On a sharp lower bound on the blow-up rate for the critical nonlinear Schrödinger equation

We consider the critical nonlinear Schrödinger equation with initial condition in the energy space and study the dynamics of finite time blow-up solutions. In an earlier sequence of papers, the authors established for a certain class of initial data on the basis of dispersive properties in a sharp and stable upper bound on the blow-up rate: .

In an earlier paper, the authors then addressed the question of a lower bound on the blow-up rate and proved for this class of initial data the nonexistence of self-similar solutions, that is,

In this paper, we prove the sharp lower bound

by exhibiting the dispersive structure in the scaling invariant space for this log-log regime. In addition, we will extend to the pure energy space a dynamical characterization of the solitons among the zero energy solutions.

     

A conjectural extension of Hecke’s converse theorem

By using specific subsequences of two different types of generalized Stern polynomials, we obtain several related classes of finite and infinite continued fractions involving a single term \(z^{t^j}\) in their partial numerators, where z is a complex variable and t is a positive integer. This approach is extended to other, sparser, subsequences of Stern polynomials, based on certain Lucas functions; this then leads to further infinite classes of continued fractions.     

Maximal exponents of polyhedral cones (I)

Let K be a proper (i.e., closed, pointed, full convex) cone in Rn. An n×n matrix A is said to be K-primitive if there exists a positive integer k such that ; the least such k is referred to as the exponent of A and is denoted by γ(A). For a polyhedral cone K, the maximum value of γ(A), taken over all K-primitive matrices A, is called the exponent of K and is denoted by γ(K). It is proved that if K is an n-dimensional polyhedral cone with m extreme rays then for any K-primitive matrix A, γ(A)?(mA−1)(m−1)+1, where mA denotes the degree of the minimal polynomial of A, and the equality holds only if the digraph (E,P(A,K)) associated with A (as a cone-preserving map) is equal to the unique (up to isomorphism) usual digraph associated with an m×m primitive matrix whose exponent attains Wielandt's classical sharp bound. As a consequence, for any n-dimensional polyhedral cone K with m extreme rays, γ(K)?(n−1)(m−1)+1. Our work answers in the affirmative a conjecture posed by Steve Kirkland about an upper bound of γ(K) for a polyhedral cone K with a given number of extreme rays.     

Decompositions for edge-coloring join graphs and cobipartite graphs

An edge-coloring is an association of colors to the edges of a graph, in such a way that no pair of adjacent edges receive the same color. A graph G is Class 1 if it is edge-colorable with a number of colors equal to its maximum degree Δ(G). To determine whether a graph G is Class 1 is NP-complete [I. Holyer, The NP-completeness of edge-coloring, SIAM J. Comput. 10 (1981) 718-720]. First, we propose edge-decompositions of a graph G with the goal of edge-coloring G with Δ(G) colors. Second, we apply these decompositions for identifying new subsets of Class 1 join graphs and cobipartite graphs. Third, the proposed technique is applied for proving that the chromatic index of a graph is equal to the chromatic index of its semi-core, the subgraph induced by the maximum degree vertices and their neighbors. Finally, we apply these decomposition tools to a classical result [A.J.W. Hilton, Z. Cheng, The chromatic index of a graph whose core has maximum degree 2, Discrete Math. 101 (1992) 135-147] that relates the chromatic index of a graph to its core, the subgraph induced by the maximum degree vertices.     

Time-Ordering and a Generalized Magnus Expansion

Both the classical time-ordering and the Magnus expansion are well known in the context of linear initial value problems. Motivated by the noncommutativity between time-ordering and time derivation, and related problems raised recently in statistical physics, we introduce a generalization of the Magnus expansion. Whereas the classical expansion computes the logarithm of the evolution operator of a linear differential equation, our generalization addresses the same problem, including, however, directly a non-trivial initial condition. As a by-product we recover a variant of the time-ordering operation, known as ${\mathsf{T}^\ast}$ -ordering. Eventually, placing our results in the general context of Rota–Baxter algebras permits us to present them in a more natural algebraic setting. It encompasses, for example, the case where one considers linear difference equations instead of linear differential equations.     

Possible inertias for the hermitian part of A + BX

Given AεM_n (C) and BεM _n,k (C) all possible inertias occurring in the Hermitian part of A+BX are determined as X runs over M_k,n(C).     

Packing Triangles in Regular Tournaments

We prove that a regular tournament with n vertices has more than pairwise arc‐disjoint directed triangles. On the other hand, we construct regular tournaments with a feedback arc set of size less than , so these tournaments do not have pairwise arc‐disjoint triangles.     

Spontaneous Resolution by Crystallization of an Octanuclear Iron(III) Complex Using Only Racemic Reagents

The P and M enantiomers of the octanuclear [Fe₈(μ₄‐O)₄(μ‐4‐Cl‐pz)₁₂Cl₄] complex, having T symmetry, were resolved by temporary substitution of chloride ligands by racemic 4‐^sBu‐phenolates and subsequent crystallization, where the (S)‐ and (R)‐phenolates coordinate selectively to the M and P complexes, respectively. The complexes were characterized by circular dichroism analysis and X‐ray structure determination. This work constitutes a rare example of enantiomeric recognition resulting in spontaneous resolution upon crystallization.     

Stable,Reactive, and Orthogonal Tetrazines: Dispersion Forces Promote the Cycloaddition with Isonitriles

The isocyano group is a structurally compact bioorthogonal functional group that reacts with tetrazines under physiological conditions. Now it is shown that bulky tetrazine substituents accelerate this cycloaddition. Computational studies suggest that dispersion forces between the isocyano group and the tetrazine substituents in the transition state contribute to the atypical structure–activity relationship. Stable asymmetric tetrazines that react with isonitriles at rate constants as high as 57 L mol^?1 s^?1 were accessible by combining bulky and electron‐withdrawing substituents. Sterically encumbered tetrazines react selectively with isonitriles in the presence of strained alkenes/alkynes, which allows for the orthogonal labeling of three proteins. The established principles will open new opportunities for developing tetrazine reactants with improved characteristics for diverse labeling and release applications with isonitriles.