Scalar Form Research Articles

Truncation of the series expressions in the advanced ENZ-theory of diffraction integrals

The point-spread function (PSF) is used in optics for design and assessment of the imaging capabilities of an optical system. It is therefore of vital importance that this PSF can be calculated fast and accurately. In the past 12 years, the Extended Nijboer-Zernike (ENZ) approach has been developed for the purpose of semi-analytic evaluation of the PSF, for circularly symmetric optical systems, in the focal region. In the earliest ENZ-years, the Debye approximation of the diffraction integral, by which the PSF is given, was considered for the very basic situation of a low-NA optical system and relatively small defocus values, so that a scalar treatment was allowed with a focal factor comprising a quadratic function in the exponential. At present, the ENZ-method allows calculation of the PSF in low- and high-NA cases, in scalar form and for vector fields (including polarization), for large wave-front aberrations, including amplitude non-uniformities, using a quasi-spherical phase focal factor in a virtually unlimited focal range around the focal plane, and no limitations in the off-axis direction. Additionally, the application range of the method has been broadened and generalized to the calculation of aerial images of extended objects by including the finite distance of the object to the entrance pupil. Also imaging into a multi-layer is now possible by accounting for both forward and backward propagation in the layers. In the advanced ENZ-approach, the generalized, complex-valued pupil function is developed into a series of Zernike circle polynomials, with exponential azimuthal dependence (having cosine/sine azimuthal dependence as special cases). For each Zernike term, the diffraction integral reduces after azimuthal integration to an integral that can be expressed as an infinite double series involving spherical Bessel functions, accounting for the parameters of the optical system and the defocus value, and Jinc functions comprising the radial off-axis value. The contribution of the present paper is the formulation of truncation rules for these double series expressions, with a general rule valid for all circle polynomials at the same time, and a dedicated rule that takes into account the degree and the azimuthal order of the involved circle polynomials to significantly reduce computational cost in specific cases. The truncation rules are based on effective bounds and asymptotics (of the Debye type) for the mentioned spherical Bessel functions and Jinc functions, and show feasibility of computation of practically all diffraction integrals that one encounters in the ENZ-practice. Thus it can be said that the advanced ENZ-theory is more or less completed from the computational point of view by the achievements of the present paper.

STUDY OF THE RARE DECAY K± → π±γγ AND HIGH PRECISION MEASUREMENT OF THE FORM FACTORS OF THE SEMILEPTONIC DECAYS K± → π0ℓ±ν

The NA48/2 experiment presents preliminary measurements of the form factors of the charged kaon semileptonic decays. The Ke3 and Kμ3 results are in good agreement allowing for a combined result which matches the precision of the current world average on the vector and scalar form factors. Recent study of the rare decay K± → π±γγ from the NA48/2 and NA62 data samples is also reported.

B s → K (*)ℓ$ \overline{\nu} $, angular analysis, S-wave contributions and |V ub |

Abstract We analyse the $ \overline{B}_s^0\to {K^{+}}{l^{-}}\overline{\nu} $ and $ \overline{B}_s^0\to {K^{*+ }}\left( {\to K\pi } \right){\ell^{-}}\overline{\nu} $ decays that are valuable for extracting the CKM matrix element |V ub|. We calculate the differential and integrated partial widths in units of |V ub|2 based on various calculations of hadronic form factors and in particular the latest Lattice QCD calculation of the B s → K * form factors. For the decay $ \overline{B}_s^0\to K\pi \ell \overline{\nu} $ , we formulate the general angular distributions with the inclusion of the various partial-wave Kπ contributions. Using the results for the Kπ scalar form factor calculated from unitarized chiral perturbation theory, we explore the S-wave effects on angular distribution variables and demonstrate that they may not be negligible, considering the high precision expected in future measurements. We also briefly discuss the impact of the S-wave ππ contributions in the $ {B^{-}}\to {\pi^{+}}{\pi^{-}}\ell \overline{v} $ decay and provide estimates for the mode $ {B^{-}}\to {K^{+}}{K^{-}}\ell \overline{\nu} $ . The studies of these channels in future can not only be used to determine |V ub|, but may also provide valuable information on the Kπ and ππ phase shifts.

Analysis of the pion scalar form factor provides model independent values off0(500) andf0(980) meson parameters

An existence of the scalar meson f0(500) is unambiguously confirmed by the pion scalar form factor analysis. The same is concerned also of the f0(980) scalar meson, though it is placed on the tail of the elastic region to be investigated in the analysis under consideration, therefore with less precise parameters values.

