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Algebra i Logika. Seminar
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 Algebra Logika: Year: Volume: Issue: Page: Find

 Representation of Lie algebras in radical ringsL. A. Bokut' 5 On an error in the paper of I. I. Žegalkin “The solvability problem on finite classes”V. F. Kostyrko 31 On solvable groups of finite rankM. I. Kargapolov 37 On the equation $zxyx^{-1}y^{-1}z^{-1}= aba^{-1}b^{-1}$ in a free groupA. I. Mal'tsev 45


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• ### The maximum mass of dark matter existing in compact stars based on the self-interacting fermionic model(1805.01314)

May 3, 2018 nucl-th, astro-ph.CO
By assuming that only gravitation acts between dark matter (DM) and normal matter (NM), we studied DM admixed neutron stars (DANSs) using the two-fluid TOV equations. The NM and DM of compact stars are simulated by the relativistic mean field (RMF) theory and non-self-annihilating self-interacting fermionic model, respectively. The effects of the particle mass of fermionic DM $m_f$ and the interaction strength parameter $y$ on the properties of DANSs are investigated in detail. $m_f$ and $y$ are considered as the free parameters due to the lack of information about the particle nature of DM so far. For a DANS, we suggest a simple universal relationship $M_D^{\max}=(0.267 y +0.627-3.21\frac{M_N}{\M_{\odot}})( \frac{1\GeV}{{m_f}})^2 \M_{\odot}$ for $y>100$, where $M_D^{\max}$ is the maximum mass of DM existing in DANSs and $M_N$ is the mass of the neutron star without DM. For free fermion DM model ($y$=0), the relationship becomes $M_D^{\max}=(0.627-0.027\frac{M_N^2}{\M_{\odot}^2}) ( \frac{1\GeV}{{m_f}})^2 \M_{\odot}$. The radius of DM $R_D$ shows a linear relationship with $M_D^{\max}$ in DANSs, namely $R_D=(7.02 \frac{M_D^{\max}}{ \M_{\odot}}+1.36)$~km. These conclusions are independent of the different NM EOSs from RMF theory. Such a kind of universal relationship connecting the nature of DM particle and mass of stars might shed light on the constraining the nature of the DM by indirect method.
• ### PICOSEC: Charged particle timing at sub-25 picosecond precision with a Micromegas based detector(1712.05256)

March 14, 2018 physics.ins-det
The prospect of pileup induced backgrounds at the High Luminosity LHC (HL-LHC) has stimulated intense interest in developing technologies for charged particle detection with accurate timing at high rates. The required accuracy follows directly from the nominal interaction distribution within a bunch crossing ($\sigma_z\sim5$ cm, $\sigma_t\sim170$ ps). A time resolution of the order of 20-30 ps would lead to significant reduction of these backgrounds. With this goal, we present a new detection concept called PICOSEC, which is based on a "two-stage" Micromegas detector coupled to a Cherenkov radiator and equipped with a photocathode. First results obtained with this new detector yield a time resolution of 24 ps for 150 GeV muons, and 76 ps for single photoelectrons.
• ### Note on fast spinning neutron stars with unified equations of states(1610.08770)

Jan. 20, 2018 nucl-th, astro-ph.SR, astro-ph.HE
For the propose of confronting updated pulsar observations with developed neutron star equation of states (EoSs), we employ four unified EoSs for both the core and the crust, namely BCPM, BSk20, BSk21, Shen-TM1, as well as two non-unified EoSs widely used in the literature, i.e. APR and GM1 EoS, which are commonly matched with the Negele-Vautherin and the Baym-Pethick-Sutherland crust EoS. All the core EoSs satisfy the recent observational constraints of the two massive pulsars whose masses are precisely measured. We show that the NS mass-equatorial radius relations are slightly affected by the smoothness at the core-crust matching interface. Moreover, the uncertainties in the crust EoS and the matching interface bring insignificant changes, even at maximally rotating (Keplerian) configurations. We also find that for all four unified EoSs, rotation can increase the star's gravitational mass up to $18\%-19\%$ and the equatorial radius by $29\%-36\%$, which are consistent with the previous calculations using non-unified EoSs. For stars as heavy as 1.4 M$_{\odot}$, the radius increase is more pronounced, reaching $41\%-43\%$, i.e. $5-6$ km. Moreover, by confronting the results using unified EoSs, which give the correct empirical values at saturation density, with two controversial determinations of the radius for the fast rotator 4U 1608-52, we address that a small radius may be better justified for this source.
• ### A key factor to the spin parameter of uniformly rotating compact stars: crust structure(1408.1654)

We study the key factor to determine the dimensionless spin parameter $j\equiv cJ/(GM^2)$ of different kinds of uniformly rotating compact stars, including the traditional neutron stars, hyperonic neutron stars, and hybrid stars, and check the reliability of the results on various types of equations of state of dense matter. The equations of state from the relativistic mean field theory and the MIT bag model are adopted to simulate compact stars. Numerical calculations of rigidly rotating neutron stars are performed using the RNS code in the framework of general relativity by solving the Einstein equations for stationary axis-symmetric spacetime. The crust structure of compact stars is found to be a key factor to determine the maximum value of the spin parameter $j_{\rm max}$. For the stars with inclusion of the crust, $j_{\rm max}\sim 0.7$ is sustained for various kinds of compact stars with $M>0.5 M_{\odot}$, and is found to be insensitive to the mass of star and selected equations of state. For the traditional neutron stars, hyperonic neutron stars and hybrid stars without crust structure, the value $j_{\rm max}$ lies in the range of $[0.7, 1.0]$. Thus, not $j>0.7$ but $j>1$ could be treated as the candidate criterion to distinguish the strange quark stars from the other kinds of compact stars. Furthermore, a universal formula $j=0.48(f/f_K)^3-0.42(f/f_K)^2+0.63(f/f_K)$ is suggested to calculate the spin parameter at any rotational frequency for all kinds of compact stars with crust structure and $M>0.5M_{\odot}$.
• ### Evidence for Multiple Chiral Doublet Bands in $^{133}$Ce(1302.0401)

April 28, 2013 nucl-ex, nucl-th
Two distinct sets of chiral-partner bands have been identified in the nucleus $^{133}$Ce. They constitute a multiple chiral doublet (M$\chi$D), a phenomenon predicted by relativistic mean field (RMF) calculations and observed experimentally here for the first time. The properties of these chiral bands are in good agreement with results of calculations based on a combination of the constrained triaxial RMF theory and the particle-rotor model.
• ### The first candidate for chiral nuclei in the $A\sim80$ mass region: $^{80}$Br(1107.5109)

July 26, 2011 nucl-ex, nucl-th
Excited states of $^{80}$Br have been investigated via the $^{76}$Ge($^{11}$B, $\alpha$3n) and $^{76}$Ge($^{7}$Li, 3n) reactions and a new $\Delta I$ = 1 band has been identified which resides $\sim$ 400 keV above the yrast band. Based on the experimental results and their comparison with the triaxial particle rotor model calculated ones, a chiral character of the two bands within the $\pi g_{9/2}\otimes \nu g_{9/2}$ configuration is proposed, which provides the first evidence for chirality in the $A\sim80$ region.
• ### Chirality in odd-$A$ Rh isotopes within triaxial particle rotor model(1102.2271)

Feb. 11, 2011 nucl-ex, nucl-th
Adopting the fully quantal triaxial particle rotor model, the candidate chiral doublet bands in odd-$A$ nuclei $^{103}$Rh and $^{105}$Rh with $\pi g_{9/2}^{-1}\otimes\nu h^{2}_{11/2}$ configuration are studied. For the doublet bands in both nuclei, agreement is excellent for the observed energies over entire spin range and $B(M1)/B(E2)$ at higher spin range. The evolution of the chiral geometry with angular momentum is discussed in detail by the angular momentum components and their probability distributions. Chirality is found to change from chiral vibration to nearly static chirality at spin $I=37/2$ and back to another type of chiral vibration at higher spin. The influence of the triaxial deformation $\gamma$ is also studied.
• ### Chiral geometry of higher excited bands in triaxial nuclei with particle-hole configuration(1008.4865)

Aug. 28, 2010 nucl-th
The lowest six rotational bands have been studied in the particle-rotor model with the particle-hole configuration $\pi h^1_{11/2}\otimes\nu h^{-1}_{11/2}$ and different triaxiality parameter $\gamma$. Both constant and spin-dependent variable moments of inertial (CMI and VMI) are introduced. The energy spectra, electromagnetic transition probabilities, angular momentum components and $K$-distribution have been examined. It is shown that, besides the band 1 and band 2, the predicted band 3 and band 4 in the calculations of both CMI and VMI for atomic nuclei with $\gamma=30^\circ$ could be interpreted as chiral doublet bands.
• ### Candidate MKiD nucleus 106Rh in triaxial relativistic mean-field approach with time-odd fields(0905.3785)

May 23, 2009 nucl-th
The configuration-fixed constrained triaxial relativistic mean-field approach is extended by including time-odd fields and applied to study the candidate multiple chiral doublets (MKiD) nucleus 106Rh. The energy contribution from time-odd fields and microscopical evaluation of center-of-mass correction as well as the modification of triaxial deformation parameters beta, gamma due to the time-odd fields are investigated. The contributions of the time-odd fields to the total energy are 0.1-0.3 MeV and they modify slightly the gamma values. However, the previously predicted multiple chiral doublets still exist.
• ### Chirality in odd-$A$ nucleus $^{135}$Nd in particle rotor model(0812.4597)

Dec. 25, 2008 nucl-ex, nucl-th
A particle rotor model is developed which couples several valence protons and neutrons to a rigid triaxial rotor core. It is applied to investigating the chirality in odd-$A$ nucleus $^{135}$Nd with $\pi h_{11/2}^2\otimes\nu h^{-1}_{11/2}$ configuration for the first time in a fully quantal approach. For the two chiral sister bands, the observed energies and the $B(M1)$ and $B(E2)$ values for the in-band as well as interband transitions are reproduced excellently. Root mean square values of the angular momentum components and their probability distributions are used for discussing in detail the chiral geometry of the aplanar rotation and its evolution with angular momentum. Chirality is found to change from a soft chiral vibration to nearly static chirality at spin $I=39/2$ and back to another type of chiral vibration at higher spin.
• ### Signature splitting in $^{173}$W with triaxial particle rotor model(0806.2407)

June 15, 2008 nucl-th
A particle rotor model with a quasi-neutron coupled with a triaxially deformed rotor is applied to study signature splitting for bands with intruder orbital $\nu7/2^{+} [633]$ and non-intruder orbital $\nu5/2^{-}[512]$ in $^{173}$W. Excellent agreement with the observed energy spectra has been achieved for both bands. Signature splitting for band $\nu7/2^{+} [633]$, and band $\nu5/2^{-}[512]$ before the onset of signature inversion, is satisfactorily reproduced by introducing the $\gamma$ degree of freedom. The phase and amplitude of signature splitting in band $\nu5/2^{-}[512]$ is attributed to strong competition between $2f_{7/2}$ and $1h_{9/2}$ components. However, the explanation of signature inversion in band $\nu5/2^{-}[512]$ self-consistently is beyond the present one quasi-neutron coupled with a triaxially deformed rotor.
• ### Chiral bands for quasi-proton and quasi-neutron coupling with a triaxial rotor(nucl-th/0703047)

