For the Sylvester continued fraction expansions of real numbers,FAN et al.(2007)proved that,for almost all real numbers,the nth partial quotient grows exponentially with respect to the product of the first n-1 partial...For the Sylvester continued fraction expansions of real numbers,FAN et al.(2007)proved that,for almost all real numbers,the nth partial quotient grows exponentially with respect to the product of the first n-1 partial quotients.In this paper,we establish the Hausdorff dimension of the exceptional set where the growth rate is a general function.展开更多
In this paper,the class of starlike functions of complex order γ(γ∈ℂ−{0})is extended from the case on unit disk U=(z∈C:|z|<1)to the case on the unit ball B in a complex Banach space or the unit polydisk U^(n) i...In this paper,the class of starlike functions of complex order γ(γ∈ℂ−{0})is extended from the case on unit disk U=(z∈C:|z|<1)to the case on the unit ball B in a complex Banach space or the unit polydisk U^(n) in C^(n).Let g be a convex function in U. We mainly establish the sharp bounds of all terms of homogeneous polynomial expansions for a subclass of g-parametric starlike mappings of complex order γ on B (resp.U^(n))when the mappings f are k-fold symmetric, k ∈ N. Our results partly solve the Bieberbach conjecture in several complex variables and generalize some prior works.展开更多
In renewing tissues,mutations conferring selective advantage may result in clonal expansions1-4.In contrast to somatic tissues,mutations driving clonal expansions in spermatogonia(CES)are also transmitted to the next ...In renewing tissues,mutations conferring selective advantage may result in clonal expansions1-4.In contrast to somatic tissues,mutations driving clonal expansions in spermatogonia(CES)are also transmitted to the next generation.This results in an effective increase of de novo mutation rate for CES drivers5-8.CES was originally discovered through extreme recurrence of de novo mutations causing Apert syndrome5.Here,we develop a systematic approach to discover CES drivers as hotspots of human de novo mutation.Our analysis of 54,715 trios ascertained for rare conditions9-13,6,065 control trios12,14-19 and population variation from 807,162 mostly healthy individuals20 identifies genes manifesting rates of de novo mutations inconsistent with plausible models of disease ascertainment.We propose 23 genes hypermutable at loss-of-function(LoF)sites as candidate CES drivers.An extra 17 genes feature hypermutable missense mutations at individual positions,suggesting CES acting through gain of function.CES increases the average mutation rate roughly 17-fold for LoF genes in both control trios and sperm and roughly 500-fold for pooled gain-of-function sites in sperm21.Positive selection in the male germline elevates the prevalence of genetic disorders and increases polymorphism levels,masking the effect of negative selection in human populations.展开更多
In this paper,we study asymptotic power series of the composition f(x)=h(g(x)),where g(x)=∑_(n=0)^(∞)b_(n)x^(-n),b_(n)∈R,and h is a given elementary function.The asymptotic expansions have been obtained for the com...In this paper,we study asymptotic power series of the composition f(x)=h(g(x)),where g(x)=∑_(n=0)^(∞)b_(n)x^(-n),b_(n)∈R,and h is a given elementary function.The asymptotic expansions have been obtained for the composition with an exponential or logarithmic function.Using the re-cursive method,we present the asymptotic expansions for the composition with seven trigonometric functions,respectively.As an application,the asymptotic expansions of roots of some equations are given.Computational results show that our recursive formula is more efficient than the method of Lagrange's inverse theorem.展开更多
Let m ≥ 1 be an integer,1 〈 β ≤ m ＋ 1.A sequence ε1ε2ε3 … with εi ∈{0,1,…,m} is called a β-expansion of a real number x if x = Σi εi/βi.It is known that when the base β is smaller than the generalized...Let m ≥ 1 be an integer,1 〈 β ≤ m ＋ 1.A sequence ε1ε2ε3 … with εi ∈{0,1,…,m} is called a β-expansion of a real number x if x = Σi εi/βi.It is known that when the base β is smaller than the generalized golden ration,any number has uncountably many expansions,while when β is larger,there are numbers which has unique expansion.In this paper,we consider the bases such that there is some number whose unique expansion is purely periodic with the given smallest period.We prove that such bases form an open interval,moreover,any two such open intervals have inclusion relationship according to the Sharkovskii ordering between the given minimal periods.We remark that our result answers an open question posed by Baker,and the proof for the case m = 1 is due to Allouche,Clarke and Sidorov.展开更多
