New Rates of Convergence for the Iterates of Some Positive Linear Operators

In this paper, we give quantitative results for the convergence of the iterates of some positive linear operators which preserve the functions \(e_0\) and \(e_1\), respectively, \(e_0\) and \(e_2\). We obtain estimates in terms of moduli of smoothness and we improve some previous results. We give examples for the q-Bernstein, the q-Bernstein-Durrmeyer, the q-Stancu, the q-Meyer-Konig–Zeller, the King type of the q-Bernstein and the King type of the q-genuine-Bernstein-Durrmeyer operators. We observe that some q-operators (\(0<q<1\)) provide better convergence for the iterates than the corresponding classical operators (\(q=1\)).     

On a Characterization of Idempotent Distributions on Discrete Fields and on the Field of <Emphasis Type="Italic">p</Emphasis>-Adic Numbers

We prove the following theorem. Let X be a discrete field, and \(\xi \) and \(\eta \) be independent identically distributed random variables with values in X and distribution \(\mu \). The random variables \(S=\xi +\eta \) and \(D=(\xi -\eta )^2\) are independent if and only if \(\mu \) is an idempotent distribution. A similar result is also proved in the case when \(\xi \) and \(\eta \) are independent identically distributed random variables with values in the field of p-adic numbers \({\mathbf {Q}}_p\), where \(p>2\), assuming that the distribution \(\mu \) has a continuous density.     

A group of continuous self-maps on a topological groupoid

The group of bisections of groupoids plays an important role in the study of Lie groupoids. In this paper another construction is introduced. Indeed, for a topological groupoid G, the set of all continuous self-maps f on G such that (x, f(x)) is a composable pair for every \(x\in G\), is denoted by \(S_G\). We show that \(S_G\) by a natural binary operation is a monoid. \(S_G(\alpha )\), the group of units in \(S_G\) precisely consists of those \(f\in S_G\) such that the map \(x\mapsto xf(x)\) is a bijection on G. Similar to the group of bisections, \(S_G(\alpha )\) acts on G from the right and on the space of continuous self-maps on G from the left. It is proved that \(S_G(\alpha )\) with the compact- open topology inherited from C(G, G) is a left topological group. For a compact Hausdorff groupoid G it is proved that the group of bisections of \(G^2\) is isomorphic to the group \(S_G(\alpha )\) and the group of transitive bisections of G, \(Bis_T(G)\), is embedded in \(S_G(\alpha )\), where \(G^2\) is the groupoid of all composable pairs.     

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 \(\mathrm{ind}\,F^{[h M]}_z=\mathrm{ind}\,F^M_z\).     

A new theorem on the prime-counting function

For \(x>0\), let \(\pi (x)\) denote the number of primes not exceeding x. For integers a and \(m>0\), we determine when there is an integer \(n>1\) with \(\pi (n)=(n+a)/m\). In particular, we show that, for any integers \(m>2\) and \(a\leqslant \lceil e^{m-1}/(m-1)\rceil \), there is an integer \(n>1\) with \(\pi (n)=(n+a)/m\). Consequently, for any integer \(m>4\), there is a positive integer n with \(\pi (mn)=m+n\). We also pose several conjectures for further research; for example, we conjecture that, for each \(m=1,2,3,\ldots \), there is a positive integer n such that \(m+n\) divides \(p_m+p_n\), where \(p_k\) denotes the k-th prime.     

<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 \(\varphi \).     

Anti-Ramsey Numbers of Graphs with Small Connected Components

The anti-Ramsey number, AR(n, G), for a graph G and an integer \(n\ge |V(G)|\), is defined to be the minimal integer r such that in any edge-colouring of \(K_n\) by at least r colours there is a multicoloured copy of G, namely, a copy of G that each of its edges has a distinct colour. In this paper we determine, for large enough \(n,\, AR(n,L\cup tP_2)\) and \(AR(n,L\cup kP_3)\) for any large enough t and k, and a graph L satisfying some conditions. Consequently, we determine AR(n, G), for large enough n, where G is \(P_3\cup tP_2\) for any \(t\ge 3,\, P_4\cup tP_2\) and \(C_3\cup tP_2\) for any \(t\ge 2,\, kP_3\) for any \(k\ge 3,\, tP_2\cup kP_3\) for any \(t\ge 1,\, k\ge 2\), and \(P_{t+1}\cup kP_3\) for any \(t\ge 3,\, k\ge 1\). Furthermore, we obtain upper and lower bounds for AR(n, G), for large enough n, where G is \(P_{k+1}\cup tP_2\) and \(C_k\cup tP_2\) for any \(k\ge 4,\, t\ge 1\).     

Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.     

Forcing Posets with Large Dimension to Contain Large Standard Examples

The dimension of a poset P, denoted \(\dim (P)\), is the least positive integer d for which P is the intersection of d linear extensions of P. The maximum dimension of a poset P with \(|P|\le 2n+1\) is n, provided \(n\ge 2\), and this inequality is tight when P contains the standard example \(S_n\). However, there are posets with large dimension that do not contain the standard example \(S_2\). Moreover, for each fixed \(d\ge 2\), if P is a poset with \(|P|\le 2n+1\) and P does not contain the standard example \(S_d\), then \(\dim (P)=o(n)\). Also, for large n, there is a poset P with \(|P|=2n\) and \(\dim (P)\ge (1-o(1))n\) such that the largest d so that P contains the standard example \(S_d\) is o(n). In this paper, we will show that for every integer \(c\ge 1\), there is an integer \(f(c)=O(c^2)\) so that for large enough n, if P is a poset with \(|P|\le 2n+1\) and \(\dim (P)\ge n-c\), then P contains a standard example \(S_d\) with \(d\ge n-f(c)\). From below, we show that \(f(c)={\varOmega }(c^{4/3})\). On the other hand, we also prove an analogous result for fractional dimension, and in this setting f(c) is linear in c. Here the result is best possible up to the value of the multiplicative constant.     

The non-abelian tensor square of residually finite groups

Let m, n be positive integers and p a prime. We denote by \(\nu (G)\) an extension of the non-abelian tensor square \(G \otimes G\) by \(G \times G\). We prove that if G is a residually finite group satisfying some non-trivial identity \(f \equiv ~1\) and for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) such that \([x,y^{\varphi }]^q = 1\), then the derived subgroup \(\nu (G)'\) is locally finite (Theorem A). Moreover, we show that if G is a residually finite group in which for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) dividing \(p^m\) such that \([x,y^{\varphi }]^q\) is left n-Engel, then the non-abelian tensor square \(G \otimes G\) is locally virtually nilpotent (Theorem B).     

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.     

Cliques in $$C_4$$-free graphs of large minimum degree

A graph G is called \(C_4\)-free if it does not contain the cycle \(C_4\) as an induced subgraph. Hubenko, Solymosi and the first author proved (answering a question of Erd?s) a peculiar property of \(C_4\)-free graphs: \(C_4\)-free graphs with n vertices and average degree at least cn contain a complete subgraph (clique) of size at least \(c'n\) (with \(c'= 0.1c^2\)). We prove here better bounds \(\big ({c^2n\over 2+c}\) in general and \((c-1/3)n\) when \( c \le 0.733\big )\) from the stronger assumption that the \(C_4\)-free graphs have minimum degree at least cn. Our main result is a theorem for regular graphs, conjectured in the paper mentioned above: 2k-regular \(C_4\)-free graphs on \(4k+1\) vertices contain a clique of size \(k+1\). This is the best possible as shown by the kth power of the cycle \(C_{4k+1}\).     

Hyperinvariant Subspace Problem for Some Classes of Operators

An n-normal operator may be defined as an \(n \times n\) operator matrix with entries that are mutually commuting normal operators and an operator \(T \in \mathcal {B(H)}\) is quasi-nM-hyponormal (for \(n \in \mathbb {N}\)) if it is unitarily equivalent to an \(n \times n\) upper triangular operator matrix \((T_{ij})\) acting on \(\mathcal {K}^{(n)}\), where \(\mathcal {K}\) is a separable complex Hilbert space and the diagonal entries \(T_{jj}\) \((j = 1,2,\ldots , n)\) are M-hyponormal operators in \(\mathcal {B(K)}\). This is an extended notion of n-normal operators. We prove a necessary and sufficient condition for an \(n \times n\) triangular operator matrix to have Bishop’s property \((\beta )\). This leads us to study the hyperinvariant subspace problem for an \(n \times n\) triangular operator matrix.     

