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# zbMATH — the first resource for mathematics

Semigroups containing proximal linear maps. (English) Zbl 0845.22004
Let $$V$$ be a finite dimensional vector space. A linear automorphism $$g$$ of $$V$$ is called a proximal element if $$g$$ has a unique eigenvalue $$\lambda = \lambda (g)$$ of maximal absolute value and the corresponding weight space $$V_n = \{v \in V; (g - \lambda E)^n v = 0$$ for some $$n\}$$ is one-dimensional. Goldsheid and Margulis noticed in 1981 that if a subgroup $$G$$ of $$GL(V)$$ contains a proximal element then so does every Zariski dense subsemigroup $$H$$ of $$G$$, provided $$V$$, considered as a $$G$$-module, is strongly irreducible. The authors show further that if $$H$$ is a strongly irreducible subsemigroup of $$GL(V)$$ and if the algebraic closure $$G$$ of $$H$$ contains a proximal element, then there is an $$\varepsilon > 0$$ and for every $$r \geq 1$$ there is a finite subset $$M$$ of $$H$$ such that for every $$g \in GL(V)$$ there is an $$r \in M$$ such that $$\gamma g$$ is $$(r, \varepsilon)$$-proximal. Thus, under the above hypotheses, the subsemigroup $$H$$ provides a rich supply of proximal elements. Extensions and refinements of the above result are obtained in the following directions: a quantitative version of proximality; reducible representations; several eigenvalues of maximal modules. The authors also point out that proximal elements play a crucial role in the dynamics of linear maps because of their simple structure. Generalizations of linear proximal maps to Lyapunov filtrations and to several irreducible summands are obtained.

##### MSC:
 22A15 Structure of topological semigroups
Full Text:
##### References:
 [1] [A] L. Auslander,The structure of compact locally affine manifolds, Topology3 (1964), 131–139. · Zbl 0136.43102 · doi:10.1016/0040-9383(64)90012-6 [2] [BL] Y. Benoist and F. Labourie,Sur les difféomorphismes d’Anosov affines à feuilletages stables et instables différentiables, Preprint. [3] [B] A. Borel,Introduction aux groupes arithmétiques, Hermann, Paris, 1969. · Zbl 0186.33202 [4] [BT] A. Borel and J. Tits,Groupes réductifs, Publications de Mathématiques de l’IHES27 (1965), 55–151. [5] [D] T. Drumm,Fundamental polyhedra for Margulis space-times, Doctoral dissertation University of Maryland, 1990. · Zbl 0773.57008 [6] [DG] T. Drunm and W. Goldman,Complete flat Lorentz 3-manifolds with free fundamental group, International Journal of Mathematics1 (1990), 149–161. · Zbl 0704.57026 · doi:10.1142/S0129167X90000101 [7] [FG] D. Fried and W. M. Goldman,Three-dimensional affine crystallographic groups, Advances in Mathematics47 (1983), 1–49. · Zbl 0571.57030 · doi:10.1016/0001-8708(83)90053-1 [8] [F] H. Furstenberg,Boundary theory and stochastic processes on homogeneous spaces, Proceedings of Symposia in Pure Mathematics (Williamstown, MA, 1972), American Mathematical Society, 1973, pp. 193–229. [9] [GK] W. M. Goldman and Y. Kamishima,The fundamental group of a compact flat Lorentz space form is virtually polycyclic, Journal of Differential Geometry19 (1984), 233–240. · Zbl 0546.53039 [10] [GM] I. Ya. Goldsheid and G. A. Margulis,Lyapunov exponents of a product of random matrices, Uspekhi matematicheskikh Nauk44 (1989), 13–60. English translation in Russian Mathematics Surveys44 (1989), 11–71. [11] [G] I. Ya. Goldsheid,Quasi-projective transformations, Preprint 220, SFB 237 Bochum, 1994. [12] [GrM] F. Grünewald and G. A. Margulis,Transitive and quasitransitive actions of affine groups preserving a generalized Lorentz-structure, Journal of Geometry and Physics5 (1988), 493–531. · Zbl 0706.57022 · doi:10.1016/0393-0440(88)90017-4 [13] [GR] Y. Guivar’ch and A. Raugi,Propriétés de contraction d’un semigroupe de matrices inversibles. Coefficients de Liapunoff d’un produit de matrices aléatoires indépendantes, Israel Journal of Mathematics65 (1989), 165–196. · Zbl 0677.60007 · doi:10.1007/BF02764859 [14] [Ma1] G. A. Margulis,Free properly discontinuous groups of affine transformations, Doklady Akademii Nauk SSSR272 (1983), 937–940. [15] [Ma2] G. A. Margulis,Complete affine locally flat manifolds with a free fundamental group, Journal of Soviet Mathematics134 (1987), 129–134, translated from Zapiski NS LOMI134 (1984), 190–205. · Zbl 0611.57023 · doi:10.1007/BF01104978 [16] [Ma3] G. A. Margulis,On the Zariski closure of the linear part of a properly discontinuous group of affine transformations, Preprint (1991). [17] [Ma4] G. A. Margulis,Discrete Subgroups of Semisimple Lie Groups, Springer, Berlin, 1991. [18] [MS] G. A. Margulis and G. A. Soifer,Maximal subgroups of infinite index in finitely generated linear groups, Journal of Algebra69 (1981), 1–23. · Zbl 0457.20046 · doi:10.1016/0021-8693(81)90123-X [19] [Me] G. Mess,Flat Lorentz spacetimes, Preprint, February 1990. [20] [Mi] J. Milnor,On fundamental groups of complete affinely flat manifolds, Advances in Mathematics25 (1977), 178–187. · Zbl 0364.55001 · doi:10.1016/0001-8708(77)90004-4 [21] [P] G. Prasad, $$\mathbb{R}$$-regular elements in Zariski-dense subgroups, Oxford Quarterly Journal of Mathematics, to appear. · Zbl 0828.22010 [22] [PR] G. Prasad and M. S. Raghunathan,Cartan subgroups and lattices in semi-simple groups, Annals of Mathematics96 (1972), 296–317. · Zbl 0245.22013 · doi:10.2307/1970790 [23] [Ra] M. S. Raghunathan,Discrete Subgroups of Lie Groups, Springer, Berlin, 1972. · Zbl 0254.22005 [24] [S] G. A. Soifer,Affine discontinuous groups, inAlgebra and Analysis (L. A. Bokut’, M. Hazewinkel and Yu. Reshetnyak, eds.), Proceedings of the First Siberian School, Lake Baikal, American Mathematical Society, 1992. [25] [Ti] J. Tits,Free subgroups in linear groups, Journal of Algebra20 (1972), 250–270. · Zbl 0236.20032 · doi:10.1016/0021-8693(72)90058-0 [26] [To] G. Tomanov,The virtual solvability of the fundamental group of a generalized Lorentz space form, Journal of Differential Geometry32 (1990), 539–547. · Zbl 0681.57027
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text:
User Details

User Since
Nov 18 2019, 7:30 PM (36 w, 6 d)
Availability
Available
LDAP User
Mstyles
MediaWiki User

Jun 30 2020

Mstyles added a comment to T251514: UI for SPARQL Endpoint for Commons.

