Review on the Fabrication of Surface Functional Structures for Enhancing Bioactivity of Titanium and Titanium Alloy Implants

Titanium and its alloys have been widely applied in many biomedical fields because of its excellent mechanical properties,corrosion resistance and good biocompatibility.However,problems such as rejection,shedding and infection will occur after titanium alloy implantation due to the low biological activity of titanium alloy surface.The structures with specific functions,which can enhance osseointegration and antibacterial properties,are fabricated on the surface of titanium implants to improve the biological activity between the titanium implants and human tissues.This paper presents a comprehensive review of recent developments and applications of surface functional structure in titanium and titanium alloy implants.The applications of surface functional structure on different titanium and titanium alloy implants are introduced,and their manufacturing technologies are summarized and compared.Furthermore,the fabrication of various surface functional structures used for titanium and titanium alloy implants is reviewed and analyzed in detail.Finally,the challenges affecting the development of surface functional structures applied in titanium and titanium alloy implants are outlined,and recommendations for future research are presented.

Structural and functional connectivity of the whole brain and subnetworks in individuals with mild traumatic brain injury:predictors of patient prognosis

Patients with mild traumatic brain injury have a diverse clinical presentation,and the underlying pathophysiology remains poorly understood.Magnetic resonance imaging is a non-invasive technique that has been widely utilized to investigate neuro biological markers after mild traumatic brain injury.This approach has emerged as a promising tool for investigating the pathogenesis of mild traumatic brain injury.G raph theory is a quantitative method of analyzing complex networks that has been widely used to study changes in brain structure and function.However,most previous mild traumatic brain injury studies using graph theory have focused on specific populations,with limited exploration of simultaneous abnormalities in structural and functional connectivity.Given that mild traumatic brain injury is the most common type of traumatic brain injury encounte red in clinical practice,further investigation of the patient characteristics and evolution of structural and functional connectivity is critical.In the present study,we explored whether abnormal structural and functional connectivity in the acute phase could serve as indicators of longitudinal changes in imaging data and cognitive function in patients with mild traumatic brain injury.In this longitudinal study,we enrolled 46 patients with mild traumatic brain injury who were assessed within 2 wee ks of injury,as well as 36 healthy controls.Resting-state functional magnetic resonance imaging and diffusion-weighted imaging data were acquired for graph theoretical network analysis.In the acute phase,patients with mild traumatic brain injury demonstrated reduced structural connectivity in the dorsal attention network.More than 3 months of followup data revealed signs of recovery in structural and functional connectivity,as well as cognitive function,in 22 out of the 46 patients.Furthermore,better cognitive function was associated with more efficient networks.Finally,our data indicated that small-worldness in the acute stage could serve as a predictor of longitudinal changes in connectivity in patients with mild traumatic brain injury.These findings highlight the importance of integrating structural and functional connectivity in unde rstanding the occurrence and evolution of mild traumatic brain injury.Additionally,exploratory analysis based on subnetworks could serve a predictive function in the prognosis of patients with mild traumatic brain injury.

3D printing of poly(ethyleneimine)-functionalized Mg-Al mixed metal oxide monoliths for direct air capture of CO_(2)

Direct air capture(DAC)of CO_(2)plays an indispensable role in achieving carbon-neutral goals as one of the key negative emission technologies.Since large air flows are required to capture the ultradilute CO_(2)from the air,lab-synthesized adsorbents in powder form may cause unacceptable gas pressure drops and poor heat and mass transfer efficiencies.A structured adsorbent is essential for the implementation of gas-solid contactors for cost-and energy-efficient DAC systems.In this study,efficient adsorbent poly(ethyleneimine)(PEI)-functionalized Mg-Al-CO_(3)layered double hydroxide(LDH)-derived mixed metal oxides(MMOs)are three-dimensional(3D)printed into monoliths for the first time with more than 90%adsorbent loadings.The printing process has been optimized by initially printing the LDH powder into monoliths followed by calcination into MMO monoliths.This structure exhibits a 32.7%higher specific surface area and a 46.1%higher pore volume,as compared to the direct printing of the MMO powder into a monolith.After impregnation of PEI,the monolith demonstrates a large adsorption capacity(1.82 mmol/g)and fast kinetics(0.7 mmol/g/h)using a CO_(2)feed gas at 400 ppm at 25℃,one of the highest values among the shaped DAC adsorbents.Smearing of the amino-polymers during the post-printing process affects the diffusion of CO_(2),resulting in slower adsorption kinetics of pre-impregnation monoliths compared to post-impregnation monoliths.The optimal PEI/MeOH ratio for the post-impregnation solution prevents pores clogging that would affect both adsorption capacity and kinetics.

