User qhliu - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T20:40:18Z http://mathoverflow.net/feeds/user/8933 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/127623/curvature-dependence-of-the-laplacian-operator-acting-on-a-n-1-dimensional-compa/127666#127666 Answer by QHLIU for Curvature dependence of the Laplacian operator acting on a n-1 dimensional compact submanifold in the n-dimensional Euclidian space QHLIU 2013-04-16T01:28:47Z 2013-04-16T01:28:47Z <p>Dear Rbega, Thank you! For a two dimensional surface, I can prove a much simpler relation by direct computations: \begin{equation*} 1/\sqrt{g}\partial _{\mu }g^{\mu \nu }\sqrt{g}\partial _ {\nu }\mathbf{R}=2 \mathbf{\mathbf{H}_{R}}\text{.} \end{equation*}</p> <p>Now, Let me calculate explicitly in general according to your formula, with use of the Einstein summation convention. Since we deal with a $n-1$ dimensional surface \begin{equation*} {\mathbf{R}} = ({ X_ {1},X_ {2},...,X_ {n} } )= X_ {j}\mathbf{i}_{Xj} \end{equation*} with $\mathbf{i}_ {X_{j}}$ denoting the unit normal along $j$-th Cartesian coordinate, we would have $\Delta _ {\mathbf{R}^{n}}$ $\mathbf{R}$ $=\partial _ {X_{i}}$ $\partial _ {X_{i}} \mathbf{R} =0,$ and \begin{equation*} \nabla _ {\mathbf{R}^{n}} \mathbf{R} \end{equation*} \begin{equation*} \equiv (\mathbf{i} _ {X_{i}} \partial _ {X_{i}}) (X_{j}\mathbf{i} _ {X_{j}}) \end{equation*} \begin{equation*} = \mathbf{i} _ {X_{i}} \delta _ {ij} \mathbf{i} _ {X_{j}}, \end{equation*} and so \begin{equation*} \mathbf{H}_{R}\mathbf{\cdot }\nabla _{\mathbf{R}^{n}}\mathbf{R=\mathbf{H} _{R}.} \end{equation*} If I\ am correct, please tell me what is $f(\mathbf{n},\mathbf{n})$ in our problem \begin{equation*} \mathbf{R}=X_{j}\mathbf{i}_{Xj}\text{,} \end{equation*} and what is $\nabla _{\mathbf{R}^{n}}^{2}$? In physics, we usually use two forms: one is the \begin{equation*} 1/\sqrt{g}\partial _{\mu }g^{\mu \nu }\sqrt{g}\partial _{\nu } \end{equation*} on the surface, and another is $\partial _ {X_{i}} \partial _ {X_{i}}$ in the $n$ dimensional Euclidean space, what is the $\nabla _{\mathbf{R} ^{n}}^{2}$? </p> http://mathoverflow.net/questions/127623/curvature-dependence-of-the-laplacian-operator-acting-on-a-n-1-dimensional-compa Curvature dependence of the Laplacian operator acting on a n-1 dimensional compact submanifold in the n-dimensional Euclidian space QHLIU 2013-04-15T14:03:36Z 2013-04-16T01:28:47Z <p>Possibly a simple question in differential geometry (maybe not accurate but understandable in mathematical terms): Given an compact surface $\mathbf {R}$ in $n$ Euclidean space parameterized by $n-1$ variables $(x_1,x_2,...,x_{n-1})$ in the following:</p> <p>$\mathbf {R}$={ $X_1,X_2,X_3,...,X_n$ }, ($X_i=X_i(x_1,x_2,...,x_{n-1}$ ) is the $i$-th Cartesian coordinate)</p> <p>Then, what is the result of Laplacian operator $∇^2=(1/(\sqrt{g})\partial_{μ}g^{μυ}\sqrt{g} \partial_{υ}$ acting on the $\mathbf {R}$ as $∇^2 \mathbf {R}$ ? I think that it should be a result that purely depends on the extrinsic curvatures, and also a geometric invariant. Please offer me the result together with a reference which is accessible to a physicist. Thanks.