Since the gui directory is overwritten by scap via symlinks, Guillaume proposed that the config files live in /etc/config and that they be symlinked to the gui directory. See more discussion here:https://gerrit.wikimedia.org/r/c/operations/puppet/+/606297. I want to remove the custom config file that lives in the gui-deploy repo to avoid any confusion of config files: https://gerrit.wikimedia.org/r/c/wikidata/query/gui-deploy/+/606545

Jun 18 2020

Mstyles updated the task description for T147505: [EPIC][Recurring task] CirrusSearch: what is updated during re-indexing.

May 29 2020

We need to disable testing profiles on the beta cluster. Could be changed in SearchSatisfaction or don't distribute the testing config to the beta cluster

Mstyles added projects to T253612: Excessive HTML entity encoding in title tag: I18n, Language-Team.

It looks like this might be an issue with the translation, https://translatewiki.net/w/i.php?title=MediaWiki:Searchresults-title/wa&diff=9392337&oldid=1189285

May 23 2020

Mstyles added a comment to T251500: oAuth authentication for SPARQL Endpoint for Commons.

Some notes for when/if we do oauth
Docs -> https://www.mediawiki.org/wiki/OAuth/For_Developers
we might be able to use this npm package for auth with mediawiki
This is an example of a tools project that uses auth and restricts who can log into the tool

May 18 2020

Mstyles updated subscribers of T251515: Automate data reload for SPARQL Endpoint for Commons.

As discussed in email with @Zbyszko the script should do the following

May 5 2020

Deployed and the example links work. Marking this as done

Mstyles updated the task description for T147505: [EPIC][Recurring task] CirrusSearch: what is updated during re-indexing.
Mstyles updated the task description for T147505: [EPIC][Recurring task] CirrusSearch: what is updated during re-indexing.

May 4 2020

[maven-release-plugin] prepare release discovery-maven-tool-configs-1.13
[maven-release-plugin] prepare for next development iteration
[maven-release-plugin] prepare release discovery-parent-pom-1.39
[maven-release-plugin] prepare for next development iteration
[maven-release-plugin] prepare for next development iteration
[maven-release-plugin] prepare release discovery-parent-pom-1.38
Mstyles committed rDPOMf050072db70a: Rename deploy archiva profile repository Id (authored by Mstyles).
Rename deploy archiva profile repository Id
Mstyles updated subscribers of T247123: Migrate wikidata-query-rdf-release-silent release job to Docker.

Talked to @Gehel and the issue is probably that the archiva credentials that come from analytics have different server ID's than the ones we use. https://github.com/wikimedia/wikimedia-discovery-discovery-parent-pom/blob/master/pom.xml#L917 vs https://github.com/wikimedia/analytics-refinery-source/blob/master/pom.xml#L96. We can either change our pom to match analytics or add a separate credential in jenkins. @Gehel prefers changing our pom to match analytics for uniformity

I know what that is, I forgot to add the archiva deployment profile. I'll put a patch out

Apr 29 2020

@hashar the jenkins job failed due to no git auth to push. you can see it here: https://integration.wikimedia.org/ci/job/wikidata-query-rdf-maven-release-docker-wdqs/3/

Apr 15 2020

This will still send a lot of logs next time we do a reindex. We should be doing one in the near future. I can leave this open until the extra logs are removed.

Mstyles added a comment to P10980 (An Untitled Masterwork).

ended up tweaking it just a little bit so I'll leave that here just on the off chance someone else (or me) needs it

import elasticsearch
import sys

Apr 13 2020

Mstyles added a comment to T222669: Normalize homoglyphs in mixed-script tokens when possible.

From the analysis chain analysis comparing the chain with and without the homoglyph token filter on a sample of 10,000 random articles for each language:

Discussed and this might be a task better served by SRE tooling and possibly for a future Search Platform SRE person

Apr 8 2020

I tested and everything works. Thanks @elukey so much for all of your help getting this done! I'm going to go ahead and mark this as closed.

Mar 18 2020

Mstyles renamed T246961: Add Kibana to Relforge from Add kibana to Relforge to Add Kibana to Relforge.

Mar 11 2020

Instead of filtering the query string queries, we want to move off of query string for spaceless languages and on to using the full text simple match query builder. This will help when we upgrade elastic search and no longer use query strings. In order to the make this move, the FTSM query builder has to be tested in relforge for Japanese. There's currently an upgrade going on with relforge, so this task will be paused until the python upgrade for relforge is complete.

Feb 13 2020

Mstyles added a comment to T219534: Test MLR models for zhwiki, jawiki and kowiki.

Config changes have been deployed but due to a configuration conflict with jawiki using the default for wgCirrusSearchFullTextQueryBuilderProfile (see https://www.mediawiki.org/wiki/User:TJones_(WMF)/Notes/Spaceless_Writing_Systems_and_Wiki-Projects), some further changes will have to be made to have the model name show up here

Mstyles edited projects for T219534: Test MLR models for zhwiki, jawiki and kowiki, added: Discovery-Search (Current work); removed Discovery-Search.

Feb 10 2020

Mstyles added a comment to T241291: Simplify WDQS Packaging.

@Jdforrester-WMF WMDE will be taking on responsibility for any new deployment methods. That work will be tracked in T192006 and T210286.

Feb 7 2020

Mstyles updated the task description for T147505: [EPIC][Recurring task] CirrusSearch: what is updated during re-indexing.
Mstyles updated the task description for T147505: [EPIC][Recurring task] CirrusSearch: what is updated during re-indexing.

Jan 21 2020

Mstyles added a comment to T241291: Simplify WDQS Packaging.

@Addshore that's correct, after removing the gui submodule, I won't be doing any further work

Jan 17 2020

very exciting to see it work here: https://gerrit.wikimedia.org/r/c/search/extra/+/563267. I know @Gehel mentioned trying to refactor where the same job runs for both pre-merge and post-merge but after chatting with @Jdforrester-WMF, it seems that convention is to have separate pre and post merge jobs. I would be happy to call this done.

Tabling this for now as it's not urgent

Mstyles added a comment to T241291: Simplify WDQS Packaging.

After a bunch of discussion with the team, it's been decided that removing the gui submodule from the RDF repository will suffice for now. That will fix our broken build issues (see https://phabricator.wikimedia.org/T242640) @Ladsgroup I definitely think you should work on that patch and getting things going with service runner if you have the bandwidth.

Jan 16 2020

@kostajh do we still need to separate sonar args for master vs non master branches then? it seems that we should be able to send all of the same sonar args whether or not the branch is master. I'm not sure how to tell the bot if something is pre or post merge.

Jan 13 2020

Mstyles added a comment to T241291: Simplify WDQS Packaging.

Had a quick sync meeting with WMDE. The outcome of that was to use this node patch as a starting point for service runner. It's unclear whether or not blubber needs to be involved in this process. Also, ideally the public image for the WDQS UI would be eliminated in favor of the new image used for this new build process.

Jan 10 2020

@Gehel I think we can consider this closed unless someone is able to reproduce

Jan 9 2020

Mstyles added a comment to T241291: Simplify WDQS Packaging.

@akosiaris Could we possibly use miscweb in front of a VM as an interim to serve up the static files before moving to service template?

