The Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions. Part II: Algebraic Operations and Types of Exponential Variation

In this second part, we thoroughly examine the types of higher-order asymptotic variation of a function obtained by all possible basic algebraic operations on higher-order varying functions. The pertinent proofs are somewhat demanding except when all the involved functions are regularly varying. Next, we give an exposition of three types of exponential variation with an exhaustive list of various asymptotic functional equations satisfied by these functions and detailed results concerning operations on them. Simple applications to integrals of a product and asymptotic behavior of sums are given. The paper concludes with applications of higher-order regular, rapid or exponential variation to asymptotic expansions for an expression of type f(x+r(x)).

The Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions. Part I: Higher-Order Regular, Smooth and Rapid Variation

Motivated by a general theory of finite asymptotic expansions in the real domain for functions f of one real variable, a theory developed in a previous series of papers, we present a detailed survey on the classes of higher-order asymptotically-varying functions where “asymptotically” stands for one of the adverbs “regularly, smoothly, rapidly, exponentially”. For order 1 the theory of regularly-varying functions (with a minimum of regularity such as measurability) is well established and well developed whereas for higher orders involving differentiable functions we encounter different approaches in the literature not linked together, and the cases of rapid or exponential variation, even of order 1, are not systrematically treated. In this semi-expository paper we systematize much scattered matter concerning the pertinent theory of such classes of functions hopefully being of help to those who need these results for various applications. The present Part I contains the higher-order theory for regular, smooth and rapid variation.

BSDES IN GAMES,COUPLED WITH THE VALUE FUNCTIONS.ASSOCIATED NONLOCAL BELLMAN-ISAACS EQUATIONS

We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs(HJB-Isaacs)equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair(W, U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs’ condition.

Na^(+)/K^(+)-ATPase:ion pump,signal transducer,or cytoprotective protein,and novel biological functions

Na^(+)/K^(+)-ATPase is a transmembrane protein that has important roles in the maintenance of electrochemical gradients across cell membranes by transporting three Na^(+)out of and two K^(+)into cells.Additionally,Na^(+)/K^(+)-ATPase participates in Ca^(2+)-signaling transduction and neurotransmitter release by coordinating the ion concentration gradient across the cell membrane.Na^(+)/K^(+)-ATPase works synergistically with multiple ion channels in the cell membrane to form a dynamic network of ion homeostatic regulation and affects cellular communication by regulating chemical signals and the ion balance among different types of cells.Therefo re,it is not surprising that Na^(+)/K^(+)-ATPase dysfunction has emerged as a risk factor for a variety of neurological diseases.However,published studies have so far only elucidated the important roles of Na^(+)/K^(+)-ATPase dysfunction in disease development,and we are lacking detailed mechanisms to clarify how Na^(+)/K^(+)-ATPase affects cell function.Our recent studies revealed that membrane loss of Na^(+)/K^(+)-ATPase is a key mechanism in many neurological disorders,particularly stroke and Parkinson's disease.Stabilization of plasma membrane Na^(+)/K^(+)-ATPase with an antibody is a novel strategy to treat these diseases.For this reason,Na^(+)/K^(+)-ATPase acts not only as a simple ion pump but also as a sensor/regulator or cytoprotective protein,participating in signal transduction such as neuronal autophagy and apoptosis,and glial cell migration.Thus,the present review attempts to summarize the novel biological functions of Na^(+)/K^(+)-ATPase and Na^(+)/K^(+)-ATPase-related pathogenesis.The potential for novel strategies to treat Na^(+)/K^(+)-ATPase-related brain diseases will also be discussed.

BOUNDARY VALUE PROBLEMS OF CONJUGATE AND GENERALIZED K-HOLOMORPHIC FUNCTIONS IN C^(2)

In this paper,conjugate k-holomorphic functions and generalized k-holomorphic functions are defined in the two-dimensional complex space,and the corresponding Riemann boundary value problems and the inverse problems are discussed on generalized bicylinders.By the characteristics of the corresponding functions and boundary properties of the Cauchy type singular integral operators with conjugate k-holomorphic kernels,the general solutions and special solutions of the corresponding boundary value problems are studied in a detailed fashion,and the integral expressions of the solutions are obtained.

