., we prove that, for some point \((L_1,\ldots ,L_5)\in {\mathscr {L}}\), it is not possible to find a neighborhood \(U\subset {\mathbb {R}}^5\) and an analytic set \(A\subset {\mathbb {R}}^5\) such that \({\mathscr {L}}\cap U=A\cap U\). We interpret this result as an obstacle for finding an existence theorem for a compact convex polytope with prescribed directions and perimeters of the faces.     

On the eigenvalues and spectral radius of starlike trees

Let \(k\ge 1\) and \(n_1,\ldots ,n_k\ge 1\) be some integers. Let \(S(n_1,\ldots ,n_k)\) be a tree T such that T has a vertex v of degree k and \(T{\setminus } v\) is the disjoint union of the paths \(P_{n_1},\ldots ,P_{n_k}\), that is \(T{\setminus } v\cong P_{n_1}\cup \cdots \cup P_{n_k}\) so that every neighbor of v in T has degree one or two. The tree \(S(n_1,\ldots ,n_k)\) is called starlike tree, a tree with exactly one vertex of degree greater than two, if \(k\ge 3\). In this paper we obtain the eigenvalues of starlike trees. We find some bounds for the largest eigenvalue (for the spectral radius) of starlike trees. In particular we prove that if \(k\ge 4\) and \(n_1,\ldots ,n_k\ge 2\), then \(\frac{k-1}{\sqrt{k-2}}<\lambda _1(S(n_1,\ldots ,n_k))<\frac{k}{\sqrt{k-1}}\), where \(\lambda _1(T)\) is the largest eigenvalue of T. Finally we characterize all starlike trees that all of whose eigenvalues are in the interval \((-2,2)\).     

Lower order eigenvalues of a system of equations of the drifting Laplacian on the metric measure spaces

Let \(\Omega \) be a bounded domain with smooth boundary in an n-dimensional metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and let \(\mathbf {u}=(u^1, \ldots , u^n)\) be a vector-valued function from \(\Omega \) to \(\mathbb {R}^n\). In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of the drifting Laplacian: \(\mathbb {L}_{\phi } \mathbf {u} + \alpha [ \nabla (\mathrm {div}\mathbf { u}) -\nabla \phi \mathrm {div} \mathbf {u}]= - \widetilde{\sigma } \mathbf {u}\), in \( \Omega \), and \(u|_{\partial \Omega }=0,\) where \(\mathbb {L}_{\phi } = \Delta - \nabla \phi \cdot \nabla \) is the drifting Laplacian and \(\alpha \) is a nonnegative constant. We establish some universal inequalities for lower order eigenvalues of this problem on the metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and the Gaussian shrinking soliton \((\mathbb {R}^n, \langle ,\rangle _{\mathrm {can}}, e^{-\frac{|x|^2}{4}}dv, \frac{1}{2})\). Moreover, we give an estimate for the upper bound of the second eigenvalue of this problem in terms of its first eigenvalue on the gradient product Ricci soliton \((\Sigma \times \mathbb {R}, \langle ,\rangle , e^{-\frac{\kappa t^2}{2}}dv, \kappa )\), where \( \Sigma \) is an Einstein manifold with constant Ricci curvature \(\kappa \).     

Conflict free colorings of (strongly) almost disjoint set-systems

\(f\: \cup {\mathcal {A}}\to {\rho}\) is called a conflict free coloring of the set-system\({\mathcal {A}}\)(withρcolors) if

$\forall A\in {\mathcal {A}}\ \exists\, {\zeta}<{\rho} (|A\cap f^{-1}\{{\zeta}\}|=1).$

The conflict free chromatic number\(\operatorname {\chi _{\rm CF}}\, ({\mathcal {A}})\) of \({\mathcal {A}}\) is the smallest ρ for which \({\mathcal {A}}\) admits a conflict free coloring with ρ colors.

\({\mathcal {A}}\) is a (λ,κ,μ)-system if \(|{\mathcal {A}}| = \lambda\), |A|=κ for all \(A \in {\mathcal {A}}\), and \({\mathcal {A}}\) is μ-almost disjoint, i.e. |A∩A′|<μ for distinct \(A, A'\in {\mathcal {A}}\). Our aim here is to study

$\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,\mu) = \sup \{\operatorname {\chi _{\rm CF}}\, ({\mathcal {A}})\: {\mathcal {A}}\mbox{ is a } (\lambda,\kappa,\mu)\mbox{-system}\}$

for λ≧κ≧μ, actually restricting ourselves to λ≧ω and μ≦ω.