@kostajh everything has been merged, and the code health job runs with sonar analysis after a patch for java projects. However we're not seeing any results from bots in the test patch in search extra (https://gerrit.wikimedia.org/r/563250) with analysis here: https://sonarcloud.io/project/activity?id=org.wikimedia.search%3Aextra-parent. Does the bot know about java projects?

you're right, it's a typo. It should be /run-java.sh. Pushing up a patch now

Mstyles added a comment to T237165: LDF server has 404 errors for JS and CSS resources.

for clarification the correct response will contain a list that looks like this

@prefix schema: <http://schema.org/> .
@prefix pq:    <http://www.wikidata.org/prop/qualifier/> .
@prefix pr:    <http://www.wikidata.org/prop/reference/> .
@prefix ps:    <http://www.wikidata.org/prop/statement/> .

and the incorrect response is HTML that looks similar to

<!DOCTYPE html><meta charset="utf-8"><meta http-equiv="X-UA-Compatible" content="IE=edge"><meta name="viewport" content="width=device-width,initial-scale=1,user-scalable=yes"><link rel="stylesheet" href="css/style.min.6c0e4865f687302c4d99.css"><link id="favicon" rel="shortcut icon">

Jan 8 2020

Mstyles added a comment to T237165: LDF server has 404 errors for JS and CSS resources.

the following curls return the correct data
curl localhost:80/bigdata/ldf -> direct to nginx server on host
curl localhost:9999/bigdata/ldf -> direct to query service on host
but curl https://query.wikidata.org/bigdata/ldf is not working indicating some problem with routing traffic.
This could be the recent switch from varnish to apache.

Dec 20 2019

I am all for reducing duplication but in this case, perhaps we can see if we get it working first and then try to reduce the duplication?

Dec 18 2019

What I was trying to say is that all of the projects are currently sending their analysis to SonarQube, so I didn't want to change any postmerge jobs. I put what I thought in the patch

I think specifically the updates are around this ticket, https://phabricator.wikimedia.org/T235833

format: plain


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# American Institute of Mathematical Sciences

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Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter
March  2013, 12(2): 803-813. doi: 10.3934/cpaa.2013.12.803

## Some results on two-dimensional Hénon equation with large exponent in nonlinearity

 1 Department of Mathematics, East China Normal University, Shanghai 200241, China

Received  September 2011 Revised  December 2011 Published  September 2012

The Hénon equation on a bounded domain in $R^2$ with large exponent in the nonlinear term is studied in this paper. We investigate positive solution obtained by the variational method and give its asymptotic behavior as the nonlinear exponent gets large.
Citation: Chunyi Zhao. Some results on two-dimensional Hénon equation with large exponent in nonlinearity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 803-813. doi: 10.3934/cpaa.2013.12.803
##### References:
 [1] Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity,, Proc. Amer. Math. Soc., 132 (2004), 1013. doi: 10.1090/S0002-9939-03-07301-5. Google Scholar [2] V. Barutello, S. Secchi and E. Serra, A note on the radial solutions for the supercritical Hénon equation,, J. Math. Anal. Appl., 341 (2008), 720. doi: 10.1016/j.jmaa.2007.10.052. Google Scholar [3] J. Byeon and Z.-Q. Wang, On the Hénon equation: asymptotic profile of ground states,, I. Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 23 (2006), 803. doi: 10.1016/j.anihpc.2006.04.001. Google Scholar [4] J. Byeon and Z.-Q. Wang, On the Hénon equation: asymptotic profile of ground states. II,, J. Differential Equations, 216 (2005), 78. doi: 10.1016/j.jde.2005.02.018. Google Scholar [5] W. X. Chen and C. M. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar [6] G. Chen, W.-M. Ni and J.X. Zhou, Algorithms and visualization for solutions of nonlinear elliptic equations,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1565. doi: 10.1142/S0218127400001006. Google Scholar [7] D. M. Cao and S. J. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation,, J. Math. Anal. Appl., 278 (2003), 1. doi: 10.1016/S0022-247X(02)00292-5. Google Scholar [8] D. M. Cao, S. J. Peng and S. S. Yan, Asymptotic behaviour of ground state solutions for the Hénon equation,, IMA J. Appl. Math., 74 (2009), 468. doi: 10.1093/imamat/hxn035. Google Scholar [9] M. Calanchi, S. Secchi and E. Terraneo, Multiple solutions for Hénon-like equation on the annulus,, J. Differential Equations, 245 (2008), 1507. doi: 10.1016/j.jde.2008.06.018. Google Scholar [10] P. Esposito, A. Pistoia and J. C. Wei, Concentrating solutions for the Hénon equation in $\mathbb R^2$,, J. Anal. Math., 100 (2006), 249. doi: 10.1007/BF02916763. Google Scholar [11] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125. Google Scholar [12] M. Gazzini and E. Serra, The Neumann problem for the Hénon equation, trace inequalities and Steklov eigenvalues,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 25 (2008), 281. doi: 10.1016/j.anihpc.2006.09.003. Google Scholar [13] M. Hénon, Numerical experiments on the stability of spherical stellar systems,, Astronom. Astrophys., 24 (1973), 229. Google Scholar [14] S. J. Li and S. J. Peng, Asymptotic behavior on the Hénon equation with supercritical exponent,, Sci. China Ser. A, 52 (2009), 2185. doi: 10.1007/s11425-009-0094-7. Google Scholar [15] W.-M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications,, Indiana Univ. Math. J., 31 (1982), 801. doi: 10.1512/iumj.1982.31.31056. Google Scholar [16] W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem,, Comm. Pure Appl. Math., 44 (1991), 819. doi: 10.1002/cpa.3160440705. Google Scholar [17] S. J. Peng, Multiple boundary concentrating solutions to Dirichlet problem of Hénon equation,, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 137. doi: 10.1007/s10255-005-0293-0. Google Scholar [18] J. Prajapat and G. Tarantello, On a class of elliptic problem in $\mathbb R^2$: symmetry and uniqueness results,, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 967. doi: 10.1017/S0308210500001219. Google Scholar [19] A. Pistoia and E. Serra, Multi-peak solutions for the Hénon equation with slightly subcritical growth,, Math. Z., 256 (2007), 75. doi: 10.1007/s00209-006-0060-9. Google Scholar [20] X. F. Ren and J. C. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity,, Trans. Amer. Math. Soc., 343 (1994), 749. doi: 10.1090/S0002-9947-1994-1232190-7. Google Scholar [21] X. F. Ren and J. C. Wei, Single-point condensation and least-energy solutions,, Proc. Amer. Math. Soc., 124 (1996), 111. doi: 10.1090/S0002-9939-96-03156-5. Google Scholar [22] E. Serra, Non radial positive solutions for the Hénon equation with critical growth,, Calc. Var. Partial Differential Equations, 23 (2005), 301. doi: 10.1007/s00526-004-0302-9. Google Scholar [23] D. Smets, J. B. Su and M. Willem, Non-radial ground states for the Hénon equation,, Commun. Contemp. Math., 4 (2002), 467. doi: 10.1142/S0219199702000725. Google Scholar [24] D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problems,, Calc. Var. Partial Differential Equations, 18 (2003), 57. doi: 10.1007/s00526-002-0180-y. Google Scholar