March 15, 2007 nucl-ex, nucl-th
A particle rotor model (PRM) with a quasi-proton and a quasi-neutron coupled with a triaxial rotor is developed and applied to study chiral doublet bands with configurations of a $h_{11/2}$ proton and a $h_{11/2}$ quasi-neutron. With pairing treated by the BCS approximation, the present quasi-particle PRM is aimed at simulating one proton and many neutron holes coupled with a triaxial rotor. After a detailed analysis of the angular momentum orientations, energy separation between the partner bands, and behavior of electromagnetic transitions, for the first time we find aplanar rotation or equivalently chiral geometry beyond the usual one proton and one neutron hole coupled with a triaxial rotor.
• ### Examining the chiral geometry in 104Rh and 106Rh(nucl-th/0702002)

Feb. 1, 2007 nucl-th
The criteria for chiral doublet bands based on one particle and one hole coupled to a triaxial rotor have been summarized. Two representative cases in $A\sim100$ odd-odd nuclei, nearly degenerate $\Delta I =1$ doublet bands in $^{104}$Rh and $^{106}$Rh are checked against these chiral criteria. It is shown that $^{106}$Rh possesses better chiral geometry than $^{104}$Rh, although the energy near degeneracy is achieved in $^{104}$Rh in comparison with the constant energy separation of doublet bands in $^{106}$Rh.
• ### Doublet bands in $^{126}$Cs in the triaxial rotor model coupled with two quasiparticles(nucl-th/0702003)

Feb. 1, 2007 nucl-th
The positive parity doublet bands based on the $\pi h_{11/2}\otimes\nu h_{11/2}$ configuration in $^{126}$Cs have been investigated in the two quasi-particles coupled with a triaxial rotor model. The energy spectra $E(I)$, energy staggering parameter $S(I)=[E(I)-E(I-1)]/2I$, $B(M1)$ and $B(E2)$ values, intraband $B(M1)/B(E2)$ ratios, $B(M1)_{\textrm{in}}/B(M1)_{\textrm{out}}$ ratios, and orientation of the angular momentum for the rotor as well as the valence proton and neutron are calculated. After including the pairing correlation, good agreement has been obtained between the calculated results and the data available, which supports the interpretation of this positive parity doublet bands as chiral bands.
• ### Practical Decoy State for Quantum Key Distribution(quant-ph/0503005)

May 10, 2005 quant-ph
Decoy states have recently been proposed as a useful method for substantially improving the performance of quantum key distribution. Here, we present a general theory of the decoy state protocol based on only two decoy states and one signal state. We perform optimization on the choice of intensities of the two decoy states and the signal state. Our result shows that a decoy state protocol with only two types of decoy states--the vacuum and a weak decoy state--asymptotically approaches the theoretical limit of the most general type of decoy state protocols (with an infinite number of decoy states). We also present a one-decoy-state protocol. Moreover, we provide estimations on the effects of statistical fluctuations and suggest that, even for long distance (larger than 100km) QKD, our two-decoy-state protocol can be implemented with only a few hours of experimental data. In conclusion, decoy state quantum key distribution is highly practical.

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# Lower bounds for chromatic number of a graph

I am trying to find a good lower bound for chromatic number of one family of graphs. I'm curious what are the known lower bounds for chromatic number. There are two obvious: $\chi(G) \geq \omega(G)$ and $\chi(G) \geq n / \alpha(G)$. One can also employ fancy Lovasz theta-function.

Chromatic number of Kneser graph can be obtained by the means of topological combinatorics (particularly, using Borsuk-Ulam theorem). But it is not clear if this method is general enough.

So, what are the more or less general techniques for lowerbounding chromatic number?

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Just one graph? Can't you just explicitly compute the chromatic polynomial? –  Qiaochu Yuan Jul 29 '10 at 17:48
I'm sorry, one family of graphs. Anyway, they are way too big. –  ilyaraz Jul 29 '10 at 18:15
The theta function can be found in polynomial time, unlike the two other bounds you suggest. As Suresh commented, it can be expressed as the solution of a semi-definite program, and can therefore be found using linear algebra. Moreover, theta is also the Shannon capacity of the (channel corresponding to the) graph. So I'd hesitate in dismissing it as "fancy"... –  András Salamon Jul 30 '10 at 10:35

Lovasz's topological lower bounds on chromatic number were extended by Babson and Kozlov -- they have a series of articles, all available on the arXiv. A nice place to start is "Complexes of Graph Homomorphisms."

There are also lower bounds on chromatic number coming from statistical physics -- see Brightwell and Winkler's "Graph homomorphisms and long range action."

All that said, it seems that one has to be a bit lucky for these methods to be applicable. See for example, "Neighborhood complex of a random graph," where it shown that topological lower bounds are very far from chromatic number for almost all graphs.

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Thank you. And do you happen to know any general techniques to estimate theta-function? –  ilyaraz Jul 29 '10 at 19:43
I don't know any general techniques to estimate the theta-function. However there are other, easier to compute, spectral lower bounds on chromatic number. In particular, see Theorem 5.2 of these notes: tinyurl.com/297vtgk –  Matthew Kahle Jul 29 '10 at 22:23
Since the theta function can be written as the solution to a semidefinite program (www-math.mit.edu/~goemans/…) maybe that can help in your problem ? –  Suresh Venkat Jul 30 '10 at 2:24
For details about how to compute the theta function, see L. Lovász: Semidefinite programs and combinatorial optimization, in: Recent Advances in Algorithms and Combinatorics 11, Springer (2003), 137-194. cs.elte.hu/~lovasz/semidef.ps –  András Salamon Jul 30 '10 at 10:41
Matthew, thank you for these references. –  András Salamon Aug 1 '10 at 2:28

Hoffman's bound states that $\chi(G) \geq 1 - \frac{\lambda_1(G)}{\lambda_n(G)}$ where $\lambda_1(G), \lambda_n(G)$ denote the largest and smallest eigenvalues of the adjacency matrix of $G$. (Note that $\lambda_n(G)$ is negative.)

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For a quick discussion with references, see Yonatan Bilu, Tales of Hoffman: Three extensions of Hoffman's bound on the graph chromatic number, Journal of Combinatorial Theory, Series B 96 (2006) dx.doi.org/10.1016/j.jctb.2005.10.002 –  András Salamon Jul 30 '10 at 11:22
Thank you. Miserably, this bound is too rough. Anyway, it is nice. –  ilyaraz Jul 30 '10 at 17:08

I discussed this issue with Stefano Gualandi a few days ago while setting up this page: http://sites.google.com/site/graphcoloring/vertex-coloring

As evidenced from the results in the web page, for arbitrary graphs of large size the best lower bound is often found by the fractional chromatic number.

This number is found by solving the linear relaxation of the integer programming formulation of the chromatic number problem. You find it, for example, here: http://www.optimization-online.org/DB_HTML/2005/12/1257.html eq. 11-13.

Computationally, you have to find a collection of maximal stable sets that covers all vertices (some may be included more than once), and then solve the corresponding LP relaxation of the chromatic number problem. The computational burden lays in the fact that the collection contains an exponential number of elements. The fractional chromatic number problem is also NP-hard.

However, there are techniques that can avoid generating all maximal stable sets. See for example: http://www.optimization-online.org/DB_HTML/2010/03/2568.html Alternatively, some stable sets chosen heuristically can be used but the result of the LP will not be the fractional chromatic number in this case.

If you need computational results and cannot find a way to obtain them yourself you can write to Stefano (who has all programs).

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It seems interesting. The first graph of my family is of 560 vertices. Is fractional chromatic bound tractable on such graph sizes? –  ilyaraz Jul 30 '10 at 17:10
But computing the fractional chromatic number is also NP-hard. –  Chris Godsil Jul 30 '10 at 17:51
In this article, table 4, optimization-online.org/DB_HTML/2010/03/2568.html there are results for a time bound of 3600 seconds. Whether the fractional chromatic number problem for a graph of 560 vertices can be solved within this time bound depends on the type of graph. From the table it seems that denser graphs are more likely to be solved earlier. –  Marco Chiarandini Aug 1 '10 at 16:25

Short of the two lower bounds you gave, there is only one other I could think of, and it may be hard to use, depending on what your family of graphs is... It is the following : Build a coloring of your graph by first taking a maximum independent set, then taking another maximum independent set in the rest of the graph, then taking another one, ....

You will obtain this way a non-optimal coloring, but it can be proved (consequence of the log(n) approximation of the set cover problem : http://en.wikipedia.org/wiki/Set_cover_problem ), that the number of classes you finally use it at most log(n) times the Chromatic number.

Well, if your graphs are nice enough for you to compute explicitly a coloring this way using k classes, you will be able to say $\chi(G)\geq k/log(n)$. Of course, there is no meaning in doing this if you do not expect the chromatic number to be very large !

Could you give some other information on your family of graphs, or are they too "hand-made" for that ? :-)

Nathann

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I don't understand the link with Set Cover? You are essentially proposing a greedy algorithm, and this is known to require up to $n$ colours for crowns with $n^2$ vertices (which are bipartite). For crowns, this would yield a lower bound of $n/\log n$ instead of the actual value of 2. en.wikipedia.org/wiki/Greedy_coloring –  András Salamon Jul 30 '10 at 10:59
If it was right, there would be no result about inapproximability within $n^{1 - \epsilon}$ factor. Anyway, thank you. –  ilyaraz Jul 30 '10 at 17:36
This algorithm requires to find at each step a "Maximum Independent Set", which is a NP-Complete problem ! This is a contradiction only if you can do that in polynomial time ;-) Nathann –  Nathann Cohen Jul 31 '10 at 2:36
Aha, Nathann you are not proposing maximal independent sets, but actually going to the trouble of finding the maximum ones (I misinterpreted what you meant as it seemed too audacious). Interesting idea! If one wants algorithms, then is computing successive independent sets actually going to beat out just computing a best colouring? Finding a maximum independent set can be done in something like $O(2^{0.304n})$ time if I remember correctly. Perhaps to make progress here ilyaraz really needs to say more about the problem -- is a closed-form bound sought, or a computable one? –  András Salamon Aug 1 '10 at 2:19
If one is just interested in theoretical bounds, then perhaps his family is nice enough to compute or bound somewhat efficiently the size of Max Independent Sets. If he is interested in patrice, well..I have been working on Sage (sagemath.org) for a year, and we have inside a software called Cliquer (users.tkk.fi/pat/cliquer.html) which I find very fast. The Coloring algorithms sadly, are way slower.**Anyone having an efficient implementation of one, or the idea of an algorithm in mind has my immediate attention, my respect, and perhaps even some money if he/she asks!** –  Nathann Cohen Aug 1 '10 at 6:12

Determining the chromatic number of an $n$-vertex graph is NP-Complete, and even approximating the chromatic number within an $NP^{1-\epsilon}$ ratio is $NP$-hard for every fixed $\epsilon>0$ (see http://en.wikipedia.org/wiki/Graph_coloring#Computational_complexity ). This means that assuming $P\ne NP$ (which is consider a very reasonable conjecture by many in complexity theory) that no lower bound algorithm exists that is - general, as it works on all graphs - efficient, as it gives a lower bound in runtime polynomial in the size of the graph - good, as in it comes close to the true lower bound

So in a way, the best algorithm that computes a lower-bound efficiently is the trivial algorithm that always outputs 1 or 2 (as we can decide 2-colorability).

