Formula For Fourier Research Articles

Fast object imaging and classification based on circular harmonic Fourier moment detection.

Limited by the number of illumination fields and the speed of a spatial light modulator, single-pixel imaging (SPI) cannot realize real-time imaging and fast classification of an object. In this paper, we proposed the circular harmonic Fourier single-pixel imaging (CHF-SPI) for the first time to realize fast imaging and classification of objects. The light field distribution satisfies the circular harmonic Fourier formula, and the light intensity values of the single-pixel detector are equivalent to the circular harmonic Fourier moments. Then the target can be reconstructed under low sampling ratio by inverse transformation. Through simulation and experimental verification, clear imaging can be performed at a sampling ratio of 0.9%. In addition, circular harmonic Fourier moments are used to construct multi-distortion invariant to classify objects with rotation and scale change. The scale change multiples of objects can be calculated and the objects can be classified by using 10 light fields. It is of great significance to classify objects quickly without imaging.

Study on designing measurement system of thermal conductivity based on one-dimensional steady-state method

Thermal conductivity has wide application in various fields such as in agricultural and food sectors. In a food industry, measuring the thermal conductivity is designed to optimize the thermal process such as drying, sterilization, and freezing. In this study, a system to measure thermal conductivity is designed according to one-dimensional steady state method through Fourier formula. The experiment uses samples consisting of rice-based product (rice grain, rice powder, rice flour, cooked rice) and whole milk powder. The analysis will be performed in the range of 15-30 minutes on its steady condition. The results are verified with the literature data and evaluated through degree equivalence calculation. As a result, the system can distinguish the samples which has different texture and composition. Furthermore, the system can figure the increasing trend of thermal conductivity with the raise of moisture content. However, in most case, thermal conductivities of the measured samples have higher value compared to the literature data (2-4 times higher). The disparity due to the variation of heating temperature, sample condition, measurement technique, and analysis. Several finding to gain more reliable system can be done by improving the insulation or evaluating the mathematical calculation into a two-dimension approximation. In general, the design of the system is plausible for measuring the thermal conductivity.

Shape optimization for the strong routing of light in periodic diffraction gratings

In the quest for the development of faster and more reliable technologies, the ability to control the propagation, confinement, and emission of light has become crucial. The design of guide mode resonators and perfect absorbers has proven to be of fundamental importance. In this project, we consider the shape optimization of a periodic dielectric slab aiming at efficient directional routing of light to reproduce similar features of a guide mode resonator. For this, the design objective is to maximize the routing efficiency of an incoming wave. That is, the goal is to promote wave propagation along the periodic slab. A Helmholtz problem with a piecewise constant and periodic refractive index medium models the wave propagation, and an accurate Robin-to-Robin map models an exterior domain. We propose an optimal design strategy that consists of representing the dielectric interface by a finite Fourier formula and using its coefficients as the design variables. Moreover, we use a high order finite element (FE) discretization combined with a bilinear Transfinite Interpolation formula. This setting admits explicit differentiation with respect to the design variables, from where an exact discrete adjoint method computes the sensitivities. We show in detail how the sensitivities are obtained in the quasi-periodic discrete setting. The design strategy employs gradient-based numerical optimization, which consists of a BFGS quasi-Newton method with backtracking line search. As a test case example, we present results for the optimization of a so-called single port perfect absorber. We test our strategy for a variety of incoming wave angles and different polarizations. In all cases, we efficiently reach designs featuring high routing efficiencies that satisfy the required criteria.

