The goal of this paper is to investigate the theory of Noether solvability for Volterra singular integral equations(VSIEs)with convolution and Cauchy kernels in a more general function class.To obtain the analytic sol...The goal of this paper is to investigate the theory of Noether solvability for Volterra singular integral equations(VSIEs)with convolution and Cauchy kernels in a more general function class.To obtain the analytic solutions,we transform such equations into boundary value problems with discontinuous coefficients by the properties of Fourier analysis.In view of the analytical Riemann-Hilbert method,the generalized Liouville theorem and Sokhotski-Plemelj formula,we get the uniqueness and existence of solutions for such problems,and study the asymptotic property of solutions at nodes.Therefore,this paper improves the theory of singular integral equations and boundary value problems.展开更多
Let M(u) be an N function, A=D r+∑r-1k=0a k(x)D k a linear differential operator and W M(A) the Sobolev Orlicz class defined by M(u) and A. In this paper we give the asymptotic estimates...Let M(u) be an N function, A=D r+∑r-1k=0a k(x)D k a linear differential operator and W M(A) the Sobolev Orlicz class defined by M(u) and A. In this paper we give the asymptotic estimates of the n K width d n(W M(A),L 2[0,1]) .展开更多
Let M(u) be an N function, A=D r+∑r-1k=0a k(x)D k a linear differential operator and W M(A) the Sobolev Orlicz class defined by M(u) and A. In this paper we give the asymptotic estimates...Let M(u) be an N function, A=D r+∑r-1k=0a k(x)D k a linear differential operator and W M(A) the Sobolev Orlicz class defined by M(u) and A. In this paper we give the asymptotic estimates of the n K width d n(W M(A),L 2[0,1]) .展开更多
For two subsets W and V of a Banach space X, let Kn（W, V, X） denote the relative Kolmogorov n-width of W relative to V defined by Kn（W,V,X）：=inf sup inf/Ln f∈W g∈V∩Ln‖f-g‖x, where the infimum is taken over ...For two subsets W and V of a Banach space X, let Kn（W, V, X） denote the relative Kolmogorov n-width of W relative to V defined by Kn（W,V,X）：=inf sup inf/Ln f∈W g∈V∩Ln‖f-g‖x, where the infimum is taken over all n-dimensional linear subspaces Ln of X. Let W2（△^τ） denote the class of 2π-periodic functions f with d-variables satisfying ∫[-π, π]^d｜△^τf（x）｜^2dx≤1, while △^τ is the r-iterate of Laplace operator △. This article discusses the relative Kolmogorov n-width of W2（△^τ） relative to W2（△^τ） in Lq（[-π, π]^d） （1≤ q ≤ ∞）, and obtain its weak asymptotic result.展开更多
Using the method of construction, with the help of inequalities, we research the Muntz rational approximation of two kinds of special function classes, and give the corresponding estimates of approximation rates of th...Using the method of construction, with the help of inequalities, we research the Muntz rational approximation of two kinds of special function classes, and give the corresponding estimates of approximation rates of these classes under widely con- ditions. Because of the Orlicz Spaces is bigger than continuous function space and the Lp space, so the results of this paper has a certain expansion significance.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
In this paper we study the convergence nf a class of means on H^p(G)(0<p<1),the means take the Bochner-Riesz means in[1],the generalized Bochner-Riesz means in[2],and the operators T^(Φ_r)in[3]as special cases....In this paper we study the convergence nf a class of means on H^p(G)(0<p<1),the means take the Bochner-Riesz means in[1],the generalized Bochner-Riesz means in[2],and the operators T^(Φ_r)in[3]as special cases.We obtain weak-type estimates for the associated maximal operators and the maximal mean boundedness for the means.展开更多
In this paper, we research the Miintz rational approximation of two kinds of spe- cial function classes, and give the corresponding estimates of approximation rates of these classes.
