User xl - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:52:30Z http://mathoverflow.net/feeds/user/14024 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/131674/any-closed-form-for-series-like-fx-sigma-ip-inftyxp-p-is-prime Any closed form for series like $F(x)=\Sigma_{i=p}^{\infty}x^p$,p is prime$? XL 2013-05-24T00:59:32Z 2013-05-24T23:34:23Z <p>Any closed form for series like $$F(x)=\Sigma_{i=2}^{\infty}x^p,\text{p is prime}$$ or $$F(x)=\Sigma_{i=0}^{\infty}x^{i!}$$?</p> <p>More generally,we can obtain a power series from decimal expansion of a number r(0&lt; r&lt;1 ) by replacing $$(\frac{1}{10})^i$$ with $$x^i$$ like $$\frac{1}{3}=3(\frac{1}{10})^1+3(\frac{1}{10})^2+\cdots 3(\frac{1}{10})^i+\cdots$$, we obtain : $$f(x)==\Sigma_{i=1}^{\infty}3x^i$$</p> <p>when f(x) is convergent,what restriction do we have to put on r(if r is c.e number) to make f(x) have a closed form?</p> <p>When is f(x) algebraic ,or transcendental?</p> http://mathoverflow.net/questions/128799/any-grammar-for-the-language-l-ap-p-is-prime-number-of-mathbbn Any grammar for the language$L =a^p$,$p$is prime number of$\mathbb{N}$XL 2013-04-26T04:49:19Z 2013-04-26T11:06:38Z <p>Any grammar for the language $$L =a^p,\text{ p is prime and }p\in \mathbb{N}?$$</p> <p>Is such a grammar related to any question of number theory like RH or the conjecture of twin primes?</p> http://mathoverflow.net/questions/127034/what-is-the-computational-complexity-of-resolution-of-singularities-of-varieties what is the computational complexity of resolution of singularities of varieties over fields with characteristic 0 XL 2013-04-10T02:27:15Z 2013-04-10T02:27:15Z <p>what is the computational complexity of resolution of singularities of varieties over fields with characteristics 0?</p> http://mathoverflow.net/questions/126519/is-there-a-mathematical-definition-of-simplify/126605#126605 Answer by XL for Is there a "mathematical" definition of "simplify"? XL 2013-04-05T11:45:37Z 2013-04-05T22:55:16Z <p>There is a theory called Kolmogorov Complexity(KC,also called algorithmic information) which has been initiated by Chaitin,Solomonoff,and Kolmogorov.Roughly speaking,an object is simple if it's KC is shorter,it is related to recursive function or computability theory,or uncomputabilty ,See Kolmogorov complexity and it's application by Ming Li and Vitanyi for the exact definition and examples.or <a href="http://www.scholarpedia.org/article/Algorithmic_complexity" rel="nofollow">http://www.scholarpedia.org/article/Algorithmic_complexity</a>. It may be what you are looking for.</p> http://mathoverflow.net/questions/126627/existence-of-unknowable-algorithms/126631#126631 Answer by XL for Existence of unknowable algorithms ? XL 2013-04-05T14:50:16Z 2013-04-05T14:50:16Z <p>The answer has to be No,since we can enumerate all algorithm.</p> http://mathoverflow.net/questions/126447/any-results-or-concise-introduction-about-nonassociative-algebra-that-even-does-n Any results or concise introduction about nonassociative algebra that even does not satisify Power associativity? XL 2013-04-03T20:35:32Z 2013-04-04T08:21:39Z <p>Any results or concise introduction about nonassociative algebra that even does not satisify Power associativity?</p> http://mathoverflow.net/questions/126444/linkage-between-singularities-of-algebraic-varieties-and-continued-fractions Linkage between singularities of algebraic varieties and continued fractions XL 2013-04-03T20:25:11Z 2013-04-04T06:58:27Z <p>I have an impression that there is linkage or relation between singulariry of algebraic variety and continued fraction when I read some book on resolution of singularity or algebraic geometry.Could any one give some reference for that?