ЕКСПЕРИМЕНТАЛЬНІ ДОСЛІДЖЕННЯ ТЕПЛОПРОВІДНОСТІ ТЕПЛОІЗОЛЯЦІЙНИХ МАТЕРІАЛІВ ІЗ МІНЕРАЛЬНОЇ ВАТИ

According to the analysis of domestic and foreign literature sources, it is noted that reducing energy consumption to create an optimal microclimate of buildings involves reducing heat loss through external enclosing structures. Construction of new buildings and thermal modernization of existing ones is carried out with the use of different properties of thermal insulation materials, passport data of manufacturers on their characteristics need to be clarified in determining the energy efficiency of buildings. Mineral wool was selected for experimental studies of thermal insulation material used to improve the thermal insulation shell. The energy efficiency of mineral wool was studied using an installation with a climate chamber with a homogeneous heat flux over the cross-sectional area of ​​the sample. To register the change in the amount of heat flux, thermocouples were used, which are located at different points of the climate chamber with a sample of mineral wool. Data on the change in temperature during the observation period before the stabilization of the heat flux was determined by the automatic registration unit. The change in time of the temperature regime at different points of the mineral wool sample is obtained. The thermal conductivity, which characterizes the efficiency of the thermal insulation material made of mineral wool, was calculated according to the known Fourier formula for stationary thermal regime from the values ​​of the temperature difference in the characteristic cross sections of the sample. The discrepancy between the values ​​of thermal conductivity of the investigated sample and the passport data of thermal conductivity of mineral wool provided by its manufacturer is revealed.

A New Conformal Cooling System for Plastic Collimators Based on the Use of Complex Geometries and Optimization of Temperature Profiles.

The paper presents a new design of conformal cooling channels, for application in collimator-type optical plastic parts. The conformal channels that are presented exceed the thermal and dynamic performance of traditional and standard conformal channels, since they implement new sections of complex topology, capable of meeting the high geometric and functional specifications of the optical part, as well as the technological requirements of the additive manufacturing of the mold cavities. In order to evaluate the improvement and efficiency of the thermal performance of the solution presented, a transient numerical analysis of the cooling phase has been carried out, comparing the traditional cooling with the new geometry that is proposed. The evolution of the temperature profile versus the thickness of the part in the collimating core with greater thickness and temperature, has been evaluated in a transient mode. The analysis of the thermal profiles, the calculation of the integral mean ejection temperature at each time of the transient analysis, and the use of the Fourier formula, show great improvement in the cycle time in comparison with the traditional cooling. The application of the new conformal design reduces the manufacturing cycle time of the collimator part by 10 s, with this value being 13% of the total manufacturing cycle of the plastic part. As a further improvement, the use of the new cooling system reduces the amount of thickness in the collimator core, which is above the ejection temperature of the plastic material. The improvement in the thermal performance of the design of the parametric cooling channels that are presented not only has a significant reduction in the cycle time, but also improves the uniformity in the temperature map of the collimating part surface, the displacement field, and the stresses that are associated with the temperature gradient on the surface of the optical part.

Solution of non-linear time fractional telegraph equation with source term using B-spline and Caputo derivative

Abstract This article focusses on the implementation of cubic B-spline approach to investigate numerical solutions of inhomogeneous time fractional nonlinear telegraph equation using Caputo derivative. L1 formula is used to discretize the Caputo derivative, while B-spline basis functions are used to interpolate the spatial derivative. For nonlinear part, the existing linearization formula is applied after generalizing it for all positive integers. The algorithm for the simulation is also presented. The efficiency of the proposed scheme is examined on three test problems with different initial boundary conditions. The influence of parameter α on the solution profile for different values is demonstrated graphically and numerically. Moreover, the convergence of the proposed scheme is analyzed and the scheme is proved to be unconditionally stable by von Neumann Fourier formula. To quantify the accuracy of the proposed scheme, error norms are computed and was found to be effective and efficient for nonlinear fractional partial differential equations (FPDEs).

Effects of general imperfect interface/interphase on the in-plane conductivity of thermal composites

The effects of general imperfect interphase/interface models on effective and local thermal conductive responses of fiber reinforced-composites are investigated. A physical interphase model is firstly introduced and then extended to the weakly conducting (WC) and highly conducting (HC) interfaces by satisfying the discontinuity conditions of temperature and normal heat flux, respectively. To simulate different geometric arrangements of fibers, hexagonal and square repeating unit cells (RUCs) are used in this paper. Regarding simulation technique, the locally-exact homogenization theory (LEHT) is extended by considering the aforementioned interphase/interface models to re-produce the localized temperature and heat flux distributions in the unit cells and then to predict the macroscopic thermal conductivities. The present method does not require mesh discretization and pre-processing, but directly obtains the accurate analytical expression of internal domains after solving the thermal differential equations. The unknown coefficients are obtained through applying imperfect interphase/interface conditions and weak-form periodic boundary conditions. Finally, the effective Fourier's formula is established to obtain the overall thermal conductivities of composites. The LEHT is advantageous in fast convergence, numerical stability and computational efficiency, thus providing a good numerical tool in studying the influence of interface parameters on composites. The generated numerical results are in good agreement with the analytical or numerical simulations in the literature. On this basis, the effects of interfacial geometric and physical parameters on the macroscopic coefficients and local thermal responses of the composites are tested. Finally, the thickness of a physical interphase is varied to test its applicability in generating equivalent effect with WC interface and HC interface. Generally, the imperfect interfacial effects play important roles in the thermal conductive response of composite materials. In the meantime, the proposed method is of great significance for the study of thermal conductions of composite microstructures.

