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Geometry  | enregistrements trouvés : 57

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Post-edited  Microlocal analysis for Kerr-de Sitter black holes Vasy, Andras (Auteur de la Conférence) | CIRM (Editeur )

In this lecture I will describe a framework for the Fredholm analysis of non-elliptic problems both on manifolds without boundary and manifolds with boundary, with a view towards wave propagation on Kerr-de-Sitter spaces, which is the key analytic ingredient for showing the stability of black holes (see Peter Hintz' lecture). This lecture focuses on the general setup such as microlocal ellipticity, real principal type propagation, radial points and generalizations, as well as (potentially) normally hyperbolic trapping, as well as the role of resonances. In this lecture I will describe a framework for the Fredholm analysis of non-elliptic problems both on manifolds without boundary and manifolds with boundary, with a view towards wave propagation on Kerr-de-Sitter spaces, which is the key analytic ingredient for showing the stability of black holes (see Peter Hintz' lecture). This lecture focuses on the general setup such as microlocal ellipticity, real principal type propagation, radial points ...

Post-edited  A hitchhiker's guide to Khovanov homology - Part I Turner, Paul (Auteur de la Conférence) | CIRM (Editeur )

There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what such a guide should look like.
It is quite a risky undertaking because it is all too easy to offend by omission, misrepresentation or other. I have not attempted a complete literature survey and inevitably these notes reflects my personal view, jaundiced as it may often be. My apologies for any offence caused.
I would like to express my warm thanks to Lukas Lewark, Alex Shumakovitch, Liam Watson and Ben Webster.
There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what ...

Post-edited  Persistence modules and Hamiltonian diffeomorphisms - Part 1 Polterovich, Leonid (Auteur de la Conférence) | CIRM (Editeur )

Theory of persistence modules is a rapidly developing field lying on the borderline between algebra, geometry and topology. It provides a very useful viewpoint at Morse theory, and at the same time is one of the cornerstones of topological data analysis. In the course I'll review foundations of this theory and focus on its applications to symplectic topology. In parts, the course is based on a recent work with Egor Shelukhin arXiv:1412.8277

Post-edited  Virtual fundamental cycles and contact homology Pardon, John (Auteur de la Conférence) | CIRM (Editeur )

I will discuss work in progress aimed towards defining contact homology using "virtual" holomorphic curve counting techniques.

Post-edited  Differential forms and the Hölder equivalence problem - Part 1 Pansu, Pierre (Auteur de la Conférence) | CIRM (Editeur )

A sub-Riemannian distance is obtained when minimizing lengths of paths which are tangent to a distribution of planes. Such distances differ substantially from Riemannian distances, even in the simplest example, the 3-dimensional Heisenberg group. This raises many questions in metric geometry: embeddability in Banach spaces, bi-Lipschitz or bi-Hölder comparison of various examples. Emphasis will be put on Gromov's results on the Hölder homeomorphism problem, and on a quasisymmetric version of it motivated by Riemannian geometry. A sub-Riemannian distance is obtained when minimizing lengths of paths which are tangent to a distribution of planes. Such distances differ substantially from Riemannian distances, even in the simplest example, the 3-dimensional Heisenberg group. This raises many questions in metric geometry: embeddability in Banach spaces, bi-Lipschitz or bi-Hölder comparison of various examples. Emphasis will be put on Gromov's results on the Hölder ...

Post-edited  Parking spaces and Catalan combinatorics for complex reflection groups Lanini, Martina (Auteur de la Conférence) | CIRM (Editeur )

Recently, Armstrong, Reiner and Rhoades associated with any (well generated) complex reflection group two parking spaces, and conjectured their isomorphism. This has to be seen as a generalisation of the bijection between non-crossing and non-nesting partitions, both counted by the Catalan numbers. In this talk, I will review the conjecture and discuss a new approach towards its proof, based on the geometry of the discriminant of a complex reflection group. This is an ongoing joint project with Iain Gordon. Recently, Armstrong, Reiner and Rhoades associated with any (well generated) complex reflection group two parking spaces, and conjectured their isomorphism. This has to be seen as a generalisation of the bijection between non-crossing and non-nesting partitions, both counted by the Catalan numbers. In this talk, I will review the conjecture and discuss a new approach towards its proof, based on the geometry of the discriminant of a complex ...

Post-edited  Endomorphisms, train track maps, and fully irreducible monodromies Kapovich, Ilya (Auteur de la Conférence) | CIRM (Editeur )

An endomorphism of a finitely generated free group naturally descends to an injective endomorphism on the stable quotient. We establish a geometric incarnation of this fact : an expanding irreducible train track map inducing an endomorphism of the fundamental group determines an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group G depends only on the component of the BNS invariant $\sum \left ( G \right )$ containing the associated homomorphism to the integers. In particular, it follows that if G is the mapping torus of an atoroidal fully irreducible automorphism of a free group and if the union of $\sum \left ( G \right )$ and $\sum \left ( G \right )$ is connected then for every splitting of $G$ as a (f.g. free)-by-(infinite cyclic) group the monodromy is fully irreducible.
This talk is based on joint work with Spencer Dowdall and Christopher Leininger.
An endomorphism of a finitely generated free group naturally descends to an injective endomorphism on the stable quotient. We establish a geometric incarnation of this fact : an expanding irreducible train track map inducing an endomorphism of the fundamental group determines an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully ...

Post-edited  Random subgroups, totally non free actions and factor representations Grigorchuk, Rostislav (Auteur de la Conférence) | CIRM (Editeur )

I will present results of three studies, performed in collaboration with M.Benli, L.Bowen, A.Dudko, R.Kravchenko and T.Nagnibeda, concerning the invariant and characteristic random subgroups in some groups of geometric origin, including hyperbolic groups, mapping class groups, groups of intermediate growth and branch groups. The role of totally non free actions will be emphasized. This will be used to explain why branch groups have infinitely many factor representations of type $II_1$. I will present results of three studies, performed in collaboration with M.Benli, L.Bowen, A.Dudko, R.Kravchenko and T.Nagnibeda, concerning the invariant and characteristic random subgroups in some groups of geometric origin, including hyperbolic groups, mapping class groups, groups of intermediate growth and branch groups. The role of totally non free actions will be emphasized. This will be used to explain why branch groups have infinitely ...

