User john bentin - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T07:14:01Z http://mathoverflow.net/feeds/user/7458 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/126171/which-compositions-have-these-sum-like-and-product-like-properties-on-the-positiv Which compositions have these sum-like and product-like properties on the positive reals? John Bentin 2013-04-01T11:24:12Z 2013-04-02T10:59:19Z <p>Consider a binary composition $\star:\Bbb R^2_{>0}\rightarrow \Bbb R_{>0}:(x,y)\mapsto x\star y$ with the following properties.</p> <p>(Commutativity)$\quad x\star y=y\star x\;$for all $x,y\in\Bbb R_{>0};$</p> <p>(Associativity)$\quad(x\star y)\star z=x\star(y\star z)\;$for all $x,y,z\in\Bbb R_{>0};$</p> <p>(Continuity)$\quad x\mapsto a\star x\;(x\in\Bbb R_{>0})\;$is continuous, for each $a\in \Bbb R_{>0};$</p> <p>(Upper monotony)$\quad x\mapsto a\star x\;(x\in\Bbb R_{>0})\;$is strictly increasing, for each $a\in \Bbb R_{>0};$</p> <p>(Surjectivity)$\quad \star\;$is unbounded above and below in $\Bbb R_{>0}.$</p> <p>These properties hold for addition and multiplication in $\Bbb R_{>0}.$ Do they hold for any other composition in $\Bbb R_{>0}$? If so, is an example of such a composition known? If not, is the above list redundant---that is, can any of the five conditions be weakened or removed?</p> http://mathoverflow.net/questions/125478/which-real-scalings-of-the-natural-numbers-approximately-accommodate-the-unbounde Which real scalings of the natural numbers approximately accommodate the unbounded powers of a noninteger? John Bentin 2013-03-24T18:43:53Z 2013-03-26T21:03:10Z <p>That is, what are the possible values of a real number $\lambda$ for which there exists a nonintegral real $\alpha >1$ such that, given any $\varepsilon >0,$ all but finitely many powers of $\alpha$ lie within $\varepsilon$ of an integral multiple of $\lambda$? As an example, we may take $\lambda =\sqrt 5$ with $\alpha =\frac 12+{\frac 12} \sqrt5.$</p> <p>Formally, define<code>\begin{align}\Lambda &amp;:=\\\{\lambda &amp;\in \Bbb R_{&gt;0}: (\exists\,\alpha\in \Bbb R_{&gt;1}{\setminus}\Bbb N)(\forall\,\varepsilon\in\Bbb R_{&gt;0})(\exists\,k\in\Bbb N)(\forall\, n\in\Bbb N_{&gt;k})(\exists\,m\in\Bbb N)\;\; |\alpha^n-\lambda m|&lt; \varepsilon\}.\end{align}</code>What are the known properties of $\Lambda$? In particular, can $\Lambda$ possess a rational or a transcendental element?</p> http://mathoverflow.net/questions/111173/how-long-can-a-primal-egyptian-fraction-be-that-optimally-approaches-unity How long can a primal egyptian fraction be, that optimally approaches unity? John Bentin 2012-11-01T16:40:41Z 2013-01-20T23:36:51Z <p>Thus, do there exist $n$ distinct primes whose summed reciprocals fall short of $1$ by the reciprocal of their product, for some $n\geqslant6$? I can get as far as $n=5$: $$\dfrac{1}{2}=1-\dfrac{1}{2},$$ $$\dfrac{1}{2}+\dfrac{1}{3}=1-\dfrac{1}{2\cdot3},$$ $$\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{7}=1-\dfrac{1}{2\cdot3\cdot7},$$ $$\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{7}+\dfrac{1}{43}=1-\dfrac{1}{2\cdot3\cdot7\cdot43},$$ $$\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{11}+\dfrac{1}{23}+\dfrac{1}{31}=1-\dfrac{1}{2\cdot3\cdot11\cdot23\cdot31}.$$ But I can't see the way beyond that. If there is a fraction for some $n\geqslant6$, then one may naturally ask: is there a bound on such $n$?</p> http://mathoverflow.net/questions/114245/i-know-that-you-know/114319#114319 Answer by John Bentin for I know that you know... John Bentin 2012-11-24T08:07:05Z 2012-11-24T08:07:05Z <p>The article <a href="http://plato.stanford.edu/entries/common-knowledge/" rel="nofollow">http://plato.stanford.edu/entries/common-knowledge/</a> is more extensive than wikipedia's. However, it doesn't deal with the issue of infinite information.