The purpose of this study was to evaluate whether the cranial and circumaxillary sutures react differently to maxillary expansion （ME） and alternate maxillary expansions and constrictions （Alt-MEC） in a rat model....The purpose of this study was to evaluate whether the cranial and circumaxillary sutures react differently to maxillary expansion （ME） and alternate maxillary expansions and constrictions （Alt-MEC） in a rat model. Twenty-two male Sprague-Dawley rats （6 weeks old） were used and divided into three groups. In ME group （n=9）, an expander was activated for 5 days. In Alt-MEC group （9 animals）, an al- ternate expansion and constriction protocol （5-day expansion and 5-day constriction for one cycle） was conducted for 2.5 cycles （25 days total）. The control group comprised 4 animals with no appliances used, each of two sacrificed on day 5 and day 25 respectively. Midpalatal suture expansion or constriction levels were assessed qualitatively and quantitatively by bite-wing X-rays and cast models. Distances between two central incisors and two maxillary first molars were measured on cast models after each activation. Circumaxillary sutures （midpalatal, maxillopalatine, premaxillary, zygomaticotemporal and frontonasal suture） in each group were characterized histologically. Results showed that midpalatal suture was wid- ened and restored after each expansion and constriction. At the end of activation, the widths between both central incisors and first molars in Alt-MEC group were significantly larger than those in ME group （P〈0.05）. Histologically, all five circumaxillary sutures studied were widened in multiple zones in Alt- MEC group. However, only midpalatal suture was expanded with cellular fibrous tissue filling in ME group. Significant osteoclast hyperplasia was observed in all circumaxillary sutures after alternate expan- sions and constrictions, but osteoclast count increase was only observed in midpalatal suture in ME group. These results suggested that cranial and circumaxillary sutures were actively reconstructed after Alt-MEC, while only midpalatal suture had active reaction after ME.展开更多
After studying finite asymptotic expansions in real powers, we have developed a general theory for expansions of type (*) ,x → x0 where the ordered n-tuple forms an asymptotic scale at x0 , i.e. as x → x0, 1 ≤ i ≤...After studying finite asymptotic expansions in real powers, we have developed a general theory for expansions of type (*) ,x → x0 where the ordered n-tuple forms an asymptotic scale at x0 , i.e. as x → x0, 1 ≤ i ≤ n – 1, and is practically assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x o. As in previous papers by the author concerning polynomial, real-power and two-term theory, the locution “factorizational theory” refers to the special approach based on various types of factorizations of a differential operator associated to . Moreover, the guiding thread of our theory is the property of formal differentiation and we aim at characterizing some n-tuples of asymptotic expansions formed by (*) and n -1 expansions obtained by formal applications of suitable linear differential operators of orders 1,2,…,n-1. Some considerations lead to restrict the attention to two sets of operators naturally associated to “canonical factorizations”. This gives rise to conjectures whose proofs build an analytic theory of finite asymptotic expansions in the real domain which, though not elementary, parallels the familiar results about Taylor’s formula. One of the results states that to each scale of the type under consideration it remains associated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion(*), if valid, is automatically formally differentiable n-1 times in two special senses.展开更多
In this paper we estimate the degree of approximation of wavelet expansions. Our result shows that the degree has the exponential decay for function f(x)∈L2 continuous in a finite interval (a, b) which is much superi...In this paper we estimate the degree of approximation of wavelet expansions. Our result shows that the degree has the exponential decay for function f(x)∈L2 continuous in a finite interval (a, b) which is much superior to those of approximation by polynomial operators and by expansions of classical orthogonal series.展开更多
