α-times Integrated Regularized Cosine Functions and Second Order Abstract Cauchy Problems

In this paper, α-times integrated C-regularized cosine functions and mild α-times integrated C-existence families of second order are introduced. Equivalences are proved among α-times integrated C-regularized cosine function for a linear operator A, C-wellposed of (α+1)-times abstract Cauchy problem and mild a -times integrated C-existence family of second order for A when the commutable condition is satisfied. In addition, if A = C-1AC, they are also equivalent to A generating the α -times integrated C-regularized cosine function. The characterization of an exponentially bounded mild α -times integrated C-existence family of second order is given out in terms of a Laplace transform.

POLYNOMIALLY BOUNDED COSINE FUNCTIONS

We characterize polynomial growth of cosine functions in terms of the resolvent of its generator and give a necessary and sufficient condition for a cosine function with an infinitesimal generator which is polynomially bounded.

On Local Integrated Cosine Functions

Suppose X is a Banach space, and A is a closed operator. We give some equivalent conditions between A generating a local integrated cosine functions and the existence of solutions of abstract Cauchy problems.

α-times Integrated Cosine Operator Functions with Growth ω

For a continuous,increasing functionω:[0,∞)→C of finite exponential type,we establish a Hille-Yosida type theorem for strongly continuous α-times(α>0)integrated cosine operator functions with O(ω).It includes the corresponding results for n-times integrated cosine operator functions that are polynomially bounded and exponentially bounded.

STRONGLY CONTINUOUS INTEGRATED COSINE OPERATOR FUNCTIONS WITH GROWTHω

For a continuous increasing function ω ： [0, ∞） → （0, ∞） of finite exponetial type, we establish a Hille-Yosida type theorem for strongly continuous integrated cosine operator functions with O（ω）. It includes the well-known polynomially bounded and exponentially bounded cases.

On Dual Cosine Operator Function

Some basic properties of daul cosine operator function are given. The concept and characterization of θ-reflexivity with respect to cosine operator function are first studied.

Perturbation Theory for Dual Cosine Function（Ⅱ）—Time－dependent Perturbation in the Sun－reflexive Case

In this paper,we proved that the infinitesimal generator of a strongly continuous cosine operator function is preserved under the time-dependent perturbation in the sun-reflexive case,where the perturbed operator is a bounded linear operator from X into a bigger space Xθ(not X),then the corresponding 2-order abstract Cauchy problem is uniformly well-posed.

THE PROBABILISTIC REPRESENTATION FORMULAE FOR COSINE OPERATOR FUNCTIONS

In this paper we obtained general representation formulae for strongly continuous cosine operator functions via probabilistic approach,which include Webb's[1]and Shaw's[2]formulae and some new one as special cases.We also give the quantitative estimations for the general formulae.

Whole Perfect Vectors and Fermat’s Last Theorem

A naïve discussion of Fermat’s last theorem conundrum is described. The present theorem’s proof is grounded on the well-known properties of sums of powers of the sine and cosine functions, the Minkowski norm definition, and some vector-specific structures.

The Impact of the Earth’s Movement through the Space on Measuring the Velocity of Light

Goal of this experiment is basically measuring the velocity of light. As usual we will measure two-way velocity of light (from A to B and back). In contrast to the similar experiments we will not assume that speeds of light from A to B and from B to A are equal. To achieve this we will take into account Earth’s movement through the space, rotation around its axis and apply “least squares method for cosine function”, which will be explained in Section 9. Assuming that direction East-West is already known, one clock, a source of light and a mirror, is all equipment we need for this experiment.

Subspaces for weak mild solutions of the second order abstract differential equation

The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/（dr^2）u（t,x）=Au（t,x）;u（0,x）=x,d/（dt）u（0,x）=0,x∈X is studied, where A is a closed operator on Banach space X. The case that the problem is ill-posed is treated, and two subspaces Y（A, k） and H（A, ω） are introduced. Y（A, k） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（ t, x） such that ess sup{（1＋t）^-k｜d/（dt）〈v（t,x）,x^＊〉｜：t≥0,x^＊∈X^＊,｜x^＊‖≤1}〈＋∞. H（A, ω） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（t,x）such that ess sup{e^-ωl｜d/（dt）〈v（t,x）,x^＊）｜：t≥0,x^＊∈X^＊,‖x^＊‖≤1}〈＋∞. The following conclusions are proved that Y（A, k） and H（A, ω） are Banach spaces, and both are continuously embedded in X; the restriction operator A ｜ Y（A,k） generates a once-integrated cosine operator family { C（t） }t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖Y（A,k）≤M（1＋t）^k,arbitary t≥0; the restriction operator A ｜H（A,ω） generates a once- integrated cosine operator family {C（t）}t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖H（A,ω）≤≤Me^ωt,arbitary t≥0.