</p> http://mathoverflow.net/questions/126322/how-to-work-out-a-grammar-if-we-know-the-language How to work out a grammar if we know the language? XL 2013-04-02T22:42:04Z 2013-04-03T07:18:30Z <p>How to work out a grammar if we know the language? Or at least How to work out a grammar if we know the language that is restricted to a special kind like CFL or CSL? For example,we know $$L=\{a^nb^nc^n \mid n \in \mathbb{N}\},$$ how can we get the grammar?Is there any algorithm?</p> <p><strong>EDIT</strong>: Language here means at least the recursively enumerable one, or computably enumerable one.</p> http://mathoverflow.net/questions/125412/is-there-any-algebraic-function-that-has-a-specific-relation-to-transcendental-on is there any algebraic function that has a specific relation to transcendental one? XL 2013-03-24T00:24:39Z 2013-03-24T06:44:48Z <p>given transcendental function $$F(x)=\sum_0^{\infty}a_i x^i,a_i\in \mathcal{N} \bigcup 0,\exists M \space a_i \leq M^i$$.</p> <p>is there algebraic function $$A(x)=\sum_0^{\infty}b_i x^i,b_i\in \mathcal{N} \bigcup 0,$$,such that$a_i =b_i$if$a_i = 0$;$a_i \leq b_i$otherwise?</p> http://mathoverflow.net/questions/124661/are-the-two-models-describing-sir-equivalent are the two models describing SIR equivalent? XL 2013-03-16T00:38:12Z 2013-03-16T00:38:12Z <p>SIR is Standard convention labels these three compartments S (for susceptible), I (for infectious) and R (for recovered). There are an ODE and a game-theoritic model to describe it,are they equivalent?</p> http://mathoverflow.net/questions/123937/is-any-cfl-intersection-union-of-cfls-that-are-not-inherently-ambiguous Is any CFL intersection,union of CFLs that are not inherently ambiguous? XL 2013-03-08T02:43:55Z 2013-03-08T02:43:55Z <p>Is any CFL intersection,union of CFLs that are not inherently ambiguous?</p> http://mathoverflow.net/questions/123860/any-other-definition-for-algebraic-number-than-the-root-of-algebraic-equation Any other definition for algebraic number than the root of algebraic equation? XL 2013-03-07T12:14:28Z 2013-03-07T17:15:25Z <p>Any other definition for algebraic number than the root of algebraic equation?</p> http://mathoverflow.net/questions/119990/condition-and-algorithm-for-decomposition-of-formal-power-series Condition and algorithm for Decomposition of formal power series XL 2013-01-27T05:16:31Z 2013-01-27T10:52:17Z <p>$$F(x)= \Sigma_0^{\infty} a_i x^i$$ is formal power series,$a_i\in N\bigcup 0$,N is the set of natural number,under what condition may it be decomposed into a system of equations terms of which are polynomials of multivariables(EDIT:decomposed means we can get F(x)expressed by x by solving the system of equations)?</p> <p>Decomposition is like: $$F(x)= \Sigma_1^{\infty} x^{3i}$$ may be decomposed into:</p> <pre><code>F(x)=F(x)B(x)C(x)x+B(x)C(x)x B(x)=x C(x)=x </code></pre> <p>If it may,is there algorithm to do that?</p> <p>these power series may be regarded as a complex function with convergence radius. Question: when is F(x) a algebraic function,or a transcendental function?</p> <p>Please do not downvote it if it is not very clear</p> http://mathoverflow.net/questions/2917/where-does-a-math-person-go-to-learn-quantum-mechanics/119366#119366 Answer by XL for Where does a math person go to learn quantum mechanics? XL 2013-01-19T21:45:17Z 2013-01-19T21:45:17Z <p>Quantum Mechanics, Enrico Fermi's work is helpful.I think mathematicians study quantum mechanics,to learn how physics is processed with math,the way and the insight physicists do work with math,and the physical interpretation as well.It is an excellent book although the content is somehow oldfashioned.