User jjcale - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:37:40Z http://mathoverflow.net/feeds/user/17261 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/125049/open-problems-in-pdes-dynamical-systems-mathematical-physics/125101#125101 Answer by jjcale for Open problems in PDEs, dynamical systems, mathematical physics jjcale 2013-03-20T22:18:13Z 2013-03-20T22:18:13Z <p>For a list of 15 open problems in mathematical physics (in 2000) see Simon's Problems, <a href="http://mathworld.wolfram.com/SimonsProblems.html" rel="nofollow">http://mathworld.wolfram.com/SimonsProblems.html</a>.</p> <p>Some of these problems are solved, as mentioned in the link.</p> http://mathoverflow.net/questions/119686/on-an-inequality-of-brezis-lieb/119688#119688 Answer by jjcale for on an inequality of Brezis-Lieb jjcale 2013-01-23T19:46:51Z 2013-01-23T19:46:51Z <p>No, choose $\Omega={z \in \mathbb{C} : |z| \leq 1 } ,\ f(z)=Re\ z^{n}$ and let $n\rightarrow \infty$ .</p> http://mathoverflow.net/questions/114555/does-physics-need-non-analytic-smooth-functions/114600#114600 Answer by jjcale for Does Physics need non-analytic smooth functions? jjcale 2012-11-26T22:53:50Z 2012-11-26T22:53:50Z <p>In the BCS-theory of superconductivity the energy gap that seperates the ground state from the excited states is a non-analytic function of the exchange energy.</p> <p>Here one doesn't use a taylor expansion of the interacting Hamiltonian around the non-interacting Hamiltonian. Instead one uses a Bogoliubov transformation.</p> http://mathoverflow.net/questions/111558/projections-in-banach-spaces/112754#112754 Answer by jjcale for Projections in Banach spaces jjcale 2012-11-18T11:41:52Z 2012-11-18T11:41:52Z <p>Here a simple example :</p> <p>Let X be the cartesian product of $L^{\infty}$ and $L^{1}$ on the interval $[0,1]$, let $P_{t}$ the canonical projection on the subspace of functions with support $[0,t]$ and choose $Q(f_{1},f_{2}) = (0,f_{1})$ .</p> http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/95895#95895 Answer by jjcale for Examples of theorems with proofs that have dramatically improved over time jjcale 2012-05-03T16:40:53Z 2012-05-03T16:40:53Z <p>Example of a bounded linear operator on a Banach space without non-trivial closed invariant subspace.</p> <p>The first example was given bei Enfo in 1975. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987 (see <a href="http://en.wikipedia.org/wiki/Per_Enflo" rel="nofollow">http://en.wikipedia.org/wiki/Per_Enflo</a>). Simpler examples were constucted for example by Beauzamy and Charles Read.</p> http://mathoverflow.net/questions/3764/does-there-exist-a-continuous-function-of-compact-support-with-fourier-transform/95678#95678 Answer by jjcale for Does there exist a continuous function of compact support with Fourier transform outside L^1? jjcale 2012-05-01T16:54:14Z 2012-05-01T16:54:14Z <p>Choose $f(x)=\frac{(\frac{1}{2} -x)}{log(x)}$ for $0 &lt; x \leq \frac{1}{2}$ and $0$ otherwise.</p> <p>Then $Im \int_{0}^{\infty} dk \ e^{-\varepsilon k} \int_{-\infty}^{\infty} dx \ f(x) e^{-ikx} = \int_{0}^{\frac{1}{2}} dx \ \frac{(\frac{1}{2} -x)}{log(x)} \cdot \frac{-x}{\varepsilon^{2}+x^{2}} \rightarrow \infty$ for $\varepsilon \downarrow 0$ .</p> <p>Therefore the Fourier transform of $f$ is not in $L^{1}$ .</p> http://mathoverflow.net/questions/85135/quasinilpotent-example/85147#85147 Answer by jjcale for Quasinilpotent example jjcale 2012-01-07T20:07:34Z 2012-01-07T20:07:34Z <p>In <a href="http://www.jstor.org/pss/2047905" rel="nofollow">http://www.jstor.org/pss/2047905</a> you can find a weighted shift operator that has this property.</p> http://mathoverflow.net/questions/83043/on-the-paley-wiener-theorem/83171#83171 Answer by jjcale for On the Paley-Wiener theorem jjcale 2011-12-11T10:15:34Z 2011-12-11T10:15:34Z <p>The answer is yes :<br> Let $h$ be an even real valued Schwartz function whose Fourier transform has compact support. Then choose $f(y) = \int_{-\infty}^y x h(x)^{2} dx$ .