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.     

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 \((u,\theta )\) .     

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.     

Domination and factorization theorems for positive strongly p-summing operators

The aim of this work is to contribute to the theory of \((p,q)\) -summing operators. We focus on positive \((p,q)\) -summing operators, introduced by Blasco (Collect Math 37(1):13–22, 1986). We characterize their conjugates and provide new domination/factorization theorems for these classes. As an application, it is also shown that certain known results on \((p,q)\) -concave operators from Banach lattices can be lifted to a class of \((q,p)\) -convex operators.     

Recovering a Constant in the Two-Dimensional Navier–Stokes System with No Initial Condition

We deal with the \(2D\) -Navier–Stokes system endowed with Cauchy boundary conditions, but with no initial condition. We assume that the right-hand side is of the form \(\beta f_0+f_1\) , where \(\beta \in \mathbb {R}\) is an unknown constant. To determine \(\beta \) we are given a functional involving the velocity field \(y\) . First we prove uniqueness for the pair \((y,\beta )\) , via suitable weak Carleman estimates, and then we show the locally Lipschitz-continuous dependence of \((y,\beta )\) on the data.     

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.     

Spherical functions associated with the three-dimensional sphere

In this paper, we determine all irreducible spherical functions \(\Phi \) of any \(K \) -type associated with the pair \((G,K)=(\mathrm{SO }(4),\mathrm{SO }(3))\) . This is accomplished by associating with \(\Phi \) a vector-valued function \(H=H(u)\) of a real variable \(u\) , which is analytic at \(u=0\) and whose components are solutions of two coupled systems of ordinary differential equations. By an appropriate conjugation involving Hahn polynomials, we uncouple one of the systems. Then, this is taken to an uncoupled system of hypergeometric equations, leading to a vector-valued solution \(P=P(u)\) , whose entries are Gegenbauer’s polynomials. Afterward, we identify those simultaneous solutions and use the representation theory of \(\mathrm{SO }(4)\) to characterize all irreducible spherical functions. The functions \(P=P(u)\) corresponding to the irreducible spherical functions of a fixed \(K\) -type \(\pi _\ell \) are appropriately packaged into a sequence of matrix-valued polynomials \((P_w)_{w\ge 0}\) of size \((\ell +1)\times (\ell +1)\) . Finally, we prove that \(\widetilde{P}_w={P_0}^{-1}P_w\) is a sequence of matrix orthogonal polynomials with respect to a weight matrix \(W\) . Moreover, we show that \(W\) admits a second-order symmetric hypergeometric operator \(\widetilde{D}\) and a first-order symmetric differential operator \(\widetilde{E}\) .     

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 \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)      

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.     

A New Topological Helly Theorem and Some Transversal Results

We prove that for a topological space \(X\) with the property that \( H_{*}(U)=0\) for \(*\ge d\) and every open subset \(U\) of \(X\) , a finite family of open sets in \(X\) has nonempty intersection if for any subfamily of size \(j,\,1\le j\le d+1,\) the \((d-j)\) -dimensional homology group of its intersection is zero. We use this theorem to prove new results concerning transversal affine planes to families of convex sets.     

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.     

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.     

Counting and Sampling Minimum Cuts in Genus g Graphs

Let \(G\) be a directed graph with \(n\) vertices embedded on an orientable surface of genus \(g\) with two designated vertices \(s\) and \(t\) . We show that computing the number of minimum \((s,t)\) -cuts in \(G\) is fixed-parameter tractable in \(g\) . Specifically, we give a \(2^{O(g)} n^2\) time algorithm for this problem. Our algorithm requires counting sets of cycles in a particular integer homology class. That we can count these cycles is an interesting result in itself as there are no prior results that are fixed-parameter tractable and deal directly with integer homology. We also describe an algorithm which, after running our algorithm to count minimum cuts once, can sample an \((s,t)\) -minimum cut uniformly at random in \(O(n \log n)\) time per sample.     