</p> http://mathoverflow.net/questions/119294/generating-function-of-a-regular-language/119307#119307 Answer by XL for Generating function of a regular language XL 2013-01-19T06:14:45Z 2013-01-19T06:14:45Z <p>Please see <a href="http://algo.inria.fr/flajolet/Publications/books.html" rel="nofollow">http://algo.inria.fr/flajolet/Publications/books.html</a> ,the book Analytic combinatorics's first several chapters.</p> http://mathoverflow.net/questions/71124/density-of-formal-language density of formal language？ XL 2011-07-24T12:48:55Z 2012-09-19T02:00:24Z <p>let$\sum_0^n l_i x^i$and$\sum_0^n 2^i x^i$be generating function of L a given language and the closure over alphabet$\Sigma= \{0,1 \}$when$n\to\infty$. let$$D=\frac{\sum_0^n l_i }{\sum_0^n 2^i }$$,$$d=\frac{ l_i x^i}{ 2^i x^i}=\frac{ l_i }{ 2^i }$$.</p> <p>Obviously,$0 \leq D,d \leq 1$.when(under what condition such as the class of the language or language with what feature) does$lim_{n\to \infty} D$and$lim_{n\to \infty} d$exist?</p> <p>If the limit does not exist,how does the$D,d$vibrates or how about the$f(D,n)$and$f(d,n)$relating to class of language or feature of language?</p> <p>Any result of questions above?</p> http://mathoverflow.net/questions/107383/any-result-or-conjecture-of-computaional-complexity-of-formal-languange-with-rati Any result or conjecture of computaional complexity of formal languange with rational generating function? XL 2012-09-17T14:18:02Z 2012-09-18T15:25:12Z <p>As we know that context-free language is in P,any result or conjecture of computaional complexity of formal languange with rational generating function?And more,any result or conjecture of computaional complexity of formal languange with algebraic generating function?</p> http://mathoverflow.net/questions/73553/when-may-function-meromorphic-be-expanded-as-power-series-with-coefficients-of When may Function (meromorphic) be expanded as power series with coefficients of integers XL 2011-08-24T09:39:08Z 2012-08-04T10:30:49Z <p>Let F be meromorphic function,with what properties may it expanded as power series with coefficients of integers in such a form:</p> <p>$$F=\sum_0^{\infty}a_i x^i,a_i\in \mathcal{N} \bigcup 0,\exists M \space a_i \leq M^i$$.</p> <p>and when the coefficients consist of a sequence of computably enumerable relation.</p> <p>If the question is ambiguous ,please tell me but please do not downvote it.</p> <p>When may Function (meromorphic) be expanded as power series with coefficients of integers</p> http://mathoverflow.net/questions/101041/in-what-probability-does-cospectra-of-cayley-graph-imply-isomorphism-of-the-corre In what probability does cospectra of Cayley graph imply isomorphism of the corresponding group XL 2012-07-01T04:50:26Z 2012-07-01T08:31:04Z <p>In what probability does cospectra of adjacent matrix of Cayley graph imply isomorphism of the corresponding group? Further more,In what probability does cospectra of adjacent matrix imply isomorphism of the corresponding graphs</p> http://mathoverflow.net/questions/79299/any-given-c-e-set-has-number-m-whose-power-bounds-the-corresponding-elements-of any given c.e.set has number M whose power bounds the corresponding elements of S? XL 2011-10-27T19:10:05Z 2011-12-25T21:07:44Z <p>For S ,any given c.e.set,does there exist a M (integer) and a partially computable function outputing every element of S the c.e.set ,such that$\forall x\in S,\exists n x=f(n)$and$x=f(n)\leq M^n$?.</p> http://mathoverflow.net/questions/79558/how-to-compute-the-change-in-entropy-of-s-system-when-the-temperature-is-decreasi How to compute the change in entropy of s system when the temperature is decreasing? XL 2011-10-30T23:22:00Z 2011-10-30T23:45:38Z <p>How to compute the change in entropy of s system when the temperature is decreasing? Such a question may be too general,then we may ask a special case:How to compute the change in entropy of Annealing system when such a process may be regarded as Markov Chain ?