A characterization of Inoue surfaces with $$p_g=0$$ and $$K^2=7$$K2=7

Inoue constructed the first examples of smooth minimal complex surfaces of general type with \(p_g=0\) and \(K^2=7\). These surfaces are finite Galois covers of the 4-nodal cubic surface with the Galois group, the Klein group \(\mathbb {Z}_2\times \mathbb {Z}_2\). For such a surface S, the bicanonical map of S has degree 2 and it is composed with exactly one involution in the Galois group. The divisorial part of the fixed locus of this involution consists of two irreducible components: one is a genus 3 curve with self-intersection number 0 and the other is a genus 2 curve with self-intersection number \(-\,1\). Conversely, assume that S is a smooth minimal complex surface of general type with \(p_g=0\), \(K^2=7\) and having an involution \(\sigma \). We show that, if the divisorial part of the fixed locus of \(\sigma \) consists of two irreducible components \(R_1\) and \(R_2\), with \(g(R_1)=3, R_1^2=0, g(R_2)=2\) and \(R_2^2=-\,1\), then the Klein group \(\mathbb {Z}_2\times \mathbb {Z}_2\) acts faithfully on S and S is indeed an Inoue surface.     

The distortion dimension of $$\mathbb $$-rank 1 lattices

Let \(X=G/K\) be a symmetric space of noncompact type and rank \(k\ge 2\). We prove that horospheres in X are Lipschitz \((k-2)\)-connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible \(\mathbb {Q}\)-rank-1 lattice \(\Gamma \) in a linear, semisimple Lie group G of \(\mathbb R\)-rank k is \(k-1\). That is, given \(m< k-1\), a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) \(\Gamma \), and a \((m+1)\)-ball B in X (or G) filling S, there is a \((m+1)\)-ball \(B'\) in \(\Gamma \) filling S such that \({{\mathrm{vol}}}B'\sim {{\mathrm{vol}}}B\). In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension \(k-1\).     

Bergman interpolation on finite Riemann surfaces. Part II: Poincaré-Hyperbolic Case

Let M be a 3-manifold with torus boundary components \(T_{1}\) and \(T_2\). Let \(\phi :T_{1} \rightarrow T_{2}\) be a homeomorphism, \(M_\phi \) the manifold obtained from M by gluing \(T_{1}\) to \(T_{2}\) via the map \(\phi \), and T the image of \(T_{1}\) in \(M_\phi \). We show that if \(\phi \) is “sufficiently complicated” then any incompressible or strongly irreducible surface in \(M_\phi \) can be isotoped to be disjoint from T. It follows that every Heegaard splitting of a 3-manifold admitting a “sufficiently complicated” JSJ decomposition is an amalgamation of Heegaard splittings of the components of the JSJ decomposition.     

Characterization of the Gamma semigroup

The Gamma semigroup with parameter \(b>0\) on \(L^p(\mathbb R^+)\) is defined by

$$\begin{aligned} W_b(t)f(x)=\frac{1}{\Gamma (t)}\int _0^x(x-y)^{t-1}e^{-b(x-y)}f(y)\,dy. \end{aligned}$$

Let S denote the multiplication operator \(f(x)\rightarrow xf(x)\) with maximal domain D(S) in \(L^p(\mathbb R^+)\). The bounded operator V on \(L^p(\mathbb R^+)\) is S-Volterra if D(S) is V-invariant and \([S,V]=V^2\) on D(S). For \(1<p<\infty \), we characterize the Gamma semigroup as the unique regular semigroup \(V(\cdot )\) on \(L^p(\mathbb R^+)\) with imaginary type less than \(\pi \), such that V(1) is S-Volterra and \(V(1)u^b=Su^b\), where \(u^b(x):=e^{-bx}\).

     

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 \( \overline{\mathscr {R}(A^*)}\).     

Upper Bounds for the Paired-Domination Numbers of Graphs

A set \(S\subseteq V\) is a paired-dominating set if every vertex in \(V{\setminus } S\) has at least one neighbor in S and the subgraph induced by S contains a perfect matching. The paired-domination number of a graph G, denoted by \(\gamma _{pr}(G)\), is the minimum cardinality of a paired-dominating set of G. A conjecture of Goddard and Henning says that if G is not the Petersen graph and is a connected graph of order n with minimum degree \(\delta (G)\ge 3\), then \(\gamma _{pr}(G)\le 4n/7\). In this paper, we confirm this conjecture for k-regular graphs with \(k\ge 4\).     

