SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION

This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B^(4),under the smallest regularity assumptions of V,ω,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the Lp type regularity theory of[10],and generalizes the work of Du,Kang and Wang[4]on a second order problem to our fourth order problems.

China:In the fourth quarter of 2023,exports of yarn and fabric to ASEAN ceased declining and rebounded

On January 8th,the China Chamber of Commerce for Import and Export of Textiles released data showing that in the fourth quarter,China’s textile and clothing exports gradually stabilised.The proportion of intermediate exports in textile and clothing exports continued to rise,providing a new impetus for the substantial expansion of exports to traditional markets and international supply chain cooperation.

Existence of Monotone Positive Solution for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function

This paper is concerned with the following fourth-order three-point boundary value problem , where , we discuss the existence of positive solutions to the above problem by applying to the fixed point theory in cones and iterative technique.

Fenlong Cultivation-the Fourth Set of Farming Methods Invented in China

The basic theory and effect of the new farming method of ＂Fenlong＂ cultivation which has been included in the main extension technology of Ministry of Agriculture of the People＇s Republic of China is fully illustrated for the first time, and it is the fourth set （generation） of farming modes and methods following manpower, animal and mechanical （tractor） farming. It follows the natural law to achieve soil activation, water saving, oxygen increase, warming and desalination through the active use of natural resources like soil, rainfall and solar energy, thereby promoting a new round of natural agricultural production and quality improvement and water con- servation, which has crop yield increase by 10%-30%, quality improvement of 5%, natural precipitation retaining increase by100%. The characteristics and mechanism are the use of spiral drill for one-time completion of the land preparation by drilling vertically to 30-50 cm of soil layer through high speed peeling. After instant high temperature and many fierce impacts, mechanical frictions, it could achieve the multiplication of the number of loose soil, soil physical modification and expansion of the soil nutrients, reservoirs, oxygen, microorganisms （＂Four pools＂）. Fenlong cultivation can give birth to new farming culture and civilization, and it can achieve the physical ＂desalinized＂ transformation and utilization of saline soil. The formation of Fenlong green farming technology system makes it possible to invent the farming tools of ＂serf-propelled Fenlong machinery＂ that has got the patent, and it is the method for farmland （dry land, paddy field） Fenlong cultivation, saline-alkali soil smash-ridging cultivation and for the abundance of grass ecology on degraded grassland. The application of Fenlong ＂4＋1＂ （arable, saline-alkali soil, grasslands, Sponge City＋rivers） green development in China can achieve the ＂double safety＂ of food and living space.

ON STABILITY OF SOLUTIONS OF CERTAIN FOURTH-ORDER DELAY DIFFERENTIAL EQUATIONS

By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established result is less restrictive than those reported in the literature.

Stokes'first problem for the fourth order fluid in a porous half space

In this study, the flow of a fourth order fluid in a porous half space is modeled. By using the modified Darcy＇s law, the flow over a suddenly moving flat plate is studied numerically. The influence of various parameters of interest on the velocity profile is revealed.

ON THE COEFFICIENTS OF DIFFERENTIATED EXPANSIONS AND DERIVATIVES OF CHEBYSHEV POLYNOMIALS OF THE THIRD AND FOURTH KINDS

Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved. Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given. Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced. As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems, two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.

MULTIPLICITY RESULTS FOR FOURTH ORDER ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE

In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-（a＋bfR^3｜↓△u｜^2dx）△u＋V（x）u=f（x,u）,x∈R^3, u∈H^2（R3）,where a, b 〉 0 are constants and the primitive of the nonlinearity f is of superlinear growth near infinity in u and is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the Ambrosetti- Rabinowitz type condition.

Heat transfer on peristaltic flow of fourth grade fluid in inclined asymmetric channel with partial slip

In this paper, the effects of slip and heat transfer are studied on the peristaltic transport of a magnetohydrodynamic （MHD） fourth grade fluid. The governing equations are modeled and solved under the long wavelength approximation by using a regular perturbation method. Explicit expressions of solutions for the stream function, the velocity, the pressure gradient, the temperature, and the heat transfer coefficient are presented. Pumping and trapping phenomena are analyzed for increasing the slip parameter. Further, the temperature profiles and the heat transfer coefficient are observed for various increasing parameters. It is found that these parameters considerably affect the considered flow characteristics. Comparisons with published results for the no-slip case are found in close agreement.