This paper, divided into three parts (Part II-A, Part II-B and Part II-C), contains the detailed factorizational theory of asymptotic expansions of type (?)?, , , where the asymptotic scale?, , is assumed to be an ext...This paper, divided into three parts (Part II-A, Part II-B and Part II-C), contains the detailed factorizational theory of asymptotic expansions of type (?)?, , , where the asymptotic scale?, , is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of . It follows two pre-viously published papers: the first, labelled as Part I, contains the complete (elementary but non-trivial) theory for;the second is a survey highlighting only the main results without proofs. All the material appearing in §2 of the survey is here reproduced in an expanded form, as it contains all the preliminary formulas necessary to understand and prove the results. The remaining part of the survey—especially the heuristical considerations and consequent conjectures in §3—may serve as a good introduction to the complete theory.展开更多
A systematic study of the phase formation, structure and magnetic properties of the R3Fe29-xTxcompounds (R=Y, Ce, Nd, Sm, Gd, Tb, and Dy; T=V and Cr) has been performed uponhydrogenation. The lattice constants and the...A systematic study of the phase formation, structure and magnetic properties of the R3Fe29-xTxcompounds (R=Y, Ce, Nd, Sm, Gd, Tb, and Dy; T=V and Cr) has been performed uponhydrogenation. The lattice constants and the unit cell volume of R3Fe29-xTxHy decrease withincreasing R atomic number from Nd to Dy, except for Ce, reflecting the lanthanide contraction.Regular anisotropic expansions mainly along the a- and b-axis rather than along the c-axis areobserved for all of the compounds upon hydrogenation. Hydrogenation leads to an increase inthe Curie temperature and a corresponding increase in the saturation magnetization at roomtemperature for each compound. First order magnetization processes (FOMP) occur in theexternal magnetic fields for Nd3Fe24.5Cr4.5H5.0, Tb3Fe27.0Cr2.0H2.8, and Gd3Fe28.0Cr1.0H4.2compounds展开更多
Part II-B of our work continues the factorizational theory of asymptotic expansions of type (*) , , where the asymptotic scale , , is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of ...Part II-B of our work continues the factorizational theory of asymptotic expansions of type (*) , , where the asymptotic scale , , is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x0. The main result states that to each scale of this type it remains as-sociated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion (*), if valid, is automatically formally differentiable n ? 1 times in the two special senses characterized in Part II-A. A second result shows that formal applications of ordinary derivatives to an asymptotic expansion are rarely admissible and that they may also yield skew results even for scales of powers.展开更多
A new method is presented for dealing with asymmetrical Abel inversion in this paper.We separate the integrated quantity into odd and even parts using Yasutomo's method. The asymmetric local value is expressed as ...A new method is presented for dealing with asymmetrical Abel inversion in this paper.We separate the integrated quantity into odd and even parts using Yasutomo's method. The asymmetric local value is expressed as the product of a weight function and a symmetric local value. The symmetric distribution is expanded into Fourier-Bessel series. The coefficients of the series are determined by the use of a least-square-fitting method.展开更多
In this paper,we study a special class of fractal interpolation functions,and give their Haar-wavelet expansions.On the basis of the expansions,we investigate the H(o|¨)lder smoothness of such functions and their...In this paper,we study a special class of fractal interpolation functions,and give their Haar-wavelet expansions.On the basis of the expansions,we investigate the H(o|¨)lder smoothness of such functions and their logical derivatives of order α.展开更多
Let mu be a locally uniformly alpha-dimensional measure on R-n and P-t(fd mu) be the Able Poisson means of Hermite expansions for f is an element of L-p(d mu), it is studied that the asymptotis properties of P-t(fd mu...Let mu be a locally uniformly alpha-dimensional measure on R-n and P-t(fd mu) be the Able Poisson means of Hermite expansions for f is an element of L-p(d mu), it is studied that the asymptotis properties of P-t(fd mu) as t --> 1_. Analogue of Wiener's theorem is obtained. Author also establishs the boundedness of the alpha-dimensional maximal conjugate Poisson integral operators from L-p(d mu) to the Lebesgne p-power integrable function spaces L-p(dx), and this derives directly the boundedness of Riesz transforms.展开更多