The extremal solution for the fractional Laplacian

We study the extremal solution for the problem \((-\Delta )^s u=\lambda f(u)\) in \(\Omega \) , \(u\equiv 0\) in \(\mathbb R ^n\setminus \Omega \) , where \(\lambda >0\) is a parameter and \(s\in (0,1)\) . We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities we prove that the extremal solution is bounded in dimensions \(n<4s\) . We also show that, for exponential and power-like nonlinearities, the extremal solution is bounded whenever \(n<10s\) . In the limit \(s\uparrow 1\) , \(n<10\) is optimal. In addition, we show that the extremal solution is \(H^s(\mathbb R ^n)\) in any dimension whenever the domain is convex. To obtain some of these results we need \(L^q\) estimates for solutions to the linear Dirichlet problem for the fractional Laplacian with \(L^p\) data. We prove optimal \(L^q\) and \(C^\beta \) estimates, depending on the value of \(p\) . These estimates follow from classical embedding results for the Riesz potential in \(\mathbb R ^n\) . Finally, to prove the \(H^s\) regularity of the extremal solution we need an \(L^\infty \) estimate near the boundary of convex domains, which we obtain via the moving planes method. For it, we use a maximum principle in small domains for integro-differential operators with decreasing kernels.     

Simplicity of normal subgroups and conjugacy class sizes

Given a finite group \(G\) which possesses a non-abelian simple normal subgroup \(N\) having exactly four \(G\) -class sizes, we prove that \(N\) is isomorphic to PSL \((2, 2^a)\) with \(a\ge 2\) . Thus, we obtain an extension for normal subgroups of the classic N. Itô’s theorem which characterizes those finite simple groups with exactly four class sizes.     

Cyclic parallel CR-submanifolds of maximal CR-dimension in a complex space form

We first classify \((2n-1)\) -dimensional cyclic parallel CR-submanifold \(M\) with CR-dimension \(n-1\) in a non-flat complex space form of constant holomorphic sectional curvature \(4c\) . Then, we prove that \(||\nabla h||^2\ge 4(n-1)c^2\) , where \(h\) is the second fundamental form on \(M\) . We also completely classify \((2n-1)\) -dimensional CR-submanifolds with CR-dimension \(n-1\) in a non-flat complex space form which satisfy the equality case of this inequality. This generalizes an inequality for real hypersurfaces in a non-flat complex space form obtained by Maeda (J Math Soc Jpn 28:529–540; 1976) and Chen et al. (Algebras Groups Geom 1:176–212; 1984) for complex projective and hyperbolic spaces, respectively.     

Some model theory of fibrations and algebraic reductions

Let \(p= \mathrm{tp }(a/A)\) be a stationary finite rank type in an arbitrary stable theory and \({\mathbb {P}}\) an \(A\) -invariant family of partial types. The following property is introduced and characterised: whenever \(c\) is definable over \((A,a)\) and \(a\) is not algebraic over \((A,c)\) , then \(\mathrm{tp }(c/A)\) is almost internal to \({\mathbb {P}}\) . The characterisation involves among other things an apparently new notion of “descent” for stationary types. Motivation comes partly from results in Sect. 2 of (Campana et al. in J Differ Geom 85(3):397–424, 2010) where structural properties of generalised hyperkähler manifolds are given. The model-theoretic results obtained here are applied back to the complex-analytic setting to prove that the algebraic reduction of a nonalgebraic (generalised) hyperkähler manifold does not descend. The results are also applied to the theory of differentially closed fields, where examples coming from differential-algebraic groups are given.     

Existence of the generalized Fučík spectrum for nonhomogeneous elliptic operators

By variational methods and Morse theory, we prove the existence of uncountably many \((\alpha ,\beta )\in \mathbb R ^2\) for which the equation \(-\mathrm{div}\, A(x, \nabla u)=\alpha u_+^{p-1} -\beta u_-^{p-1}\) in \(\Omega \) , has a sign changing solution under the Neumann boundary condition, where a map \(A\) from \(\overline{\Omega }\times \mathbb R ^N\) to \(\mathbb R ^N\) satisfying certain regularity conditions. As a special case, the above equation contains the \(p\) -Laplace equation. However, the operator \(A\) is not supposed to be \((p-1)\) -homogeneous in the second variable. In particular, it is shown that generally the Fu?ík spectrum of the operator \(-\mathrm{div}\, A(x, \nabla u)\) on \(W^{1,p}(\Omega )\) contains some open unbounded subset of \(\mathbb R ^2\) .     

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.     

Continuous pursuit curves on cat (K) spaces

We prove theorems of existence, uniqueness and rate of convergence for continuous pursuit curves in cat \((K)\) spaces. We prove that these pursuit curves have a regularity property that serves as a replacement for \(C^{1,1}\) regularity in smooth spaces.