NORMAL FAMILIES AND SHARED VALUES OF HOLOMORPHIC FUNCTIONS

Let F be a family of holomorphic functions in a domain D, k be a positive integer, a, b（≠0）, c（≠0） and d be finite complex numbers. If, for each f∈F, all zeros of f-d have multiplicity at least k, f^（k） = a whenever f=0, and f=c whenever f^（k） = b, then F is normal in D. This result extends the well-known normality criterion of Miranda and improves some results due to Chen-Fang, Pang and Xu. Some examples are provided to show that our result is sharp.

NORMAL FAMILIES OF MEROMORPHIC FUNCTIONS WITH SHARED VALUES

We obtain some normality criteria of families of meromorphic functions sharing values related to Hayman conjecture, which improves some earlier related results.

Normal Reference Value of Red Blood Cell Count of Chinese Presenile Men and Geographical Factors

This paper aims at providing a scientific basis for unifying the normal reference value standards of red blood cell count of Chinese presenile men. The paper, using microscopical counting method, studies the relationship between the normal reference values of 38,061 samples of red blood cell count of presenile men and eight geographical factors in 297 units in China. It is found that the correlation of geographical factors and the normal reference value of red blood cell count of presenile men is quite significant （F=303.00, P=-0.000）. By using the method of stepwise regression analysis, one regression equation is inferred. It is concluded that if geographical data are obtained in a certain area, the normal reference value of red blood cell count of presenile men in this area can be reckoned by using the regression analysis. Furthermore, according to the geographical factors, China can be divided into eight regions： Northeast China Region, North China Region, Shanxi-Shaanxi-Irmer Mongolia Region, Middle and Lower Reaches of the Changjiang River Region, Southeast China Region, Northwest China Region, Southwest China Region and Qinghai-Tibet Plateau Region.

Normality and Shared Value Concerning Differential Polynomials

Let F be a family of functions meromorphic in a domain D, let m, n k , k be three positive integers and b be a finite nonzero complex number. Suppose that, (1) for eachf∈F, all zeros of f have multiplicities at least k ; (2) for each pair of functions f, g ∈F,P(f)H(f) and P(g)H(g) share b, where P(f) and H(f) were defined as (1.1) and (1.2) and nk ≥ max 1≤i≤k-1 {n i }; (3) m ≥ 2 or nk ≥ 2, k ≥ 2, then F is normal in D.

Normal families and shared values of meromorphic functions

We studied the normality conditions in families of meromorphic functions, improved the results of Fang and Zalcman [Fang ML, Zalcman L, Normal families and shared values of meromorphic functions, Computational Methods and Function Theory, 2001, 1 （1）： 289-299], and generalized two new normality criterions. Let F be a family of meromorphic functions in a domain D, a a non-zero finite complex number, B a positive real number, and k and m two positive integers satisfying m〉2k＋4. If every function denoted by f belonging to F has only zeros with multiplicity at least k and satisfies f^m（z）f^（k）（Z）=α→ ｜^f（k）（z）｜ ≤B or f^m（z）f^（k）（z）=α→｜f（z）｜ ≥, then F is normal in D.

INTERSECTION POINT RULE OF THE RETENTION VALUE AND NORMAL BOILING POINT OF THE HOMOLOGUES IN REVERSED-PHASE HIGH-PERFORMANCE LIQUID CHROMATOGRAPHY

In this paper, a linear relationship between the logarithm of capacity factor k and normal boiling point to of the homologues has been derived, based on the basic retention equation of liquid chromatography according to statistical thermodyoamics proposed by professor Ln Peizhang and others, This equation has been verified by a large number of experimental data, all the strsight lines of lnk- of bumologues for different mobile phass coaiposltion cross each other at the same point, So the intereection point equation van proposed, wbich was used to prodict the retention valu, the result was satisfactory.

Riemann Boundary Value Problem of Non-Normal Type on the Infinite Straight Line

Various kinds of Riemann boundary value problems (BVPs) for analytic functions on closed curves or on open arc, doubly periodic Riemann BVPs, doubly quasi-periodic Riemann BVPs, and BVPs for polyanalytic functions have been widely investigated in [1-8]. The main ap- proach is to use the decomposition of polyanalytic functions and their generalization to transform the boundary value problems to their corresponding boundary value problems for analytic functions. Recently, inverse Riemann BVPs for generalized analytic functions or bianalytic functions have been investigated in [9-12]. In this paper, we consider a kind of Riemann BVP of non-normal type on the infinite straight line and discuss the solvable conditions and the general solution for it.

