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
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
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
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 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
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
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
Let N be a closed, nonorientable surface, M be a simply connected 4-manifold. f: N→M is an embedding with normal bundle v<sub>f</sub>. The normal Euler class e（v<sub>f</sub>） of f is an elem...Let N be a closed, nonorientable surface, M be a simply connected 4-manifold. f: N→M is an embedding with normal bundle v<sub>f</sub>. The normal Euler class e（v<sub>f</sub>） of f is an element in H<sup>2</sup>（N,）, where is the local coefficient determined by w<sub>1</sub>（v<sub>f</sub>） =w<sub>1</sub>（N）. It is very important to determine e（v<sub>f</sub>）[N] for all embeddings. This problem is closely related to whether a two-dimensional homology class can be represented by a smooth embedded sphere. In this note, we determine all the possible normal Euler numbers of embedding real projective plane into indefinite 4-manifolds.展开更多
Let N be a closed,orientable 4-manifold satisfying H<sub>1</sub>（N,Z）=0,and M be a closed,connected,nonorientable surface embedded in N with normal bundle v.The Euler class e（v）ofv is an element of H&l...Let N be a closed,orientable 4-manifold satisfying H<sub>1</sub>（N,Z）=0,and M be a closed,connected,nonorientable surface embedded in N with normal bundle v.The Euler class e（v）ofv is an element of H<sub>2</sub>（M,（?））,where （?） denotes the twisted integer coefficients determined byw<sub>1</sub>（v）=w<sub>1</sub>（M）.We study the possible values of e（v）[M],and prove H<sub>1</sub>（N-M）=Z<sub>2</sub> or 0.Underthe condition of H<sub>1</sub>（N-M,Z）=Z<sub>2</sub>,we conclude that e（v）[M]can only take the followingvalues:2σ（N）-2（n+β<sub>2</sub>）,2σ（N）-2（n+β<sub>2</sub>-2）,2σ（N）-2（n+β<sub>2</sub>-4）,…,2σ（N）+2（n+β<sub>2</sub>）,where σ（N） is the usual index of N,n the nonorientable genus of M and β<sub>2</sub> the 2nd real Bettinumber.Finally,we show that these values can be actually attained by appropriate embeddingfor N=homological sphere.In the case of N=S<sup>4</sup>.this is just the well-known Whitney conjectureproved by W.S.Massey in 1969.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.
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.
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.
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.
文摘Let N be a closed, nonorientable surface, M be a simply connected 4-manifold. f: N→M is an embedding with normal bundle v<sub>f</sub>. The normal Euler class e（v<sub>f</sub>） of f is an element in H<sup>2</sup>（N,）, where is the local coefficient determined by w<sub>1</sub>（v<sub>f</sub>） =w<sub>1</sub>（N）. It is very important to determine e（v<sub>f</sub>）[N] for all embeddings. This problem is closely related to whether a two-dimensional homology class can be represented by a smooth embedded sphere. In this note, we determine all the possible normal Euler numbers of embedding real projective plane into indefinite 4-manifolds.
文摘Let N be a closed,orientable 4-manifold satisfying H<sub>1</sub>（N,Z）=0,and M be a closed,connected,nonorientable surface embedded in N with normal bundle v.The Euler class e（v）ofv is an element of H<sub>2</sub>（M,（?））,where （?） denotes the twisted integer coefficients determined byw<sub>1</sub>（v）=w<sub>1</sub>（M）.We study the possible values of e（v）[M],and prove H<sub>1</sub>（N-M）=Z<sub>2</sub> or 0.Underthe condition of H<sub>1</sub>（N-M,Z）=Z<sub>2</sub>,we conclude that e（v）[M]can only take the followingvalues:2σ（N）-2（n+β<sub>2</sub>）,2σ（N）-2（n+β<sub>2</sub>-2）,2σ（N）-2（n+β<sub>2</sub>-4）,…,2σ（N）+2（n+β<sub>2</sub>）,where σ（N） is the usual index of N,n the nonorientable genus of M and β<sub>2</sub> the 2nd real Bettinumber.Finally,we show that these values can be actually attained by appropriate embeddingfor N=homological sphere.In the case of N=S<sup>4</sup>.this is just the well-known Whitney conjectureproved by W.S.Massey in 1969.