</p> <p>If any clarification is needed ,please point in your comment.</p> http://mathoverflow.net/questions/79555/what-are-the-application-of-the-integral-and-differential-in-times-from-integer-t What are the application of the integral and differential in times from integer to rational ,real and complex XL 2011-10-30T22:59:40Z 2011-10-30T23:33:55Z <p>For instance,if$\int f=\frac{1}{a}e^{ax}$is regard as integral in one time,we use notation$T(1)\int f=\frac{1}{a}e^{ax}$,we may extend it to fractional ,real,or complex time in such a way:$$T(\alpha)\int f=\frac{1}{a^{\alpha}}e^{ax},\alpha \in N,Z,Q,R,or C$$.</p> <p>We even may extend$\alpha$to a more general field than C.</p> <p>First question :with what class of function f can the intergral be extended in such a way? every integralable one?</p> <p>Second question:what is the application of such extension in math?Anyone gives any example?</p> <p>Third question:can we explain such extension or operation by geometry as usual integral?How should we explain such extension or operation by geometry?It can not intepreted in geometry?</p> <p>Fourth question:what are the more general extension of such operation than C?</p> http://mathoverflow.net/questions/79422/do-all-power-series-with-coefficients-a-n-in-c-e-set-converge-and-be-represe do all power series with coefficients$a_n \in c.e.$set converge and be represented as automorphic function? XL 2011-10-28T18:43:05Z 2011-10-28T18:58:43Z <p>given any c.e.set S outputed by an partially computable function$f(n)$, a power series F in the form$F=\Sigma_0^{\infty} a_n x^n$corresponding to the set S,whose coefficient$a_n=f(n)$if$f(n)$is defined,otherwise$a_n=0$. when does F be an automorphic function?</p> http://mathoverflow.net/questions/71845/what-is-fraction-of-two-series-constructed-with-rational-and-algebraic-number-in What is fraction of two series constructed with rational and algebraic number in an interval XL 2011-08-02T00:47:43Z 2011-08-02T01:13:02Z <p>What is fraction of two series constructed with rational and algebraic number in an interval</p> <p>Let$E,C,R,N $be sets of computably enumerable number,computable numbers,rational numbers, natural numbers in interval(0,1] respectively.let $$\sigma_{S_x^i}=\sum_{x\in S}x^i$$ where$S$is$E,C,R$,or$N $,and let $$F(P|U)=\frac{\sigma_{P_x^i}}{\sigma_{U_x^i}}$$ .what is$F(C|E),F(R|E),F(R|C)$if the fractions exist? And when do the fractions exist?</p> <p>Let$T$,$A$be the sets of computably enumerable and transcendental numbers and algebraic set in the interval (0,1],what are$F(T|E),F(T|C),F(A|C),F(R|A)$?</p> <p>Those are questions that I think are strongly related to my two posts of conditional probability of subset of numbers and class of languages,but obviously are different from them.</p> http://mathoverflow.net/questions/71797/what-is-the-conditional-probability-of-some-subset-of-numbers What is the conditional probability of some subset of numbers XL 2011-08-01T13:23:42Z 2011-08-01T14:47:04Z <p>What is the conditional probability of some subset of numbers</p> <p>Let$E,C,R,N $be sets of computably enumerable number,computable numbers,rational numbers, natural numbers respectively.$E$is sets of computably enumerable numbers and it's subset of$E$is the Cantor space,take the uniform probability measure on the Cantor space,then what is conditional probability$P(C),P(S),P(R),P(N),P(R|C),P(N|C),\cdots,P(N|R)$?</p> <p>Let$T$be set of sets of computably enumerable and transcendental numbers,what is$P(T),P(T|C)$?</p> http://mathoverflow.net/questions/70949/what-is-the-conditional-probability-or-probablity-of-classes-of-languages What is the conditional probability or probablity of classes of languages? XL 2011-07-22T02:23:21Z 2011-07-22T03:13:06Z <p>What is the conditional probability or probability of classes of languages?