In this paper. we study the average n-K width of the convolution class B_(pq)(G)(or B_(?)(G)), for which the kernel G(x) is a PF density, in the metric L_q(R)(or L_(qp)(R)) for the case 1≤q<p ≤∞, and obtain some...In this paper. we study the average n-K width of the convolution class B_(pq)(G)(or B_(?)(G)), for which the kernel G(x) is a PF density, in the metric L_q(R)(or L_(qp)(R)) for the case 1≤q<p ≤∞, and obtain some exact results.展开更多
Introduction: The Six-Minute Walk Test (6MWT) is an inexpensive method to objectively evaluate physical capacity or limitation and stratify prognosis in patients with Heart Failure (HF). Since the clinical p...Introduction: The Six-Minute Walk Test (6MWT) is an inexpensive method to objectively evaluate physical capacity or limitation and stratify prognosis in patients with Heart Failure (HF). Since the clinical perception of symptoms may be adapted or compromised, regular evaluation from medical interviews often fails to determine functional classification. This study aimed to assess the correlation between New York Heart Association Functional Class (NYHA-FC) and the distance walked in the 6MWT. Methods: We conducted a cross-sectional observational study that included patients with HF with reduced ejection fraction followed up at an outpatient service of a teaching hospital, from August 2018 to April 2019. Patients in NYHA-FC I, II, or III were included. We compared NYHA-FC subjectively obtained during the consultation with the 6MWT performed after medical consultation, and the correlation between these two parameters was assessed. Results: The study included 70 patients with HF, 41 (58.6%) of whom were female. The mean age was 61.2 ± 12.7 years. The most prevalent etiologies were dilated idiopathic cardiomyopathy (35.7%) followed by ischemic cardiomyopathy (25.7%). The mean ejection fraction was 34.1% ± 9.8%. The average distance walked in the 6MWT by NYHA-FC I patients was 437.8 ± 95.8 meters, NYHA-FC II 360.1 ± 96.4, and NYHA-FC III 248.4 ± 98.3. Functional class measured by the 6MWT was different than that estimated by NYHA-FC in 34 patients (48.6%), 23 (32.9%) for a higher functional class and 11 (15.7%) for a lower one (p = 0.07). Pearson’s correlation coefficient between NYHA-FC and the 6MWT was -0.55. Conclusion: There was a moderate correlation between the subjective NYHA-FC and the 6MWT. The 6MWT revealed a different classification from NYHA-FC in almost half of the patients. Among those who presented discrepancies between methods, 6MWT reclassification towards a higher functional class was more common.展开更多
Introduction: Borderline Class II malocclusion due to deficient mandible can be treated either by orthodontic camouflage, fixed functional appliances or by orthodontics followed by surgical mandibular advancement. Met...Introduction: Borderline Class II malocclusion due to deficient mandible can be treated either by orthodontic camouflage, fixed functional appliances or by orthodontics followed by surgical mandibular advancement. Methodology: A prospective study was designed on young adults with Class II malocclusion on account of a deficient mandible. A total of 45 subjects were divided into three groups of 15 individuals each. The patients were treated either by camouflage, fixed functional appliances or by orthognathic surgery. Pre and post treatment cephalograms were used to assess the skeletal, dental and soft tissue changes. Pre and post treatment profile photographs were assessed on a Visual Analogue Scale (VAS) by orthodontists, oral surgeons and laypersons. Results: Each group achieved a reduction in facial convexity, but the results obtained from the surgical group were more pronounced than the camouflage and the fixed functional group. Conclusion: The reduction in convexity in the camouflage group was by retracting the upper anteriors, which increases the nasolabial angle. In the fixed functional appliance a combination of skeletal and dentoalveolar changes can be observed. However the most appropriate reduction in profile convexity can be obtained by combined orthodontic and surgical treatment of malocclusion.展开更多
Let Γ be a regular curve and Lp (Γ), 1<p<+∞. be the class of all complex - valued functions f de-fined on Γ which are such that |f|p is mtegrabie in sense of Lebesgue. In this work, we define the k th p-Fabc...Let Γ be a regular curve and Lp (Γ), 1<p<+∞. be the class of all complex - valued functions f de-fined on Γ which are such that |f|p is mtegrabie in sense of Lebesgue. In this work, we define the k th p-Fabcrpolynomial F k,p (z),the kth p-Faber principle part F k.p (1/z) for Γ , and defined the nth p-Fcber- Laurent rational function Rn.p (f, z) and p- generalized modulus of continuity Ωp(f, t) of a function f of Lp(Γ) We inves-tigate some properties of Fk,p (z) and Fk.p (1/z). And then we prove a direct theorem characterizing the degree of approximation with respect to Ω (. , t) in the mean of functions of Lp(Γ) by the rational junctions Rn.p (. . z).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