QUASIPERIODICITY OF TRANSCENDENTAL MEROMORPHIC FUNCTIONS

This paper is devoted to considering the quasiperiodicity of complex differential polynomials,complex difference polynomials and complex delay-differential polynomials of certain types,and to studying the similarities and differences of quasiperiodicity compared to the corresponding properties of periodicity.

FROM WAVE FUNCTIONS TO TAU-FUNCTIONS FOR THE VOLTERRA LATTICE HIERARCHY

For an arbitrary solution to the Volterra lattice hierarchy,the logarithmic derivatives of the tau-function of the solution can be computed by the matrix-resolvent method.In this paper,we define a pair of wave functions of the solution and use them to give an expression of the matrix resolvent;based on this we obtain a new formula for the k-point functions for the Volterra lattice hierarchy in terms of wave functions.As an application,we give an explicit formula of k-point functions for the even GUE(Gaussian Unitary Ensemble)correlators.

Neuropeptide cholecystokinin:a key neuromodulator for hippocampal functions

Spatial memory is crucial for survival within external surroundings and wild environments.The hippocampus,a critical hub for spatial learning and memory formation,has received extensive investigations on how neuromodulators shape its functions(Teixeira et al.,2018;Zhang et al.,2024).However,the landscape of neuromodulations in the hippocampal system remains poorly understood because most studies focus on classical monoamine neuromodulators,such as acetylcholine,serotonin,dopamine,and noradrenaline.The neuropeptides,comprising the most abundant neuromodulators in the central nervous system,play a pivotal role in neural information processing in the hippocampal system.Cholecystokinin(CCK),one of the most abundant neuropeptides,has been implicated in regulating various physiological and neurobiological statuses(Chen et al.,2019).CCK-A receptor(CCK-AR)and CCK-B receptors(CCK-BR)are two key receptors mediating the biological functions of CCK,both of which belong to class-A sevenfold transmembrane G protein-coupled receptors(Nishimura et al.,2015).CCK-AR preferentially reacts to sulfated CCK,whereas CCK-BR binds both CCK and gastrin with similar affinities(Ding et al.,2022).The expression patterns of CCK-AR and CCK-BR are distinct,implying that CCK has various functions in target regions.For instance,CCK-AR is widely expressed in the GI and brain subregions and is hence implicated in the control of digestive function and satiety regulation.Conversely,CCK-BR is abundantly and widely distributed in the central nervous system,which majorly regulates anxiety,learning,and memory(Ding et al.,2022).However,the roles of endogenous CCK and CCK receptors in regulating hippocampal function at electrophysiological and behavioral levels have received less attention.

The biological functions and metabolic pathways of valine in swine

Valine is an essential amino acid and a type of branched-chain amino acid. Due to the involvement of branchedchain amino acids in various metabolic pathways, there has been a surge of interests in valine nutrition and its role in animal physiology. In pigs, the interactions between valine and other branched-chain amino acids or aromatic amino acids are complex. In this review, we delve into the interaction mechanism, metabolic pathways, and biological functions of valine. Appropriate valine supplementation not only enhances growth and reproductive performances, but also modulates gut microbiota and immune functions. Based on past observations and interpretations, we provide recommended feed levels of valine for weaned piglets, growing pigs, gilts, lactating sows, barrows and entire males. The summarized valine nutrient requirements for pigs at different stages offer valuable insights for future research and practical applications in animal husbandry.

A Cauchy integral formula for(p,q)-monogenic functions withα-weight

Firstly,some properties for(p,q)-monogenic functions withα-weight in Clifford analysis are given.Then,the Cauchy-Pompeiu formula is proved.Finally,the Cauchy integral formula and the Cauchy integral theorem for(p,q)-monogenic functions withα-weight are given.

