ENTIRE SOLUTIONS OF HIGHER ORDER DIFFERENTIAL EQUATIONS WITH ENTIRE COEFFICIENTS HAVING THE SAME ORDER

In this paper,we consider entire solutions of higher order homogeneous differential equations with the entire coefficients having the same order,and prove that the entire solutions are of infinite lower order.The properties on the radial distribution,the limit direction of the Julia set and the existence of a Baker wandering domain of the entire solutions are also discussed.

A Class of Potentials for Hyperbolic Transcendental Entire Maps

We identify a class of transcendental entire maps of finite order, of disjoint-type, satisfying the rapid derivative growth condition. Within this class, we show that there exist hyperbolic transcendental entire maps that generate a large class of potentials which intersect the so-called tame potentials and form a distinct class of potentials. The methods and techniques derived from the thermodynamic formalism are applied to these potentials for transcendental entire maps acting on some subset of the Julia set which is conjugated to the shift map over a code space with a countable alphabet endowed with the euclidean induced metric on the complex plane.

An Introduction to the Theory of Field Extensions

This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to algebraic extensions. Finally, we construct finite extensions of Q and finite extensions of the function field over finite field Fp using the notion of field completion, analogous to field extensions. With the study of field extensions, considering any polynomial with coefficients in the field, we can find the roots of the polynomial, and with the notion of algebraically closed fields, we have one field, F, where we can find the roots of any polynomial with coefficients in F.

The Factors of Both Romanticism and Transcendentalism in Emily Dickinson's Character

Emily Dickinson is one of the most widely read and well-known American poets. Although there were just seven poems published during her lifetime, she was brought to many people's attention. Some people believe that Dickinson is a real romantic but others think she is a transcendentalist. However, both the factors of romanticism and transcendentalism can be seen in her character indeed. Romantic factors include individualism in her personality and her escaping from society. Transcendental factors are oversoul and mysticalness. If all the factors are taken into account, Emily Dickinson can fall into the category of both romanticism and transcendentalism.

A Review of Hawthorne: A New Voice of American Pessimistic Humanism in 19^(th) Century

A penetrating analysis into Hawthorne's writings of different periods and different angles brings us not only an astonishment at his unique understanding of evil, but also a definite awareness of his opposite but staunch position as a pessimist.Meanwhile, following the discussions about Melville and other writers, his influences of pessimistic ideas and belief in evil allow of no doubt. Subsequently, through an inquisition to the roots of this tragic attitude, his standpoint is found to be inevitable.

Analytic solutions to a finite width strip with a single edge crack of two-dimensional quasicrystals

In this paper, we investigate the well-known problem of a finite width strip with a single edge crack, which is useful in basic engineering and material science. By extending the configuration to a two-dimensional decagonal quasicrystal, we obtain the analytic solutions of modesⅠand Ⅱ using the transcendental function conformal mapping technique. Our calculation results provide an accurate estimate of the stress intensity factors K_Ⅰ and K_Ⅱ, which can be expressed in a quite simple form and are essential in the fracture theory of quasicrystals. Meanwhile, we suggest a generalized cohesive force model for the configuration to a two-dimensional decagonal quasicrystal. The results may provide theoretical guidance for the fracture theory of two-dimensional decagonal quasicrystals.

A SIGNIFICANT IMPROVEMENT ON NEWTON’S ITERATIVE METHOD

For solving nonlinear and transcendental equation f(x)=0 , a singnificant improvement on Newton's method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov's methods of dynamic system. These new methods preserve quadratic convergence and computational efficiency of Newton's method, and remove the monotoneity condition imposed on f(x):f′(x)≠0 .

ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS

In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines.In this paper we shall consider the distribution problem of Julia sets of meromorphic maps.We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines.Meanwhile,examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines.Moreover,we shall show that the Julia set of a transcendental analytic self-map of C*can neither contain a free Jordan arc nor be contained in any finite set of straight lines.

Wavelet multiresolution interpolation Galerkin method for nonlinear boundary value problems with localized steep gradients

The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix.The wavelet multiresolution interpolation Galerkin method that applies this interpolation to represent the unknown function and nonlinear terms independently is proposed to solve the boundary value problems with the mixed Dirichlet-Robin boundary conditions and various nonlinearities,including transcendental ones,in which the discretization process is as simple as that in solving linear problems,and only common two-term connection coefficients are needed.All matrices are independent of unknown node values and lead to high efficiency in the calculation of the residual and Jacobian matrices needed in Newton’s method,which does not require numerical integration in the resulting nonlinear discrete system.The validity of the proposed method is examined through several nonlinear problems with interior or boundary layers.The results demonstrate that the proposed wavelet method shows excellent accuracy and stability against nonuniform grids,and high resolution of localized steep gradients can be achieved by using local refined multiresolution grids.In addition,Newton’s method converges rapidly in solving the nonlinear discrete system created by the proposed wavelet method,including the initial guess far from real solutions.

Further Extension of a Theorem of Values Distribution of Meromorphic Function f^((k))f^n

In this paper we derive the fundamental inequality of the theorem of meromorphic functions. It extends some results of Yi Hong-xun et al. As one of its application, we then study the value distribution of f^（k）f^n-c（z）.