</p> http://mathoverflow.net/questions/113879/does-strict-order-preservation-of-powerset-curtail-the-candidates-for-violation-o Does strict order-preservation of powerset curtail the candidates for violation of CH? John Bentin 2012-11-19T22:43:36Z 2012-11-20T14:51:59Z <p>Thus, let <code>$\mathrm{OPP}$</code> be the axiom that $|A|\lt|B| \Rightarrow |2^A|\lt|2^B|$ for any sets $A$ and $B$; and, for any ordinal $\alpha$, let <code>$\mathrm{CH}_\alpha$</code> be the hypothesis that $\aleph_\alpha=\frak c$ (so that <code>$\mathrm{CH}_1=\mathrm{CH}$</code>) . Define $S$ to be the set of those ordinals $\alpha\in\frak c$ such that <code>$\mathrm{CH}_\alpha$</code> does not provably (within <code>$\mathrm{ZFC}$</code>) violate <code>$\mathrm{ZFC}$</code> (for example, it is known that $\omega\backslash${<code>$0$</code>}$\subseteq S$ and $\omega \notin S$); and let $S'$ be the set of those $\alpha\in\frak c$ such that <code>$\mathrm{CH}_\alpha$</code> does not provably (within <code>$\mathrm{ZFC}$</code>) violate <code>$\mathrm{ZFC}$</code> <code>$\&amp;$</code> <code>$\mathrm{OPP}$</code>. Clearly $S'\subseteq S$. But is $S'= S$ ? Or are any elements of $S$ known to be not in $S'$ ?</p> <p>My guess is that <code>$\mathrm{OPP}$</code> can't restrict the possibilities for violations of <code>$\mathrm{CH}$</code> because the sets it talks about in the consequent---especially $2^B$--- are too big to be relevant; but I'm not sure of my footing here.</p> http://mathoverflow.net/questions/104606/what-is-known-about-the-polynomial-factorization-of-power-series What is known about the polynomial factorization of power series? John Bentin 2012-08-13T10:01:21Z 2012-08-16T16:19:15Z <p>Some power series factorize; $1+\sum_{n=1}^\infty x^n=\prod_{n=1}^\infty (1+x^{2^n})$ and $1+\sum_{n=1}^\infty x^{2n}/(2n+1)!=\prod_{x=1}^\infty (1+x^2/n^2\pi^2)$ for example; while others do not----in particular, $1+\sum_{n=1}^\infty x^n/n!$. Specifically the question is: what is known about necessary and sufficient conditions on the coefficients of the power series $1+\sum_{n=1}^\infty a_nx^n$ (assuming a positive radius of convergence) for it to factorize as $\prod_{n=1}^\infty (1+ b_nx+c_nx^2)$, where the coefficients $a_n$, $b_n$, and $c_n$ are real constants?</p> <p><strong>Remarks</strong> The Weierstrass factorization theorem doesn't directly answer the question, since here we allow Taylor series of non-entire functions (e.g. the first example above) and exclude non-polynomial factors. The roots of the question are algebraic: loosely stated, under what conditions can the fundamental theorem of algebra be pushed to infinity? A positive radius of convergence is supposed because formal power series with no functional meaning make me uncomfortable, and admitting them might complicate the answer. I previously posted an unanswered version of this question on Math.StackExchange.</p> <p><strong>Edit</strong> Thanks to juan and Douglas Zare for showing my error in the preamble of the question by citing the interesting factorization of the exponential series by Gingold et al. As Aaron Meyerowitz indicates, restriction to linear and quadratic factors leads respectively to quite different situations. Alexandre Eremenko answered the linear case. Allowing polynomial factors of unrestricted degree opens a wide vista. The rather simple power series $1$ includes among its polynomial product representations, valid for $|x| \lt r_0$, such forms as $$\left(1-\dfrac{p(x)}{r} \right) \prod_{n=0}^\infty \left[1+\left(\dfrac{p(x)}{r} \right)^{2^n} \right],$$ where $r_0$ is an arbitrary positive constant, $p(x)$ is an arbitrary polynomial, and $r$ is any real constant strictly exceeding $|p(x)|$ whenever $|x|\leqslant r_0$. Myriad expansions like these may be woven into any polynomial product expansion of any power series. I find this vista daunting and so will stick to the specified quadratic case, which remains unanswered so far.</p> http://mathoverflow.net/questions/103874/do-approximately-the-same-polynomials-have-approximately-the-same-roots/103879#103879 Answer by John Bentin for Do approximately the same polynomials have approximately the same roots? John Bentin 2012-08-03T16:18:30Z 2012-08-05T11:28:22Z <p>For the headline question, no. See, for example, Wilkinson's polynomial: <a href="http://en.wikipedia.org/wiki/Wilkinson" rel="nofollow">http://en.wikipedia.org/wiki/Wilkinson</a>'s_polynomial.