show all references

##### References:
 [1] Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity,, Proc. Amer. Math. Soc., 132 (2004), 1013. doi: 10.1090/S0002-9939-03-07301-5. Google Scholar [2] V. Barutello, S. Secchi and E. Serra, A note on the radial solutions for the supercritical Hénon equation,, J. Math. Anal. Appl., 341 (2008), 720. doi: 10.1016/j.jmaa.2007.10.052. Google Scholar [3] J. Byeon and Z.-Q. Wang, On the Hénon equation: asymptotic profile of ground states,, I. Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 23 (2006), 803. doi: 10.1016/j.anihpc.2006.04.001. Google Scholar [4] J. Byeon and Z.-Q. Wang, On the Hénon equation: asymptotic profile of ground states. II,, J. Differential Equations, 216 (2005), 78. doi: 10.1016/j.jde.2005.02.018. Google Scholar [5] W. X. Chen and C. M. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar [6] G. Chen, W.-M. Ni and J.X. Zhou, Algorithms and visualization for solutions of nonlinear elliptic equations,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1565. doi: 10.1142/S0218127400001006. Google Scholar [7] D. M. Cao and S. J. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation,, J. Math. Anal. Appl., 278 (2003), 1. doi: 10.1016/S0022-247X(02)00292-5. Google Scholar [8] D. M. Cao, S. J. Peng and S. S. Yan, Asymptotic behaviour of ground state solutions for the Hénon equation,, IMA J. Appl. Math., 74 (2009), 468. doi: 10.1093/imamat/hxn035. Google Scholar [9] M. Calanchi, S. Secchi and E. Terraneo, Multiple solutions for Hénon-like equation on the annulus,, J. Differential Equations, 245 (2008), 1507. doi: 10.1016/j.jde.2008.06.018. Google Scholar [10] P. Esposito, A. Pistoia and J. C. Wei, Concentrating solutions for the Hénon equation in $\mathbb R^2$,, J. Anal. Math., 100 (2006), 249. doi: 10.1007/BF02916763. Google Scholar [11] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125. Google Scholar [12] M. Gazzini and E. Serra, The Neumann problem for the Hénon equation, trace inequalities and Steklov eigenvalues,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 25 (2008), 281. doi: 10.1016/j.anihpc.2006.09.003. Google Scholar [13] M. Hénon, Numerical experiments on the stability of spherical stellar systems,, Astronom. Astrophys., 24 (1973), 229. Google Scholar [14] S. J. Li and S. J. Peng, Asymptotic behavior on the Hénon equation with supercritical exponent,, Sci. China Ser. A, 52 (2009), 2185. doi: 10.1007/s11425-009-0094-7. Google Scholar [15] W.-M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications,, Indiana Univ. Math. J., 31 (1982), 801. doi: 10.1512/iumj.1982.31.31056. Google Scholar [16] W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem,, Comm. Pure Appl. Math., 44 (1991), 819. doi: 10.1002/cpa.3160440705. Google Scholar [17] S. J. Peng, Multiple boundary concentrating solutions to Dirichlet problem of Hénon equation,, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 137. doi: 10.1007/s10255-005-0293-0. Google Scholar [18] J. Prajapat and G. Tarantello, On a class of elliptic problem in $\mathbb R^2$: symmetry and uniqueness results,, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 967. doi: 10.1017/S0308210500001219. Google Scholar [19] A. Pistoia and E. Serra, Multi-peak solutions for the Hénon equation with slightly subcritical growth,, Math. Z., 256 (2007), 75. doi: 10.1007/s00209-006-0060-9. Google Scholar [20] X. F. Ren and J. C. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity,, Trans. Amer. Math. Soc., 343 (1994), 749. doi: 10.1090/S0002-9947-1994-1232190-7. Google Scholar [21] X. F. Ren and J. C. Wei, Single-point condensation and least-energy solutions,, Proc. Amer. Math. Soc., 124 (1996), 111. doi: 10.1090/S0002-9939-96-03156-5. Google Scholar [22] E. Serra, Non radial positive solutions for the Hénon equation with critical growth,, Calc. Var. Partial Differential Equations, 23 (2005), 301. doi: 10.1007/s00526-004-0302-9. Google Scholar [23] D. Smets, J. B. Su and M. Willem, Non-radial ground states for the Hénon equation,, Commun. Contemp. Math., 4 (2002), 467. doi: 10.1142/S0219199702000725. Google Scholar [24] D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problems,, Calc. Var. Partial Differential Equations, 18 (2003), 57. doi: 10.1007/s00526-002-0180-y. Google Scholar
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2018 Impact Factor: 0.925


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# BORIS FX V10.0.1 WIN64 – XFORCE [NEW]

BORIS FX V10.0.1 WIN32 – XFORCE Autodesk® Civil 3D® 2006 – Win32 BORIS FX V10.0.1 WIN64 – XFORCE Autodesk® INSTANTANEOUS.The incident took place in the district on Monday evening, police said, adding that an investigation has been initiated. A case of dowry harassment was registered at the Namajipura police station. The victim’s family members alleged that the accused used to throw stones at their house, and the woman was pressurised to get married. The family members of the victim alleged that the police took more than three hours to register the complaint. The victim’s family were pressurised to get married. A case of dowry harassment was registered at the Namajipura police station. The victim’s family alleged that the accused used to throw stones at their house, and the woman was pressurised to get married. The family members of the victim alleged that the police took more than three hours to register the complaint. The victim’s sister said that the family had previously informed the police but they did not take any action. However, they informed the police only when the accused began to throw stones at their house, she said. The investigating officer refused to give any information and said that the case is under investigation, she added.Q: How to prove that they are geometric sequences? I have a number of different questions on how to prove that something is a geometrically increasing sequence. I think they are all similar, so I will start with the first. Question: $a_1=17$ and $a_{n+1}=\frac {1}{3} a_n$. Find $a_n$. A: The recurrence relation becomes $$a_{n+1} = \frac{1}{3}a_n$$ Using that $$\frac{a_{n+1}}{a_n}=\frac{1}{3}$$ Substituting $a_n$ in $a_{n+1}$ for an arbitrary $a_n$ we get a_{n+1}=\left(\frac{1}{3


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# Introduction to Trigonometric Ratios of a Triangle

• Last Updated : 15 Jun, 2022

Trigonometry is all about triangles or to be more precise about the relation between the angles and sides of a right-angled triangle. There are three sides of a triangles named as Hypotenuse, Adjacent, and Opposite. The ratio between these sides based on the angle between them are called Trigonometric Ratios.

As given in the figure in a right angle triangle

• The side opposite to the right angle is called the hypotenuse
• The side opposite to an angle is called the opposite side
• For angle C opposite side is AB
• For angle A opposite side is BC
• The side adjacent to an angle is called the adjacent side
• For angle C adjacent side is BC
• For angle A adjacent side is AB

### Trigonometric ratios

There are 6 basic trigonometric relations that form the basics of trigonometry. These 6 trigonometric relations are ratios of all the different possible combinations in a right-angled triangle.

These trigonometric ratios are called

• Sine
• Cosine
• Tangent
• Cosecant
• Secant
• Cotangent

The mathematical symbol θ is used to denote the angle.

A. Sine (sin)

Sine of an angle is defined by the ratio of lengths of sides which is opposite to the angle and the hypotenuse. It is represented as sinθ

B. Cosine (cos)

Cosine of an angle is defined by the ratio of lengths of sides which is adjacent to the angle and the hypotenuse. It is represented as cosθ

C. Tangent(tan)

Tangent of an angle is defined by the ratio of length of sides which is opposite to the angle and the side which is adjacent to the angle. It is represented as tanθ

D. Cosecant(csc)

Cosecant of an angle is defined by the ratio of length of the hypotenuse and the side opposite the angle. It is represented as cscθ

E. Secant(sec)

Secant of an angle is defined by the ratio of length of the hypotenuse and the side and the side adjacent to the angle. It is represented as secθ

F. Cotangent(cot)

Cotangent  of an angle is defined by the ratio of length of sides which is adjacent to the angle and the side which is opposite to the angle. It is represented as cotθ.

