Gradient Estimates for Commutators of Square Roots of Elliptic Operators with Complex Bounded Measurable Coefficients

Let \(L=-\mathrm{div}(A\nabla )\) be a second order divergence form elliptic operator and A an accretive \(n\times n\) matrix with bounded measurable complex coefficients in \({\mathbb R}^n\). Let \(\nabla b\in L^n({\mathbb R}^n)\,(n>2)\). In this paper, we prove that the commutator generated by b and the square root of L, which is defined by \([b,\sqrt{L}]f(x)=b(x)\sqrt{L}f(x)-\sqrt{L}(bf)(x)\), is bounded from the homogenous Sobolev space \({\dot{L}}_1^2({\mathbb R}^n)\) to \(L^2({\mathbb R}^n)\).     

Packing Chromatic Number of Base-3 Sierpiński Graphs

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that there exists a k-vertex coloring of G in which any two vertices receiving color i are at distance at least \(i+1\). Let \(S^n\) be the base-3 Sierpiński graph of dimension n. It is proved that \(\chi _{\rho }(S^1) = 3\), \(\chi _{\rho }(S^2) = 5\), \(\chi _{\rho }(S^3) = \chi _{\rho }(S^4) = 7\), and that \(8\le \chi _\rho (S^n) \le 9\) holds for any \(n\ge 5\).     

Closed range and nonclosed range adjointable operators on Hilbert $$C^*$$-modules

In this paper, we show that for a positive operator A on a Hilbert \(C^*\)-module \( \mathscr {E} \), the range \( \mathscr {R}(A) \) of A is closed if and only if \( \mathscr {R}(A^\alpha ) \) is closed for all \(\alpha \in (0,1)\cup (1,+\,\infty )\), and this occurs if and only if \( \mathscr {R}(A)=\mathscr {R}(A^\alpha ) \) for all \(\alpha \in (0,1)\cup (1,+\,\infty )\). As an application, we prove that for an adjontable operator A if \(\mathscr {R}(A)\) is nonclosed, then \(\dim \left( \overline{\mathscr {R}(A)}/\mathscr {R}(A)\right) =+\,\infty \). Finally, we show that for an adjointable operator A if \( \overline{\mathscr {R}(A^*) } \) is orthogonally complemented in \( \mathscr {E} \), then under certain coditions there exists an idempotent C and a unique operator X such that \( XAX=X, AXA=CA, AX=C \) and \( XA=P_{A^*} \), where \( P_{A^*} \) is the orthogonal projection of \( \mathscr {E} \) onto \( \overline{\mathscr {R}(A^*)}\).     

Couplings of Brownian Motions of Deterministic Distance in Model Spaces of Constant Curvature

We consider the model space \(\mathbb {M}^{n}_{K}\) of constant curvature K and dimension \(n\ge 1\) (Euclidean space for \(K=0\), sphere for \(K>0\) and hyperbolic space for \(K<0\)), and we show that given a function \(\rho :[0,\infty )\rightarrow [0, \infty )\) with \(\rho (0)=\mathrm {dist}(x,y)\) there exists a coadapted coupling (X(t), Y(t)) of Brownian motions on \(\mathbb {M}^{n}_{K}\) starting at (x, y) such that \(\rho (t)=\mathrm {dist}(X(t),Y(t))\) for every \(t\ge 0\) if and only if \(\rho \) is continuous and satisfies for almost every \(t\ge 0\) the differential inequality

$$\begin{aligned} -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) \le \rho '(t)\le -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) +\tfrac{2(n-1)\sqrt{K}}{\sin (\sqrt{K}\rho (t))}. \end{aligned}$$

In other words, we characterize all coadapted couplings of Brownian motions on the model space \(\mathbb {M}^{n}_{K}\) for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of \(\rho \) satisfying the above hypotheses.

     

Some new Kirkman signal sets

A decomposition of the blocks of an \(\textsf {STS}(v)\) into partial parallel classes of size m is equivalent to a Kirkman signal set \(\textsf {KSS}(v,m)\). We give decompositions of \(\textsf {STS}(4v-3)\) into classes of size \(v-1\) when \(v \equiv 3 \pmod {6}\), \(v \not = 3\). We also give decompositions of \(\textsf {STS}(v)\) into classes of various sizes when v is a product of two arbitrary integers that are both congruent to \(3 \pmod {6}\). These results produce new families of \(\textsf {KSS}(v,m)\).     

