We classify Dehn surgeries on (p, q, r) pretzel knots that result in a manifold of finite fundamental group. The only hyperbolic pretzel knots that admit non-trivial finite surgeries are (−2, 3, 7) and (−2, 3, 9). Agol and Lackenby’s 6-theorem reduces the argument to knots with small indices p, q, r. We treat these using the Culler-Shalen norm of the SL(2, C)-character variety. In particular, w...

Several inviscid models in hydrodynamics and geophysics such as the incompressible Euler vorticity equations, the surface quasi-geostrophic equation and the Boussinesq equations are not known to have even local well-posedness in the corresponding borderline Sobolev spaces. Here H is referred to as a borderline Sobolev space if the L∞-norm of the gradient of the velocity is not bounded by the H-...

We calculate the R-R zero-norm states of type II string spectrum. To fit these states into the right symmetry charge parameters of the gauge transformations of the R-R tensor forms, one is forced to T-dualize some type I open string space-time coordinates and thus to introduce D-branes into the theory. We also demonstrate that the scalar T-dual R-R 10-form zero-norm state, together with the NS-...

The operator norm of an n × n complex matrix A is its norm as a linear operator on the Euclidean space I C; i.e., ‖A‖ = sup{‖Ax‖ : x ∈ I C, ‖x‖ = 1}. The Frobenius norm of A is defined as ‖A‖2 = ( ∑ i,j |aij|) 1 2 = (trA∗A) 1 2 . Both these norms are used frequently in analysis of matrices. They can also be described in terms of the singular values of A − the square roots of the eigenvalues of ...

We consider a family of Schrödinger equations with unbounded Hamiltonian quadratic nonlinearities on generic tori dimension d≥1. study the behavior high Sobolev norms Hs, s≫1, solutions initial conditions in Hs whose Hρ-Sobolev norm, 1≪ρ≪s, is smaller than ε≪1. provide control Hs-norm over time interval order O(ε−2). Due to lack conserved quantities controlling norms, key ingredient proof const...

In recent developments, norms have become important entities that are considered in agent-based systems’ designs. Norms are not only able to organize and coordinate the actions and behaviour of agents but have a direct impact on the achievement of agents’ goals. Consequently, an agent in a multiagent system requires a mechanism that detects specific norms for adoption while rejecting others. Th...

In the paper aggregations of fuzzy relations using functions of n variables are considered. After recalling properties of fuzzy relations, aggregation functions which preserve: reflexivity, irreflexivity, T -asymmetry, T -antisymmetry, symmetry, S-connectedness, T -transitivity, negative Stransitivity, T -S-semitransitivity and T -S-Ferrers property of fuzzy relations, where T is a t-norm and S...

We summarize the basic properties of continuous tensor product systems of Hilbert spaces and their role in non-commutative dynamics. These are lecture notes, not intended for publication. 1. Concrete Product Systems In these notes we assume the reader is familiar with the definition and basic properties of E0-semigroups, and has some familiarity with their role in noncommutative dynamics. Our p...

Among other things we show that for each n-tuple of positive rational numbers (a 1 ,. .. , a n) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a 1 x 1 +· · ·+a n x n = 1 with x 1 ,. .. , x n S-units are not contained in fewer than exp((4 + o(1))s 1/2 (log s) −1/2) proper linear subspaces of C n. This generalizes a result of Erd˝ os, Stewart...