User math-player - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T22:58:27Z http://mathoverflow.net/feeds/user/16974 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103475/riemann-hypothesis-for-zeta-function-of-definable-sets-over-finite-fields Riemann hypothesis for zeta function of definable sets over finite fields Math-player 2012-07-29T20:32:19Z 2013-06-06T21:02:20Z <p>Hi, </p> <p>Consider the zeta function of a definable set over a finite field. More precisely, let $\varphi$ be a formula in the language of fields and let $X$ be a definable subset of $F^n$ given by $X=\{(a_1,\dots,a_n) \in F^n| F\models \varphi(a_1,\dots,a_n) \}$ where $F$ is a finite field of characteristic $p$, then $$Z_X(t)= \exp \sum_{i=1}^ \infty \frac{N_i(X)}{i} t^i,$$ where $N_i(X)= card \{ (a_1,\dots,a_n) \in F_i^n| F_i \models \varphi[a_1,\dots,a_n]\}$ with $F_i$ is the unique extension of $F$ of degree $i$.</p> <p>My question is: </p> <p>Does the Riemann hypothesis for $Z_X(t)$ hold, i.e. if we make the change of variables $t=q^{-s}$ does it follow that all zeroes of $Z_X(q^{-s})$ lie on the line $Re(s)=1/2$? </p> <p>Thank you</p> http://mathoverflow.net/questions/125890/barwise-compactness-theorem Barwise compactness theorem Math-player 2013-03-29T09:12:17Z 2013-03-30T01:32:47Z <p>In "Admissible Sets and Structures" page 101 theorem 5.8 Barwise introduces a weird form of his compactness theorem in which there are two theories $T$ and $T'$ both $\Sigma_1$, such that every $\varphi \in T$ is a pure set (while sets (or formulas) of $T'$ are allowed to be not pure sets, so that they may involve {\it urelements}). Also, he assumes that the theory $T$ is a set of {\it finitary} formulas (because of the key assumption $o(\mathbb{A}_{\mathfrak{M}})=\omega$). </p> <p>Then assuming that for every finite $T_0 \subset T$, $T_0 \cup T'$ has a model then necessarily $T\cup T'$ has a model.</p> <p>It seems to me that this leads to a contradiction because of the following: Consider the set $\omega$ (set of finite ordinals) and assume it is the pure part of $\mathbb{A}_{\mathfrak{M}}$. </p> <p>So clearly $o(\mathbb{A}_{\mathfrak{M}})=\omega$ (where by $o(...)$ we mean the ordinal rank of the set of pure sets (i.e. not involving urelements) of a set). Also, let $\mathfrak{M}=(M )$. $M$ is set of urelements containing a copy of $\omega$. Assume $T'=\lbrace {\rm There \; exists \; a \; surjection \; from \; a \; finite\; ordinal \; to \;} N \subset M \rbrace \cup T''$. </p> <p>Then $T'$ is a set of one infinitary sentence plus $T''$ where $T''$ is a set of formulas specifying that there exists a map from $\omega$ into $N$, plus a set of sentences specifying that two elements of $\omega$ map to the same element of $N$ only if they are congruent by some equivalence relation $\equiv$. Also, let $T$ be the set of (finitary!) formulas specifying that there exists a more than $n$ distinct equivalence classes for $\equiv$, a sentence for each $n \in \omega$. </p> <p>Then every $T_0 \cup T'$ has a model where $T_0 \subset T$ finite, (since $N$ is assumed finite) but $T\cup T'$ does not have a model. How could that be?</p> <p>Thank you</p> http://mathoverflow.net/questions/124632/riemann-roch-theorem-for-arbitrary-1-dim-schemes Riemann-Roch theorem for arbitrary 1-dim schemes Math-player 2013-03-15T17:22:53Z 2013-03-15T17:53:54Z <p>Please I am looking for a reference on Riemann-Roch theorem for 1-dimensional schemes (and not necessarily varieties) if there is any.</p> <p>Thank you</p> http://mathoverflow.net/questions/119138/extension-of-a-function Extension of a function Math-player 2013-01-17T05:54:50Z 2013-01-17T06:33:28Z <p>Hello,</p> <p>Given a $\mathcal{C}^\infty$ function $\varphi$ defined on a portion of a surface $\Sigma^-$ and let $\Sigma$ be a closed surface or union of surfaces bounding a compact volume $\Omega \subset \mathbb{R}^3$ such that $\Sigma^- \subset \Sigma$. When is it possible to extend the function $\varphi$ to the whole domain $\Omega$ such that the following condition is satisfied:</p> <p>$\varphi$ is $\mathcal{C}^\infty$ on $\Omega$ and $\frac{\partial \varphi}{\partial {\mathbf n}}=0$ on $\Sigma$ with ${\mathbf n}$ the normal vector to $\Sigma$?