Post-edited  My favorite groups Ghys, Etienne (Auteur de la Conférence) | CIRM (Editeur )

The world of groups is vast and meant for wandering! During this week, I will give seven short talks describing seven groups, or class of groups, that I find fascinating. These seven talks will be independent and I will have no intention of being exhaustive (this would be silly since there are uncountably many groups, even finitely generated!). In each talk, I will introduce the hero, state one or two results, and formulate one or two conjectures. The world of groups is vast and meant for wandering! During this week, I will give seven short talks describing seven groups, or class of groups, that I find fascinating. These seven talks will be independent and I will have no intention of being exhaustive (this would be silly since there are uncountably many groups, even finitely generated!). In each talk, I will introduce the hero, state one or two results, and formulate one or two c...

Post-edited  Some new inequalities for the Cheeger constant Fragalà, Ilaria (Auteur de la Conférence) | CIRM (Editeur )

We discuss some new results for the Cheeger constant in dimension two, including:
- a polygonal version of Faber-Krahn inequality;
- a reverse isoperimetric inequality for convex bodies;
- a Mahler-type inequality in the axisymmetric setting;
- asymptotic behaviour of optimal partition problems.
Based on some recent joint works with D.Bucur,
and for the last part also with B.Velichkov and G.Verzini.

Post-edited  A microlocal toolbox for hyperbolic dynamics Dyatlov, Semyon (Auteur de la Conférence) | CIRM (Editeur )

I will discuss recent applications of microlocal analysis to the study of hyperbolic flows, including geodesic flows on negatively curved manifolds. The key idea is to view the equation $(X + \lambda)u = f$ , where $X$ is the generator of the flow, as a scattering problem. The role of spatial infinity is taken by the infinity in the frequency space. We will concentrate on the case of noncompact manifolds, featuring a delicate interplay between shift to higher frequencies and escaping in the physical space. I will show meromorphic continuation of the resolvent of $X$; the poles, known as Pollicott-Ruelle resonances, describe exponential decay of correlations. As an application, I will prove that the Ruelle zeta function continues meromorphically for flows on non-compact manifolds (the compact case, known as Smale's conjecture, was recently settled by Giulietti-Liverani- Pollicott and a simple microlocal proof was given by Zworski and the speaker). Joint work with Colin Guillarmou. I will discuss recent applications of microlocal analysis to the study of hyperbolic flows, including geodesic flows on negatively curved manifolds. The key idea is to view the equation $(X + \lambda)u = f$ , where $X$ is the generator of the flow, as a scattering problem. The role of spatial infinity is taken by the infinity in the frequency space. We will concentrate on the case of noncompact manifolds, featuring a delicate interplay between ...

Post-edited  Ancient solutions of geometric flows Daskalopoulos, Panagiota (Auteur de la Conférence) | CIRM (Editeur )

We will give a survey of recent research progress on ancient or eternal solutions to geometric flows such as the Ricci flow, the Mean Curvature flow and the Yamabe flow.
We will address the classification of ancient solutions to parabolic equations as well as the construction of new ancient solutions from the gluing of two or more solitons.

53C44

Post-edited  Autour de la géométrie stochastique : polytopes aléatoires et autres modèles Calka, Pierre (Auteur de la Conférence) | CIRM (Editeur )

La géométrie stochastique est l'étude d'objets issus de la géométrie euclidienne dont le comportement relève du hasard. Si les premiers problèmes de probabilités géométriques ont été posés sous la forme de casse-têtes mathématiques, le domaine s'est considérablement développé depuis une cinquantaine d'années de part ses multiples applications, notamment en sciences expérimentales, et aussi ses liens avec l'analyse d'algorithmes géométriques. L'exposé sera centré sur la description des polytopes aléatoires qui sont construits comme enveloppes convexes d'un ensemble aléatoire de points. On s'intéressera plus particulièrement aux cas d'un nuage de points uniformes dans un corps convexe fixé ou d'un nuage de points gaussiens et on se focalisera sur l'étude asymptotique de grandeurs aléatoires associées, en particulier via des calculs de variances limites. Seront également évoqués d'autres modèles classiques de la géométrie aléatoire tels que la mosaïque de Poisson-Voronoi. La géométrie stochastique est l'étude d'objets issus de la géométrie euclidienne dont le comportement relève du hasard. Si les premiers problèmes de probabilités géométriques ont été posés sous la forme de casse-têtes mathématiques, le domaine s'est considérablement développé depuis une cinquantaine d'années de part ses multiples applications, notamment en sciences expérimentales, et aussi ses liens avec l'analyse d'algorithmes géométriques. ...

Multi angle  Curvature measures of random sets Zähle, Martina (Auteur de la Conférence) | CIRM (Editeur )

A survey on some developments in curvature theory for random sets will be given. We first consider previous models with classical singularities like polyconvex sets or unions of sets with positive reach. The main part of the talk concerns extensions to certain classes of random fractals which have been investigated in the last years. In these cases limits of rescaled versions for suitable approximations are used.

Multi angle  Pseudo-Anosov braids are generic Wiest, Bert (Auteur de la Conférence) | CIRM (Editeur )

We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with $n\geq 3$ strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius $l$ tends to $1$ exponentially quickly as $l$ tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic pairs of elements of the braid group, the conjugacy search problem can be solved in quadratic time. The idea behind both results is that generic braids can be conjugated ''easily'' into a rigid braid.
braid groups - Garside groups - Nielsen-Thurston classification - pseudo-Anosov - conjugacy problem
We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with $n\geq 3$ strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius $l$ tends to $1$ exponentially quickly as $l$ tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic pairs of elements of the braid group, the ...

Multi angle  A hitchhiker's guide to Khovanov homology - Part II Turner, Paul (Auteur de la Conférence) | CIRM (Editeur )

There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what such a guide should look like. It is quite a risky undertaking because it is all too easy to offend by omission, misrepresentation or other. I have not attempted a complete literature survey and inevitably these notes reflects my personal view, jaundiced as it may often be. My apologies for any offence caused. I would like to express my warm thanks to Lukas Lewark, Alex Shumakovitch,Liam Watson and Ben Webster. There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what ...

Multi angle  A hitchhiker's guide to Khovanov homology - Part III Turner, Paul (Auteur de la Conférence) | CIRM (Editeur )

There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what such a guide should look like.
It is quite a risky undertaking because it is all too easy to offend by omission, misrepresentation or other. I have not attempted a complete literature survey and inevitably these notes reflects my personal view, jaundiced as it may often be. My apologies for any offence caused.
I would like to express my warm thanks to Lukas Lewark, Alex Shumakovitch, Liam Watson and Ben Webster.
There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what ...

