T-AND HAYMAN T-POINTS OF MEROMORPHIC FUNCTIONS FOR SMALL FUNCTIONS IN THE UNIT DISK

In this article, we consider the singular points of meromorphic functions in the unit disk. We prove the second fundamental theorem for the Ahlfors-Shimizu＇s characteristic in the unit disk in terms of Nevanlinna theory in the angular domains, and obtain the existence of T-points and Hayman T-points dealing with small functions as target.

TWO ENTIRE FUNCTIONS SHARING FOUR SMALL FUNCTIONS IN THE SENSE OF _(k))(β,f)=_(k))(β,g)

This paper proves a result that if two entire functions f(z) and g(z) share four small functions aj(z) (j = 1,2,3,4) in the sense of Ek)(aj, f) = Ek)(aj,g), (j = 1,2,3,4) (k ≥ 11), then there exists f(z) = g(z).

TWO MEROMORPHIC FUNCTIONS SHARE SOME PAIRS OF SMALL FUNCTIONS WITH TRUNCATED MULTIPLICITIES

Using the techniques proposed in [3], we prove that two nonconstant meromorphic functions f and g on C must be linked by a quasi-Mbius transformation if they share some pairs of small functions with more precise truncated multiplicities, which improve and extend the results of Duc Quang Si.

A Remark on n-small-functions Theorem of Steinmetz

This paper studies the Nevanlinna second fundamental theorem, and the extra tem T(r,f) in the n-small-functions theorem of Steinmetz is removed.

Uniqueness of Meromorphic Functions Sharing One or Two Small Functions IM

In this paper, the uniqueness problems of two meromorphic functions sharing one or two small functions IM are studied. By imposing certain conditions on the counting functions of four or three other small functions and constructing auxiliary function, some general uniqueness results are obtained.

Analyzing the Effect of the Intra-Pixel Position of Small PSFs for Optimizing the PL of Optical Subpixel Localization

Subpixel localization techniques for estimating the positions of point-like images captured by pixelated image sensors have been widely used in diverse optical measurement fields.With unavoidable imaging noise,there is a precision limit(PL)when estimating the target positions on image sensors,which depends on the detected photon count,noise,point spread function(PSF)radius,and PSF’s intra-pixel position.Previous studies have clearly reported the effects of the first three parameters on the PL but have neglected the intra-pixel position information.Here,we develop a localization PL analysis framework for revealing the effect of the intra-pixel position of small PSFs.To accurately estimate the PL in practical applications,we provide effective PSF(e PSF)modeling approaches and apply the Cramér–Rao lower bound.Based on the characteristics of small PSFs,we first derive simplified equations for finding the best PL and the best intra-pixel region for an arbitrary small PSF;we then verify these equations on real PSFs.Next,we use the typical Gaussian PSF to perform a further analysis and find that the final optimum of the PL is achieved at the pixel boundaries when the Gaussian radius is as small as possible,indicating that the optimum is ultimately limited by light diffraction.Finally,we apply the maximum likelihood method.Its combination with e PSF modeling allows us to successfully reach the PL in experiments,making the above theoretical analysis effective.This work provides a new perspective on combining image sensor position control with PSF engineering to make full use of information theory,thereby paving the way for thoroughly understanding and achieving the final optimum of the PL in optical localization.

On Some Sums Involving Small Arithmetic Functions

Let f be any arithmetic function and define S_(f)(x):=Σ_(n≤x)f([x/n]).If the function f is small,namely,f(n)﹤﹤n^(ε),then the error term E_(f)(x)in the asymptotic formula of S_f(x)has the form O(x^(1/2+ε)).In this paper,we shall study the mean square of E_(f)(x)and establish some new results of E_(f)(x)for some special functions.

Uniqueness Problems for Meromorphic Functions that Share Three Values

This paper investigate the uniqueness problems for meromorphic functions that share three values CM and proves a uniqueness theorem on this topic which can be used to improve some previous related results.

The Uniqueness of the Entire Functions Whose n-th Powers Share a Small Function with Their Derivatives

In this paper, we prove the following result： Let f（z） be a transcendental entire function, Q（z） ≡ 0 be a small function of f（z）, and n ≥ 2 be a positive integer. If fn（z） and（fn（z）） share Q（z） CM, then f（z） = ce 1 nz, where c is a nonzero constant. This result extends Lv＇s result from the case of polynomial to small entire function.

Further Extension on a Theorem of the Value Distribution of f′f^n

By using small function method, the following result is obtained. If f（z） is transcendental meromorphic and that ψ（z） is non-zero meromorphic and that T（r,ψ） = S（r, f）, then（n＋1）T（r,f）≤N^-（r,1/f＇f^n-ψ）＋2N^-（r,1/f）＋N^-（r,f）＋S（r,f）.