</p> http://mathoverflow.net/questions/24672/perturbations-of-an-operator-that-disconnect-the-spectrum/81760#81760 Answer by jjcale for Perturbations of an operator that disconnect the spectrum jjcale 2011-11-23T23:13:31Z 2011-11-23T23:13:31Z <p>For Hilbert spaces, the conjecture follows from fact 4 and the answer to question <a href="http://mathoverflow.net/questions/81652/complement-of-a-subspace-which-is-a-cartesian-product" rel="nofollow">http://mathoverflow.net/questions/81652/complement-of-a-subspace-which-is-a-cartesian-product</a> applied to the kernel of the map $H\times H\ni (v,w) \mapsto Av + (I-A)w \in X$ . </p> http://mathoverflow.net/questions/81652/complement-of-a-subspace-which-is-a-cartesian-product Complement of a subspace which is a cartesian product jjcale 2011-11-22T20:24:54Z 2011-11-23T20:38:37Z <p>Let $H$ be a Hilbert space and $U$ a closed subspace of $H\times H$ . Does then exist closed subspaces $V$ and $W$ of $H$ such that $H\times H = U \oplus (V\times W)$ ? </p> <p>See also <a href="http://mathoverflow.net/questions/24672/perturbations-of-an-operator-that-disconnect-the-spectrum" rel="nofollow">http://mathoverflow.net/questions/24672/perturbations-of-an-operator-that-disconnect-the-spectrum</a> .</p> http://mathoverflow.net/questions/2147/most-helpful-math-resources-on-the-web/81107#81107 Answer by jjcale for Most helpful math resources on the web jjcale 2011-11-16T19:39:24Z 2011-11-16T19:39:24Z <p>For people who are interested in prime factorization : www.mersenneforum.org</p> http://mathoverflow.net/questions/78164/quasinilpotence-and-finite-spectrum-ii quasinilpotence and finite spectrum II jjcale 2011-10-14T19:14:38Z 2011-10-14T19:14:38Z <p>Let A be a quasinilpotent operator on a Hilbert space and let every operator of the algebra generated by $A$ and $A^{*}$ have finite spectrum. Does then follow, that A is nilpotent ? </p> <p>See also <a href="http://mathoverflow.net/questions/77177/quasinilpotence-and-finite-spectrum" rel="nofollow">http://mathoverflow.net/questions/77177/quasinilpotence-and-finite-spectrum</a> and <a href="http://mathoverflow.net/questions/35207/finite-dimensional-subalgebras-of-c-star-algebras" rel="nofollow">http://mathoverflow.net/questions/35207/finite-dimensional-subalgebras-of-c-star-algebras</a></p> http://mathoverflow.net/questions/77177/quasinilpotence-and-finite-spectrum quasinilpotence and finite spectrum jjcale 2011-10-04T21:19:28Z 2011-10-06T18:44:48Z <p>Let A be a quasinilpotent operator on a Hilbert space and let $A^{*}A$ have finite spectrum.<br> Does then follow, that A is nilpotent ?</p> http://mathoverflow.net/questions/77177/quasinilpotence-and-finite-spectrum/77384#77384 Answer by jjcale for quasinilpotence and finite spectrum jjcale 2011-10-06T18:44:48Z 2011-10-06T18:44:48Z <p>I found a counterexample : </p> <p>Let $e_{1},e_{2},...$ be ON basis of the Hilbert space and define A by $Ae_{2n-1} = \sqrt{1-\frac{1}{n^{2}}} \ e_{2n} \ + \ \frac{1}{n} \ e_{2n+1}$ , $\ \$n=1,2,3,... ,<br> $Ae_{2n} = 0$ , $\ \$n=1,2,3,... </p> <p>Then A is a partial isometry and therefore $A^{*}A$ a projection.<br> Furthermore A is quasinilpotent but not nilpotent. </p> http://mathoverflow.net/questions/51531/theorems-that-are-obvious-but-hard-to-prove/73802#73802 Answer by jjcale for Theorems that are 'obvious' but hard to prove jjcale 2011-08-26T20:46:00Z 2011-08-26T20:46:00Z <p>A differentiable manifold M that is homeomorphic to the n-sphere is also diffeomorphic to the n-sphere . Obvious, but wrong ! (But right for 1-, 2-, 3-, 5-, 6- and 12-spheres).