This paper deals with the radial symmetry of positive solutions to the nonlocal problem(-Δ)_(γ)~su=b(x)f(u)in B_(1){0},u=h in R~N B_(1),where b:B_1→R is locally Holder continuous,radially symmetric and decreasing i...This paper deals with the radial symmetry of positive solutions to the nonlocal problem(-Δ)_(γ)~su=b(x)f(u)in B_(1){0},u=h in R~N B_(1),where b:B_1→R is locally Holder continuous,radially symmetric and decreasing in the|x|direction,F:R→R is a Lipschitz function,h:B_1→R is radially symmetric,decreasing with respect to|x|in R^(N)/B_(1),B_(1) is the unit ball centered at the origin,and(-Δ)_γ~s is the weighted fractional Laplacian with s∈(0,1),γ∈[0,2s)defined by(-△)^(s)_(γ)u(x)=CN,slimδ→0+∫R^(N)/B_(δ)(x)u(x)-u(y)/|x-y|N+2s|y|^(r)dy.We consider the radial symmetry of isolated singular positive solutions to the nonlocal problem in whole space(-Δ)_(γ)^(s)u(x)=b(x)f(u)in R^(N)\{0},under suitable additional assumptions on b and f.Our symmetry results are derived by the method of moving planes,where the main difficulty comes from the weighted fractional Laplacian.Our results could be applied to get a sharp asymptotic for semilinear problems with the fractional Hardy operators(-Δ)^(s)u+μ/(|x|^(2s))u=b(x)f(u)in B_(1)\{0},u=h in R^(N)\B_(1),under suitable additional assumptions on b,f and h.展开更多
Recognizing various traffic signs,especially the popular circular traffic signs,is an essential task for implementing advanced driver assistance system.To recognize circular traffic signs with high accuracy and robust...Recognizing various traffic signs,especially the popular circular traffic signs,is an essential task for implementing advanced driver assistance system.To recognize circular traffic signs with high accuracy and robustness,a novel approach which uses the so-called improved constrained binary fast radial symmetry(ICBFRS) detector and pseudo-zernike moments based support vector machine(PZM-SVM) classifier is proposed.In the detection stage,the scene image containing the traffic signs will be converted into Lab color space for color segmentation.Then the ICBFRS detector can efficiently capture the position and scale of sign candidates within the scene by detecting the centers of circles.In the classification stage,once the candidates are cropped out of the image,pseudo-zernike moments are adopted to represent the features of extracted pictogram,which are then fed into a support vector machine to classify different traffic signs.Experimental results under different lighting conditions indicate that the proposed method has robust detection effect and high classification accuracy.展开更多
This paper presents an eye and iris detection algorithm for human facial images. The authors combine three features of the eye to develop the algorithm:1) the pixels surrounding the eyes are more variable than other...This paper presents an eye and iris detection algorithm for human facial images. The authors combine three features of the eye to develop the algorithm:1) the pixels surrounding the eyes are more variable than other parts of the face; 2) eye pixels are darker than their neighbors; 3) eyes often exhibit radial symmetric properties. Through the first feature,two rough regions of both eyes are detected on the face. Eye masks are then formed based on the second feature,and a fast radial symmetry transform is applied to the two rough regions of both eyes. Finally,accurate iris centers are located by searching the maximum value of the radial symmetry transform results. Using 450 human facial images from the Caltech face database,experiments show that the success rate of the proposed method is 91.7%. The effectiveness of the method was also verified through detection of video frames.展开更多
The paper generalizes the direct method of moving planes to the Logarithmic Laplacian system.Firstly,some key ingredients of the method are discussed,for example,Narrow region principle and Decay at infinity.Then,the ...The paper generalizes the direct method of moving planes to the Logarithmic Laplacian system.Firstly,some key ingredients of the method are discussed,for example,Narrow region principle and Decay at infinity.Then,the radial symmetry of the solution of the Logarithmic Laplacian system is obtained.展开更多