УСТАНОВКА ДЛЯ ДОСЛІДЖЕННЯ ТЕПЛОПРОВІДНОСТІ ЕНЕРГОЕФЕКТИВНИХ ТЕПЛОІЗОЛЯЦІЙНИХ МАТЕРІАЛІВ РОСЛИННОГО ПОХОДЖЕННЯ

One of the key point criteria of any thermal insulation material that determine its energy efficiency is thermal conductivity. Determination of thermal conductivity of materials of plant origin, as well as any other, can be reliably determined only experimentally. The proposed original design of an inexpensive installation for studying the thermal conductivity of heat-insulating materials of plant origin contains a heat-insulated empty body for placing the test sample and a heat source. The heat flow is created by a heating element, which consists of an upper and lower metal plate of a given thickness with a known coefficient of thermal conductivity. The volume created between the plates is filled with a layer of quartz sand to create additional homogeneous heat flow. This design of the heating element provides a homogeneous distribution of heat flux over the cross-sectional area of the test sample of the insulating material. To register the change in the magnitude of the heat flux, thermocouples are used, which are located on the inner surface of the upper plate of the heat source and on the cold side of the investigated sample of heat-insulating material. Data on temperature changes are registered by the automatic registration block. A tubular electric heater of appropriate configuration is symmetrically attached to the lower plate along the cross-sectional area. The tubular electric heater is connected to the voltage control unit. Between the lower plane of the lower plate and the housing of the installation there is an air layer to minimize heat loss from the heating element. The coefficient of thermal conductivity, which characterizes the efficiency of the insulating material, is calculated by the known Fourier formula for the steady-state temperature regime from the input information about the temperature difference in the characteristic cross sections of the sample.

HENSTOCK INTEGRAL IN HARMONIC ANALYSIS

The feasibility of Henstock approach to integration theory is demonstrated by application of Henstock-type integrals with respect to different derivation bases to the problem of recovering the coefficients of orthogonal series by generalized Fourier formulae.

Thermal Transport in Graphene Oxide Films: Theoretical Analysis and Molecular Dynamics Simulation.

As a derivative material of graphene, graphene oxide films hold great promise in thermal management devices. Based on the theory of Fourier formula, we deduce the analytical formula of the thermal conductivity of graphene oxide films. The interlaminar thermal property of graphene oxide films is studied using molecular dynamics simulation. The effect of vacancy defect on the thermal conductance of the interface is considered. The interfacial heat transfer efficiency of graphene oxide films strengthens with the increasing ratio of the vacancy defect. Based on the theoretical model and simulation results, we put forward an optimization model of the graphene oxide film. The optimal structure has the minimum overlap length and the maximum thermal conductivity. An estimated optimal overlap length for the GO (graphene-oxide) films with degree of oxidation 10% and density of vacancy defect 2% is 0.33 μm. Our results can provide effective guidance to the rationally designed defective microstructures on engineering thermal transport processes.

A modified Fourier approach for analysis of nanofluid heat generation within a semi-circular enclosure subjected to MFD viscosity

In this study, the heat transfer behavior in a semi-circular cavity filled with Fe3O4-H2O nanofluid is analyzed using the modified Fourier formula. MFD (magnetic field dependent) viscosity is considered for ferrofluid viscosity. The salient features elaborating transportation of heat namely the heat sink/source aspect is accounted. The control volume finite element method (CVFEM) is employed for numerical computations of the nonlinear system. The effects of different parameters such as Hartmann number, heat source/sink parameter, thermal relaxation parameter, Rayleigh number, nanoparticles volume fraction and shape factor on heat transfer and fluid flow properties are investigated. Based on the active parameters of the study, a correlation was obtained for the average Nusselt number. According to the results, the average Nusselt number is proportional to nanoparticles volume fraction, thermal relaxation factor, and Rayleigh number. The results also indicate that the magnetic field can be applied as a regulatory factor to the system. It is also concluded that the spherical shape nanoparticles provide lesser heat exchange while the platelet nanoparticles shape represents better heat exchange.