In of itself, that above isn't terrible for your purposes because in a sense all one needs for a lower-bound proof is a certificate that the given family of graphs has high chromatic number. However, assuming $NP\ne coNP$ (another plausible conjecture), there cannot be a proof system such that: - general, as it works for all graphs - small, in the sense that it is of size polynomial in the size of the graph - good, as for all graphs with chromatic number >=4 the proof system lower bounds the chromatic number likewise

(This last point follows from the fact that even determining if a graph is 3-colorable is $NP$-Complete.)

So in a sense, there can't be "general lower-bound technique" but rather only a collection of ad-hoc methods.

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Since the various bounds are all decidable most mathematicians would perhaps be happy with them, even if they are hard to compute? –  András Salamon Jul 30 '10 at 11:14
As was pointed out to me later, the $n^{1-\epsilon}$ inapproximability result also extends to the certificate case, so even approximate lower-bound techniques are ruled out by the assumption that $NP\ne coNP$. One caveat is this machinery is for graphs with large chromatic number (ie. growing as $n^{\delta}$ for some $\delta>0$), it is not currently known how hard it is to distinguish 3-colorable graphs from 100-colorable graphs. –  miforbes Jul 30 '10 at 14:25
To address And\'{a}s' comment, even though everything is still decidable in the above, and so in principle could be used for a mathematical proof, the fact of the matter is that most proof techniques in combinatorics (that I know of) yield certificates that are "small" with respect to the problem. Another perspective however is that enumeration over all possible colorings is a valid way of deciding if a graph is $k$-colorable, but is practically useless for the purpose of a proof because it takes exponential time and has little structure to work with. –  miforbes Jul 30 '10 at 15:52
I suppose then one of the various ad-hoc methods might work, but it seems unlikely that an (as-yet-unknown) "general" method might work. As an aside, that is a fair tight bound that you have already - how did you prove it? Does that method not give the "correct" answer already? –  miforbes Jul 31 '10 at 14:27
I have a $\Theta(\log n)$-gap and what I want is to close it. These clique and independent set bounds are obvious in this case, and upper bound is obtained by the means of the probabilistic method. –  ilyaraz Aug 1 '10 at 22:09

Knowing more about your graph class would be very helpful. I know Bill Cook and Stephan Held are doing some work on lower-bounding $\chi$ using LP duality and branch-and-bound. Basically they look for a lower bound on the fractional chromatic number by finding a reasonably good feasible solution for the dual LP, i.e. a fractional clique.

A fractional clique is just a non-negative vertex weighting on the graph so that no stable set has weight more than 1. The total weight of a fractional clique is a lower bound for the fractional chromatic number, and so is in turn a lower bound for the chromatic number.

Of course, even with only 560 vertices this will not necessarily get you very far in a short amount of time. You can do tricks to help yourself with the time cost. Obviously you can start by throwing away vertices with degree lower than the bound you're hoping for. You can also partition the vertices of the graph into dense subgraphs, then try to bound the chromatic number of these subgraphs individually. Doing this using my RNSC (restricted neighbourhood search clustering) algorithm, which was originally used to find clusters in biological networks, helped a little bit with what Cook and Held were doing, but not too much.

I'm copy-pasting an abstract from a talk that Bill Cook gave this winter, which includes a link to their colouring page:

DATE: Tuesday, February 9

SPEAKER: Bill Cook (Georgia Tech)

TITLE: Computing the chromatic number of graphs

ABSTRACT: It can be very difficult in practice to optimally color a graph. For example, a set of randomly-generated test instances introduced by David Johnson in 1989 remain unsolved, the smallest example having only 125 vertices. We discuss the use of linear-programming methods to compute safe lower bounds on the chromatic number. Our methods do not depend on the floating-point accuracy of linear- programming software. This talk is based on joint work with Stephan Held (University of Bonn). Computational results and computer codes are freely available at site: http://code.google.com/p/exactcolors/

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By the way, you mention the Kneser graph. This is one graph class that can be used as a nasty example showing that the chromatic number is not necessarily upper-bounded by any function of the fractional chromatic number, meaning that in the worst case, the fractional chromatic number will give you a really lousy approximation. –  Andrew D. King Sep 2 '10 at 18:23


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# Euler's Formula/Examples/e^2 k i pi

## Example of Use of Euler's Formula

$\forall k \in \Z: e^{2 k i \pi} = 1$

## Proof

 $\ds e^{2 k i \pi}$ $=$ $\ds \cos 2 k \pi + i \sin 2 k \pi$ Euler's Formula $\ds$ $=$ $\ds 1 + i \times 0$ Cosine of Multiple of $\pi$, Sine of Multiple of $\pi$ $\ds$ $=$ $\ds 1$

$\blacksquare$


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# IB DP Maths Topic 9.2 Series that converge absolutely. HL Paper 3

## Question

Find the set of values of k for which the improper integral $$\int_2^\infty {\frac{{{\text{d}}x}}{{x{{(\ln x)}^k}}}}$$ converges.

[6]
a.

Show that the series $$\sum\limits_{r = 2}^\infty {\frac{{{{( – 1)}^r}}}{{r\ln r}}}$$ is convergent but not absolutely convergent.

[5]
b.

## Markscheme

consider the limit as $$R \to \infty$$ of the (proper) integral

$$\int_2^R {\frac{{{\text{d}}x}}{{x{{(\ln x)}^k}}}}$$     (M1)

substitute $$u = \ln x,{\text{ d}}u = \frac{1}{x}{\text{d}}x$$     (M1)

obtain $$\int_{\ln 2}^{\ln R} {\frac{1}{{{u^k}}}{\text{d}}u = \left[ { – \frac{1}{{k – 1}}\frac{1}{{{u^{k – 1}}}}} \right]_{\ln 2}^{\ln R}}$$     A1

Note: Ignore incorrect limits or omission of limits at this stage.

or $$[\ln u]_{\ln 2}^{\ln R}$$ if k = 1     A1

Note: Ignore incorrect limits or omission of limits at this stage.

because $$\ln R{\text{ }}({\text{and }}\ln \ln R) \to \infty {\text{ as }}R \to \infty$$     (M1)

converges in the limit if k > 1     A1

[6 marks]

a.

C: $${\text{terms}} \to 0{\text{ as }}r \to \infty$$     A1

$$\left| {{u_{r + 1}}} \right| < \left| {{u_r}} \right|$$ for all r     A1

convergence by alternating series test     R1

AC: $${(x\ln x)^{ – 1}}$$ is positive and decreasing on $$[2,\,\infty )$$     A1

not absolutely convergent by integral test using part (a) for k = 1     R1

[5 marks]

b.

## Examiners report

A good number of candidates were able to find the integral in part (a) although the vast majority did not consider separately the integral when k = 1. Many candidates did not explicitly set a limit for the integral to let this limit go to infinity in the anti – derivative and it seemed that some candidates were “substituting for infinity”. This did not always prevent candidates finding a correct final answer but the lack of good technique is a concern. In part (b) many candidates seemed to have some knowledge of the relevant test for convergence but this test was not always rigorously applied. In showing that the series was not absolutely convergent candidates were often not clear in showing that the function being tested had to meet a number of criteria and in so doing lost marks.

a.

A good number of candidates were able to find the integral in part (a) although the vast majority did not consider separately the integral when k = 1. Many candidates did not explicitly set a limit for the integral to let this limit go to infinity in the anti – derivative and it seemed that some candidates were “substituting for infinity”. This did not always prevent candidates finding a correct final answer but the lack of good technique is a concern. In part (b) many candidates seemed to have some knowledge of the relevant test for convergence but this test was not always rigorously applied. In showing that the series was not absolutely convergent candidates were often not clear in showing that the function being tested had to meet a number of criteria and in so doing lost marks.

b.

## Question

Consider the infinite series $$S = \sum\limits_{n = 0}^\infty {{u_n}}$$ where $${u_n} = \int_{nx}^{(n + 1)\pi } {\frac{{\sin t}}{t}{\text{d}}t}$$.

Explain why the series is alternating.

[1]
a.

(i)     Use the substitution $$T = t – \pi$$ in the expression for $${u_{n + 1}}$$ to show that $$\left| {{u_{n + 1}}} \right| < \left| {{u_n}} \right|$$.

(ii)     Show that the series is convergent.

[9]
b.

Show that $$S < 1.65$$.

[4]
c.

## Markscheme

as $$t$$ moves through the intervals $$[0,{\text{ }}\pi ],{\text{ }}[\pi ,{\text{ }}2\pi ],{\text{ }}[2\pi ,{\text{ }}3\pi ],{\text{ }}[3\pi ,{\text{ }}4\pi ]$$, etc, the sign of $$\sin t$$, (and therefore the sign of the integral) alternates $$+ ,{\text{ }} – ,{\text{ }} + ,{\text{ }} –$$, etc, so that the series is alternating     R1

Note:     Award R1 only if it includes a clear reason that justifies that the sign of the integrand alternates between − and + and this pattern is valid for all the terms.

The change of signs can be justified by a labelled graph of $$y = \sin (x)$$ or $$y = \frac{{\sin x}}{x}$$ that shows the intervals $$[0,{\text{ }}\pi ],{\text{ }}[\pi ,{\text{ }}2\pi ],{\text{ }}[2\pi ,{\text{ }}3\pi ],{\text{ }} \ldots$$

[1 mark]

a.

(i)     $${u_{n + 1}} = \int_{(n + 1)\pi }^{(n + 2)\pi } {\frac{{\sin t}}{t}{\text{d}}t}$$

(M1)

put $$T = t–\pi$$ and $${\text{d}}T = {\text{d}}t$$     (M1)

the limits change to $$n\pi ,{\text{ }}(n + 1)\pi$$

$$\left| {{u_{n + 1}}} \right| = \int_{n\pi }^{(n + 1)\pi } {\frac{{\left| {\sin (T + \pi )} \right|}}{{T + \pi }}{\text{d}}T}$$ (or equivalent)     A1

$$\left| {\sin (T + \pi )} \right| = \left| {\sin (T)} \right|$$ or $$\sin (T + \pi ) = – \sin (T)$$     (M1)

$$= \int_{n\pi }^{(n + 1)\pi } {\frac{{\left| {\sin T} \right|}}{{T + \pi }}{\text{d}}T}$$

$$< \int_{n\pi }^{(n + 1)\pi } {\frac{{\left| {\sin T} \right|}}{T}{\text{d}}T = \left| {{u_n}} \right|}$$    A1AG

(ii)     $$\left| {{u_n}} \right| = \int_{n\pi }^{(n + 1)\pi } {\frac{{\sin t}}{t}{\text{d}}t}$$

$$< \int_{n\pi }^{(n + 1)\pi } {\frac{1}{t}{\text{d}}t}$$    M1

$$= [\ln t]_{n\pi }^{(n + 1)\pi }$$    A1

$$= \ln \left( {\frac{{n + 1}}{n}} \right)$$    A1

$$\to \ln 1 = 0$$ as $$n \to \infty$$

from part (i) $$\left| {{u_n}} \right|$$ is a decreasing sequence and since $$\mathop {\lim }\limits_{n \to \infty } \left| {{u_n}} \right| = 0$$,     R1

the series is convergent     AG

[9 marks]

b.

attempt to calculate the partial sums $$\sum\limits_{i = 0}^{n – 1} {{u_i} = \int_0^{n\pi } {\frac{{\sin t}}{t}{\text{d}}t} }$$     (M1)

the first partial sums are

two consecutive partial sums for $$n \geqslant 4$$     A1A1

(eg $${S_4} = 1.49$$ and $${S_5} = 1.63$$ or $${S_{100}} = 1.567 \ldots$$ and $${S_{101}} = 1.573 \ldots$$)

Note:     These answers must be given to a minimum of 3 significant figures.

the sum to infinity lies between any two consecutive partial sums,

eg between 1.49 and 1.63     R1

so that $$S < 1.65$$     AG

Note:     Award A1A1R1 to candidates who calculate at least two partial sums for only odd values of $$n$$ and state that the upper bound is less than these values.

