A Quantitative Evaluation of Vector Transcendental Functions on ARMv8-Based Processors

Transcendental functions are important functions in various high performance computing applications.Because these functions are time-consuming and the vector units on modern processors become wider and more scalable,there is an increasing demand for developing and using vector transcendental functions in such performance-hungry applications.However,the performance of vector transcendental functions as well as their accuracy remain largely unexplored.To address this issue,we perform a comprehensive evaluation of two Single Instruction Multiple Data(SIMD)intrinsics based vector math libraries on two ARMv8 compatible processors.We first design dedicated microbenchmarks that help us understand the performance behavior of vector transcendental functions.Then,we propose a piecewise,quantitative evaluation method with a set of meaningful metrics to quantify their performance and accuracy.By analyzing the experimental results,we find that vector transcendental functions achieve good performance speedups thanks to the vectorization and algorithm optimization.Moreover,vector math libraries can replace scalar math libraries in many cases because of improved performance and satisfactory accuracy.Despite this,the implementations of vector math libraries are still immature,which means further optimization is needed,and our evaluation reveals feasible optimization solutions for future vector math libraries.

On the Julia Sets of Permutable Transcendental Entire Functions

Let f and g be two permutable transcendental entire functions. In this paper, we first prove that J(fg)=J(f n g m) for any positive integers n and m . Then we prove that the function h(p(z))+az ∈/ B , where h(z) is any transcendental entire function with h′(z)=0 having infinitely many solutions, p(z) is a polynomial with deg p ≥2 and a(≠0) ∈ C .

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.

Julia sets of semi-conjugated transcendental entire functions

Suppose that f and g are two transcendental entire functions, and h is a non-constant periodic entire function. We denote the Julia set and Fatou set off by J（f） and F（f）, respectively, lffand g are semiconjugated, that is, h · f = g · h, in this paper, we will show that z ∈ J（f） if and only if h（z） ∈ J（g） （ similarly, z F（f） if and only ifh（z） ∈ F（g））, and this extends a result of Bergweiler.

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.

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.

The Dynamics of Semigroups of Transcendental Meromorphic Functions

This paper considers the dynamics associated with an arbitrary semigroup of transcendental meromorphic functions. Fatou-Julia theory was used to investigate the dynamics of these semigroups. Some results of the dynamics of a rational mapping on the Riemann sphere were extended to the case.

Bounded Fatou Components of Transcendental Entire Functions with Order Less than 1/2

Let f be a transcendental entire function with order ρ 〈 1/2 and let a be a sufficiently large constant. We prove that if there exists r0 〉 1 such that, for all r 〉 r0 and any small ε 〉0,M（r^σ,f）≥M（r,f）σ＋ε.then every component of the Fatou set F（f） is bounded.

On the value distribution of f^nf^(k)

This paper presents an inequality, by use of which some results about the value distribution of f nf (k) are proved, where n and k are two positive integers.

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）.

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.

Entire Solutions of Differential Equations with Finite Order Transcendental Entire Coefficients

In this paper, we investigate the complex oscillation of the differential equation fk+Ak-1fk-1+…+AOf=F where Ak-1.…, Ao F 0 are finite order transcendental entire functions, such that there exists an Ad（0≤d≤k-1） being dominant in the sense that either it has larger order than any other Aj（j=0.…. d-l. d+l.…. k-1）, or it is the only transcendental function. We obtain some precise estimates of the exponent of convergence of the zero-sequence of solutions to the above equation.

Design of a reconfigurable transcendental function generator

In order to take into account the computing efficiency and flexibility of calculating transcendental functions, this paper proposes one kind of reconfigurable transcendental function generator. The generator is of a reconfigurable array structure composed of 30 processing elements （PEs）. The coordinate rotational digital computer （CORDIC） algorithm is implemented on this structure. Different functions, such as sine, cosine, inverse tangent, logarithmic, etc., can be calculated based on the structure by reconfiguring the functions of PEs. The functional simulation and field programmable gate array （FPGA） verification show that the proposed method obtains great flexibility with acceptable performance.

INTEGRABILITY OF SECOND ORDER AUTONOMOUS SYSTEM

The Liouville's integrability of the second order autonomous system is studied. It isproved that a second order polynomial system is Liouville integrable if and only if thereis an integral factor μ(x, y), such that or a rational function in x and y.