where L is the left regular representation of G. From this it can be deduced that for any v in V and f in C c (G) the vector π(f) is smooth, and more precisely that if X lies in U(g) then π(X)π(f)v = π(LXf)v. This implies that V ∞ is dense in V , since if {fn} is a Dirac sequence on G then π(fn)v → v. The subspace of V ∞ spanned by the π(f)v with f in C c (G) is called the Gårding subspace of V...

Call a function f : F 2 → {0, 1} odd-cycle-free if there are no x1, . . . , xk ∈ Fn2 with k an odd integer such that f(x1) = · · · = f(xk) = 1 and x1 + · · · + xk = 0. We show that one can distinguish odd-cycle-free functions from those ǫ-far from being odd-cycle-free by making poly(1/ǫ) queries to an evaluation oracle. To obtain this result, we use connections between basic Fourier analysis an...

Abstract A topological space is iso-dense if it has a dense set of isolated points, and scattered each its non-empty subspaces an point. In $$\textbf{ZF}$$ ZF (i.e. Zermelo–Fraenkel theory without the Axiom Choice ( $$\textbf{AC}$$ AC )), basic properties spaces are investigated. new permutat...

This work studies two interrelated problems online robust PCA (RPCA) and online lowrank matrix completion (MC). In recent work by Candès et al., RPCA has been defined as a problem of separating a low-rank matrix (true data), L := [`1, `2, . . . `t, . . . , `tmax ] and a sparse matrix (outliers), S := [x1, x2, . . . xt, . . . , xtmax ] from their sum, M := L + S. Our work uses this definition of...

We prove there is a countable dense homogeneous subspace of R of size א1. The proof involves an absoluteness argument using an extension of the Lω1ω(Q) logic obtained by adding predicates for Borel sets. A separable topological space X is countable dense homogeneous (CDH) if, given any two countable dense subsets D and D′ of X, there is a homeomorphism h of X such that h[D] = D′. The main purpo...

We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X −→ X such that the set A = {x ∈ X : ||Rx|| → ∞} is non-empty and nowhere dense in X. Moreover, if x ∈ X \ A then some subsequence of (Rx)n=1 converges weakly to x. This answers in the negative a recent conjecture of Prǎjiturǎ. The result can be extended to any Banach s...

This paper designs and evaluates a practical algorithm, called Prac-ReProCS, for recovering a time sequence of sparse vectors St and a time sequence of dense vectors Lt from their sum, Mt := St + Lt, when any subsequence of the Lt’s lies in a slowly changing low-dimensional subspace. A key application where this problem occurs is in video layering where the goal is to separate a video sequence ...

In this article, the separability of real normed spaces and its properties are mainly formalized. In the first section, it is proved that a real normed subspace is separable if it is generated by a countable subset. We used here the fact that the rational numbers form a dense subset of the real numbers. In the second section, the basic properties of the separable normed spaces are discussed. It...

The rational QZ method generalizes the by implicitly supporting subspace iteration. In this paper we extend introducing shifts and poles of higher multiplicity in Hessenberg pencil, which is a pencil consisting two matrices. result multishift, multipole iteration on block pencils allows one to stick real arithmetic for input pencil. combination with optimally packed aggressive early deflation a...