Gradient Models in Molecular Biophysics: Progress, Challenges, Opportunities.

In the interest of developing a bridge between researchers modeling materials and those modeling biological molecules, we survey recent progress in developing nonlocal-dielectric continuum models for studying the behavior of proteins and nucleic acids. As in other areas of science, continuum models are essential tools when atomistic simulations (e.g. molecular dynamics) are too expensive. Because biological molecules are essentially all nanoscale systems, the standard continuum model, involving local dielectric response, has basically always been dubious at best. The advanced continuum theories discussed here aim to remedy these shortcomings by adding features such as nonlocal dielectric response, and nonlinearities resulting from dielectric saturation. We begin by describing the central role of electrostatic interactions in biology at the molecular scale, and motivate the development of computationally tractable continuum models using applications in science and engineering. For context, we highlight some of the most important challenges that remain and survey the diverse theoretical formalisms for their treatment, highlighting the rigorous statistical mechanics that support the use and improvement of continuum models. We then address the development and implementation of nonlocal dielectric models, an approach pioneered by Dogonadze, Kornyshev, and their collaborators almost forty years ago. The simplest of these models is just a scalar form of gradient elasticity, and here we use ideas from gradient-based modeling to extend the electrostatic model to include additional length scales. The paper concludes with a discussion of open questions for model development, highlighting the many opportunities for the materials community to leverage its physical, mathematical, and computational expertise to help solve one of the most challenging questions in molecular biology and biophysics.

Model independent [formula omitted] and [formula omitted] meson parameters by pion scalar form factor analysis

The f0(500) and f0(980) meson parameters are determined by the pion scalar form factor analysis, in which all existing S-wave iso-scalar ππ-scattering phase shift data in the elastic region are utilized.

Lqbounds on restrictions of spectral clusters to submanifolds for low regularity metrics

We prove $L^q$ bounds on the restriction of spectral clusters to submanifolds in Riemannian manifolds equipped with metrics of $C^{1,\alpha}$ regularity for $0 \leq \alpha \leq 1$. Our results allow for Lipschitz regularity when $\alpha =0$, meaning they give estimates on manifolds with boundary. When $0< \alpha \leq 1$, the scalar second fundamental form for a codimension 1 submanifold can be defined, and we show improved estimates when this form is negative definite. This extends results of Burq-G\'erard-Tzvetkov and Hu to manifolds with low regularity metrics.

Chiral dynamics and S-wave contributions in semileptonic B decays

The flavor-changing neutral current process b → sl + l − is beneficial to testing the standard model and hunting for new physics scenarios. In exclusive decay modes like B → K * (892)l + l −, the S-wave effects may not be negligible and thus have to be reliably estimated. Using the scalar form factors derived from dispersion relations in two channels and matched to Chiral Perturbation Theory, we investigate the S-wave contributions in $ {{\overline{B}}^0}\to {K^{-}}{\pi^{+}}{l^{+}}{l^{-}} $ , with the Kπ invariant mass lying in the vicinity of the mass of K * (892), and B s → K − K + l + l − with m KK ~ m ϕ . We find that the S-wave will modify differential decay widths by about 10% in the process of $ {{\overline{B}}^0}\to {K^{-}}{\pi^{+}}{l^{+}}{l^{-}} $ and about 5% in B s → K − K + l + l −. A forward-backward asymmetry for the charged kaon in the final state arises from the interference between the S-wave and P-wave contributions. The measurement of this asymmetry offers a new way to determine the variation of the Kπ S-wave phase versus the invariant mass.

Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations

We establish nonlinear stability and asymptotic behavior of traveling periodic waves of viscous conservation laws under localized perturbations or nonlocalized perturbations asymptotic to constant shifts in phase, showing that long-time behavior is governed by an associated second-order formal Whitham modulation system. A key point is to identify the way in which initial perturbations translate to initial data for this formal system, a task accomplished by detailed estimates on the linearized solution operator about the background wave. Notably, our approach gives both a common theoretical treatment and a complete classification in terms of “phase-coupling” or “-decoupling” of general systems of conservation or balance laws, encompassing cases that had previously been studied separately or not at all. At the same time, our refined description of solutions gives the new result of nonlinear asymptotic stability with respect to localized perturbations in the phase-decoupled case, further distinguishing behavior in the different cases. An interesting technical aspect of our analysis is that for systems of conservation laws the Whitham modulation description is of system rather than scalar form, as a consequence of which renormalization methods such as have been used to treat the reaction-diffusion case in general do not seem to apply.

