User michael renardy - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T11:49:22Z http://mathoverflow.net/feeds/user/12120 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/130080/functional-equations/130084#130084 Answer by Michael Renardy for Functional equations Michael Renardy 2013-05-08T14:48:22Z 2013-05-08T14:48:22Z <p>If you assume smoothness of f, both identities lead to f being constant. For the first one, take the derivative with respect to x and y. You find $$f_{xy}=0.$$ Now take the derivative with respect to x and z. This yields zero on the left, and on the right, since you already know $$f_{xz}=0,$$ you find $$f_z=0.$$ So we have $$f(x,y)=a(x)+b(y)$$ with b'=0, and f is constant.</p> <p>For the second problem, let $$g=\ln f,$$ and you find $$g(x,y)+g(y,z)+g(x,z)=0.$$ Again you find $$g_{xy}=0,$$ so $$g(x,y)=a(x)+b(y),$$ and $$2a(x)+a(y)+b(y)+2b(z)=0.$$ Taking derivatives with respect to x and z, we find that a and b must be constant.</p> http://mathoverflow.net/questions/128640/volume-of-a-convex-set/128643#128643 Answer by Michael Renardy for Volume of a convex set Michael Renardy 2013-04-24T18:12:40Z 2013-04-24T18:12:40Z <p>If I understand the question correctly, no. In particular, for a curve of constant width, the width does not determine the area.</p> http://mathoverflow.net/questions/128520/double-integral-of-plane-wave-squared-over-a-circular-domain/128530#128530 Answer by Michael Renardy for Double Integral of plane wave squared over a circular domain Michael Renardy 2013-04-23T19:46:38Z 2013-04-23T19:46:38Z <p>I used Mathematica, doing first the theta integration. The hypergeometric function that arises is actually a Bessel function, but Mathematica does not find that on its own. The final result I get is $${\pi a^2\over 2}(1+{J_1(2a\sqrt{v_x^2+v_y^2})\over a\sqrt{v_x^2+v_y^2}}).$$</p> http://mathoverflow.net/questions/125598/riemann-lebesgue-lemma-for-measures/125605#125605 Answer by Michael Renardy for Riemann-Lebesgue lemma for measures Michael Renardy 2013-03-26T08:29:40Z 2013-03-26T08:29:40Z <p>The following web site has a review article on work related to this question: <a href="http://mypage.iu.edu/~rdlyons/pdf/seventy.pdf" rel="nofollow">http://mypage.iu.edu/~rdlyons/pdf/seventy.pdf</a></p> http://mathoverflow.net/questions/124942/finding-an-optimal-p-such-that-u-in-lp/124989#124989 Answer by Michael Renardy for Finding an optimal $p$ such that $u \in L^p$ Michael Renardy 2013-03-19T17:24:36Z 2013-03-19T17:24:36Z <p>This is not a full answer, but it shows you can do better than p=6. In the following, subscripts x and y refer to the x and y dependence. You have $$u \in H^{2/3}_x ( L^2_y )\cap L^2_x( H^1_y ).$$ By interpolation, you find $$u\in H^{2\alpha/3}_x(H^{1-\alpha}_y).$$ For $\alpha=3/5$, we find $$u\in H^{2/5}_x(H^{2/5}_y).$$ In one dimension $H^{2/5}$ embeds into $L^{10}$, so you have at least $p=10$. Since this does not use your last condition, it is probably not optimal.</p> http://mathoverflow.net/questions/121573/spherical-bessel-functions/124340#124340 Answer by Michael Renardy for Spherical Bessel functions Michael Renardy 2013-03-12T18:42:26Z 2013-03-14T01:16:23Z <p>In Abramowitz and Stegun, Handbook of Mathematical Functions, you will find the formula 10.1.52 $$\sum_0^\infty j_n^2(x)={Si(2x)\over 2x}.$$ Consequently $$|j_n(x)|\le {1\over\sqrt{x}}\sqrt{{\max_{[0,\infty)}|Si(x)|\over 2}}\sim {0.962\over\sqrt{x}}.$$</p> http://mathoverflow.net/questions/122833/weak-compactness-of-unit-ball-in-equivalent-norm/122854#122854 Answer by Michael Renardy for weak*-compactness of unit ball in equivalent norm Michael Renardy 2013-02-25T02:59:48Z 2013-02-25T02:59:48Z <p>On the sequence space $l^1$, define the equivalent norm $$\Vert x \Vert =\sum |x_i|+2|\sum x_i|.