The Gray image of constacyclic codes over the finite chain ring $$F_{p^m}[u]/\langle u^k\rangle $$

Let \(\mathbb {F}_{p^m}\) be a finite field of cardinality \(p^m\), where p is a prime, and k, N be any positive integers. We denote \(R_k=F_{p^m}[u]/\langle u^k\rangle =F_{p^m}+uF_{p^m}+\cdots +u^{k-1}F_{p^m}\) (\(u^k=0\)) and \(\lambda =a_0+a_1u+\cdots +a_{k-1}u^{k-1}\) where \(a_0, a_1,\ldots , a_{k-1}\in F_{p^m}\) satisfying \(a_0\ne 0\) and \(a_1=1\). Let r be a positive integer satisfying \(p^{r-1}+1\le k\le p^r\). First we define a Gray map from \(R_k\) to \(F_{p^m}^{p^r}\), then prove that the Gray image of any linear \(\lambda \)-constacyclic code over \(R_k\) of length N is a distance preserving linear \(a_0^{p^r}\)-constacyclic code over \(F_{p^m}\) of length \(p^rN\). Furthermore, the generator polynomials for each linear \(\lambda \)-constacyclic code over \(R_k\) of length N and its Gray image are given respectively. Finally, some optimal constacyclic codes over \(F_{3}\) and \(F_{5}\) are constructed.     

Heinz mean curvature estimates in warped product spaces $$M\times _{e^{\psi }}N$$

If a graph submanifold (x, f(x)) of a Riemannian warped product space \((M^m\times _{e^{\psi }}N^n,\tilde{g}=g+ e^{2\psi }h)\) is immersed with parallel mean curvature H, then we obtain a Heinz-type estimation of the mean curvature. Namely, on each compact domain D of M, \(m\Vert H\Vert \le \frac{A_{\psi }(\partial D)}{V_{\psi }(D)}\) holds, where \(A_{\psi }(\partial D)\) and \(V_{\psi }(D)\) are the \({\psi }\)-weighted area and volume, respectively. In particular, \(H=0\) if (M, g) has zero-weighted Cheeger constant, a concept recently introduced by Impera et al. (Height estimates for killing graphs. arXiv:1612.01257, 2016). This generalizes the known cases \(n=1\) or \(\psi =0\). We also conclude minimality using a closed calibration, assuming \((M,g_*)\) is complete where \(g_*=g+e^{2\psi }f^*h\), and for some constants \(\alpha \ge \delta \ge 0\), \(C_1>0\) and \(\beta \in [0,1)\), \(\Vert \nabla ^*\psi \Vert ^2_{g_*}\le \delta \), \(\mathrm {Ricci}_{\psi ,g_*}\ge \alpha \), and \({\mathrm{det}}_g(g_*)\le C_1 r^{2\beta }\) holds when \(r\rightarrow +\infty \), where r(x) is the distance function on \((M,g_*)\) from some fixed point. Both results rely on expressing the squared norm of the mean curvature as a weighted divergence of a suitable vector field.     

<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 \(\varphi \).     

Partial permutation decoding for binary linear and $$Z_4$$-linear Hadamard codes

In this paper, s-\({\text {PD}}\)-sets of minimum size \(s+1\) for partial permutation decoding for the binary linear Hadamard code \(H_m\) of length \(2^m\), for all \(m\ge 4\) and \(2 \le s \le \lfloor {\frac{2^m}{1+m}}\rfloor -1\), are constructed. Moreover, recursive constructions to obtain s-\({\text {PD}}\)-sets of size \(l\ge s+1\) for \(H_{m+1}\) of length \(2^{m+1}\), from an s-\({\text {PD}}\)-set of the same size for \(H_m\), are also described. These results are generalized to find s-\({\text {PD}}\)-sets for the \({\mathbb {Z}}_4\)-linear Hadamard codes \(H_{\gamma , \delta }\) of length \(2^m\), \(m=\gamma +2\delta -1\), which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type \(2^\gamma 4^\delta \). Specifically, s-PD-sets of minimum size \(s+1\) for \(H_{\gamma , \delta }\), for all \(\delta \ge 3\) and \(2\le s \le \lfloor {\frac{2^{2\delta -2}}{\delta }}\rfloor -1\), are constructed and recursive constructions are described.     