Consider the following Neumann problem d△u- u + k(x)u^p = 0 and u 〉 0 in B1, δu/δv =0 on OB1,where d 〉 0, B1 is the unit ball in R^N, k(x) = k(|x|) ≠ 0 is nonnegative and in C(-↑B1), 1 〈 p 〈 N+2/N...Consider the following Neumann problem d△u- u + k(x)u^p = 0 and u 〉 0 in B1, δu/δv =0 on OB1,where d 〉 0, B1 is the unit ball in R^N, k(x) = k(|x|) ≠ 0 is nonnegative and in C(-↑B1), 1 〈 p 〈 N+2/N-2 with N≥ 3. It was shown in [2] that, for any d 〉 0, problem (*) has no nonconstant radially symmetric least energy solution if k(x) ≡ 1. By an implicit function theorem we prove that there is d0 〉 0 such that (*) has a unique radially symmetric least energy solution if d 〉 d0, this solution is constant if k(x) ≡ 1 and nonconstant if k(x) ≠ 1. In particular, for k(x) ≡ 1, do can be expressed explicitly.展开更多
We discuss the properties of solutions for the following elliptic partial differential equations system in Rn,where 0 〈α〈 n, pi and qi (i = 1, 2) satisfy some suitable assumptions. Due to equivalence between the ...We discuss the properties of solutions for the following elliptic partial differential equations system in Rn,where 0 〈α〈 n, pi and qi (i = 1, 2) satisfy some suitable assumptions. Due to equivalence between the PDEs system and a given integral system, we prove the radial symmetry and regularity of positive solutions to the PDEs system via the method of moving plane in integral forms and Regularity Lifting Lemma. For the special case, when p1 + p2= q1 + q2 = n+α/n-α, we classify the solutions of the PDEs system.展开更多
We consider the blow-up solutions to the following coupled nonlinear Schr¨odinger equations{iu_(t)+Δu+(|u|^(2p)+|u|^(p−1)|v|^(p+1))u=0,iv_(t)+Δv+(|v|^(2p)+|v|^(p−1)|u|^(p+1))v=0,u(0,x)=u0(x),v(0,x)=v0(x),x 2 R ...We consider the blow-up solutions to the following coupled nonlinear Schr¨odinger equations{iu_(t)+Δu+(|u|^(2p)+|u|^(p−1)|v|^(p+1))u=0,iv_(t)+Δv+(|v|^(2p)+|v|^(p−1)|u|^(p+1))v=0,u(0,x)=u0(x),v(0,x)=v0(x),x 2 R N,t0.On the basis of the conservation of mass and energy,we establish two sufficient conditions to obtain the existence of a blow-up for radially symmetric solutions.These results improve the blow-up result of Li and Wu[10]by dropping the hypothesis of finite variance((|x|u_(0),|x|v_(0))∈ L^(2)(R^(N))×L^(2)(R^(N))).展开更多
In this paper,we first establish narrow region principle and decay at infinity theorems to extend the direct method of moving planes for general fractional p-Laplacian systems.By virtue of this method,we investigate t...In this paper,we first establish narrow region principle and decay at infinity theorems to extend the direct method of moving planes for general fractional p-Laplacian systems.By virtue of this method,we investigate the qualitative properties of positive solutions for the following Schrodinger system with fractional p-Laplacian{(-△)^(s)_(p)u+au^(p-1)=f(u,v),(-△)^(t)_(p)v+bv(p-1)=g(u,v),where 0<s,t<1 and 2<p<∞.We obtain the radial symmetry in the unit ball or the whole space R^(N)(N≥2),the monotonicity in the parabolic domain and the nonexistence on the half space for positive solutions to the above system under some suitable conditions on f and g,respectively.展开更多
In this article, we consider the fractional Laplacian equation {(-△)α/2u=k(x)f(u),x∈Rn+, u=0, x Rn+, where 0 〈α 〈 2,En+:= {x = (x1,x2,… ,xn)|xn〉 0}. When K is strictly decreasing with respect to ...In this article, we consider the fractional Laplacian equation {(-△)α/2u=k(x)f(u),x∈Rn+, u=0, x Rn+, where 0 〈α 〈 2,En+:= {x = (x1,x2,… ,xn)|xn〉 0}. When K is strictly decreasing with respect to |x'|, the symmetry of positive solutions is proved, where x' = (x1, x2,…, xn-1) ∈Rn- 1. When K is strictly increasing with respect to xn or only depend on xn, the nonexistence of positive solutions is obtained.展开更多
