scholarly journals Annihilator, Completeness and Convergence of Wavelet System

2007 ◽  
Vol 188 ◽  
pp. 59-105 ◽  
Author(s):  
Kwok-Pun Ho
Keyword(s):  
Function Space ◽  
Dual Space ◽  
Linear Span ◽  
Dual Frame ◽  
Wavelet System ◽  
One Dimensional ◽  
Trivial Intersection ◽  
The One ◽  
Definition Of ◽  
Dyadic Cubes

AbstractWe show that if is a frame and {ψＱ}Ｑ∈Q ∈ ∩ Mα(ℝn) is its dual frame (for the definition of Mα(ℝn), see Definition 2.1), where Q is the collection of dyadic cubes, then for any f ∈ S′(ℝn), there exists a sequence of polynomials, PL,L′,L″, such that(0.1) in the topology of S′(ℝn), where δ(i) = max(2i, 1). We prove this result by explicitly constructing the polynomials PL,L′,L″. Furthermore, using the above result, we assert that the linear span of the one-dimensional wavelet system is dense in a function space if and only if the dual space of this function space has an trivial intersection with the set of polynomials. This is proved by using the annihilator of the one-dimensional wavelet system.

scholarly journals Fractional negative binomial and Pólya processes

2018 ◽  
Vol 38 (1) ◽  
pp. 77-101
Author(s):  
Palaniappan Vellai Samy ◽  
Aditya Maheshwari
Keyword(s):  
Poisson Process ◽  
Negative Binomial ◽  
Random Variable ◽  
Infinitely Divisible ◽  
One Dimensional ◽  
Fractional Poisson Process ◽  
Negative Binomial Process ◽  
The One ◽  
Definition Of ◽  
Binomial Process

In this paper, we define a fractional negative binomial process FNBP by replacing the Poisson process by a fractional Poisson process FPP in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process SFPP is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.

scholarly journals Comments on open string with ‘massive’ boundary term

2020 ◽  
Vol 476 (2236) ◽  
pp. 20190748
Author(s):  
Arkady A. Tseytlin
Keyword(s):  
Scattering Amplitudes ◽  
Partition Function ◽  
Gauge Field ◽  
Path Integral ◽  
Open String ◽  
One Dimensional ◽  
Conformally Invariant ◽  
Additional Constant ◽  
The One ◽  
Definition Of

We discuss possible definition of open string path integral in the presence of additional boundary couplings corresponding to the presence of masses at the ends of the string. These couplings are not conformally invariant implying that as in a non-critical string case one is to integrate over the one-dimensional metric or reparametrizations of the boundary. We compute the partition function on the disc in the presence of an additional constant gauge field background and comment on the structure of the corresponding scattering amplitudes.

Theory of thermal explosions with simultaneous parallel reactions I. Foundations and the one-dimensional case

1984 ◽  
Vol 393 (1804) ◽  
pp. 85-100 ◽  
Keyword(s):  
Activation Energy ◽  
Dimensionless Parameter ◽  
Exothermic Reactions ◽  
One Dimensional ◽  
Temperature Excess ◽  
Parallel Reactions ◽  
The One ◽  
Definition Of ◽  
Thermal Explosions ◽  
Thermal Explosion Theory

Many real, exothermic systems involve more than one simultaneous reaction. Even when they are chemically independent, interactions must arise through their several responses to the collective generation of heat. A simple and unifying approach is possible to the behaviour of such systems below and up to criticality. It introduces a communal activation energy E as the basis for dimensionless quantities ( θ, δ, ϵ and so on) but otherwise involves only familiar ideas from basic thermal explosion theory. The definition of E is E = RT 2 d (In Z )/d T , where Z = Ʃ Z i . Here, Z is the rate of energy release per unit volume (the power density) by the whole system and Z i is the contribution of the constituent i . This enables us to define and use the conventional dimensionless parameter δ for the whole system and for its constituent reactions. We illustrate affairs by considering a pair of concurrent, exothermic reactions; heat is transferred solely by conduction towards the faces (temperature T a ) of an infinite slab of thickness 2 a and conductivity k . For a constituent reaction ( i = 1, 2 here) δ i = ( Ea 2 / k RT 2 a ) Z i ( T a ) and for the whole system δ = δ 1 + δ 2 (+...) for two (or more) reactions. We find that the condition δ > δ cr guarantees instability, where δ cr is always less than 0.878. The bounds 0.65 < δ cr < 0.878 are good enough for a substantial range of relative sizes of activation energy 0.2 < E 1 / E 2 < 5. We also pursue the problem numerically and present solutions for critical δ and critical central temperature excess over the whole composition range for a pair of simultaneous exothermic reactions.