</p> http://mathoverflow.net/questions/91444/what-is-parameterization-of-the-trefoil-knot-surface-in-r what is parameterization of the Trefoil knot surface in R³? QHLIU 2012-03-17T05:06:18Z 2012-03-23T15:15:17Z <p>what is parameterization e.g., (x(u,v),y(u,v),z(u,v)), of the Trefoil knot surface in R³ whose cross section of the surface can be circular, or in general elliptic?</p> <p>Thanks!</p> http://mathoverflow.net/questions/37704/are-these-operators-defined-on-2d-surfaces-self-adjoint Are these operators defined on 2D surfaces self-adjoint? QHLIU 2010-09-04T09:11:54Z 2010-09-11T14:03:35Z <p>My research group finds/proposes a fundamental operator in quantum mechanics, the Cartesian momentum as I called (I think for mathematician the ref. 2007 is sufficient). However, I do not know whether it is self-adjoint or not (we are all physicists). If a mathematician can give a definite answer to it for even simple surfaces such as cylindrical and spherical, he has then a nice paper.</p> <p>The standard representation of the curved smooth surface $M$ embedded\ in $R^{3}$ is,</p> <p>$\mathbf{r}(\xi ,\zeta )\mathbf{=}\left( x(\xi ,\zeta ),y(\xi ,\zeta ),z(\xi ,\zeta )\right)$.</p> <p>The covariant derivatives of $\mathbf{r}$ are $\mathbf{r}_{\mu }=\partial \mathbf{r}/ \partial x^{\mu }$ .</p> <p>The contravariant derivatives </p> <p>$\mathbf{r}^{\mu }\equiv g^{\mu \upsilon }\mathbf{r}_{\upsilon }$ </p> <p>is the generalized inverse of the covariant ones $\mathbf{r}_{\mu }$.</p> <p>The unit normal vector at point $(\xi ,\zeta )$ is $\mathbf{n=r}^{\xi } \times \mathbf{r}^{\zeta }/ \sqrt{g}$. </p> <p>The Hermitian Cartesian momentum $\mathbf{p}$ takes a compact form,</p> <p>$\mathbf{p=}-i\hbar (\mathbf{r}^{\mu }\partial _{\mu }+H\mathbf{n),}$</p> <p>where $H$ is the mean curvature of the surface. When the motion is constraint-free or in a flat plane, i.e., when $H=0$, the constraint induced terms $H\mathbf{n}$ vanish. Then the Cartesian momentum operator reproduces its usual form as, $\mathbf{p=}-i\hbar \nabla$.</p> <p>For a particle moves on the surface of a sphere of radius $r$, $x=r\sin \theta \cos \varphi ,\text{ }y=r\sin \theta \sin \varphi ,\text{ }z=r\cos \theta$,</p> <p>the hermitian operators for Cartesian momenta $p_{i}$ are respectively, </p> <p>$p_{x} =-\frac{i\hbar }{r}(\cos \theta \cos \varphi \frac{\partial }{\partial \theta }-\frac{\sin \varphi }{\sin \theta }\frac{\partial }{\partial \varphi }-\sin \theta \cos \varphi ),$</p> <p>$p_{y} =-\frac{i\hbar }{r}(\cos \theta \sin \varphi \frac{\partial }{\partial \theta }+\frac{\cos \varphi }{\sin \theta }\frac{\partial }{\partial \varphi }-\sin \theta \sin \varphi ),$</p> <p>$p_{z} =\frac{i\hbar }{r}(\sin \theta \frac{\partial }{\partial \theta }+\cos \theta ).$</p> <p>On the spherical surface, the complete set of the spherical harmonics defines the Hilbert space.</p> <hr> <p>Refs.</p> <p>2003, Liu Q H and Liu T G, Int. Quantum Hamiltonian for the Rigid Rotator, J. Theoret. Phys. 42(2003)2877.</p> <p>2004, Liu Q H, Hou J X, Xiao Y P and Li L X, Quantum Motion on 2D Surface of Nonspherical Topology, Int. J. Theoret. Phys. 43(2004)1011.</p> <p>2005, Xiao Y P, Lai M M, Hou J X, Chen X W and Liu Q H, A Secondary Operator Ordering Problem for a Charged Rigid Planar Rotator in Uniform Magnetic Field, Comm. Theoret. Phys. 44(2005)49.