On Representation of Riesz–space–valued Functions by Fourier Series on Multiplicative Systems

The result that was established in [1] for Walsh series with coefficients belonging to a Riesz–space is generalized for multiplicative systems. Conditions for recovering the coefficients of a convergent series on the multiplicative systems from its sum by Fourier formulae are considered. For these formulae the Henstock–Kurzweil integral for Riesz–space–valued functions defined with respect to the special basis will be used.

Gaussian covariance matrices for anisotropic galaxy clustering measurements

Measurements of the redshift-space galaxy clustering have been a prolific source of cosmological information in recent years. Accurate covariance estimates are an essential step for the validation of galaxy clustering models of the redshift-space two-point statistics. Usually, only a limited set of accurate N-body simulations is available. Thus, assessing the data covariance is not possible or only leads to a noisy estimate. Further, relying on simulated realisations of the survey data means that tests of the cosmology dependence of the covariance are expensive. With these points in mind, this work presents a simple theoretical model for the linear covariance of anisotropic galaxy clustering observations with synthetic catalogues. Considering the Legendre moments (`multipoles') of the two-point statistics and projections into wide bins of the line-of-sight parameter (`clustering wedges'), we describe the modelling of the covariance for these anisotropic clustering measurements for galaxy samples with a trivial geometry in the case of a Gaussian approximation of the clustering likelihood. As main result of this paper, we give the explicit formulae for Fourier and configuration space covariance matrices. To validate our model, we create synthetic HOD galaxy catalogues by populating the haloes of an ensemble of large-volume N-body simulations. Using linear and non-linear input power spectra, we find very good agreement between the model predictions and the measurements on the synthetic catalogues in the quasi-linear regime.

Mathematical model of heat transfer in a fluid with account for its relaxation properties

Using the terms that take account for the temporal and spatial nonlocality (time variation of the heat flux and the temperature gradient) in the formula of Fourier’s law for the heat flux a differential equation for a fluid in motion is derived that contains the second time derivative and themixed derivative with respect to the spatial and temporal variables. Numerical solution of the problem of heat transfer in the laminar fluid flow in a plane channel demonstrates that, in view of the lag in the time variation of the heat flux from zero to a certain maximum value, the boundary condition of the first kind (thermal shock) cannot be instantaneously realized. The process of its stabilization on the wall is characterized by a certain time interval, whose duration is determined by the relaxation properties of the fluid. At large values of the dimensionless coefficients of the heat flux relaxation and the temperature gradient the boundary condition of the first kind can be realized only as the steady state is attainted, as Fo→∞. In this case, the flow does not contain temperature jumps and negative temperature values.

Fabrication, Microstructure, Thermal and Electrical Properties of Copper Heat Sink Composites

Copper as well as copper base composites reinforced with coated and uncoated 1 wt% diamond, graphite particles or short carbon fibers are prepared by powder metallurgy process. The reinforcement particles were encapsulated with silver as well as copper layer by using the electroless deposition technique to investigate the influence of the reinforcement surface coating on the microstructure, density, electrical and thermal properties of the sintered samples. The coated and the uncoated powders were cold compacted at 600 MPa, and then sintered at 1173 K (900°C) for 2 h under hydrogen atmosphere. The phase composition, morphology and microstructure of the prepared powders as well as the copper base sintered composites were investigated using X-ray diffraction analysis (XRD) and Scanning Electron Microscope (SEM) equipped with an Energy Dispersive Spectrometer (EDS) respectively. The density of the sintered composites was measured by Archimedes method. The copper base consolidated composites had a density up to 96% and the reinforcement coated particles were distributed uniformly within the copper matrix better than the uncoated one. The electrical resistivity at room temperature and the heat transfer conduction of the produced samples were measured in a temperature range between 323 K (50°C) and 393 K (120°C). The results observed that the sintered materials prepared from the coated powder have lower electrical resistivity than the sintered materials prepared from the mixed powders. On the other hand the thermal conductivity values were calculated using the heat transfer conduction values by means of the Fourier formula. The results observed that the thermal conductivity of copper is (391 W/m·K), 1 wt% diamond/Cu is (408 W/m·K), 1 wt% graphite coated silver/Cu is (393 W/m·K), 1 wt% Cu coated short carbon fiber/Cu is (393 W/m·K), graphite/Cu is (383 W/m·K) and short carbon fiber/Cu is (382 W/m·K). The obtained composites are expected to be suitable for heat sink applications. The heat transfer testing experiments were done. The forced convection of the present work was done and compared with the previous work in the literature, and satisfactory agreement was achieved.