Normal Functions Concerning Shared Values

In this paper we discuss normal functions concerning shared values. We obtain the follow result. Let F be a family of meromorphic functions in the unit disc A, and a be a nonzero finite complex number. If for any f ∈F, the zeros of f are of multiplicity, f and f′ share a, then there exists a positive number M such that for any f∈F1（1-｜z｜^2） ｜f′（z）｜/1＋｜f（z）｜^2≤M.

Normal reference value of forced vital capacity of Chinese younger women and geographical factors

Objective To supply a scientific basis for standardizing the normal reference value of forced vital capacity(FVC)of Chinese younger women.Methods We studied the relationship between the normal reference value of 21767 samples of FVC of younger women and eight geographical factors in 157 areas in China.Results It was found that the correlation between geographical factors and the normal reference value of FVC of younger women was quite significant(F=5.884,P=0.000).By adopting the method of mathematical regression analysis,one regression equation was inferred:=3.146+0.00006919X1+0.01315X4-0.006966X6+0.09524X8±0.254.In the above equation,is the normal reference value of younger women’ FVC(L);X1 is the altitude(m);X4 is the annual mean air temperature(℃);X6 is the annual mean relative humidity(%);X8 is the annual mean wind speed(m/s);0.254 is the value of the residual standard deviation.Conclusion If geographical values are obtained in a certain area,the normal reference value of FVC of younger women in this area can be obtained by using the regression equation.Furthermore,according to the geographical factors,China can be divided into eight regions:Northeast China Region,North China Region,Shanxi-Shaanxi-Inner Mongolia Region,middle and lower reaches of the Yangzte River Region,Southeast China Region,Northwest China Region,Southwest China Region,and Qinghai-Tibet Plateau Region.

The Value Distribution and Normality Criteria of a Class of Meromorphic Functions

In this article,we use Zalcman Lemma to investigate the normal family of meromorphic functions concerning shared values,which improves some earlier related results.

STUDYING THE FOCAL VALUE OF ORDINARY DIFFERENTIAL EQUATIONS BY NORMAL FORM THEORY

We present a new method to calculate the focal value of ordinary differential equation by applying the theorem defined the relationship between the normal form and focal value,with the help of a symbolic computation language M ATHEMATICA,and extending the matrix representation method.This method can be used to calculate the focal value of any high order terms.This method has been verified by an example.The advantage of this method is simple and more readily applicable.the result is directly obtained by substitution.

Normal Criteria and Shared Values by Differential Polynomials

For a family of meromorphic functions on a domain D, it is discussed whether F is normal on D if for every pair functions f(z),g∈F , f'–afnand g'–agn share value d on D when n=2,3, where a, b are two complex numbers, a≠0,∞,b≠∞.Finally, the following result is obtained:Let F be a family of meromorphic functions in D, all of whose poles have multiplicity at least 4 , all of whose zeros have multiplicity at least 2. Suppose that there exist two functions a(z) not idendtically equal to zero, d(z) analytic in D, such that for each pair of functions f and in F , f'–a(z)f2 and g'–a(z)g2 share the function d(z) . If a(z) has only a multiple zeros and f(z)≠∞ whenever a(z)=0 , then F is normal in D.

Singular Value Inequalities for Compact Normal Operators

We give singular value inequality to compact normal operators, which states that if is compact normal operator on a complex separable Hilbert space, where is the cartesian decomposition of , then Moreover, we give inequality which asserts that if?is compact normal operator, then .Several inequalities will be proved.

Normal values of flash visual evoked potentials in rabbits: Differences between the sexes and the eyes

