The purpose of this paper is to define the generalized Euler numbers and the generalized Euler numbers of higher order, their recursion formula and some properties were established, accordingly Euler numbers and Euler...The purpose of this paper is to define the generalized Euler numbers and the generalized Euler numbers of higher order, their recursion formula and some properties were established, accordingly Euler numbers and Euler numbers of higher order were extended.展开更多
Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying...Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying the linear load is generalized,and then it is expanded into the corresponding Fourier series.With the obtained summation results of the infinite series,it is found that they are related to Bernoulli num-bers and π. The recurrent formula of Bernoulli numbers is presented. The relationships among the coefficients of the beam,Bernoulli numbers and Euler numbers are found,and the relative mathematical formulas are presented.展开更多
Let p 〉 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and...Let p 〉 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and hypergeometric series. We prove that ∑k=0^p-1（k^2k/2k）≡（-1）^（p-1）/2-p^2Ep-3（modp^3） ∑k=1^（p-1）/2（k^2k）/k≡（-1）^（p＋1）/2 8/3pEp-3（mod p^2）,∑k=0^（p-1）/2（k^2k）^2/16k≡（-1）^（p-1）/2＋p^2Ep-3（mod p^3）,where E0, E1, E2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for π2, π-2 and the constant K ：= ∑k=1^∞（k/3）/k^2 （with （-） the Jacobi symbol）, two of which are ∑k=1^∞（10k-3）8k/k2（k^2k）^2（k^3k）=π^2/2and ∑k=1^∞（15k-4）（-27）^k-1/k^3（k^2k）^2（k^3k）=K.展开更多
The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbe...The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.展开更多
Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynom...Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynomials, Bernoulli numbers;its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained;the formulae for obtaining all π<sup>m</sup> as series on k<sup>-m</sup> and for expanding functions into series of Euler polynomials.展开更多
In this paper, one introduces the polynomials R<sub>n</sub>(x) and numbers R<sub>n</sub> and derives some interesting identities related to the numbers and polynomials: R<sub>n</sub>...In this paper, one introduces the polynomials R<sub>n</sub>(x) and numbers R<sub>n</sub> and derives some interesting identities related to the numbers and polynomials: R<sub>n</sub> and R<sub>n</sub>(x). We also give relation between the Stirling numbers, the Bell numbers, the R<sub>n</sub> and R<sub>n</sub>(x).展开更多
In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and dedu...In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and deduce the corresponding special cases.展开更多
The identification of objects in binary images is a fundamental task in image analysis and pattern recognition tasks. The Euler number of a binary image is an important topological measure which is used as a feature i...The identification of objects in binary images is a fundamental task in image analysis and pattern recognition tasks. The Euler number of a binary image is an important topological measure which is used as a feature in image analysis. In this paper, a very fast algorithm for the detection and localization of the objects and the computation of the Euler number of a binary image is proposed. The proposed algorithm operates in one scan of the image and is based on the Image Block Representation (IBR) scheme. The proposed algorithm is more efficient than conventional pixel based algorithms in terms of execution speed and representation of the extracted information.展开更多
This paper summarizes the works of numerous inspiring hard-working mathematicians in chronological order who have toiled the fields of mathematics to bring forward the harvest of Eulers number, also known as Napiers n...This paper summarizes the works of numerous inspiring hard-working mathematicians in chronological order who have toiled the fields of mathematics to bring forward the harvest of Eulers number, also known as Napiers number or more infamously, e.展开更多