</p> <p>Of course I assume that $\Sigma^-$ and $\Sigma$ are $\mathcal{C}^\infty$ 2-dim real manifolds.</p> http://mathoverflow.net/questions/80633/textbook-for-etale-cohomology Textbook for Etale Cohomology Math-player 2011-11-10T20:53:38Z 2013-01-07T11:08:25Z <p>What is the best textbook (or book) for studying Etale cohomology?</p> http://mathoverflow.net/questions/86336/exercises-for-group-theory-for-physics Exercises for group theory for physics Math-player 2012-01-21T22:18:43Z 2012-12-27T20:46:20Z <p>I teach a course on group theory for physics at the level of senior undergraduates. I follow basically the book by Georgi "Lie algebra in particle physics". So I teach them, the groups SU(2), SU(3) and other related subjects. However there are too little exercises in this book, and I couldn't find enough exercises on the net. Do you know where can I find exercises on group theory FOR physicists?</p> http://mathoverflow.net/questions/116531/the-unreasonable-effectiveness-of-physics-in-mathematics-why-what-how-to-catch/116534#116534 Answer by Math-player for The Unreasonable Effectiveness of Physics in Mathematics. Why ? What/how to catch? Math-player 2012-12-16T16:44:25Z 2012-12-23T21:13:43Z <p>I think that since theoretical physics is really near the edge in its quest to understand the building blocks of our universe, it chooses the "right" mathematical structures for its development. Quantum field theory and String theory are a laboratory of great mathematical ideas, so the inappropriate mathematical structures (those which are not suitable for the description of reality) are put aside and by natural selection the appropriate and useful mathematical structures, tools and ideas are emphasized. Your question transforms itself then to the following: what is the interplay between the world of mathematics and the real world, and does mathematics really exist in some way? </p> <p>It follows from the above considerations that there is no way to absorb the physical ideas into the framework of mathematics, unless mathematics itself changes its objective from seeking the absolute truth to seeking the truth about nature, i.e. unless mathematics transforms into mathematical physics.</p> http://mathoverflow.net/questions/77072/definable-measure-preserving-isomorphisms-of-p-adic-semialgebraic-sets Definable measure preserving isomorphisms of $p$-adic semialgebraic sets Math-player 2011-10-03T20:01:58Z 2012-11-15T20:21:59Z <p>Hi,</p> <p>Consider a $p$-adic field $K$ (finite extension $\DeclareMathOperator{\bQ}{\mathbb{Q}}$of $\bQ_p$) in Macintyre language $\DeclareMathOperator{\cL}{\mathcal{L}}$ $\cL_{\rm Mac}$. Let $Z$ be a definable (i.e. semialgebraic) analytic subset $Z$ of $K^n$ of $p$-adic dimension $d$ and let $C$ be a definable subset of $Z$ of strictly lower dimension.</p> <p>My question is:</p> <p>Is there an analytic definable isomorphism $f: Z \rightarrow Z \setminus C$ such that $\mid {\rm Jac} f \mid=1$?</p> <p>I know the answer for $n=1$ is no but I need to know whether it is still no for $n>1$.</p> <p>Translation with no model theory:by a semialgebraic set we mean a set union of sets of the form $$\lbrace x \in {\bQ}_p^m \mid f(x) =0, g_1(x) \in P_{n_1}, \dots, g_k(x) \in P_{n_k} \rbrace ,$$ where $P_n$ is the set $\lbrace y \in {\bQ}_p^\times \mid \exists x \in {\bQ}_p^\times, y=x^n \rbrace$. A semialgebraic function is a function whose graph is a semialgebraic set.