A geometric approach to alternating <Emphasis Type="Italic">k</Emphasis>-linear forms

Denote by \({{\mathcal {G}}}_k(V)\) the Grassmannian of the k-subspaces of a vector space V over a field \({\mathbb {K}}\). There is a natural correspondence between hyperplanes H of \({\mathcal {G}}_k(V)\) and alternating k-linear forms on V defined up to a scalar multiple. Given a hyperplane H of \({{\mathcal {G}}_k}(V)\), we define a subspace \(R^{\uparrow }(H)\) of \({{\mathcal {G}}_{k-1}}(V)\) whose elements are the \((k-1)\)-subspaces A such that all k-spaces containing A belong to H. When \(n-k\) is even, \(R^{\uparrow }(H)\) might be empty; when \(n-k\) is odd, each element of \({\mathcal {G}}_{k-2}(V)\) is contained in at least one element of \(R^{\uparrow }(H)\). In the present paper, we investigate several properties of \(R^{\uparrow }(H)\), settle some open problems and propose a conjecture.     

Constructive Approximation in de Branges–Rovnyak Spaces

In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function f can be approximated in norm by its dilates \(f_r(z):=f(rz)~(r<1)\). We show that this is not the case for the de Branges–Rovnyak spaces \(\mathcal{H}(b)\). More precisely, we exhibit a space \(\mathcal{H}(b)\) in which the polynomials are dense and a function \(f\in \mathcal{H}(b)\) such that \(\lim _{r\rightarrow 1^-}\Vert f_r\Vert _{\mathcal{H}(b)}=\infty \). On the positive side, we prove the following approximation theorem for Toeplitz operators on general de Branges–Rovnyak spaces \(\mathcal{H}(b)\). If \((h_n)\) is a sequence in \(H^\infty \) such that \(\Vert h_n\Vert _{H^\infty }\le 1\) and \(h_n(0)\rightarrow 1\), then \(\Vert T_{\overline{h}_n}f-f\Vert _{\mathcal{H}(b)}\rightarrow 0\) for all \(f\in \mathcal{H}(b)\). Using this result, we give the first constructive proof that, if b is a nonextreme point of the unit ball of \(H^\infty \), then the polynomials are dense in \(\mathcal{H}(b)\).     

Eldan’s Stochastic Localization and Tubular Neighborhoods of Complex-Analytic Sets

Let \(f: \mathbb {C}^n \rightarrow \mathbb {C}^k\) be a holomorphic function and set \(Z = f^{-1}(0)\). Assume that Z is non-empty. We prove that for any \(r > 0\),

$$\begin{aligned} \gamma _n(Z + r) \ge \gamma _n(E + r), \end{aligned}$$

where \(Z + r\) is the Euclidean r-neighborhood of Z; \(\gamma _n\) is the standard Gaussian measure in \(\mathbb {C}^n\), and \(E \subseteq \mathbb {C}^n\) is an \((n-k)\)-dimensional, affine, complex subspace whose distance from the origin is the same as the distance of Z from the origin.

     

Plethysm and Lattice Point Counting

We apply lattice point counting methods to compute the multiplicities in the plethysm of \(\textit{GL}(n)\). Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old conjecture of Howe on the leading term of plethysm. For any partition \(\mu \) of 3, 4, or 5, we obtain an explicit formula in \(\lambda \) and k for the multiplicity of \(S^\lambda \) in \(S^\mu (S^k)\).     

The crank moments weighted by the parity of cranks

In this note, we introduce the 2kth crank moment \(\mu _{2k}(-1,n)\) weighted by the parity of cranks and show that \((-1)^n \mu _{2k}(-1,n)>0\) for \(n\ge k \ge 0\). When \(k=0\), the inequality \((-1)^n \mu _{2k}(-1,n)>0\) reduces to Andrews and Lewis’s inequality \((-1)^n(M_e(n)-M_o(n))>0\) for \(n\ge 0\), where \(M_e(n)\) (resp. \(M_o(n)\)) denotes the number of partitions of n with even (resp. odd) crank. Several generating functions of \(\mu _{2k}(-1,n)\) are also studied in order to show the positivity of \((-1)^n\mu _{2k}(-1,n)\).     

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 \((-2,2)\).     

