On the Weyl’s Lemma for Triharmonic Functions

In this paper,by choosing some appropriate test functions,we prove the Weyl’s lemma for triharmonic functions based on the new type of mean value formulas.

A Set of Almost Automorphic Functions and Applications

For a set S of real numbers, we introduce the concept of S-almost automorphic functions valued in a Banach space. It generalizes in particular the space of Z-almost automorphic functions. Considering the space of S-almost automorphic functions, we give sufficient conditions of the existence and uniqueness of almost automorphic solutions of a differential equation with a piecewise constant argument of generalized type. This is done using the Banach fixed point theorem.

Deep brain implantable microelectrode arrays for detection and functional localization of the subthalamic nucleus in rats with Parkinson’s disease

The subthalamic nucleus(STN)is considered the best target for deep brain stimulation treatments of Parkinson’s disease(PD).It is difficult to localize the STN due to its small size and deep location.Multichannel microelectrode arrays(MEAs)can rapidly and precisely locate the STN,which is important for precise stimulation.In this paper,16-channel MEAs modified with multiwalled carbon nanotube/poly(3,4-ethylenedioxythiophene):poly(styrene sulfonate)(MWCNT/PEDOT:PSS)nanocomposites were designed and fabricated,and the accurate and rapid identification of the STN in PD rats was performed using detection sites distributed at different brain depths.These results showed that nuclei in 6-hydroxydopamine hydrobromide(6-OHDA)-lesioned brains discharged more intensely than those in unlesioned brains.In addition,the MEA simultaneously acquired neural signals from both the STN and the upper or lower boundary nuclei of the STN.Moreover,higher values of spike firing rate,spike amplitude,local field potential(LFP)power,and beta oscillations were detected in the STN of the 6-OHDA-lesioned brain,and may therefore be biomarkers of STN localization.Compared with the STNs of unlesioned brains,the power spectral density of spikes and LFPs synchronously decreased in the delta band and increased in the beta band of 6-OHDA-lesioned brains.This may be a cause of sleep and motor disorders associated with PD.Overall,this work describes a new cellular-level localization and detection method and provides a tool for future studies of deep brain nuclei.

Functions of Allusions from Leppihalme’s model

The general function of allusions is often thought to add clarity and significance to ideas and descriptions.However,it would be difficult to establish an exhaustive list of mutually exclusive category

SOME INEQUALITIES OF HERMITE-HADAMARD TYPE FOR s-CONVEX FUNCTIONS

In this paper several inequalities of the left-hand side of Hermite-Hadamard’s inequality are obtained for s-convex functions.

Green's functions for uniformly distributed loads acting on an inclined line in a poroelastic layered site

Based on one type of practical Biot＇s equation and the dynamic-stiffness matrices ofa poroelastic soil layer and half-space, Green＇s functions were derived for unitformly distributed loads acting on an inclined line in a poroelastie layered site. This analysis overcomes significant problems in wave scattering due to local soil conditions and dynamic soil-structure interaction. The Green＇s functions can be reduced to the case of an elastic layered site developed by Wolf in 1985. Parametric studies are then carried out through two example problems.

Integral Inequalities of Hermite-Hadamard Type for Functions Whose 3rd Derivatives Are <i>s</i>-Convex

In the paper, the authors find some new inequalities of Hermite-Hadamard type for functions whose third derivatives are s-convex and apply these inequalities to discover inequalities for special means.

ON THE LOWER BOUND FOR A CLASS OF HARMONIC FUNCTIONS IN THE HALF SPACE

The main objective is to derive a lower bound from an upper one for harmonic functions in the half space, which extends a result of B. Y. Levin from dimension 2 to dimension n 〉 2. To this end, we first generalize the Carleman＇s formula for harmonic functions in the half plane to higher dimensional half space, and then establish a Nevanlinna＇s representation for harmonic functions in the half sphere by using HSrmander＇s theorem.

Decoupling coefficients of dilatational wave for Biot's dynamic equation and its Green's functions in frequency domain

Green＇s functions for Blot＇s dynamic equation in the frequency domain can be a highly useful tool for the investigation of dynamic responses of a saturated porous medium. Its applications are found in soil dynamics, seismology, earthquake engineering, rock mechanics, geophysics, and acoustics. However, the mathematical work for deriving it can be daunting. Green＇s functions have been presented utilizing an analogy between the dynamic thermoelasticity and the dynamic poroelasticity in the frequency domain using the u-p formulation. In this work, a special term ＂decoupling coefficient＂ for the decomposition of the fast and slow dilatational waves is proposed and expressed to present a new methodology for deriving the poroelastodynamic Green＇s functions. The correct- ness of the solution is demonstrated by numerically comparing the current solution with Cheng＇s previous solution. The separation of the two waves in the present methodology allows the more accurate evaluation of Green＇s functions, particularly the solution of the slow dilatational wave. This can be advantageous for the numerical implementation of the boundary element method （BEM） and other applications.