#### Trigonometric table

Trigonometric Table for Specific Angles:

### Solving for a side in right triangles with trigonometry

This is one of the most basic and useful use of trigonometry using the trigonometric ratios mentioned is to find the length of a side of a right-angled triangle But to do, so we must already know the length of the other two sides or an angle and length of one side.

Steps to follow if one side and one angle are known:

1. Choose a trigonometric ratio which contains the given side and the unknown side
2. Use algebra to find the unknown side

#### Example:

Solution:

Steps to follow if two sides are known:

1. Mark the known sides as adjacent, opposite or hypotenuse with respective to anyone of the acute angles in the triangle.
2. Decide on which trigonometric ratio can be found out from the above table.
3. Find the angle (X)
4. Use an trigonometric ratio with respect to X which is a ratio of a known side and an unknown side.
5. Use algebra to find the unknown side.

#### Example:

Solution:

My Personal Notes arrow_drop_up


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## Buon Natale e Felice Anno Nuovo

Gentilissimi/e Lettori e Lettrici, Dear Reader, Sehr geehrte Leserinnen und Leser,

grazie mille per aver fatto tappa durante le vostre peregrinazioni cibernautiche nel mio sito web e per aver dedicato un po’ del vostro tempo nel leggere i miei ariticoli. Spero che li avete trovati tanto interessanti e utili da continuate a tornare a leggermi.
Voglio anticipare alcune delle prossime publicazioni.
Tra breve usciranno nuovi titoli:

• The Logistic Map and the Feigenbaum Constants: a Retro Programming Inspired Excursion
• L’integrazione numerica di equazioni differenziali, parte II: 50 anni fa l’uomo ha messo piede sulla Luna
• Retro Programming: Acid-base Titration
• Retro Programming: Plant evolution

Per il momento auguro a tutti voi di trascorrere con le vostri cari un felice Natale e di avere un nuovo anno pieno di buone notizie.

thank you so much for stopping by during your cyber-wanderings on my website and for taking some time to read my articles. I hope you found them so interesting and useful that you keep coming back to read me.
I want to anticipate some of the next publications.
These new titles will be released shortly:

• The Logistic Map and the Feigenbaum Constants: a Retro Programming Inspired Excursion
• The numerical integration of differential equations, part II: 50 years ago man set foot on the moon
• Retro Programming: Acid-base Titration
• Retro Programming: Plant evolution

For the moment I wish you all to spend a happy Christmas with your loved ones and to have a new year full of good news.

Sehr geehrte Leserinnen und Leser,

vielen Dank für Ihren Besuch auf meiner Website und dafür, dass Sie sich die Zeit genommen haben, meine Artikel zu lesen. Ich hoffe, Sie fanden sie so interessant und nützlich, dass Sie immer wieder zurueckkommen, um meinen Block weiter zu lesen.
Ich möchte einige der nächsten Veröffentlichungen vorwegnehmen.
Diese neue Titel werden in Kürze veröffentlicht:

• Retro-Programmierung: Die Logistikkarte und die Feigenbaum-Konstanten
• The numerical integration of differential equations, part II: 50 years ago man set foot on the moon
• Retro Programming: Acid-base Titration
• Retro Programming: Plant evolution

Ich wuensche Ihnen allen ein frohes Weihnachtsfest mit Ihren Lieben und ein gesundes neues Jahr voller guter Nachrichten.

## A personal tribute to the founder of MD simulation of biological molecules: Prof Herman J.C. Berendsen (1934-2019)

On the 7 October 2019, Prof Dr Herman Johan Christiaan Berendsen passed away just shortly after his 85 birthday. Prof Berendsen is considered the founder of the molecular dynamics simulation of biological system: the area of theoretical research that also shaped my scientific career. He was working at the University of Groningen in the pictoresque northen part of the Netherlands. It was there that I meet him the first time as it allowed me to conduct research in his lab during the last year of my doctorate researches training at the University of Rome “La Sapienza”. After I completed my doctorate, Herman gave me the opportunity to continue working in his group with a postdoc position within the “Protein Folding” EU Training network. This happen just two years before his retirement and therefore I was also one of his last postdocs. After retirement, Herman dedicated himself to write two books that distillate all his experience in the area of molecular simulation [1] and in the education [2]. He stated in a project on the social scientific network Researchgate that “I am retired and work occasionally on methods for multiscale simulations.”

Posted in Research, What is new | 2 Comments

## The First 150 Years of the Periodic Table of the Elements

This year marks the 150th anniversary of the periodic table of the elements (TPE) which currently has 118 entries, the latest arrival (the Tennessium) was discovered 10 years ago (2009), and I feel obliged as a chemist to give some a small informative contribution to celebrate this important event.

## Retro programming nostalgia III: the MSX Microcomputer and the Orbit of the Planets in the Solar System

In a recent article, I have explained the Euler’s method for solving ordinary differential equations using as a motivation the fictionalized version in the film Hidden Figures of the scientific contribution of Katherine Goble and her two colleagues to the NASA space program. As an example of application, I have also shown a program written in the awk programming language for calculating the orbits of planets of the solar system. However, my interest in astrodynamics come back to my juvenile age, when still going to high school, my parents decided to gift me a more sophisticated microcomputer than my previous one (the celebrated Commodore VIC 20). So I became a programmer of a Philips MSX VG 8010 that I still jelously own in its original box. So, powered by the versatile Federico Faggin’s Zilog Z80 processor with a clock 3.58 MHz, with an impressive (for a previous owner of a VIC20 with a mere 3.583 kB!) memory of 32 kB RAM , 16kB of video RAM and a dedicated tape-record device as storage system, I started to write more sophisticated in MSX Basic. At that time, I was eagerly following the department “Ricreazioni al Computer” by the famous computer scientist A. K. Dewdney on the magazine “Le Scienze”, the Italian edition of Scientific American. The new microcomputer allowed me to experiment with the fascinating computational topics that Dewdney was offering every month. One of these topics was dedicated to the simulation of stars using the algorithm based on the Euler integration of the Newton equation. Following the instruction of Dewney, I managed to write a small program in MSX basic and this was the starting of my interest in computational astronomy.

## The Logistic Map and the Feigenbaum Constants: a Retro Programming Inspired Excursion

“… Mitchell Feigenbaum was an unusual case. He had exactly one published article to his name, and he was working on nothing that seemed to have any particular promise. His hair was a ragged mane, sweeping back from his wide brow in the style of busts of German composers… At the age of twenty he had already become a savant among savants, an ad hoc consultant [at Los Alamos National Laboratory, USA] whom scientist would go to see about any expecially intractable problem.”

James Gleick, Chaos: the amazing science of the unpredectable.