Sandwich semigroups in locally small categories I: foundations

Fix (not necessarily distinct) objects i and j of a locally small category S, and write \(S_{ij}\) for the set of all morphisms \(i\rightarrow j\). Fix a morphism \(a\in S_{ji}\), and define an operation \(\star _a\) on \(S_{ij}\) by \(x\star _ay=xay\) for all \(x,y\in S_{ij}\). Then \((S_{ij},\star _a)\) is a semigroup, known as a sandwich semigroup, and denoted by \(S_{ij}^a\). This article develops a general theory of sandwich semigroups in locally small categories. We begin with structural issues such as regularity, Green’s relations and stability, focusing on the relationships between these properties on \(S_{ij}^a\) and the whole category S. We then identify a natural condition on a, called sandwich regularity, under which the set \({\text {Reg}}(S_{ij}^a)\) of all regular elements of \(S_{ij}^a\) is a subsemigroup of \(S_{ij}^a\). Under this condition, we carefully analyse the structure of the semigroup \({\text {Reg}}(S_{ij}^a)\), relating it via pullback products to certain regular subsemigroups of \(S_{ii}\) and \(S_{jj}\), and to a certain regular sandwich monoid defined on a subset of \(S_{ji}\); among other things, this allows us to also describe the idempotent-generated subsemigroup \(\mathbb E(S_{ij}^a)\) of \(S_{ij}^a\). We also study combinatorial invariants such as the rank (minimal size of a generating set) of the semigroups \(S_{ij}^a\), \({\text {Reg}}(S_{ij}^a)\) and \(\mathbb E(S_{ij}^a)\); we give lower bounds for these ranks, and in the case of \({\text {Reg}}(S_{ij}^a)\) and \(\mathbb E(S_{ij}^a)\) show that the bounds are sharp under a certain condition we call MI-domination. Applications to concrete categories of transformations and partial transformations are given in Part II.     

Fredholmness of fringe operators over the bidisk

Let M be an invariant subspace of \(H^2\) over the bidisk. Associated with M, we have the fringe operator \(F^M_z\) on \(M\ominus w M\). For \(A\subset H^2\), let [A] denote the smallest invariant subspace containing A. Assume that \(F^M_z\) is Fredholm. If h is a bounded analytic function on \(\mathbb {D}^2\) satisfying \(h(0,0)\not =0\), then \(F^{[h M]}_z\) is Fredholm and \(\mathrm{ind}\,F^{[h M]}_z=\mathrm{ind}\,F^M_z\).     

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 \((-2,2)\).     

Generalized Drazin invertibility and local spectral theory

If \(A\in B(\mathcal{X})\) is an upper triangular Banach space operator with diagonal \((A_1,A_2)\), \(A_1\) invertible and \(A_2\) quasinilpotent, then \(A_1^{-1}\oplus A_2\) satisfies either of the single-valued extension property, Dunford’s condition (C), Bishop’s property \((\beta )\), decomposition property \((\delta )\) or is decomposable if and only if \(A_1\) has the property. The operator \(A^{-1}_1\oplus 0\) is subscalar (resp., left polaroid, right polaroid) if and only if \(A_1\) is subscalar (resp., left polaroid, right polaroid). For Drazin invertible operators A, with Drazin inverse B, this implies that B satisfies any one of these properties if and only if A satisfies the property.     

Algebraic properties of Zappa–Szép products of semigroups and monoids

Direct, semidirect and Zappa–Szép products provide tools to decompose algebraic structures, with each being a natural generalisation of its predecessor. In this paper we examine Zappa–Szép products of monoids and semigroups and investigate generalised Greens relations \({\mathcal R}^{*},\, {\mathcal L}^{*},\, \widetilde{\mathcal {R}}_E\) and \(\widetilde{\mathcal {L}}_E\) for these Zappa–Szép products. We consider a left restriction semigroup S with semilattice of projections E and define left and right actions of S on E and E on S, respectively, to form the Zappa–Szép product \(E \bowtie S\). We further investigate properties of \(E \bowtie S\) and show that S is a retract of \(E\bowtie S\). We also find a subset T of \(E \bowtie S\) which is left restriction.     