A generalization of total graphs

Let R be a commutative ring with nonzero identity, \(L_{n}(R)\) be the set of all lower triangular \(n\times n\) matrices, and U be a triangular subset of \(R^{n}\), i.e., the product of any lower triangular matrix with the transpose of any element of U belongs to U. The graph \(GT^{n}_{U}(R^n)\) is a simple graph whose vertices consists of all elements of \(R^{n}\), and two distinct vertices \((x_{1},\dots ,x_{n})\) and \((y_{1},\dots ,y_{n})\) are adjacent if and only if \((x_{1}+y_{1}, \ldots ,x_{n}+y_{n})\in U\). The graph \(GT^{n}_{U}(R^n)\) is a generalization for total graphs. In this paper, we investigate the basic properties of \(GT^{n}_{U}(R^n)\). Moreover, we study the planarity of the graphs \(GT^{n}_{U}(U)\), \(GT^{n}_{U}(R^{n}{\setminus } U)\) and \(GT^{n}_{U}(R^n)\).     

Eichler’s commutation relation and some other invariant subspaces of Hecke operators

Let k be an odd positive integer, L a lattice on a regular positive definite k-dimensional quadratic space over \(\mathbb {Q}\), \(N_L\) the level of L, and \(\mathscr {M}(L)\)  be the linear space of \(\theta \)-series attached to the distinct classes in the genus of L. We prove that, for an odd prime \(p|N_L\), if \(L_p=L_{p,1}\,\bot \, L_{p,2}\), where \(L_{p,1}\) is unimodular, \(L_{p,2}\) is (p)-modular, and \(\mathbb {Q}_pL_{p,2}\) is anisotropic, then \(\mathscr {M}(L;p):=\) \(\mathscr {M}(L)\) \(+T_{p^2}.\) \(\mathscr {M}(L)\)  is stable under the Hecke operator \(T_{p^2}\). If \(L_2\) is isometric to \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle 2\varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}1\\ 1&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) with \(\varepsilon \in \mathbb {Z}_2^{\times }\) and \(\kappa :=\frac{k-1}{2}\), then \(\mathscr {M}(L;2):=T_{2^2}.\mathscr {M}(L)+T_{2^2}^2.\,\mathscr {M}(L)\) is stable under the Hecke operator \(T_{2^2}\). Furthermore, we determine some invariant subspaces of the cusp forms for the Hecke operators.     

Relationships Between the 2-Metric Dimension and the 2-Adjacency Dimension in the Lexicographic Product of Graphs

Given a connected simple graph \(G=(V(G),E(G))\), a set \(S\subseteq V(G)\) is said to be a 2-metric generator for G if and only if for any pair of different vertices \(u,v\in V(G)\), there exist at least two vertices \(w_1,w_2\in S\) such that \(d_G(u,w_i)\ne d_G(v,w_i)\), for every \(i\in \{1,2\}\), where \(d_G(x,y)\) is the length of a shortest path between x and y. The minimum cardinality of a 2-metric generator is the 2-metric dimension of G, denoted by \(\dim _2(G)\). The metric \(d_{G,2}: V(G)\times V(G)\longmapsto {\mathbb {N}}\cup \{0\}\) is defined as \(d_{G,2}(x,y)=\min \{d_G(x,y),2\}\). Now, a set \(S\subseteq V(G)\) is a 2-adjacency generator for G, if for every two vertices \(x,y\in V(G)\) there exist at least two vertices \(w_1,w_2\in S\), such that \(d_{G,2}(x,w_i)\ne d_{G,2}(y,w_i)\) for every \(i\in \{1,2\}\). The minimum cardinality of a 2-adjacency generator is the 2-adjacency dimension of G, denoted by \({\mathrm {adim}}_2(G)\). In this article, we obtain closed formulae for the 2-metric dimension of the lexicographic product \(G\circ H\) of two graphs G and H. Specifically, we show that \(\dim _2(G\circ H)=n\cdot {\mathrm {adim}}_2(H)+f(G,H),\) where \(f(G,H)\ge 0\), and determine all the possible values of f(G, H).     

On the density of images of the power maps in Lie groups

Let G be a connected Lie group. In this paper, we study the density of the images of individual power maps \(P_k:G\rightarrow G:g\mapsto g^k\). We give criteria for the density of \(P_k(G)\) in terms of regular elements, as well as Cartan subgroups. In fact, we prove that if \(\mathrm{Reg}(G)\) is the set of regular elements of G, then \(P_k(G)\cap \mathrm{Reg}(G)\) is closed in \(\mathrm{Reg}(G)\). On the other hand, the weak exponentiality of G turns out to be equivalent to the density of all the power maps \(P_k\). In linear Lie groups, weak exponentiality reduces to the density of \(P_2(G)\). We also prove that the density of the image of \(P_k\) for G implies the same for any connected full rank subgroup.     