$$ Let $e^n$ be the nth unit vector, and define $x^n=e^1-e^n$. Then $\Vert x^n\Vert=2$. But the weak-* limit of $x^n$ is $e^1$, and $\Vert e^1\Vert=3$.</p> http://mathoverflow.net/questions/122202/nonnegative-fourier-transform/122252#122252 Answer by Michael Renardy for nonnegative Fourier Transform Michael Renardy 2013-02-19T02:20:53Z 2013-02-19T02:26:12Z <p>If $\hat f$ is nonnegative, then (up to a factor), $$f(0)=\int \hat f=\Vert \hat f \Vert_1 = \Vert f \Vert_\infty.$$</p> http://mathoverflow.net/questions/120848/control-of-the-c1-norm-of-a-diffeomorphism/120867#120867 Answer by Michael Renardy for Control of the $C^1$ norm of a diffeomorphism Michael Renardy 2013-02-05T15:37:30Z 2013-02-05T15:37:30Z <p>Basically, you are asking if $L^\infty$ bounds for first derivatives can be controlled by $L^2$ bounds for second derivatives. This works in one dimension, but not two (Sobolev imbedding theorem).</p> http://mathoverflow.net/questions/116691/weak-derivative-and-continuous-function/116816#116816 Answer by Michael Renardy for weak derivative and continuous function Michael Renardy 2012-12-19T20:27:41Z 2012-12-19T20:27:41Z <p>Multiplication of a distribution by a smooth function is defined in the way you indicate. So there is nothing to prove.</p> http://mathoverflow.net/questions/116265/mollification-with-prescribed-boundary-values/116270#116270 Answer by Michael Renardy for Mollification with prescribed boundary values Michael Renardy 2012-12-13T12:02:33Z 2012-12-13T12:02:33Z <p>Let $p_n$ be any sequence of $C^\infty$ functions converging to $f$. Let $q_n$ be any sequence of $C^\infty$ functions such that $q_n=f$ and $\partial q_n/\partial n$ converges to $\partial f/\partial n$ on $\partial B$. Let $\phi_n$ be a $C^\infty$ cutoff function with support contracting towards $\partial B$. Finally, let $g_n=\phi_nq_n+(1-\phi_n)p_n$.</p> http://mathoverflow.net/questions/116112/continuity-of-critical-points-with-respect-to-a-parameterisation/116113#116113 Answer by Michael Renardy for Continuity of critical points with respect to a parameterisation. Michael Renardy 2012-12-11T19:56:21Z 2012-12-11T19:56:21Z <p>If you know that P''>0, then the implicit function theorem should be applicable to give you continuity.</p> http://mathoverflow.net/questions/116107/continuity-of-an-extension-map/116111#116111 Answer by Michael Renardy for Continuity of an extension map Michael Renardy 2012-12-11T19:39:26Z 2012-12-11T19:39:26Z <p>Yes, this extension works. Your extended function is clearly in $H^1$ on the annulus as well as in $R^n\backslash B_2$. Since the traces on $\partial B_2$ agree, it is then in $H^1$ on $R^n\backslash B_r$.</p> http://mathoverflow.net/questions/115526/odes-without-a-lipschitz-condition/115529#115529 Answer by Michael Renardy for ODE's without a Lipschitz condition Michael Renardy 2012-12-05T18:54:26Z 2012-12-05T18:54:26Z <p>It is well known that existence holds if f is continuous (Peano's existence theorem). This is documented in lots of textbooks.</p> http://mathoverflow.net/questions/115160/fourier-transform-for-entire-function/115173#115173 Answer by Michael Renardy for Fourier Transform, for entire function Michael Renardy 2012-12-02T11:24:56Z 2012-12-02T11:24:56Z <p>There is a definition of Fourier transforms for distributions, not just tempered distributions. The Fourier transform of a test function is an entire function of exponential growth, and the Fourier transforms of distributions are defined by duality. The Fourier transforms of distributions are known as analytic functionals. You may find an exposition of this topic in the monograph by Gelfand and Shilov.