</p> <p>2006a, Lai M M, Wang X, Xiao Y P and Liu Q H, Gauge Transformation and Constraint Induced Operator Ordering for Charged Rigid Planar Rotator in Uniform Magnetic Field, Comm. Theoret. Phys. 46(2006) 843. </p> <p>2006b, Wang X, Xiao Y P, Liu T G, Lai M M and Rao, Quantum Motion on 2D Surfaces of Spherical Topology, Int. J. Theoret. Phys. 45(2006)2509. </p> <p>2006c, Liu Q H, Universality of Operator Ordering in Kinetic Energy Operator for Particles Moving on two Dimensional Surfaces, Int. J. Theoret. Phys. 45(2006)2167.</p> <p>2007, Liu Q H., Tong C L., Lai M M., Constraint-induced mean curvature dependence of Cartesian momentum operators J. Phys. A 40(2007)4161.</p> <p>2010, Zhu X M, Xu M and Liu Q H, Wave packets on spherical surface viewed from expectation values of Cartesian variables, Int. J. Geom. Meth. Mod. Phys., 7(2010)411-423.</p> http://mathoverflow.net/questions/38401/an-asymptotic-expression-for-the-solution-to-the-squares-problem-suggested-by-sta An asymptotic expression for the solution to the squares problem suggested by statistical mechanics QHLIU 2010-09-11T10:52:09Z 2010-09-11T13:07:37Z <p>The $s$ squares problem is to count the number $r_s (n)$ of integer solutions $(x_1,x_2,...,x_s)$ of the Diophantine equation $x_{1}^{2}+x_{2}^{2}+...+x_{s}^{2}=n$ in which changing the sign or order of the $x_i$ ’s gives distinct solutions. If looking into the statistical mechanics for classical ideal gas in 3D, we meet with the same thing with $s=3N$, $N$ is the number of particles. But now the $3N$ squares problem is to count the number of the microstates in the so-called microscope ensemble. The following asymptotic expression of $r_{3N}(n)$ is experimentally validated, so it is physically proved: </p> <p>$r_{3N}(n)\approx \frac{{\pi}^{3N/2}}{\Gamma (3N/2)} {{n}^{3N/2-1}}$, in thermodynamic limit $n/N=const.$ and $n \to \infty$ .</p> <p>My question is: How to give an estimate of the error, and does anyone know such a formula in mathematical literature?</p> <hr> <p>Ref. </p> <p>S.C.Miline, New infinite families of exact sums of squares formulas, Jacobi elliptic functions and Ramanujan’s tau function, Proc. Natl. Acad. Sci. USA, 1996, 93:15004-15008, and references cited therein.</p> http://mathoverflow.net/questions/37360/complete-estimates-of-the-error-for-a-well-known-asymptotic-expression-of-partiti complete estimates of the error for a well-known asymptotic expression of partition p(n,m) QHLIU 2010-09-01T09:43:21Z 2010-09-02T04:14:36Z <p>Let $p(n,m)$ be the number of partitions of an integer $n$ into integers $\le m$, we have a well-known asymptotic expression: </p> <p>For a fixed $m$ and $n\to\infty$, $$p(n,m)=\frac{n^{m-1}}{m!(m-1)!} (1+O(1/n))$$</p> <p>My question is: why the error $O(1/n)$ is independent of $m$? Or how can it be extended for $m$ growing slowly with $n$? Please help me to find the answer or the references. Thanks. </p> http://mathoverflow.net/questions/37360/complete-estimates-of-the-error-for-a-well-known-asymptotic-expression-of-partiti/37364#37364 Answer by QHLIU for complete estimates of the error for a well-known asymptotic expression of partition p(n,m) QHLIU 2010-09-01T10:38:43Z 2010-09-01T10:49:42Z <p>Thank Robin Chapman very much for editing.