Usually image assessment methods could be classified into two categories: subjective as-sessments and objective ones. The latter are judged by the correlation coefficient with subjective quality measurement MOS (Mean ...Usually image assessment methods could be classified into two categories: subjective as-sessments and objective ones. The latter are judged by the correlation coefficient with subjective quality measurement MOS (Mean Opinion Score). This paper presents an objective quality assessment algorithm special for binary images. In the algorithm, noise energy is measured by Euclidean distance between noises and signals and the structural effects caused by noise are described by Euler number change. The assessment on image quality is calculated quantitatively in terms of PSNR (Peak Signal to Noise Ratio). Our experiments show that the results of the algorithm are highly correlative with subjective MOS and the algorithm is more simple and computational saving than traditional objective assessment methods.展开更多
Submerged gas injection into liquid leads to complex multiphase flow, in which nozzle geometries are crucial important for the operational expenditure in terms of pressure drop. The influence of the nozzle geometry on...Submerged gas injection into liquid leads to complex multiphase flow, in which nozzle geometries are crucial important for the operational expenditure in terms of pressure drop. The influence of the nozzle geometry on pressure drop between nozzle inlet and outlet has been experimentally studied for different gas flow rates and bath depths. Nozzles with circular, gear-like and four-leaf cross-sectional shape have been studied. The results indicate that, besides the hydraulic diameter of the outlet, the orifice area and the perimeter of the nozzle tip also play significant roles. For the same superficial gas velocity, the average pressure drop from the four-leaf-shaped geometry is the least. The influence of bath depth was found negligible. A correlation for the modified Euler number considering the pressure drop is proposed depending on nozzle geometric parameter and on the modified Froude number with the hydraulic diameter of the nozzle do as characteristic length.展开更多
We investigate the quantum metric and topological Euler number in a cyclically modulated Su-Schrieffer-Heeger(SSH)model with long-range hopping terms.By computing the quantum geometry tensor,we derive exact expression...We investigate the quantum metric and topological Euler number in a cyclically modulated Su-Schrieffer-Heeger(SSH)model with long-range hopping terms.By computing the quantum geometry tensor,we derive exact expressions for the quantum metric and Berry curvature of the energy band electrons,and we obtain the phase diagram of the model marked by the first Chern number.Furthermore,we also obtain the topological Euler number of the energy band based on the Gauss-Bonnet theorem on the topological characterization of the closed Bloch states manifold in the first Brillouin zone.However,some regions where the Berry curvature is identically zero in the first Brillouin zone result in the degeneracy of the quantum metric,which leads to ill-defined non-integer topological Euler numbers.Nevertheless,the non-integer"Euler number"provides valuable insights and an upper bound for the absolute values of the Chern numbers.展开更多
A fundamental problem in four dimensional differential topology is to find a surface with minimal genus which represents a given homology class. This problem was considered by many people for closed 4 manifolds. In th...A fundamental problem in four dimensional differential topology is to find a surface with minimal genus which represents a given homology class. This problem was considered by many people for closed 4 manifolds. In this paper,we consider this problem for four manifold with boundary.展开更多
Labeling connected components and holes and computing the Euler number in a binary image are necessary for image analysis, pattern recognition, and computer （robot） vision, and are usually made independently of each...Labeling connected components and holes and computing the Euler number in a binary image are necessary for image analysis, pattern recognition, and computer （robot） vision, and are usually made independently of each other in conventional methods. This paper proposes a two-scan algorithm for labeling connected components and holes simultaneously in a binary image by use of the same data structure. With our algorithm, besides labeling, we can also easily calculate the number and the area of connected components and holes, as well as the Euler number. Our method is very simple in principle, and experimental results demonstrate that our method is much more efficient than conventional methods for various kinds of images in cases where both labeling and Euler number computing are necessary.展开更多
The authors compute the(rational) Betti number of real toric varieties associated to Weyl chambers of type B, and furthermore show that their integral cohomology is p-torsion free for all odd primes p.