A Hybrid Level Set Optimization Design Method of Functionally Graded Cellular Structures Considering Connectivity

With the continuous advancement in topology optimization and additive manufacturing(AM)technology,the capability to fabricate functionally graded materials and intricate cellular structures with spatially varying microstructures has grown significantly.However,a critical challenge is encountered in the design of these structures–the absence of robust interface connections between adjacent microstructures,potentially resulting in diminished efficiency or macroscopic failure.A Hybrid Level Set Method(HLSM)is proposed,specifically designed to enhance connectivity among non-uniform microstructures,contributing to the design of functionally graded cellular structures.The HLSM introduces a pioneering algorithm for effectively blending heterogeneous microstructure interfaces.Initially,an interpolation algorithm is presented to construct transition microstructures seamlessly connected on both sides.Subsequently,the algorithm enables the morphing of non-uniform unit cells to seamlessly adapt to interconnected adjacent microstructures.The method,seamlessly integrated into a multi-scale topology optimization framework using the level set method,exhibits its efficacy through numerical examples,showcasing its prowess in optimizing 2D and 3D functionally graded materials(FGM)and multi-scale topology optimization.In essence,the pressing issue of interface connections in complex structure design is not only addressed but also a robust methodology is introduced,substantiated by numerical evidence,advancing optimization capabilities in the realm of functionally graded materials and cellular structures.

Functionally graded structure of a nitride-strengthened Mg_(2)Si-based hybrid composite

The ex-situ incorporation of the secondary SiC reinforcement,along with the in-situ incorporation of the tertiary and quaternary Mg_(3)N_(2) and Si_(3)N_(4) phases,in the primary matrix of Mg_(2)Si is employed in order to provide ultimate wear resistance based on the laser-irradiation-induced inclusion of N_(2) gas during laser powder bed fusion.This is substantialized based on both the thermal diffusion-and chemical reactionbased metallurgy of the Mg_(2)Si–SiC/nitride hybrid composite.This study also proposes a functional platform for systematically modulating a functionally graded structure and modeling build-direction-dependent architectonics during additive manufacturing.This strategy enables the development of a compositional gradient from the center to the edge of each melt pool of the Mg_(2)Si–SiC/nitride hybrid composite.Consequently,the coefficient of friction of the hybrid composite exhibits a 309.3%decrease to–1.67 compared to–0.54 for the conventional nonreinforced Mg_(2)Si structure,while the tensile strength exhibits a 171.3%increase to 831.5 MPa compared to 485.3 MPa for the conventional structure.This outstanding mechanical behavior is due to the(1)the complementary and synergistic reinforcement effects of the SiC and nitride compounds,each of which possesses an intrinsically high hardness,and(2)the strong adhesion of these compounds to the Mg_(2)Si matrix despite their small sizes and low concentrations.

Sound Transmission Loss Analysis of a Double Plate-Acoustic Cavity Coupling System with In-Plane Functionally Graded Materials

In this paper, the isogeometric analysis (IGA) is employed to develop an acoustic radiation model for a double plate-acoustic cavity coupling system, with a focus on analyzing the sound transmission loss (STL). The functionally graded (FG) plate exhibits a different material properties in-plane, and the power-law rule is adopted as the governing principle for material mixing. To validate the harmonic response and demonstrate the accuracy and convergence of the isogeometric modeling, ANASYS is utilized to compare with numerical examples. A plane wave serves as the acoustic excitation, and the Rayleigh integral is applied to discretize the radiated plate. The STL results are compared with the literature, confirming the reliability of the coupling system. Finally, the investigation is conducted to study impact of cavity depth and power-law parameter on the STL.

Changes of structure and functional properties of rice protein in the fresh edible rice during the seed development

It has been reported that fresh edible rice has more bioactive compounds and its protein is easier to digest and has lower hypoallergenic than mature rice. In this paper, the changes in structure and functional properties of proteins at five different stages, including early milky stage(EMS), middle milky stage(MMS), late milky stage(LMS), waxy ripe stage(WS)and ripening stage(RS), during the seed development were investigated. It was found that with the seed developing, the molecular weight of fresh rice protein gradually become larger while the secondary structure changed from the highest content of disordered structure at MMS to the highest content of ordered structure at RS, which affect the surface hydrophobicity and then the functional properties of proteins, including foaming properties, emulsifying properties and oil holding capacity. Fresh rice protein at MMS has the strongest surface hydrophobicity while fresh edible rice protein at RS has the strongest oil holding capability. The results of our study can provide a theoretical basis for the application of fresh rice protein in the food industry and help to develop new fresh edible rice food.

