Representation and Characterization of Non-Stationary Processes by Dilation Operators and Induced Shape Space Manifolds

We have introduce a new vision of stochastic processes through the geometry induced by the dilation. The dilation matrices of a given processes are obtained by a composition of rotations matrices, contain the measure information in a condensed way. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, it coincides with the Naimark Dilation and only one rotation matrix is computed, when the process is non-stationary, a set of rotation matrices are computed. In particular, the periodicity of the correlation function that may appear in some classes of signal is transmitted to the set of dilation matrices. These rotation matrices, which can be arbitrarily close to each other depending on the sampling or the rescaling of the signal are seen as a distinctive feature of the signal. In order to study this sequence of matrices, and guided by the possibility to rescale the signal, the correct geometrical framework to use with the dilation's theoretic results is the space of curves on manifolds, that is the set of all curve that lies on a base manifold. To give a complete sight about the space of curve, a metric and the derived geodesic equation are provided. The general results are adapted to the more specific case where the base manifold is the Lie group of rotation matrices. The notion of the shape of a curve can be formalized as the set of equivalence classes of curves given by the quotient space of the space of curves and the increasing diffeomorphisms. The metric in the space of curve naturally extent to the space of shapes and enable comparison between shapes.Comment: 19 pages, draft pape

Proton Decay, Yukawa Couplings and Underlying Gauge Symmetry in String Theory

In string theory, massless particles often originate from a symmetry breaking of a large gauge symmetry G to its subgroup H. The absence of dimension-4 proton decay in supersymmetric theories suggests that (\bar{D},L) are different from \bar{H}(\bar{\bf 5}) in their origins. In this article, we consider a possibility that they come from different irreducible components in $\mathfrak{g}/\mathfrak{h}$. Requiring that all the Yukawa coupling constants of quarks and leptons be generated from the super Yang--Mills interactions of G, we found in the context of Georgi--Glashow H=SU(5) unification that the minimal choice of G is E_7 and E_8 is the only alternative. This idea is systematically implemented in Heterotic String, M theory and F theory, confirming the absence of dimension 4 proton decay operators. Not only H=SU(5) but also G constrain operators of effective field theories, providing non-trivial information.Comment: 73 page

Off-the-Wall Higgs in the Universal Randall-Sundrum Model

We outline a consistent Randall-Sundrum (RS) framework in which a fundamental 5-dimensional Higgs doublet induces electroweak symmetry breaking (EWSB). In this framework of a warped Universal Extra Dimension, the lightest Kaluza-Klein (KK) mode of the bulk Higgs is tachyonic leading to a vacuum expectation value (vev) at the TeV scale. The consistency of this picture imposes a set of constraints on the parameters in the Higgs sector. A novel feature of our scenario is the emergence of an adjustable bulk profile for the Higgs vev. We also find a tower of non-tachyonic Higgs KK modes at the weak scale. We consider an interesting implementation of this ``Off-the-Wall Higgs'' mechanism where the 5-dimensional curvature-scalar coupling alone generates the tachyonic mode responsible for EWSB. In this case, additional relations among the parameters of the Higgs and gravitational sectors are established. We discuss the experimental signatures of the bulk Higgs in general, and those of the ``Gravity-Induced'' EWSB in particular.Comment: 27 pages, 4 figure

Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems

We consider the Gopakumar-Ooguri-Vafa correspondence, relating ${\rm U}(N)$ Chern-Simons theory at large $N$ to topological strings, in the context of spherical Seifert 3-manifolds. These are quotients $\mathbb{S}^{\Gamma} = \Gamma\backslash\mathbb{S}^3$ of the three-sphere by the free action of a finite isometry group. Guided by string theory dualities, we propose a large $N$ dual description in terms of both A- and B-twisted topological strings on (in general non-toric) local Calabi-Yau threefolds. The target space of the B-model theory is obtained from the spectral curve of Toda-type integrable systems constructed on the double Bruhat cells of the simply-laced group identified by the ADE label of $\Gamma$. Its mirror A-model theory is realized as the local Gromov-Witten theory of suitable ALE fibrations on $\mathbb{P}^1$, generalizing the results known for lens spaces. We propose an explicit construction of the family of target manifolds relevant for the correspondence, which we verify through a large $N$ analysis of the matrix model that expresses the contribution of the trivial flat connection to the Chern-Simons partition function. Mathematically, our results put forward an identification between the $1/N$ expansion of the $\mathrm{sl}_{N + 1}$ LMO invariant of $\mathbb{S}^\Gamma$ and a suitably restricted Gromov-Witten/Donaldson-Thomas partition function on the A-model dual Calabi-Yau. This $1/N$ expansion, as well as that of suitable generating series of perturbative quantum invariants of fiber knots in $\mathbb{S}^\Gamma$, is computed by the Eynard-Orantin topological recursion.Comment: 65 page