In this paper,we are concerned with the regularity and symmetry of positive solutions of the following nonlinear integral system u(x) = ∫R n G α(x-y)v(y) q/|y|β dy,v(x) = ∫R n G α(x-y)u(y) p/|y|β...In this paper,we are concerned with the regularity and symmetry of positive solutions of the following nonlinear integral system u(x) = ∫R n G α(x-y)v(y) q/|y|β dy,v(x) = ∫R n G α(x-y)u(y) p/|y|β dy for x ∈ R n,where G α(x) is the kernel of Bessel potential of order α,0 ≤β 〈 α 〈 n,1 〈 p,q 〈 n-β/β and 1/p + 1 + 1/q + 1 〉 n-α + β/n.We show that positive solution pairs(u,v) ∈ L p +1(R n) × L q +1(R n) are Ho¨lder continuous,radially symmetric and strictly decreasing about the origin.展开更多
We classify all positive solutions for the following integral system:{ui(x)=∫Rn1/│x-y│^n-α fi(u(y))dy,x∈R^n,i=1,…,m,0〈α〈n,and u(x)=(u1(x),u2(x)…,um(x)).Here fi(u), 1 ≤ i ≤m, monotone non...We classify all positive solutions for the following integral system:{ui(x)=∫Rn1/│x-y│^n-α fi(u(y))dy,x∈R^n,i=1,…,m,0〈α〈n,and u(x)=(u1(x),u2(x)…,um(x)).Here fi(u), 1 ≤ i ≤m, monotone nondecreasing are real-valued functions of homogeneous degree n+α/n-α and are monotone nondecreasing with respect to all the independent variables U1, u2, ..., urn.In the special case n ≥ 3 and α = 2. we show that the above system is equivalent to thefollowing elliptic PDE system:This system is closely related to the stationary SchrSdinger system with critical exponents for Bose-Einstein condensate展开更多
In this paper, we consider the following integral system: u(x) = R n v q (y) | x y | nα dy, v(x) = R n u p (y) | x y | nμ dy, (0.1) where 0 〈 α, μ 〈 n; p, q ≥ 1. Using the method of moving planes...In this paper, we consider the following integral system: u(x) = R n v q (y) | x y | nα dy, v(x) = R n u p (y) | x y | nμ dy, (0.1) where 0 〈 α, μ 〈 n; p, q ≥ 1. Using the method of moving planes in an integral form which was recently introduced by Chen, Li, and Ou in [2, 4, 8], we show that all positive solutions of (0.1) are radially symmetric and decreasing with respect to some point under some general conditions of integrability. The results essentially improve and extend previously known results [4, 8].展开更多
In this article, we study positive solutions to the system{Aαu(x) = Cn,αPV∫Rn(a1(x-y)(u(x)-u(y)))/(|x-y|n+α)dy = f(u(x), Bβv(x) = Cn,βPV ∫Rn(a2(x-y)(v(x)-v(y))/(|x-y|n+β)dy ...In this article, we study positive solutions to the system{Aαu(x) = Cn,αPV∫Rn(a1(x-y)(u(x)-u(y)))/(|x-y|n+α)dy = f(u(x), Bβv(x) = Cn,βPV ∫Rn(a2(x-y)(v(x)-v(y))/(|x-y|n+β)dy = g(u(x),v(x)).To reach our aim, by using the method of moving planes, we prove a narrow region principle and a decay at infinity by the iteration method. On the basis of these results, we conclude radial symmetry and monotonicity of positive solutions for the problems involving the weighted fractional system on an unit ball and the whole space. Furthermore, non-existence of nonnegative solutions on a half space is given.展开更多
In this paper, we are concerned with the following Hardy-Sobolev type system{(-?)^(α/2) u(x) =v^q(x)/|y|^(t_2) (-?)α/2 v(x) =u^p(x)/|y|^(t_1),x =(y, z) ∈(R ~k\{0}) × R^(n-k),(0.1)where 0 < α < n, 0 <...In this paper, we are concerned with the following Hardy-Sobolev type system{(-?)^(α/2) u(x) =v^q(x)/|y|^(t_2) (-?)α/2 v(x) =u^p(x)/|y|^(t_1),x =(y, z) ∈(R ~k\{0}) × R^(n-k),(0.1)where 0 < α < n, 0 < t_1, t_2 < min{α, k}, and 1 < p ≤ τ_1 :=(n+α-2t_1)/( n-α), 1 < q ≤ τ_2 :=(n+α-2 t_2)/( n-α).We first establish the equivalence of classical and weak solutions between PDE system(0.1)and the following integral equations(IE) system{u(x) =∫_( R^n) G_α(x, ξ)v^q(ξ)/|η|t^2 dξ v(x) =∫_(R^n) G_α(x, ξ)(u^p(ξ))/|η|^(t_1) dξ,(0.2)where Gα(x, ξ) =(c n,α)/(|x-ξ|^(n-α))is the Green's function of(-?)^(α/2) in R^n. Then, by the method of moving planes in the integral forms, in the critical case p = τ_1 and q = τ_2, we prove that each pair of nonnegative solutions(u, v) of(0.1) is radially symmetric and monotone decreasing about the origin in R^k and some point z0 in R^(n-k). In the subcritical case (n-t_1)/(p+1)+(n-t_2)/(q+1)> n-α,1 < p ≤ τ_1 and 1 < q ≤ τ_2, we derive the nonexistence of nontrivial nonnegative solutions for(0.1).展开更多