Plant functional trait diversity and structural diversity co-underpin ecosystem multifunctionality in subtropical forests

Tree species diversity is assumed to be an important component in managing forest ecosystems because of effects on multiple functions or ecosystem multifunctionality.However,the importance of tree diversity in determining multifunctionality in structurally complex subtropical forests relative to other regulators(e.g.,soil microbial diversity,stand structure,and environmental conditions)remains uncertain.In this study,effects of aboveground(species richness and functional and structural diversity)and belowground(bacterial and fungal diversity)biodiversity,functional composition(community-weighted means of species traits),stand structure(diameter at breast height and stand density),and soil factors(pH and bulk density)on multifunctionality(including biomass production,carbon stock,and nutrient cycling)were examined along a tree diversity gradient in subtropical forests.The community-weighted mean of tree maximum height was the best predictor of ecosystem multifunctionality.Functional diversity explained a higher proportion of the variation in multifunctionality than that of species richness and fungal diversity.Stand structure-played an important role in modulating the effects of tree diversity on multifunctionality.The work highlights that species composition and maximizing forest structural complexity are effective strategies to increase forest multifunctionality while also conserving biodiversity in the management of multifunctional forests under global environmental changes.

Structures and Functions of Cuticular Wax in Postharvest Fruit and Its Regulation:A Comprehensive Review with Future Perspectives

Cuticular wax plays a major role in the growth and storage of plant fruits.The cuticular wax coating,which covers the outermost layer of a fruit’s epidermal cells,is insoluble in water.Cuticular wax is mainly composed of very long-chain fatty acids(VLCFAs);their derivatives,including esters,primary alcohols,secondary alcohols,aldehydes,and ketones;and triterpenoids.This complex mixture of lipids is probably biosynthesized in the epidermal cells of most plants and exuded onto the surface.Cuticular wax not only makes the fruit less susceptible to microbial infection but also reduces mechanical damage to the fruit,thereby maintaining the fruit’s commodity value.To date,research has mostly focused on the changes,function,and regulation of fruit wax before harvest,while ignoring the changes and functions of wax in fruit storage.This paper reviews on the composition,structure,and metabolic regulation of cuticular wax in fruits.It also focuses on postharvest factors affecting wax composition,such as storage temperature,relative humidity(RH),gas atmosphere,and as exogenous hormones;and the effects of wax on fruit postharvest quality,including water dispersion,fruit softening,physiological disorders,and disease resistance.These summaries may be of assistance in better understanding the changes in cuticular wax in postharvest fruit and the resulting effects on fruit quality.

Structural and functional alterations in the brains of patients with anisometropic and strabismic amblyopia:a systematic review of magnetic resonance imaging studies

Amblyopia is the most common cause of vision loss in children and can persist into adulthood in the absence of effective intervention.Previous clinical and neuroimaging studies have suggested that the neural mechanisms underlying strabismic amblyopia and anisometropic amblyopia may be different.Therefore,we performed a systematic review of magnetic resonance imaging studies investigating brain alterations in patients with these two subtypes of amblyopia;this study is registered with PROSPERO(registration ID:CRD42022349191).We searched three online databases(PubMed,EMBASE,and Web of Science) from inception to April 1,2022;39 studies with 633 patients(324patients with anisometropic amblyo pia and 309 patients with strabismic amblyopia) and 580 healthy controls met the inclusion criteria(e.g.,case-control designed,pee r-reviewed articles) and were included in this review.These studies highlighted that both strabismic amblyopia and anisometropic amblyopia patients showed reduced activation and distorted topological cortical activated maps in the striate and extrastriate co rtices during tas k-based functional magnetic resonance imaging with spatial-frequency stimulus and retinotopic representations,respectively;these may have arisen from abnormal visual experiences.Compensations for amblyopia that are reflected in enhanced spontaneous brain function have been reported in the early visual cortices in the resting state,as well as reduced functional connectivity in the dorsal pathway and structural connections in the ventral pathway in both anisometro pic amblyopia and strabismic amblyopia patients.The shared dysfunction of anisometro pic amblyopia and strabismic amblyopia patients,relative to controls,is also chara cterized by reduced spontaneous brain activity in the oculomotor co rtex,mainly involving the frontal and parietal eye fields and the cerebellu m;this may underlie the neural mechanisms of fixation instability and anomalous saccades in amblyopia.With regards to specific alterations of the two forms of amblyo pia,anisometropic amblyo pia patients suffer more microstructural impairments in the precortical pathway than strabismic amblyopia patients,as reflected by diffusion tensor imaging,and more significant dysfunction and structural loss in the ventral pathway.Strabismic amblyopia patients experience more attenuation of activation in the extrastriate co rtex than in the striate cortex when compared to anisometropic amblyopia patients.Finally,brain structural magnetic resonance imaging alterations tend to be lateralized in the adult anisometropic amblyopia patients,and the patterns of brain alterations are more limited in amblyopic adults than in childre n.In conclusion,magnetic resonance imaging studies provide important insights into the brain alterations underlying the pathophysiology of amblyopia and demonstrate common and specific alte rations in anisometropic amblyo pia and strabismic amblyopia patients;these alterations may improve our understanding of the neural mechanisms underlying amblyopia.