</p> <p>It was shown that (Scowcroft &amp; van den Dries) and Cluckers (see R. Cluckers: Classification of semi-algebraic p-adic sets up to semi-algebraic bijection, Journal für die reine und angewandte Mathematik, 540, 105 - 114 (2001) math.LO/0311434. ) that dimension is a semialgebraic invariant which means that two semialgebraic sets have a semialgebraic bijection between them if and only if they have the same dimension. (Semialgebraic sets of dimension $n$ have a semialgebraic bijection with an open subset of ${\bQ}_p^n$.) But they did not constrain the semialgebraic bijection to have $p$-adic jacobian 1 which I do now.</p> <p>Thank you</p> http://mathoverflow.net/questions/109205/existence-of-weakly-compact-cardinals Existence of weakly compact cardinals Math-player 2012-10-09T07:51:43Z 2012-10-09T16:03:20Z <p>I looked on the web to search for weakly compact cardinals. The web sources indicate many properties of weakly compact cardinals and say that their existence is not entailed by the axioms of ZFC. However it is not indicated whether their existence is {\it consistent} with the axioms of ZFC.</p> <p>So my question is: Is it known that the axiom of existence of weakly compact cardinals consistent with ZFC?</p> <p>Thanks</p> http://mathoverflow.net/questions/107539/characterizing-mathbbq-among-number-fields Characterizing $\mathbb{Q}$ among number fields Math-player 2012-09-19T08:35:14Z 2012-09-19T09:44:18Z <p>Is there an $\mathcal{L}_{\omega_1\omega}$ formula or set of formulas that characterizes the rationals $\mathbb{Q}$ among other number fields?</p> <p>EDIT: My formula must not contain an infinite number of constants from $\mathbb{Q}$. </p> http://mathoverflow.net/questions/107448/birch-and-swinnerton-dyer-conjecture-in-positive-characteristic Birch and Swinnerton-Dyer conjecture in positive characteristic Math-player 2012-09-18T08:47:58Z 2012-09-18T11:22:29Z <p>Is the Birch and Swinnerton-Dyer conjecture known in positive characteristic?</p> http://mathoverflow.net/questions/106745/characteristic-zero-and-characteristic-p-in-algebraic-geometry Characteristic zero and characteristic $p$ in algebraic geometry Math-player 2012-09-09T18:30:58Z 2012-09-11T06:34:51Z <p>Are there non-trivial (i.e. excluding concepts that can be defined only for $p>0$) statements in algebraic geometry that hold for all fields of characteristic $p$ for all prime $p$ but are known to be false in characteristic zero?</p> http://mathoverflow.net/questions/105467/admissible-sets-and-infinitary-languages Admissible sets and infinitary languages Math-player 2012-08-25T13:58:48Z 2012-08-29T00:24:38Z <p>Consider a first-order language $L$ and the infinitary language $L(\omega_1,\omega)$ obtained from $L$ by allowing infinite conjonctions and disjonctions. Let $X$ be an infinite set of sentences, including (possibly infinitely many) infinitary formulae and infinitely many finite formulae (let $\Delta$ be an infinite set of finite formulas, $\Delta \subset X$). Let $\mathcal{A}$ be the least admissible set containing $X$ ($X \subset \mathcal{A}$). Is it possible that $\mathcal{A}$ does not contain $\Delta$ as an element? ($\Delta \notin \mathcal{A}$). </p> http://mathoverflow.net/questions/104958/does-normal-riemann-hypothesis-follow-from-extended-riemann-hypothesis Does normal Riemann hypothesis follow from extended Riemann hypothesis Math-player 2012-08-18T00:09:06Z 2012-08-18T00:29:16Z <p>Does normal Riemann hypothesis for $\zeta_{\mathbb{Q}}$ follows from the extended Riemann hypothesis for some $K \neq \mathbb{Q}$ (i.e. the statement that all zeroes of the Dedekind zeta function $\zeta_K$ for $K$ a number field in the critical strip lie on the axis $\mathfrak{R}(s)=1/2$)?</p> http://mathoverflow.net/questions/104797/how-to-know-that-a-set-is-sigma How to know that a set is \Sigma? Math-player 2012-08-15T21:53:16Z 2012-08-17T01:36:32Z <p>Let $\mathcal{A}$ be an admissible set, or for simplicity a model of ZF without the power set axiom. Let $X \subset \mathcal{A}$. Is there an easy way to show $X$ is $\Sigma$ on $\mathcal{A}$?