On entire solutions of some Fermat type differential-difference equations

On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].

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.

ENTIRE SOLUTIONS OF LOTKA-VOLTERRA COMPETITION SYSTEMS WITH NONLOCAL DISPERSAL

This paper is mainly concerned with entire solutions of the following two-species Lotka-Volterra competition system with nonlocal(convolution)dispersals:{u_(t)=k*u-u+u(1-u-av),x∈R,t∈R,vt=d(k*v-v)+rv(1-v-bu),c∈R,t∈R.(0.1)Here a≠1,b≠1,d,and r are positive constants.By studying the eigenvalue problem of(0.1)linearized at(ϕc(ξ),0),we construct a pair of super-and sub-solutions for(0.1),and then establish the existence of entire solutions originating from(ϕc(ξ),0)as t→−∞,whereϕc denotes the traveling wave solution of the nonlocal Fisher-KPP equation ut=k*u−u+u(1−u).Moreover,we give a detailed description on the long-time behavior of such entire solutions as t→∞.Compared to the known works on the Lotka-Volterra competition system with classical diffusions,this paper overcomes many difficulties due to the appearance of nonlocal dispersal operators.

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.

A chaotic hierarchical encryption/watermark embedding scheme for multi-medical images based on row-column confusion and closed-loop bi-directional diffusion

Security during remote transmission has been an important concern for researchers in recent years.In this paper,a hierarchical encryption multi-image encryption scheme for people with different security levels is designed,and a multiimage encryption(MIE)algorithm with row and column confusion and closed-loop bi-directional diffusion is adopted in the paper.While ensuring secure communication of medical image information,people with different security levels have different levels of decryption keys,and differentiated visual effects can be obtained by using the strong sensitivity of chaotic keys.The highest security level can obtain decrypted images without watermarks,and at the same time,patient information and copyright attribution can be verified by obtaining watermark images.The experimental results show that the scheme is sufficiently secure as an MIE scheme with visualized differences and the encryption and decryption efficiency is significantly improved compared to other works.

Evaluation of hybrid closed-loop insulin delivery system in type 1 diabetes in real-world clinical practice:One-year observational study

BACKGROUND In 2016,the Food and Drug Administration approved the first hybrid closed-loop(HCL)insulin delivery system for adults with type 1 diabetes(T1D).There is limited information on the impact of using HCL systems on patient-reported outcomes(PROs)in patients with T1D in real-world clinical practice.In this independent study,we evaluated glycemic parameters and PROs over one year of continuous use of Medtronic’s 670G HCL in real-world clinical practice.AIM To assess the effects of hybrid closed loop system on glycemic control and quality of life in adults with T1D.METHODS We evaluated 71 patients with T1D(mean age:45.5±12.1 years;59%females;body weight:83.8±18.7 kg,body mass index:28.7±5.6 kg/m2,A1C:7.6%±0.8%)who were treated with HCL at Joslin Clinic from 2017 to 2019.We measured A1C and percent of glucose time-in-range(%TIR)at baseline and 12 months.We measured percent time in auto mode(%TiAM)for the last two weeks preceding the final visit and assessed PROs through several validated quality-of-life surveys related to general health and diabetes management.RESULTS At 12 mo,A1C decreased by 0.3%±0.1%(P=0.001)and%TIR increased by 8.1%±2.5%(P=0.002).The average%TiAM was only 64.3%±32.8%and was not associated with A1C,%TIR or PROs.PROs,provided at baseline and at the end of the study,showed that the physical functioning submodule of 36Item Short-Form Health Survey increased significantly by 22.9%(P<0.001).Hypoglycemia fear survey/worry scale decreased significantly by 24.9%(P<0.000);Problem Areas In Diabetes reduced significantly by-17.2%(P=0.002).The emotional burden submodules of dietary diversity score reduced significantly by-44.7%(P=0.001).Furthermore,analysis of Clarke questionnaire showed no increase in awareness of hypoglycemic episodes.WHO-5 showed no improvements in subject’s wellbeing among participants after starting the 670G HCL system.Finally,analysis of Pittsburgh Sleep Quality Index showed no difference in sleep quality,sleep latency,or duration of sleep from baseline to 12 mo.CONCLUSION The use of HCL in real-world clinical practice for one year was associated with significant improvements in A1C,%TIR,physical functioning,hypoglycemia fear,emotional distress,and emotional burden related to diabetes management.However,these changes were not associated with time in auto mode.

On the growth of transcendental entire solutions of algebraic differential equations

In this paper, we investigate the growth of transcendental entire solutionsof the following algebraic differential equation a(z)f'~2 +(b_2(z)f^2 +b_1(z)f +b_0(z))f'=d_3(z)f^3+d_2(z)f^2 +d_1(z)f +d_0(z), where a(z), b_i(z) (0<- i <=2) and d_j (z) (0<=j<= 3) are allpolynomials, and this equation relates closely to the following well-known algebraic differentialequation C(z,w)w'~2 + B(z,w)w' + A(z,w) =0, where G(z,w)not ident to 0, B(z,w) and A(z,w) are threepolynomials in z and w. We give relationships between the growth of entire solutions and the degreesof the above three polynomials in detail.

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 .

UNIQUENESS OF ENTIRE FUNCTIONS SHARING ONE VALUE

We deal with the problem of entire functions sharing one value weakly. Moreover, we improve and generalize some former results obtained by J.-F.Chen, et al. [6], Y.Xu and H.L.Qiu [4], M.L. Fang [5], C.C. Yang, and X.H. Hua [3].

ON ENTIRE SOLUTIONS OF TWO TYPES OF SYSTEMS OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS

In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.