Pectin fractions extracted sequentially from Cerasus humilis:Their compositions,structures,functional properties and antioxidant activities

Three pectin fractions(water-soluble fraction(WSF),chelator-soluble fraction(CSF),and sodium carbonatesoluble fraction(NSF))were obtained from Chinese dwarf cherry(Cerasus humilis)fruits.All of them were branched low methoxylated pectins with an amorphous or partially nanocrystalline nature and eight neutral monosaccharides(arabinose and galactose were most abundant).WSF,CSF and NSF had a degree of methylation(DM)of 35.82%,14.85%and 7.13%,uronic acid(UA)content of 76.02%,83.71%and 69.01%,and total protein content of 2.4%,2.1%and 8.8%,respectively.Their molecular weights were 340.31,330.16 and 141.31 kg/mol,respectively(analyzed by gel permeation chromatography(GPC)).WSF,CSF and NSF exhibited good rheological,thermal,emulsifying,emulsion-stabilizing,water-adsorbing,oil-binding,cholesterol-binding and antioxidant properties.NSF had the highest emulsifying,emulsion stabilizing,water-/oil-/cholesterol-binding and antioxidant capacities,followed by CSF.NSF had the highest viscosity(406.77 m Pa·s),flowability,and resistance to heat-induced changes/damage,which may be related to its lowest polydispersity index,DM and UA content and highest protein content.The three pectin fractions with desirable characteristics can be used as food additives/ingredients and dietary supplements.

The action mechanisms and structures designs of F-containing functional materials for high performance oxygen electrocatalysis

Non-renewable fossil fuels have led to serious problems such as global warming,environmental pollution,etc.Oxygen electrocatalysis including oxygen reduction reaction(ORR)and oxygen evolution reaction(OER)plays a central role in clean energy conversion,enabling a number of sustainable processes for future air battery technologies.Fluorine,as the most electronegative element(4.0)not only can induce more efficient regulation for the electronic structure,but also can bring more abundant defects and other novel effects in materials selection and preparation for favorable catalysis with respect to the other nonmetal elements.However,an individual and comprehensive overview of fluorine-containing functional materials for oxygen electrocatalysis field is still blank.Therefore,it is very meaningful to review the recent progresses of fluorine-containing oxygen electrocatalysts.In this review,we first systematically summarize the controllable preparation methods and their possible development directions based on fluorine-containing materials from four preparation methods.Due to the strong electron-withdrawing properties of fluorine,its control of the electronic structure can effectively enhance the oxygen electrocatalytic activity of the materials.In addition,the catalytic enhancement effect of fluorine on carbonbased materials also includes the prevent oxidation and the layer peeling,and realizes the precise atomic control.And the catalytic improvement mechanism of fluorine containing metal-based compounds also includes the hydration of metal site,the crystal transformation,and the oxygen vacancy induction.Then,based on their various dimensions(0D–3D),we also have summarized the advantages of different morphologies on oxygen electrocatalytic performances.Finally,the prospects and possible future researching direction of F-containing oxygen electrocatalysts are presented(e.g.,novel pathways,advanced methods for measurement and simulation,field assistance and multi-functions).The review is considered valuable and helpful in exploring the novel designs and mechanism analyses of advanced fluorine-containing electrocatalysts.