Multi angle  A hitchhiker's guide to Khovanov homology - Part IV Turner, Paul (Auteur de la Conférence) | CIRM (Editeur )

There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what such a guide should look like.
It is quite a risky undertaking because it is all too easy to offend by omission, misrepresentation or other. I have not attempted a complete literature survey and inevitably these notes reflects my personal view, jaundiced as it may often be. My apologies for any offence caused.
I would like to express my warm thanks to Lukas Lewark, Alex Shumakovitch, Liam Watson and Ben Webster.
There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what ...

Codes MSC

Nuage de mots clefs ici

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CharacterData

The `CharacterData` abstract interface represents a `Node` object that contains characters. This is an abstract interface, meaning there aren't any object of type `CharacterData`: it is implemented by other interfaces, like `Text`, `Comment`, or `ProcessingInstruction` which aren't abstract.

Properties

Inherits properties from its parent, `Node`, and implements the `ChildNode` and `NonDocumentTypeChildNode` interface.

`CharacterData.data`
Is a `DOMString` representing the textual data contained in this object.
`CharacterData.length` Read only
Returns an `unsigned long` representing the size of the string contained in `CharacterData.data`.
`NonDocumentTypeChildNode.nextElementSibling` Read only
Returns the `Element` immediately following the specified one in its parent's children list, or `null` if the specified element is the last one in the list.
`NonDocumentTypeChildNode.previousElementSibling` Read only
Returns the `Element` immediately prior to the specified one in its parent's children list, or `null` if the specified element is the first one in the list.

Methods

Inherits methods from its parent, `Node`, and implements the `ChildNode` and `NonDocumentTypeChildNode` interface.

`CharacterData.appendData()`
Appends the given `DOMString` to the `CharacterData.data` string; when this method returns, `data` contains the concatenated `DOMString`.
`CharacterData.deleteData()`
Removes the specified amount of characters, starting at the specified offset, from the `CharacterData.data` string; when this method returns, `data` contains the shortened `DOMString`.
`CharacterData.insertData()`
Inserts the specified characters, at the specified offset, in the `CharacterData.data` string; when this method returns, `data` contains the modified `DOMString`.
`ChildNode.remove()`
Removes the object from its parent children list.
`CharacterData.replaceData()`
Replaces the specified amount of characters, starting at the specified offset, with the specified `DOMString`; when this method returns, `data` contains the modified `DOMString`.
`CharacterData.substringData()`
Returns a `DOMString` containing the part of `CharacterData.data` of the specified length and starting at the specified offset.

Specifications

Specification Status Comment
DOM
The definition of 'CharacterData' in that specification.
Living Standard Added implemention of the `ChildNode` and `NonDocumentTypeChildNode` interface.
Document Object Model (DOM) Level 3 Core Specification
The definition of 'CharacterData' in that specification.
Obsolete No change from Document Object Model (DOM) Level 2 Core Specification.
Document Object Model (DOM) Level 2 Core Specification
The definition of 'CharacterData' in that specification.
Obsolete No change from Document Object Model (DOM) Level 1 Specification.
Document Object Model (DOM) Level 1 Specification
The definition of 'CharacterData' in that specification.
Obsolete Initial definition.

Browser compatibility

FeatureChromeEdgeFirefoxInternet ExplorerOperaSafari
Basic support1 Yes16 Yes Yes
Implements `ChildNode` Interface Yes Yes251 No Yes No
`appendData` ? ? ? ? ? ?
`data` ? ? ? ? ? ?
`deleteData` ? ? ? ? ? ?
`insertData` ? ? ? ? ? ?
`length` ? ? ? ? ? ?
`replaceData` ? ? ? ? ? ?
`substringData` ? ? ? ? ? ?
FeatureAndroid webviewChrome for AndroidEdge mobileFirefox for AndroidOpera AndroidiOS SafariSamsung Internet
Basic support Yes Yes Yes Yes Yes Yes ?
Implements `ChildNode` Interface ? Yes Yes252 Yes ? ?
`appendData` ? ? ? ? ? ? ?
`data` ? ? ? ? ? ? ?
`deleteData` ? ? ? ? ? ? ?
`insertData` ? ? ? ? ? ? ?
`length` ? ? ? ? ? ? ?
`replaceData` ? ? ? ? ? ? ?
`substringData` ? ? ? ? ? ? ?

1. Two properties, `nextElementSibling` and `previousElementSibling`, have been moved to the `NonDocumentTypeChildNode` interface, also implemented by `CharacterData`.

format: plain


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# Start a new discussion

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## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeOct 19th 2009
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 21st 2009
This comment is invalid XHTML+MathML+SVG; displaying source.

I added at Cauchy complete category the direct hyperlink to free cocompletion.

I have a question: why does it say there

 ... preserves small coproducts and coequalizers.

and not just "preserves small colimits". Do you mean "...preserves small coproducts and ALL coequalizers"?

I am also not sure if I understand the "all presheaves that are connected and projective". What does "connected" mean here? I think that should have a hyperlink.

I'll add to tiny object the example: every representable in a presheaf category.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 21st 2009

Please have a look at the example section I wrote at tiny object, pointing to Cauchy completion.

I haven't really looked into this Cauchy completion business. I'd be interested in seeing at Cauchy completion one or two examples spelled out, or at least indicated. I understand and appreciate how metric spaces are enriched categories, but I am not sure yet about the Cauchy completion part (of couse I know that i can read about it in Lawvere's article, if you force me to).

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeOct 21st 2009

An object X is connected if hom(X, -) preserves coproducts (as explained at connected space). An object is projective if hom(X, -) preserves coequalizers of parallel pairs of maps.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeOct 21st 2009

I have now added material to Cauchy complete category, with particular detailed attention to the classical case of metric spaces. I think it might be time to remove the old discussion I had with Toby, since we apparently came to a consensus (and I'm tired of rereading my snippy response (-: ).

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeOct 21st 2009

Are you sure you want to use "projective" that way? The usual meaning of projective object is that hom(X,-) preserves some kind of epimorphism. But I think even preserving regular epimorphisms doesn't mean that it preserves coequalizers.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeOct 21st 2009

Some edits and a query at tiny object.

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeOct 21st 2009
• (edited Oct 21st 2009)

 I think there's every likelihood that Mike is right about preservation of coequalizers not meaning the same thing as preservation of regular epis (off hand not even sure it's true if pullbacks are preserved). So we might want to reconsider using "projective" that way, although I thought I had a vague memory that it was sometimes used that way.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeOct 21st 2009
That's wonderfully written. Thanks, Todd.
• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeOct 22nd 2009
Yet another question about the relation between Cauchy completion and ideal completion.
• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeOct 22nd 2009

I also wanted to say: Thanks, Todd, that's an impressive entry. I need to keep this stuff in mind, it clearly points towards something big and important.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeOct 22nd 2009
• (edited Oct 22nd 2009)

I have added a bunch of hyperlinks now to Cauchy complete category.