</p> <p>Let$E,C,S,F,R $be the class of computably enumerable languages,computable languagesl,context-sensitive anguages,context-free languages and regular languages respectively.$E$is class of all computably enumerable languages and it's subset of$E$is the Cantor space,take the uniform probability measure on the Cantor space,then what is the probability or conditional probability$P(C),P(S),P(F),P(R),P(S|C),P(F|C),P(R|C),\cdots,P(R|F)$?suppose L is selected randomly under the fair-coin (Lebesgue-Cantor) measure on$2^{\omega}$.</p> http://mathoverflow.net/questions/70346/transformation-of-diophantine-equation-into-another-one-or-a-series Transformation of Diophantine equation into another one or a series XL 2011-07-14T16:27:22Z 2011-07-15T01:25:39Z <p>Two questions: First,Given a Diophantine equation$P(y,x_1,\cdot \cdot \cdot,x_j)=0$where$y \in S,S \subseteq \mathbb{Z}$is set of the solutions of the Diophantine equation when$x_1,\cdot \cdot \cdot,x_j \in \mathbb{Z}$,how much is the number of variables of a Diophantine equation equivalent to it (that is they just have the same set of solutions S of n over$\mathbb{Z}$)with the least number of variables ? 11?or much smaller or bigger? </p> <p>Second,Given a Diophantine equation$P(y,x_1,\cdot \cdot \cdot,x_j)=0$where$y \in S,S \subseteq \mathbb{Z}$is set of the solutions of the Diophantine equation when$x_1,\cdot \cdot \cdot,x_j \in \mathbb{Z}$,is there series in the form$f(y)=\Sigma_0^{\infty}a_i y^i$where$a_i \in \mathbb{Z}$or$a_i \in \mathbb{Q}$equivalent to the diopantine equaton (that is the series have just the same set of solutions (set of zero ) S over$\mathbb{Z}$as the Diophantine equation S.)?</p> <p>The second question seems to be true,but I do not know how to prove it.</p> http://mathoverflow.net/questions/63803/relation-between-partially-computable-function-and-complex-function Relation between partially computable function and complex function XL 2011-05-03T12:40:16Z 2011-05-03T16:15:33Z <p>Given a partially computable function, is there an analytic complex function which is equal to it at every point of it's domain? Or under what condition does a partially computable function correspond to the restriction of an analytic complex function?</p> http://mathoverflow.net/questions/63814/which-discrete-formula-and-its-continuous-counterpart-do-you-think-are-most-in which discrete formula and it's continuous counterpart do you think are most interesting? XL 2011-05-03T15:00:29Z 2011-05-03T15:14:15Z <p>For example,$N!$和$\Gamma(z)$are discrete formula and it's continuous counterpart .Maybe,the word continuous is not appropriate here</p> http://mathoverflow.net/questions/61621/what-is-the-measure-of-productive-and-immune-sets-in-the-cantor-space What is the measure of productive and immune sets in the Cantor space? XL 2011-04-14T01:51:57Z 2011-04-14T05:01:02Z <p>We view subsets of the natural numbers as their characteristic functions, which are elements of the Cantor space$2^\mathbb{N}\$. We take the uniform probability measure on the Cantor space. Under this view, what is the measure of the family of all productive sets (in the sense of computability theory)? Immune sets? Sets which are neither immune nor productive nor computably enumerable?