Investigation of the structural, electronic and mechanical properties of CaO–SiO_(2) compound particles in steel based on density functional theory

CaO–SiO_(2)compounds compromise one of the most common series of oxide particles in liquid steels, which could significantly affect the service performance of the steels as crack initiation sites. However, the structural, electronic, and mechanical properties of the compounds in CaO–SiO_(2)system are still not fully clarified due to the difficulties in the experiments. In this study, a thorough investigation of these properties of CaO–SiO_(2)compound particles in steels was conducted based on first-principles density functional theory. Corresponding phases were determined by thermodynamic calculation, including gamma dicalcium silicate(γ-C2S), alpha-prime(L) dicalcium silicate(αL′-C2S), alpha-prime(H) dicalcium silicate(αH′-C2S), alpha dicalcium silicate(α-C2S), rankinite(C3S2), hatrurite(C3S), wollastonite(CS), and pseudowollastonite(Ps-CS). The results showed that the calculated crystal structures of the eight phases agree well with the experimental results. All the eight phases are stable according to the calculated formation energies, and γ-C2S is the most stable. O atom contributes the most to the reactivity of these phases. The Young’s modulus of the eight phases is in the range of 100.63–132.04 GPa. Poisson’s ratio is in the range of0.249–0.281. This study provided further understanding concerning the CaO–SiO_(2)compound particles in steels and fulfilled the corresponding property database, paving the way for inclusion engineering and design in terms of fracture-resistant steels.

GREEN'S FUNCTION AND THE POINTWISE BEHAVIORS OF THE VLASOV-POISSON-FOKKER-PLANCK SYSTEM

The pointwise space-time behaviors of the Green’s function and the global solution to the Vlasov-Poisson-Fokker-Planck(VPFP)system in three dimensional space are studied in this paper.It is shown that the Green’s function consists of the diffusion waves decaying exponentially in time but algebraically in space,and the singular kinetic waves which become smooth for all(t,x,v)when t>0.Furthermore,we establish the pointwise space-time behaviors of the global solution to the nonlinear VPFP system when the initial data is not necessarily smooth in terms of the Green’s function.

Restraining migration and dissolution of transition-metal-ions via functionalized separator for Li-rich Mn-based cathode with high-energy-density

Lithium-rich manganese-based materials(LRMs) are promising cathode for high-energy-density lithiumion batteries due to their high capacity,low toxicity,and low cost.However,LRMs suffer from serious voltage decay and capacity fade due to continual migration and dissolution of transition metal ions(TMs) during cycling process.Herein,a novel strategy is proposed to inhibit the TMs migration of LRMs through a modified separator by means of functionalized carbon coating layer,which depends on the chemical constraint of the abundant functional groups in the modified super P.In addition,it has been found that the dissolution of TMs can be restrained based on the Le Chatelier's principle.Moreover,the modified separator owns good wettability toward the electrolyte.As a result,the LRMs cathode with the modified separator delivers a high discharge capacity of 329.93 mA h g-1 at 0.1 C,and achieves good cyclic performance,the enhanced reaction kinetics and low voltage decay.Therefore,this work provides a new idea to promote the comprehensive electrochemical performances of Li-ion batteries with LRMs cathode through a strategy of separator modification.

Photon Structure and Wave Function from the Vector Potential Quantization

A photon structure is advanced based on the experimental evidence and the vector potential quantization at a single photon level. It is shown that the photon is neither a point particle nor an infinite wave but behaves rather like a local “wave-corpuscle” extended over a wavelength, occupying a minimum quantization volume and guided by a non-local vector potential real wave function. The quantized vector potential oscillates over a wavelength with circular left or right polarization giving birth to orthogonal magnetic and electric fields whose amplitudes are proportional to the square of the frequency. The energy and momentum are carried by the local wave-corpuscle guided by the non-local vector potential wave function suitably normalized.