• CommentRowNumber13.
• CommentAuthorTobyBartels
• CommentTimeOct 22nd 2009

I changed references to 'contractive maps' into 'short maps'. The former term seems to vary in the literature; for example, Wikipedia defines it (presumably following the reference that it gives) as any Lipschitz map, while the next Google hit defines it as a map with a Lipschitz constant strictly less than $1$ (and which therefore has a fixed point).

• CommentRowNumber14.
• CommentAuthorTobyBartels
• CommentTimeOct 22nd 2009

I wrote connected object, although only for objects of an extensive category, since only this was explained at connected space.

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeJul 26th 2011

I have added to Cauchy complete category an Example-subsection on Cauchy-completion of prosets and posets in regular and exact categories, citing an article by Rosolini.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeJul 27th 2011

I have added to Cauchy complete category a fairly extensive list of explicit definitions, propositions and theorems from Borceux-Dejean (linked to there).

Nothing that was not already indicated in the Idea-section, but I found it worthwhile to spell it all out in a bit more detail.

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeJul 28th 2011

I have added to Cauchy complete category a further subsection: In terms of essential geometric morphisms.

This is supposed to contain the statements that Todd was pointing out in another thread here.

• CommentRowNumber18.
• CommentAuthorTim Campion
• CommentTimeMar 2nd 2014

I’ve added a few examples based on this nCafe discussion. They’re not worked out at all, but I wanted to highlight the sheer variety of different things that Cauchy completion can mean. One thing this list could use is some references – I really wasn’t sure about these.

• CommentRowNumber19.
• CommentAuthorTim_Porter
• CommentTimeMar 2nd 2014

There is an early paper by Street (1974) mentioned in Kelly’s enriched monograph. Kelly discusses Cauchy completeness to some extent, so perhaps that could be put as a starting point. I do not know that stuff well enough to be sure.

• CommentRowNumber20.
• CommentAuthorMike Shulman
• CommentTimeMar 19th 2021


text:
# Molecular dynamics¶

Molecular dynamics (MD) can be used to simulate the evolution of a system in time.

Examples

To perform a MD simulation, first select the corresponding Task:

Task MolecularDynamics

All aspects of the simulation can then be configured using the MolecularDynamics block.

MolecularDynamics
Barostat
BulkModulus float
Duration integer_list
Equal [None | XYZ | XY | YZ | XZ]
Pressure float_list
Scale [XYZ | Shape | X | Y | Z | XY | YZ | XZ]
Tau float
Type [None | Berendsen | MTK]
End
CalcPressure [True | False]
Checkpoint
Frequency integer
End
InitialVelocities
File string
Temperature float
Type [Zero | Random | FromFile | Input]
Values # Non-standard block. See details.
...
End
End
NSteps integer
Preserve
AngularMomentum [True | False]
CenterOfMass [True | False]
Momentum [True | False]
End
Print
System [True | False]
Velocities [True | False]
End
Restart string
Thermostat
BerendsenApply [Local | Global]
ChainLength integer
Duration integer_list
FirstAtom integer
LastAtom integer
Tau float
Temperature float_list
Type [None | Berendsen | NHC]
End
TimeStep float
Trajectory
SamplingFreq integer
End
End

## General¶

The time evolution of the system is simulated by numerically integrating the equations of motion. A velocity Verlet integrator is used with a time step set by the TimeStep key. The MD driver will perform NSteps timesteps in total.

Because the overall computational cost depends on NSteps but not on TimeStep, it is desirable to set the timestep as large as possible to maximize the sampled timescale with a given computational budget. However, numerical integration errors grow rapidly as the timestep increases. These errors will cause a loss of energy conservation, crashes, and other artifacts. It is thus important to set the TimeStep value carefully, as its optimal value strongly depends on the studied system and simulated conditions.

As a rule of thumb, reasonable timesteps for systems not undergoing chemical reactions are 10-20 times lower than the period of the fastest vibration mode. Systems containing hydrogen atoms at room temperature can thus be accurately simulated using a 1 fs timestep. Longer timesteps can be safely used for systems containing only heavy atoms (vibration periods scale with the square root of the atomic mass). Conversely, the timestep needs to be made shorter for high-temperature simulations. The same also applies to simulations of chemical reactions, which are usually accompanied by significant transient local heating. The default timestep of 0.25 fs should work for most of these cases.

MolecularDynamics
NSteps
Type: Integer 1000 The number of steps to be taken in the MD simulation.
TimeStep
Type: Float 0.25 Femtoseconds The time difference per step.

During a long simulation, numerical integration errors will cause some system-wide quantities to drift from their exact values. For example, the system may develop a nonzero net linear velocity, causing an overall translation or flow. Non-periodic (molecular) systems may also develop nonzero angular momentum (overall rotation) and a Brownian motion of their center of mass through space. These problems are corrected by periodically removing any accumulated drift. This feature can be controlled using the Preserve key.

MolecularDynamics
Preserve
Type: Block Periodically remove numerical drift accumulated during the simulation to preserve different whole-system parameters.
AngularMomentum
Type: Bool True Remove overall angular momentum of the system. This option is ignored for 3D-periodic systems.
CenterOfMass
Type: Bool False Translate the system to keep its center of mass at the coordinate origin. This option is not very useful for 3D-periodic systems.
Momentum
Type: Bool True Remove overall (linear) momentum of the system.

## (Re-)Starting a simulation¶

The state of a system at the beginning of a simulation is defined by the positions and momenta of all atoms. The positions can be set in the input or loaded from a file as described under System definition. Initial velocities are then supplied using the InitialVelocities block.

Probably the most common way to start up a simulation is to draw the initial velocities from a Maxwell-Boltzmann distribution by setting Type=Random and Temperature to a suitable value. Alternatively, velocities can be loaded from an ams.rkf file produced by an earlier simulation using Type=FromFile and File. This is the recommended way to start a production simulation from the results of a short preparation/equilibration run.

Velocities of all atoms in units of Å/fs can also be explicitly defined in the Values block after setting Type=Input. This is mainly useful to repeat or extend simulations done by other programs. For example, velocities can be extracted from the vels or moldyn.vel files used by the standalone ReaxFF program. A simple AWK script is supplied in scripting/standalone/reaxff-ams/vels2ams.awk to help with the conversion.