</p> <p><strong>Edit:</strong> From the response to this answer, it seems that some expansion is needed. The OP asks two questions. In the the title, and repeated in the text, the question is: Do approximately the same polynomials have approximately the same roots? The answer to this question is no, as Wilkinson shows with his monic degree-20 polynomial, which has well separated roots of moderate size, namely the integers 1 to 20. A decrement of $2^{-23}$ in the coefficient of the degree-19 term (which is $-210$)---a proportional change of less than 1 in 1.7 billion---induces substantial changes in the roots: for example, the root pair $16.5\pm 0.5$ becomes the root pair $16.73024\pm 2.81262\mathrm i$ (to 5 d.p.). The second question asked by the OP is whether the coefficients-to-roots map is continuous, which was already answered (affirmatively) by others. </p> http://mathoverflow.net/questions/97501/is-there-a-good-comparative-study-of-the-banach-integral Is there a good comparative study of the Banach integral? John Bentin 2012-05-20T19:54:11Z 2012-05-21T12:45:43Z <p>The Banach integral is elegant in its definition, and I am intrigued as to why it is so rarely seen. Is it in practice difficult to calculate from the definition? And are there any other problems with it? I would also be interested to see examples of functions that are Banach-integrable, but not Lebesgue-integrable, and functions (if any) whose Banach and Lebesgue integrals over some interval are defined but different. <strong>Added in response to comments:</strong> The Banach integral is defined in <a href="http://books.google.co.uk/books?id=azS2ktxrz3EC&amp;pg=PA1167&amp;lpg=PA1167&amp;dq=banach+integral&amp;source=bl&amp;ots=MZbE-yvuoN&amp;sig=iZirxsujVkKuCkN0YlvpGqNRZkE&amp;hl=en&amp;sa=X&amp;ei=GQO6T4-HJsWP8gOlqey0Cg&amp;ved=0CNQDEOgBMAE#v=onepage&amp;q=banach%20integral&amp;f=false" rel="nofollow"><em>The Encyclopaedic Dictionary of Mathematics</em></a>. Like the familiar Riemann integral, it's an integral for real functions over a real interval, and not applied to general Banach spaces (except perhaps in some generalization that I know nothing about). </p> http://mathoverflow.net/questions/87627/fraktur-symbols-for-lie-algebras-mathfrakg-etc/87646#87646 Answer by John Bentin for Fraktur symbols for Lie algebras: $\mathfrak{g}$, etc. John Bentin 2012-02-06T09:53:42Z 2012-02-06T09:53:42Z <p>Fraktur was the standard font for cardinal numbers, for example, in writing the continuum hypothesis as $\aleph_1=\frak c$.</p> http://mathoverflow.net/questions/80461/which-squares-succeed-a-factorial Which squares succeed a factorial? John Bentin 2011-11-09T08:49:52Z 2011-11-09T08:49:52Z <p>One notices that $5^2=4!+1$, $11^2=5!+1$, and $71^2=7!+1$. Are there others?</p> http://mathoverflow.net/questions/73711/the-concept-of-duality/73746#73746 Answer by John Bentin for The concept of Duality John Bentin 2011-08-26T08:32:13Z 2011-08-27T07:03:04Z <p>Finite-dimensional linear spaces. A particular feature in this case is that the (algebraic) dual of a finite-dimensional vector space, namely the space of linear maps from the vector space into the base field, is isomorphic to the original space (since it is of the same dimensionality) but not <em>canonically</em> so. In contrast, the bi-dual (the dual of the dual) is <em>canonically</em> isomorphic to the original space, and so may be identified with it. </p> http://mathoverflow.net/questions/50952/can-sierpinskis-anisotropic-bicolouring-of-the-plane-assuming-the-continuum-hyp Can Sierpinski's anisotropic bicolouring of the plane, assuming the continuum hypothesis (CH), be extended to three dimensions? John Bentin 2011-01-02T21:02:41Z 2011-07-22T13:30:43Z <p>Sierpinski showed that, on the assumption of CH (in fact, equivalently to it), each point in the plane can be coloured (say) black or white so that every section of the plane parallel to the $x$ axis is "almost" white $-$ in the sense that all but countably many points of it are white $-$ while every section parallel to the $y$ axis is almost black. Equivalently, given CH, every line through the origin can be almost white, while every circle centred on the origin is almost black. Is there a corresponding result for three-dimensional space? For example, assuming CH, can we bicolour space so that every plane through the origin is almost white, while every sphere centred at the origin is almost black?</p> http://mathoverflow.net/questions/69242/what-bounds-the-ratio-of-summed-partial-harmonic-means-to-a-sum What bounds the ratio of summed partial harmonic means to a sum? John Bentin 2011-07-01T10:00:39Z 2011-07-09T20:36:51Z <p>For positive real $x_1$ , $x_2$ ,..., define their $k$th partial harmonic mean as $h_k = k/(1/x_1 +\cdots+1/x_k)$ for $k = 1, 2, ...,$ and let</p> <p>$\alpha=\sup_{x_1,x_2,... \geqslant0}\: \lim_{n\rightarrow\infty}\dfrac{h_1+\cdots+h_n}{x_1+\cdots+x_n}.