</p> <p>There is a nice asymptotic expression for partition $q(n,M)$ that denotes the number of partitions of $n$ with $M$ parts all distinct: As $n\to\infty$,</p> <p>$$q(n,M)\approx \frac{(n-1)!}{M!(M-1)!(n-M)!}\left( 1+O\left( \frac{M^{3}}{n} \right) \right)$$</p> <p>Isn't there no similar asymptotic expression for partition $p(n,m)$?</p> http://mathoverflow.net/questions/127623/curvature-dependence-of-the-laplacian-operator-acting-on-a-n-1-dimensional-compa/127625#127625 Comment by QHLIU QHLIU 2013-04-16T03:07:50Z 2013-04-16T03:07:50Z my clarification sees in the form of answer below. http://mathoverflow.net/questions/127623/curvature-dependence-of-the-laplacian-operator-acting-on-a-n-1-dimensional-compa/127625#127625 Comment by QHLIU QHLIU 2013-04-15T15:39:07Z 2013-04-15T15:39:07Z Thank you for your answer. But in my question, the Laplacian(-Beltrami) operator takes a definite form, corresponding to your $\Delta$. Then what does the difference between $\Delta$ and $\Delta _ {\mathbb {R}^n}$? It appears a compact form of the result for the definite $\mathbf{R}=\{X_{1},X_{2},...,X_{n}\}$. http://mathoverflow.net/questions/91444/what-is-parameterization-of-the-trefoil-knot-surface-in-r/91459#91459 Comment by QHLIU QHLIU 2012-03-18T00:26:23Z 2012-03-18T00:26:23Z @Jyrki Lahtonen Trefoil knot surface in R&#179; seems quite inevitable in physics world, e.g. in the string theory, electromagnetism, fluid membrane, condensed matter physics, nanostructure, ..., cf. Faddeev model (<a href="http://hopfion.com/faddeev.html" rel="nofollow">hopfion.com/faddeev.html</a>), Linked and knotted beams of light (<a href="http://www.nature.com/nphys/journal/v4/n9/abs/nphys1056.html" rel="nofollow">nature.com/nphys/journal/v4/n9/abs/nphys1056.html</a>), etc. What I am interested if quantum constrained motion on the surface possibly results in some novel consequence, cf. Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere (<a href="http://pra.aps.org/abstract/PRA/v84/i4/e042101" rel="nofollow">pra.aps.org/abstract/PRA/v84/i4/e042101</a>). http://mathoverflow.net/questions/91444/what-is-parameterization-of-the-trefoil-knot-surface-in-r/91459#91459 Comment by QHLIU QHLIU 2012-03-17T13:52:52Z 2012-03-17T13:52:52Z Thank you very much! http://mathoverflow.net/questions/37704/are-these-operators-defined-on-2d-surfaces-self-adjoint/38408#38408 Comment by QHLIU QHLIU 2010-09-12T03:22:33Z 2010-09-12T03:22:33Z @ Martin Gisser: <a href="http://iopscience.iop.org/1751-8121/40/15/007" rel="nofollow">iopscience.iop.org/1751-8121/40/15/007</a> or <a href="http://arxiv.org/abs/0910.0769" rel="nofollow">arxiv.org/abs/0910.0769</a> http://mathoverflow.net/questions/38401/an-asymptotic-expression-for-the-solution-to-the-squares-problem-suggested-by-sta/38406#38406 Comment by QHLIU QHLIU 2010-09-11T13:55:58Z 2010-09-11T13:55:58Z fedja: I admit that physical way of reasoning usually loses the mathematical rigor, but it is supported by the experiments. Besides, physical reasoning enriches the mathematical studies, such as the Fourier series, distribution theory, etc. All primitive physical results are then mathematically proved correct; no exception