In the last lecture we discussed how distributional JL implies Gordon's theorem, and began our discussion of sparse JL. We wrote Πx 2 = σ T A T x A x σ and bounded the expression using Hanson-Wright in terms of the Frobenius norm. In this lecture we'll bound that Frobenius norm and then discuss applications to fast nearest neighbors. Note that we defined B x = A T x A x as the center of the pro...

We provide a Frobenius type existence result for finite-dimensional invariant submanifolds for stochastic equations in infinite dimension, in the spirit of Da Prato and Zabczyk [5]. We recapture and make use of the convenient calculus on Fréchet spaces, as developed by Kriegl and Michor [16]. Our main result is a weak version of the Frobenius theorem on Fréchet spaces. As an application we char...

It is known that the Sylow subgroups of a Frobenius complement are cyclic or generalized quaternion. In this paper it is shown that there are no restrictions at all on the structure of the Sylow subgroups of the FrobeniusWielandt complements that appear in the well-known Wielandt's generalization of Frobenius' Theorem. Some examples of explicit constructions are also given. 0. Introduction Let ...

Various forms of the extension problem are discussed for linear codes de ned over nite rings. The extension theorem for symmetrized weight compositions over nite Frobenius rings is proved. As a consequence, an extension theorem for weight functions over certain nite commutative rings is also proved. The proofs make use of the linear independence of characters as well as the linear independence ...

Let R be a finite commutative ring. We prove that a MacWilliams type relation between the m-spotty weight enumerators of a linear code over R and its dual hold, if and only if, R is a Frobenius (equivalently, Quasi-Frobenius) ring, if and only if, the number of maximal ideals and minimal ideals of R are the same, if and only if, for every linear code C over R, the dual of the dual C is C itself...

The models of collective decision-making considered in this paper are nonlinear interconnected cooperative systems with saturating interactions. These systems encode the possible outcomes of a decision process into different steady states of the dynamics. In particular, they are characterized by two main attractors in the positive and negative orthant, representing two choices of agreement amon...

The classical Frobenius–Schur indicators for finite groups are character sums defined for any representation and any integer m ≥ 2. In the familiar case m = 2, the Frobenius–Schur indicator partitions the irreducible representations over the complex numbers into real, complex, and quaternionic representations. In recent years, several generalizations of these invariants have been introduced. Bu...

Different kinds of reduction for ordinary differential equations, such as λ –symmetry and σ reductions, are recovered particular cases Frobenius theorem distribution vector fields. This general approach provides some hints to tackle the reconstruction problem solve it under suitable assumptions on involved in process.

We introduce a relative Gurevich pressure for random countable topologically mixing Markov shifts. It is shown that the relative variational principle holds for this notion of pressure. We also prove a relative RuellePerron-Frobenius theorem which enables us to construct a wealth of invariant Gibbs measures for locally fiber Hölder continuous functions. This is accomplished via a new constructi...

Let X be a smooth projective variety over an algebraically field k with char(k) = p > 0 and F : X → X1 be the relative Frobenius morphism. When dim(X) = 1, we prove that F∗W is a stable bundle for any stable bundle W (Theorem 2.3). As a step to study the question for higher dimensional X , we generalize the canonical filtration (defined by Joshi-Ramanan-Xia-Yu for curves) to higher dimensional ...