</p> http://mathoverflow.net/questions/114814/distributional-limits-concerning-the-regularity-of-maxwells-equations/114831#114831 Answer by Michael Renardy for Distributional limits concerning the regularity of Maxwells equations Michael Renardy 2012-11-29T01:13:16Z 2012-11-29T01:13:16Z <p>Yes, there is a way of interpreting this integral. Decompose E into a component $E_t$ tangential to level sets of $\epsilon$ and a component $E_n$ normal to them. Moreover, let us set $D=\epsilon E$, and decompose it in the same way. You can write your integrand as $E_t^2\nabla \epsilon-D_n^2\nabla(1/\epsilon)$. Now if you take the limit of a discontinuous material boundary, then both $E_t$ and $D_n$ are continuous, so there is no problem definining the integral in this form.</p> http://mathoverflow.net/questions/111655/invariant-set-of-lotka-volterra-equation/111664#111664 Answer by Michael Renardy for Invariant set of Lotka-Volterra equation Michael Renardy 2012-11-06T16:39:11Z 2012-11-06T16:39:11Z <p>If you substitute $x=e^u$, $y=e^v$, then your system becomes Hamiltonian.</p> http://mathoverflow.net/questions/111456/well-posedness-of-euler-poisson-system-for-semiconductors/111484#111484 Answer by Michael Renardy for Well-posedness of Euler-Poisson system for semiconductors Michael Renardy 2012-11-04T20:32:35Z 2012-11-04T20:32:35Z <p>Peter Markowich has worked extensively on this type of problem. Just look up his publications on Math Reviews.</p> http://mathoverflow.net/questions/111476/a-gronwall-type-inequality/111478#111478 Answer by Michael Renardy for A Gronwall-type inequality. Michael Renardy 2012-11-04T16:59:28Z 2012-11-04T16:59:28Z <p>Consider g=0, c=2, f(t)=t.</p> <p>By the way, what did you mean by "if you need some assumptions?" Are there folks out there who prove things without them?</p> http://mathoverflow.net/questions/109501/a-linear-equation-related-to-camassa-holm-equation/109523#109523 Answer by Michael Renardy for A linear equation related to Camassa-Holm equation Michael Renardy 2012-10-13T10:11:08Z 2012-10-13T10:11:08Z <p>Set $v=u-u_{xx}$. Then your equation is of the form $v_t=-\gamma v_x+Av+f$, where $A$ is a bounded operator. The rest is routine.</p> http://mathoverflow.net/questions/28983/characterizing-convex-polynomials/107346#107346 Answer by Michael Renardy for Characterizing convex polynomials Michael Renardy 2012-09-16T20:45:38Z 2012-09-16T20:45:38Z <p>p must be of even order, and its leading coefficient must be positive. Moreover, p is strictly convex (convex) if p'' has no real zeros (no real zeros of odd multiplicity). The number of real zeros of a polynomial is characterized by Sturm's theorem (http://en.wikipedia.org/wiki/Sturm's_theorem#Number_of_real_roots).</p> http://mathoverflow.net/questions/106042/question-about-the-closed-form-of-a-function/106058#106058 Answer by Michael Renardy for question about the closed form of a function Michael Renardy 2012-08-31T17:54:31Z 2012-08-31T17:54:31Z <p>If $\theta>2$ then $\phi(\theta)=16$. For $\theta&lt;2$, let $$t(x)=\frac{4(x^3-4x^2-6x-4)}{x^2(1+2x)},$$ $$g(x)=f(x,t(x)).$$ The function t is invertible from $[x_m,\infty)$ to $[0,2)$, where $x_m$ is the root of t(x)=0, approximately 5.28. We have $\phi(\theta)=g(t^{-1}(\theta))$.</p> http://mathoverflow.net/questions/105388/higher-order-divided-differences-and-derivatives/105407#105407 Answer by Michael Renardy for Higher order divided differences and derivatives Michael Renardy 2012-08-24T18:15:32Z 2012-08-24T18:15:32Z <p>In general the answer is no. For instance, if f is any odd function, then $$\lim_{h\to 0}\frac{f(h)+f(-h)-2f(0)}{h^2}=0,$$ without any assumptions on differentiability of f. So it certainly does not follow that f''(0) exists.</p> http://mathoverflow.net/questions/105193/implications-of-a-hypothetical-blow-up-of-navier-stokes-for-the-mathematical-mode/105242#105242 Answer by Michael Renardy for Implications of a hypothetical blow-up of Navier-Stokes for the mathematical model Michael Renardy 2012-08-22T14:43:54Z 2012-08-22T14:43:54Z <p>If the velocity becomes infinite, it exceeds the speed of sound, so incompressibility is no longer a valid assumption. On the other hand, we should consider the fact that the pressure also reaches minus infinity in a hypothetical blow-up solution. The practical consequence of this is cavitation.