2-Domination number of generalized Petersen graphs

Let \(G=(V,E)\) be a graph. A subset \(S\subseteq V\) is a k-dominating set of G if each vertex in \(V-S\) is adjacent to at least k vertices in S. The k-domination number of G is the cardinality of the smallest k-dominating set of G. In this paper, we shall prove that the 2-domination number of generalized Petersen graphs \(P(5k+1, 2)\) and \(P(5k+2, 2)\), for \(k>0\), is \(4k+2\) and \(4k+3\), respectively. This proves two conjectures due to Cheng (Ph.D. thesis, National Chiao Tung University, 2013). Moreover, we determine the exact 2-domination number of generalized Petersen graphs P(2k, k) and \(P(5k+4,3)\). Furthermore, we give a good lower and upper bounds on the 2-domination number of generalized Petersen graphs \(P(5k+1, 3), P(5k+2,3)\) and \(P(5k+3, 3).\)     

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 \(n \gg 0\).     

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 \(t \downarrow 0\).     

Coloring Complete and Complete Bipartite Graphs from Random Lists

Assign to each vertex v of the complete graph \(K_n\) on n vertices a list L(v) of colors by choosing each list independently and uniformly at random from all f(n)-subsets of a color set \([n] = \{1,\dots , n\}\), where f(n) is some integer-valued function of n. Such a list assignment L is called a random (f(n), [n])-list assignment. In this paper, we determine the asymptotic probability (as \(n \rightarrow \infty \)) of the existence of a proper coloring \(\varphi \) of \(K_n\), such that \(\varphi (v) \in L(v)\) for every vertex v of \(K_n\). We show that this property exhibits a sharp threshold at \(f(n) = \log n\). Additionally, we consider the corresponding problem for the line graph of a complete bipartite graph \(K_{m,n}\) with parts of size m and n, respectively. We show that if \(m = o(\sqrt{n})\), \(f(n) \ge 2 \log n\), and L is a random (f(n), [n])-list assignment for the line graph of \(K_{m,n}\), then with probability tending to 1, as \(n \rightarrow \infty \), there is a proper coloring of the line graph of \(K_{m,n}\) with colors from the lists.     

The Terwilliger algebra of the incidence graph of the Hamming graph

Let \(\varGamma \) be a distance-semiregular graph on Y, and let \(D^Y\) be the diameter of \(\varGamma \) on Y. Let \(\varDelta \) be the halved graph of \(\varGamma \) on Y. Fix \(x \in Y\). Let T and \(T'\) be the Terwilliger algebras of \(\varGamma \) and \(\varDelta \) with respect to x, respectively. Assume, for an integer i with \(1 \le 2i \le D^Y\) and for \(y,z \in \varGamma _{2i}(x)\) with \(\partial _{\varGamma }(y,z)=2\), the numbers \(|\varGamma _{2i-1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) and \(|\varGamma _{2i+1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) depend only on i and do not depend on the choice of y, z. The first goal in this paper is to show the relations between T-modules of \(\varGamma \) and \(T'\)-modules of \(\varDelta \). Assume \(\varGamma \) is the incidence graph of the Hamming graph H(D, n) on the vertex set Y and the set \({\mathcal {C}}\) of all maximal cliques. Then, \(\varGamma \) satisfies above assumption and \(\varDelta \) is isomorphic to H(D, n). The second goal is to determine the irreducible T-modules of \(\varGamma \). For each irreducible T-module W, we give a basis for W the action of the adjacency matrix on this basis and we calculate the multiplicity of W.     

Hyers–Ulam Stability of Differential Operators on Reproducing Kernel Function Spaces

In this paper, we investigate the Hyers–Ulam stability of the differential operators \(T_\lambda \) and D on the weighted Hardy spaces \(H_\beta ^2\) with the reproducing property. We obtain a necessary and sufficient condition in order that D is stable on \(H_\beta ^2\), and construct an example concerning the stability of \(T_\lambda \) on \(H_\beta ^2\). Moreover, we also investigate the Hyers–Ulam stability of the partial differential operators \(D_i\) on the several variables reproducing kernel space \(H_f^2(\mathbb {B}_d)\).     

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.