Nanoscale roughness and morphology affect the IsoElectric Point of titania surfaces

We report on the systematic investigation of the role of surface nanoscale roughness and morphology on the charging behaviour of nanostructured titania (TiO2) surfaces in aqueous solutions. IsoElectric Points (IEPs) of surfaces have been characterized by direct measurement of the electrostatic double layer interactions between titania surfaces and the micrometer-sized spherical silica probe of an atomic force microscope in NaCl aqueous electrolyte. The use of a colloidal probe provides well-defined interaction geometry and allows effectively probing the overall effect of nanoscale morphology. By using supersonic cluster beam deposition to fabricate nanostructured titania films, we achieved a quantitative control over the surface morphological parameters. We performed a systematical exploration of the electrical double layer properties in different interaction regimes characterized by different ratios of characteristic nanometric lengths of the system: the surface rms roughness Rq, the correlation length {\xi} and the Debye length {\lambda}D. We observed a remarkable reduction by several pH units of IEP on rough nanostructured surfaces, with respect to flat crystalline rutile TiO2. In order to explain the observed behavior of IEP, we consider the roughness-induced self-overlap of the electrical double layers as a potential source of deviation from the trend expected for flat surfaces.Comment: 63 pages, including 7 figures and Supporting Informatio

Brane-localized Kinetic Terms in the Randall-Sundrum Model

We examine the effects of boundary kinetic terms in the Randall-Sundrum model with gauge fields in the bulk. We derive the resulting gauge Kaluza-Klein (KK) state wavefunctions and their corresponding masses, as well as the KK gauge field couplings to boundary fermions, and find that they are modified in the presence of the boundary terms. In particular, for natural choices of the parameters, these fermionic couplings can be substantially suppressed compared to those in the conventional Randall-Sundrum scenario. This results in a significant relaxation of the bound on the lightest gauge KK mass obtained from precision electroweak data; we demonstrate that this bound can be as low as a few hundred GeV. Due to the relationship between the lightest gauge KK state and the electroweak scale in this model, this weakened constraint allows for the electroweak scale to be near a TeV in this minimal extension of the Randall-Sundrum model with bulk gauge fields, as opposed to the conventional scenario.Comment: 24 pages, 8 figures, LaTex. Discussion and figure added addressing the effects of this analysis on the hierarch

Validation of in situ applicable measuring techniques for analysis of the water adsorption by stone

As the water adsorbing behaviour (WAB) of stone is a key factor for most degradation processes, its analysis is a decisive aspect when monitoring deterioration and past conservation treatments, or when selecting a proper conservation treatment. In this study the performance of various non-destructive methods for measuring the WAB are compared, with the focus on the effect of the variable factors of the methods caused by their specific design. The methods under study are the contact-sponge method (CSM), the Karsten tube (KT) and the Mirowski pipe (MIR). Their performance is compared with the standardized capillary rise method (CR) and the results are analysed in relation to the open porosity of different lithotypes. Furthermore the effect of practical encumbrances which could limit the application of these methods was valuated. It was found that KT and CSM have complementary fields of investigation, where CSM is capable of measuring the initial water uptake of less porous materials with a high precision, while KT was found commodious for measuring longer contact times for more porous lithotypes. MIR showed too many discommodities, leading to unreliable results. To adequately compare the results of the different methods, the size of the contact area appears to be the most influential factor, whereas the contact material and pressure on the surface do not indicate a significant influence on the results. The study of these factors is currently being extended by visualization of the water adsorption process via X-ray and neutron radiography in combination with physico-mathematical models describing the WAB