</p> http://mathoverflow.net/questions/73603/how-to-calculate-a-fredholm-index-numerically How to calculate a Fredholm index numerically jjcale 2011-08-24T19:37:33Z 2011-08-25T07:02:04Z <p>How can one calculate the index of a Fredholm operator numerically ?</p> <p>In numerically calculations one uses always finte dimensional spaces. But linear operators on finite dimensional spaces have always index zero. </p> http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics/73597#73597 Answer by jjcale for Examples of non-rigorous but efficient mathematical methods in physics jjcale 2011-08-24T18:56:36Z 2011-08-24T18:56:36Z <p>The Hypernetted-chain approximation used in statistical mechanics.</p> <p>Was for instance used in the theory of the fractional quantum hall effect by Laughlin in order to estimate the energies of elementary excitations of Laughlins wave function. </p> http://mathoverflow.net/questions/40590/does-a-quantitative-version-of-fredholm-theory-exist/73595#73595 Answer by jjcale for Does a quantitative version of Fredholm theory exist? jjcale 2011-08-24T18:17:51Z 2011-08-24T18:17:51Z <p>The second resolvent equation you are studying is an important tool in quantum scattering theory.</p> <p>See M. Reed and B. Simon, "Methods of Modern Mathematical Physics", Vol IV, Academic Press </p> http://mathoverflow.net/questions/20726/bounding-a-spectral-gap-what-proof-techniques-exist/73440#73440 Answer by jjcale for Bounding a spectral gap: what proof techniques exist? jjcale 2011-08-22T21:19:10Z 2011-08-22T21:19:10Z <p>For VBS quantum antiferromagnets in one dimension see also :</p> <p>Ian Affleck, Tom Kennedy, Elliott H. Lieb and Hal Tasaki, Valence bond ground states in isotropic quantum antiferromagnets. Comm. Math. Phys., Volume 115, Number 3 (1988) </p> <p>and</p> <p>Stefan Knabe, Energy gaps and elementary excitations for certain VBS-quantum antiferromagnets, Journal of Statistical Physics Volume 52, Numbers 3-4, 1988</p> http://mathoverflow.net/questions/16074/how-is-the-physical-meaning-of-an-irreducible-representation-justified/73435#73435 Answer by jjcale for How is the physical meaning of an irreducible representation justified? jjcale 2011-08-22T20:12:45Z 2011-08-22T20:12:45Z <p>You can use group representation theory to reduce the dimensions of the problem. And you can explain energy degeneracies . That's all !</p> http://mathoverflow.net/questions/73242/existence-of-a-special-density Existence of a special density jjcale 2011-08-19T20:53:32Z 2011-08-21T05:44:47Z <p>Does the following function <code>$f:\mathcal{P}(\mathbb{N})\rightarrow\{0,1\}$</code> exist :<br> $f(\mathbb{N})=1$,<br> $f(A\cup B)=f(A)+f(B)$ for $A\cap B=\emptyset$,<br> $f(A)=0$ for finite $A$ </p> http://mathoverflow.net/questions/114743/is-the-poincare-action-on-the-klein-gordon-quantum-field-strongly-continuous Comment by jjcale jjcale 2013-06-01T17:37:42Z 2013-06-01T17:37:42Z Your argument only shows that the action is not norm continous, and this means that the generators of the group action are unbounded operators. http://mathoverflow.net/questions/131725/whats-wrong-with-this-arithmetic-model-for-the-change-in-the-perception-of-numb Comment by jjcale jjcale 2013-05-24T18:58:36Z 2013-05-24T18:58:36Z See <a href="http://en.wikipedia.org/wiki/Measurement_problem" rel="nofollow">en.wikipedia.org/wiki/Measurement_problem</a>. Or read Roger Penrose's &quot;The road to reality&quot;. http://mathoverflow.net/questions/131583/is-there-anyway-to-rewrite-a-partial-differential-equation-using-language-of-diff Comment by jjcale jjcale 2013-05-23T17:32:21Z 2013-05-23T17:32:21Z An example are maxwells equations, see <a href="https://en.wikipedia.org/wiki/Maxwell%27s_equations" rel="nofollow">en.wikipedia.org/wiki/Maxwell%27s_equations</a>. http://mathoverflow.net/questions/113959/a-function-whose-fixed-points-are-the-primes Comment by jjcale jjcale 2013-05-15T18:19:11Z 2013-05-15T18:19:11Z @Rodrigo: I used PARI/GP : f(n)=n+n/vecmin(factor(n)[,1])-1 g(n)=while(1,my(m);m=f(n);if(n==m,break);print(m);n=m;) g2(n)={my(m);my(i=0);while(1,m=f(n);i++;if(n==m,break);n=m;);[m,i]} g2max(n)={ my(m=[0,0]); for(i=2,n, my(p=g2(i)); if(p[2] &gt; m[2],m=[i,p[2]]); ); m; } http://mathoverflow.net/questions/113959/a-function-whose-fixed-points-are-the-primes Comment by jjcale jjcale 2013-05-05T07:09:03Z 2013-05-05T07:09:03Z Computed for n up to 10000000 : record holder 6730914 with orbit length 226 . http://mathoverflow.net/questions/121565/mathematicians-whose-works-were-criticized-by-contemporaries-but-became-widely-ac/121572#121572 Comment by jjcale jjcale 2013-02-13T19:44:34Z 2013-02-13T19:44:34Z See also <a href="http://physics.stackexchange.com/questions/18193/theoretical-proof-forbidding-loschmidt-reversal" rel="nofollow" title="theoretical proof forbidding loschmidt reversal">physics.stackexchange.com/questions/18193/&hellip;</a> . http://mathoverflow.net/questions/114018/fastest-way-to-factor-integers-260 Comment by jjcale jjcale 2012-11-21T21:04:56Z 2012-11-21T21:04:56Z Maybe you should ask this question in the forum <a href="http://www.mersenneforum.org/" rel="nofollow">mersenneforum.org</a> . http://mathoverflow.net/questions/111558/projections-in-banach-spaces/112754#112754 Comment by jjcale jjcale 2012-11-18T12:32:49Z 2012-11-18T12:32:49Z @fedja : Because $\|(P_{t+\varepsilon}-P_{t})Q(f_{1},f_{2})\| \leq \varepsilon \|f_{1}\|_{\infty} \leq \varepsilon \|(f_{1},f_{2})\|$ http://mathoverflow.net/questions/111558/projections-in-banach-spaces Comment by jjcale jjcale 2012-11-17T20:27:05Z 2012-11-17T20:27:05Z The restriction of $Q$ to $(I-P_{0})P_{1}X$ must be quasinilpotent. http://mathoverflow.net/questions/110405/resonance-of-schrodinger-operator Comment by jjcale jjcale 2012-10-24T19:35:23Z 2012-10-24T19:35:23Z @Delio Mugnolo : The Resolvent is only analytic outside the spectrum. If for instance the spectrum consists of the non negative real numbers, then it might be possible that on a subspace of the Hilbert space the resolvent can be analytically continued from the upper half plane to the lower half plane across the spectrum. If A is self adjoint then the resolvent itself can never have a pole outside the real axis, but the analytic continuation may have a pole there, and that's a resonance. And the imaginary part of the pole is interpreted as the inverse of the lifetime of the state http://mathoverflow.net/questions/110405/resonance-of-schrodinger-operator Comment by jjcale jjcale 2012-10-23T19:28:46Z 2012-10-23T19:28:46Z @Delio Mugnolo : To my knowledge a resonance is not a pole of the resolvent but a pole of an analytic continuation of the resolvent. http://mathoverflow.net/questions/109612/matrices-that-are-1-in-a-sense Comment by jjcale jjcale 2012-10-14T16:53:18Z 2012-10-14T16:53:18Z A normal matrix N is in this class iff all absolute values of its eigenvalues are at least 1. http://mathoverflow.net/questions/107100/weyl-quantization-and-convexity Comment by jjcale jjcale 2012-09-13T19:14:09Z 2012-09-13T19:14:09Z Here is a reference : <a href="http://arxiv.org/pdf/1007.1796.pdf" rel="nofollow">arxiv.org/pdf/1007.1796.pdf</a> http://mathoverflow.net/questions/105906/common-eigenvector Comment by jjcale jjcale 2012-08-30T19:28:17Z 2012-08-30T19:28:17Z Every nonzero element of the spectrum of a compact operator on a Banach space is an eigenvalue of this operator. So this is a classical result in the infinite dimensional case. http://mathoverflow.net/questions/36094/notable-mathematics-during-world-war-ii Comment by jjcale jjcale 2012-08-29T19:45:32Z 2012-08-29T19:45:32Z Arne Beurling did important work during World War II : He deciphered a german secret teletypewriter, see <a href="http://en.wikipedia.org/wiki/Arne_Beurling" rel="nofollow">en.wikipedia.org/wiki/Arne_Beurling</a> .