[4 marks]

c.

## Examiners report

Very few candidates presented a valid reason to justify the alternating nature of the series. In most cases candidates just reformulated the wording of the question by saying that it changed signs and completely ignored the interval over which the expression had to be integrated to obtain each term.

a.

(i) Most candidates achieved 1 or 2 marks for attempting the given substitution; in most cases candidates failed to find the correct limits of integration for the new variable and then relate the expressions of the consecutive terms of the series. In part (ii) very few correct attempts were seen; in some cases candidates did recognize the conditions for the alternating series to be convergent but very few got close to establish that the limit of the general term was zero.

b.

A few good attempts to use partial sums were seen although once again candidates showed difficulties in identifying what was needed to show the given answer. In most cases candidates just verified with GDC that in fact for high values of n the series was indeed less than the upper bound given but could not provide a valid argument that justified the given statement.

c.

## Question

Consider the infinite spiral of right angle triangles as shown in the following diagram.

The $$n{\text{th}}$$ triangle in the spiral has central angle $${\theta _n}$$, hypotenuse of length $${a_n}$$ and opposite side of length 1, as shown in the diagram. The first right angle triangle is isosceles with the two equal sides being of length 1.

Consider the series $$\sum\limits_{n = 1}^\infty {{\theta _n}}$$.

Using l’Hôpital’s rule, find $$\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{\arcsin \left( {\frac{1}{{\sqrt {(x + 1)} }}} \right)}}{{\frac{1}{{\sqrt x }}}}} \right)$$.

[6]
a.

(i) Find $${a_1}$$ and $${a_2}$$ and hence write down an expression for $${a_n}$$.

(ii) Show that $${\theta _n} = \arcsin \frac{1}{{\sqrt {(n + 1)} }}$$.

[3]
b.

Using a suitable test, determine whether this series converges or diverges.

[6]
c.

## Markscheme

$$\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{\arcsin \left( {\frac{1}{{\sqrt {(x + 1)} }}} \right)}}{{\frac{1}{{\sqrt x }}}}} \right)$$ is of the form $$\frac{0}{0}$$

and so will equal the limit of $$\frac{{\frac{{\frac{{ – 1}}{2}{{(x + 1)}^{ – \frac{3}{2}}}}}{{\sqrt {1 – \left( {\frac{1}{{x + 1}}} \right)} }}}}{{\frac{{ – 1}}{2}{x^{ – \frac{3}{2}}}}}$$     M1M1A1A1

Note: M1 for attempting differentiation of the top and bottom, M1A1 for derivative of top (only award M1 if chain rule is used), A1 for derivative of bottom.

$$= \mathop {\lim }\limits_{x \to \infty } \frac{{{{\left( {\frac{x}{{(x + 1)}}} \right)}^{\frac{3}{2}}}}}{{\sqrt {\frac{x}{{x + 1}}} }} = \mathop {\lim }\limits_{x \to \infty } \left( {\frac{x}{{x + 1}}} \right)$$    M1

Note: Accept any intermediate tidying up of correct derivative for the method mark.

$$= 1$$    A1

[6 marks]

a.

(i)     $${a_1} = \sqrt 2 ,{\text{ }}{a_2} = \sqrt 3$$     A1

$${a_n} = \sqrt {n + 1}$$    A1

(ii)     $$\sin {\theta _n} = \frac{1}{{{a_n}}} = \frac{1}{{\sqrt {n + 1} }}$$     A1

Note: Allow $${\theta _n} = \arcsin \left( {\frac{1}{{{a_n}}}} \right)$$ if $${a_n} = \sqrt {n + 1}$$ in b(i).

so $${\theta _n} = \arcsin \frac{1}{{\sqrt {(n + 1)} }}$$     AG

[3 marks]

b.

for $$\sum\limits_{n = 1}^\infty {\arcsin \frac{1}{{\sqrt {(n + 1)} }}}$$ apply the limit comparison test (since both series of positive terms)     M1

with $$\sum\limits_{n = 1}^\infty {\frac{1}{{\sqrt n }}}$$     A1

from (a) $$\mathop {\lim }\limits_{n \to \infty } \frac{{\arcsin \frac{1}{{\sqrt {(n + 1)} }}}}{{\frac{1}{{\sqrt n }}}} = 1$$, so the two series either both converge or both diverge     M1R1

$$\sum\limits_{n = 1}^\infty {\frac{1}{{\sqrt 2 }}}$$ diverges (as is a $$p$$-series with $$p = \frac{1}{2}$$)     A1

hence $$\sum\limits_{n = 1}^\infty {{\theta _n}}$$ diverges     A1

[6 marks]

c.

[N/A]

a.

[N/A]

b.

[N/A]

c.

## Question

Given that $$n > {\text{ln}}\,n$$ for $$n > 0$$, use the comparison test to show that the series $$\sum\limits_{n = 0}^\infty {\frac{1}{{{\text{ln}}\left( {n + 2} \right)}}}$$ is divergent.

[3]
a.

Find the interval of convergence for $$\sum\limits_{n = 0}^\infty {\frac{{{{\left( {3x} \right)}^n}}}{{{\text{ln}}\left( {n + 2} \right)}}}$$.

[7]
b.

## Markscheme

METHOD 1

$${\text{ln}}\left( {n + 2} \right) < n + 2$$     (A1)

$$\Rightarrow \frac{1}{{{\text{ln}}\left( {n + 2} \right)}} > \frac{1}{{n + 2}}$$ (for $$n \geqslant 0$$)     A1

Note: Award A0 for statements such as $$\sum\limits_{n = 0}^\infty {\frac{1}{{{\text{ln}}\left( {n + 2} \right)}}} > \sum\limits_{n = 0}^\infty {\frac{1}{{n + 2}}}$$. However condone such a statement if the above A1 has already been awarded.

$$\sum\limits_{n = 0}^\infty {\frac{1}{{n + 2}}}$$ (is a harmonic series which) diverges     R1

Note: The R1 is independent of the A1s.

Award R0 for statements such as “$$\frac{1}{{n + 2}}$$ diverges”.

so $$\sum\limits_{n = 0}^\infty {\frac{1}{{{\text{ln}}\left( {n + 2} \right)}}}$$ diverges by the comparison test     AG

METHOD 2

$$\frac{1}{{{\text{ln}}\,n}} > \frac{1}{n}$$ (for $$n \geqslant 2$$)     A1

Note: Award A0 for statements such as $$\sum\limits_{n = 2}^\infty {\frac{1}{{{\text{ln}}\,n}}} > \sum\limits_{n = 2}^\infty {\frac{1}{n}}$$. However condone such a statement if the above A1 has already been awarded.

a correct statement linking $$n$$ and $$n + 2$$ eg,

$$\sum\limits_{n = 0}^\infty {\frac{1}{{{\text{ln}}\left( {n + 2} \right)}}} = \sum\limits_{n = 2}^\infty {\frac{1}{{{\text{ln}}\,n}}}$$ or $$\sum\limits_{n = 0}^\infty {\frac{1}{{n + 2}}} = \sum\limits_{n = 2}^\infty {\frac{1}{n}}$$     A1

Note: Award A0 for $$\sum\limits_{n = 0}^\infty {\frac{1}{n}}$$

$$\sum\limits_{n = 2}^\infty {\frac{1}{n}}$$ (is a harmonic series which) diverges

(which implies that $$\sum\limits_{n = 2}^\infty {\frac{1}{{{\text{ln}}\,n}}}$$ diverges by the comparison test)      R1

Note: The R1 is independent of the A1s.

Award R0 for statements such as $$\sum\limits_{n = 0}^\infty {\frac{1}{n}}$$ deiverges and “$${\frac{1}{n}}$$ diverges”.

Award A1A0R1 for arguments based on $$\sum\limits_{n = 1}^\infty {\frac{1}{n}}$$.

so $$\sum\limits_{n = 0}^\infty {\frac{1}{{{\text{ln}}\left( {n + 2} \right)}}}$$ diverges by the comparison test      AG

[3 marks]

a.

applying the ratio test $$\mathop {{\text{lim}}}\limits_{n \to \infty } \left| {\frac{{{{\left( {3x} \right)}^{n + 1}}}}{{{\text{ln}}\left( {n + 3} \right)}} \times \frac{{{\text{ln}}\left( {n + 2} \right)}}{{{{\left( {3x} \right)}^n}}}} \right|$$     M1

$$= \left| {3x} \right|$$ (as $$\mathop {{\text{lim}}}\limits_{n \to \infty } \left| {\frac{{{\text{ln}}\left( {n + 2} \right)}}{{{\text{ln}}\left( {n + 3} \right)}}} \right| = 1$$     A1

Note: Condone the absence of limits and modulus signs.

Note: Award M1A0 for $$3{x^n}$$. Subsequent marks can be awarded.

series converges for $$– \frac{1}{3} < x < \frac{1}{3}$$

considering $$x = – \frac{1}{3}$$ and $$x = \frac{1}{3}$$     M1

Note: Award M1 to candidates who consider one endpoint.

when $$x = \frac{1}{3}$$, series is $$\sum\limits_{n = 0}^\infty {\frac{1}{{{\text{ln}}\left( {n + 2} \right)}}}$$ which is divergent (from (a))      A1

Note: Award this A1 if $$\sum\limits_{n = 0}^\infty {\frac{1}{{{\text{ln}}\left( {n + 2} \right)}}}$$ is not stated but reference to part (a) is.

when $$x = – \frac{1}{3}$$, series is $$\sum\limits_{n = 0}^\infty {\frac{{{{\left( { – 1} \right)}^n}}}{{{\text{ln}}\left( {n + 2} \right)}}}$$     A1

$$\sum\limits_{n = 0}^\infty {\frac{{{{\left( { – 1} \right)}^n}}}{{{\text{ln}}\left( {n + 2} \right)}}}$$ converges (conditionally) by the alternating series test      R1

(strictly alternating, $$\left| {{u_n}} \right| > \left| {{u_{n + 1}}} \right|$$ for $$n \geqslant 0$$ and $$\mathop {{\text{lim}}}\limits_{n \to \infty } \left( {{u_n}} \right) = 0$$)

so the interval of convergence of S is $$– \frac{1}{3} \leqslant x < \frac{1}{3}$$     A1

Note: The final A1 is dependent on previous A1s – ie, considering correct series when $$x = – \frac{1}{3}$$ and $$x = \frac{1}{3}$$ and on the final R1.

Award as above to candidates who firstly consider $$x = – \frac{1}{3}$$ and then state conditional convergence implies divergence at $$x = \frac{1}{3}$$.