A proper vertex coloring of a graph G is acyclic if there is no bicolored cycles in G.A graph G is acyclically k-choosable if for any list assignment L={L(v):v∈V(G)}with|L(v)|≥k for each vertex v∈V(G),there exists ...A proper vertex coloring of a graph G is acyclic if there is no bicolored cycles in G.A graph G is acyclically k-choosable if for any list assignment L={L(v):v∈V(G)}with|L(v)|≥k for each vertex v∈V(G),there exists an acyclic proper vertex coloringφof G such thatφ(v)∈L(v)for each vertex v∈V(G).In this paper,we prove that every graph G embedded on the surface with Euler characteristic numberε=-1 is acyclically 11-choosable.展开更多
Euler Number is one of the most important characteristics in topology. In twodimension digital images, the Euler characteristic is locally computable. The form of Euler Number formula is different under 4-connected an...Euler Number is one of the most important characteristics in topology. In twodimension digital images, the Euler characteristic is locally computable. The form of Euler Number formula is different under 4-connected and 8-connected conditions. Based on the definition of the Foreground Segment and Neighbor Number, a formula of the Euler Number computing is proposed and is proved in this paper. It is a new idea to locally compute Euler Number of 2D image.展开更多
In this paper,we prove two supercongruences conjectured by Z.-W.Sun via the Wilf-Zeilberger method.One of them is,for any prime p>3,4F3[7/6 1/2 1/2 1/2 1/6 1 1]-1/8]-1/2≡p(-2/p)+p^(3)/4(2/p)Ep-3 (mod p^(4))where(&...In this paper,we prove two supercongruences conjectured by Z.-W.Sun via the Wilf-Zeilberger method.One of them is,for any prime p>3,4F3[7/6 1/2 1/2 1/2 1/6 1 1]-1/8]-1/2≡p(-2/p)+p^(3)/4(2/p)Ep-3 (mod p^(4))where(·/p)stands for the Legendre symbol,and E_(n)is the n-th Euler number.展开更多
In this paper, we show that, without involving the C~*-bundle theory, elementary differ- ential topology can do the classification of homogeneous C~*-crossed product C(X)×α Z, where X is a compact differentiable...In this paper, we show that, without involving the C~*-bundle theory, elementary differ- ential topology can do the classification of homogeneous C~*-crossed product C(X)×α Z, where X is a compact differentiable manifold of low dimension and α is a diffeomorphism of X. The motivation of this work is the Shultz’s theorem which states that a C~*-algebra can be identified with its pure state space carrying w~*-topology and certain geometric structure.展开更多
We consider the finite element based computation of topological quantities of implicitly represented surfaces within a diffuse interface framework.Utilizing an adaptive finite element implementation with effective gra...We consider the finite element based computation of topological quantities of implicitly represented surfaces within a diffuse interface framework.Utilizing an adaptive finite element implementation with effective gradient recovery techniques,we discuss how the Euler number can be accurately computed directly from the numerically solved phase field functions or order parameters.Numerical examples and applications to the topological analysis of point clouds are also presented.展开更多
基金Supported by the NNSF of China(10001016) SF for the Prominent Youth of Henan Province
文摘The purpose of this paper is to define the generalized Euler numbers and the generalized Euler numbers of higher order, their recursion formula and some properties were established, accordingly Euler numbers and Euler numbers of higher order were extended.
基金Supported by the National Natural Science Foundation of China(51276017)
文摘Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying the linear load is generalized,and then it is expanded into the corresponding Fourier series.With the obtained summation results of the infinite series,it is found that they are related to Bernoulli num-bers and π. The recurrent formula of Bernoulli numbers is presented. The relationships among the coefficients of the beam,Bernoulli numbers and Euler numbers are found,and the relative mathematical formulas are presented.
基金supported by the National Natural Science Foundation of China(GrantNo.10871087)the Overseas Cooperation Fund of China(Grant No.10928101)
文摘Let p 〉 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and hypergeometric series. We prove that ∑k=0^p-1（k^2k/2k）≡（-1）^（p-1）/2-p^2Ep-3（modp^3） ∑k=1^（p-1）/2（k^2k）/k≡（-1）^（p＋1）/2 8/3pEp-3（mod p^2）,∑k=0^（p-1）/2（k^2k）^2/16k≡（-1）^（p-1）/2＋p^2Ep-3（mod p^3）,where E0, E1, E2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for π2, π-2 and the constant K ：= ∑k=1^∞（k/3）/k^2 （with （-） the Jacobi symbol）, two of which are ∑k=1^∞（10k-3）8k/k2（k^2k）^2（k^3k）=π^2/2and ∑k=1^∞（15k-4）（-27）^k-1/k^3（k^2k）^2（k^3k）=K.
基金the Guangdong Provincial Natural Science Foundation (No.05005928)the National Natural Science Foundation (No.10671155) of P.R.China
文摘The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.
文摘Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynomials, Bernoulli numbers;its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained;the formulae for obtaining all π<sup>m</sup> as series on k<sup>-m</sup> and for expanding functions into series of Euler polynomials.