BACKGROUND: The wave form, latency and wave amplitude of visual evoked potentials (VEP) are obviously affected by the stimulative parameters, physiological status of the subjects and anesthetics, thus there are greater normal variations and individual differences. The features of flash VEP (F-VEP) are to be observed. OBJECTIVE: To observe and compare the differences of F-VEP latencies and wave amplitudes between eyes in rabbits, and investigate the correlation with sex and the side of eyes. DESIGN: A comparative animal experiment. SETTING: Department of Neurosurgery, General Hospital of Chinese PLA. MATERIALS: The experiment was carried out in the neurophysiological laboratory of the Institute of Neurosciences, General Hospital of Chinese PLA from September 2004 to February 2005. Thirty big-ear rabbits of clean degree, 15 males and 15 females, weighing 2.0-2.5 kg, were provided by the animal center of the General Hospital of Chinese PLA. METHODS: Viking-IV perioperative monitor and flash stimulator for special use were applied. The rabbits were anesthetized with intramuscular injection of compound ketamine. The recording electrode was placed at 3 mm anterior to exoccipital tuberosity (onion, Oz), and the reference electrode was placed at the ear edge of the same side. The stimulative frequency was 1.9 Hz, and the amplifier was 50 μV; The range of wave filter was 5 Hz for high pass and 100 Hz for low pass; The average overlapping was 200 times, and the analytical time was 250 ms. MAIN OUTCOME MEASURES: Comparison of F-VEP wave forms; F-VEP latencies and wave amplitudes. RESULTS: All the 30 rabbits were involved in the analysis of results. ① Comparison of F-VEP wave forms in rabbits: The F-VEP waves mainly manifested as positive-negative-positive (PNP). The F-VEP manifestations to light stimulations were extremely similar between left and right eyes, and the wave amplitudes of both eyes were obviously increased. ② Determinations of F-VEP latencies and wave amplitudes: There were no significant differences in the latencies and wave amplitudes of F-VEP between the two groups (P > 0.05), the latencies of P1, N1 and P2 waves of left eye were (32.59±2.01), (43.85±4.35) and (66.20±8.13) ms, respectively, and the wave amplitudes of P1, N1 and P2 waves of left eye were (16.45±10.22), (7.93±3.56) and (17.62±8.18) μV. F-VEP latencies were stable with small changes, whereas amplitudes had greater changes. CONCLUSION: ① The latency and wave amplitudes F-VEP were stable with good repetition. ② The latencies and wave amplitudes had no significant differences between males and females, as well as between left and right eyes. Further investigation should be undertaken on the animal and clinical field.

Analysis on preserving or increasing the value of social security funds under the new normal economy in China

The ＂new normal＂ is the current stage of development of China＇s economy. The ＂new＂ in new normal means different to past; the ＂normal＂ means relatively stable. For a period of time in the future, the decrease of potential growth rate has become an indisputable fact. Social security fund is an important source of funds for the protection of people＇s livelihood, whose every movement often attract much attention. Under the situation of new normal economy, how to ensure the preserving or increasing value of social security funds on the premise of security is the core task in the operations management of social security fund investment. At present, there are some difficulties in the development of the social security funds, such as the management system and operation mechanism are not perfect, some institutional measures have not been fully implemented and the progress of the law and regulations is slow. In this paper, we will put forward some corresponding improvement measures, such as expand the scope of social security fund investment, accelerate the implementation of the nominal account system and strengthen the actuarial management of social security fund.

PRELIMINARY EXPLORATION OF NORMAL VALUE OF BA ON ADULT IN HEBEI REGION

目的:探讨河北省地区成年人的脑电地形图的正常值。方法:对60例正常成人通过BA-Ⅲ型脑电地形图仪进行了检查。结果:60例成年人的脑电地形图各频段的正常平均值被初步确定。结论:60例成年人的脑电地形图的正常值是有参考价值的。

PRELIMINARY EXPLORATION OF NORMAL VALUE OF BA ON CHILDREN IN HEBEI REGION

目的:探讨河北省地区儿童的脑电地形图的正常值。方法:对30例正常儿童通过BA-Ⅲ型脑电地形图仪进行了检查。结果:30例正常儿童地形图各频段的平均正常值被初步确定。结论:30例正常儿童的脑电地形图的正常值是有参考价值的。

SOME NORMALITY CRITERIA OF MEROMORPHIC FUNCTIONS

Let F be a family of functions meromorphic in a domain D, let n ≥ 2 be a positive integer, and let a ≠ 0, b be two finite complex numbers. If, for each f ∈ F, all of whose zeros have multiplicity at least k ＋ 1, and f ＋ a（f^（k））^n≠b in D, then F is normal in D.

ON HILBERT BOUNDARY VALUE PROBLEMS WITH RADICAL

The Hilbert boundary value problem Re{λ（t） p√ψ＋（t）} = c（t）, t∈L of normal type with Holder continuous coefficients is discussed, where L is the unit circle ｜t｜ = 1,p ≥2 is any definite integer,ψ^＋（t） is the boundary value of the unknown function ψ（z） holomorphic in ｜z｜ 〈 1 with single-valued continuous p√ψ＋（t） on L.

THE NORMAL FAMILITY OF MERMORPHIC FUNCTIONS

Let F be a family of mermorphic functions in a domain D, and let a, b, c be complex numbers, a ≠ b. If for each f ∈ F, the zeros of f-c are of multiplicity ≥ k ＋ 1, and -↑Ef（k）（a） belong to -↑Ef （a）, -↑Ef（k）（b）belong to -↑Ef （b）, then F is normal in D.