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Frames of translates with prescribed fine structure in shift invariant spaces
For a given finitely generated shift invariant (FSI) subspace \cW\subset
L^2(\R^k) we obtain a simple criterion for the existence of shift generated
(SG) Bessel sequences E(\cF) induced by finite sequences of vectors \cF\in
\cW^n that have a prescribed fine structure i.e., such that the norms of the
vectors in \cF and the spectra of S_{E(\cF)} is prescribed in each fiber of
\text{Spec}(\cW)\subset \T^k. We complement this result by developing an
analogue of the so-called sequences of eigensteps from finite frame theory in
the context of SG Bessel sequences, that allows for a detailed description of
all sequences with prescribed fine structure. Then, given we characterize the finite sequences \cF\in\cW^n such
that , for , and such that the fine spectral
structure of the shift generated Bessel sequences E(\cF) have minimal spread
(i.e. we show the existence of optimal SG Bessel sequences with prescribed
norms); in this context the spread of the spectra is measured in terms of the
convex potential P^\cW_\varphi induced by \cW and an arbitrary convex
function .Comment: 31 pages. Accepted in the JFA. This revised version has several
changes in the notation and the organization of the text. There exists text
overlap with arXiv:1508.01739 in the preliminary section
Cell-like resolutions preserving cohomological dimensions
We prove that for every compactum X with dim_Z X = 2 there is a
cell-like resolution r: Z --> X from a compactum Z onto X such that dim Z <= n
and for every integer k and every abelian group G such that dim_G X = 2
we have dim_G Z <=k. The latter property implies that for every simply
connected CW-complex K such that e-dim X <= K we also have e-dim Z <= K.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-46.abs.htm
On the cohomology algebra of a fiber
Let f:E-->B be a fibration of fiber F. Eilenberg and Moore have proved that
there is a natural isomorphism of vector spaces between H^*(F;F_p) and
Tor^{C^*(B)}(C^*(E),F_p). Generalizing the rational case proved by Sullivan,
Anick [Hopf algebras up to homotopy, J. Amer. Math. Soc. 2 (1989) 417--453]
proved that if X is a finite r-connected CW-complex of dimension < rp+1 then
the algebra of singular cochains C^*(X;F_p) can be replaced by a commutative
differential graded algebra A(X) with the same cohomology. Therefore if we
suppose that f:E-->B is an inclusion of finite r-connected CW-complexes of
dimension < rp+1, we obtain an isomorphism of vector spaces between the algebra
H^*(F;F_p) and Tor^{A(B)}(A(E),F_p) which has also a natural structure of
algebra. Extending the rational case proved by Grivel-Thomas-Halperin [PP
Grivel, Formes differentielles et suites spectrales, Ann. Inst. Fourier 29
(1979) 17--37] and [S Halperin, Lectures on minimal models, Soc. Math. France
9-10 (1983)] we prove that this isomorphism is in fact an isomorphism of
algebras. In particular, $H^*(F;F_p) is a divided powers algebra and p-th
powers vanish in the reduced cohomology \mathaccent "707E {H}^*(F;F_p).Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-36.abs.htm
Conformal GUT inflation, proton lifetime and non-thermal leptogenesis
In this paper, we generalize Coleman-Weinberg (CW) inflation in grand unified
theories (GUTs) such as and by means of
considering two complex singlet fields with conformal invariance. In this
framework, inflation emerges from a spontaneously broken conformal symmetry.
The GUT symmetry implies a potential with a CW form, as a consequence of
radiative corrections. The conformal symmetry flattens the above VEV branch of
the CW potential to a Starobinsky plateau. As a result, we obtain and for e-foldings.
Furthermore, this framework allow us to estimate the proton lifetime as
years, whose decay is mediated by the superheavy
gauge bosons. Moreover, we implement a type I seesaw mechanism by weakly
coupling the complex singlet, which carries two units of lepton number, to the
three generations of singlet right handed neutrinos (RHNs). The spontaneous
symmetry breaking of global lepton number amounts to the generation of neutrino
masses. We also consider non-thermal leptogenesis in which the inflaton
dominantly decays into heavy RHNs that sources the observed baryon asymmetry.
We constrain the couplings of the inflaton field to the RHNs, which gives the
reheating temperature as GeV.info:eu-repo/semantics/publishedVersio
The results of crossbreeding between chios and the local fat-tail awassi
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