Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (III)

This is the last of a series of papers which give a necessary and sufficient condition for two essential simple loops on a 2-bridge sphere in a 2-bridge link complement to be homotopic in the link complement. The first paper of the series treated the case of the 2-bridge torus links, and the second paper treated the case of 2-bridge links of slope \(n/(2n+1)\) and \((n+1)/(3n+2)\) , where \(n \ge 2\) is an arbitrary integer. In this paper, we first treat the case of 2-bridge links of slope \(n/(mn+1)\) and \((n+1)/((m+1)n+m)\) , where \(m \ge 3\) is an arbitrary integer, and then treat the remaining cases by induction.     

On cycles for the doubling map which are disjoint from an interval

Let \(T:[0,1]\rightarrow [0,1]\) be the doubling map and let \(0 . We say that an integer \(n\ge 3\) is bad for \((a,b)\) if all \(n\) -cycles for \(T\) intersect \((a,b)\) . Let \(B(a,b)\) denote the set of all \(n\) which are bad for \((a,b)\) . In this paper we completely describe the sets: $$\begin{aligned} D_2=\{(a,b) : B(a,b)\,\text {is finite}\} \end{aligned}$$ and $$\begin{aligned} D_3=\{(a,b) : B(a,b)=\varnothing \}. \end{aligned}$$ In particular, we show that if \(b-a<\frac{1}{6}\) , then \((a,b)\in D_2\) , and if \(b-a\le \frac{2}{15}\) , then \((a,b)\in D_3\) , both constants being sharp.     

Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (II)

This is the second of a series of papers which give a necessary and sufficient condition for two essential simple loops on a 2-bridge sphere in a 2-bridge link complement to be homotopic in the link complement. The first paper of the series treated the case of the \(2\) -bridge torus links. In this paper, we treat the case of \(2\) -bridge links of slope \(n/(2n+1)\) and \((n+1)/(3n+2)\) , where \(n \ge 2\) is an arbitrary integer.     

Nonexistence of some quantum jump codes with specified parameters

Quantum jump codes (JC) are quantum codes which correct errors caused by quantum jumps. A spontaneous emission error design (SEED) has been introduced to construct quantum JC. In this paper the nonexistence of some \((n,m,t)_k\) -JCs is proved, where \(m={n-t\atopwithdelims ()k-t},\, k\ge t+1,\, \text{ and }\, (n,t,k)=(2k+1,1,k), (2k,2,k),(7,2,3),(8,3,4)\) . It is also shown that, for these parameters with \(t=1,2\) , the existence of an \((n,m,t)_k\) -JC is equivalent to that of a \(t-(n,k;m)\) -SEED.     

Additive mappings satisfying algebraic conditions in rings

Let \(R\) be any \((n+1)!\) -torsion free ring and \(F,D: R\rightarrow R\) be additive mappings satisfying \(F(x^{n+1})=(\alpha (x))^nF(x)+\sum \nolimits _{i=1}^n (\alpha (x))^{n-i}(\beta (x))^iD(x)\) for all \(x\in R\) , where \(n\) is a fixed integer and \(\alpha \) , \(\beta \) are automorphisms of \(R\) . Then, \(D\) is Jordan left \((\alpha , \beta )\) -derivation and \(F\) is generalized Jordan left \((\alpha , \beta )\) -derivation on \(R\) and if additive mappings \(F\) and \(D\) satisfying \(F(x^{n+1})=F(x)(\alpha (x))^n+\sum \nolimits _{i=1}^n (\beta (x))^iD(x)(\alpha (x))^{n-i}\) for all \(x\in R\) . Then, \(D\) is Jordan \((\alpha , \beta )\) -derivation and \(F\) is generalized Jordan \((\alpha , \beta )\) -derivation on \(R\) . At last some immediate consequences of the above theorems have been given.     