This year, on June 30th 2019, Mitchell J. Feigenbaum died at the age of 74. Feigenbaum was an American mathematician that became famous with the discovery of the universal constants that bear his name. In the middle of the ’80, reading Le Scienze magazine (the Italian edition of Scientific American) I got to know of the contribution to the chaos theory of this charismatic mathematician. In particular, I was delighted by reading the Douglas Hofstadter’s article in the rubric “Temi Metamagici” ( Methamagical themes) (Scientific American, November 1981). The article explained the emergence of the chaos in the iteration map of the logistic equation, the same equation deeply studied by Feigenbaum. The full story about the Mitchell Feigenbaum and his discovery of his universal constants is delightly narrated in the beautiful book Chaos:the amazing science of the unpredectable by J. Gleick [1]. Here it is just another small extract:

“… in the summer of 1975, at a gathering in Aspen, Colorado, he heard Steve Smale [another key mathematicial in the developing of the chaos theory, NDA] talk about some of the mathematical qualities of the same quadratic difference equation [the same studied by Robert May, NDA]. Smale seemed to think that there were some interesting open questions about the exact point at which the mapping changes from periodic to chaotic. As always, Smale had a sharp instinct for questions worth exploring. Feigenbaum decided to look into it once more.”[1]

## Numerical Integration of Differential Equations. Part I.: Katherine Goble and the Euler’s Method.

This article was inspired by the beautiful 2016 movie Hidden Figures (based on the book of the same name by M. L. Shetterley) which tell the dramatic story of three talented black women scientist that worked as “human computers” for NASA in 1961 for the Mercury project.

In the movie, the mathematician Katherine Goble (interpreted by Taraji P. Henson), had a brilliant intuition on how to numerically solve the complex problem to find the transfer trajectory for the reentry into the Earth atmosphere of the Friendship 7 capsule with the astronaut John Glenn on board. In the particular scene, she was standing together with other engineers and the director of the Langley Research Center (a fictional character interpreted by Kevin Coster) in front of the vast blackboard looking to graph and equations when she says that the solution might be in the “old math” and she runs to take an old book from a bookshelf with the description of the Euler method. The scene is also nicely described in the youtube video lesson by Prof. Alan Garfinkel of the UCLA. A detailed description of the numerical solution based on the original derivation of K. Globe is in the Wolfram blog website.

Katherine Globe was using for these complex calculation her brilliant brain with the support of a mechanical calculator (the Friden STW-10, in the movie, this machine is visible in different scenes). In a scene of the film, she revealed that her typical computing performance was of 10000 calculations per day and probably for calculations, she was not referring to single arithmetic operations! These exceptional mathematical skills have given a significative contribution at the beginning of the American space program, but it became insufficient to handle the more complex mathematics necessary to land the man on the Moon, and the other fantastic NASA achievements.

## The Calculation of the Lattice Energy: The Born-Haber Cycle

My blog in italian on this topics is very popular and for this reason I decided to add an English translation (when I have some free time, I will also translate the text in the Figure and Table). So be tune and more will come!

The stability of a crystal lattice at constant T and P conditions is linked to the Gibbs free energy of lattice formation by the relations

$M^+ (g) + X^- (g) \rightarrow MX (s) \hfill (1)$

$\displaystyle \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \hfill (2)$

If ${\Delta G^{\circ}}$ is more negative for the formation of the ${I}$ structure than for the ${II}$ structure, the ${II \rightarrow I}$ transition will be spontaneous and the solid will have that structure.


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# Dielectric constant formula in terms of electric field

## 100 best questions for Dielectric constant formula in terms of electric field

We've collected 100 popular questions in the «Dielectric constant formula in terms of electric field» category so you can quickly find the answer to your question!

How the electric field varies with the dielectric constant?
The dielectric constant of a material determines the amount of energy that a capacitor can …
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Why does dielectric decrease electric field. When a dielectric material is placed in an electric …
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Why does dielectric decrease electric field When a dielectric material is placed in an electric …
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Why does dielectric decrease electric field When a dielectric material is placed in an electric …
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In electromagnetism, a dielectric (or dielectric material) is an electrical insulator that can be polarized …
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The external field induces dipole moment by stretching or re-orienting the molecules of dielectric. Net …
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A dielectric material gets polarized when it placed in an electric field. The field produce …
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Does a dielectric increase electric field? Introducing a dielectric into a capacitor decreases the electric …
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To study the effect of dielectric constant on the antenna radiation efficiency, dielectric constant is …
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The presence of a dielectric between the two plates actually decreases the electric field inside …
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In electromagnetism, a dielectric (or dielectric material) is an electrical insulator that can be polarized …
Does a dielectric increase electric field?
Introducing a dielectric into a capacitor decreases the electric field, which decreases the voltage, which …
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The field produce due to the polarization of material minimize the effect of external field. …
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The polarization of the dielectric in the capacitor does reduce the effective electric field of …
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Answer:The dielectric material get polarized when it is placed in an electric field. The field …
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The electric field E in the dielectric is equal to the total surface charge density …
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In this case the magnitude of the electric field is 20 5 = 4 N/C …
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A constant electric field can create a constant electric current (composed of electric monopoles, typically …
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Constant Electric Field Energy Materials. Vincent Ming Hong Ng,...... If d Xj =0, …
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Magnetic fields are created by electric currents. A constant electric field can create a constant …
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Suppose we want to calculate the macroscopic electric field Er( ) GG at some point, …
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The planar charge creates an electric field which is felt in all of space: it …
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Capacitance is the ratio of charge to voltage. Introducing a dielectric into a capacitor decreases …
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(b) The dielectric reduces the electric field strength inside the capacitor, resulting in a smaller …
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As you correctly observed, the electric field stays the same in the capacitor after insertion …
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Assuming a constant electric field and only an electric field, there is a constant acceleration …
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E = Eo/k k is dielectric constant
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Work and Voltage: Constant Electric Field The case of a constant electric field , as …
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Work and Voltage: Constant Electric Field The case of a constant electric field , as …
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This produces a layer of opposite charge on the surface of the dielectric that attracts …
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The Electric Field formula is expressed by. If q and Q are two charges separated …
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Formula of Electric Field The Electric Field formula is expressed by If q and Q …
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The electric field E in the dielectric is equal to the total surface charge density …
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Article: what is electric field constant Thinking What Is Electric Field Constant to Eat? We've …
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where ϕ is the potential function due to nearby charges. Related to your specific question, …
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Energy Density Formula In the case of electric field or capacitor, the energy density formula …
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This means that the dielectric itself has an induced field of E (1 − ϵ …
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The electric field is constant between the two oppositely charged plates of the capacitor except …
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Work and Voltage: Constant Electric Field The case of a constant electric field , as …
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It basically measures how fast the potential varies as the coordinate is changed (but the …
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My book says that a cell (or a battery) creates a constant electric field inside …
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327. 1) No, the electric field from a single infinite plate is constant as well. …
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text:
# Geometric intuition for derivatives of basic trig functions

I was inspired by this question to try and come up with geometric proofs for the derivatives of basic trig functions--basically, those that have simple representations on the unit circle ($\sin, \cos, \tan, \sec, \csc, \cot$):

I was initially a bit skeptical about how easy it might be, but then I found this very simple proof for $\sin$ and $\cos$; the basic insight can be seen in this picture from an alternative version of the proof I found later:

Basically, we use the fact arc $PQ$ and segment $PQ$ are the same as $\Delta\theta\rightarrow 0$, and the former has measure $\Delta\theta$.

Nevertheless, I've had no luck so far getting a proof for $\sec$; I have a feeling the proofs of $sec$ and $tan$ are very closely related, as are the $\csc, \cot$ proofs.