TIME-HARMONIC DYNAMIC GREEN'S FUNCTIONS FOR ONE-DIMENSIONAL HEXAGONAL QUASICRYSTALS

Quasicrystals have additional phason degrees of freedom not found in conventional crystals. In this paper, we present an exact solution for time-harmonic dynamic Green＇s function of one-dimensional hexagonal quasicrystals with the Laue classes 6/mh and 6/mhmm. Through the introduction of two new functions φ and ψ, the original problem is reduced to the determination of Green＇s functions for two independent Helmholtz equations. The explicit expressions of displacement and stress fields are presented and their asymptotic behaviors are discussed. The static Green＇s function can be obtained by letting the circular frequency approach zero.

In-plane dynamic Green's functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic half-space

The dynamic stiffness method combined with the Fourier transform is utilized to derive the in-plane Green’s functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic（TI）half-space.The loaded layer is fixed to obtain solutions restricted in it and the corresponding reactions forces,which are then applied to the total system with the opposite sign.By adding solutions restricted in the loaded layer to solutions from the reaction forces,the global solutions in the wavenumber domain are obtained,and the dynamic Green’s functions in the space domain are recovered by the inverse Fourier transform.The presented formulations can be reduced to the isotropic case developed by Wolf（1985）,and are further verified by comparisons with existing solutions in a uniform isotropic as well as a layered TI halfspace subjected to horizontally distributed loads which are special cases of the more general problem addressed.The deduced Green’s functions,in conjunction with boundary element methods,will lead to significant advances in the investigation of a variety of wave scattering,wave radiation and soil-structure interaction problems in a layered TI site.Selected numerical results are given to investigate the influence of material anisotropy,frequency of excitation,inclination angle and layered on the responses of displacement and stress,and some conclusions are drawn.

Effect of vegetation on soil bacteria and their potential functions for ecological restoration in the Hulun Buir Sandy Land,China

To date,much of research on revegetation has focused on soil microorganisms due to their contributions in the formation of soil and soil remediation process.However,little is known about the soil bacteria and their functions respond to the diverse vegetational types in the process of vegetation restoration.Effects of dominated vegetation,i.e.,Artemisia halodendron Turcz Ex Bess,Caragana microphylla Lam.,Hedysarum fruticosum Pall.and Pinus sylvestris L.on bacterial community structures and their potential functions in the Hulun Buir Sandy Land,China were determined using high-throughput 16S rRNA gene sequencing and phylogenetic investigation of communities by reconstruction of unobserved states(PICRUSt)in 2015.Although the dominant phyla of soil bacterial community among different types of vegetation,including Proteobacteria,Actinobacteria,Acidobacteria,Bacteroidetes and Firmicutes,were similar,the relative abundance of these dominant groups significantly differed,indicating that different types of vegetation might result in variations in the composition of soil bacterial community.In addition,functional genes of bacterial populations were similar among different types of vegetation,whereas its relative abundance was significantly differed.Most carbon fixation genes showed a high relative abundance in P.sylvestris,vs.recalcitrant carbon decomposition genes in A.halodendron,suggesting the variations in carbon cycling potential of different types of vegetation.Abundance of assimilatory nitrate reduction genes was the highest in P.sylvestris,vs.dissimilatory nitrate reduction and nitrate reductase genes in A.halodendron,indicating higher nitrogen gasification loss and lower nitrogen utilization gene functions in A.halodendron.The structures and functional genes of soil bacterial community showed marked sensitivities to different plant species,presenting the potentials for regulating soil carbon and nitrogen cycling.