</p> <p>PS: $X$ is $\Sigma$ on $\mathcal{A}$ if it is definable in $\mathcal{A}$ by a $\Sigma$ formula i.e. a formula built from atomic formulas and their negations using only $\wedge,\vee, \forall x \in y, \exists x$. To be specific assume $\mathcal{A}=L_{\omega_1^{CK}}$. Also, I need to consider the specific case of $X$ being a set of (codes of) $\mathcal{L}(\omega_1,\omega)$-sentences or to be precise: $X$ is a set of $\mathcal{L}$-sentences + infinitary sentences specifying that certain quotient sets are finite. Here $\mathcal{L}$ is some first order language.</p> http://mathoverflow.net/questions/104788/abstract-number-ring-of-any-characteristic Abstract number ring of any characteristic Math-player 2012-08-15T20:36:56Z 2012-08-15T22:36:08Z <p>Let $(n_1,\ldots, n_i,\ldots)$ be an infinite tuple of nonnegative integers. Is there an abstract number ring $D$ of a given characteristic $p>0$ and $I_1,\dots, I_n , \ldots$ its nonzero ideals (by assumption $D/I_i$ are finite) such that #$D/I_i=n_i$ for each $i$?</p> http://mathoverflow.net/questions/103556/a-unified-description-of-zeta-functions-of-a-curve-over-mathbbf-q-and-rieman A unified description of zeta functions of a curve over $\mathbb{F}_q$ and Riemann $\zeta$ function Math-player 2012-07-30T22:19:47Z 2012-07-30T23:18:30Z <p>Is there a unified description (or a set of axioms) of the zeta function of an algebraic curve over a finite field $\mathbb{F}_q$ and the Riemann zeta function? </p> http://mathoverflow.net/questions/102729/axioms-for-zeta-function-of-a-scheme Axioms for zeta function of a scheme Math-player 2012-07-20T09:55:01Z 2012-07-20T09:55:01Z <p>Is there any set of axioms that characterize completely the zeta function of a scheme over a finite field of characteristic $p$?</p> http://mathoverflow.net/questions/102536/axioms-for-riemann-zeta-function Axioms for Riemann $\zeta$ function Math-player 2012-07-18T12:43:02Z 2012-07-19T01:05:13Z <p>Are there any set of axioms that completely characterize the Riemann zeta function? i.e. like Ressayre axioms for the exponential function in an exponential field or functional equations.</p> http://mathoverflow.net/questions/102338/is-it-possible-to-reconstruct-zeta-function-knowing-its-zeroes Is it possible to reconstruct $\zeta$-function knowing its zeroes? Math-player 2012-07-16T10:07:35Z 2012-07-16T17:07:39Z <p>Hello, Is it possible to reconstruct the Riemann zeta function given the precise location of its infinitely many zeroes? Thanks</p> http://mathoverflow.net/questions/101978/how-to-define-zeta-function-for-curves-over-a-number-field How to define zeta function for curves over a number field Math-player 2012-07-11T18:25:02Z 2012-07-11T18:45:27Z <p>How to define zeta function for a curve over $\mathbb{Z}$ or $\mathbb{Q}$?</p> http://mathoverflow.net/questions/81654/etale-cohomology-in-the-p-adic-setting Etale cohomology in the $p$-adic setting Math-player 2011-11-22T20:40:26Z 2012-06-15T13:02:43Z <p>Can we hope for application of Etale cohomology techniques in proving results concerning semialgebraic subsets of $\mathbb{Q}_p^n$?</p> <p>Recall that semialgebraic subsets are obtained from $p$-adic algebraic varieties using boolean connectives and coordinate projection.</p> <p>The results I am interested in are a sort of algebraic topology over the $p$-adics. But of course one cannot use the standard topology on the $p$-adics since it is totally disconnected. So I hope to show for instance that a ball with one smaller ball removed cannot be diffeomorphic to a ball, etc .