Parity–time(PT) symmetry has been demonstrated in the frame of classic optics. Its applications in laser science have resulted in unconventional control and manipulation of resonant modes. PT-symmetric periodic circu...Parity–time(PT) symmetry has been demonstrated in the frame of classic optics. Its applications in laser science have resulted in unconventional control and manipulation of resonant modes. PT-symmetric periodic circular Bragg lasers were previously proposed. Analyses with a transfer-matrix method have shown their superior properties of reduced threshold and enhanced modal discrimination between the radial modes. However, the properties of the azimuthal modes were not analyzed, which restricts further development of circular Bragg lasers. Here, we adopt the coupled-mode theory to design and analyze chirped circular Bragg lasers with radial PT symmetry. The new structures possess more versatile modal control with further enhanced modal discrimination between the azimuthal modes. We also analyze azimuthally modulated circular Bragg lasers with radial PT symmetry, which are shown to achieve even higher modal discrimination.展开更多
One of the challenges of remote sensing and computer vision lies in the three-dimensional(3-D)reconstruction of individual trees by using automated methods through very high-resolution(VHR)data sets.However,a successf...One of the challenges of remote sensing and computer vision lies in the three-dimensional(3-D)reconstruction of individual trees by using automated methods through very high-resolution(VHR)data sets.However,a successful and complete 3-D reconstruction relies on precise delineation of the trees in two dimensions.In this paper,we present an original approach to detect and delineate citrus trees using unmanned aerial vehicles based on photogrammetric digital surface models(DSMs).The symmetry of the citrus trees in a DSM is handled by an orientationbased radial symmetry transform which is computed in a unique way.Next,we propose an efficient strategy to accurately build influence regions of each tree,and then we delineate individual citrus trees through active contours by taking into account the influence region of each canopy.We also present two efficient strategies to filter out erroneously detected canopy regions without having any height thresholds.Experiments are carried out on eight test DSMs composed of different types of citrus orchards with varying densities and canopy sizes.Extensive comparisons to the state-of-the-art approaches reveal that our proposed approach provides superior detection and delineation performances through supporting a nice balance between precision and recall measures.展开更多
基金supported by the NSFC(12001252)the Jiangxi Provincial Natural Science Foundation(20232ACB211001)。
文摘This paper deals with the radial symmetry of positive solutions to the nonlocal problem(-Δ)_(γ)~su=b(x)f(u)in B_(1){0},u=h in R~N B_(1),where b:B_1→R is locally Holder continuous,radially symmetric and decreasing in the|x|direction,F:R→R is a Lipschitz function,h:B_1→R is radially symmetric,decreasing with respect to|x|in R^(N)/B_(1),B_(1) is the unit ball centered at the origin,and(-Δ)_γ~s is the weighted fractional Laplacian with s∈(0,1),γ∈[0,2s)defined by(-△)^(s)_(γ)u(x)=CN,slimδ→0+∫R^(N)/B_(δ)(x)u(x)-u(y)/|x-y|N+2s|y|^(r)dy.We consider the radial symmetry of isolated singular positive solutions to the nonlocal problem in whole space(-Δ)_(γ)^(s)u(x)=b(x)f(u)in R^(N)\{0},under suitable additional assumptions on b and f.Our symmetry results are derived by the method of moving planes,where the main difficulty comes from the weighted fractional Laplacian.Our results could be applied to get a sharp asymptotic for semilinear problems with the fractional Hardy operators(-Δ)^(s)u+μ/(|x|^(2s))u=b(x)f(u)in B_(1)\{0},u=h in R^(N)\B_(1),under suitable additional assumptions on b,f and h.
