An Approximation Problem in Multiplicatively Invariant Spaces

Let $\mathcal{H}$ be Hilbert space and $(\Omega,\mu)$ a $\sigma$-finite measure space. Multiplicatively invariant (MI) spaces are closed subspaces of $L^2(\Omega, \mathcal{H})$ that are invariant under point-wise multiplication by functions in a fix subset of $L^{\infty}(\Omega).$ Given a finite set of data $\mathcal{F}\subseteq L^2(\Omega, \mathcal{H}),$ in this paper we prove the existence and construct an MI space $M$ that best fits $\mathcal{F}$, in the least squares sense. MI spaces are related to shift invariant (SI) spaces via a fiberization map, which allows us to solve an approximation problem for SI spaces in the context of locally compact abelian groups. On the other hand, we introduce the notion of decomposable MI spaces (MI spaces that can be decomposed into an orthogonal sum of MI subspaces) and solve the approximation problem for the class of these spaces. Since SI spaces having extra invariance are in one-to-one relation to decomposable MI spaces, we also solve our approximation problem for this class of SI spaces. Finally we prove that translation invariant spaces are in correspondence with totally decomposable MI spaces.Comment: 18 pages, To appear in Contemporary Mathematic

Luminance-dependent hue shift in protanopes

For normal trichromats, the hue of a light can change as its luminance varies. This Bezold-Brücke (B-B) hue shift is commonly attributed to nonlinearity in the blue–yellow opponent system. In the present study, we questioned whether protanopes experience analogous changes. Two protanopes (Ps) viewed spectral lights at six luminance levels across three log steps. Two normal trichromats (NTs) were tested for comparison. A variant of the color-naming method was used, with an additional “white” term. To overcome the difficulty of Ps’ idiosyncratic color naming, we converted color-naming functions into individual color spaces, by way of interstimulus similarities and multidimensional scaling (MDS). The color spaces describe each stimulus in terms of spatial coordinates, so that hue shifts are measured geometrically, as displacements along specific dimensions. For the NTs, a B-B shift derived through MDS agreed well with values obtained directly by matching color-naming functions. A change in color appearance was also observed for the Ps, distinct from that in perceived brightness. This change was about twice as large as the B-B shift for NTs and combined what the latter would distinguish as hue and saturation shifts. The protanopic analogue of the B-B shift indicates that the blue–yellow nonlinearity persists in the absence of a red–green signal. In addition, at mesopic levels (# 38 td), the Ps’ MDS solution was two dimensional at longer wavelengths, suggesting rod input. Conversely, at higher luminance levels (76 td–760 td) the MDS solution was essentially one dimensional, placing a lower limit on S-cone input at longer wavelengths

Algebra properties for Sobolev spaces- Applications to semilinear PDE's on manifolds

In this work, we aim to prove algebra properties for generalized Sobolev spaces $W^{s,p} \cap L^\infty$ on a Riemannian manifold, where $W^{s,p}$ is of Bessel-type $W^{s,p}:=(1+L)^{-s/m}(L^p)$ with an operator $L$ generating a heat semigroup satisfying off-diagonal decays. We don't require any assumption on the gradient of the semigroup. To do that, we propose two different approaches (one by a new kind of paraproducts and another one using functionals). We also give a chain rule and study the action of nonlinearities on these spaces and give applications to semi-linear PDEs. These results are new on Riemannian manifolds (with a non bounded geometry) and even in the Euclidean space for Sobolev spaces associated to second order uniformly elliptic operators in divergence form.Comment: 29 page

Local tropical linear spaces

In this paper we study general tropical linear spaces locally: For any basis B of the matroid underlying a tropical linear space L, we define the local tropical linear space L_B to be the subcomplex of L consisting of all vectors v that make B a basis of maximal v-weight. The tropical linear space L can then be expressed as the union of all its local tropical linear spaces, which we prove are homeomorphic to Euclidean space. Local tropical linear spaces have a simple description in terms of polyhedral matroid subdivisions, and we prove that they are dual to mixed subdivisions of Minkowski sums of simplices. Using this duality we produce tight upper bounds for their f-vectors. We also study a certain class of tropical linear spaces that we call conical tropical linear spaces, and we give a simple proof that they satisfy Speyer's f-vector conjecture.Comment: 13 pages, 1 figure. Some results are stated in a bit more generality. Minor corrections were also mad

The Hodge bundle on Hurwitz spaces

In 2009 Kokotov, Korotkin and Zograf gave a formula for the class of the Hodge bundle on the Hurwitz space of admissible covers of genus g and degree d of the projective line. They gave an analytic proof of it. In this note we give an algebraic proof and an extension of the result.Comment: 8 pages; misprints correcte