Symbolic and Graphical Computations of a Class of Slightly Perturbed Equations

In this paper, a class of slightly perturbed equations of the form F(x)= ξ -x+αΦ(x) will be treated graphically and symbolically, where Φ(x) is an analytic function of x. For graphical developments, we set up a simple graphical method for the real roots of the equation F(x)=0 illustrated by four transcendental equations. In fact, the graphical solution usually provides excellent initial conditions for the iterative solution of the equation. A property avoiding the critical situations between divergent to very slow convergent solutions may exist in the iterative methods in which no good initial condition close to the root is available. For the analytical developments, literal analytical solutions are obtained for the most celebrated slightly perturbed equation which is Kepler’s equation of elliptic orbit. Moreover, the effect of the orbital eccentricity on the rate of convergence of the series is illustrated graphically.

Theoretical Study of the Energies of the Oscillating System with a Well-Distributed Mass of the Spring

The energy of a spring with a well-distributed mass ms is theoretically studied in this paper.The solution of the wave equation is derived in detail,and then the kinetic energy and potential energy of the spring are studied with the wave equation,as well as the kinetic energy of the oscillating mass M.The kinetic energy and potential energy of the spring,and total energy are numerically simulated for different ratios ms/M with considering the spring’s mass,which makes the property of energy of the oscillating system understood easily.

Finite Type Transcendental Entire Functions Whose Buried Points Set Contains Unbounded Positive Real Interval

Let fμ(z)=z·ep(z)+μ with p(z) being real coefficient polynomial and it's leading coefficient be positive, μ∈R+, when p(z) and μ satisfy two certain conditions, buried point set of fμ(z) contains unbounded positive real interval.

Three Evolutions of the Landscape Cognitive Model from the Perspective of Philosophy

Looking back at the history of landscape design, there are various styles and schools as well as numerous design concepts. From the perspective of philosophical epistemology, landscape design has experienced three periods, namely, apriorism, empiricism and transcendentalism. Combining the design phenomena in the history of garden design, this paper sorted out the evolution of landscape design in terms of methodology, epistemology and instrumentalism, in order to speculate on the macroscopic trajectory of the evolution of landscape cognitive model.

HERMAN RINGS WITH SMALL PERIODS AND OMITTED VALUES

All possible arrangements of cycles of three periodic as well as four periodicHerman rings of transcendental meromorphic functions having at least one omitted value aredetermined. It is shown that if p = 3 or 4, then the number of p-cycles of Herman rings isat most one. We have also proved a result about the non-existence of a 3-cycle and a 4-cycleof Herman rings simultaneously. Finally some examples of functions having no Herman ringare discussed.

Transcendental Meromorphic Functions Whose First Order Derivatives Have Finitely Many Zeros

Let f be a meromorphic function in C. If the order of f is greater than 2,has finitely many zeros and f takes a non-zero finite value finitely times, and then ?is unbounded.

Exploring the Relation between Karma, Qi, Spirituality, and Subjective Well-Being among People in Taiwan

In addition to recognizing well-being as a multidimensional construct,the potential indicators of well-being also have been recognized as multivariate.Specifically,some recent studies have suggested that various indices of social beliefs,religious attitudes and spiritual practices have been said to be robustly associated with well-being.Therefore,given the overwhelming need for promoting well-being and an increasing recognition of the multivariate indicators of well-being,this research project attempted to evaluate the relation of Qi,karma,transcendental experiences,and spiritual practices with the subjective well-being of people in Taiwan.The relevant data from the national sample of 1,933 participants were subjected to factor analysis to extract relevant factors,which included social and religious attitudes of karma and Qi,spiritual indicators of transcendental experiences and spiritual practices,and positive indicators of subjective well-being.In addition to significant intercorrelations,the hierarchical regression analyses after controlling for demographics suggested that Qi had the highest contribution to subjective well-being,followed by spiritual practices,karma,and transcendental experiences.These results are discussed,their implications are elucidated,and the directions for future research are suggested.

Analysis of the Relationship Between Humans and Nature Through Emerson,Thoreau and Hawthorne's Views

Although Emerson, Thoreau and Hawthorne were all influenced by transcendentalism, they have different understanding of and attitudes towards nature and the concept of transcendentalism because of their cultural background. Through analyzing similarities and differences of their points of view of nature, people can find that they understand nature with an increasingly deeper tendency. Firstly, Emerson only knew that people could get comfort from nature in the spiritual world, and then Thoreau found out that people could make use of nature from a more substantial way. At last, Hawthorne set a dialectical view of how people should live with nature. As a result, people learn what kind of relationship people should set with nature. Not only can people benefit from nature in their spiritual world, but also they make use of resources in it. This is the reason why people should protect nature during the process of utilizing it.

The Lebesgue Measure of the Julia Sets of Permutable Transcendental Entire Functions

In 1958, Baker posed the question that if f and g are two permutable transcendental entire functions, must their Julia sets be the same? In order to study this problem of permutable transcendental entire functions, by the properties of permutable transcendental entire functions, we prove that if f and g are permutable transcendental entire functions, then mes (J(f)) = mes (J(g)). Moreover, we give some results about the zero measure of the Julia sets of the permutable transcendental entire functions family.

Application of Lambert <i>W</i>Function to Planck Spectral Radiance Frequencies

Planck’s radiation law provides an equation for the intensity of the electromagnetic radiation from a physical body as a function of frequency and temperature. The frequency that corresponds to the maximum intensity is a function of temperature. At a specific temperature, for the frequencies correspond to much less than the maximum intensity, an equation was derived in the form of the Lambert W function. Numerical calculations validate the equation. A new form of solution for the Euler’s transcendental equation was derived in the form of the Lambert W function with logarithmic argument. Numerical solutions to the Euler’s equation were determined iteratively and iterative convergences were investigated. Numerical coincidences with physical constants were explored.