</p> http://mathoverflow.net/questions/102663/inequality-of-a-function/102669#102669 Answer by Michael Renardy for inequality of a function Michael Renardy 2012-07-19T14:13:51Z 2012-07-19T14:13:51Z <p>It is true for any positive $\lambda$. If $x\le\alpha$, the right hand side is greater or equal to $\lambda^2x^2$. Hence it suffices to show that $u-1+\exp(-u)-u^2\le 0$ for positive $u$. It is easy to show that the function $f(u)=u-1+\exp(-u)-u^2$ is monotone decreasing and concave for $u\ge 0$.</p> http://mathoverflow.net/questions/45185/pseudonyms-of-famous-mathematicians/99949#99949 Answer by Michael Renardy for Pseudonyms of famous mathematicians Michael Renardy 2012-06-18T22:07:10Z 2012-06-18T22:07:10Z <p>D'Alembert's name was in a sense a "pseudonym." D'Alembert was abandoned as an infant. However, d'Alembert was neither the name of his birth parents nor his adoptive parents. He made it up when he was a student.</p> http://mathoverflow.net/questions/80024/boundary-regularity-for-the-dirichlet-problem/98306#98306 Answer by Michael Renardy for Boundary regularity for the Dirichlet problem Michael Renardy 2012-05-29T20:39:37Z 2012-05-29T20:39:37Z <p>Grisvard's book has an extensive discussion of 2d elliptic problems with corners. Your problem is singular at the rim, and the singularity there is essentially the same one as a 2d problem with a 360 degree corner.</p> http://mathoverflow.net/questions/98260/a-question-about-matrices-with-more-details/98304#98304 Answer by Michael Renardy for A question about matrices with more details Michael Renardy 2012-05-29T20:30:28Z 2012-05-29T20:30:28Z <p>Let $$A=\pmatrix{1&amp;0\cr 0&amp;0},\quad B=\pmatrix{0&amp;1\cr 0&amp;0}.$$ Then $A^2=A$, $AB=B$, $BA=0$, $B^2=0$. It follows that $$A\prod (A+t_iB)B=B,$$ $$B\prod (A+t_iB)A=0.$$</p> http://mathoverflow.net/questions/12828/inverse-gamma-function/98301#98301 Answer by Michael Renardy for Inverse gamma function? Michael Renardy 2012-05-29T19:43:07Z 2012-05-29T19:43:07Z <p>This article addresses the question: <a href="http://www.ams.org/journals/proc/2012-140-04/S0002-9939-2011-11023-2/S0002-9939-2011-11023-2.pdf" rel="nofollow">http://www.ams.org/journals/proc/2012-140-04/S0002-9939-2011-11023-2/S0002-9939-2011-11023-2.pdf</a></p> http://mathoverflow.net/questions/97740/inverse-closed-matrix-spaces/97786#97786 Answer by Michael Renardy for inverse-closed matrix spaces Michael Renardy 2012-05-23T19:35:16Z 2012-05-23T19:35:16Z <p>Matrices for which a given vector is an eigenvector are inverse closed. This subspace has dimension $n^2-n+1$.</p> http://mathoverflow.net/questions/130071/asymptotics-of-a-function/130086#130086 Comment by Michael Renardy Michael Renardy 2013-05-08T15:21:58Z 2013-05-08T15:21:58Z The last term is $n^{-3n}$, which is certainly not dominant! The asymptotics is probably obtainable by comparing with $$\int_0^\infty x^n n^{-4x}\,dx=\frac{n!}{(4\ln n)^n}.$$ http://mathoverflow.net/questions/128806/is-f-continuous Comment by Michael Renardy Michael Renardy 2013-04-26T12:13:51Z 2013-04-26T12:13:51Z Why was this tagged special functions? I am going to remove this tag. http://mathoverflow.net/questions/128640/volume-of-a-convex-set/128643#128643 Comment by Michael Renardy Michael Renardy 2013-04-25T12:33:07Z 2013-04-25T12:33:07Z Well, yes, but I thought you were looking for a formula given by an integral in terms of w. http://mathoverflow.net/questions/128640/volume-of-a-convex-set/128643#128643 Comment by Michael Renardy Michael Renardy 2013-04-25T10:55:27Z 2013-04-25T10:55:27Z I