MolecularDynamics
InitialVelocities
Type: Block Sets the frequency for printing to stdout and storing the molecular configuration on the .rkf file.
File
Type: String AMS RKF file containing the initial velocities.
Temperature
Type: Float Kelvin Sets the temperature for the Maxwell-Boltzmann distribution when the type of the initial velocities is set to random, in which case specifying this key is mandatory. ADFinput will use the thermostat temperature as default.
Type
Type: Multiple Choice Random [Zero, Random, FromFile, Input] Specifies the initial velocities to assign to the atoms. Three methods to assign velocities are available. Zero: All atom are at rest at the beginning of the calculation. Random: Initial atom velocities follow a Maxwell-Boltzmann distribution for the temperature given by the [MolecularDynamics%InitialVelocities%Temperature] keyword. FromFile: Load the velocities from a previous ams result file. Input: Atom’s velocities are set to the values specified in the [MolecularDynamics%InitialVelocities%Values] block, which can be accessed via the Expert AMS panel in ADFinput.
Values
Type: Non-standard block This block specifies the velocity of each atom when [MolecularDynamics%InitialVelocities%Type] is set to Input. Each row must contain three floating point values (corresponding to the x,y,z component of the velocity vector) and a number of rows equal to the number of atoms must be present, given in the same order as the [System%Atoms] block.

The MD module also supports exact restarts of interrupted simulations by pointing the Restart key to an ams.rkf file. This will restore the entire state of the MD module from the last available checkpoint (if the previous simulation was interrupted) or from the final state (if the previous simulation ended after NSteps). An earlier trajectory can thus be seamlessly extended by increasing NSteps and using Restart.

Note

Restart should be combined with LoadSystem from the same ams.rkf to restore the atomic positions.

Warning

The Restart feature is only intended for exact restarts, so the rest of the MolecularDynamics settings should be the same as in the original run. Only NSteps and engine settings (contents of the Engine block) can always be changed safely across restarts.

Although some MD settings (such as the trajectory sampling options) can in practice be changed without problems, changing others (such as thermostat or barostat settings) will cause the restart to fail or produce physically incorrect results. It is thus strongly recommended to only use Restart for exact continuation and InitialVelocities Type=FromFile together with LoadSystem otherwise.

MolecularDynamics
Restart
Type: String The path to the ams.rkf file from which to restart the simulation.

## Thermostats and barostats¶

By default, the MD simulation samples the microcanonical (NVE) ensemble. Although this is useful to check energy conservation and other basic physical properties, it does not directly map to common experimental conditions. The canonical (NVT) ensemble can be sampled instead by applying a Thermostat, which serves as a simulated heat bath around the system, keeping its average temperature at a set value.

AMS offers two thermostats with drastically different properties, mode of operation, and applicability, selected using the Type key:

Berendsen

The Berendsen friction thermostat drives the system to a particular target temperature by rescaling the velocities of all atoms in each step. This ensures rapid (exponential) convergence of the temperature with a time constant Tau. However, this thermostat produces an incorrect velocity distribution and should thus be avoided in all situations where correct energy fluctuations are important. Additionally, using a too short time constant Tau tends to cause incorrect equipartition of energy between different degrees of freedom in the system, leading to the “flying ice cube” phenomenon. The time constant Tau should thus be set as large as possible to limit these artifacts while still providing sufficient temperature control. Common values of Tau for condensed-phase systems lie between 100 fs (strong damping, rapid convergence) and 10 ps (weak coupling with minimal artifacts).

This thermostat is mainly useful for systems far from equilibrium, for example during the initial preparation and equilibration phase of a simulation. The NHC thermostat should be preferred where possible.

NHC
This enables a chain of coupled Nosé-Hoover thermostats. This method introduces artificial degrees of freedom representing the heat bath and ensures correct sampling of the canonical ensemble. The combined total energy of the system and the heat bath is conserved and shown in the GUI as Conserved Energy. Checking this quantity for drift and artifacts thus offers a valuable test of the correctness of the simulation. This thermostat exhibits oscillatory relaxation with a period of Tau. It is thus not well suited for systems starting far from equilibrium, because the oscillations may take long to settle. The time constant Tau should be at least comparable to the period of some natural oscillation of the system to ensure efficient energy transfer. It is commonly on the order of hundreds of femtoseconds, although higher values may be used if weak coupling is desired.

Multiple independent thermostats can be used to separately control different regions of the system at the same time. This is done by specifying the Thermostat block multiple times and setting the FirstAtom and/or LastAtom keys to the desired range of atoms. Care needs to be taken to avoid defining thermostats with overlapping atom ranges.

MolecularDynamics
Thermostat
Type: Block True This block allows to specify the use of a thermostat during the simulation. Depending on the selected thermostat type, different additional options may be needed to characterize the specific thermostat’ behavior.
BerendsenApply
Type: Multiple Choice Global [Local, Global] Select how to apply the scaling correction for the Berendsen thermostat: - per-atom-velocity (Local) - on the molecular system as a whole (Global).
ChainLength
Type: Integer 10 Number of individual thermostats forming the NHC thermostat
Duration
Type: Integer List Specifies how many steps should a transition from a particular temperature to the next one in sequence take.
FirstAtom
Type: Integer 1 Index of the first atom to be thermostatted
LastAtom
Type: Integer 0 Index of the last atom to be thermostatted. A value of zero means the last atom in the system.
Tau
Type: Float Femtoseconds The time constant of the thermostat.
Temperature
Type: Float List Kelvin The target temperature of the thermostat.
Type
Type: Multiple Choice None [None, Berendsen, NHC] Selects the type of the thermostat.

Just like using a Thermostat to control the temperature of the system, a Barostat can be applied to keep the pressure constant by adjusting the volume. This enables sampling the isenthalpic-isobaric (NpH) ensemble by using only a barostat or the isothermal-isobaric (NpT) ensemble by combining a barostat and a thermostat. Unlike thermostats, a barostat always applies to the entire system and there can thus be at most one barostat defined.