Has anyone seen a proof for the four remaining basic functions anywhere? Perhaps I just haven't drawn the right picture yet.

Another possible avenue is this representation:

• Related: This is my three-dimensional "trigonograph" of the derivatives of sine and cosine. As for the other functions ... I have no solution to offer here, but a thought: I've always found it interesting that, just as sine and cosine are derivatives of each other (sign changes notwithstanding), secant and tangent are as well ... except for an "extra" factor of secant: $$\frac{d}{d\theta}\color{red}{\sec\theta} = \color{blue}{\tan\theta}\cdot\sec\theta \qquad \frac{d}{d\theta}\color{blue}{\tan\theta} = \color{red}{\sec\theta}\cdot\sec\theta$$ – Blue May 26 '15 at 16:12
• @Blue the answers below give you the tie you've been looking for--basically, the extra $\sec\theta$ comes from the radius of the circle used in the proof; $\csc\theta$ and $\cot\theta$ show the same switch from the circle of radius $\csc\theta$. And the reason the pairings are like that can be tied back to the Pythagorean trig identities--$\sin^2\theta+\cos^2\theta=1$, $1+\tan^2\theta=\sec^2\theta$, and $1+\cot^2\theta=\csc^2\theta$. – MichaelChirico May 27 '15 at 17:25
• Indeed, I'm familiar with Jim's solution. (Your "summary graphic" is a nice touch.) My "thought" had been intended as a hint in that direction, since I didn't have a postable image readily available. In retrospect, the hinty-ness c/should've been clearer, say, by explicitly suggesting that you seek a "$d\sec$-$d\tan$-$\sec\theta d\theta$" right triangle (which arises pretty naturally from your second "avenue"). I think I got caught-up in colorizing my equations and lost focus. :) – Blue May 27 '15 at 19:30

Yes, there are nice geometric explanations of the derivative formulas for all six basic trig functions, which ought to be much more widely known. (I hesitate to use the word "proof" for an argument that uses infinitesimals.) They are all based on the following fact about isosceles triangles:

For an isosceles triangle with small vertex angle $d\theta$, the base length $ds$ satisfies $$ds \;\approx\; r\,d\theta$$ where $r$ is the length of the legs.

As you already pointed out, there is a nice geometric explanation of the derivative formulas for $\sin \theta$ and $\cos \theta$ that uses this fact. The following picture shows a right triangle version of the explanation, as opposed to the unit circle version you gave above:

The following picture shows a geometric explanation of the derivative formulas for $\sec \theta$ and $\tan \theta$, again using right triangles.

Finally, the following picture shows a geometric explanation of the derivative formulas for $\csc \theta$ and $\cot \theta$.

This is a direct corollary of @JimBelk's wonderful answer, but I thought it might be nice to have a concise summary graphic which gets all the derivatives at once, which it turns out is pretty simple to do from the second unit circle picture I posted with my question:

The blue triangle shows the derivatives of $\sin\theta$ and $\cos\theta$, the yellow triangle shows the derivatives of $\sec\theta$ and $\tan\theta$, and the salmon triangle shows the derivatives of $\csc\theta$ and $\cot\theta$.

It should be straightforward to verify this using similar triangles and a limit argument to show the green segments are equal in length to the arcs of 3 circles of appropriate radii.


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## January 9, 2022 Single Round Match 821 Editorials

#### AssignPoints

We just need to execute the point allocation algorithm described in the statement.

Instead of sorting all contestants according to their time, we can also note that the range of possible times is pretty small, so we can instead iterate through all times: assign points to everyone who finished in 1 second, then everyone who finished in 2 seconds, and so on.

public int[] assign(int N, int[] results) {
int[] points = new int[N];
int nextScore = N;
for (int t=1; t<=10_000; ++t) {
int current = 0;
for (int n=0; n<N; ++n) if (results[n] == t) {
points[n] = nextScore;
++current;
}
nextScore -= current;
}
return points;
}

#### OneBattleship

Imagine that we color all cells in the grid red, green or blue based on the value (row+column) modulo 3.

Regardless of how we place the battleship, it will always contain one red, one green, and one blue cell.

Thus, we can count the number of ‘.’ cells of each color, choose the color with the smallest number of occurrences (which has to be at most one third of all unknown cells), and shoot at all ‘.’ cells of that color. We are guaranteed to hit the battleship.

public String[] hit(String[] grid) {
int R = grid.length;
int C = grid[0].length();

int[] counts = new int[3];
for (int r=0; r<R; ++r) for (int c=0; c<C; ++c) {
if (grid[r].charAt(c) == '.') ++counts[ (r+c)%3 ];
}

int where = 0;
for (int i=0; i<3; ++i) if (counts[i] < counts[where]) where = i;

for (int r=0; r<R; ++r) {
for (int c=0; c<C; ++c) {
if (grid[r].charAt(c) == 'W') { answer[r] += 'W'; continue; }
if ((r+c)%3 == where) { answer[r] += '*'; continue; }
}
}
}

#### ConnectTheWorld

We are given a forest: an acyclic undirected graph. The graph is guaranteed to have more than one connected component. Our task is to add edges to make the graph connected, without increasing the maximum degree of the graph.

If the current maximum degree is 0 (isolated vertices only), there is clearly no solution.

If the current maximum degree is 1, this means that we have at least three vertices: at least two that are already connected by an edge, and at least one in another component than those two. But then the connected graph that contains all of them must clearly contain a vertex of degree more than 1, and thus there is again no solution.

In all other cases (i.e., the given graph has maximum degree 2 or more) a solution exists. One way to show this (and also an algorithm to construct one) is as follows:

Initially, all connected components of our graph are trees. Each tree contains a vertex of degree at most 1. (If it didn’t, the sum of its degrees would be at least twice its number of vertices, which would imply that it has at least as many edges as vertices.) Thus, we can take two trees, find one such vertex in each of them, and connect those by an edge. This will decrease the number of connected components by 1. The new component is clearly again a tree. (The new edge could not have created any cycle.) Repeating this process will eventually connect the entire graph.

As the input graph is small, we can afford to use a somewhat inefficient implementation that is easier and with no special cases. We will proceed as follows:

• Use a hashmap to assign numbers 0 to N-1 to city names.
• Build an adjacency matrix for the current graph.
• Use Floyd-Warshall to construct its transitive closure (i.e., find connected components).
• Assign colors 0 to C-1 to the individual connected components of our graph.
• Compute degrees and check the maximum degree to see whether a solution exists.
• For each color x from 0 to C-2: connect components with colors x and C-1.
public String[] connect(String[] terminalA,
String[] terminalB,
String[] isolated)
{
int N = 0;
HashMap<String, Integer> ID = new HashMap<String, Integer>();
int M = terminalA.length;
int[][] edges = new int[M][2];
for (int m=0; m<M; ++m) {
if (!ID.containsKey(terminalA[m])) { ID.put(terminalA[m],N); ++N; }
if (!ID.containsKey(terminalB[m])) { ID.put(terminalB[m],N); ++N; }
edges[m][0] = ID.get( terminalA[m] );
edges[m][1] = ID.get( terminalB[m] );
}
for (String x : isolated) { ID.put(x,N); ++N; }

int[] degree = new int[N];
for (int m=0; m<M; ++m) {
++degree[ edges[m][0] ];
++degree[ edges[m][1] ];
}

int MAX = 0;
for (int x : degree) MAX = Math.max( MAX, x );

if (MAX <= 1) return new String[0]; // no solution

boolean[][] reachable = new boolean[N][N];
for (int n=0; n<N; ++n) reachable[n][n] = true;
for (int m=0; m<M; ++m) reachable[edges[m][0]][edges[m][1]] = true;
for (int m=0; m<M; ++m) reachable[edges[m][1]][edges[m][0]] = true;

for (int k=0; k<N; ++k) for (int i=0; i<N; ++i) for (int j=0; j<N; ++j)
if (reachable[i][k] && reachable[k][j]) reachable[i][j] = true;

int[] color = new int[N];
for (int n=0; n<N; ++n) color[n] = -1;
int C = 0;

for (int n=0; n<N; ++n) if (color[n] == -1) {
for (int a=0; a<N; ++a) if (reachable[n][a]) color[a] = C;
++C;
}