Existence of solutions and positive solutions to a fourth-order two-point BVP with second derivative

Several existence theorems were established for a nonlinear fourth-order two-point boundary value problem with second derivative by using Leray-Schauder fixed point theorem, equivalent norm and technique on system of integral equations. The main conditions of our results are local. In other words, the existence of the solution can be determined by considering the height of the nonlinear term on a bounded set. This class of problems usually describes the equilibrium state of an elastic beam which is simply supported at both ends.

Thermophoresis and concentration effects in a fourth grade peristaltic flow with convective walls

In this investigation,we have studied the peristaltic fluid flow in an asymmetric channel with convective walls.Fourth grade fluid model has been utilized in view of the fact that the results of all other differential type models can be deduced as the special case.Combined effects of heat and mass transfer are considered.The thermophoresis effects occur in the energy equation.Convective heat and mass boundary conditions have been given special attention.Long wave length and low Reynolds number approximations are utilized for the simplifications.Approximate analytic solutions for the velocity,temperature and concentration profiles are calculated using perturbation technique and elaborated in the form of graphical observations for various physical quantities.

Effects of the fourth component and undercooling on morphology of primary Mg-Zn-Y icosahedral quasicrystal phase under normal casting conditions

The paper presents some results of the investigation on effects of the fourth component(Ti,C,Sb or Cu)and undercooling on the morphology,size and forming process of primary Mg-Zn-Y icosahedral quasicrystal phase(I-phase)under normal casting conditions.The result shows that the addition of certain amount of fourth component can transform I-phase morphology from petal-like to spherical.However,I-phase will grow up to petal-like if superfluous addition of the fourth component applied.It is also found that the solidified morphology of I-phase depends on the stability of spherical I-phase during the subsequent growth,and critical radius of maintaining the spherical I-phase interface relatively stable.Further,mini-sized spherical I-phase can be produced with high content of the fourth component by undercooling.Such findings are beneficial for industrializing Mgbased quasicrystals.

Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem—Part III. An Original Solution of Hilbert’s Fourth Problem

This article refers to the “Mathematics of Harmony” by Alexey Stakhov [1], a new interdisciplinary direction of modern science. The main goal of the article is to describe two modern scientific discoveries—New Geometric Theory of Phyl-lotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci -Goniometry ( is a given positive real number). Although these discoveries refer to different areas of science (mathematics and theoretical botany), however they are based on one and the same scien-tific ideas—The “golden mean,” which had been introduced by Euclid in his Elements, and its generalization—The “metallic means,” which have been studied recently by Argentinian mathematician Vera Spinadel. The article is a confirmation of interdisciplinary character of the “Mathematics of Harmony”, which originates from Euclid’s Elements.

Hybrid Effects of Thermal and Concentration Convection on Peristaltic Flow of Fourth Grade Nanofluids in an Inclined Tapered Channel: Applications of Double-Diffusivity

This article brings into focus the hybrid effects of thermal and concentration convection on peristaltic pumping of fourth grade nanofluids in an inclined tapered channel.First,the brief mathematical modelling of the fourth grade nanofluids is provided along with thermal and concentration convection.The Lubrication method is used to simplify the partial differential equations which are tremendously nonlinear.Further,analytical technique is applied to solve the differential equations that are strongly nonlinear in nature,and exact solutions of temperature,volume fraction of nanoparticles,and concentration are studied.Numerical and graphical findings manifest the influence of various physical flow-quantity parameters.It is observed that the nanoparticle fraction decreases because of the increasing values of Brownian motion parameter and Dufour parameter,whereas the behaviour of nanoparticle fraction is quite opposite for thermophoresis parameter.It is also noted that the temperature profile decreases with increasing Brownian motion parameter values and rises with Dufour parameter values.Moreover,the concentration profile ascends with increasing thermophoresis parameter and Soret parameter values.

Effective administration of cranial drilling therapy in the treatment of fourth degree temporal,facial and upper limb burns at high altitude:A case report

BACKGROUND Fourth degree burns damage the full thickness of the skin and affect underlying tissues.Skin grafting after debridement is often used to cover the wounds of salvageable severe burns.A granulation wound can be formed by drilling the skull to the barrier layer to solve the problem of skull exposure.Low oxygen levels present at high altitudes aggravate ischemia and hypoxia which can negatively impact wound healing.The impaired healing in such cases can be ameliorated by hyperbaric oxygen therapy.CASE SUMMARY We describe a patient who presented with fourth degree burns to the left temporal and facial regions upon admission in December 2018.The periosteum of the skull and the deep fascia of the face were exposed.After the first stage of debridement and skin grafting,the temporal skin did not survive well.Granulation was induced by cranial drilling,and then a local flap was transferred to cover the wound.The left temporal and facial wounds were completely covered and the patient recovered well.CONCLUSION Skin grafting and flap transfer after early debridement to cover the wound and control infection were of great significance.In the later stages of the patient's treatment,survival of the skin graft and skin flap was observed.The second stage repair was performed to achieve successful skin grafting by cranial granulation.Granulation was formed by drilling the skull,and then the wound was closed,which is suitable for cases with skull exposure and wounds with poor blood supply.We consider that hyperbaric oxygen treatment and improving tissue oxygen supply were beneficial in this patient.