A new method for identifying nonlinear time varying systems with unknown structure is presented. The method extends the application area of basis sequence identification. The essential idea is to utilize the learning ...A new method for identifying nonlinear time varying systems with unknown structure is presented. The method extends the application area of basis sequence identification. The essential idea is to utilize the learning and nonlinear approximating ability of neural networks to model the non linearity of the system, characterize time varying dynamics of the system by the time varying parametric vector of the network, then the parametric vector of the network is approximated by a weighted sum of known basis sequences. Because of black box modeling ability of neural networks, the presented method can identify nonlinear time varying systems with unknown structure. In order to improve the real time capability of the algorithm, the neural network is trained by a simple fast learning algorithm based on local least squares presented by the authors. The effectiveness and the performance of the method are demonstrated by some simulation results.展开更多
We call “asymptotic mean” (at +∞) of a real-valued function the number, supposed to exist, , and highlight its role in the geometric theory of asymptotic expansions in the real domain of type (*) where the comparis...We call “asymptotic mean” (at +∞) of a real-valued function the number, supposed to exist, , and highlight its role in the geometric theory of asymptotic expansions in the real domain of type (*) where the comparison functions , forming an asymptotic scale at +∞, belong to one of the three classes having a definite “type of variation” at +∞, slow, regular or rapid. For regularly varying comparison functions we can characterize the existence of an asymptotic expansion (*) by the nice property that a certain quantity F(t) has an asymptotic mean at +∞. This quantity is defined via a linear differential operator in f and admits of a remarkable geometric interpretation as it measures the ordinate of the point wherein that special curve , which has a contact of order n - 1 with the graph of f at the generic point t, intersects a fixed vertical line, say x = T. Sufficient or necessary conditions hold true for the other two classes. In this article we give results for two types of expansions already studied in our current development of a general theory of asymptotic expansions in the real domain, namely polynomial and two-term expansions.展开更多
In this paper,the multiple stochastic integral with respect to a Wiener D'-process is defined.And also it is shown that for a D'-valued nonlinear random functional there exists a sequence of multiple integral ...In this paper,the multiple stochastic integral with respect to a Wiener D'-process is defined.And also it is shown that for a D'-valued nonlinear random functional there exists a sequence of multiple integral kernels such that the nonlinear functional can be expanded by series of multiple Wiener integrals of the integral kernels with respect to the Wiener D'-process.展开更多
One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs.Anot...One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs.Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems,particularly with high dimensional random parameters.We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems.The first task in this two-step process is to employ the procedure developed in[1]to construct an"arbitrary"polynomial chaos expansion basis using a finite number of statistical moments of the random inputs.The second step is a novel procedure to effect sparse approximation via l1 minimization in order to quantify the forward uncertainty.To enhance the performance of the preconditioned l1 minimization problem,we sample from the so-called induced distribution,instead of using Monte Carlo(MC)sampling from the original,unknown probability measure.We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures(such as the equilibrium measure)when we have incomplete information about the distribution.We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions,and on a Kirchoff plating bending problem with random Young’s modulus.展开更多