Leading logarithms in N-flavour mesonic Chiral Perturbation Theory

We extend earlier work on leading logarithms in the massive nonlinear O(n) sigma model to the case of SU(N)×SU(N)/SU(N) which coincides with mesonic Chiral Perturbation Theory for N flavours of light quarks. We discuss the leading logarithms for the mass and decay constant to six loops and for the vacuum expectation value 〈q¯q〉 to seven loops. For dynamical quantities the expressions grow extremely large much faster such that we only quote the leading logarithms to five loops for the vector and scalar form factor and for meson–meson scattering. The last quantity we consider is the vector–vector to meson–meson amplitude where we quote results up to four loops for a subset of quantities, in particular for the pion polarizabilities. As a side result we provide an elementary proof that the factors of N appearing at each loop order are odd or even depending on the order and the remaining traces over external flavours.

Seismic and aseismic deformation along the East African Rift System from a reanalysis of the GPS velocity field of Africa

The improvement of the geodetic coverage within the African Plate over the last decade together with an extended GPS position time-series allows improved accuracy in determining the velocity field than prior geodetic studies. Using this new velocity field of the whole African continent, the best model proposed here remains consistent with previous studies including the existence of two small plates along the East African Rift System (EARS, Victoria and Rovuma). We focus specifically on the velocities along this plate boundary by estimating both the geodetic and the seismic moment rate. Whereas we use a scalar form of the Kostrov relation to calculate the geodetic moment rate, the seismic moment rate is obtained by integrating the cumulative truncated Gutenberg–Richter earthquake distribution of local events in the 39-yr-long worldwide catalogue, using a maximum likelihood method. This statistical method allows us to take into account the probable incompleteness of the existing catalogue and to assume the seismic moment rate calculated from this short catalogue to be representative of the long-term seismic deformation. The comparison of geodetic and seismic energy release sheds light on the variations of mechanical behaviour related to intracontinental extension along the EARS. The southward increase, observed along the rift, of the proportion of geodetic moment seismically accommodated suggests a significant control of the thermal structure associated with different states of rifting evolution.

Optimal FPGA implementation of CL multiwavelets architecture for signal denoising application

Wavelet transform is considered one of the efficient transforms of this decade for real time signal processing. Due to implementation constraints scalar wavelets do not possess the properties such as compact support, regularity, orthogonality and symmetry, which are desirable qualities to provide a good signal to noise ratio (SNR) in case of signal denoising. This leads to the evolution of the new dimension of wavelet called ‘multiwavelets’, which possess more than one scaling and wavelet filters. The architecture implementation of multiwavelets is an emerging area of research. In real time, the signals are in scalar form, which demands the processing architecture to be scalar. But the conventional Donovan Geronimo Hardin Massopust (DGHM) and Chui-Lian (CL) multiwavelets are vectored and are also unbalanced. In this article, the vectored multiwavelet transforms are converted into a scalar form and its architecture is implemented in FPGA (Field Programmable Gate Array) for signal denoising application. The architecture is compared with DGHM multiwavelets architecture in terms of several objective and performance measures. The CL multiwavelets architecture is further optimised for best performance by using DSP48Es. The results show that CL multiwavelet architecture is suited better for the signal denoising application.

Scalar and Pseudo-Scalar Form Factors in Electro- and Weak-Production of Pion Near Threshold

We investigated pion production near threshold by the weak current in terms of multipole amplitudes. By exploiting the chiral Ward identity based on the QCD Lagrangian, we derived relevant multipole amplitudes in closed forms and presented their numerical results. In the amplitudes, scalar and pseudo scalar (PS) form factors, which represent the scalar and the PS quark density distributions, manifest by themselves. We applied these amplitudes to the cross sections for the weak- and electro-production near threshold. Both pion and PS form factor contributions are shown to account for the t-channel contribution in the charged pion electro-production near threshold. The asymmetry on the pion production by the neutrino and anti-neutrino is also discussed with their longitudinal and transverse cross sections.