[7 marks]

b.

[N/A]

a.

[N/A]

b.


text:
perm filename FILMS.TEX[BIB,CSR] blob sn#542947 filedate 1980-11-12 generic text, type C, neo UTF8
COMMENT ⊗   VALID 00003 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	\input macro[tex,zm]
C00003 00003	Prints of the  following films are  available for short-term  loan to
C00017 ENDMK
C⊗;
\input macro[tex,zm]

\chapter{Film Reports}{Stanford Department of Computer Science}

Prints of the  following films are  available for short-term  loan to
interested groups without charge.   They may be shown only  to groups
that have paid no admission fee.  To make a reservation, write to:
Film Services, Department of Computer Science, Stanford University,
Stanford, California 94305 USA.

\bpar{{\ic Ellis D. Kroptechev and Zeus, his Marvelous Time-Sharing System}
The  advantages of  time-sharing over  standard batch  processing are
revealed through the good offices of the Zeus time-sharing  system on
a PDP-1 computer.  Our hero,  Ellis, is saved from a fate  worse than
death.  Recommended for mature audiences only.
(This film was done by Art Eisenson and Gary Feldman.
It's 16mm black and white with  sound, runs approximately 15
minutes, and was produced in March 1967.)}

\bpar{{\ic Butterfinger}
Describes  the  state  of  the  hand-eye  system  at  the  Artificial
Intelligence Project in the fall of 1967.  The PDP-6 computer getting
visual  information  from  a  television  camera  and  controlling an
electrical-mechanical  arm  solves  simple  tasks  involving stacking
blocks.  The techniques of recognizing the blocks and their positions
as well as controlling the arm are briefly presented.  Rated G''.
(This film was done by Gary Feldman.
It's 16mm  color with sound, runs approximately 8 minutes,
and was produced in March 1968.)}

\bpar{{\ic Hear Here}
Describes the state  of the speech  recognition project as  of Spring
1969.  A discussion of the problems of speech recognition is followed
by two  real time  demonstrations of the  current system.   The first
shows the computer learning to recognize phrases and the second shows
how  the  hand-eye  system  may  be  controlled  by  voice  commands.
Commands as  complicated as  Pick up  the small  block in  the lower
lefthand corner,'' are recognized and the tasks are carried out by the
computer controlled arm.
(This film was done by Raj Reddy, Dave Espar  and Art Eisenson.
It's 16mm color with sound,
runs approximately 15 minutes, and was produced in March 1969.)}

\bpar{{\ic Avoid}
An illustration of Peiper's Ph.D. thesis.  The problem is to move the
computer controlled mechanical arm through a space filled with one or
more known obstacles.  The film shows the arm as it is moving through
varous cluttered environments with fairly good success.
(This film was done by Gary  Feldman and  Donald Peiper.  It's 16mm silent color,
runs approximately 5 minutes, and was produced in March 1969.)}

\bpar{{\ic Instant Insanity}
Shows  the  hand-eye  system  solving  the  puzzle {\ic Instant Insanity}.
Se\-quen\-ces include  finding and  recognizing cubes,  color recognition
and object manipulation.  (Made to accompany a paper presented at the
1971 IJCAI.  May be hard to understand without a narrator.)
(This film was done by
Richard Paul  and Karl  Pingle.
It's 16mm silent color, runs approximately 6 minutes,
and was produced in August 1971.)}

\bpar{{\ic Motion and Vision}
A  technical presentation  of  three research  projects  completed in
1972:   advanced arm control by R.P. Paul (AIM-177),  visual feedback
control by A. Gill  (AIM-178), and representation and  description of
curved objects by G. Agin (AIM-173).  Drags a bit.
(This film was done by Suzanne Kandra.
It's 16mm  color with  sound, runs approximately 22 minutes,
and was produced in November 1972.)}

\bpar{{\ic Computer Interactive Picture Processing  [MARS  Project]}
This film describes  an automated picture differencing  technique for
analyzing the variable surface  features on Mars using  data returned
by the Mariner 9 spacecraft.  The system uses a time-shared, terminal
oriented PDP-10  computer.  The film  proceeds at a  breathless pace.
Don't blink, or you will miss an entire scene.
(This film was done by
Larry Ward.  It's 16mm color with sound, runs approximately 8 minutes,
and was produced in the Fall of 1972.)}

\bpar{{\ic Display Simulations of 6-Legged Walking}
A  display  simulation of  a  6-legged ant-like  walker  getting over
various ob\-sta\-cles.  The research  is aimed at a planetary  rover that
would  get around  by walking.   This cartoon  favorate  beats Mickey
Mouse hand down....or rather, feet'' down.
(This film was done by D.I.  Okhotsimsky et al. of the Institute  of  Applied
Mathematics,  USSR  Academy  of Science.  The
titles were translated by Stanford AI Lab and edited by  Suzanne Kandra.
It's a 16mm silent black and white, and runs approximately 10 minutes.
There are two versions:  one done in 1972, and one done in 1976.)}

\bpar{{\ic Automated Pump Assembly}
Shows the  hand-eye system  assembly a simple  pump, using  vision to
locate  the  pump  body  and to  check  for  errors.   The  parts are
assembled and screws inserted, using some special tools  designed for
the arm.  Some titles are included to help explain the film.
(This film was done by Richard  Paul,  Karl  Pingle  and Bob
Bolles.  It's 16mm silent color, runs approximately 7 minutes
at sound speed, and was produced in April 1973.)}

\bpar{{\ic Dialog with a Robot}
Presents  a natural  language dialog  with a  simulated  robot block-
manipulation  system.  The  dialog is  substantially the  same  as in
Understanding Natural Language'' (T. Winograd, Academic Press, 1972).
No explanatory or narrative material is on the film.
(This film was done by
Terry Winograd when he was at the MIT A.I. Lab.
It's 16mm silent black and white, runs approximately 20 minutes,
and was produced in 1971.)}

\bpar{{\ic Programmable Assembly, Three Short Examples}
The  first  segment  demonstrates the  arm's  ability  to dynamically
adjust for position and orientation changes.  The task is to  mount a
bearing and seal  on a crankshaft.  Next,  the arm is  shown changing
tools and  recovering from  a run-time  error.  Finally,  a cinematic
first:  TWO arms cooperating to assemble a hinge.
(This film was done by Karl  Pingle, Lou
Paul and Bob Boles.  It's 16mm color with sound,
runs approximately 8 minutes, and was produced in October 1974.)}

\bpar{{\ic Display Terminals at Stanford}
Although there are many  effective programs to use  display terminals
for  special   graphics  applications,   very  few   general  purpose
timesharing systems provide good support for using  display terminals
in normal text display applications.  This film shows a session using
the  display system  at  the Stanford  A.I. Lab,  explaining  how the
display support features in  the Stanford monitor enhance  the user's
control over his job and facilitate the writing  of display-effective
user programs.
(This film was done by Brian  Harvey.  It's 16mm  black and
white with sound, runs approximaely 13 minutes, and was produced in May 1975.)}

\bpar{{\ic Pointy - an Interactive System for Assembly}
POINTY is  an interactive programming  system that uses  a mechanical
ma\-nip\-u\-lator  as  a  measuring  tool  to  determine  the  position and
orientation of various parts  laid out in a work  station.  Positions
may be determined precisely by means of a sharp pointed tool  held in
the manipulator hand, or by using the finger touch sensors and moving
the  arm to  the  desired points  either manually  or  under computer
control.  Arbitrary orientations may be determined from  the location
of three points.  The data generated may be referred to symbolically,
so that the programmer is freed from having to think in terms  of the
numerical vlues of object locations.  The data is saved in a computer
file for later use in a program to assemble the parts.

This film illustrates the  use of POINTY instructions to  collect the
poisition data of two parts of a water valve assembly.  It  shows the
use  of  multiple  points to  determine  orientations,  the procedure
followed to obtain the data, and how the programmer may refer  to the
data symbolically.   Finally, the arm  is shown putting  together the
water valve assembly.
(This film was done by
M. Shahid Mujtaba.  It's 16mm color with sound,
runs approximately 10 minutes, and was produced in December 1977.)}

Alternatively,  prints may  be purchased  at cost  (typically  $\$40$to$\$200$).  This is handled by individual arrangement.  Contact the
Publications Coordinator (Stanford Department of Computer Science) directly
if you are interested in purchasing any of these films.
%from: Photo Tech Laboratories (phone: 213/462-5457).

\vfill\end


text:
# Create a JDBC Enrichment Workflow

To set up JDBC Enrichment in Moogsoft AIOps, configure a workflow in the JDBC Enrichment Workflow Engine to trigger the query you set up in the JDBC Enrichment Integration. See Configure the JDBC Enrichment Integration. This topic covers the final step in the JDBC enrichment example Enrich Alerts Using a JDBC Data Source.

The following diagram illustrates the process to enrich alert data from an external database:

In the JDBC Enrichment Workflow Engine, define a workflow with an action that uses the getJDBCEnrichment function. Use the databaseDefName and tableDefName arguments to identify specify the database definition and the table definition you set up in the JDBC Enrichment integration. See Configure the JDBC Enrichment Integration.

The value1 and value2 arguments let you map fields in your alert to the $$params.value1 or $$params.value2 variables in your query. This way different workflows can send different values to the same table definition query. For more information, see the Query section in JDBC Enrichment Reference.

## Step 4 example: create a JDBC Enrichment workflow

In the example scenario, you want to enrich all alerts with information from the CMDB. Therefore, create a new workflow in the JDBC Enrichment Workflow Engine as follows:

1. Create a workflow called "Enrich from CMDB" with the description "Query CMDB and add data to custom_info ". Leave the rest of the fields blank or unchecked.

2. Add an action into the workflow called "Query CMDB" to specify the database definition you created earlier. See Configure the JDBC Enrichment Integration:

• Function: getJDBCEnrichment

• Arguments

databaseDefName: "cmdb"

tableDefName: "server"

3. Add an action into the workflow called "Forward to Maintenance Window Manager" to send the alert to the next stage for data processing:

• Function: forward

• Arguments

moolet: MaintenanceWindowManager

Forwarding Behavior: Always Forward

When this workflow is active, it queries the CMDB according to the JDBC Enrichment integration configuration. Given the following alert data:

{ ... "source":"sflinux101", ...}

The Workflow Engine updates the custom_info for the alert as follows:

"custom_info":
{"enrichment":
{"cmdb_info": {
"support_group": "SF NOC",
"location": "1265 Battery ST, SF, CA"              }
},
"mooghandling": {"isEnriched": true}
}


text:
# How does the Taylor Series converge at all points for certain functions

The way my professor defined Taylor polynomials is: the $$n^{th}$$ degree Taylor polynomial $$p(x)$$ of $$f(x)$$ is a polynomial that satisfies $$\lim_{x\to 0}{f(x)-p(x) \over x^n} = 0$$. This is actually the little-o notation $$o(x^n)$$, which means $$(f(x)-p(x)) \ll x^n$$ as $$x$$ approaches $$0$$. From this I have got the intuition that Taylor Polynomials work only for $$|x| < 1$$ because $$x^n$$ gets smaller as $$n$$ gets bigger only when $$|x| < 1$$. And the textbook seemed to agree with my intuition, because the textbook says “Taylor polynomial near the origin” (probably implying $$|x| < 1$$).