超大城市外围乡村地域功能的演化路径及定位建议

掌握功能演化规律,合理确定功能定位,有助于乡村多元价值的挖掘和实现,是乡村地域高质量发展的前提。该研究以超大城市北京的外围乡村地域作为典型案例,基于乡村地域功能的内涵和分类,构建了适用于长时间序列乡村地域功能强度测度的评价指标体系,测度和识别了2000—2020年北京不同乡村地域的功能强度及功能结构,分析和总结了研究期间北京乡村地域的功能结构演化过程及形成路径。结果表明:1)超大城市乡村地域不同功能的强度呈现出显著的区域差异,功能结构整体呈现从单一优势功能主导向多种优势功能主导的演化趋势;2)功能结构能够反映不同类型乡村地域发展的差异,基于功能结构的演化特征,可以将超大城市外围乡村地域的功能结构演化过程归纳为“功能强化升级式路径”、“功能调整转换式路径”“功能综合完善式路径”3类发展路径;3)强化生活服务功能是当前城乡融合发展目标下推动超大城市外围乡村地域功能多元化和高质量发展的首要举措;4)通过分析优势功能及功能结构的演化过程,可以明确不同乡村地域的发展特征并指导其功能定位的制定,有利于分类推动超大城市外围乡村地域的多元化发展。研究结果丰富了乡村多功能理论的典型地域案例,可为超大城市外围乡村地域的功能定位设定及发展路径的决策和实践提供参考。

食用菌多糖结构与功能研究进展

食用菌具有较高的营养价值和功能价值,多糖作为食用菌最主要的活性成分之一,不仅能够为机体供给能量,而且可以参与生物合成反应及细胞的各项生命活动。大多数食用菌多糖是α-葡聚糖、β-葡聚糖、混合α,β-葡聚糖,以及由果糖、半乳糖、甘露糖等多种单糖组成的杂多糖,其中,β-葡聚糖是食用菌中最具生物活性的多糖。食用菌多糖具有多种生物活性。为了为食用菌多糖在功能性食品中的应用提供理论基础,综述了食用菌多糖的结构及其抗氧化、抗衰老、调节免疫、抗炎、抗肿瘤、降血糖、降血脂、抗病毒、抗辐射、抗突变、抑菌、抗疲劳、抗凝血等方面的功能及机理,并对其进行了展望。

生物炭对微塑料污染下普通白菜生长和土壤细菌群落结构的影响

采用盆栽试验,以四季小白菜为试材,研究了施用生物炭对不同微塑料类型(PP、PE、PVC)污染下普通白菜生长、土壤细菌群落结构和丰度的影响,并进行了功能预测分析。结果表明,微塑料污染对普通白菜鲜质量和叶片数有抑制作用,施用生物炭对微塑料污染下的普通白菜鲜质量、株高和根长均有促进作用。微塑料及其与生物炭共存处理显著增加了土壤细菌丰度,其中PER处理土壤细菌丰度最大;PE、PVC及其与生物炭共存处理均显著促进土壤细菌Chao1指数和ACE指数增加。在门水平上,放线菌门、变形菌门、绿弯菌门、厚壁菌门、酸杆菌门为优势类群;微塑料及其生物炭共存处理促进了放线菌门相对丰度的增加,抑制了变形菌门、酸杆菌门相对丰度。土壤p H以及有机碳、铵态氮、硝态氮、速效钾含量是土壤细菌群落结构变化的主要影响因素。PICRUSt分析表明,施用生物炭提高了微塑料污染下土壤新陈代谢、环境信息处理和有机系统等有益细菌代谢通路的相对丰度。综上,施用生物炭对微塑料污染下土壤微生物影响较大,但其影响程度因微塑料类型而异。

Spatial Agglomeration and Diffusion of Population Based on a Regional Density Function Approach:A Case Study of Shandong Province in China

Population density functions have long been used to describe the spatial structure of regional population distributions.Several studies have been conducted to examine the population distribution in Shandong Province,China,but few have applied regional density functions to the analysis.Therefore,based on the 2000,2010,and 2020 population censuses,this study used monocentric and polycentric regional density functions to study the characteristics of population agglomeration and diffusion in Shandong.This is followed by an in-depth discussion based on population growth rate data and hot-and cold-spot analyses.The results showed that the Shandong Province population was spatially unevenly distributed.Population growth rates were higher in urban centers and counties,with more significant changes in population size in the eastern coastal areas than in the inland areas.As verified in this study,the logarithmic form of the single-center regional density function R2 was greater than 0.8,which was in line with the population spatial structure of Shandong Province.During the study period,the estimated population density of the regional center and the absolute value of the regional population density gradient both increased,indicating a clear and increasing trend of centripetal agglomeration of regional centers over the study period.Overall,the R2 value of the multicenter region density function was higher than that of the single-center region density function.The polycentric regional density function showed that the population density gradient of some centers had a downward trend,which reflected the spatial development trend of outward diffusion in these centers.Meanwhile,the variation in the estimated population density and the population density gradient exhibited differences in the central population distribution patterns at different levels.