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）.

Nischarin-siRNA delivered by polyethyleniminealginate nanoparticles accelerates motor function recovery after spinal cord injury

A previous study by our group found that inhibition of nischarin promotes neurite outgrowth and neuronal regeneration in Neuro-2 a cells and primary cortical neurons.In recent years,more and more studies have shown that nanomaterials have good prospects in treatment of spinal cord injury.We proposed that small interfering RNA targeting nischarin（Nis-si RNA） delivered by polyethyleneimine-alginate（PEIALG） nanoparticles promoted motor function recovery in rats with spinal cord injury.Direct microinjection of 5 μL PEI-ALG/Nis-si RNA into the spinal cord lesion area of spinal cord injury rats was performed.From day 7 after surgery,Basso,Beattie and Bresnahan score was significantly higher in rats from the PEI-ALG/Nis-si RNA group compared with the spinal cord injury group and PEI-ALG/Control-si RNA group.On day 21 after injection,hematoxylin-eosin staining showed that the necrotic area was reduced in the PEI-ALG/Nis-si RNA group.Immunohistochemistry and western blot assay results confirmed successful inhibition of nischarin expression and increased protein expression of growth-associated protein-43 in the PEI-ALG/Nis-si RNA group.These findings suggest that a complex of PEI-ALG nanoparticles and Nis-si RNA effectively suppresses nischarin expression,induces expression of growth-associated protein-43,and accelerates motor function recovery after spinal cord injury.

Entire functions sharing some values with their difference operators

We investigate a uniqueness topic of entire functions with finite order sharing some values with their difference operators and obtain one result that if△f(≠0)and entire function f share a,b CM(counting multiplicities),then△f≡f.This result partially confirms a conjecture due to Chen and Yi(2013).

ABSORPTIVE FUNCTION OF SEGMENTAL ALLOTRANSPLANTED SMALL INTESTINES IN PIGS

Ten outbred pigs were each operated on for three times. First, a 130 cm length of terminal ileum of each pig was isolated on its vascular pedicle as a Thiry-Vella loop. One week later, the solitary ileal segments were transplanted heterotopically in two pigs. And after 28 days of heterotopic transplantation, the transplanted intestine was interposed into continuity of host intestine as orthotopic transplant. During the experiment, tests were made on 6th day after the first operation (period 1), the 14th (II), 28th (III) day after heterotopic transplantation, and 3 weeks after interposition (IV) respectively for the levels of glucose, palmitate and leucine. Additionally, at period I, III, and IV, a 3 cm length of intestinal mucosa was excised for morphologic observation and determination of DNA, RNA and protein contents. After heterotopic transplantation, the absorptive function of transplanted intestine was severely impaired for two weeks. The absorption of glucose! and palmitate was partially recovered by period III, at which time leucine level had return to normal. At period IV, the absorptive function of glucose and leucine had surpassed normal levels, while palmitate had risen to the level of pretransplantation. After transplantation, at period III, DNA, RNA and protein contents were well below normal. Three weeks after orthotopic transplantation, RNA and protein had risen to normal level, while DNA content remained below normal, The morphologic changes during the experiment were correlated with the changes of contents in RNA, protein and DNA. The area, height, width of villi and the area, depth, width:of crypt were below normal at III and recovered by 3 weeks after orthotopic transplantation (period IV), but were still lower than the levels at pretransplantation. Crypt depths were deeper than those of pretransplantation.

On Meromorphic Solutions of a Certain Type of Nonlinear Differential Equations

We consider transcendental merornorphic solutions with N（r, f） = S（r, f） of the following type of nonlinear differential equations：f^n ＋ Pn-2（f） = p1（z）e~α1（z） ＋ p2（z）^α2（z）,where n ≥2 is an integer, Pn-2（f） is a differential polynomial in f of degree not greater than n-2 with small functions of f as its coefficients, p1（z）, p2（z） are nonzero small functions of f1 and α1（z）, α2（z） are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms of the above equation. Our results extend and improve some known results obtained most recently.

Oscillation Result for Some Non-homogeneous Linear Differential Equations and Its Application

This paper investigates solutions of some non-homogeneous linear differential equations, which have non-homogeneous term as the small function of solution. Using the similar method, we can generalize the result of G.Gundersen and L.Z.Yang.

Julia Sets of Skew Products for Class MA_p

Let fj M (j = 1, 2, …, m; m1) and %f be the skew product associated with the generator system {f1, f2, …, fm}. Then F(%f) is completely invariant under (%f); J(%f) is completely invariant under%f; J(%f) is perfect; J(%f) has interior points if and only if F(%f) =; if fj MAp (p5), j = 1, 2, …, m, then the set of the repelling fixed points of%fof all orders are dense in J(%f).