Erratum to: Abstract commensurability and quasi-isometry classification of hyperbolic surface group amalgams

A fundamental result by Gromov and Thurston asserts that, if M is a closed hyperbolic n-manifold, then the simplicial volume \(\Vert M\Vert \) of M is equal to \(\mathrm{Vol}(M)/v_n\), where \(v_n\) is a constant depending only on the dimension of M. The same result also holds for complete finite-volume hyperbolic manifolds without boundary, while Jungreis proved that the ratio \(\mathrm{Vol}(M)/\Vert M\Vert \) is strictly smaller than \(v_n\) if M is compact with nonempty geodesic boundary. We prove here a quantitative version of Jungreis’ result for \(n\ge 4\), which bounds from below the ratio \(\Vert M\Vert /\mathrm{Vol}(M)\) in terms of the ratio \(\mathrm{Vol}(\partial M)/\mathrm{Vol}(M)\). As a consequence, we show that, for \(n\ge 4\), a sequence \(\{M_i\}\) of compact hyperbolic n-manifolds with geodesic boundary satisfies \(\lim _i \mathrm{Vol}(M_i)/\Vert M_i\Vert =v_n\) if and only if \(\lim _i \mathrm{Vol}(\partial M_i)/\mathrm{Vol}(M_i)=0\). We also provide estimates of the simplicial volume of hyperbolic manifolds with geodesic boundary in dimension 3.     

Arithmetic properties of 7-regular partitions

Let \(b_{\ell }(n)\) denote the number of \(\ell \)-regular partitions of n. By employing the modular equation of seventh order, we establish the following congruence for \(b_{7}(n)\) modulo powers of 7: for \(n\ge 0\) and \(j\ge 1\),

$$\begin{aligned} b_{7}\left( 7^{2j-1}n+\frac{3\cdot 7^{2j}-1}{4}\right) \equiv 0 \pmod {7^j}. \end{aligned}$$

We also find some infinite families of congruences modulo 2 and 7 satisfied by \(b_{7}(n)\).

     

On height isotopy classes of embeddings in the plane of a Morse function of a circle

Let \(\mathrm{SM}_{2n}(S^1,\mathbb {R})\) be a set of stable Morse functions of an oriented circle such that the number of singular points is \(2n\in \mathbb {N}\) and the order of singular values satisfies the particular condition. For an orthogonal projection \(\pi :\mathbb {R}^2\rightarrow \mathbb {R}\), let \({\tilde{f}}_0\) and \({\tilde{f}}_1:S^1\rightarrow \mathbb {R}^2\) be embedding lifts of f. If there is an ambient isotopy \(\tilde{\varphi }_t:\mathbb {R}^2\rightarrow \mathbb {R}^2\) \((t\in [0,1])\) such that \({\pi \circ \tilde{\varphi }}_t(y_1,y_2)=y_1\) and \(\tilde{\varphi }_1\circ {\tilde{f}}_0={\tilde{f}}_1\), we say that \({\tilde{f}}_0\) and \({\tilde{f}}_1\) are height isotopic. We define a function \(I:\mathrm{SM}_{2n}(S^1,\mathbb {R})\rightarrow \mathbb {N}\) as follows: I(f) is the number of height isotopy classes of embeddings such that each rotation number is one. In this paper, we determine the maximal value of the function I equals the n-th Baxter number and the minimal value equals \(2^{n-1}\).     

<Emphasis Type="Italic">q</Emphasis>-Frequent hypercyclicity in spaces of operators

We provide conditions for a linear map of the form \(C_{R,T}(S)=RST\) to be q-frequently hypercyclic on algebras of operators on separable Banach spaces. In particular, if R is a bounded operator satisfying the q-frequent hypercyclicity criterion, then the map \(C_{R}(S)=RSR^*\) is shown to be q-frequently hypercyclic on the space \(\mathcal {K}(H)\) of all compact operators and the real topological vector space \(\mathcal {S}(H)\) of all self-adjoint operators on a separable Hilbert space H. Further we provide a condition for \(C_{R,T}\) to be q-frequently hypercyclic on the Schatten von Neumann classes \(S_p(H)\). We also characterize frequent hypercyclicity of \(C_{M^*_\varphi ,M_\psi }\) on the trace-class of the Hardy space, where the symbol \(M_\varphi \) denotes the multiplication operator associated to \(\varphi \).