Compactness and sequential completeness in some spaces of operators

Let \(X\) be a completely regular Hausdorff space and \(C_b(X)\) be the Banach lattice of all real-valued bounded continuous functions on \(X\) , endowed with the strict topologies \(\beta _\sigma ,\) \(\beta _\tau \) and \(\beta _t\) . Let \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) \((z=\sigma ,\tau ,t)\) stand for the space of all \((\beta _z,\xi )\) -continuous linear operators from \(C_b(X)\) to a locally convex Hausdorff space \((E,\xi ),\) provided with the topology \(\mathcal{T}_s\) of simple convergence. We characterize relative \(\mathcal{T}_s\) -compactness in \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) in terms of the representing Baire vector measures. It is shown that if \((E,\xi )\) is sequentially complete, then the spaces \((\mathcal{L}_{\beta _z,\xi }(C_b(X),E),\mathcal{T}_s)\) are sequentially complete whenever \(z=\sigma \) ; \(z=\tau \) and \(X\) is paracompact; \(z=t\) and \(X\) is paracompact and ?ech complete. Moreover, a Dieudonné–Grothendieck type theorem for operators on \(C_b(X)\) is given.     

Scattered linear sets generated by collineations between pencils of lines

Let \(A\) and \(B\) be two points of \(\mathrm{{PG}}(2,q^n)\) , and let \(\Phi \) be a collineation between the pencils of lines with vertices \(A\) and \(B\) . In this paper, we prove that the set of points of intersection of corresponding lines under \(\Phi \) is either the union of a scattered \(\mathrm{{GF}}(q)\) -linear set of rank \(n+1\) with the line \(AB\) or the union of \(q-1\) scattered \(\mathrm{{GF}}(q)\) -linear sets of rank \(n\) with \(A\) and \(B\) . We also determine the intersection configurations of two scattered \(\mathrm{{GF}}(q)\) -linear sets of rank \(n+1\) of \(\mathrm{{PG}}(2,q^n)\) both meeting the line \(AB\) in a \(\mathrm{{GF}}(q)\) -linear set of pseudoregulus type with transversal points \(A\) and \(B\) .     

Combinatorial solutions providing improved security for the generalized Russian cards problem

We present the first formal mathematical presentation of the generalized Russian cards problem, and provide rigorous security definitions that capture both basic and extended versions of weak and perfect security notions. In the generalized Russian cards problem, three players, Alice, Bob, and Cathy, are dealt a deck of \(n\) cards, each given \(a\) , \(b\) , and \(c\) cards, respectively. The goal is for Alice and Bob to learn each other’s hands via public communication, without Cathy learning the fate of any particular card. The basic idea is that Alice announces a set of possible hands she might hold, and Bob, using knowledge of his own hand, should be able to learn Alice’s cards from this announcement, but Cathy should not. Using a combinatorial approach, we are able to give a nice characterization of informative strategies (i.e., strategies allowing Bob to learn Alice’s hand), having optimal communication complexity, namely the set of possible hands Alice announces must be equivalent to a large set of \(t-(n, a, 1)\) -designs, where \(t=a-c\) . We also provide some interesting necessary conditions for certain types of deals to be simultaneously informative and secure. That is, for deals satisfying \(c = a-d\) for some \(d \ge 2\) , where \(b \ge d-1\) and the strategy is assumed to satisfy a strong version of security (namely perfect \((d-1)\) -security), we show that \(a = d+1\) and hence \(c=1\) . We also give a precise characterization of informative and perfectly \((d-1)\) -secure deals of the form \((d+1, b, 1)\) satisfying \(b \ge d-1\) involving \(d-(n, d+1, 1)\) -designs.     