$</p> <p>What is this bound, and for which $x_1$ , $x_2$ ,... is it attained? All I can do is show $\alpha \geqslant 2$ by taking $x_k = 1/k\;\;$ ($k = 1, 2, ...$).</p> http://mathoverflow.net/questions/35524/how-fast-can-the-base-bumping-function-in-goodsteins-theorem-grow How fast can the base-bumping function in Goodstein's theorem grow? John Bentin 2010-08-13T19:22:06Z 2011-05-30T14:16:01Z <p>In the usual presentation of Goodstein's theorem, the base is bumped up by the "add 1" function. Does the theorem still hold when we replace this function by a fast-growing one (e.g. Ackermann or busy beaver)? How far can we push this? For example, let's define $g_0(n)$ to be the number of Goodstein iterations needed to reach 0 when we start with base 2 and seed $n$ (so that $g_0(0)$ = 0). Then we can build a hierarchy of functions by defining $g_{k+1}(n)$ as the number of Goodstein iterations needed to reach 0 with seed $n$ and base-bumping function $g_k$ ($k$ = 0, 1, ...), continuing through the ordinals by diagonalization at each limit ordinal. Surely it's got to break down when we go past $\epsilon_0$, if not long before that! </p> http://mathoverflow.net/questions/58466/is-the-following-statement-a-correct-formulation-of-the-much-doubted-p-np-con Is the following statement a correct formulation of the (much doubted) P = NP conjecture? John Bentin 2011-03-14T19:31:01Z 2011-03-14T23:02:42Z <p>"Call a Turing machine $A$ a <em>d-machine</em> if, for some polynomial $p(\cdot)$, when $A$ starts with any input string, say of length $n$, in its alphabet on its (otherwise blank) tape, it will halt in a number of steps bounded by $p(n)$ with its tape either blank or bearing just a string of length $n$. Then each d-machine $A$ corresponds to a d-machine $B$ which, when started with any input string in its alphabet, will halt with a blank tape, if $A$ halts with a blank tape for every input the length of $B$'s input, and otherwise will halt with an output tape that, input to $A$, will lead to $A$ halting with a non-blank tape."</p> <p><strong>Interpretation:</strong> Input for $A$ codes candidate "solutions" while blank/non-blank output indicates just refutation/verification. For $B$, input marks only length while output codes an $A$-verifiable solution.</p> http://mathoverflow.net/questions/56379/how-aspherical-can-a-gomboc-be How aspherical can a Gömböc be? John Bentin 2011-02-23T10:44:12Z 2011-02-23T11:34:51Z <p>A <a href="http://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c" rel="nofollow">Gömböc</a> is a homogeneous massive convex solid that can rest on a horizontal plane in just two positions of equilibrium under gravity: one stable and the other unstable. How small a proportion of the volume bounded by its circumscribing sphere can a Gömböc occupy?</p> http://mathoverflow.net/questions/53134/what-sums-of-equal-powers-of-consecutive-natural-numbers-are-powers-of-the-same-o What sums of equal powers of consecutive natural numbers are powers of the same order? John Bentin 2011-01-24T21:42:03Z 2011-02-11T12:13:13Z <p>Trivially $n^1=n^1$, and everyone knows that $3^2+4^2=5^2$. <a href="http://mathoverflow.net/questions/53048" rel="nofollow">Denis Serre</a> quoted $3^3+4^3+5^3=6^3$ in a recent MathOverflow question (which prompted this one). Are any other examples known? </p> http://mathoverflow.net/questions/54961/webpages-for-specialized-communities/55000#55000 Answer by John Bentin for Webpages for specialized communities John Bentin 2011-02-10T07:26:35Z 2011-02-10T07:26:35Z <p>Foundations of mathematics: <a href="http://cs.nyu.edu/pipermail/fom/" rel="nofollow">http://cs.nyu.edu/pipermail/fom/</a></p> http://mathoverflow.net/questions/54426/could-the-cluster-set-of-a-rational-sequence-be-of-intermediate-cardinality-give Could the cluster set of a rational sequence be of intermediate cardinality (given not-CH)? John Bentin 2011-02-05T18:23:34Z 2011-02-05T18:33:05Z <p>It's easy to construct a sequence of rational numbers whose set of cluster (or accumulation or limit) points is of any finite cardinality, or is countably infinite, or has the cardinality of the continuum. Thus there appears no obvious barrier to forming a rational sequence with a cluster set of any cardinality up to $\mathfrak{c}$. Now suppose that we reject the continuum hypothesis. Is it consistent with $\mathrm{ZFC}$ that there could be an oracular sequence $(a_0 , a_1 , \dots ) \in \{0, 1\}^\mathbb{N}$ such that the cluster set of the sequence $(n_{2i - 1}/n_{2i} : i = 1, 2, \dots)$, where $n_0$ , $n_1$ , ... are the elements of $\{n \in \mathbb{N} : a_n = 1\}$ in natural order, has cardinality strictly between $\omega$ and $\mathfrak{c}$ ?