In this paper,by exponential complete Bell polynomials,we establish a(general)harmonic number asymptotic expansion,and give the corresponding recurrence of the coefficient sequence in the expansion.By the methods of t...In this paper,by exponential complete Bell polynomials,we establish a(general)harmonic number asymptotic expansion,and give the corresponding recurrence of the coefficient sequence in the expansion.By the methods of the generating functions and summation transformations,we also present an explicit expression for the coefficient sequence of the expansion.Moreover,we establish two(general)lacunary harmonic number asymptotic expansions,which contain only even or odd power terms in the logarithmic term.展开更多
Let ε ： N →4 R be a parameter function satisfying the condition ε （k） ＋ k ＋ 1 〉 0 and let Tε ： （0, 1] → （0, 1] be a transformation defined by Tε（x）=[-1＋（k＋1）x]/[1＋k-kεx] for x∈ （ 1/（k＋1）,...Let ε ： N →4 R be a parameter function satisfying the condition ε （k） ＋ k ＋ 1 〉 0 and let Tε ： （0, 1] → （0, 1] be a transformation defined by Tε（x）=[-1＋（k＋1）x]/[1＋k-kεx] for x∈ （ 1/（k＋1）, 1/k]. Under the algorithm Tε, every x ∈ （0, 1] is attached an expansion, called generalized continued fraction （GCFε） expansion with parameters by Schweiger. Define the sequence {kn （x）}n≥l of the partial quotients of x by k1 （x） =「1/x」 and kn （x） = k1 （T^（n-1）ε （x）） for every n≥ 2. Under the restriction -k - 1 〈 ε（k） 〈 -k, define the set of non-recurring GCFε expansions asFε= {x ∈（0, 1] ： kn＋1（x） 〉 kn（x） for infinitely many n}. It has been proved by Schweiger that Fε has Lebesgue measure 0. In the present paper, we strengthen this result by showing that {dimH Fε≥1/2 ,when ε（k）=-k-1＋ρ for a constant 0 〈 ρ 〈 1;1/（s＋2）≤dimH Fε ≤1/s,when ε（k）=-k-1＋1/k^s for any s≥1 where dimH denotes the Hausdorff dimension.展开更多
基金Supported by Projects from Chongqing Municipal Science and Technology Commission(CSTB2022NSCQ-MSX0445)。
文摘For the Sylvester continued fraction expansions of real numbers,FAN et al.(2007)proved that,for almost all real numbers,the nth partial quotient grows exponentially with respect to the product of the first n-1 partial quotients.In this paper,we establish the Hausdorff dimension of the exceptional set where the growth rate is a general function.
基金supported by the National Natural Science Foundation of China(12061035)the Research Foundation of Jiangxi Science and Technology Normal University of China(2021QNBJRC003)supported by the Graduate Innovation Fund of Jiangxi Science and Technology Normal University(YC2024-X10).
文摘In this paper,the class of starlike functions of complex order γ(γ∈ℂ−{0})is extended from the case on unit disk U=(z∈C:|z|<1)to the case on the unit ball B in a complex Banach space or the unit polydisk U^(n) in C^(n).Let g be a convex function in U. We mainly establish the sharp bounds of all terms of homogeneous polynomial expansions for a subclass of g-parametric starlike mappings of complex order γ on B (resp.U^(n))when the mappings f are k-fold symmetric, k ∈ N. Our results partly solve the Bieberbach conjecture in several complex variables and generalize some prior works.
文摘In renewing tissues,mutations conferring selective advantage may result in clonal expansions1-4.In contrast to somatic tissues,mutations driving clonal expansions in spermatogonia(CES)are also transmitted to the next generation.This results in an effective increase of de novo mutation rate for CES drivers5-8.CES was originally discovered through extreme recurrence of de novo mutations causing Apert syndrome5.Here,we develop a systematic approach to discover CES drivers as hotspots of human de novo mutation.Our analysis of 54,715 trios ascertained for rare conditions9-13,6,065 control trios12,14-19 and population variation from 807,162 mostly healthy individuals20 identifies genes manifesting rates of de novo mutations inconsistent with plausible models of disease ascertainment.We propose 23 genes hypermutable at loss-of-function(LoF)sites as candidate CES drivers.An extra 17 genes feature hypermutable missense mutations at individual positions,suggesting CES acting through gain of function.CES increases the average mutation rate roughly 17-fold for LoF genes in both control trios and sperm and roughly 500-fold for pooled gain-of-function sites in sperm21.Positive selection in the male germline elevates the prevalence of genetic disorders and increases polymorphism levels,masking the effect of negative selection in human populations.
基金Supported by The Innovation Fund of Postgraduate,Sichuan University of Science&Engineering(Y2024336)NSF of Sichuan Province(2023NSFSC0065).