FINITE SYSTEMS ON PHASE SPACE

2006 ◽  
Vol 20 (11n13) ◽  
pp. 1956-1967 ◽  
Author(s):  
KURT BERNARDO WOLF
Keyword(s):  
Distribution Function ◽  
Phase Space ◽  
Hamiltonian Systems ◽  
Wigner Distribution ◽  
Wigner Distribution Function ◽  
Finite Fourier Transform ◽  
One Dimensional ◽  
Finite Systems ◽  
The One ◽  
Definition Of

This contribution summarizes work on finite, non-cyclic Hamiltonian systems —in particular the one-dimensional finite oscillator—, in conjunction with a Lie algebraic definition of the (meta-) phase space of finite systems, and a corresponding Wigner distribution function for the state vectors. The consistency of this approach is important for the strategy of fractionalization of a finite Fourier transform, and the contraction of finite unitary to continuous symplectic transformations of Hamiltonian systems.

scholarly journals Thermal processes in natural gas pipeline transport

2020 ◽  
Vol 16 (3) ◽  
pp. 260-266
Author(s):  
Vladimir A. Suleymanov ◽  
◽  
Keyword(s):  
Thermal Energy ◽  
Gas Flow ◽  
Gas Pipeline ◽  
Friction Forces ◽  
One Dimensional ◽  
Natural Gas Pipeline ◽  
Longitudinal Temperature ◽  
The One ◽  
Definition Of

A commonly used premise in pipeline hydraulics where the work of friction forces performed at the movement of real gas in the gas pipeline completely turns into thermal energy is verified in the article. By means of the integral definition of Clausius entropy, it is shown that the premise of the conversion of friction forces into thermal energy of gas flow is justified with an acceptable accuracy for engineering applications in relation to the one-dimensional formulation of the task regarding the determination of the longitudinal temperature field of gas.

Effective aperture of X-ray compound refractive lenses

2017 ◽  
Vol 24 (3) ◽  
pp. 609-614 ◽  
Author(s):  
V. G. Kohn
Keyword(s):  
Integral Intensity ◽  
Two Dimensional ◽  
One Dimensional ◽  
X Ray ◽  
The Real ◽  
A Value ◽  
Effective Aperture ◽  
The One ◽  
Definition Of ◽  
Focusing Properties

A new definition of the effective aperture of the X-ray compound refractive lens (CRL) is proposed. Both linear (one-dimensional) and circular (two-dimensional) CRLs are considered. It is shown that for a strongly absorbing CRL the real aperture does not influence the focusing properties and the effective aperture is determined by absorption. However, there are three ways to determine the effective aperture in terms of transparent CRLs. In the papers by Kohn [(2002). JETP Lett. 76, 600–603; (2003). J. Exp. Theor. Phys. 97, 204–215; (2009). J. Surface Investig. 3, 358–364; (2012). J. Synchrotron Rad. 19, 84–92; Kohn et al. (2003). Opt. Commun. 216, 247–260; (2003). J. Phys. IV Fr, 104, 217–220], the FWHM of the X-ray beam intensity just behind the CRL was used. In the papers by Lengeler et al. [(1999). J. Synchrotron Rad. 6, 1153–1167; (1998). J. Appl. Phys. 84, 5855–5861], the maximum intensity value at the focus was used. Numerically, these two definitions differ by 50%. The new definition is based on the integral intensity of the beam behind the CRL over the real aperture. The integral intensity is the most physical value and is independent of distance. The new definition gives a value that is greater than that of the Kohn definition by 6% and less than that of the Lengeler definition by 41%. A new approximation for the aperture function of a two-dimensional CRL is proposed which allows one to calculate the two-dimensional CRL through the one-dimensional CRL and to obtain an analytical solution for a complex system of many CRLs.