For instance, we prove that

? for any limit cardinal κ (or κ=ω) and integers n≧0, k>0, GCH implies

$\operatorname {\chi _{\rm CF}}\, (\kappa^{+n},t,k+1) =\begin{cases}\kappa^{+(n+1-i)}&; \text{if \ } i\cdot k < t \le (i+1)\cdot k,\ i =1,\dots,n;\\[2pt]\kappa&; \text{if \ } (n+1)\cdot k < t;\end{cases}$

? if λ≧κ≧ω>d>1, then λ<κ ^+ω implies \(\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,d) <\omega\) and λ≧? _ω (κ) implies \(\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,d) = \omega\);? GCH implies \(\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,\omega) \le \omega_{2}\) for λ≧κ≧ω ₂ and V=L implies \(\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,\omega) \le \omega_{1}\) for λ≧κ≧ω ₁;? the existence of a supercompact cardinal implies the consistency of GCH plus \(\operatorname {\chi _{\rm CF}}\,(\aleph_{\omega+1},\omega_{1},\omega)= \aleph_{\omega+1}\) and \(\operatorname {\chi _{\rm CF}}\, (\aleph_{\omega+1},\omega_{n},\omega) = \omega_{2}\) for 2≦n≦ω;? CH implies \(\operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega,\omega) = \operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega_{1},\omega) = \omega_{1}\), while \(MA_{\omega_{1}}\) implies \(\operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega,\omega) = \operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega_{1},\omega) = \omega\).     

Uniform Bounds on the Relative Error in the Approximation of Upper Quantiles for Sums of Arbitrary Independent Random Variables

Fix any \(n\ge 1\). Let \(\tilde{X}_1,\ldots ,\tilde{X}_n\) be independent random variables. For each \(1\le j \le n\), \(\tilde{X}_j\) is transformed in a canonical manner into a random variable \(X_j\). The \(X_j\) inherit independence from the \(\tilde{X}_j\). Let \(s_y\) and \(s_y^*\) denote the upper \(\frac{1}{y}{\underline{\text{ th }}}\) quantile of \(S_n=\sum _{j=1}^nX_j\) and \(S^*_n=\sup _{1\le k\le n}S_k\), respectively. We construct a computable quantity \(\underline{Q}_y\) based on the marginal distributions of \(X_1,\ldots ,X_n\) to produce upper and lower bounds for \(s_y\) and \(s_y^*\). We prove that for \(y\ge 8\)

$$\begin{aligned} 6^{-1} \gamma _{3y/16}\underline{Q}_{3y/16}\le s^*_{y}\le \underline{Q}_y \end{aligned}$$

where

$$\begin{aligned} \gamma _y=\frac{1}{2w_y+1} \end{aligned}$$

and \(w_y\) is the unique solution of

$$\begin{aligned} \Big (\frac{w_y}{e\ln (\frac{y}{y-2})}\Big )^{w_y}=2y-4 \end{aligned}$$

for \(w_y>\ln (\frac{y}{y-2})\), and for \(y\ge 37\)

$$\begin{aligned} \frac{1}{9}\gamma _{u(y)}\underline{Q}_{u(y)}<s_y \le \underline{Q}_y \end{aligned}$$

where

$$\begin{aligned} u(y)=\frac{3y}{32} \left( 1+\sqrt{1-\frac{64}{3y}}\right) . \end{aligned}$$

The distribution of \(S_n\) is approximately centered around zero in that \(P(S_n\ge 0) \ge \frac{1}{18}\) and \(P(S_n\le 0)\ge \frac{1}{65}\). The results extend to \(n=\infty \) if and only if for some (hence all) \(a>0\)

$$\begin{aligned} \sum _{j=1}^{\infty }E\{(\tilde{X}_j-m_j)^2\wedge a^2\}<\infty . \end{aligned}$$

(1)

     

One-sided invertibility of discrete operators and their applications

For \(p\in [1,\infty ]\), we establish criteria for the one-sided invertibility of binomial discrete difference operators \({{\mathcal {A}}}=aI-bV\) on the space \(l^p=l^p(\mathbb {Z})\), where \(a,b\in l^\infty \), I is the identity operator and the isometric shift operator V is given on functions \(f\in l^p\) by \((Vf)(n)=f(n+1)\) for all \(n\in \mathbb {Z}\). Applying these criteria, we obtain criteria for the one-sided invertibility of binomial functional operators \(A=aI-bU_\alpha \) on the Lebesgue space \(L^p(\mathbb {R}_+)\) for every \(p\in [1,\infty ]\), where \(a,b\in L^\infty (\mathbb {R}_+)\), \(\alpha \) is an orientation-preserving bi-Lipschitz homeomorphism of \([0,+\infty ]\) onto itself with only two fixed points 0 and \(\infty \), and \(U_\alpha \) is the isometric weighted shift operator on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f= (\alpha ^\prime )^{1/p}(f\circ \alpha )\). Applications of binomial discrete operators to interpolation theory are given.     

CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

Let \(\mathcal S\) be an abelian group of automorphisms of a probability space \((X, {\mathcal A}, \mu )\) with a finite system of generators \((A_1, \ldots , A_d).\) Let \(A^{{\underline{\ell }}}\) denote \(A_1^{\ell _1} \ldots A_d^{\ell _d}\), for \({{\underline{\ell }}}= (\ell _1, \ldots , \ell _d).\) If \((Z_k)\) is a random walk on \({\mathbb {Z}}^d\), one can study the asymptotic distribution of the sums \(\sum _{k=0}^{n-1} \, f \circ A^{\,{Z_k(\omega )}}\) and \(\sum _{{\underline{\ell }}\in {\mathbb {Z}}^d} {\mathbb {P}}(Z_n= {\underline{\ell }}) \, A^{\underline{\ell }}f\), for a function f on X. In particular, given a random walk on commuting matrices in \(SL(\rho , {\mathbb {Z}})\) or in \({\mathcal M}^*(\rho , {\mathbb {Z}})\) acting on the torus \({\mathbb {T}}^\rho \), \(\rho \ge 1\), what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on \({\mathbb {T}}^\rho \) after normalization? In this paper, we prove a central limit theorem when X is a compact abelian connected group G endowed with its Haar measure (e.g., a torus or a connected extension of a torus), \(\mathcal S\) a totally ergodic d-dimensional group of commuting algebraic automorphisms of G and f a regular function on G. The proof is based on the cumulant method and on preliminary results on random walks.     

On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where

$$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$

is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and

$$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$

then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

     

Diagonals of injective tensor products of Banach lattices with bases

Let E be a Banach lattice with a 1-unconditional basis \(\{e_i: i \in \mathbb {N}\}\). Denote by \(\Delta (\check{\otimes }_{n,\epsilon }E)\) (resp. \(\Delta (\check{\otimes }_{n,s,\epsilon }E)\)) the main diagonal space of the n-fold full (resp. symmetric) injective Banach space tensor product, and denote by \(\Delta (\check{\otimes }_{n,|\epsilon |}E)\) (resp. \(\Delta (\check{\otimes }_{n,s,|\epsilon |}E)\)) the main diagonal space of the n-fold full (resp. symmetric) injective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic. We also show that the tensor diagonal \(\{e_i\otimes \cdots \otimes e_i: i \in \mathbb {N}\}\) is a 1-unconditional basic sequence in both \(\check{\otimes }_{n,\epsilon }E\) and \(\check{\otimes }_{n,s,\epsilon }E\).     

Distance Magic Labeling in Complete 4-partite Graphs

Let G be a complete k-partite simple undirected graph with parts of sizes \(p_1\le p_2\cdots \le p_k\). Let \(P_j=\sum _{i=1}^jp_i\) for \(j=1,\ldots ,k\). It is conjectured that G has distance magic labeling if and only if \(\sum _{i=1}^{P_j} (n-i+1)\ge j{{n+1}\atopwithdelims (){2}}/k\) for all \(j=1,\ldots ,k\). The conjecture is proved for \(k=4\), extending earlier results for \(k=2,3\).     

On a class of maximality principles

We study various classes of maximality principles, \(\mathrm {MP}(\kappa ,\Gamma )\), introduced by Hamkins (J Symb Log 68(2):527–550, 2003), where \(\Gamma \) defines a class of forcing posets and \(\kappa \) is an infinite cardinal. We explore the consistency strength and the relationship of \(\textsf {MP}(\kappa ,\Gamma )\) with various forcing axioms when \(\kappa \in \{\omega ,\omega _1\}\). In particular, we give a characterization of bounded forcing axioms for a class of forcings \(\Gamma \) in terms of maximality principles MP\((\omega _1,\Gamma )\) for \(\Sigma _1\) formulas. A significant part of the paper is devoted to studying the principle MP\((\kappa ,\Gamma )\) where \(\kappa \in \{\omega ,\omega _1\}\) and \(\Gamma \) defines the class of stationary set preserving forcings. We show that MP\((\kappa ,\Gamma )\) has high consistency strength; on the other hand, if \(\Gamma \) defines the class of proper forcings or semi-proper forcings, then by Hamkins (2003), MP\((\kappa ,\Gamma )\) is consistent relative to \(V=L\).     