</p> http://mathoverflow.net/questions/53967/what-patterns-have-been-measured-in-the-graph-of-the-number-of-two-prime-sum-repr What patterns have been measured in the graph of the number of two-prime-sum representations of even numbers? John Bentin 2011-02-01T09:22:47Z 2011-02-04T05:44:59Z <p>There are remarkable patterns of density in the graph <a href="http://upload.wikimedia.org/wikipedia/commons/7/7c/Goldbach-1000000.png" rel="nofollow">http://upload.wikimedia.org/wikipedia/commons/7/7c/Goldbach-1000000.png</a> plotting the number of representations of even numbers up to a million as the sum of two primes. Has anyone measured these patterns? The referenced graph was uploaded in 2006. Given the enormous data bases of primes, it would seem to be a matter of simple checking of sums to generate the graph, with a complexity that grows not much more than linearly with the size of the even number. Thus, with the growth of computer power, some advance on a million might have been achieved over the past few years. Has this been done? </p> http://mathoverflow.net/questions/44946/why-is-abelian-infrequently-capitalized/54247#54247 Answer by John Bentin for Why is "abelian" infrequently capitalized? John Bentin 2011-02-03T22:12:38Z 2011-02-03T22:12:38Z <p>A simple rule for this is that, if the adjective has a precise mathematical meaning, it should be lower-cased: for example, abelian (= commutative) group, euclidean metric, or gaussian (= normal) distribution. If the meaning is vaguer, referring to the method, style, or approach generally associated with the originator, then capitalizing is appropriate: for example, Bayesian statistics. If the word does not carry an adjectival ending---e.g. Hilbert (space)---it should always be capitalized.</p> http://mathoverflow.net/questions/51704/lower-bound-on-the-curvature-of-the-curves-on-m/51707#51707 Answer by John Bentin for Lower bound on the curvature of the curves on $M$ John Bentin 2011-01-10T22:37:54Z 2011-01-11T11:06:22Z <p>No. For example, on a 2-sphere in $\mathbb{R}^3$, you can draw a circle as small as you like. (Answer withdrawn: this of course answers the question with the bounding in the other direction)</p> http://mathoverflow.net/questions/49946/principal-curvatures-and-curvature-directions/49978#49978 Answer by John Bentin for Principal curvatures and curvature directions John Bentin 2010-12-20T17:15:25Z 2010-12-20T17:15:25Z <p>In general, the concept of "eigenvector" is slightly misleading. The fundamental concept is <em>eigenspace</em>. However, eigenspaces of dimension greater than 1 are generally considered pathological and rather a nuisance. Mostly we would rather have one-dimensional eigenspaces, and for these any convenient nonzero element---an eigenvector--- will serve as a representative. It would be pedantically correct but encumbering to insist on talking about one-dimensional eigenspaces rather than eigenvectors. The arbitrariness of eigenvectors becomes clearer when we really have to deal with a multidimensional eigenspace.</p> http://mathoverflow.net/questions/49197/what-are-some-examples-of-journals-that-will-accept-undergraduate-student-researc/49229#49229 Answer by John Bentin for What are some examples of journals that will accept undergraduate student research? John Bentin 2010-12-13T08:54:27Z 2010-12-13T08:54:27Z <p>You might first want to read the article by Arthur White &amp; Robin Wilson: "The hunting group", <em>The Mathematical Gazette</em> <strong>79</strong>, no. 484, March 1995, which is about the group theory of change ringing.</p> http://mathoverflow.net/questions/45177/what-polynomials-biject-from-mathbbn2-to-mathbbn What polynomials biject from $\mathbb{N}^{2}$ to $\mathbb{N}$? John Bentin 2010-11-07T15:42:10Z 2010-11-07T20:23:45Z <p>Perhaps there are none with integral coefficients; so let us admit rational coefficients. The map $(x, y) \mapsto x + \frac{1}{2}(x + y)(x + y + 1)$ is well known, and swapping $x$ and $y$ in the formula yields another, so we have two for starters.