</p> http://mathoverflow.net/questions/85643/nonstandard-methods-or-model-theory-and-arithmetic-geometry/85649#85649 Answer by Math-player for Nonstandard Methods ( or Model Theory ) and Arithmetic Geometry Math-player 2012-01-14T11:18:59Z 2012-04-16T17:22:41Z <p>Try this: <a href="http://www.dpmms.cam.ac.uk/~cb496/nsag1.pdf" rel="nofollow">www.dpmms.cam.ac.uk/~cb496/nsag1.pdf</a> and also this <a href="http://wwwmath.uni-muenster.de/u/serpe/documents/ultramath2008serpe-nonstandard-handout.pdf" rel="nofollow">http://wwwmath.uni-muenster.de/u/serpe/documents/ultramath2008serpe-nonstandard-handout.pdf</a>, <a href="http://logicandanalysis.org/index.php/jla/article/view/77/29" rel="nofollow">logicandanalysis.org/index.php/jla/article/view/77/29</a> and references therein.</p> http://mathoverflow.net/questions/94014/mathematical-foundations-of-quantum-field-theory/94019#94019 Answer by Math-player for Mathematical foundations of Quantum Field Theory Math-player 2012-04-14T09:44:11Z 2012-04-14T09:44:11Z <p>There is a nice formulation of the geometry of QFT, available at <a href="http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics-impa-v01-2011.pdf" rel="nofollow">http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics-impa-v01-2011.pdf</a>.</p> http://mathoverflow.net/questions/92354/noether-normalization Noether Normalization Math-player 2012-03-27T08:48:29Z 2012-03-28T03:06:07Z <p>I teach a course of undergraduate algebraic geometry. I noticed that students have difficulty grasping the proof of Noether normalizsation Lemma (given in Reid's undergraduate algebraic geometry). Also, the proof given in Mumford red book is also non-elementary. Does anyone know of some easier proof of the result? Thank you</p> http://mathoverflow.net/questions/92356/a-infty-basic-reference $A_\infty$ basic reference Math-player 2012-03-27T09:07:15Z 2012-03-27T17:44:01Z <p>Can anyone provide me with a basic reference on $A_\infty$ categories? Thank you</p> http://mathoverflow.net/questions/89845/relation-between-metalanguage-and-object-language/89851#89851 Answer by Math-player for Relation between Metalanguage and Object Language Math-player 2012-02-29T09:41:03Z 2012-02-29T09:41:03Z <p>There are two roles for metalanguage: First, to avoid contradictions like the liar paradox, because in the liar paradox we have a statement that speaks about itself so it does not respect the hierarchy language-metalanguage. The other role is to allow us to speak freely and to use theorems of the language as meta-theorems. So if we use the scheme of deduction using the principle of excluded middle, we know that we are using a meta-theorems, but this is just a way to use the corresponding theorem in the object language without repetition. So for example the metatheorem: "If the negation of a proposition A does not hold, then A holds" can be replaced by the theorem in the object language " $\neg \neg A \Rightarrow A$".</p> http://mathoverflow.net/questions/89514/zeta-function-of-an-elliptic-curve-over-mathbbf-pt Zeta function of an elliptic curve over $\mathbb{F}_p(t)$ Math-player 2012-02-25T20:56:07Z 2012-02-25T21:28:27Z <p>What is the form of zeta function of an elliptic curve over $\mathbb{F}_p(t)$? Does it satisfy a Riemann hypothesis?</p> http://mathoverflow.net/questions/88910/2-short-article-vs-a-long-one/88924#88924 Answer by Math-player for 2 short article vs. a long one Math-player 2012-02-19T10:18:22Z 2012-02-19T10:18:22Z <p>I would say that writing two articles with the same introductory body is not the problem. Rather, are the results of both articles going in the same direction? If yes, then it should be better writing a single article, or a two-part article.; If, on the other hand the results of the two articles point in different directions then maybe you should split the article.