文摘In this paper,we study asymptotic power series of the composition f(x)=h(g(x)),where g(x)=∑_(n=0)^(∞)b_(n)x^(-n),b_(n)∈R,and h is a given elementary function.The asymptotic expansions have been obtained for the composition with an exponential or logarithmic function.Using the re-cursive method,we present the asymptotic expansions for the composition with seven trigonometric functions,respectively.As an application,the asymptotic expansions of roots of some equations are given.Computational results show that our recursive formula is more efficient than the method of Lagrange's inverse theorem.
文摘Let m ≥ 1 be an integer,1 〈 β ≤ m ＋ 1.A sequence ε1ε2ε3 … with εi ∈{0,1,…,m} is called a β-expansion of a real number x if x = Σi εi/βi.It is known that when the base β is smaller than the generalized golden ration,any number has uncountably many expansions,while when β is larger,there are numbers which has unique expansion.In this paper,we consider the bases such that there is some number whose unique expansion is purely periodic with the given smallest period.We prove that such bases form an open interval,moreover,any two such open intervals have inclusion relationship according to the Sharkovskii ordering between the given minimal periods.We remark that our result answers an open question posed by Baker,and the proof for the case m = 1 is due to Allouche,Clarke and Sidorov.
基金supported by Peking University School of Stomatology Youth Scientific Research Fund of China(No.PKUSS20120113)
文摘The purpose of this study was to evaluate whether the cranial and circumaxillary sutures react differently to maxillary expansion （ME） and alternate maxillary expansions and constrictions （Alt-MEC） in a rat model. Twenty-two male Sprague-Dawley rats （6 weeks old） were used and divided into three groups. In ME group （n=9）, an expander was activated for 5 days. In Alt-MEC group （9 animals）, an al- ternate expansion and constriction protocol （5-day expansion and 5-day constriction for one cycle） was conducted for 2.5 cycles （25 days total）. The control group comprised 4 animals with no appliances used, each of two sacrificed on day 5 and day 25 respectively. Midpalatal suture expansion or constriction levels were assessed qualitatively and quantitatively by bite-wing X-rays and cast models. Distances between two central incisors and two maxillary first molars were measured on cast models after each activation. Circumaxillary sutures （midpalatal, maxillopalatine, premaxillary, zygomaticotemporal and frontonasal suture） in each group were characterized histologically. Results showed that midpalatal suture was wid- ened and restored after each expansion and constriction. At the end of activation, the widths between both central incisors and first molars in Alt-MEC group were significantly larger than those in ME group （P〈0.05）. Histologically, all five circumaxillary sutures studied were widened in multiple zones in Alt- MEC group. However, only midpalatal suture was expanded with cellular fibrous tissue filling in ME group. Significant osteoclast hyperplasia was observed in all circumaxillary sutures after alternate expan- sions and constrictions, but osteoclast count increase was only observed in midpalatal suture in ME group. These results suggested that cranial and circumaxillary sutures were actively reconstructed after Alt-MEC, while only midpalatal suture had active reaction after ME.
文摘After studying finite asymptotic expansions in real powers, we have developed a general theory for expansions of type (*) ,x → x0 where the ordered n-tuple forms an asymptotic scale at x0 , i.e. as x → x0, 1 ≤ i ≤ n – 1, and is practically assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x o. As in previous papers by the author concerning polynomial, real-power and two-term theory, the locution “factorizational theory” refers to the special approach based on various types of factorizations of a differential operator associated to . Moreover, the guiding thread of our theory is the property of formal differentiation and we aim at characterizing some n-tuples of asymptotic expansions formed by (*) and n -1 expansions obtained by formal applications of suitable linear differential operators of orders 1,2,…,n-1. Some considerations lead to restrict the attention to two sets of operators naturally associated to “canonical factorizations”. This gives rise to conjectures whose proofs build an analytic theory of finite asymptotic expansions in the real domain which, though not elementary, parallels the familiar results about Taylor’s formula. One of the results states that to each scale of the type under consideration it remains associated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion(*), if valid, is automatically formally differentiable n-1 times in two special senses.