</p> http://mathoverflow.net/questions/42929/suggestions-for-good-notation/43036#43036 Answer by John Bentin for Suggestions for good notation John Bentin 2010-10-21T11:15:56Z 2010-10-21T11:15:56Z <p>Using $(a, b, ... )$ is handy to denote a <em>column</em> vector, which is the transpose of the row vector $[a, b, ... ]$, especially in linear text. Correspondingly, all displayed matrices should be written with brackets, not parentheses. This notation agrees with the usual identification of coordinates with column vectors.</p> http://mathoverflow.net/questions/40005/generalizing-a-problem-to-make-it-easier/40058#40058 Answer by John Bentin for Generalizing a problem to make it easier John Bentin 2010-09-26T21:58:44Z 2010-09-26T21:58:44Z <p>Here's an example in planar euclidean geometry. Consider an equilateral triangle of side $a$ and a general point in the plane distant $b$, $c$, and $d$ from the respective vertices. Then</p> <p>$3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2$.</p> <p>This is an an awful slog to get by planar trigonometry. Even harder to do by trig in three dimensions is the corresponding result for the regular tetrahedron. However, it's easy to get the $(n - 1)$-dimensional result for a regular $(n - 1)$-dimensional simplex of side $d_0$, with vertex distances $d_1$ ,..., $d_n$ :</p> <p>$n(d_0^4 + ... + d_n^4) = (d_0^2 + ... + d_n^2)^2$.</p> <p>You can do this by embedding the euclidean $(n - 1)$-dimensional space as the hyperplane of points $(x_1 ,..., x_n)$ in euclidean $n$-space such that $x_1 + ... +x_n = d_0/\sqrt2$. The vertices of the simplex can then be represented as the points $(d_0/\sqrt2)(1, 0 ,..., 0)$, ... , $(d_0/\sqrt2)(0 ,..., 0, 1)$ in the hyperplane, and the result drops out in a few lines. </p> http://mathoverflow.net/questions/31016/a-sequential-optimizing-task A sequential optimizing task John Bentin 2010-07-08T10:13:50Z 2010-09-04T09:38:24Z <p>Find distinct positive real numbers $x_1$ , $x_2$ , ... of least supremum such that, for each positive integer $n$, any two of 0, $x_1$ , $x_2$ ,..., $x_n$ differ by $1/n$ or more.</p> <p>Note that the hurdle term $1/n$ is optimal in the sense that any replacement for it would need to stay below a constant multiple of it to allow a finite supremum. By a nonconstructive proof, there is a unique solution minimal with respect to the lexicographic ordering of real sequences; so a constructed solution (while eluding me) doesn't seem impossible. Although I haven't seen this problem anywhere, it looks too simple not to have been posed before. Any pointers would be welcome. </p> http://mathoverflow.net/questions/31016/a-sequential-optimizing-task/37706#37706 Answer by John Bentin for A sequential optimizing task John Bentin 2010-09-04T09:38:24Z 2010-09-04T09:38:24Z <p>Belated thanks to Kevin O'Bryant for his pointer to discrepancy theory. This led me eventually to a source where the problem is solved: See Theorem 6.7 in Harald Niederreiter's book <em>Random Number Generation and Quasi Monte Carlo Methods</em> (SIAM 1992). The logarithmic sequence described by Tracy Hall is due to Rusza and is indeed optimal. </p> http://mathoverflow.net/questions/31016/a-sequential-optimizing-task/32753#32753 Answer by John Bentin for A sequential optimizing task John Bentin 2010-07-21T06:55:09Z 2010-07-21T16:04:45Z <p>I'll now put forward my candidate solution to the problem. It clearly satisfies the hurdle condition, but I can't prove its optimality. To get a handle on the algorithm, let's represent $x_1$ , $x_2$ , … as hotel guests numbered accordingly. Recall Hilbert's hotel, where the unfortunate guests were ever being shunted from their room to a higher-numbered room. The hotel in this case, rather than having a countably infinite number of discrete rooms, is a continuum of “rooms”, represented by the points of a closed bounded real interval; [0 <strong>,</strong> 1.4] is big enough, as it turns out. The “guests”, numbered 1, 2, … , are movable tags, each assigned to a rational point in the interval. Unlike Hotel Hilbert, which is always full, this hotel starts empty apart from the proprietor who resides permanently at 0, and the guests arrive one by one in