文摘In this paper, one introduces the polynomials R<sub>n</sub>(x) and numbers R<sub>n</sub> and derives some interesting identities related to the numbers and polynomials: R<sub>n</sub> and R<sub>n</sub>(x). We also give relation between the Stirling numbers, the Bell numbers, the R<sub>n</sub> and R<sub>n</sub>(x).
基金Supported by the PCSIRT of Education of China(IRT0621)Supported by the Innovation Program of Shanghai Municipal Education Committee of China(08ZZ24)Supported by the Henan Innovation Project for University Prominent Research Talents of China(2007KYCX0021)
文摘In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and deduce the corresponding special cases.
文摘The identification of objects in binary images is a fundamental task in image analysis and pattern recognition tasks. The Euler number of a binary image is an important topological measure which is used as a feature in image analysis. In this paper, a very fast algorithm for the detection and localization of the objects and the computation of the Euler number of a binary image is proposed. The proposed algorithm operates in one scan of the image and is based on the Image Block Representation (IBR) scheme. The proposed algorithm is more efficient than conventional pixel based algorithms in terms of execution speed and representation of the extracted information.
文摘This paper summarizes the works of numerous inspiring hard-working mathematicians in chronological order who have toiled the fields of mathematics to bring forward the harvest of Eulers number, also known as Napiers number or more infamously, e.
基金Supported by Innovation Fund for Small Technology Based Firms, China (No.04C26213301189)Science and Technology Foundation by Beijng Jiaotong University (No.2005SM009)the Key Laboratory of Advanced Information Science and Network Technology of Beijing (No.TDXX0509).
文摘Usually image assessment methods could be classified into two categories: subjective as-sessments and objective ones. The latter are judged by the correlation coefficient with subjective quality measurement MOS (Mean Opinion Score). This paper presents an objective quality assessment algorithm special for binary images. In the algorithm, noise energy is measured by Euclidean distance between noises and signals and the structural effects caused by noise are described by Euler number change. The assessment on image quality is calculated quantitatively in terms of PSNR (Peak Signal to Noise Ratio). Our experiments show that the results of the algorithm are highly correlative with subjective MOS and the algorithm is more simple and computational saving than traditional objective assessment methods.
基金Project(51676211) supported by the National Natural Science Foundation of ChinaProject(2017SK2253) supported by the Key R&D Plan of Hunan Province of China+1 种基金Project(2015zzts044) supported by Fundamental Research Funds for the Central Universities,ChinaProject(201606370092) supported by the China Scholarship Council
文摘Submerged gas injection into liquid leads to complex multiphase flow, in which nozzle geometries are crucial important for the operational expenditure in terms of pressure drop. The influence of the nozzle geometry on pressure drop between nozzle inlet and outlet has been experimentally studied for different gas flow rates and bath depths. Nozzles with circular, gear-like and four-leaf cross-sectional shape have been studied. The results indicate that, besides the hydraulic diameter of the outlet, the orifice area and the perimeter of the nozzle tip also play significant roles. For the same superficial gas velocity, the average pressure drop from the four-leaf-shaped geometry is the least. The influence of bath depth was found negligible. A correlation for the modified Euler number considering the pressure drop is proposed depending on nozzle geometric parameter and on the modified Froude number with the hydraulic diameter of the nozzle do as characteristic length.
基金Project supported by the Beijing Natural Science Foundation(Grant No.1232026)the Qinxin Talents Program of BISTU(Grant No.QXTCP C201711)+2 种基金the R&D Program of Beijing Municipal Education Commission(Grant No.KM202011232017)the National Natural Science Foundation of China(Grant No.12304190)the Research fund of BISTU(Grant No.2022XJJ32).
文摘We investigate the quantum metric and topological Euler number in a cyclically modulated Su-Schrieffer-Heeger(SSH)model with long-range hopping terms.By computing the quantum geometry tensor,we derive exact expressions for the quantum metric and Berry curvature of the energy band electrons,and we obtain the phase diagram of the model marked by the first Chern number.Furthermore,we also obtain the topological Euler number of the energy band based on the Gauss-Bonnet theorem on the topological characterization of the closed Bloch states manifold in the first Brillouin zone.However,some regions where the Berry curvature is identically zero in the first Brillouin zone result in the degeneracy of the quantum metric,which leads to ill-defined non-integer topological Euler numbers.Nevertheless,the non-integer"Euler number"provides valuable insights and an upper bound for the absolute values of the Chern numbers.