基金This work is supported by the Natural Science Foundation of Zhejiang,PR China.
文摘In this paper we estimate the degree of approximation of wavelet expansions. Our result shows that the degree has the exponential decay for function f(x)∈L2 continuous in a finite interval (a, b) which is much superior to those of approximation by polynomial operators and by expansions of classical orthogonal series.
文摘This paper, divided into three parts (Part II-A, Part II-B and Part II-C), contains the detailed factorizational theory of asymptotic expansions of type (?)?, , , where the asymptotic scale?, , is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of . It follows two pre-viously published papers: the first, labelled as Part I, contains the complete (elementary but non-trivial) theory for;the second is a survey highlighting only the main results without proofs. All the material appearing in §2 of the survey is here reproduced in an expanded form, as it contains all the preliminary formulas necessary to understand and prove the results. The remaining part of the survey—especially the heuristical considerations and consequent conjectures in §3—may serve as a good introduction to the complete theory.
文摘A systematic study of the phase formation, structure and magnetic properties of the R3Fe29-xTxcompounds (R=Y, Ce, Nd, Sm, Gd, Tb, and Dy; T=V and Cr) has been performed uponhydrogenation. The lattice constants and the unit cell volume of R3Fe29-xTxHy decrease withincreasing R atomic number from Nd to Dy, except for Ce, reflecting the lanthanide contraction.Regular anisotropic expansions mainly along the a- and b-axis rather than along the c-axis areobserved for all of the compounds upon hydrogenation. Hydrogenation leads to an increase inthe Curie temperature and a corresponding increase in the saturation magnetization at roomtemperature for each compound. First order magnetization processes (FOMP) occur in theexternal magnetic fields for Nd3Fe24.5Cr4.5H5.0, Tb3Fe27.0Cr2.0H2.8, and Gd3Fe28.0Cr1.0H4.2compounds
文摘Part II-B of our work continues the factorizational theory of asymptotic expansions of type (*) , , where the asymptotic scale , , is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x0. The main result states that to each scale of this type it remains as-sociated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion (*), if valid, is automatically formally differentiable n ? 1 times in the two special senses characterized in Part II-A. A second result shows that formal applications of ordinary derivatives to an asymptotic expansion are rarely admissible and that they may also yield skew results even for scales of powers.
文摘A new method is presented for dealing with asymmetrical Abel inversion in this paper.We separate the integrated quantity into odd and even parts using Yasutomo's method. The asymmetric local value is expressed as the product of a weight function and a symmetric local value. The symmetric distribution is expanded into Fourier-Bessel series. The coefficients of the series are determined by the use of a least-square-fitting method.
文摘In this paper,we study a special class of fractal interpolation functions,and give their Haar-wavelet expansions.On the basis of the expansions,we investigate the H(o|¨)lder smoothness of such functions and their logical derivatives of order α.
文摘Let mu be a locally uniformly alpha-dimensional measure on R-n and P-t(fd mu) be the Able Poisson means of Hermite expansions for f is an element of L-p(d mu), it is studied that the asymptotis properties of P-t(fd mu) as t --> 1_. Analogue of Wiener's theorem is obtained. Author also establishs the boundedness of the alpha-dimensional maximal conjugate Poisson integral operators from L-p(d mu) to the Lebesgne p-power integrable function spaces L-p(dx), and this derives directly the boundedness of Riesz transforms.
文摘A new method for identifying nonlinear time varying systems with unknown structure is presented. The method extends the application area of basis sequence identification. The essential idea is to utilize the learning and nonlinear approximating ability of neural networks to model the non linearity of the system, characterize time varying dynamics of the system by the time varying parametric vector of the network, then the parametric vector of the network is approximated by a weighted sum of known basis sequences. Because of black box modeling ability of neural networks, the presented method can identify nonlinear time varying systems with unknown structure. In order to improve the real time capability of the algorithm, the neural network is trained by a simple fast learning algorithm based on local least squares presented by the authors. The effectiveness and the performance of the method are demonstrated by some simulation results.