A class of distributions called Box-Cox symmetric was proposed for random variables with asymmetric distributions. This class allows through its structure an interpretation of the parameters in terms of quantiles (in ...A class of distributions called Box-Cox symmetric was proposed for random variables with asymmetric distributions. This class allows through its structure an interpretation of the parameters in terms of quantiles (in particular, the median), relative dispersion and skewness. This study presents the initial </span><span style="font-family:Verdana;">results of the computational development of basic functions of each of the</span><span style="font-family:Verdana;"> distributions that make up the Box-Cox symmetric class. Four functions have been developed to compose a routine in software R up to now. These functions are related to random numbers generation, probability density function, cumulative distribution function, and quantile function associated to a given probability. Examples of implemented functions were presented. The gamlss routine was used to check the performance of developed functions.展开更多
Objectives To compare respiratory parameters of peripheral blood mononuclear cell mitochondria and iron metabolism indicators in patients with different NYHA functional classes of ischemic heart failure(HF).Methods Th...Objectives To compare respiratory parameters of peripheral blood mononuclear cell mitochondria and iron metabolism indicators in patients with different NYHA functional classes of ischemic heart failure(HF).Methods This single center, prospective, non-blinded study enrolled 20 patients with diagnosed chronic HF of ischemic genesis with reduced and mildly reduced left ventricle ejection fraction. The maximum oxygen consumption at the peak of the exercise test(VO2peak), iron metabolism parameters and respiratory activity of peripheral blood mononuclear cell mitochondria were assessed.Results Among the patients, a half of individuals were diagnosed with iron deficiency. Subgroups of patients with different HF severity did not significant differ in VO2peak(P=0.209), serum iron(P=0.468) and ferritin(P=0.235) levels. But there was a trend in increasing in these parameters with increasing NYHA HF functional class. Respiratory control coefficient(RC) in NADdependent and FAD-dependent mitochondrial oxidation were lower in patients with NYHA HF Ⅲ functional class compared to individuals with NYHA HF I functional class(P=0.028 and P=0.040, respectively). Serum iron(P=0.026), ferritin(P=0.045)levels, transferrin saturation(P=0.006) were negatively correlated with RC in NAD-dependent mitochondrial oxidation.Conclusions In aggravation of ischemic HF NYHA FC, there is a decrease in RC of PBMC mitochondria during the oxidation of NAD-dependent and FAD-dependent substrates. In the whole sample, patients with laboratory-confirmed iron deficiency accounted a half of the total number. Iron metabolism parameters had a paradoxical inverse relationship with the level of RC in PBMC mitochondria of patients with HF.展开更多
The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-fun...The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-function, Appell function F3 and a general class of poly- nomials. The results obtained provide unification and extension of the results given by Saxena et al. [13], Srivastava and Grag [17], Srivastava et al. [20], and etc. The results are obtained in compact form and are useful in preparing some tables of operators of fractional calculus. On account of the general nature of the Saigo-Maeda operators, R-function, and a general class of polynomials a large number of new and known results involving Saigo fractional calculus operators and several special functions notably H-function, /-function, Mittag-Leffier function, generalized Wright hypergeometric function, generalized Bessel-Maitland function follow as special cases of our main findings.展开更多
Let X (t)(t∈R^N) be a d-dimensional fractional Brownian motion. A contiunous function f:R^N→R^d is called a polar function of X(t)(t∈R^N) if P{ t∈R^N\{0},X(t)=t(t)}=0. In this paper, the characteristies of the cla...Let X (t)(t∈R^N) be a d-dimensional fractional Brownian motion. A contiunous function f:R^N→R^d is called a polar function of X(t)(t∈R^N) if P{ t∈R^N\{0},X(t)=t(t)}=0. In this paper, the characteristies of the class of polar functions are studied. Our theorem 1 improves the previous results of Graversen and Legall. Theorem2 solves a problem of Legall (1987) on Brownian motion.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.11971015).