AMS offers two barostats with similar properties to the related thermostats:

Berendsen
The Berendsen friction-like isobaric ensemble method rescales the system in each step to drive the pressure towards a target value. Similarly to the Berendsen thermostat, the relaxation is exponential with a time constant Tau. Similar considerations for the choice of Tau apply as in the case of the thermostat, but the value of Tau for the barostat is usually at least several times higher than the corresponding Tau used for the thermostat. This barostat does not have any conserved quantity.
MTK
This enables the Martyna-Tobias-Klein extended Lagrangian barostat, which generates a true isobaric ensemble by integrating the cell parameters as additional degrees of freedom. This barostat is derived from the Andersen-Hoover isotropic barostat and the Parrinello-Rahman-Hoover anisotropic barostat. Like the NHC thermostat, it exhibits oscillatory relaxation unsuitable for systems far from equilibrium. This barostat must always be combined with a NHC thermostat. One instance of such thermostat coupled to the atoms as usual, while a second instance is created internally and coupled to the cell degrees of freedom.
MolecularDynamics
Barostat
Type: Block This block allows to specify the use of a barostat during the simulation.
BulkModulus
Type: Float 2200000000.0 Pascal An estimate of the bulk modulus (inverse compressibility) of the system for the Berendsen barostat. This is only used to make Tau correspond to the true observed relaxation time constant. Values are commonly on the order of 10-100 GPa (1e10 to 1e11) for solids and 1 GPa (1e9) for liquids (2.2e9 for water). Use 1e9 to match the behavior of standalone ReaxFF.
Duration
Type: Integer List Specifies how many steps should a transition from a particular pressure to the next one in sequence take.
Equal
Type: Multiple Choice None [None, XYZ, XY, YZ, XZ] Enforce equal scaling of the selected set of dimensions. They will be barostatted as one dimension according to the average pressure over the components.
Pressure
Type: Float List Pascal Specifies the target pressure.
Scale
Type: Multiple Choice XYZ [XYZ, Shape, X, Y, Z, XY, YZ, XZ] Dimensions that should be scaled by the barostat to maintain pressure. Selecting Shape means that all three dimensions and also all the cell angles are allowed to change.
Tau
Type: Float Femtoseconds Specifies the time constant of the barostat.
Type
Type: Multiple Choice None [None, Berendsen, MTK] Selects the type of the barostat.

### Temperature and pressure regimes¶

Arbitrary temperature and pressure regimes can be generated by setting Temperature or Pressure to a list of values, corresponding to the successive set points. This needs to be accompanied by a Duration key specifying the length of each regime segment in steps:

Thermostat
Temperature 0     300     300     500     500     300
Duration      100     200     100     200     100
End

Note that there is always N-1 Duration values for N Temperature values. The target temperature of the thermostat in this example will evolve as follows:

1. Increase linearly from 0 to 300 K over 100 steps.
2. Stay constant at 300 K for 200 steps.
3. Increase linearly from 300 to 500 K over 100 steps.
4. Stay constant at 500 K for 200 steps.
5. Decrease linearly from 500 to 300 K over 100 steps.
6. Stay constant at 300 K for the rest of the simulation.

## Trajectory sampling and output¶

A basic principle of the numerical integration of motion in MD is that the changes in the state of the system between successive time steps are small. This means that storing the results of every step is not useful, because all the data is strongly correlated. Instead, a snapshot of the system is taken every N steps, where N is set low enough to still capture the fastest motion of interest but high enough to avoid wasting space due to correlations. The resulting sequence of snapshots is then commonly called the trajectory.

AMS writes the trajectory to the History and MDHistory sections of ams.rkf, according to the settings in the Trajectory block. A snapshot of the system and various MD variables is stored every SamplingFreq timesteps.

The trajectory itself contains only the data needed for subsequent analysis of the dynamics of the system. However, much more data is usually generated on every integration step. This includes, for example, the internal data used by an engine when evaluating the energies and forces. This information is normally discarded after each step, because it is often very large. However, a Checkpoint containing the complete internal state of the MD driver together with a result file generated by the engine is stored every Frequency steps. An interrupted simulation can then be restarted from this checkpoint using the Restart keyword. Additionally, the engine result files called MDStep*.rkf can also be used to extract various engine-specific details about the system, such as the orbitals for QM engines.

MolecularDynamics
Trajectory
Type: Block Sets the frequency for printing to stdout and storing the molecular configuration on the .rkf file.
SamplingFreq
Type: Integer 100 Write the the molecular geometry (and possibly other properties) to the .rkf file once every N steps.
Checkpoint
Type: Block Sets the frequency for storing the entire MD state necessary for restarting the calculation.
Frequency
Type: Integer 1000 Write the MD state and engine-specific data to the respective .rkf files once every N steps.
CalcPressure
Type: Bool False Calculate the pressure in periodic systems. This may be computationally expensive for some engines that require numerical differentiation. Some other engines can calculate the pressure for negligible additional cost and will always do so, even if this option is disabled.
Print
Type: Block This block controls the printing of additional information to stdout.
System
Type: Bool False Print the chemical system before and after the simulation.
Velocities
Type: Bool False Print the atomic velocities before and after the simulation.


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## Introductory Chemistry (5th Edition)

The $I$ concentration on this equilibrium is equal to 0.015M
1. Write the equilibrium constant expression: $K_{eq} = \frac{[Products]}{[Reactants]} = \frac{[I]^2}{[I_2]}$ 2. Find the $I$ concentration: $1.1 \times 10^{-2} = \frac{[I]^2}{(0.0205)}$ $1.1 \times 10^{-2} \times 0.0205 = [I]^2$ $2.255 \times 10^{-4} = [I]^2$ $[I] = \sqrt {2.255 \times 10^{-4}} = 0.015M$


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# [Fourier and PDEs] Solving the Heat Equation

## 2. Nondimensionalizing the Heat Equation

Consider the heat equation $\pd{T}{t} = \kappa \pd{^2 T}{x^2},$

on the interval $0 \leq x \leq L$ with boundary conditions $T(0, t) = T(L, t) = 0$ and a homogeneous initial temperature $T_0$.

a) Non-dimensionalize the heat equation, being careful to show your new variables are indeed non-dimensional, to give $\pd{\tilde{T}}{\tilde{t}} = \pd{^2 \tilde{T}}{\tilde{x}^2}$

on the interval $0 \leq \tilde{x} \leq 1$ with boundary conditions $\tilde{T} = 0$ at either ends and a homogeneous initial temperature of $1$.

Spoiler

b) Find the non-dimensional temperature $\tilde{T}(\tilde{x}, t)$.

Spoiler

c) At approximately what time does the non-dimensional temperature fall below $0.5$ everywhere? What is the corresponding dimensional time? How would the itme change for an identical system which is of length $2L$?

Spoiler

## 3. Heat equation with non-uniform initial condition

Consider the heat equation $T_t = \kappa T_{xx}$ on the interval $0 \leq x \leq L$ with boundary conditions $T(0, t) = T(L, t) = 0$ and initial temperature $T(x,0) = x(L - x)$.

a) Find $T(x,t)$ at later times $t > 0$.

b) (i) Find the heat flux at $x = 0$ for $t > 0$ and (ii) what is the heat flux at $x = L/2$.

Spoiler
1) Note that you can guess that the right-hand side is $-\lambda^2$ because if it was positive instead, then solutions would grow exponentially in time.