String[] names = new String[N];
for (String x : terminalA) names[ID.get(x)] = x;
for (String x : terminalB) names[ID.get(x)] = x;
for (String x : isolated) names[ID.get(x)] = x;

for (int c=0; c<C-1; ++c) {
int x = 0; while (color[x] != c || degree[x] > 1) ++x;
int y = 0; while (color[y] != C-1 || degree[y] > 1) ++y;
++degree[x];
++degree[y];
for (int n=0; n<N; ++n) if (color[n] == c) color[n] = C-1;
}
}

#### KnapsackTweak

The general idea should not be too complicated, but there are a few caveats:

• The solution of the original knapsack that is the closest one to the target is not necessarily the one we want to tweak.
• The total price of the solution we want to tweak may be bigger than target, and even bigger than 100,000, so we have to be careful with upper bounds in our solution.

When looking at a particular collection of items, we actually care about two parameters: not just their total price P but also their count C. Once we know those two values, we can compute the amount by which we need to tweak them in order to reach the target price T: we need to take the difference abs(P-T) and split it among the group, so some of the items in the group will need to be tweaked by ceiling( abs(P-T) / C ).

Armed with this observation we can solve our task as follows:

• We will use knapsack-style dynamic programming to determine the set of currently reachable prices for each number of items separately.
• For each number of items, we will take the reachable price that is closest to the target and we’ll compute the amount of tweaking we would need.

Let M be the maximum price of a single item. Consider any fixed count C of items. Consider the collection formed by the C items with the largest prices. We can now repeatedly take the most expensive item in the collection and replace it by the cheapest of the other items. Each such change will decrease the total price of the collection by less than M. Eventually, we will reach the cheapest possible collection. The key conclusion from this is that the set of reachable collection prices is “dense”, there are never gaps bigger than the largest item.

There are only three possibilities to consider:

• All collections of C items are cheaper than the target.
• Some are cheaper, some are not. In this case it follows from the above observation that the cheapest of those that have price >= T will have price less than T+M.
• All collections of C items have cost at least T. In that case the best collection of C items is simply the collection of C cheapest items.

This is the approach chosen in the reference solution: it does DP to construct all reachable prices for each number of items and each total price up to T+M, and it also tries greedily constructing the cheapest collection for each number of items. The optimal solution must be somewhere among these. The time complexity of this solution is O(N^2*(T+M)).

If we think about the problem some more, we can show that the greedy part can be avoided completely. This is because:

• For the smallest C such that the cheapest collection of C items already has price >= T, this price does not exceed T+M and thus our DP will catch it.
• Let A be the actual real amount by which we would have to decrease each item in order to hit exactly T in the above solution.
• The target is positive, so at least some items must have positive tweaked prices. In particular, the largest price Q in that collection has to be greater than A.
• As we now increment C, we will be adding extra items that each have price at least Q, and thus greater than A.
• Thus, for each bigger C, even if we subtract A from all items in the cheapest collection, their new sum will be greater than T, and thus we need to tweak them by more than A in order to reach T exactly. But that means that among these options we will never find a better solution.

Thus the problem was quite forgiving: if you forgot to consider this option, it did not come back to bite you 🙂

#### CrossPrimesOut

In this problem we are essentially given an open-data problem. We’ll need to deal with some large primes, but we can do much of the work in the comfort of our local workstation instead of just having to do it in the submitted solution during its runtime. Or, at the very least, we can do all the testing locally to make sure everything works.

The general idea of the solution is hinted at already in Example 0 where we, after several steps of simulating the infinite process, conclude the following: “At this point it’s easy to show that in the final sequence element 0 will be 68 and element 1 will be 5654600.”

Why/how can we show that? We just processed the prime 401 and all future primes will be bigger, so the token “68” is surely safe. What about “5654600”? Well, clearly all 3- or more-digit substrings of this string are composite numbers, so there will never be any prime crossed out from this token either.

One possible approach that works really well looks as follows:

• Construct a big enough prefix of the sequence of digits. D=400 digits turns out to be more than enough for all test cases. The prefix can be found via binary search: we are looking for the biggest number P such that (P / 10^D)^2 < X, in other words, P^2 < X * 10^(2D).
• Iterate over all primes up to 10^6 and cross those out. You can get to this point quickly.
• At this point you can verify that for each valid X you already have much more than 30 space-separated tokens. As the number of tokens can only increase, we can now be sure that we have enough digits.
• From this point on, we finish the construction as follows: we look at all tokens we have, take all their substrings of 7 or more digits and test them for primality. The primes found this way are the only candidates for moments when something interesting will happen in the future. All other primes won’t influence our prefix of the text.
• The previous step is the step where the open-data aspect comes into play. If your language of choice doesn’t have a good primality test, you can outsource it: iterate over all X, when doing the primality testing do what you can (e.g., just trial division up to some bound), and log the numbers for which you aren’t sure. Then simply use an external tool (e.g., “factor” in linux) and hard-wire the outputs into your solution (“these are primes, these aren’t”).
• Caveat: only actual primes of 7 or more digits should be considered as candidates. E.g., when the 7- character string “0000002” gives you the prime 2, you don’t want to process it again.
• For each candidate, in sorted order, we process it just like we processed the small primes earlier: we check whether it still has an occurrence, and if it does, we erase the first one. (Note that we cannot just erase all candidates. Some may overlap and then only the earlier one gets erased.)

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text:
# Math Help - Probability Problem

1. ## Probability Problem

A box contains 12 red and 13 white marbles. If 8 marbles are chosen at random (without replacement) determine the probability that 6 are red. Write your answer as a fraction reduced to lowest terms.

PLease help! I thought I knew what to do but then I just kept confusing myself.

2. Originally Posted by sweeetcaroline
A box contains 12 red and 13 white marbles. If 8 marbles are chosen at random (without replacement) determine the probability that 6 are red. Write your answer as a fraction reduced to lowest terms.
PLease help! I thought I knew what to do but then I just kept confusing myself.
Why not show us what you have done that is confusing you?
We cannot help if we are in the dark.

3. Well I thought that I could use a binomial or geometic model, but that didn't take into account that of 8 chosen probability that 6 are red. And then I thought that maybe somehow using the number of possibilities, figure out how many that has 6 red if 8 are chosen but I wasn't exactly sure how to do that. Am I on the right track with either of these ideas?

4. hypergeometric

${ {12\choose 6}{13\choose 2}\over {25\choose 8}}$