ON THE BOUNDEDNESS AND THE STABILITY RESULTS FOR THE SOLUTION OF CERTAIN FOURTH ORDER DIFFERENTIAL EQUATIONS VIA THE INTRINSIC METHOD

In this paper, we first present constructing a Lyapunov function for (1. 1) and then we show the asymptotic stability in the large of the trivial solution x=0 for case p≡ 0,and the boundedness result of the solutions of (1 .1 ) for case p≠0. These results improve sveral well-known results.

Fourth Industrial Revolution:Technological Drivers,Impacts and Coping Methods

The world is marching into a new development period when the digital technology,physical technology,and biological technology have achieved an unprecedented development respectively in their own fields,and at the same time their applications are converging greatly.These are the three major technological drivers for the Fourth Industrial Revolution.This paper discusses the specific technology niches of each kind technological driver behind the Fourth Industrial Revolution,and then evaluates impacts of the Fourth Industrial Revolution on global industrial,economic,and social development.At last this paper proposes possible measures and policies for both firms and governments to cope with the changes brought by the Fourth Industrial Revolution.

An Efficient Technique for Finding the Eigenvalues of Fourth-Order Sturm-Liouville Problems

In this work, we present a computational method for solving eigenvalue problems of fourth-order ordinary differential equations which based on the use of Chebychev method. The efficiency of the method is demonstrated by three numerical examples. Comparison results with others will be presented.

Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem—Part II. A New Geometric Theory of Phyllotaxis (Bodnar’s Geometry)

This article refers to the “Mathematics of Harmony” by Alexey Stakhov in 2009, a new interdisciplinary direction of modern science. The main goal of the article is to describe two modern scientific discoveries–New Geometric Theory of Phyllotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci λ-Goniometry (λ > 0 is a given positive real number). Although these discoveries refer to different areas of science (mathematics and theoretical botany), however they are based on one and the same scientific ideas-the “golden mean,” which had been introduced by Euclid in his Elements, and its generalization—the “metallic means,” which have been studied recently by Argentinian mathematician Vera Spinadel. The article is a confirmation of interdisciplinary character of the “Mathematics of Harmony”, which originates from Euclid’s Elements.

Hydrocarbon accumulation and exploration prospect of mound-shoal complexes on the platform margin of the fourth member of Sinian Dengying Formation in the east of Mianzhu-Changning intracratonic rift, Sichuan Basin, SW China

Drilling,seismic and logging data were used to evaluate the hydrocarbon accumulation conditions of the mound-shoal complexes in the platform margin of the fourth member of Sinian Dengying Formation in the east side of the Mianzhu-Changning intracratonic rift in the Sichuan Basin.The four understandings are:(1)The platform margin belt of the Deng 4 Member can be divided into three sections,northern,middle and southern;the middle section is at the core of the Gaoshiti-Moxi paleouplift and the structural high now,while the southern and northern sections are at the slope of the paleouplift and the structural lows now;the three sections have similar development characteristics and reservoir features of platform margin mound-shoal complex.(2)In the margin of the east side of the rift,there are several faults nearly perpendicular to the platform margin belt,the faults divide the platform margin belt into rugged paleo-landform,and the high part developed platform margin mound-shoal complexes and the reservoirs are good in physical properties,while the low part developed inter-beach depression and no mound-shoal complexes,where the reservoirs are poor in physical properties.(3)The six groups of faults nearly perpendicular to the platform margin belt divide the platform margin belt into seven large mound-shoal complexes which have similar hydrocarbon accumulation conditions and accumulation evolution process and are rich in petroleum.(4)The inter shoal depressions between the mound-shoal complexes are characterized by tighter lithology,which can block the updip direction of the mounds and shoals at the lower part of the slope of the paleouplift and are favorable for the later preservation of mound-shoal gas reservoirs.This has been proved by Well Jiaotan 1 and Heshen 2 drilled successfully.The mound-shoal complexes on the platform margin of the structural slope area have a good exploration prospect.