ENTIRE FUNCTIONS SHARING ONE SMALL FUNCTION CM WITH THEIR SHIFTS AND DIFFERENCE OPERATORS

In this article, we mainly devote to proving uniqueness results for entire functionssharing one small function CM with their shift and difference operator simultaneously. Letf（z） be a nonconstant entire function of finite order, c be a nonzero finite complex constant, and n be a positive integer. If f（z）, f（z＋c）, and △n cf（z） share 0 CM, then f（z＋c）≡Af（z）, where A（≠0） is a complex constant. Moreover, let a（z）, b（z）（ O） ∈ S（f） be periodic entire functions with period c and if f（z） - a（z）, f（z ＋ c） - a（z）, △cn f（z） - b（z） share 0 CM, then f（z ＋ c） ≡ f（z）.

NON-EXISTENCE OF POSITIVE ENTIRE SOLUTIONS FOR ELLIPTIC INEQUALITIES OF P-LAPLACIAN

In this paper, the nonexistence of positive entire solutions for div(|Du| p-2 Du)q(x)f(u),x∈R N, is established, where p】1,Du=(D 1u,...,D Nu),q∶R N→(0,∞) and f∶(0,∞)→(0,∞) are continuous functions.

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).

ON ENTIRE SOLUTIONS OF SOME TYPE OF NONLINEAR DIFFERENCE EQUATIONS

In this article, the existence of finite order entire solutions of nonlinear difference equations fn＋ Pd（z, f） = p1 eα1 z＋ p2 eα2 z are studied, where n ≥ 2 is an integer, Pd（z, f） is a difference polynomial in f of degree d（≤ n-2）, p1, p2 are small meromorphic functions of ez, and α1, α2 are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.

LIMITING DIRECTIONS OF JULIA SETS OF ENTIRE SOLUTIONS TO COMPLEX DIFFERENTIAL EQUATIONS

In this paper,we mainly investigate the dynamical properties of entire solutions of complex differential equations.With some conditions on coefficients,we prove that the set of common limiting directions of Julia sets of solutions,their derivatives and their primitives must have a definite range of measure.

LIMITING DIRECTION AND BAKER WANDERING DOMAIN OF ENTIRE SOLUTIONS OF DIFFERENTIAL EQUATIONS

In this paper, we mainly investigate entire solutions of complex differential equations with coefficients involving exponential functions, and obtain the dynamical properties of the solutions, their derivatives and primitives. With some conditions on coefficients, for the solutions, their derivatives and their primitives, we consider the common limiting directions of Julia set and the existence of Baker wandering domain.

Is it necessary to use the entire root as a donor when transferring contralateral C7 nerve to repair median nerve？

If a partial contralateral C7 nerve is transferred to a recipient injured nerve, results are not satisfactory. However, if an entire contralateral C7 nerve is used to repair two nerves, both recipient nerves show good recovery. These findings seem contradictory, as the above two methods use the same donor nerve, only the cutting method of the contralateral C7 nerve is different. To verify whether this can actually result in different repair effects, we divided rats with right total brachial plexus injury into three groups. In the entire root group, the entire contralateral C7 root was transected and transferred to the median nerve of the affected limb. In the posterior division group, only the posterior division of the contralateral C7 root was transected and transferred to the median nerve. In the entire root ＋ posterior division group, the entire contralateral C7 root was transected but only the posterior division was transferred to the median nerve. After neurectomy,the median nerve was repaired on the affected side in the three groups. At 8, 12, and 16 weeks postoperatively, electrophysiological examination showed that maximum amplitude, latency, muscle tetanic contraction force, and muscle fiber cross-sectional area of the flexor digitorum superficialis muscle were significantly better in the entire root and entire root ＋ posterior division groups than in the posterior division group. No significant difference was found between the entire root and entire root ＋ posterior division groups. Counts of myelinated axons in the median nerve were greater in the entire root group than in the entire root ＋ posterior division group, which were greater than the posterior division group. We conclude that for the same recipient nerve, harvesting of the entire contralateral C7 root achieved significantly better recovery than partial harvesting, even if only part of the entire root was used for transfer. This result indicates that the entire root should be used as a donor when transferring contralateral C7 nerve.

Entire solutions of a certain type of functional-differential equations

In this paper, we shall utilize Nevanlinna value distribution theory and normality theory to study the solvability of a certain type of functional-differential equations. We also consider the solutions of some nonlinear differential equations.

ON ALGEBRAIC SOLUTIONS OF THE SECOND-ORDER COMPLEX DIFFERENTIAL EQUATIONS WITH ENTIRE ALGEBRAIC ELEMENT COEFFICIENTS

The main purpose of this paper is to study the problems on the existence of algebraic solutions for some second-order complex differential equations with entire algebraic function element coeifficients. Several theorems on the existence of solutions are obtained, which perfect the solution theory of linear complex differential equations.

Entire Catchment and Buffer Zone Approaches to Modeling Linkage Between River Water Quality and Land Cover——A Case Study of Yamaguchi Prefecture,Japan

This study investigated the linkages between river water quality and land use in river catchments in Yama- guchi Prefecture, the western Japan, in order to examine the effect of land use changes of both entire catchment and buffer zone on river water quality. Dissolved Oxygen (DO), pH, Biological Oxygen Demand (BOD), Suspended Solids (SS), Escherichia coli, Total Nitrogen (TN) and Total Phosphorus (TP) were considered as river water quality indica- tors. Satellite images were applied to generating the land use map. Multiple regression model was applied to linking the changes in the river water quality with the land uses in both entire catchment area and buffer zone. The results in- dicate that the integrative application of land use data from the entire catchment and the buffer zone could give rise to more robust model to predict the concentrations of Suspended Solids (r2=0.88) and Total Nitrogen (r2=0.90), rather than models which separately considered land use data in catchment and buffer zone.