An algorithm for performing texture transfer based on an affine transformation model is presented. Image textures are modelled by placing a single patch of texture, a prototype patch, from which the texture is derived using a set of affine transformations. Overlapping target patches covering the entire image are used and a corresponding set of transformations are found to map the prototype patc...

Naively, given a field k and some n > 0, an affine algebraic set in kn is the zero set of some finite collection of polynomial equations with n variables and coefficients in k. A morphism between two affine algebraic sets is a map defined by a tuple of polynomials. Armed with Hilbert’s Nullstellensatz, it isn’t too hard to make this precise in the case that k is algebraically closed, so we begi...

The affine synthesis operator Sc = P j>0 P k∈Zd cj,kψj,k is shown to map the mixed-norm sequence space `(`) surjectively onto L(R) under mild conditions on the synthesizer ψ ∈ L(R), 1 ≤ p < ∞, with R Rd ψ dx = 1. Here ψj,k(x) = |det aj |ψ(ajx−k), and the dilation matrices aj expand, for example aj = 2I . Affine synthesis further maps a discrete mixed Hardy space `(h) onto H(R). Therefore the H-...

Let $\mathcal{C}_d\subset \mathbb{C}^{d+1}$ be the space of non-singular, univariate polynomials degree $d$. The Vi\`{e}te map $\mathscr{V} : \mathcal{C}_d \rightarrow Sym_d(\mathbb{C})$ sends a polynomial to its unordered set roots. It is classical fact that induced $\mathscr{V}_*$ at level fundamental groups realises an isomorphism between $\pi_1(\mathcal{C}_d)$ and Artin braid group $B_d$. F...

Let H0 be a selfadjoint operator such that Tr e −βH 0 is of trace class for some β < 1, and let Xǫ denote the set of ǫ-bounded forms, i.e ‖(H0+C) X(H 0 +C)‖ < C for some C > 0. Let X := Span ∪ǫ∈(0,1/2] Xǫ. Let M denote the underlying set of the quantum information manifold of states of the form ρ X = e0X , X ∈ X . We show that if Tr e0 = 1, 1. the map Φ, Φ(X) = 1 2 Tr ( e0 + e0 ) − 1 is a quant...

We prove that every mapping torus of any free group endomorphism is residually finite. We show how to use a not yet published result of E. Hrushovski to extend our result to arbitrary linear groups. The proof uses algebraic self-maps of affine spaces over finite fields. In particular, we prove that when such a map is dominant, the set of its fixed closed scheme points is Zariski dense in the af...

We describe how to compute topological objects associated to a complex polynomial map of n > 2 variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the critical values at infinity, and the Milnor numbers at infinity for all irregular fibers. Then for a family of polynomials we detect parameters where the topolo...

In this paper we show that the Euler number of the compactified Jacobian of a rational curve C with locally planar singularities is equal to the multiplicity of the δ-constant stratum in the base of a semi-universal deformation of C. In particular, the multiplicity assigned by Yau, Zaslow and Beauville to a rational curve on a K3 surface S coincides with the multiplicity of the normalisation ma...