On matching cover of graphs

A k-matching cover of a graph \(G\) is a union of \(k\) matchings of \(G\) which covers \(V(G)\) . The matching cover number of \(G\) , denoted by \(mc(G)\) , is the minimum number \(k\) such that \(G\) has a \(k\) -matching cover. A matching cover of \(G\) is optimal if it consists of \(mc(G)\) matchings of \(G\) . In this paper, we present an algorithm for finding an optimal matching cover of a graph on \(n\) vertices in \(O(n^3)\) time (if use a faster maximum matching algorithm, the time complexity can be reduced to \(O(nm)\) , where \(m=|E(G)|\) ), and give an upper bound on matching cover number of graphs. In particular, for trees, a linear-time algorithm is given, and as a by-product, the matching cover number of trees is determined.     

On the non-existence of maximal difference matrices of deficiency 1

A \(k\times u\lambda \) matrix \(M=[d_{ij}]\) with entries from a group \(U\) of order \(u\) is called a \((u,k,\lambda )\) -difference matrix over \(U\) if the list of quotients \(d_{i\ell }{d_{j\ell }}^{-1}, 1 \le \ell \le u\lambda ,\) contains each element of \(U\) exactly \(\lambda \) times for all \(i\ne j.\) Jungnickel has shown that \(k \le u\lambda \) and it is conjectured that the equality holds only if \(U\) is a \(p\) -group for a prime \(p.\) On the other hand, Winterhof has shown that some known results on the non-existence of \((u,u\lambda ,\lambda )\) -difference matrices are extended to \((u,u\lambda -1,\lambda )\) -difference matrices. This fact suggests us that there is a close connection between these two cases. In this article we show that any \((u,u\lambda -1,\lambda )\) -difference matrix over an abelian \(p\) -group can be extended to a \((u,u\lambda ,\lambda )\) -difference matrix.     

Vertex transitive graphs from injective linear mappings

Based on a motivation coming from the study of the metric structure of the category of finite dimensional vector spaces over a finite field \(\mathbb {F}\) , we examine a family of graphs, defined for each pair of integers \(1 \le k \le n\) , with vertex set formed by all injective linear transformations \(\mathbb {F}^k \rightarrow \mathbb {F}^n\) and edges corresponding to pairs of mappings, \(f\) and \(g\) , with \(\lambda (f,g)= \dim \mathrm{Im }(f-g)=1 \) . For \(\mathbb {F}\cong \mathrm{GF }(q)\) , this graph will be denoted by \(\mathrm{INJ }_q(k,n)\) . We show that all such graphs are vertex transitive and Hamiltonian and describe the full automorphism group of each \(\mathrm{INJ }_q (k,n)\) for \(k . Using the properties of line-transitive groups, we completely determine which of the graphs \(\mathrm{INJ }_q (k,n)\) are Cayley and which are not. The Cayley ones consist of three infinite families, corresponding to pairs \((1,n),\,(n-1,n)\) , and \((n,n)\) , with \(n\) and \(q\) arbitrary, and of two sporadic examples \(\mathrm{INJ }_{2} (2,5)\) and \(\mathrm{INJ }_{2}(3,5)\) . Hence, the overwhelming majority of our graphs is not Cayley.     

Fully inert subgroups of free Abelian groups

A subgroup \(H\) of an Abelian group \(G\) is called fully inert if \((\phi H + H)/H\) is finite for every \(\phi \in \mathrm{End}(G)\) . Fully inert subgroups of free Abelian groups are characterized. It is proved that \(H\) is fully inert in the free group \(G\) if and only if it is commensurable with \(n G\) for some \(n \ge 0\) , that is, \((H + nG)/H\) and \((H + nG)/nG\) are both finite. From this fact we derive a more structural characterization of fully inert subgroups \(H\) of free groups \(G\) , in terms of the Ulm–Kaplansky invariants of \(G/H\) and the Hill–Megibben invariants of the exact sequence \(0 \rightarrow H \rightarrow G \rightarrow G/H \rightarrow 0\) .     