文摘We call “asymptotic mean” (at +∞) of a real-valued function the number, supposed to exist, , and highlight its role in the geometric theory of asymptotic expansions in the real domain of type (*) where the comparison functions , forming an asymptotic scale at +∞, belong to one of the three classes having a definite “type of variation” at +∞, slow, regular or rapid. For regularly varying comparison functions we can characterize the existence of an asymptotic expansion (*) by the nice property that a certain quantity F(t) has an asymptotic mean at +∞. This quantity is defined via a linear differential operator in f and admits of a remarkable geometric interpretation as it measures the ordinate of the point wherein that special curve , which has a contact of order n - 1 with the graph of f at the generic point t, intersects a fixed vertical line, say x = T. Sufficient or necessary conditions hold true for the other two classes. In this article we give results for two types of expansions already studied in our current development of a general theory of asymptotic expansions in the real domain, namely polynomial and two-term expansions.
文摘In this paper,the multiple stochastic integral with respect to a Wiener D'-process is defined.And also it is shown that for a D'-valued nonlinear random functional there exists a sequence of multiple integral kernels such that the nonlinear functional can be expanded by series of multiple Wiener integrals of the integral kernels with respect to the Wiener D'-process.
基金supported by the NSF of China(No.11671265)partially supported by NSF DMS-1848508+4 种基金partially supported by the NSF of China(under grant numbers 11688101,11571351,and 11731006)science challenge project(No.TZ2018001)the youth innovation promotion association(CAS)supported by the National Science Foundation under Grant No.DMS-1439786the Simons Foundation Grant No.50736。
文摘One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs.Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems,particularly with high dimensional random parameters.We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems.The first task in this two-step process is to employ the procedure developed in[1]to construct an"arbitrary"polynomial chaos expansion basis using a finite number of statistical moments of the random inputs.The second step is a novel procedure to effect sparse approximation via l1 minimization in order to quantify the forward uncertainty.To enhance the performance of the preconditioned l1 minimization problem,we sample from the so-called induced distribution,instead of using Monte Carlo(MC)sampling from the original,unknown probability measure.We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures(such as the equilibrium measure)when we have incomplete information about the distribution.We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions,and on a Kirchoff plating bending problem with random Young’s modulus.
基金the National Natural Science Foundation of China(Grant No.11501081).
文摘In this paper,by exponential complete Bell polynomials,we establish a(general)harmonic number asymptotic expansion,and give the corresponding recurrence of the coefficient sequence in the expansion.By the methods of the generating functions and summation transformations,we also present an explicit expression for the coefficient sequence of the expansion.Moreover,we establish two(general)lacunary harmonic number asymptotic expansions,which contain only even or odd power terms in the logarithmic term.
基金Supported by the National Natural Science Foundation of China(Grant No.11361025)
文摘Let ε ： N →4 R be a parameter function satisfying the condition ε （k） ＋ k ＋ 1 〉 0 and let Tε ： （0, 1] → （0, 1] be a transformation defined by Tε（x）=[-1＋（k＋1）x]/[1＋k-kεx] for x∈ （ 1/（k＋1）, 1/k]. Under the algorithm Tε, every x ∈ （0, 1] is attached an expansion, called generalized continued fraction （GCFε） expansion with parameters by Schweiger. Define the sequence {kn （x）}n≥l of the partial quotients of x by k1 （x） =「1/x」 and kn （x） = k1 （T^（n-1）ε （x）） for every n≥ 2. Under the restriction -k - 1 〈 ε（k） 〈 -k, define the set of non-recurring GCFε expansions asFε= {x ∈（0, 1] ： kn＋1（x） 〉 kn（x） for infinitely many n}. It has been proved by Schweiger that Fε has Lebesgue measure 0. In the present paper, we strengthen this result by showing that {dimH Fε≥1/2 ,when ε（k）=-k-1＋ρ for a constant 0 〈 ρ 〈 1;1/（s＋2）≤dimH Fε ≤1/s,when ε（k）=-k-1＋1/k^s for any s≥1 where dimH denotes the Hausdorff dimension.