</p> http://mathoverflow.net/questions/131674/any-closed-form-for-series-like-fx-sigma-ip-inftyxp-p-is-prime/131678#131678 Comment by XL XL 2013-05-24T09:54:11Z 2013-05-24T09:54:11Z @i707107,those hold ,provided a series is convergent with radius r,otherwise,they may not hold,am I right?For instance,$$f(x)=\frac{1-\sqrt{1-4x^2}}{2x^2}$$ http://mathoverflow.net/questions/131674/any-closed-form-for-series-like-fx-sigma-ip-inftyxp-p-is-prime/131678#131678 Comment by XL XL 2013-05-24T05:49:29Z 2013-05-24T05:49:29Z @Gerry,thank you for your reminding http://mathoverflow.net/questions/131674/any-closed-form-for-series-like-fx-sigma-ip-inftyxp-p-is-prime Comment by XL XL 2013-05-24T05:48:53Z 2013-05-24T05:48:53Z @Gerry,@i707i707,thank both of you very much,your expressions are right,they are what I intend to express http://mathoverflow.net/questions/131674/any-closed-form-for-series-like-fx-sigma-ip-inftyxp-p-is-prime/131678#131678 Comment by XL XL 2013-05-24T05:06:39Z 2013-05-24T05:06:39Z @i707i707,thank you.But it is only a partial answer to this question,especially,when r is algebraic number http://mathoverflow.net/questions/21290/whats-an-example-of-a-transcendental-power-series/21416#21416 Comment by XL XL 2013-05-01T08:01:44Z 2013-05-01T08:01:44Z Can not help saying it is so beautiful http://mathoverflow.net/questions/128799/any-grammar-for-the-language-l-ap-p-is-prime-number-of-mathbbn Comment by XL XL 2013-04-26T12:03:42Z 2013-04-26T12:03:42Z Thank you ,Emil. http://mathoverflow.net/questions/128799/any-grammar-for-the-language-l-ap-p-is-prime-number-of-mathbbn/128801#128801 Comment by XL XL 2013-04-26T09:01:35Z 2013-04-26T09:01:35Z @Luke,Thank you very much. http://mathoverflow.net/questions/128799/any-grammar-for-the-language-l-ap-p-is-prime-number-of-mathbbn/128801#128801 Comment by XL XL 2013-04-26T05:38:55Z 2013-04-26T05:38:55Z Obviously,there is a grammar that produce it,But we do not know the grammar. But I am not sure this has bearing on number theoretic questions like RH or TPC,since the grammar may be transformed into algorithm or function which may characterize the set of prime number.Anyway, thank you for your answer. http://mathoverflow.net/questions/127034/what-is-the-computational-complexity-of-resolution-of-singularities-of-varieties Comment by XL XL 2013-04-10T05:19:35Z 2013-04-10T05:19:35Z @Ulrich,Thank you a lot http://mathoverflow.net/questions/126627/existence-of-unknowable-algorithms/126631#126631 Comment by XL XL 2013-04-07T09:23:37Z 2013-04-07T09:23:37Z @Loic,what I know about the definition of Turing Machine is like:scholarpedia.org/article/Turing_machine.In the formal definition,The tape can extend in two direction,right,and left.I really don't know what &quot;Turing Machine with finite language and memory&quot; means.Sorry for that http://mathoverflow.net/questions/126627/existence-of-unknowable-algorithms/126631#126631 Comment by XL XL 2013-04-06T13:10:14Z 2013-04-06T13:10:14Z Actually,I do not understand what a Turing Machine with finite language means.And a Turing Machine usually has unlimited tape which is equivalent to memory of modern digital computer. http://mathoverflow.net/questions/126627/existence-of-unknowable-algorithms/126640#126640 Comment by XL XL 2013-04-06T11:21:38Z 2013-04-06T11:21:38Z @Goldstern,what do you think about the dovetailed computation? http://mathoverflow.net/questions/126627/existence-of-unknowable-algorithms/126631#126631 Comment by XL XL 2013-04-06T10:50:17Z 2013-04-06T10:50:17Z @Kahle and Loice,We may dovetail the computation,Is this not an algorithm to perform the required task? http://mathoverflow.net/questions/126627/existence-of-unknowable-algorithms/126640#126640 Comment by XL XL 2013-04-05T15:27:31Z 2013-04-05T15:27:31Z And one day,we will have proven RH,and constructed a algorithm. So,the question is really ambiguous http://mathoverflow.net/questions/126444/linkage-between-singularities-of-algebraic-varieties-and-continued-fractions/126445#126445 Comment by XL XL 2013-04-04T11:08:43Z 2013-04-04T11:08:43Z @Charles,it is only permissible to choose one post as answer,so I have chosen the one above.