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# SublimeHg

Issue commands to Mercurial from Sublime Text.

## Requirements

• Mercurial 1.9 or above (command server)

## Configuration

By default, the executable name for Mercurial is hg. If you need to use a different one, such as hg.bat, change the variable packages.sublime_hg.hg_exe to it in Preferences | Global Settings - User.

Example:

{
"packages.sublime_hg.hg_exe": "hg.bat"
}

## How to Use

SublimeHg can be used in two ways:

• Through a command-line interface: show_sublime_hg_cli command.

Regardless of the method used to call it, SublimeHg ultimately talks to the Mercurial command server. The command-line interface is the more flexible option, but some operations might be quicker through the menu.

Normally, you will follow these steps to use SublimeHg:

1. Open the Command Palette (Ctrl+Shift+P) and look for SublimeHg.
2. Select option
3. Select Mercurial command (or type in command line)

Alternatively, you can assign show_sublime_hg_cli and show_sublime_hg_menu their own key bindings.

### Tab Completion

While in the command-line, top level commands will be autocompleted when you press Tab.


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# What is the value of $\operatorname{arctan} \left(-\frac{12}{5}\right)$?

I am trying to find the value of $$\sin(\operatorname{arctan}(-12/5))$$ manually (without a calculator). I know I need to solve the inner portion I need to find the angle at which $$\operatorname{tan}(\theta) = -12/5$$. But this is not one of the easy angles to simply look up the value.

I also know it is a $$5, 12, 13$$ right triangle, both because I recognize the numbers and because $$x^2 +y^2 = r^2$$ and $$r = 13$$. But I don't know how to find that angle and can't find a good example.

• If you're just going to be taking the sine of the angle, then you don't have to know the angle itself. Just use your knowledge of the $5$-$12$-$13$ triangle to determine what the sine should be (with appropriate consideration for the sign). See, for instance, this question. – Blue Jan 27 at 23:04
• This angle is not a rational multiple of $\pi$. – GEdgar Jan 28 at 1:35
• Thank you @Blue That is the explanation of the answer below I was looking for. That makes perfect sense now. – Nathan Jan 28 at 1:56
• @Nathan: Glad to help! I guess I won't need this analogy: A restaurant gives-out party favors for patrons on their birthdays. On Mondays, they have blue hats w/white lettering, and white balloons w/blue lettering; on Tuesdays, red hats w/green lettering, and green balloons w/red lettering; on Wednesdays, orange hats w/brown lettering, and brown balloons w/orange lettering; etc. At work, you see a colleague with one of the hats; she says, "Oh, my birthday was last week." You don't have to know which day was her birthday to deduce what type of balloon she got; the hat tells you all you need. – Blue Jan 28 at 2:12

## 2 Answers

$$x=\operatorname{arctan}(-12/5)$$ is an angle on the fourth quadrant that has $$\operatorname{tan} x=-12/5$$. By your observation, you know that its sine is $$-12/13$$ (just draw the triangle on the fourth quadrant with hypotenuse $$13$$, adjacent (horizontal) side $$5$$ and opposite (vertical) side $$12$$).

• You are totally right, but I didn't fully understand your answer until I read @Blue's comment above. Pasted here: If you're just going to be taking the sine of the angle, then you don't have to know the angle itself. Just use your knowledge of the 5-12-13 triangle to determine what the sine should be (with appropriate consideration for the sign). See, for instance, this question – Nathan Jan 28 at 1:58

Let $$\theta = \arctan \frac {-12}{5}$$

$$\tan \theta = \frac {-12}{5} = \frac {\sin \theta}{\cos \theta}$$ which means there there is a right triangle with angle $$\theta$$ and the opposite side = $$r\sin \theta = -12$$ ($$-12$$ meaning extends below the $$x$$ axis) and an adjacent side $$r \cos \theta = 5$$ where $$r$$ is the hypotenuse.

So $$\sin \theta = \frac {-12}r$$ where $$r$$ is the hypotenuse = $$\sqrt {(-12)^2 + 5^2} = 13$$.

So $$\sin \theta = \frac {-12}{13}$$.

....

In general:

$$\sin (\arcsin \frac ab) = \frac {a}{\sqrt{a^2 + b^2}}$$

• I suppose that the last expression is $\sin (\arctan \frac ab) = \frac {a}{\sqrt{a^2 + b^2}}$ – Claude Leibovici Jan 28 at 6:08


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# How To Find The Common Ratio Of A Geometric Sequence Without The First Term