On commutator of generalized Aluthge transformations and Fuglede–Putnam theorem

Let \(A=U|A|\) be the polar decomposition of A on a complex Hilbert space \({\mathscr {H}}\) and \(0<s,t\). Then \({\widetilde{A}}_{s, t}=|A|^sU|A|^t\) and \({\widetilde{A}}_{s, t}^{(*)}=|A^*|^sU|A^*|^t\) are called the generalized Aluthge transformation and generalized \(*\)-Aluthge transformation of A, respectively. A pair (A, B) of operators is said to have the Fuglede–Putnam property (breifly, the FP-property) if \(AX=XB\) implies \(A^*X=XB^*\) for every operator X. We prove that if (A, B) has the FP-property, then \(({\widetilde{A}}_{s, t},{\widetilde{B}}_{s, t})\) and \((({\widetilde{A}}_{s, t})^{*},({\widetilde{B}}_{s, t})^{*})\) has the FP-property for every \(s,t>0\) with \(s+t=1\). Also, we prove that \(({\widetilde{A}}_{s, t},{\widetilde{B}}_{s, t})\) has the FP-property if and only if \((({\widetilde{A}}_{s, t})^{*},({\widetilde{B}}_{s, t})^{*})\) has the FP-property, where A, B are invertible and \( 0 < s, t \) with \( s + t =1\). Moreover, we prove that if \(0 < s, t\) and \({\widetilde{A}}_{s, t}\) is positive and invertible, then \(\left\| {\widetilde{A}}_{s, t}X-X{\widetilde{A}}_{s, t}\right\| \le \left\| A\right\| ^{2t}\left\| ({\widetilde{A}}_{s, t})^{-1}\right\| \left\| X\right\| \) for every operator X. Also, if \( 0 <s, t\) and X is positive, then \(\left\| |{\widetilde{A}}_{s, t}|^{2r} X-X|{\widetilde{A}}_{s, t}|^{2r}\right\| \le \frac{1}{2}\left\| |A|\right\| ^{2r}\left\| X\right\| \) for every \(r>0\).     

Tightness and Convergence of Trimmed Lévy Processes to Normality at Small Times

For nonnegative integers r, s, let \(^{(r,s)}X_t\) be the Lévy process \(X_t\) with the r largest positive jumps and the s smallest negative jumps up till time t deleted, and let \(^{(r)}\widetilde{X}_t\) be \(X_t\) with the r largest jumps in modulus up till time t deleted. Let \(a_t \in \mathbb {R}\) and \(b_t>0\) be non-stochastic functions in t. We show that the tightness of \(({}^{(r,s)}X_t - a_t)/b_t\) or \(({}^{(r)}{\widetilde{X}}_t - a_t)/b_t\) as \(t\downarrow 0\) implies the tightness of all normed ordered jumps, and hence the tightness of the untrimmed process \((X_t -a_t)/b_t\) at 0. We use this to deduce that the trimmed process \(({}^{(r,s)}X_t - a_t)/b_t\) or \(({}^{(r)}{\widetilde{X}}_t - a_t)/b_t\) converges to N(0, 1) or to a degenerate distribution as \(t\downarrow 0\) if and only if \((X_t-a_t)/b_t \) converges to N(0, 1) or to the same degenerate distribution, as \(t \downarrow 0\).     

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 convex iterative roots of non-monotonic mappings

Let I be an interval. We consider the non-monotonic convex self-mappings \(f:I\rightarrow I\) such that \(f^2\) is convex. They have the property that all iterates \(f^n\) are convex. In the class of these mappings we study three families of functions possessing convex iterative roots. A function f is said to be iteratively convex if f possesses convex iterative roots of all orders. A mapping f is said to be dyadically convex if for every \(n\ge 2\) there exists a convex iterative root \(f^{1/2^n}\) of order \(2^n\) and the sequence \(\{f^{1/2^n}\}\) satisfies the condition of compatibility, that is \( f^{1/2^n}\circ f^{1/2^n}= f^{1/2^{n-1}}.\) A function f is said to be flowly convex if it possesses a convex semi-flow of f, that is a family of convex functions \(\{f^t,t>0\}\) such that \(f^t\circ f^s=f^{t+s}, \ \ t,s >0\) and \(f^1=f\). We show the relations among these three types of convexity and we determine all convex iterative roots of non-monotonic functions.     

On the gaps between non-zero Fourier coefficients of cusp forms of higher weight

We show that if a modular cuspidal eigenform f of weight 2k is 2-adically close to an elliptic curve \(E/\mathbb {Q}\), which has a cyclic rational 4-isogeny, then n-th Fourier coefficient of f is non-zero in the short interval \((X, X + cX^{\frac{1}{4}})\) for all \(X \gg 0\) and for some \(c > 0\). We use this fact to produce non-CM cuspidal eigenforms f of level \(N>1\) and weight \(k > 2\) such that \(i_f(n) \ll n^{\frac{1}{4}}\) for all \(n \gg 0\).