Image Splicing Forgery Detection Using Feature-Based of Sonine Functions and Deep Features

The growing prevalence of fake images on the Internet and social media makes image integrity verification a crucial research topic.One of the most popular methods for manipulating digital images is image splicing,which involves copying a specific area from one image and pasting it into another.Attempts were made to mitigate the effects of image splicing,which continues to be a significant research challenge.This study proposes a new splicing detectionmodel,combining Sonine functions-derived convex-based features and deep features.Two stages make up the proposed method.The first step entails feature extraction,then classification using the“support vector machine”(SVM)to differentiate authentic and spliced images.The proposed Sonine functions-based feature extraction model reveals the spliced texture details by extracting some clues about the probability of image pixels.The proposed model achieved an accuracy of 98.93% when tested with the CASIA V2.0 dataset“Chinese Academy of Sciences,Institute of Automation”which is a publicly available dataset for forgery classification.The experimental results show that,for image splicing forgery detection,the proposed Sonine functions-derived convex-based features and deep features outperform state-of-the-art techniques in terms of accuracy,precision,and recall.Overall,the obtained detection accuracy attests to the benefit of using the Sonine functions alongside deep feature representations.Finding the regions or locations where image tampering has taken place is limited by the study.Future research will need to look into advanced image analysis techniques that can offer a higher degree of accuracy in identifying and localizing tampering regions.

Dynamical distribution of continuous service time model involving non-Maxwellian collision kernel and value functions

The distribution of continuous service time in call centers is investigated.A non-Maxwellian collision kernel combining two different value functions in the interaction rule are used to describe the evolution of continuous service time,respectively.Using the statistical mechanical and asymptotic limit methods,Fokker–Planck equations are derived from the corresponding Boltzmann-type equations with non-Maxwellian collision kernels.The steady-state solutions of the Fokker–Planck equation are obtained in exact form.Numerical experiments are provided to support our results under different parameters.

AGGREGATE SPECIAL FUNCTIONS TO APPROXIMATE PERMUTING TRI-HOMOMORPHISMS AND PERMUTING TRI-DERIVATIONS ASSOCIATED WITH A TRI-ADDITIVEψ-FUNCTIONAL INEQUALITY IN BANACH ALGEBRAS

In this paper,we define a new class of control functions through aggregate special functions.These class of control functions help us to stabilize and approximate a tri-additiveψ-functional inequality to get a better estimation for permuting tri-homomorphisms and permuting tri-derivations in unital C*-algebras and Banach algebras by the vector-valued alternative fixed point theorem.

Quantum control based on three forms of Lyapunov functions

This paper introduces the quantum control of Lyapunov functions based on the state distance, the mean of imaginary quantities and state errors.In this paper, the specific control laws under the three forms are given.Stability is analyzed by the La Salle invariance principle and the numerical simulation is carried out in a 2D test system.The calculation process for the Lyapunov function is based on a combination of the average of virtual mechanical quantities, the particle swarm algorithm and a simulated annealing algorithm.Finally, a unified form of the control laws under the three forms is given.

Crustal and uppermost mantle structure of the northeastern Qinghai-Xizang Plateau from joint inversion of surface wave dispersions and receiver functions with P velocity constraints

Lithospheric structure beneath the northeastern Qinghai-Xizang Plateau is of vital significance for studying the geodynamic processes of crustal thickening and expansion of the Qinghai-Xizang Plateau. We conducted a joint inversion of receiver functions and surface wave dispersions with P-wave velocity constraints using data from the Chin Array Ⅱ temporary stations deployed across the Qinghai-Xizang Plateau. Prior to joint inversion, we applied the H-κ-c method(Li JT et al., 2019) to the receiver function data in order to correct for the back-azimuthal variations in the arrival times of Ps phases and crustal multiples caused by crustal anisotropy and dipping interfaces. High-resolution images of vS, crustal thickness, and vP/vSstructures in the Qinghai-Xizang Plateau were simultaneously derived from the joint inversion. The seismic images reveal that crustal thickness decreases outward from the Qinghai-Xizang Plateau. The stable interiors of the Ordos and Alxa blocks exhibited higher velocities and lower crustal vP/vSratios. While, lower velocities and higher vP/vSratios were observed beneath the Qilian Orogen and Songpan-Ganzi terrane(SPGZ), which are geologically active and mechanically weak, especially in the mid-lower crust.Delamination or thermal erosion of the lithosphere triggered by hot asthenospheric flow contributes to the observed uppermost mantle low-velocity zones(LVZs) in the SPGZ. The crustal thickness, vS, and vP/vSratios suggest that whole lithospheric shortening is a plausible mechanism for crustal thickening in the Qinghai-Xizang Plateau, supporting the idea of coupled lithospheric-scale deformation in this region.