THE WETLAND TYPES,FUNCTIONS ANDCONSERVATION IN CHINA

ABSTRACT: According to preliminary statistics, there are 9. 4 ×106ha of mire, 8. 0 × 106ha of lake, 2. 1 × 106ha of salt marsh, 2.1 × 107ha of shallow sea (0 - 5m), and 3. 8 × 107ha of paddyfield, their total area amounts to 8. 45× 107ha. Wetland consists of natural wetland system and man-made wetland system. According to hydrology, landform, soil and vegetation etc., natural wetland can be divided into the following types: marine, esturine, riverine, lacustrine, palustrine subsystems. On the basis of the wetland bottom compound, waterlogged state and vegetation forms, it can be subdivided into 26 wetland classes. Man-made wetland can be subdivided into 4 wetland classes. Wetland is a unique landscape in the earth and one of the most important living environment with rich resources and many functions. At present, 262 different types of Wetland Natural Reserves have been established in China, in which 7 Wetland Nature Reserves have been listed in international important wetlands of The Wetland

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.

Comprehensive form of solution for Lamb's dynamic problem expressed by Green's functions

By the analysis for the vectors of a wave field in the cylindrical coordinate and Sommerfeld＇s identity as well as Green＇s functions of Stokes＇ solution pertaining the conventional elastic dynamic equation, the results of Green＇s function in an infinite space of an axisymmetric coordinate are shown in this paper. After employing a supplementary influence field and the boundary conditions in the free surface of a senti-space, the authors obtain the solutions of Green＇s function for Lamb＇s dynamic problem. Besides, the vertical displacement uzz and the radial displacement urz can match Lamb＇s previous results, and the solutions of the linear expansion source u^r and the linear torsional source uee are also given in the paper. The authors reveal that Green＇s function of Stokes＇ solution in the semi-space is a comprehensive form of solution expressing the dynamic Lamb＇s problem for various situations. It may benefit the investigation of deepening and development of Lamb＇s problems and solution for pertinent dynamic problems conveniently.

<i>Ratanasampil</i>(Tibetan Medicine, RNSP) Reduces <i>β</i>-Amyloid Protein (Aβ) and Pro-Inflammatory Factor Levels and Improves Cognitive Functions in Mild-to-Moderate Alzheimer’s Disease (AD) Patients Living at High Altitude

Ratanasampil (RNSP) is a traditional Tibetan medicine used for the treatment of stroke and cerebrovascular diseases. Previous discoveries that RNSP can reduce β-amyloid protein levels and increase learning and memory in Alzheimer’s mouse models (Tg2576) led us to investigate whether RNSP can improve cognitive functions in Alzheimer’s patients. In this study, 146 AD patients living in Qinghai province received either one gram or 0.33 gram daily of RNSP for 16 weeks. Placebo patients received Piracetam. Serum Aβ40 and Aβ42 levels were measured at the beginning of the study and after 4 and 16 weeks of treatment. Compared to the same group before treatment, MMSE scores, ADAS-cog scores and ADL scores were significantly improved (p 0.05, p > 0.05). After 16-week treatment, serum TNF-α, IL-1β, IL-6 and Aβ42 levels were significantly decreased (p < 0. 01) in the high-dose RNSP group, whereas no significant differences were found in the low-dose and placebo groups. The Aβ42/Aβ40 ratio was significantly decreased after 4-week and 16-week treatment in the high-dose RNSP group (p < 0. 05, p < 0.01). Furthermore, serum Aβ42 concentrations had a strong positive correlation with TNF-α, IL-1β and IL-6 levels. There were no observable adverse effects in either treatment or control groups. We conclude that further clinical trials of RNSP in Alzheimer disease are warranted.

Evaluation of Generalized Error Function via Fast-Converging Power Series

A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power series expansion developed in resolving the so-called Grandi’s paradox. Comparisons with accurate tabulated values for well-known cases such as the error function are presented using the expansions truncated at various orders.

A hybrid method integrating Green's function Monte Carlo and projected entangled pair states

This paper introduces a hybrid approach combining Green’s function Monte Carlo(GFMC)method with projected entangled pair state(PEPS)ansatz.This hybrid method regards PEPS as a trial state and a guiding wave function in GFMC.By leveraging PEPS’s proficiency in capturing quantum state entanglement and GFMC’s efficient parallel architecture,the hybrid method is well-suited for the accurate and efficient treatment of frustrated quantum spin systems.As a benchmark,we applied this approach to study the frustrated J_(1)–J_(2) Heisenberg model on a square lattice with periodic boundary conditions(PBCs).Compared with other numerical methods,our approach integrating PEPS and GFMC shows competitive accuracy in the performance of ground-state energy.This paper provides systematic and comprehensive discussion of the approach of our previous work[Phys.Rev.B 109235133(2024)].

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.

Refinements to Hadamard’s Inequality for Log-Convex Functions

In this paper we show that a log-convex function satisfies Hadamard's inequality, as well as we give an extension for this result in several directions.