文摘The goal of this paper is to investigate the theory of Noether solvability for Volterra singular integral equations(VSIEs)with convolution and Cauchy kernels in a more general function class.To obtain the analytic solutions,we transform such equations into boundary value problems with discontinuous coefficients by the properties of Fourier analysis.In view of the analytical Riemann-Hilbert method,the generalized Liouville theorem and Sokhotski-Plemelj formula,we get the uniqueness and existence of solutions for such problems,and study the asymptotic property of solutions at nodes.Therefore,this paper improves the theory of singular integral equations and boundary value problems.
文摘Let M(u) be an N function, A=D r+∑r-1k=0a k(x)D k a linear differential operator and W M(A) the Sobolev Orlicz class defined by M(u) and A. In this paper we give the asymptotic estimates of the n K width d n(W M(A),L 2[0,1]) .
文摘Let M(u) be an N function, A=D r+∑r-1k=0a k(x)D k a linear differential operator and W M(A) the Sobolev Orlicz class defined by M(u) and A. In this paper we give the asymptotic estimates of the n K width d n(W M(A),L 2[0,1]) .
基金Supported partly by National Natural Science Foundation of China (10471010)partly by the project "Representation Theory and Related Topics" of the "985 Program" of Beijing Normal University
文摘For two subsets W and V of a Banach space X, let Kn（W, V, X） denote the relative Kolmogorov n-width of W relative to V defined by Kn（W,V,X）：=inf sup inf/Ln f∈W g∈V∩Ln‖f-g‖x, where the infimum is taken over all n-dimensional linear subspaces Ln of X. Let W2（△^τ） denote the class of 2π-periodic functions f with d-variables satisfying ∫[-π, π]^d｜△^τf（x）｜^2dx≤1, while △^τ is the r-iterate of Laplace operator △. This article discusses the relative Kolmogorov n-width of W2（△^τ） relative to W2（△^τ） in Lq（[-π, π]^d） （1≤ q ≤ ∞）, and obtain its weak asymptotic result.
基金supported by the National Science Foundation of China(No.11161033)Inner Mongolia Normal University Talent Project Foundation(No.RCPY-2-2012-K-036)
文摘Using the method of construction, with the help of inequalities, we research the Muntz rational approximation of two kinds of special function classes, and give the corresponding estimates of approximation rates of these classes under widely con- ditions. Because of the Orlicz Spaces is bigger than continuous function space and the Lp space, so the results of this paper has a certain expansion significance.
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In this paper we study the convergence nf a class of means on H^p(G)(0<p<1),the means take the Bochner-Riesz means in[1],the generalized Bochner-Riesz means in[2],and the operators T^(Φ_r)in[3]as special cases.We obtain weak-type estimates for the associated maximal operators and the maximal mean boundedness for the means.
基金Supported by the National Natural Science Foundation of China(11161033)Inner Mongolia Natural Science Foundation (2009MS0105)
文摘In this paper, we research the Miintz rational approximation of two kinds of spe- cial function classes, and give the corresponding estimates of approximation rates of these classes.