scholarly journals On ℤ2-indices for ground states of fermionic chains

2020 ◽  
Vol 32 (09) ◽  
pp. 2050028 ◽  
Author(s):  
Chris Bourne ◽  
Hermann Schulz-Baldes
Keyword(s):  
Spectral Gap ◽  
Ground States ◽  
Higher Order ◽  
Spectral Flow ◽  
One Dimensional ◽  
Finite Systems ◽  
Car Algebra ◽  
The One ◽  
Definition Of ◽  
Topological Obstruction

For parity-conserving fermionic chains, we review how to associate [Formula: see text]-indices to ground states in finite systems with quadratic and higher-order interactions as well as to quasifree ground states on the infinite CAR algebra. It is shown that the [Formula: see text]-valued spectral flow provides a topological obstruction for two systems to have the same [Formula: see text]-index. A rudimentary definition of a [Formula: see text]-phase label for a class of parity-invariant and pure ground states of the one-dimensional infinite CAR algebra is also provided. Ground states with differing phase labels cannot be connected without a closing of the spectral gap of the infinite GNS Hamiltonian.

Design of an Axial Impulse Turbine for Enthalpy Drop Recovery

2014 ◽  
Author(s):  
Gaetano Morgese ◽  
Marco Torresi ◽  
Bernardo Fortunato ◽  
Sergio Mario Camporeale
Keyword(s):  
Industrial Process ◽  
Geometrical Parameters ◽  
Impulse Turbine ◽  
One Dimensional ◽  
Analytical Functions ◽  
Process Plants ◽  
The One ◽  
Definition Of ◽  
Turbine Design ◽  
Work Done

In industrial process plants, often there is the need to reduce the pressure of the operating flow. Generally this is performed by means of valves which expand the flow without any work done. The same operation could be performed by replacing these valves with turbines, with the advantage of energy recovery, hence improving the overall efficiency of the system. In this work, a simple and rapid method is shown in order to design a single stage, straight bladed, axial impulse turbine for enthalpy recovery. Assigned the desired flow rate and the minimum power output, the turbine design is performed according to a one-dimensional study into which loss effects are considered by means of appropriate coefficients. From the one-dimensional analysis the heights, the pitch angle, the inlet and outlet angles of both rotor and stator blades are obtained. Actually, the rotor and stator blade profiles are defined by means of several analytical functions. The blade design is then validated by means of CFD simulations. The definition of loss coefficients and blade geometrical parameters is clearly an iterative process, which needs to be repeated until convergence is reached. Furthermore, by means of fully 3D simulations, the effect of the rotor-stator distance is investigated in order to maximize the turbine performance.

Design and Application of Digital Filter Based on Calculus Computing Concept

2014 ◽  
Vol 513-517 ◽  
pp. 3151-3155
Author(s):  
Yan Chun Zhao
Keyword(s):  
Fractional Calculus ◽  
Digital Filter ◽  
Digital Filters ◽  
Engineering Analysis ◽  
Analysis Software ◽  
One Dimensional ◽  
The One ◽  
Definition Of ◽  
Object Of Study ◽  
The Mathematical Model

Calculus has been widely applied in engineering fields. The development of Integer order calculus theory is more mature in the project which can obtain fractional calculus theory through the promotion of integration order. It extends the flexibility of calculation and achieves the engineering analysis of multi-degree of freedom. According to fractional calculus features and the characteristics of fractional calculus, this paper treats the frequency domain as the object of study and gives the fractional calculus definition of the frequency characteristics. It also designs the mathematical model of fractional calculus digital filters using Fourier transform and Laplace transform. At last, this paper stimulates and analyzes numerical filtering of fractional calculus digital filter circuit using matlab general numerical analysis software and FDATool filter toolbox provided by matlab. It obtains the one-dimensional and two-dimensional filter curves of fractional calculus method which achieves the fractional Calculus filter of complex digital filter.

Sign in / Sign up

Export Citation Format

Share Document