Family complexity and cross-correlation measure for families of binary sequences

We study the relationship between two measures of pseudorandomness for families of binary sequences: family complexity and cross-correlation measure introduced by Ahlswede et al. in 2003 and recently by Gyarmati et al., respectively. More precisely, we estimate the family complexity of a family \((e_{i,1},\ldots ,e_{i,N})\in \{-1,+1\}^N\), \(i=1,\ldots ,F\), of binary sequences of length \(N\) in terms of the cross-correlation measure of its dual family \((e_{1,n},\ldots ,e_{F,n})\in \{-1,+1\}^F\), \(n=1,\ldots ,N\). We apply this result to the family of sequences of Legendre symbols with irreducible quadratic polynomials modulo \(p\) with middle coefficient \(0\), that is, \(e_{i,n}=\big (\frac{n^2-bi^2}{p}\big )_{n=1}^{(p-1)/2}\) for \(i=1,\ldots ,(p-1)/2\), where \(b\) is a quadratic nonresidue modulo \(p\), showing that this family as well as its dual family has both a large family complexity and a small cross-correlation measure up to a rather large order.     

On the <Emphasis Type="Italic">p</Emphasis>-Laplacian with Robin boundary conditions and boundary trace theorems

Let \(\Omega \subset \mathbb {R}^\nu \), \(\nu \ge 2\), be a \(C^{1,1}\) domain whose boundary \(\partial \Omega \) is either compact or behaves suitably at infinity. For \(p\in (1,\infty )\) and \(\alpha >0\), define

$$\begin{aligned} \Lambda (\Omega ,p,\alpha ):=\inf _{\begin{array}{c} u\in W^{1,p}(\Omega )\\ u\not \equiv 0 \end{array}}\dfrac{\displaystyle \int _\Omega |\nabla u|^p \mathrm {d} x - \alpha \displaystyle \int _{\partial \Omega } |u|^p\mathrm {d}\sigma }{\displaystyle \int _\Omega |u|^p\mathrm {d} x}, \end{aligned}$$

where \(\mathrm {d}\sigma \) is the surface measure on \(\partial \Omega \). We show the asymptotics

$$\begin{aligned} \Lambda (\Omega ,p,\alpha )=-(p-1)\alpha ^{\frac{p}{p-1}} - (\nu -1)H_\mathrm {max}\, \alpha + o(\alpha ), \quad \alpha \rightarrow +\infty , \end{aligned}$$

where \(H_\mathrm {max}\) is the maximum mean curvature of \(\partial \Omega \). The asymptotic behavior of the associated minimizers is discussed as well. The estimate is then applied to the study of the best constant in a boundary trace theorem for expanding domains, to the norm estimate for extension operators and to related isoperimetric inequalities.

     

The Difference Between a Discrete and Continuous Harmonic Measure

We consider a discrete-time, continuous-state random walk with steps uniformly distributed in a disk of radius h. For a simply connected domain D in the plane, let \(\omega _h(0,\cdot ;D)\) be the discrete harmonic measure at \(0\in D\) associated with this random walk, and \(\omega (0,\cdot ;D)\) be the (continuous) harmonic measure at 0. For domains D with analytic boundary, we prove there is a bounded continuous function \(\sigma _D(z)\) on \(\partial D\) such that for functions g which are in \(C^{2+\alpha }(\partial D)\) for some \(\alpha >0\) we have

$$\begin{aligned} \lim _{h\downarrow 0} \frac{\int _{\partial D} g(\xi ) \omega _h(0,|\mathrm{d}\xi |;D) -\int _{\partial D} g(\xi )\omega (0,|\mathrm{d}\xi |;D)}{h} = \int _{\partial D}g(z) \sigma _D(z) |\mathrm{d}z|. \end{aligned}$$

We give an explicit formula for \(\sigma _D\) in terms of the conformal map from D to the unit disk. The proof relies on some fine approximations of the potential kernel and Green’s function of the random walk by their continuous counterparts, which may be of independent interest.

     

Some remarks on energy inequalities for harmonic maps with potential

We call the \({\delta}\)-vector of an integral convex polytope of dimension d flat if the \({\delta}\)-vector is of the form \({(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)}\), where \({a \geq 1}\). In this paper, we give the complete characterization of possible flat \({\delta}\)-vectors. Moreover, for an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^N}\) of dimension d, we let \({i(\mathcal{P},n)=|n\mathcal{P}\cap \mathbb{Z}^N|}\) and \({i^*(\mathcal{P},n)=|n(\mathcal{P} {\setminus}\partial \mathcal{P})\cap \mathbb{Z}^N|}\). By this characterization, we show that for any \({d \geq 1}\) and for any \({k,\ell \geq 0}\) with \({k+\ell \leq d-1}\), there exist integral convex polytopes \({\mathcal{P}}\) and \({\mathcal{Q}}\) of dimension d such that (i) For \({t=1,\ldots,k}\), we have \({i(\mathcal{P},t)=i(\mathcal{Q},t),}\) (ii) For \({t=1,\ldots,\ell}\), we have \({i^*(\mathcal{P},t)=i^*(\mathcal{Q},t)}\), and (iii) \({i(\mathcal{P},k+1) \neq i(\mathcal{Q},k+1)}\) and \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).