Since Taylor Series is basically Taylor polynomial with $$n\to\infty$$, I intuitively thought that the Taylor Series would also only converge to the function it represents in the interval $$(-1, 1)$$.

For example, in the case of $$1\over1-x$$, it is well known that the Taylor series only converges at $$|x| < 1$$.

However, all of a sudden, the textbook says that the Taylor series of $$\cos x$$ converges for all real $$x$$. It confused me because previously I thought the Taylor series would only work for $$|x|<1$$. Now, I know that the Taylor Series is defined like this: $$f(x) = Tf(x) \Leftrightarrow \lim_{n\to\infty}R_{n}f(x) = 0$$

And I know how to get the maximum of Taylor Remainder for $$\cos x$$ using Taylor's Theorem, and I know that the limit of that Taylor Remainder is $$0$$ for all real $$x$$, which makes the Taylor Series of $$cosx$$ converge to $$\cos x$$, pointwise. However, I just can't get why my initial intuition is wrong (why taylor series converges for all $$x$$ for certain functions, like $$\cos x$$, also $$\sin x$$ and $$e^x$$, etc.)

• For the specific example $e^x$, the terms in its series are $x^n/n!$. Note that $n!$ grows much faster than $x^n$ as $n$ grows. That's one reason to believe the Taylor series of $e^x$ converges. Apr 24 '20 at 18:10
• In my mind, the key idea of calculus is the local linear approximation $f(x) \approx f(a) + f'(a)(x - a)$, which is a good approximation when $x$ is near $a$. It is natural to ask, what if we approximate $f$ locally by a quadratic or cubic function rather than by a linear function. This leads to the idea of Taylor polynomial approximation, which is indeed initially intended to be a local approximation to $f$. But when we look at the remainder term in Taylor's theorem, there's something kind of amazing, surprising that happens which is that often it's small even when $x$ is far from $a$. Apr 24 '20 at 18:11
• @trisct Yes that is true indeed. The limit of Taylor remainder for $e^x$ is also zero for all x, which further explains why the Taylor Series converges. But I still I can't see why my initial intuition is wrong.
– the
Apr 24 '20 at 18:12
• @littleO So it just happened to be that the Taylor remainder's limit is zero even for x far away?
– the
Apr 24 '20 at 18:14
• @linearAlg In my mind, yes. It is just one of the miracles of math, which makes us wonder why things have worked out more nicely than we deserved. Apr 24 '20 at 18:16

Actually, things may go wrong in $$(-1,1)$$. For instance, the Taylor series centered at $$0$$ of $$f(x)=\frac1{1-nx}$$ only converges to $$f(x)$$ on $$\left(-\frac1n,\frac1n\right)$$. And if$$f(x)=\begin{cases}e^{-1/x^2}&\text{ if }x\ne0\\0&\text{ if }x=0,\end{cases}$$then the Taylor series of $$f$$ only converges to $$f(x)$$ if $$x=0$$.
On the other hand, yes, Taylor series centered at $$0$$ are made to converge to $$f(x)$$ near $$0$$. But that's no reason to expect that they don't converge to $$f(x)$$ when $$x$$ is way from $$0$$. That would be like expecting that a non-constant power series $$a_0+a_1x+a_2x^2+\cdots$$ takes larger and larger values as the distance from $$x$$ to $$0$$. That happens often, but $$1-\frac1{2!}x^2+\frac1{4!}x^4-\cdots=\cos(x)$$, which is bounded.
In other words, you have gone from $$a\implies b$$ to $$\tilde a\implies \tilde b,$$ which you can see to be clearly false, identically. That is, it is not necessarily true for all $$a,\,b.$$
Since you already know why the series for entire functions like $$\cos x$$ converges everywhere (as you explain towards the end of your post), you should now see where your original intuition (I would say erroneous belief) misled you.


text:
# Means to an End

by Richard P. Kubelka

Mathematics Magazine
April, 2001

Subject classification(s): Calculus | Single Variable Calculus | Limits
Applicable Course(s): 3.2 Mainstream Calculus II | 3.5 Non-mainstream Calc II

The limit of the geometric mean of the first $$n$$ integers raised to the real positive power $$s$$, divided by their arithmetic mean is shown to be $$(s+1)/e^s$$. An elementary derivation of Stirling`s approximation suggested this limit for $$s=1$$.

A pdf copy of the article can be viewed by clicking below. Since the copy is a faithful reproduction of the actual journal pages, the article may not begin at the top of the first page.


text:
# Math Help - Contour lines

1. ## Contour lines

Hey guys!!!Could you help me at this exercise???
Draw some contour lines in the (x, y)–plane of the function h(x,y)=(a*(x+y))/(x^2+y^2+a^2).

2. ## Re: Contour lines

tzina, with the aid of a 3D grapher I was able to to come up with some contour lines.

let $\space$ $z = \displaystyle\frac{a(x + y)}{x^2 + a^2 + y^2}$

for each value of $\space$ $a$, $\space$ you get a unique set of contour lines, but each set of lines has similar features to them

.Set $\space$ $z$ $\space$ in increments of $\space$ $.1$. For example $\space$: $\space$ $z = 0, .1, .2, .3, ...$ $\space$ etc.

For each value of $\space$ $z$ $\space$ you can create one contour line.

When $\space$ $z = 0$ $\space$ the contour line will show up as a straight line

if $\space$ $z = 0$, $\space$ $a(x + y) = 0$.

You will have the line $\space$ $x + y = 0$

3. ## Re: Contour lines

continuing with my response:

Each succeeding contour line will be a circle where the center of each circle will fall on the line $\space$ $x - y = 0$

You will be able to calculate the center and radius of each circle by setting $\space$ $a$ $\space$as a specific constant and solving for $\space$ $x$ and $y$ $\space$ for each value of $\space$ $z$.

You will end up with a series of circles whose centers fall on the line $\space$ $x - y = 0$.

I hope this helps.

4. ## Re: Contour lines

Hi,
Here's a graphing trick that you can use to draw contour lines. Assuming your graphing software can handle implicit functions, have it graph the function sin(s*2pi*h(x,y))=0. This is then the set of curves h(x,y)=k/s for k any integer. If your grapher has sliders, you can rapidly change the step size s. Also in your case you can rapidly change the constant a. Here's some contours for your function with s=10 (step size .1) and a =3 (the contour at height 0 is as noted above the line x+y=0, it's shown in red):


text:
Database Open Access

# Treadmill Maximal Exercise Tests from the Exercise Physiology and Human Performance Lab of the University of Malaga

Published: Dec. 10, 2021. Version: 1.0.1

Mongin, D., García Romero, J., & Alvero Cruz, J. R. (2021). Treadmill Maximal Exercise Tests from the Exercise Physiology and Human Performance Lab of the University of Malaga (version 1.0.1). PhysioNet. https://doi.org/10.13026/7ezk-j442.

Mongin, D., Chabert, C., Courvoisier, D. S., García-Romero, J., & Alvero-Cruz, J. R. (2021). Heart rate recovery to assess fitness: Comparison of different calculation methods in a large cross-sectional study. Research in Sports Medicine

Goldberger, A., Amaral, L., Glass, L., Hausdorff, J., Ivanov, P. C., Mark, R., ... & Stanley, H. E. (2000). PhysioBank, PhysioToolkit, and PhysioNet: Components of a new research resource for complex physiologic signals. Circulation [Online]. 101 (23), pp. e215–e220.

## Abstract

The present database is an ensemble of the cardiorespiratory measurements acquired during 992 treadmill maximal graded exercise tests (GET) performed in the Exercise Physiology and Human Performance Lab of the University of Malaga. Heart rate, oxygen consumption, carbon dioxide generation, and pulmonary ventilation are measured on a breath-to-breath basis along with the treadmill speed during maximal effort tests. Participants are amateur and professional athletes are of ages ranging from 10 to 63 years old. The age, height, and weight of the participants are provided, as well as the temperature and humidity during the test.

## Background

Cardiac and respiratory data measured during graded exercise tests are key measures to calculate several cardio-respiratory indices used in sport science [1] and medicine [2,3]. All the parts of the oxygen consumption and heart rate dynamics during a GET are of interest: the rate of increase at exercise onset [4], the slope during exercise [5,6], the maximal values [7,8], the dynamics changes at ventilatory thresholds [9,10], the nonlinear dynamics during effort [11–13], and finally the dynamics during recovery [14,15]. The study of each of these segments of the oxygen consumption or heart rate dynamics during effort has led to indices used to characterize and predict health, fitness, or performance, such as ventilatory thresholds, heart resting rate, rate of heart rate increase, deflection point of the performance curve, etc. But comparisons between existing calculation methods for these indices are lacking, and reproducibility of the calculation is sometimes hindered by the lack of available open-source code. The goal of this dataset is then two-fold: To facilitate the publication and diffusion of calculation methods and associated code to analyze cardiorespiratory measurements of maximal exercise tests; To encourage studies comparing different calculation methods of indices derived from the cardio-respiratory measurements acquired during effort tests, such as ventilatory thresholds or heart rate recovery.

## Methods

The measurements were taken between 2008 and 2018. The athletes performed a maximal Graded Exercise Testing (GET) on a PowerJog J series treadmill connected to a CPX MedGraphics gas analyzer system (Medical Graphics, MN, USA) with breath-by-breath measurements of respiratory parameters -including oxygen consumption and pulmonary ventilation- and heart rate collected by a Mortara 12-lead ECG device.

The stress tests consisted of a continuous (ramping) or step-by-step incremental effort. Most of the exercise phases are preceded by a warmup period of walking at 5 km/h. When incremental, the step amplitudes range from 0.5 to 1 km/h. The participants were asked to go beyond exhaustion, and the test was considered maximal if the oxygen consumption was saturated. The effort was then ceased, and to avoid vasovagal syncope, the treadmill speed was set back to the initial 5 km/h speed, and the participant was asked to walk.

Exercise testing was voluntary, and before its initiation, written informed consent was obtained from the participants and the legal guardians of those under 18 years of age. All effort tests were performed under the supervision of a doctor in sport science, and their analysis was carried out according to the principles of the Declaration of Helsinki. The study protocols were approved by the Research Ethics Committee of the University of Málaga.

## Data Description

The dataset contains two files:

• subject-info.csv contains the participant info at the time of the test. The variable ID identifies a participant, whereas the variable ID_test identifies an exercise test. This file contains 992 lines, one for each test. The different variables are described in the table below, with their corresponding amount or median [Inter Quartile Range] value.
 variable value ID_test 992 ID 857 Age (years) 27.10 [21.10, 36.32] Weight (kg) 73.00 [66.00, 80.23] Height (cm) 175.00 [170.00, 180.00] Humidity (%) 47.00 [42.00, 54.00] Temperature (°C) 22.90 [20.80, 24.40] Sex = 1 (Female) (%) 149 (15.0)
• test_measure.csv, contains all the cardiorespiratory measurements taken during each effort test. The data is in long format, so the file contains one line for each breath measurement for all of the 992 effort tests, resulting in the 575087 lines present in the file. The time of each measurement is identified by the variable time indicating the seconds elapsed since the effort test start, the exercise test is identified by the ID_test variable, and the variable ID indicates the participant. These effort tests contain a median [Inter Quartile Range] of 580 [484, 673] measures, for a median duration of 1093.00 [978.75, 1208.00] seconds. The variables in this file are:  time Time since the measurement starts, in seconds Speed Speed of the treadmill, in km/h HR Heart rate, in beat per min VO2 Oxygen consumption, in mL/min VCO2 Carbon dioxide production, in mL/min RR Respiration rate, in respiration per minute VE Pulmonary ventilation, in L/min ID Participant identification ID_test Effort test identification

Note that VO2, VCO2, and VE measures are missing for 30 tests.