基金Supported by the Program for Changjiang Scholars and Innovative Research Team (2008)Program for New Centoury Excellent Talents in University(NCET-09-0045)+1 种基金the National Nat-ural Science Foundation of China (60773044,61004059)the Natural Science Foundation of Beijing(4101001)
文摘Recognizing various traffic signs,especially the popular circular traffic signs,is an essential task for implementing advanced driver assistance system.To recognize circular traffic signs with high accuracy and robustness,a novel approach which uses the so-called improved constrained binary fast radial symmetry(ICBFRS) detector and pseudo-zernike moments based support vector machine(PZM-SVM) classifier is proposed.In the detection stage,the scene image containing the traffic signs will be converted into Lab color space for color segmentation.Then the ICBFRS detector can efficiently capture the position and scale of sign candidates within the scene by detecting the centers of circles.In the classification stage,once the candidates are cropped out of the image,pseudo-zernike moments are adopted to represent the features of extracted pictogram,which are then fed into a support vector machine to classify different traffic signs.Experimental results under different lighting conditions indicate that the proposed method has robust detection effect and high classification accuracy.
基金Research Project of Pilot Fatigue Monitoring System Based on Computer Vision (No.MHR06Z16)
文摘This paper presents an eye and iris detection algorithm for human facial images. The authors combine three features of the eye to develop the algorithm:1) the pixels surrounding the eyes are more variable than other parts of the face; 2) eye pixels are darker than their neighbors; 3) eyes often exhibit radial symmetric properties. Through the first feature,two rough regions of both eyes are detected on the face. Eye masks are then formed based on the second feature,and a fast radial symmetry transform is applied to the two rough regions of both eyes. Finally,accurate iris centers are located by searching the maximum value of the radial symmetry transform results. Using 450 human facial images from the Caltech face database,experiments show that the success rate of the proposed method is 91.7%. The effectiveness of the method was also verified through detection of video frames.
基金Supported by the National Natural Science Foundation of China(11501342,12001344)。
文摘The paper generalizes the direct method of moving planes to the Logarithmic Laplacian system.Firstly,some key ingredients of the method are discussed,for example,Narrow region principle and Decay at infinity.Then,the radial symmetry of the solution of the Logarithmic Laplacian system is obtained.
基金the National Natural Science Foundation of China(No.10571174,10631030)Chinese Academy oF Sciences grant KJCX3-SYW-S03.
文摘Consider the following Neumann problem d△u- u + k(x)u^p = 0 and u 〉 0 in B1, δu/δv =0 on OB1,where d 〉 0, B1 is the unit ball in R^N, k(x) = k(|x|) ≠ 0 is nonnegative and in C(-↑B1), 1 〈 p 〈 N+2/N-2 with N≥ 3. It was shown in [2] that, for any d 〉 0, problem (*) has no nonconstant radially symmetric least energy solution if k(x) ≡ 1. By an implicit function theorem we prove that there is d0 〉 0 such that (*) has a unique radially symmetric least energy solution if d 〉 d0, this solution is constant if k(x) ≡ 1 and nonconstant if k(x) ≠ 1. In particular, for k(x) ≡ 1, do can be expressed explicitly.
基金Supported by National Natural Science Foundation of China(Grant No.11571268)the foundation of Xi’an University of Finance and Economics(Grant No.12XCK07)
文摘We discuss the properties of solutions for the following elliptic partial differential equations system in Rn,where 0 〈α〈 n, pi and qi (i = 1, 2) satisfy some suitable assumptions. Due to equivalence between the PDEs system and a given integral system, we prove the radial symmetry and regularity of positive solutions to the PDEs system via the method of moving plane in integral forms and Regularity Lifting Lemma. For the special case, when p1 + p2= q1 + q2 = n+α/n-α, we classify the solutions of the PDEs system.