Nonlinear Schrödinger equations near an infinite well potential

The paper deals with standing wave solutions of the dimensionless nonlinear Schrödinger equation where the potential \(V_\lambda :\mathbb {R}^N\rightarrow \mathbb {R}\) is close to an infinite well potential \(V_\infty :\mathbb {R}^N\rightarrow \mathbb {R}\) , i. e. \(V_\infty =\infty \) on an exterior domain \(\mathbb {R}^N\setminus \Omega \) , \(V_\infty |_\Omega \in L^\infty (\Omega )\) , and \(V_\lambda \rightarrow V_\infty \) as \(\lambda \rightarrow \infty \) in a sense to be made precise. The nonlinearity may be of Gross–Pitaevskii type. A standing wave solution of \((NLS_\lambda )\) with \(\lambda =\infty \) vanishes on \(\mathbb {R}^N\setminus \Omega \) and satisfies Dirichlet boundary conditions, hence it solves We investigate when a standing wave solution \(\Phi _\infty \) of the infinite well potential \((NLS_\infty )\) gives rise to nearby solutions \(\Phi _\lambda \) of the finite well potential \((NLS_\lambda )\) with \(\lambda \gg 1\) large. Considering \((NLS_\infty )\) as a singular limit of \((NLS_\lambda )\) we prove a kind of singular continuation type results.     

Fundamental domains for properly discontinuous affine groups

We construct a fundamental region for the action on the \(2d+1\) -dimensional affine space of some free, discrete, properly discontinuous groups of affine transformations preserving a quadratic form of signature \((d+1, d)\) , where \(d\) is any odd positive integer.     

Construction of relative difference sets and Hadamard groups

‘There exist normal \((2m,2,2m,m)\) relative difference sets and thus Hadamard groups of order \(4m\) for all \(m\) of the form $$\begin{aligned} m= x2^{a+t+u+w+\delta -\epsilon +1}6^b 9^c 10^d 22^e 26^f \prod _{i=1}^s p_i^{4a_i} \prod _{i=1}^t q_i^2 \prod _{i=1}^u \left( (r_i+1)/2)r_i^{v_i}\right) \prod _{i=1}^w s_i \end{aligned}$$ under the following conditions: \(a,b,c,d,e,f,s,t,u,w\) are nonnegative integers, \(a_1,\ldots ,a_r\) and \(v_1,\ldots ,v_u\) are positive integers, \(p_1,\ldots ,p_s\) are odd primes, \(q_1,\ldots ,q_t\) and \(r_1,\ldots ,r_u\) are prime powers with \(q_i\equiv 1\ (\mathrm{mod}\ 4)\) and \(r_i\equiv 1\ (\mathrm{mod}\ 4)\) for all \(i, s_1,\ldots ,s_w\) are integers with \(1\le s_i \le 33\) or \(s_i\in \{39,43\}\) for all \(i, x\) is a positive integer such that \(2x-1\) or \(4x-1\) is a prime power. Moreover, \(\delta =1\) if \(x>1\) and \(c+s>0, \delta =0\) otherwise, \(\epsilon =1\) if \(x=1, c+s=0\) , and \(t+u+w>0, \epsilon =0\) otherwise. We also obtain some necessary conditions for the existence of \((2m,2,2m,m)\) relative difference sets in partial semidirect products of \(\mathbb{Z }_4\) with abelian groups, and provide a table cases for which \(m\le 100\) and the existence of such relative difference sets is open.     

Optimal initial value conditions for the existence of strong solutions of the Boussinesq equations

Consider the instationary Boussinesq equations in a smooth bounded domain \(\Omega \subseteq \mathbb {R}^3\) with initial values \(u_0 \in L^2_{\sigma }(\Omega )\) , \( \theta _0 \in L^2(\Omega )\) and gravitational force \(g\) . We call \((u,\theta )\) strong solution if \((u,\theta )\) is a weak solution and additionally Serrin’s condition \(u \in L^s(0,T; L^q(\Omega ))\) holds where \( 1 satisfy \(\frac{2}{s} + \frac{3}{q} =1\) . In this paper we show that \(\int _0^{\infty } \Vert e^{-tA} u_0 \Vert _q^s \, dt < \infty \) is necessary and sufficient for the existence of such a strong solution \((u,\theta )\) in a sufficiently small interval \([0,T[\, , 0 < T\le \infty \) . Furthermore we show that strong solutions are uniquely determined and that they are smooth if the data are smooth. The crucial point is the fact that we have required no additional integrability condition for \(\theta \) in the definition of a strong solution \((u,\theta )\) .     