基金The author was supported by the National Natural Science Found of China.
文摘In this paper. we study the average n-K width of the convolution class B_(pq)(G)(or B_(?)(G)), for which the kernel G(x) is a PF density, in the metric L_q(R)(or L_(qp)(R)) for the case 1≤q<p ≤∞, and obtain some exact results.
文摘Introduction: The Six-Minute Walk Test (6MWT) is an inexpensive method to objectively evaluate physical capacity or limitation and stratify prognosis in patients with Heart Failure (HF). Since the clinical perception of symptoms may be adapted or compromised, regular evaluation from medical interviews often fails to determine functional classification. This study aimed to assess the correlation between New York Heart Association Functional Class (NYHA-FC) and the distance walked in the 6MWT. Methods: We conducted a cross-sectional observational study that included patients with HF with reduced ejection fraction followed up at an outpatient service of a teaching hospital, from August 2018 to April 2019. Patients in NYHA-FC I, II, or III were included. We compared NYHA-FC subjectively obtained during the consultation with the 6MWT performed after medical consultation, and the correlation between these two parameters was assessed. Results: The study included 70 patients with HF, 41 (58.6%) of whom were female. The mean age was 61.2 ± 12.7 years. The most prevalent etiologies were dilated idiopathic cardiomyopathy (35.7%) followed by ischemic cardiomyopathy (25.7%). The mean ejection fraction was 34.1% ± 9.8%. The average distance walked in the 6MWT by NYHA-FC I patients was 437.8 ± 95.8 meters, NYHA-FC II 360.1 ± 96.4, and NYHA-FC III 248.4 ± 98.3. Functional class measured by the 6MWT was different than that estimated by NYHA-FC in 34 patients (48.6%), 23 (32.9%) for a higher functional class and 11 (15.7%) for a lower one (p = 0.07). Pearson’s correlation coefficient between NYHA-FC and the 6MWT was -0.55. Conclusion: There was a moderate correlation between the subjective NYHA-FC and the 6MWT. The 6MWT revealed a different classification from NYHA-FC in almost half of the patients. Among those who presented discrepancies between methods, 6MWT reclassification towards a higher functional class was more common.
文摘Introduction: Borderline Class II malocclusion due to deficient mandible can be treated either by orthodontic camouflage, fixed functional appliances or by orthodontics followed by surgical mandibular advancement. Methodology: A prospective study was designed on young adults with Class II malocclusion on account of a deficient mandible. A total of 45 subjects were divided into three groups of 15 individuals each. The patients were treated either by camouflage, fixed functional appliances or by orthognathic surgery. Pre and post treatment cephalograms were used to assess the skeletal, dental and soft tissue changes. Pre and post treatment profile photographs were assessed on a Visual Analogue Scale (VAS) by orthodontists, oral surgeons and laypersons. Results: Each group achieved a reduction in facial convexity, but the results obtained from the surgical group were more pronounced than the camouflage and the fixed functional group. Conclusion: The reduction in convexity in the camouflage group was by retracting the upper anteriors, which increases the nasolabial angle. In the fixed functional appliance a combination of skeletal and dentoalveolar changes can be observed. However the most appropriate reduction in profile convexity can be obtained by combined orthodontic and surgical treatment of malocclusion.
文摘Let Γ be a regular curve and Lp (Γ), 1<p<+∞. be the class of all complex - valued functions f de-fined on Γ which are such that |f|p is mtegrabie in sense of Lebesgue. In this work, we define the k th p-Fabcrpolynomial F k,p (z),the kth p-Faber principle part F k.p (1/z) for Γ , and defined the nth p-Fcber- Laurent rational function Rn.p (f, z) and p- generalized modulus of continuity Ωp(f, t) of a function f of Lp(Γ) We inves-tigate some properties of Fk,p (z) and Fk.p (1/z). And then we prove a direct theorem characterizing the degree of approximation with respect to Ω (. , t) in the mean of functions of Lp(Γ) by the rational junctions Rn.p (. . z).