基金the National Natural Science Foundation of China(11771314)the Sichuan Science and Technology Program(2022JDTD0019)the Guizhou Province Science and Technology Basic Project(Qian Ke He Basic[2020]1Y011)。
文摘We consider the blow-up solutions to the following coupled nonlinear Schr¨odinger equations{iu_(t)+Δu+(|u|^(2p)+|u|^(p−1)|v|^(p+1))u=0,iv_(t)+Δv+(|v|^(2p)+|v|^(p−1)|u|^(p+1))v=0,u(0,x)=u0(x),v(0,x)=v0(x),x 2 R N,t0.On the basis of the conservation of mass and energy,we establish two sufficient conditions to obtain the existence of a blow-up for radially symmetric solutions.These results improve the blow-up result of Li and Wu[10]by dropping the hypothesis of finite variance((|x|u_(0),|x|v_(0))∈ L^(2)(R^(N))×L^(2)(R^(N))).
基金Supported by the National Natural Science Foundation of China(12101452,12071229,11771218)。
文摘In this paper,we first establish narrow region principle and decay at infinity theorems to extend the direct method of moving planes for general fractional p-Laplacian systems.By virtue of this method,we investigate the qualitative properties of positive solutions for the following Schrodinger system with fractional p-Laplacian{(-△)^(s)_(p)u+au^(p-1)=f(u,v),(-△)^(t)_(p)v+bv(p-1)=g(u,v),where 0<s,t<1 and 2<p<∞.We obtain the radial symmetry in the unit ball or the whole space R^(N)(N≥2),the monotonicity in the parabolic domain and the nonexistence on the half space for positive solutions to the above system under some suitable conditions on f and g,respectively.
基金supported by the Fundamental Research Founds for the Central Universities(3102015ZY069)the Natural Science Basic Research Plan in Shaanxi Province of China(2016M1008)
文摘In this article, we consider the fractional Laplacian equation {(-△)α/2u=k(x)f(u),x∈Rn+, u=0, x Rn+, where 0 〈α 〈 2,En+:= {x = (x1,x2,… ,xn)|xn〉 0}. When K is strictly decreasing with respect to |x'|, the symmetry of positive solutions is proved, where x' = (x1, x2,…, xn-1) ∈Rn- 1. When K is strictly increasing with respect to xn or only depend on xn, the nonexistence of positive solutions is obtained.
基金Chen research is supported by NSF of China (10961015)Yang research is supported by NSF of China (10961016)the GAN PO555 Program of Jiangxi
文摘In this paper,we are concerned with the regularity and symmetry of positive solutions of the following nonlinear integral system u(x) = ∫R n G α(x-y)v(y) q/|y|β dy,v(x) = ∫R n G α(x-y)u(y) p/|y|β dy for x ∈ R n,where G α(x) is the kernel of Bessel potential of order α,0 ≤β 〈 α 〈 n,1 〈 p,q 〈 n-β/β and 1/p + 1 + 1/q + 1 〉 n-α + β/n.We show that positive solution pairs(u,v) ∈ L p +1(R n) × L q +1(R n) are Ho¨lder continuous,radially symmetric and strictly decreasing about the origin.
基金supported by NSF Grant DMS-0604638Li partially supported by NSF Grant DMS-0401174
文摘We classify all positive solutions for the following integral system:{ui(x)=∫Rn1/│x-y│^n-α fi(u(y))dy,x∈R^n,i=1,…,m,0〈α〈n,and u(x)=(u1(x),u2(x)…,um(x)).Here fi(u), 1 ≤ i ≤m, monotone nondecreasing are real-valued functions of homogeneous degree n+α/n-α and are monotone nondecreasing with respect to all the independent variables U1, u2, ..., urn.In the special case n ≥ 3 and α = 2. we show that the above system is equivalent to thefollowing elliptic PDE system:This system is closely related to the stationary SchrSdinger system with critical exponents for Bose-Einstein condensate
基金Supported by National Natural Science Foundation of China-NSAF (10976026)
文摘In this paper, we consider the following integral system: u(x) = R n v q (y) | x y | nα dy, v(x) = R n u p (y) | x y | nμ dy, (0.1) where 0 〈 α, μ 〈 n; p, q ≥ 1. Using the method of moving planes in an integral form which was recently introduced by Chen, Li, and Ou in [2, 4, 8], we show that all positive solutions of (0.1) are radially symmetric and decreasing with respect to some point under some general conditions of integrability. The results essentially improve and extend previously known results [4, 8].