Hydrodynamics and Heat Transfer Analysis of Airflow in a Sinusoidally Curved Channel

For heat transfer enhancement in heat exchangers,different types of channels are often tested.The performance of heat exchangers can be made better by considering geometry composed of sinusoidally curved walls.This research studies the modeling and simulation of airflow through a 2πunits long sinusoidally curved wavy channel.For the purpose,two-dimensional Navier Stokes equations along with heat equations are under consideration.To simulate the fluid flow problem,the finite element-based software COMSOL Multiphysics 5.4 is used.The parametric study for Reynolds number from Re=100 to Re=1000 and the period of vibration P from 0 to 5 are observed.The surface plots,streamline patterns,contours,and graphs are presented for the velocity field magnitude,temperature,and pressure against the Reynolds number as well as period of vibration.The results are compared with various literature.It is found that due to the creation of periodic contraction regions the velocity magnitude of the flow is continuously increasing with the increase of Reynolds number,on the contrary the pressure is decreasing from inlet to outlet of the channel.Also,a periodic variation in the pressure distribution along the vibrating boundaries has been found with an average increase of 500%for the high Reynolds number.A novel work was done by expressing the rotation rate per second in terms of local Reynolds number for the recirculating regions found due to the periodic oscillation of the boundaries.The average temperature near the outlet where a fixed temperature is imposed initially is decreasing with an increase in Reynolds number.The convection process is weakened due to an increase of periodic vibration of boundaries.

Some Results on Dual Cosine Operator Function

Some basic properties of dual cosine operator function are given. The concept and characterization of θ reflexivity with respect to cosine operator function are first studied.

Shape Modification of Parametric Curves

A technique of shape modification and deformation for parametric curvesthrough cosine extension function (CEF) is presented. First, a special extension function isdefined, based on which a shape operator matrix is constructed. Then combining such matrix with thecenter of extension and principal directions, two kinds of deformation matrices are defined.Finally, curve deformation is achieved through multiplying its position vector in a local coordinatesystem by deformation matrix or adding the multiplication of a vector field and quasi-deformationmatrix to its position vector in the original coordinate system. Since CEF contains several variableparameters, each of which generates a different effect of shape modification such as controllingthe degree of continuity of the modified part of curve with the unchanged part, ideal deformationeffects can be got fairly and easily. Examples of theoretical analysis show that the method ispotentially useful for geometric modeling, computer graphics and so on.

PERIODIC DEFORMATION OF PARAMETRIC SURFACE

Deformation is a powerful tool for geometric modeling and design. Such a toolcan be used to create a new shape from existing shape without restarting whole design process. Anew mathematical model for producing controllable periodic deformations is proposed. By introducingcosine extension functions construct a shape operator matrix and then use the matrix to transformthe position vector of some points on the object surface so as to create the deformation effects.Because the cosine extension functions have a number of variable parameters with differentproperties, the method has corresponding interactive control means. The user can manipulate thoseparameters to get desirable periodic deformation effects. Experimental results show that the methodis feasible and applicable to engineering and research fields such as sheet metal forming bystamping and CAD.

MAXIMAL SUBSPACES FOR SOLUTIONS OF THE SECOND ORDER ABSTRACT CAUCHY PROBLEM

For a continuous, increasing function ω ： R^＋ →R^＋／{0} of finite exponential type, this paper introduces the set Z（A, ω） of all x in a Banach space X for which the second order abstract differential equation （2） has a mild solution such that [ω（t）]^-1u（t,x） is uniformly continues on R^＋, and show that Z（A, ω） is a maximal Banach subspace continuously embedded in X, where A ∈ B（X） is closed. Moreover, A[z（A,ω） generates an O（ω（t）） strongly continuous cosine operator function family.

Shape Modification and Deformation of Parametric Surfaces

A new method for shape modification or deformation of all kinds of parametric surfaces is presented. Instead of using any control mesh and designing any auxiliary tools for modification or deformation, this method only needs to multiply the equation of the original surface by several shape operator matrixes so that deformation effects are achieved. Due to introducing interactive control parameters with different properties those parameters can be changed to get more ideal effects of deformation or shape modification. The core of the method is that so-called cosine extension function is proposed by which the shape operator matrix can be created. Experiment shows that the method is simple, intuitive and easy to control, and diversified local deformations or shape modifications can easily be made via small and random variations of interactive control parameters.

On the Stability of the Generalized Sine Functional Equations

The aim of this paper is to study the stability problem of the generalized sine functional equations as follows：g（x）f（y）=f（x＋y/2）^2-f（x-y/2）^2 f（x）g（y）=f（x＋y/2）^2-f（x-y/2）^2,g（x）g（y）=f（x＋y/2）^-f（x-y/2）^2Namely, we have generalized the Hyers Ulam stability of the （pexiderized） sine functional equation.

REGULARIZED SEMIGROUPS GENERATED BY PSEUDODIFFERENTIAL OPERATORS ON C~α

Let Op(f) be a pseudodifferential operator with symbol f∈ S m ρ,0 having constant coefficients. We prove that Op(f) generates a regularized semigroup or cosine function on C α (R n) (0<α<1) when the symbol f(ξ) and its derivatives satisfy certain growth conditions.

MULTIVARIATE FOURIER TRANSFORM METHODS OVER SIMPLEX AND SUPER-SIMPLEX DOMAINS

In this paper we propose the well-known Fourier method on some non-tensor product domains in Rd, including simplex and so-called super-simplex which consists of （d ＋ 1）! simplices. As two examples, in 2-D and 3-D case a super-simplex is shown as a parallel hexagon and a parallel quadrilateral dodecahedron, respectively. We have extended most of concepts and results of the traditional Fourier methods on multivariate cases, such as Fourier basis system, Fourier series, discrete Fourier transform （DFT） and its fast algorithm （FFT） on the super-simplex, as well as generalized sine and cosine transforms （DST, DCT） and related fast algorithms over a simplex. The relationship between the basic orthogonal system and eigen-functions of a LaDlacian-like operator over these domains is explored.