is so far identified. Why isn’t it wrong? The approximate solution to the squares problem is easily attainable by treating the lattice (hyper-)spherical surface as a smooth one, and the number of the solutions is then nothing but the area of the surface area. http://mathoverflow.net/questions/38401/an-asymptotic-expression-for-the-solution-to-the-squares-problem-suggested-by-sta Comment by QHLIU QHLIU 2010-09-11T12:20:11Z 2010-09-11T12:20:11Z Robin Chapman: Thanks a lot. Some English and presentation problems are fixed. http://mathoverflow.net/questions/38401/an-asymptotic-expression-for-the-solution-to-the-squares-problem-suggested-by-sta Comment by QHLIU QHLIU 2010-09-11T11:56:18Z 2010-09-11T11:56:18Z Robin Chapman: yes, it is a mathematical problem that has not been proved yet. In physics, such a formula is experimentally verifed so physically correct. A ref: P. K. Pathria, Statistical Mechanics, 2nd ed. (1997) p.17 http://mathoverflow.net/questions/38401/an-asymptotic-expression-for-the-solution-to-the-squares-problem-suggested-by-sta Comment by QHLIU QHLIU 2010-09-11T11:42:00Z 2010-09-11T11:42:00Z Robin Chapman: &quot;Experimentally proved&quot; amounts to &quot;mathematically proved&quot;? No. In mathematics, we have to know estimates of the error, and it is a mathematical problem. http://mathoverflow.net/questions/38401/an-asymptotic-expression-for-the-solution-to-the-squares-problem-suggested-by-sta Comment by QHLIU QHLIU 2010-09-11T11:37:23Z 2010-09-11T11:37:23Z In statistical mechanics for classical ideal gas, the so-called thermodynamic limit must be imposed. So, $n$ and $N$ must be linearly dependent and $n to \infty$. However, I do not know if the asymptotic expression holds without the thermodynamic limit, nor the error of the expression http://mathoverflow.net/questions/38401/an-asymptotic-expression-for-the-solution-to-the-squares-problem-suggested-by-sta Comment by QHLIU QHLIU 2010-09-11T11:23:59Z 2010-09-11T11:23:59Z To Robin Chapman: true, it is an asymptotic expression in limit $N\to\infty$. In statistical mechanics, it is also a famous problem. If there is such a mathematical literature, I am sure no physicist has been aware of its existence. It's hard to accept a fact that physicists in this circle have all overlooked such a fundamental result. http://mathoverflow.net/questions/38401/an-asymptotic-expression-for-the-solution-to-the-squares-problem-suggested-by-sta Comment by QHLIU QHLIU 2010-09-11T11:15:06Z 2010-09-11T11:15:06Z To Gjergji Zaimi: not inside a sphere, but on the surface of it. http://mathoverflow.net/questions/38389/do-you-agree-with-c-n-yangs-comment-on-modern-math-books Comment by QHLIU QHLIU 2010-09-11T05:18:34Z 2010-09-11T05:18:34Z Yang's comment is absolutely true! http://mathoverflow.net/questions/37360/complete-estimates-of-the-error-for-a-well-known-asymptotic-expression-of-partiti/37453#37453 Comment by QHLIU QHLIU 2010-09-04T08:20:54Z 2010-09-04T08:20:54Z Please be noted that in G. Szekeres' two papers in 1951 and 1953, the asymptotic formulae for $p(n,m)$ valid only under strong condition that $m$ is related to $n^2$. In physics, no asymptotic formula not useful unless $m \alpha n$ ($\alpha &gt;0$ and as $m$ is large).