基金Supported by National Natural Science Foundation of China(11771354)
文摘In this article, we study positive solutions to the system{Aαu(x) = Cn,αPV∫Rn(a1(x-y)(u(x)-u(y)))/(|x-y|n+α)dy = f(u(x), Bβv(x) = Cn,βPV ∫Rn(a2(x-y)(v(x)-v(y))/(|x-y|n+β)dy = g(u(x),v(x)).To reach our aim, by using the method of moving planes, we prove a narrow region principle and a decay at infinity by the iteration method. On the basis of these results, we conclude radial symmetry and monotonicity of positive solutions for the problems involving the weighted fractional system on an unit ball and the whole space. Furthermore, non-existence of nonnegative solutions on a half space is given.
基金supported by the NNSF of China(11371056)partly supported by the NNSF of China(11501021)+1 种基金the China Postdoctoral Science Foundation(2013M540057)partly supported by Scientific Research Fund of Jiangxi Provincial Education Department(GJJ160797)
文摘In this paper, we are concerned with the following Hardy-Sobolev type system{(-?)^(α/2) u(x) =v^q(x)/|y|^(t_2) (-?)α/2 v(x) =u^p(x)/|y|^(t_1),x =(y, z) ∈(R ~k\{0}) × R^(n-k),(0.1)where 0 < α < n, 0 < t_1, t_2 < min{α, k}, and 1 < p ≤ τ_1 :=(n+α-2t_1)/( n-α), 1 < q ≤ τ_2 :=(n+α-2 t_2)/( n-α).We first establish the equivalence of classical and weak solutions between PDE system(0.1)and the following integral equations(IE) system{u(x) =∫_( R^n) G_α(x, ξ)v^q(ξ)/|η|t^2 dξ v(x) =∫_(R^n) G_α(x, ξ)(u^p(ξ))/|η|^(t_1) dξ,(0.2)where Gα(x, ξ) =(c n,α)/(|x-ξ|^(n-α))is the Green's function of(-?)^(α/2) in R^n. Then, by the method of moving planes in the integral forms, in the critical case p = τ_1 and q = τ_2, we prove that each pair of nonnegative solutions(u, v) of(0.1) is radially symmetric and monotone decreasing about the origin in R^k and some point z0 in R^(n-k). In the subcritical case (n-t_1)/(p+1)+(n-t_2)/(q+1)> n-α,1 < p ≤ τ_1 and 1 < q ≤ τ_2, we derive the nonexistence of nontrivial nonnegative solutions for(0.1).
基金Hong Kong Research Grants Council Early Career Scheme(24208915)National Natural Science Foundation of China(NSFC)Research Grants Council of Hong Kong Joint Research Scheme(N_CUHK415/15)
文摘Parity–time(PT) symmetry has been demonstrated in the frame of classic optics. Its applications in laser science have resulted in unconventional control and manipulation of resonant modes. PT-symmetric periodic circular Bragg lasers were previously proposed. Analyses with a transfer-matrix method have shown their superior properties of reduced threshold and enhanced modal discrimination between the radial modes. However, the properties of the azimuthal modes were not analyzed, which restricts further development of circular Bragg lasers. Here, we adopt the coupled-mode theory to design and analyze chirped circular Bragg lasers with radial PT symmetry. The new structures possess more versatile modal control with further enhanced modal discrimination between the azimuthal modes. We also analyze azimuthally modulated circular Bragg lasers with radial PT symmetry, which are shown to achieve even higher modal discrimination.
基金This work was supported by the Scientific and Technical Research Council of Turkey(TUBITAK)[grant number 114Y671].
文摘One of the challenges of remote sensing and computer vision lies in the three-dimensional(3-D)reconstruction of individual trees by using automated methods through very high-resolution(VHR)data sets.However,a successful and complete 3-D reconstruction relies on precise delineation of the trees in two dimensions.In this paper,we present an original approach to detect and delineate citrus trees using unmanned aerial vehicles based on photogrammetric digital surface models(DSMs).The symmetry of the citrus trees in a DSM is handled by an orientationbased radial symmetry transform which is computed in a unique way.Next,we propose an efficient strategy to accurately build influence regions of each tree,and then we delineate individual citrus trees through active contours by taking into account the influence region of each canopy.We also present two efficient strategies to filter out erroneously detected canopy regions without having any height thresholds.Experiments are carried out on eight test DSMs composed of different types of citrus orchards with varying densities and canopy sizes.Extensive comparisons to the state-of-the-art approaches reveal that our proposed approach provides superior detection and delineation performances through supporting a nice balance between precision and recall measures.