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A class of distributions called Box-Cox symmetric was proposed for random variables with asymmetric distributions. This class allows through its structure an interpretation of the parameters in terms of quantiles (in particular, the median), relative dispersion and skewness. This study presents the initial </span><span style="font-family:Verdana;">results of the computational development of basic functions of each of the</span><span style="font-family:Verdana;"> distributions that make up the Box-Cox symmetric class. Four functions have been developed to compose a routine in software R up to now. These functions are related to random numbers generation, probability density function, cumulative distribution function, and quantile function associated to a given probability. Examples of implemented functions were presented. The gamlss routine was used to check the performance of developed functions.
基金supported by Russian Science Foundation,RSF 23-75-00009(part of the study corresponding to finding 1)Part of the study corresponding to finding 2 was carried out within the state assignment,FSR No.:122020300045-5(03.02.2022).
文摘Objectives To compare respiratory parameters of peripheral blood mononuclear cell mitochondria and iron metabolism indicators in patients with different NYHA functional classes of ischemic heart failure(HF).Methods This single center, prospective, non-blinded study enrolled 20 patients with diagnosed chronic HF of ischemic genesis with reduced and mildly reduced left ventricle ejection fraction. The maximum oxygen consumption at the peak of the exercise test(VO2peak), iron metabolism parameters and respiratory activity of peripheral blood mononuclear cell mitochondria were assessed.Results Among the patients, a half of individuals were diagnosed with iron deficiency. Subgroups of patients with different HF severity did not significant differ in VO2peak(P=0.209), serum iron(P=0.468) and ferritin(P=0.235) levels. But there was a trend in increasing in these parameters with increasing NYHA HF functional class. Respiratory control coefficient(RC) in NADdependent and FAD-dependent mitochondrial oxidation were lower in patients with NYHA HF Ⅲ functional class compared to individuals with NYHA HF I functional class(P=0.028 and P=0.040, respectively). Serum iron(P=0.026), ferritin(P=0.045)levels, transferrin saturation(P=0.006) were negatively correlated with RC in NAD-dependent mitochondrial oxidation.Conclusions In aggravation of ischemic HF NYHA FC, there is a decrease in RC of PBMC mitochondria during the oxidation of NAD-dependent and FAD-dependent substrates. In the whole sample, patients with laboratory-confirmed iron deficiency accounted a half of the total number. Iron metabolism parameters had a paradoxical inverse relationship with the level of RC in PBMC mitochondria of patients with HF.
基金NBHM Department of Atomic Energy,Government of India,Mumbai for the finanicai assistance under PDF sanction no.2/40(37)/2014/R&D-II/14131
文摘The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-function, Appell function F3 and a general class of poly- nomials. The results obtained provide unification and extension of the results given by Saxena et al. [13], Srivastava and Grag [17], Srivastava et al. [20], and etc. The results are obtained in compact form and are useful in preparing some tables of operators of fractional calculus. On account of the general nature of the Saigo-Maeda operators, R-function, and a general class of polynomials a large number of new and known results involving Saigo fractional calculus operators and several special functions notably H-function, /-function, Mittag-Leffier function, generalized Wright hypergeometric function, generalized Bessel-Maitland function follow as special cases of our main findings.
文摘Let X (t)(t∈R^N) be a d-dimensional fractional Brownian motion. A contiunous function f:R^N→R^d is called a polar function of X(t)(t∈R^N) if P{ t∈R^N\{0},X(t)=t(t)}=0. In this paper, the characteristies of the class of polar functions are studied. Our theorem 1 improves the previous results of Graversen and Legall. Theorem2 solves a problem of Legall (1987) on Brownian motion.