## Usage Notes

ID_test, the variable identifying the GET, is named from the ID of the participant paired with the GET index. For example, ID_date = 245_3 is the third exercise test of participant ID = 245. An example of calculation in R [16] of simple heart rate recovery and cardiorespiratory index from these files is provided in an associated GitHub repository [17].

## Acknowledgements

The authors would like to thank all past and present members of the School of Sports Medicine of University of Málaga and Research Group CTS-132. Junta de Andalucía (Exercise Physiology).

## Conflicts of Interest

The authors have no conflicts of interest to declare.

## References

1. Medicine AC of S. ACSM’s Guidelines for Exercise Testing and Prescription. Edición: 9th edition. Philadelphia: Lippincott Williams and Wilkins; 2013.
2. Cole CR, Blackstone EH, Pashkow FJ, Snader CE, Lauer MS. Heart-Rate Recovery Immediately after Exercise as a Predictor of Mortality. New England Journal of Medicine. 1999 Oct 28;341(18):1351–7.
3. Ross Robert, Blair Steven N., Arena Ross, Church Timothy S., Després Jean-Pierre, Franklin Barry A., et al. Importance of Assessing Cardiorespiratory Fitness in Clinical Practice: A Case for Fitness as a Clinical Vital Sign: A Scientific Statement From the American Heart Association. Circulation. 2016 Dec 13;134(24):e653–99.
4. Bellenger CR, Thomson RL, Howe PRC, Karavirta L, Buckley JD. Monitoring athletic training status using the maximal rate of heart rate increase. Journal of Science and Medicine in Sport. 2016 Jul 1;19(7):590–5.
5. Bodner ME, Rhodes EC. A review of the concept of the heart rate deflection point. Sports Med. 2000 Jul;30(1):31–46.
6. Swain DP, Leutholtz BC, King ME, Haas LA, Branch JD. Relationship between % heart rate reserve and % VO2 reserve in treadmill exercise. Med Sci Sports Exerc. 1998 Feb;30(2):318–21.
7. Smith TP, McNaughton LR, Marshall KJ. Effects of 4-wk training using Vmax/Tmax on VO2max and performance in athletes. Med Sci Sports Exerc. 1999 Jun;31(6):892–6.
8. Karvonen J, Vuorimaa T. Heart rate and exercise intensity during sports activities. Practical application. Sports Med. 1988 May;5(5):303–11.
9. Binder RK, Wonisch M, Corra U, Cohen-Solal A, Vanhees L, Saner H, et al. Methodological approach to the first and second lactate threshold in incremental cardiopulmonary exercise testing. Eur J Cardiovasc Prev Rehabil. 2008 Dec;15(6):726–34.
10. Conconi F, Ferrari M, Ziglio PG, Droghetti P, Codeca L. Determination of the anaerobic threshold by a noninvasive field test in runners. J Appl Physiol Respir Environ Exerc Physiol. 1982 Apr;52(4):869–73.
11. Artiga Gonzalez A, Bertschinger R, Brosda F, Dahmen T, Thumm P, Saupe D. Kinetic analysis of oxygen dynamics under a variable work rate. Hum Mov Sci. 2019 Aug;66:645–58.
12. Mongin D, Chabert C, Uribe Caparros A, Collado A, Hermand E, Hue O, et al. Validity of dynamical analysis to characterize heart rate and oxygen consumption during effort tests. Scientific Reports. 2020 Jul 24;10(1):12420.
13. Mongin D, Chabert C, Caparros AU, Guzmán JFV, Hue O, Alvero-Cruz JR, et al. The complex relationship between effort and heart rate: a hint from dynamic analysis. Physiol Meas. 2020 Nov;41(10):105003.
14. Borresen J, Lambert MI. Autonomic control of heart rate during and after exercise : measurements and implications for monitoring training status. Sports Med. 2008;38(8):633–46.
15. Bellenger CR, Fuller JT, Thomson RL, Davison K, Robertson EY, Buckley JD. Monitoring Athletic Training Status Through Autonomic Heart Rate Regulation: A Systematic Review and Meta-Analysis. Sports Med. 2016 Oct;46(10):1461–86.
16. R Core Team. R: A Language and Environment for Statistical Computing [Internet]. Vienna, Austria: R Foundation for Statistical Computing; 2019. Available from: https://www.R-project.org
17. Denis Mongin / HRR_comparison [Internet]. GitLab. [cited 2021 Mar 19]. Available from: https://gitlab.com/dmongin/hrr_comparison

##### Access

Access Policy:
Anyone can access the files, as long as they conform to the terms of the specified license.

PhysioNet Contributor Review Health Data License 1.5.0

Data Use Agreement:
PhysioNet Contributor Review Health Data Use Agreement 1.5.0

##### Corresponding Author
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## Files

Total uncompressed size: 22.1 MB.

##### Access the files
wget -r -N -c -np https://physionet.org/files/treadmill-exercise-cardioresp/1.0.1/


text:
# gravitational waves and inflation theory

I am not a technical guy and I have no scientific knowloedge in physics but I have been reading books, watching videos in order to understand our cosmology and existance. Forgive me if I ask nonsense questions, if I do I'd do it because of my lack of knowledge.

As we all know gravitational waves have been found. I have also been reading Hidden Reality of Brian Greene which I think a little bit confusing (it is funny that his videos are much more understandable than his books). I understand that GW occur on very high levels of energy (although I don't why it requires such energy levels). Based on this, what I want to ask are:

• Why GW requires such energy levels?
• Why attractive gravity doesn't create GW?
• removed

I hope I explained me and my questions clear.

Regards

-
What you have here are three questions that could each stand on their own. If I might, I recommend asking one question covering your first two questions and separate the third question to be on its own. The reason for the end of inflation is fairly distinct and deserving of a stand-alone answer. –  Jim the Enchanter Jun 25 '14 at 13:02
That said, these are very good questions. –  Jim the Enchanter Jun 25 '14 at 13:02
Thank you @Jim for your positive comments. The fact that I did not separate them is, although I first searched the topics and could not find answers, because I did not want to bother people with possibly duplicate questions and I wanted group them under the explanation that I am not a science guy. Nevertheless, I might better separate the third one as you suggested. Thanks again. –  iso_9001_ Jun 25 '14 at 13:05

Gravitational waves have never been directly detected.

Gravitational waves are predicted by general relativity and have been inferred from other observations.

Strong evidence of gravitational waves is the change in period of the Hulse-Talyor binary star system. Energy is being lost from the system at a rate consistent with radiation of gravitational waves.

The March 17th 2014 report of evidence of primordial gravitational waves is not a direct observation of the waves. Instead photons were being observed and gravitation waves inferred from the photons. Now that the research group (BICEP2) has published their work in a peer reviewed journal, they state that dust may in fact be responsible for the effect attributed to gravitational waves. Quoting their published paper "models are not sufficiently constrained ... to exclude the possibility of dust emission bright enough to explain the entire excess signal". So it is not yet confirmed that primordial gravitation waves have been detected.

•Why GW requires such energy levels?

There is no threshold energy for producing gravitation waves. For example a rotating rod, rotating perpendicualr to the axis of the rod, would produce gravitation waves. However gravitation is relatively weak force and it is very difficult, so far impossible, to detect gravitational waves.

•Why attractive gravity doesn't create GW?

At the risk of oversimplifying, time varying gravitational fields produce gravitational waves. Or more technically, the nth time derivative of the n-multipole moment (such as the 3rd derivative of the quadrople moment) of the stress-energy tensor must be non-zero for gravitation waves to be produced. Attractive gravity (such a two objects attracting each other and orbiting their common center of mass) can cause graviational waves if there is time dependence such as orbitting.

-
I recall reading about gravity waves detected from a binary pulsar or something like that. It could be what you referred to, but I'm going to double check on that nonetheless. Even in March they were saying that hose gravity waves would not be the first detection; merely the first detection of primordial GWs. –  Jim the Enchanter Jun 25 '14 at 13:39
the Hulse-Talyor binary is a pulsar. Gravitational waves are inferred from the change in period of the pulsar. The waves are not directly observed. –  DavePhD Jun 25 '14 at 13:41
Well beyond some sort of gravity-sensing telescope, I can't imagine what would qualify as direct observation. –  Jim the Enchanter Jun 25 '14 at 13:46
@Jim The Laser Interferometer Gravitation Wave Observatory (LIGO) ligo.caltech.edu is one experiment that attempts direct observation –  DavePhD Jun 25 '14 at 13:51
I'd say that qualifies as a gravity-sensing telescope just as VLBI qualifies as a tectonic motion-sensing telescope. Pretty cool though –  Jim the Enchanter Jun 25 '14 at 13:56

From the title I'd guess you're thinking about the reports that gravitational waves had been detected by the BICEP2 team. They measured the polarisation of the cosmic microwave background and found patterns that they concluded must have been caused by primordial gravitational waves during inflation. However it now seems likely that they misjudged the background noise in their data, and it's no longer clear whether they did detect GW or not. If anyone's interested their paper is here, or Google BICEP2 for many, many blog articles on the subject.

So at the moment the only secure GW detection is the indirect detection from measurements of the Hulse-Taylor binary, as Dave mentioned in his answer. However this doesn't invalidate your question and we can still address the points you raised.

Why GW requires such energy levels?

Cosmic inflation happened early in the evolution of the universe when energies were enormously high, and (assuming our theories are correct) it would have generated copious amounts of gravitational radiation. But gravitational waves don't require a high energy. As an analogy consider creating sound waves: if you clap your hands it makes a sound people can hear from a few yards away, while if you let off a hydrogen bomb it makes a sound people can hear from miles away! It doesn't take high energies to make a sound, but high energies make a very loud sound.

In principle you and I generate gravity waves just by moving around, but the intensity of these waves is absurdly small. The Hulse-Taylor binary is vastly more energetic than you and I, but even that generates such small amounts of gravitational waves that it requires exceedingly careful measurements to detect them. It isn't that the very high energies involved at the time of inflation were needed to generate gravitational waves, but rather that such high energies were needed to generate gravitational waves that we can still detect 13.7 billion years later.

Why attractive gravity doesn't create GW?

It isn't the attractiveness or otherwise of gravity that matters. Gravity waves are created by any distribution of matter or energy that has an oscillating quadropole moment. This would apply to exotic matter and negative matter (if they existed, which they probably don't) in the same way it applies to normal matter.

And finally:

If GW supports Inflation Theory, why the exponential growth of the universe stopped (or did it ever stop?)

No-one knows the answer to this because we don't have a fundamental theory of inflation. The various theories of inflation postulate a scalar field that causes it, but without giving any concrete description of the physics that creates the field. Generally the field will be selected so inflation ends after the required number of e-foldings.

There are attempts to provide fundamental descriptions of the scalar field, e.g. linking it to the Higgs particle, but as yet no widely accepted theory has been found.