PRINCIPLE OF MINIMUM EXTENT IN SPATIAL SPECTRUM EXTRAPOLATION PROBLEMS OF COMPLEX-VALUED SOURCES

This paper considers a problem of spatial spectrum extrapolation for complex-valued sources, whose position is a priori unknown. This problem was solved using the principle of minimum spatial extent of the radiation source. The quasi-extent functional is being considered in its scalar and vector form, which allowed to formulate the respective problems of non-linear optimization. The gradient optimization approach is suggested as a solution. Numerical simulation was performed for a single and five complex-valued sources

ChPT calculations of pionic formfactors

An overview on chiral perturbation theory calculations of form factors is presented. The main focus is given on the form factors related to the lightest meson, pion, namely: pion decay constant, pion vector and scalar form factor, radiative pion decay and transition form factor. A pure calculation within the effective theory can be extended using further methods, as resonance chiral theory and leading logarithm calculations.

Morpho-Taxonomic Studies on the Genus Spirogyra Link (Cholorophyta) Occuring in Fresh Water Bodies of Jammu, Jammu And Kashmir

Sixteen species of Spirogyra Link 1820 (Zygnemophyceae, Chlorophyceae) were collected during 2008 to 2010 from different freshwater habitats of three districts of Jammu province viz., Samba district, Jammu district and Udhampur district. They were taxonomically determined on the basis of vegetative structure and reproductive structure. Their reproduction was mostly studied during winters and spring seasons. Both lateral conjugation and scalar form conjugation were observed. The scalariform conjugation was most common type of conjugation method among species. S. calcarea, S rectispire, S submarina, S hollandiae and S buchetii were taxonomically determined and have been described for first time in India and Jammu. Their reproduction was observed to occur mostly in winter and spring season. All sixteen species were found abundantly in both lentic and lotic water habitat.

On the determinant representations of Gaudin models’ scalar products and form factors

We propose alternative determinant representations of certain form factors and scalar products of states in rational Gaudin models realized in terms of compact spins. We use alternative pseudo-vacuums to write overlaps in terms of partition functions with domain wall boundary conditions. Contrarily to Slavnov's determinant formulas, this construction does not require that any of the involved states be solutions to the Bethe equations; a fact that could prove useful in certain non-equilibrium problems. Moreover, by using an atypical determinant representation of the partition functions, we propose expressions for the local spin raising and lowering operators form factors which only depend on the eigenvalues of the conserved charges. These eigenvalues define eigenstates via solutions of a system of quadratic equations instead of the usual Bethe equations. Consequently, the current work allows important simplifications to numerical procedures addressing decoherence in Gaudin models.

Improved dispersive analysis of the scalar form factor of the nucleon

We present a coupled system of integral equations for the pp → ¯NN and ¯K K → ¯N N S-waves derived from Roy–Steiner equations for pion–nucleon scattering. We discuss the solution of the corresponding two-channel Muskhelishvili–Omnes problem and apply the results to a dispersive analysis of the scalar form factor of the nucleon fully including ¯KK intermediate states. In particular, we determine the corrections Ds and DD, which are needed for the extraction of the pion– nucleon s term from pN scattering, and show that the difference DD −Ds = (−1.8±0.2)MeV is insensitive to the input pN parameters.

Finite strings from non-chiral Mumford forms

We show that there is an infinite class of partition functions with world-sheet metric, space-time coordinates and first order systems, that correspond to volume forms on the moduli space of Riemann surfaces and are free of singularities at the Deligne-Mumford boundary. An example is the partition function with 4=2(c_2+c_3+c_4-c_5) space-time coordinates, a $b$-$c$ system of weight 3, one of weight 4 and a beta-gamma system of weight 5. Such partition functions are derived from the mapping of the Mumford forms to non-factorized scalar forms on M_g introduced in arXiv:1209.6049.

WIMP-nucleus scattering in chiral effective theory

We discuss long-distance QCD corrections to the WIMP-nucleon(s) interactions in the framework of chiral effective theory. For scalar-mediated WIMP-quark interactions, we calculate all the next-to-leading-order corrections to the WIMP-nucleus elastic cross-section, including two-nucleon amplitudes and recoil-energy dependent shifts to the single-nucleon scalar form factors. As a consequence, the scalar-mediated WIMP-nucleus cross-section cannot be parameterized in terms of just two quantities, namely the neutron and proton scalar form factors at zero momentum transfer, but additional parameters appear, depending on the short-distance WIMP-quark interaction. Moreover, multiplicative factorization of the cross-section into particle, nuclear and astro-particle parts is violated. In practice, while the new effects are of the natural size expected by chiral power counting, they become very important in those regions of parameter space where the leading order WIMP-nucleus amplitude is suppressed, including the so-called "isospin-violating dark matter" regime. In these regions of parameter space we find order-of-magnitude corrections to the total scattering rates and qualitative changes to the shape of recoil spectra.