the order 1, 2, … . The proprietor operates the strict rule that, when there are a total of $m$ guests in the hotel, there must be a space of at least $1/m$ between the residents (including himself), for $m$ = 1, 2, … . Guest 1 is assigned to the point 1. Guest 2 is placed at 1/2. When guest 3 arrives, she is put at 1/3, while guests 1 and 2 are moved up by a distance 2/3 – 2/(3 + 1) = 1/6. When guest 4 comes, he is allotted to 1/2 + 1/6 + 1/4 = 11/12. The general rule is as follows. For $k$ = 1, 2, … , after $2^k$ guests have been accommodated, the next $2^k$ are put consecutively into the $2^k$ spaces between the proprietor and the first $2^k$ guests, in left-to-right order: An odd-numbered arrival, say guest $2n$ – $1$, is assigned to the point below her right-hand neighbour (guest $n$) that is 1/($2n$ – $1$) above her left-hand neighbour, while all the guests to her right are moved up by a distance 2/($2n$ – 1) – $1/n$; when the next, even-numbered, guest arrives (guest $2n$), he can just go to the midpoint of the next space up, at a distance 1/$2n$ above his left-hand neighbour (guest $n$), and mercifully no resident has to move until the next (odd-numbered) guest arrives. The result is that the guests, now identified with their limiting room positions, are each the sum of two parts: The first part is a sum of a finite number of distinct fractions of the form $1/n$, while the second is an infinite series whose terms are of the form 1/$n$($2n$ – 1), where $n$ is a positive integer. Generally an infinite number of terms of the latter type are absent; only in the case of $x_1$ is the series free of gaps, and then the first “sum” has only one term. Thus $x_1$ = 1 + ∑{1/$j$($2j$ – 1) : $j$ = 2, 3, …} = ln 4, and this is also the supremum of the sequence.</p> <p>An alternative characterization, suggested by Gerhard Paseman, is as follows: For j from 2^(k-1)+1 to 2^k, you will arrange to place guest (2j-1) to the left of guest j and guest 2j to the right of guest j. Since space to the left of guest j has previously been guaranteed to be 1/j from his/her lefthand neighbor, guest (2j-1) needs more since it needs 1/(2j-1) space to his/her left and right. So add the difference delta_j = (2/(2j-1) – 1/j) to the left of guest j and shift guest j and every guest on the right of guest j by this difference delta_j. Since guest 2j does not need more than 1/j = (2/2j) space, no such adjustment is needed for guest 2j. This gives guest 1 infinitely many adjustments; 1 ends up at place 1 + sum(j > 1) delta_j = 1 + sum(j > 1) {2 [ 1/(2j-1) - 1/2j ] } = 2 ln(2).</p> http://mathoverflow.net/questions/131435/why-dont-more-mathematicians-improve-wikipedia-articles/131515#131515 Comment by John Bentin John Bentin 2013-05-23T13:53:25Z 2013-05-23T13:53:25Z @Douglas Zare: Good editing! However, there is still room for small improvements in the punctuation (which I don't have the rep points to do). (1) In line 1, commas are needed after &quot;Wikipedia&quot; and &quot;small&quot;, with the comma after &quot;good&quot; preferably deleted. (2) In line 3, a comma should follow &quot;edits&quot;. (3) The comma splice in line 6 needs attention, perhaps by replacing &quot;click&quot; with &quot;by clicking&quot;. http://mathoverflow.net/questions/131047/how-to-determine-the-number-of-a-cube-within-a-bigger-cube Comment by John Bentin John Bentin 2013-05-18T17:42:07Z 2013-05-18T17:42:07Z Your latter cube is indeed a cube, but the first-mentioned &quot;cube&quot; is not. Such a rectangular body is called a <i>cuboid</i>. http://mathoverflow.net/questions/129759/modern-mathematical-achievements-accessible-to-undergraduates Comment by John Bentin John Bentin 2013-05-05T20:05:54Z 2013-05-05T20:05:54Z Do you mean the <i>symmetric</i> matrix associated with the quadratic form? http://mathoverflow.net/questions/129110/number-of-distinct-sums-of-integers Comment by John Bentin John Bentin 2013-04-30T06:57:43Z 2013-04-30T06:57:43Z You talk about &quot;subsets&quot; of $S$, but then you have discarded the multiset structure of $S$. Do you mean <i>sub-multisets</i> of $S$? http://mathoverflow.net/questions/126171/which-compositions-have-these-sum-like-and-product-like-properties-on-the-positiv/126202#126202 Comment by John Bentin John Bentin 2013-04-02T08:50:31Z 2013-04-02T08:50:31Z This is a beautifully clear