</p> http://mathoverflow.net/questions/62818/the-sets-in-mathematical-logic/88744#88744 Answer by Math-player for The sets in mathematical logic Math-player 2012-02-17T18:02:24Z 2012-02-17T18:02:24Z <p>I think this problem shows itself at many stages of the human thinking. So we can form the expression "This expression is wrong" and immediately the liar paradox appears. So really this has nothing to do with mathematics proper, but with logic itself. And the solution is multistaged itself: at the basic level we can use Russell's (naive) type theory as a remedy. In fact when we talk about mathematics we already make sure we don't break certain rules on the metalanguage level. Next when we want to do set theory, we use the axioms of Zermelo-Fraenkel. Problem solved. Of course we can formalize the metalanguage and we can use again a ZF set theory in this task instead of type theory, but in doing so, we use an informal meta-meta-language for which no way to ignore type theory. So, type theory is the ultimate building block of "human logic" and not just "mathematical logic". </p> http://mathoverflow.net/questions/125890/barwise-compactness-theorem/125962#125962 Comment by Math-player Math-player 2013-03-30T08:16:09Z 2013-03-30T08:16:09Z Thank you very much. http://mathoverflow.net/questions/124773/shall-i-send-this-paper-to-the-arxiv Comment by Math-player Math-player 2013-03-17T13:17:00Z 2013-03-17T13:17:00Z Thank you for the advice. http://mathoverflow.net/questions/124632/riemann-roch-theorem-for-arbitrary-1-dim-schemes/124635#124635 Comment by Math-player Math-player 2013-03-15T18:00:21Z 2013-03-15T18:00:21Z Thank you for the reference. http://mathoverflow.net/questions/119370/vector-fields-on-2-sphere Comment by Math-player Math-player 2013-01-19T23:44:45Z 2013-01-19T23:44:45Z Yes the closed curve condition is just put for illumination purposes. http://mathoverflow.net/questions/119138/extension-of-a-function Comment by Math-player Math-player 2013-01-17T10:28:26Z 2013-01-17T10:28:26Z @Willie: If we know that $\Sigma^_$ is relatively closed in $\Sigma$ does this solve the problem? http://mathoverflow.net/questions/119138/extension-of-a-function Comment by Math-player Math-player 2013-01-17T09:42:35Z 2013-01-17T09:42:35Z @Willie: thank you for the comment. http://mathoverflow.net/questions/80633/textbook-for-etale-cohomology/118266#118266 Comment by Math-player Math-player 2013-01-07T11:37:28Z 2013-01-07T11:37:28Z There is no &#233;tale cohomology inside. http://mathoverflow.net/questions/117883/who-is-mathoverflows-quid Comment by Math-player Math-player 2013-01-02T18:48:25Z 2013-01-02T18:48:25Z Sorry, I didn't mean to offence anybody. Couldn't delete the post. http://mathoverflow.net/questions/116881/is-it-ok-to-send-drafts Comment by Math-player Math-player 2012-12-20T16:56:53Z 2012-12-20T16:56:53Z He is very diplomatic so he might not say no but may be annoyed. http://mathoverflow.net/questions/116881/is-it-ok-to-send-drafts Comment by Math-player Math-player 2012-12-20T16:51:20Z 2012-12-20T16:51:20Z A DRAFT in my opinion is a very rough preliminary version. Other mathematician is my previous thesis advisor. http://mathoverflow.net/questions/108098/grothendieck-ring-of-varieties-carrying-a-function Comment by Math-player Math-player 2012-12-01T19:30:03Z 2012-12-01T19:30:03Z Why did you choose $f$ to be valued algebraic and not constructible? http://mathoverflow.net/questions/114344/13-months-and-not-even-one-report-what-would-you-do/114355#114355 Comment by Math-player Math-player 2012-11-24T18:16:08Z 2012-11-24T18:16:08Z what happened then? http://mathoverflow.net/questions/114348/is-the-field-with-one-element-still-alive Comment by Math-player Math-player 2012-11-24T17:10:42Z 2012-11-24T17:10:42Z What do you mean? http://mathoverflow.net/questions/109214/known-results-in-characteristic-p Comment by Math-player Math-player 2012-10-10T06:02:19Z 2012-10-10T06:02:19Z And do you have a reference for Kunneth decomposition? http://mathoverflow.net/questions/109214/known-results-in-characteristic-p Comment by Math-player Math-player 2012-10-10T05:32:14Z 2012-10-10T05:32:14Z It does not matter, if the theory has models in char $p$ that is ok, don't need to be true for all fields of char $p$.