-
This is really thorough, thank you. May I ask, why the universe needed inflationary (I mean exponential) growth? –  iso_9001_ Jun 25 '14 at 14:58
@iso_9001_: the universe doesn't need inflation, we physicists need inflation for our theories to work :-) Without it we have no way of explaining away the flatness problem. It's still possible we might find some other explanation and inflation didn't happen. This seems pretty unlikely though - for example the spectrum of fluctuations in the CMB agrees with the predicitoons of inflation, though this is not a proof. –  John Rennie Jun 25 '14 at 15:01
Now I understood clearly what you meant :) I was pushing myself to understand the necessity, the absolute requirement for Inflation but could not figure it out. Now I see that it might be the case, but it doesn't have to be. There is one last question I want to ask. I read somewhere that GW clarifies the last point of Einstein's General Relativity which I believe is Repulsive Gravity. However I could not make sure. Is it really about Repulsive Gravity? –  iso_9001_ Jun 25 '14 at 15:06
@iso_9001_: By repulsive gravity do you mean dark energy or the cosmological constant? If so I don't know of any connection to gravitational waves. –  John Rennie Jun 25 '14 at 15:13
Actually, I am not sure what I am talking about. In Brian Greene's book Hidden Reality, he says it is cosmological constant. However, I read before that Einstein accepted this cosmological constant as his biggest mistake so I am confused here. What I was trying to ask is, while reading news about potential discovery of GW, people said This also sheds light to the one last undiscovered aspect of General Relativity and I don't know what that last aspect is. I though it would be Repulsive Gravity that pushes matter rahter attracting them. –  iso_9001_ Jun 25 '14 at 15:21


text:
Free Version
Moderate

# Continuity of Piecewise Cosine Functions

APCALC-0NB5QF

Discuss the continuity of:
$$f(x)=\begin{Bmatrix} \cos(x)\quad & x<-\pi \\\ |\cos(x)| & -\pi \le x\le \frac { \pi }{ 2 } \\\ { \cos }^{ 2 }(x) & x>\frac { \pi }{ 2 } \end{Bmatrix}$$

A

$f(x)$ is has a jump discontinuity at $x=\cfrac {\pi }{2}$.

B

$f(x)$ is continuous on $[-2\pi , 0]$.

C

$f(x)$ is continuous on $[-\pi , \pi ]$.

D

$f(x)$ is continuous for all real numbers.


text:
Document Type : Original Manuscript

Authors

1 Department of Mathematics, Shahid Chamran University of Ahvaz, P.O. Box 6135783151, Ahvaz, Iran.

2 Department of Science, Petroleum University of Technology, P.O. Box 6318714317, Ahvaz, Iran.

Abstract

Let $C_c(X)$ (resp., $C_c^*(X)$) denote the functionally
countable subalgebra of $C(X)$ (resp., $C^*(X)$),
consisting of all functions (resp., bounded functions) with countable image.
$C_c(X)$ (resp., $C_c^*(X)$) as a topological ring via $m_c$-topology (resp., $m^*_c$-topology) and $u_c$-topology (resp., $u^*_c$-topology) is investigated and the equality of the latter two topologies is characterized.
Topological spaces which are called $N$-spaces are introduced and studied.
It is shown that the $m_c$-topology on $C_c(X)$ and its relative topology as a subspace of $C(X)$ (with $m$-topology) coincide if and only if $X$ is an $N$-space. We also show that $X$ is pseudocompact if and only if it is both a countably pseudocompact, and an $N$-space.

Keywords

###### ##### References
1. S. K. Acharyya, R. Bharati, and A. D. Ray, Rings and subrings of continuous functions with countable range, Queast. Math., 44(6) (2021) 829–848.
2. F. Azarpanah, O. A. S. Karamzadeh, Z. Keshtkar, and A. R. Olfati, On maximal ideals of Cc(X) and the uniformity of its localization, Rocky Mt. J. Math., 48(2) (2018), 1–9.
3. F. Azarpanah, F. Manshoor, and R. Mohamadian, Connectedness and compactness in C(X) with the m-topology and generalized m-topology, Topology Appl., 159 (2012) 3486–3493.
4. E. van Douwen, Nonnormality or hereditary paracompactness of some spaces of real functions, Topology Appl., 39 (1991) 3–32.
5. R. Engelking, General Topology, Berlin, Germany, Heldermann Verlag, 1989.
6. M. Ghadermazi, O. A. S. Karamzadeh, and M. Namdari, On the functionally countable subalgebra of C(X), Rend. Sem. Mat. Univ. Padova, 129 (2013), 47–69.
7. M. Ghadermazi, O. A. S. Karamzadeh, and M. Namdari, C(X) versus its functionally countable subalgebra, Bull. Iran. Math. Soc., 245 (2019), 173– 187.
8. L. Gillman and M. Jerison, Rings of continuous functions, Springer-Verlag, 1976.
9. J. Gomez-Pérez and W. W. McGovern, The m-topology on Cm(X) revisited, Topology Appl., 153 (2006) 1838–1848.
10. E. Hewitt, Rings of real-valued continuous functions I, Trans. Amer. Math. Soc., 48(64) (1948) 54–99.
11. G. Di Maio, L’. Holá, D. Holý, and D. McCoy, Topology on the space of continuous functions, Topology Appl., 86 (1998) 105–122.
12. A. Veisi, On the mc-topology on the functionally countable subalgebra of C(X), J. Algebr. Syst., 9(2) (2022) 335–345.


text:
#### Iteration and Recursion One Mark Questions

11th Standard

Reg.No. :
•
•
•
•
•
•

Computer Science

Time : 00:30:00 Hrs
Total Marks : 10
10 x 1 = 10
1. A loop invariant need not be true

(a)

at the start of the loop

(b)

at the start of each iteration

(c)

at the end of each iteration

(d)

at the start of the algorithm

2. We wish to cover a chessboard with dominoes, the number of black squares and the number of white squares covered by dominoes, respectively, placing a domino can be modeled by

(a)

b:= b + 2

(b)

w:= w + 2

(c)

b, w := b1+, w1+

(d)

b:= w

3. If m x a + n x b is an invariant for the assignment a, b : = a + 8, b + 7, the values of m and n are

(a)

m=8,n=7

(b)

m=7,n=-8

(c)

m=7,n=8

(d)

m =8,n=-7

4. If Fibonacci number is defined recursively as$f(n)=\begin{cases} 0n=0 \\ 1n=1 \\ f(n-1)+f(n-2)otherwise \end{cases}$to evaluate F(4), how many times F() is applied?

(a)

3

(b)

4

(c)

8

(d)

9

5. Using this recursive definition${ a }^{ n }=\begin{cases} 1\quad \quad \quad \quad \quad \quad ifn=0 \\ a\times { a }^{ n }-1\quad otherwise \end{cases}$how many multiplications are needed to calculate a10?

(a)

11

(b)

10

(c)

9

(d)

8

6. Which of the following algorithm design techniques to execute the same action repeatedly

(a)

Assignment

(b)

Iteration

(c)

Recursion

(d)

Both b and c

7. Which statements executed repeatedly as long as the loop condition is true?

(a)

Sequential

(b)

Abstraction

(c)

Iteration

(d)

Assignment

8. Which of the following is updated when each time the loop body is executed?

(a)

Data

(b)

Variables

(c)

Function

(d)

All of these

9. The unchanged variables of the loop body is called loop

(a)

condition

(b)

loop statement

(c)

loop variable

(d)

loop invariant

10. Which of the following is the key to construct iterative algorithms

(a)

loop invariant

(b)

loop updation

(c)

loop variable

(d)

lop condition


text:
# thraxil.org:

## Wacko Jacko Humor

For some time-wasting Michael Jackson Humor:

http://www.cagle.com/news/MichaelJacksonTrial/main.asp

I wonder how much time MJ spends in AOL chat rooms :-)

## The Legendary Couch-Moving Epic

I would like to share with the readers of thraxil my amazing adventure from last week:

http://www.ericmattes.com/couch/

TAGS: couch

## Microsoft Passport Exploit

I just found out about this today. An unthinkably simple exploit allows anybody to change anybody else’s password to whatever they want. Microsoft found out about the problem after somebody posted it to a public message board. The FTC is considering a ridiculously large fine to punish Microsoft. See article here:

http://theregister.co.uk/content/6/30620.html

Detailed report here:

http://securitytracker.com/alerts/2003/May/1006728.html

This brings to mind the issue… MS seems to be suffering from their own popularity. I’ve heard the excuse “nobody ever got fired for choosing microsoft.” It’s true, and I think this is why they can never be dethroned. Unfortunately there are legions of hackers who have devoted themselves to exploiting flaws in MS products and punishing those who use them. So… they will always be number 1 despite the fact that they are notoriously flawed. Obviously there are flaws in competitor’s software, but nobody is trying to write viruses for those programs! It’s sad that so many people depend on MS software.

MS should really hire a team of hackers to hack their software before it is released or else they will never see the end of stuff like this.

TAGS: microsoft

## Rock, Paper, Scissors

There’s a whole world of RPS out there that I never knew about! Although it’s a pretty trivial game for most people, computer scientists (being who they are) have been exploring the strategy behind this seemingly random game. However, you’ve got to admit that when you play, you do have a strategy of some sort. It is this fact that the computer RPS opponents will attempt to exploit. Check out this page for the current champion:

http://ofb.net/~egnor/iocaine.html

also check this one out for some real-time action that you can play on the web:

http://www.csua.berkeley.edu/~emin/writings/lz_rps/

-E


text:
## How to Calculate and Solve for the Area, Length and Height of a Trapezium | The Calculator Encyclopedia

The image above is a Trapezium

To compute the area of a trapezium, three essential parameter is needed and they are the length of the top side (a), length of the bottom side (b) and the height of the trapezium (h).

The formula for calculating the area of a trapezium is;

A = 0.5[a + b]h

Where;
A = Area of a trapezium
a = length of the top side of the trapezium
b = length of the bottom side of the trapezium
h = height of the trapezium

Let’s solve an example;
Find the area of a trapezium where length of top side (a) is 7 cm, length of bottom side (b) is 12 cm and height of trapezium (h) is 10 cm.

This implies that;
a = Length of top side of the trapezium = 7 cm
b = Length of bottom side of the trapezium = 12 cm
h = Height of the trapezium = 10 cm

A = 0.5[a + b]h
A = 0.5[7 + 12]10
A = 0.5[19]10
A = 95

Therefore, the area of a trapezium is 95 cm2

How to Calculate the Height of a Trapezium when the Area, Length of top side and Length of bottom side of the Trapezium is given.

The formula is h = A / 0.5(a + b)

Where;
A = Area of the trapezium
a = Length of the top side of the trapezium
b = Length of the bottom side of the trapezium

Let’s solve for an example;
Given that the length of top side (a) is 10 cm, length of bottom side (b) is 14 cm and the area of the trapezium is 20 cm2 Find the height of the trapezium?

This implies that;

A = Area of the trapezium = 20 cm2
a = Length of top side of the trapezium = 10 cm
b = Length of bottom side of the trapezium = 14 cm

h = A / 0.5(a + b)
h = 20 / 0.5(10 + 14)
h = 20 / 0.5(24)
h = 20 / 12
h = 1.667

Therefore, the height of the trapezium is 1.667 cm.