文摘A fundamental problem in four dimensional differential topology is to find a surface with minimal genus which represents a given homology class. This problem was considered by many people for closed 4 manifolds. In this paper,we consider this problem for four manifold with boundary.
基金supported in part by the Grant-in-Aid for Scientific Research (C) of the Ministry of Education, Science, Sports and Culture of Japan under Grant No. 23500222
文摘Labeling connected components and holes and computing the Euler number in a binary image are necessary for image analysis, pattern recognition, and computer （robot） vision, and are usually made independently of each other in conventional methods. This paper proposes a two-scan algorithm for labeling connected components and holes simultaneously in a binary image by use of the same data structure. With our algorithm, besides labeling, we can also easily calculate the number and the area of connected components and holes, as well as the Euler number. Our method is very simple in principle, and experimental results demonstrate that our method is much more efficient than conventional methods for various kinds of images in cases where both labeling and Euler number computing are necessary.
基金supported by the Basic Science Research Program through the National Research Foundation of Korea(Nos.NRF-2016R1D1A1A09917654,NRF-2015R1C1A1A01053495)
文摘The authors compute the(rational) Betti number of real toric varieties associated to Weyl chambers of type B, and furthermore show that their integral cohomology is p-torsion free for all odd primes p.
基金NSFC(Grant Nos.12101285,12171222)Basic and Applied Basic Research Foundation and Jointof Guangdong Province,China(Grant No.2019A1515110324)+1 种基金Guangdong Basic and Applied Basic Research Foundation(Natural Science Foundation of Guangdong Province,China,Grant No.2021A1515010254)Foundation of Lingnan Normal University(Grant Nos.ZL2021017,ZL1923)。
文摘A proper vertex coloring of a graph G is acyclic if there is no bicolored cycles in G.A graph G is acyclically k-choosable if for any list assignment L={L(v):v∈V(G)}with|L(v)|≥k for each vertex v∈V(G),there exists an acyclic proper vertex coloringφof G such thatφ(v)∈L(v)for each vertex v∈V(G).In this paper,we prove that every graph G embedded on the surface with Euler characteristic numberε=-1 is acyclically 11-choosable.
文摘Euler Number is one of the most important characteristics in topology. In twodimension digital images, the Euler characteristic is locally computable. The form of Euler Number formula is different under 4-connected and 8-connected conditions. Based on the definition of the Foreground Segment and Neighbor Number, a formula of the Euler Number computing is proposed and is proved in this paper. It is a new idea to locally compute Euler Number of 2D image.
基金Supported by the National Natural Science Foundation of China(Grant No.12001288)。
文摘In this paper,we prove two supercongruences conjectured by Z.-W.Sun via the Wilf-Zeilberger method.One of them is,for any prime p>3,4F3[7/6 1/2 1/2 1/2 1/6 1 1]-1/8]-1/2≡p(-2/p)+p^(3)/4(2/p)Ep-3 (mod p^(4))where(·/p)stands for the Legendre symbol,and E_(n)is the n-th Euler number.
文摘In this paper, we show that, without involving the C~*-bundle theory, elementary differ- ential topology can do the classification of homogeneous C~*-crossed product C(X)×α Z, where X is a compact differentiable manifold of low dimension and α is a diffeomorphism of X. The motivation of this work is the Shultz’s theorem which states that a C~*-algebra can be identified with its pure state space carrying w~*-topology and certain geometric structure.
基金supported in part by US NSF-DMS 1016073,NSFC 11271350 and 91130019Special Research Funds for State Key Laboratories Y22612A33S+1 种基金China 863 project 2010AA012301 and 2012AA01A304China 973 project 2011CB309702.
文摘We consider the finite element based computation of topological quantities of implicitly represented surfaces within a diffuse interface framework.Utilizing an adaptive finite element implementation with effective gradient recovery techniques,we discuss how the Euler number can be accurately computed directly from the numerically solved phase field functions or order parameters.Numerical examples and applications to the topological analysis of point clouds are also presented.