answer to the question I <i>should have</i> asked. It all seems so simple now! Just one query: In the second line of the proof of lemma 2, should the initial term of the inequalities be $y$, not $x$? http://mathoverflow.net/questions/126171/which-compositions-have-these-sum-like-and-product-like-properties-on-the-positiv Comment by John Bentin John Bentin 2013-04-01T17:28:30Z 2013-04-01T17:28:30Z @Todd: Yes, your example works well with, say, $f(x)=x^2$. I would happily take this as the &quot;accepted answer&quot;. http://mathoverflow.net/questions/126171/which-compositions-have-these-sum-like-and-product-like-properties-on-the-positiv Comment by John Bentin John Bentin 2013-04-01T17:13:24Z 2013-04-01T17:13:24Z @Gerry, @Gerald: I think that Gerry's composition satisfies all the conditions except upper monotony. Take, for example, $a=1/\mathrm e$. http://mathoverflow.net/questions/125478/which-real-scalings-of-the-natural-numbers-approximately-accommodate-the-unbounde/125600#125600 Comment by John Bentin John Bentin 2013-03-26T08:45:28Z 2013-03-26T08:45:28Z Thank you very much for this answer. Just a few queries concerning your 4th paragraph: (1) I can't read &quot;$\sum_1^{\infty}\|\frac{1][\sqrt{5}} \Phi^n\| \lt \infty$&quot;; (2) I guess the equality sign in &quot;$\|x=y\| \le \|x\|+\|y\|$&quot; should be a plus sign; (3) By &quot;$\lambda$ is any real $\lambda=\frac{a}{\sqrt{5}}+b+c\Phi$&quot; do you mean &quot;$\lambda$ is any real of the form $\lambda=\frac{a}{\sqrt{5}}+b+c\Phi$ with integral $a$, $b$, and $c$&quot;? http://mathoverflow.net/questions/125478/which-real-scalings-of-the-natural-numbers-approximately-accommodate-the-unbounde/125487#125487 Comment by John Bentin John Bentin 2013-03-25T14:19:59Z 2013-03-25T14:19:59Z @Douglas: Thank you for all your work on this. However, it seems to me that your answer addresses a question different from the one I posed, namely with the order of quantification of $\alpha$ and $\varepsilon$ inverted. In my question, the order is expressed as $(\exists\,\alpha\in \Bbb R_{&gt;1}{\setminus}\Bbb N)(\forall\,\varepsilon\in\Bbb R_{&gt;0})$, while your $\alpha$ is dependent on a previously chosen $\epsilon$. http://mathoverflow.net/questions/125478/which-real-scalings-of-the-natural-numbers-approximately-accommodate-the-unbounde/125487#125487 Comment by John Bentin John Bentin 2013-03-24T23:00:37Z 2013-03-24T23:00:37Z Sorry to be so dense, but I can't see how a method to construct a number whose powers, modulo $1$, are uniformly distributed over $(0, 1)$ can be applied to finding one whose power sequence converges, modulo $1$, to $0$. http://mathoverflow.net/questions/125478/which-real-scalings-of-the-natural-numbers-approximately-accommodate-the-unbounde/125487#125487 Comment by John Bentin John Bentin 2013-03-24T20:37:08Z 2013-03-24T20:37:08Z Thank you. Could you please give me a reference (or suitable search terms) for &quot;the usual construction of non-integral real numbers greater than 1 whose powers are close to integers&quot;? http://mathoverflow.net/questions/118254/usage-of-set-theory-in-undergraduate-studies/118261#118261 Comment by John Bentin John Bentin 2013-01-07T13:49:36Z 2013-01-07T13:49:36Z +1, especially for &quot;We should always, always distinguish a function $f$ from its value $f(x)$ at $x$&quot;. But I don't get your next point: If we define (say) $y$ := $x^2+1$, then what is wrong with saying that (the variable) $y$ depends on (the variable) $x$? http://mathoverflow.net/questions/117735/approximating-erf-by-tanh/117743#117743 Comment by John Bentin John Bentin 2013-01-01T09:06:38Z 2013-01-01T09:06:38Z Thanks, Aryeh. I was confusing erf with a cumulative distribution function. http://mathoverflow.net/questions/117735/approximating-erf-by-tanh/117743#117743 Comment by John Bentin John Bentin 2013-01-01T08:30:20Z 2013-01-01T08:30:20Z Yes. But the function $f$ here doesn't tend to zero at <i>both</i> ends of the interval $[0, \infty)$, because $f(0)=\mathrm{erf}\, 0-\mathrm{tanh}\,0=\frac{1}{2}-0=\frac{1}{2}$. http://mathoverflow.net/questions/117735/approximating-erf-by-tanh/117743#117743 Comment by John Bentin John Bentin 2012-12-31T21:20:50Z 2012-12-31T21:20:50Z Sorry, I can only see that, on your supposition, $f'$ has $2$ (or more) positive zeros. But anyway that's enough for the rest of the argument to work.