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

In this paper, we propose a fast and accurate numerical method based on Fourier transforms to solve Kolmogorov forward equations of symmetric scalar Lévy processes. The method is based on the accurate numerical formulas for Fourier transforms proposed by Ooura. These formulas are combined with nonuniform fast Fourier transforms (FFT) and fractional FFT to speed up the numerical computations. Moreover, we propose a formula for numerical indefinite integration on equispaced grids as a component of the method. The proposed integration formula is based on the sinc-Gauss sampling formula, which is a function approximation formula. This integration formula is also combined with the FFT. Therefore, all steps of the proposed method are executed using the FFT and its variants. The proposed method allows us to be free from some special treatments for a nonsmooth initial condition and numerical time integration. The numerical solutions obtained by the proposed method appear to be exponentially convergent on the interval if the corresponding exact solutions do not have sharp cusps. Furthermore, the real computational times are approximately consistent with the theoretical estimates.

On Choosing Fourier Transforms for Practical Geoscience Applications

The variety of definitions of Fourier transforms can create confusion for practical applications. This paper examines the choice of formulas for Fourier transforms and determines the appropriate choices for geoscience applications. One set of Discrete Fourier Transforms can be defined that approximate Fourier integrals and provide transforms between sampled continuous functions in both domains. For applications involving transforms between a continuous function and a discrete function a second set of Discrete Fourier Transforms is needed with different scaling factors. Two classes of application are presented: those where either form of transforms can be used and those where it is necessary to use a particular transform to obtain the correct results.

Universal calculation formula and calibration method in Fourier transform profilometry

We propose a universal calculation formula of Fourier transform profilometry and give a strict theoretical analysis about the phase-height mapping relation. As the request on the experimental setup of the universal calculation formula is unconfined, the projector and the camera can be located arbitrarily to get better fringe information, which makes the operation flexible. The phase-height calibration method under the universal condition is proposed, which can avoid measuring the system parameters directly. It makes the system easy to manipulate and improves the measurement velocity. A computer simulation and experiment are conducted to verify its validity. The calculation formula and calibration method have been applied to measure an object of 22.00 mm maximal height. The relative error of the measurement result is only 0.59%. The experimental results prove that the three-dimensional shape of tested objects can be reconstructed exactly by using the calculation formula and calibration method, and the system has better universality.

A Study on Value-at-Risk and Lévy Processes

The Basle 2 Capital Accord issued by the Basle Committee on banking supervision has proposed a multiplier superior to 3 on banks' internal 99% 10-day Value-at-Risk calculated for market risk exposure. This ad hoc factor has not been fully explained and is poorly justified by arguing that the standard classical models of stock price dynamics do not adequately capture actual market risks. More generally, the current crisis has questioned a lot risk management practices about the determination of appropriate VaRs. In this paper, we revisit the computation of Value-at-Risk by introducing a new method based on Levy processes. After a brief preliminary study where jump tests are performed in order to confirm the need for jump processes in financial modeling, we provide a new presentation of Variance Gamma Processes with Drift, that are reconstructed in an original way starting from the exponential distribution. Then, we display a fit of these processes on the American and French markets, before providing a new general Fourier formula that allows to compute VaR quickly and efficiently. Based on this formula, we conduct a study of the term structure of VaR, and provide a discussion of the Basle 2 and forthcoming Solvency 2 agreements.

Extension theorems for spheres in the finite field setting

In this paper we study the boundedness of extension operators associated with spheres in vector spaces over finite fields. In even dimensions, we estimate the number of incidences between spheres and points in the translated set from a subset of spheres. As a result, we improve the Tomas-Stein exponents, previous results by the authors in [Iosevich and Koh, Illinois J. of Mathematics: 2008]. The analytic approach and the explicit formula for Fourier transform of the characteristic function on spheres play an important role to get good bounds for exponential sums.