see a formula for the perimeter there, not for the area. http://mathoverflow.net/questions/128634/finding-center-of-union-of-circles Comment by Michael Renardy Michael Renardy 2013-04-24T17:43:43Z 2013-04-24T17:43:43Z From the figure in the link, I am guessing that you want the center of mass of the area common to all the circles. You can always decompose this area into a finite union of polygons and lens-shaped areas which are bounded by a straight line and a circular arc. For each of those parts, you can find the area and center of mass by elementary means. http://mathoverflow.net/questions/128634/finding-center-of-union-of-circles Comment by Michael Renardy Michael Renardy 2013-04-24T17:32:00Z 2013-04-24T17:32:00Z I take back my earlier comment. I think the way the problem is intended, the mass density where the circles overlap is supposed to be 1 rather than adding up the masses of the circles. With that interpretation, the problem does not seem trivial. http://mathoverflow.net/questions/128634/finding-center-of-union-of-circles Comment by Michael Renardy Michael Renardy 2013-04-24T16:53:14Z 2013-04-24T16:53:14Z For one thing, you need to define &quot;center&quot; for something that is not a circle. I presume you mean the center of gravity. Now consider what happens if you concentrate the mass of each circle at its midpoint ... This site is for research level questions. Voting to close. http://mathoverflow.net/questions/128368/stability-of-a-matrix Comment by Michael Renardy Michael Renardy 2013-04-22T17:11:20Z 2013-04-22T17:11:20Z @Betrand: It does not matter, does it? Either way, this is hardly a research level question. http://mathoverflow.net/questions/124942/finding-an-optimal-p-such-that-u-in-lp/124989#124989 Comment by Michael Renardy Michael Renardy 2013-03-21T15:54:37Z 2013-03-21T15:54:37Z The monograph of Lions and Magenes, for instance. http://mathoverflow.net/questions/124942/finding-an-optimal-p-such-that-u-in-lp/124989#124989 Comment by Michael Renardy Michael Renardy 2013-03-19T18:30:33Z 2013-03-19T18:30:33Z It is supposed to mean $H^{2/3}$ (as a function of x) with values in $L^2$ (as a function of y). http://mathoverflow.net/questions/120885/what-is-the-solution-of-u-tu-xx-frac1xu-x Comment by Michael Renardy Michael Renardy 2013-02-05T18:32:53Z 2013-02-05T18:32:53Z The ODE you give can be solved in terms of Bessel functions. But, perhaps more fundamentally, you did not specify boundary conditions for your problem. http://mathoverflow.net/questions/120862/boundedness-of-a-given-boundary-value-problem Comment by Michael Renardy Michael Renardy 2013-02-05T15:34:49Z 2013-02-05T15:34:49Z Homework, voting to close. http://mathoverflow.net/questions/120198/generator-of-a-generated-c-0-semigroup Comment by Michael Renardy Michael Renardy 2013-01-29T16:45:56Z 2013-01-29T16:45:56Z If U is bounded, this is rather obvious. However, if U is unbounded, there is in general an issue of domains. It is easy to come up with example where $U\rho$ is an unbounded operator, but $A(\rho)$ is bounded. Hence the generator of $P_t$ will in general be an extension of $A$. http://mathoverflow.net/questions/118317/show-series-converges-to-match-geometric-series Comment by Michael Renardy Michael Renardy 2013-01-08T00:00:53Z 2013-01-08T00:00:53Z This is not true, as can be seen by plugging in z=0. Moreover, Taylor expansion around 0 does not yield anything that looks like the geometric series. Mathematica finds $1+z+2z^2+z^3+2z^4+... http://mathoverflow.net/questions/118301/are-all-irrational-elementary-numbers-conjectured-to-be-normal Comment by Michael Renardy Michael Renardy 2013-01-07T20:29:57Z 2013-01-07T20:29:57Z Since it seems to unknown even whether such popular numbers as e or$\sqrt{2}\$ are normal, what would be the point of formulating such a conjecture?