Extraction of Acoustic Normal Mode Depth Functions Using Range-Difference Method with Vertical Linear Array Data

Data-derived normal mode extraction is an effective method for extracting normal mode depth functions in the absence of marine environmental data.However,when the corresponding singular vectors become nonunique when two or more singular values obtained from the cross-spectral density matrix diagonalization are nearly equal,this results in unsatisfactory extraction outcomes for the normal mode depth functions.To address this issue,we introduced in this paper a range-difference singular value decomposition method for the extraction of normal mode depth functions.We performed the mode extraction by conducting singular value decomposition on the individual frequency components of the signal's cross-spectral density matrix.This was achieved by using pressure and its range-difference matrices constructed from vertical line array data.The proposed method was validated using simulated data.In addition,modes were successfully extracted from ambient noise.

Comparison of brain functions between healthy participants and methamphetamine users with various addiction histories:Data analysis based on EEG and fNIRS

The electroencephalogram(EEG)rhythm and functional near-infrared spectroscopy(fNIRS)activation levels have not been compared between a healthy control group(HCG)and methamphetamine user group(MUG)with different addiction histories.This study used 64-electrode EEG and fNIRS to conduct an experiment that analyzed the resting and craving states.The EEG and fNIRS data of 56 participants were collected,including 14 healthy participants,14 methamphetamine users with an addiction history of 0.5–5 years,14 users with an addiction history of 5–10 years,and 14 users with an addiction history of 10–15 years.Isolated effective coherence(iCoh)within the brain network was used to process the EEG data.Statistical analysis was performed to compare differences in iCoh among the delta,theta,alpha,beta,and gamma bands and explore oxyhemoglobin activation levels in the ventrolateral prefrontal cortex,dorsolateral prefrontal cortex,orbitofrontal cortex,and frontopolar prefrontal cortex(FPC)of the control group.Finally,the Kmeans,Gaussian mixed model(GMM),linear discriminant analysis(LDA),support vector machine(SVM),Bayes,and convolutional neural networks(CNN)algorithms were used to classify methamphetamine users based on drug and neutral images.A 3-class accuracy was achieved.Changes in EEG and fNIRS activation levels of HCG and MUG with varied addiction histories were demonstrated.

Bioactive material promotes longdistance regeneration of optic nerve to restore visual functions

Visual system is vital to human beings.Unfortunately,the optic nerve lacks the ability to regenerate after injury.Therefo re,long-distance regeneration of the optic nerve is a major unsolved medical problem in the world(Laha et al.,2017).Recently,Li and So groups' study showed that the bioactive material(ciliary neurotrophic factor[CNTF]-chitosan) could promote long-distance regeneration of the completely transected optic nerve in adult rats and partially restored the visual functions(Liu et al.,2023).This study sheds light on the clinical potential for repairing the severely injured optic nerve.

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.

FINITE RANGE SET WITH TRUNCATED MULTIPLICITY FOR MEROMORPHIC FUNCTIONS ON SOME COMPLEX DISC

In this paper,we consider the truncated multiplicity finite range set problem of meromorphic functions on some complex disc.By using the value distribution theory of meromorphic functions,we establish a second main theorem for meromorphic functions with finite growth index which share meromorphic functions(may not be small functions).As its application,we also extend the result of a finite range set with truncated multiplicity.