The M-principal graph of a commutative ring

Let \(R\) be a commutative ring and \(M\) be an \(R\) -module. In this paper, we introduce the \(M\) -principal graph of \(R\) , denoted by \(M-PG(R)\) . It is the graph whose vertex set is \(R\backslash \{0\}\) , and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(xM=yM\) . In the special case that \(M=R, M-PG(R)\) is denoted by \(PG(R)\) . The basic properties and possible structures of these two graphs are studied. Also, some relations between \(PG(R)\) and \(M-PG(R)\) are established.     

Uncertainty principles and sum complexes

Let \(p\) be a prime and let \(A\) be a nonempty subset of the cyclic group \(C_p\) . For a field \({\mathbb F}\) and an element \(f\) in the group algebra \({\mathbb F}[C_p]\) let \(T_f\) be the endomorphism of \({\mathbb F}[C_p]\) given by \(T_f(g)=fg\) . The uncertainty number \(u_{{\mathbb F}}(A)\) is the minimal rank of \(T_f\) over all nonzero \(f \in {\mathbb F}[C_p]\) such that \(\mathrm{supp}(f) \subset A\) . The following topological characterization of uncertainty numbers is established. For \(1 \le k \le p\) define the sum complex \(X_{A,k}\) as the \((k-1)\) -dimensional complex on the vertex set \(C_p\) with a full \((k-2)\) -skeleton whose \((k-1)\) -faces are all \(\sigma \subset C_p\) such that \(|\sigma |=k\) and \(\prod _{x \in \sigma }x \in A\) . It is shown that if \({\mathbb F}\) is algebraically closed then $$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{k :\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$ The main ingredient in the proof is the determination of the homology groups of \(X_{A,k}\) with field coefficients. In particular it is shown that if \(|A| \le k\) then \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)      

Frequent Hypercyclicity of Random Entire Functions for the Differentiation Operator

We study the random entire functions defined as power series \(f(z) = \sum _{n=0}^\infty (X_n/n!) z^n\) with independent and identically distributed coefficients \((X_n)\) and show that, under very weak assumptions, they are frequently hypercyclic for the differentiation operator \(D: H({\mathbb {C}}) \rightarrow H({\mathbb {C}}),\,f \mapsto Df = f'\) . This gives a very simple probabilistic construction of \(D\) -frequently hypercyclic functions in \(H({\mathbb {C}})\) . Moreover we show that, under more restrictive assumptions on the distribution of the \((X_n)\) , these random entire functions have a growth rate that differs from the slowest growth rate possible for \(D\) -frequently hypercyclic entire functions at most by a factor of a power of a logarithm.     

Carleson Type Measures for Fock–Sobolev Spaces

We describe the \((p,q)\) Fock–Carleson measures for weighted Fock–Sobolev spaces in terms of the objects \((s,t)\) -Berezin transforms, averaging functions, and averaging sequences on the complex space \(\mathbb{C }^n\) . The main results show that while these objects may have growth not faster than polynomials to induce the \((p,q)\) measures for \(q\ge p\) , they should be of \(L^{p/(p-q)}\) integrable against a weight of polynomial growth for \(q<p\) . As an application, we characterize the bounded and compact weighted composition operators on the Fock–Sobolev spaces in terms of certain Berezin type integral transforms on \(\mathbb{C }^n\) . We also obtained estimation results for the norms and essential norms of the operators in terms of the integral transforms. The results obtained unify and extend a number of other results in the area.