The sum of the numbers in a geometric progression is also known as a geometric series. Use the discriminant to determine the number and type of roots of a quadratic equation. and common difference d is given by: Ex) The nth term of an arithmetic sequence with first term 2 and common difference 3. (3) (d) Find the sum to infinity of the sequence. • Find the sums of infinite geometric sequences. Find the common ratio and the first term. Geometric progression - Wikipedia wikipedia. 8 , r = −5 16) a 1 = 1, r = 2 Given the first term and the common ratio of a geometric sequence find the recursive formula and the three terms in the sequence after the last one given. A Geometric Sequence has a constant ratio, denoted by r. The number multiplied (or divided) at each stage of a geometric sequence is called the "common ratio" r, because if you divide (that is, if you find the ratio of) successive terms, you'll always get this common value. a n a 1 (n 1)d Formula for the nth term a 6 16 (6 1)d n 6, a 1 16 91 16 5d a 6 91 75 5d Subtract 16 from each side. Geometric Progression, Series & Sums Introduction. Find the value of r. Best Answer: You are trying to find the 8th term of the geometric sequence (a8 means the 8th term). Retrieve Document. a our beginning term of our sequence, and r our common ratio. The tribonacci numbers are similar to the Fibonacci numbers, except that each term is the sum of the three previous terms in the sequence. I believe our approach is justified, and in order to explain why – consistent with the project of laying out the basic worldview and epistemology behind our research – I find myself continually returning to the distinction between what I call “sequence thinking” and “cluster thinking. StepsIdentify the first term in the sequence, call this number a. How many terms are in the geometric sequence having a first term 2, a last term 32, and a common ratio −2? 11. This number is called the common ratio for the geometric sequence and corresponds to r. Problems and Notes for MTHT 466 Make up an arithmetic sequence such that the first term is 23 and the 15th term is 51. Geometric Progressions. A1 and r may be entered as an integer, a decimal or a fraction. Geometric sequences are a little more complicated, depending on the ratio $$\rho$$ (recall Theorem 2. the sum to infinity is 27. Calculate the possible values of the first term and of the common ratio of the progression. determine the common ratio of a geometric sequence whose first term is 2 and whose fourth term is 16? Answer, The nth term of a geometric series is given by the formula, n th term = , where r is the common ratio and a is the first term. Geometric Sequence - Find the COMMON RATIO Added Jan 29, 2014 by DrVB in Mathematics Given any two terms in a geometric sequence, find the common ratio r, which is given by r = X(n) / X(n-1). Then find a 11. Note that the first few terms of the series are. Input: There are three inputs: the number of terms, the first term, and the common ratio of a geometric progression. note: in fibonacci sequence, the next number in. This Geometric Sequence Calculator is used to calculate the nth term and the sum of the first n terms of a geometric sequence. geometric sequence, respectively. Finding a Term of a Geometric Sequence Find the 12th term of the geometric sequence Solution The common ratio of this sequence is Because the first term is you can determine the 12th term to be Formula for geometric sequence Substitute 5 for 3 for and 12 for Use a calculator. Geometric Sequence Formula The geometric sequence formula refers to determining the n th term of a geometric sequence. determine the common ratio of a geometric sequence whose first term is 2 and whose fourth term is 16? Answer, The nth term of a geometric series is given by the formula, n th term = , where r is the common ratio and a is the first term. That is, if the common ratio is r, a 2 = a 1 x rSimilarly the third term (a 3) is obtained by multiplying a fixed quantity (the common ratio) to the second term (a 2). Consider the sum of the first six terms of the GP with first term 2 and common ratio 4. a n a 1 (n 1)d Formula for the nth term a 6 16 (6 1)d n 6, a 1 16 91 16 5d a 6 91 75 5d Subtract 16 from each side. Continue using each term to find the next term. How to Find Any Term of a Geometric Sequence. (2) (Total 9 marks) 8. A sequence a 1, a 2, a 3, ,a n is said to be geometric is the ratio between consecutive terms remains. • Find the sixth term in the sequence, a 6. This means that in order to get the next element in the sequence we multiply the ratio $$r$$ by the previous element in the sequence. •Find the sum of an infinite geometric series. ind the common ratio and write out the first four terms of the geometric sequence {3^n-1/6} Common ratio is a1=, a2=, a3=, a4= The answer: A geometric sequence has the (general) form: a_n = a_1 * (r)^(n - 1) a_n = a with a subscript of n (this is the nth term in the sequence) a_1 = a with a subscript of 1 (this is the 1st term in the sequence). Use the discriminant to determine the number and type of roots of a quadratic equation. where r ≠ 0 is the common ratio and a is a scale factor, equal to the sequence's start value. Find the Sum of the Series 3 , 3/4 , 3/16 , 3/64 This is a geometric sequence since there is a common ratio between each term. Keep reading for a detailed definition, the formula for determining the common ratio and some example problems. You may know that the 50th term of an arithmetic sequence is 300, and you know that the terms have been increasing by 7 (the "common difference"), but you want to find out what the first term of the sequence was. Find the first term of the sequence. The third term of a geometric sequence is 324 and the sixth term is 96 (a) Show that the common ratio of the sequence is (2) (b) Find the first term of the sequence. nth term of a geometric sequence. TSW find the nth term and the nth partial sum of a sequence given the first term and common difference. Click Create Assignment to assign this modality to your LMS. Known as either as geometric sequence or geometric progression, multiplying or dividing on each occasion to obtain a successive term produces a number sequence. You’ve got a very straight forward way of writing an infinite string of numbers. This C Program allows the user to enter first value, total. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power. What is the common ratio of the series? Task #10 – Derive Formula for Infinite Geometric Series – (continued). Really clear math lessons (pre-algebra, algebra, precalculus), cool math games, online graphing calculators, geometry art, fractals, polyhedra, parents and teachers areas too. Practice Problem 1: Find the ninth term of the geometric sequence whose first term is 4 and whose common ratio is ½. common ratio r is given by: an alrn The nth term of a geometric sequence with a first term of 3 and common ratio 2 is given by. so, 16 = divide both sides by 2 ==> 16/3 =. The first few terms are 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, and 1334961. To determine the long-term effect of Warfarin, we considered a finite geometric series of $$n$$ terms, and then considered what happened as $$n$$ was allowed to grow without bound. determine the common ratio of a geometric sequence whose first term is 2 and whose fourth term is 16? Answer, The nth term of a geometric series is given by the formula, n th term = , where r is the common ratio and a is the first term. There are two ways of finding the common ratio of a geometric sequence: (1) The first one is to divide the number and the number after it. Geometric Series is a sequence of terms in which next element is obtained by multiplying common ration to previous element. it doesn't have a common difference or a common ratio. There is no common difference. We can use this formula to find the first and second set of number. Code users may find the text of provisions in effect on any given date in the past by using the appropriate List of CFR Sections Affected (LSA). The nth term of an arithmetic PPT. 3 2 3 4 3 8 3 16. For example, in the sequence above, you would not be expect-ed to know that the 6th term is 33 without being given. n 2 Find the first and the 10th terms. Find the seventh term of the geometric sequence, given the first term and common ratio. The common ratio is 3. The first term of a geometric progression exceeds the second term by $$2$$, and the sum of the second and third terms is $$\frac{4}{3}$$. The formula of common ratio is dividing second term with the first one. In the explicit formula for a geometric sequence, the variable r represents the common ratio for the sequence. • Use geometric sequences to model and solve real-life problems. Find the common ratio. For example, for the sequence 2 5 10 17 26 37 how would the 7th term be found?. The common ratio of the given sequence is -2. Maths Sequence & Series part 12 (Sum of ters of GP Geometric Progression) CBSE class 11 Maths 4. Then find a20. In order for an infinite geometric series to have a sum, the common ratio r must be between − 1 and 1. You can use the nth term formula to find the common difference. The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. Find the value of a) the common ratio; b) the first term. Different numbers x, y and z are the first three terms of a geometric progression with common ratio r, and also the first, second and fourth terms of an arithmetic progression. The first term is given, a 1 = 5. Where, a is the first term and r is the common ratio. Studyres contains millions of educational documents, questions and answers, notes about the course, tutoring questions, cards and course recommendations that will help you learn and learn. = No sum, since. This example highlights a limitation to the similarities between geometric sequences and exponential functions. And the second term is 9^1. Common Ratio Next Term N-th Term Value given Index Index. The first term a1 is 3, and n 2. Find the first term, common ratio, and an explict rule for the nth term. Geometric Mean. An infinite geometric series has first term -3/2 and sums to twice the common ratio. The (n+1) th term of GP can be calculated as (n+1) th = n th x R where R is the common ratio (n+1) th /n th The formula to calculate N th term of GP : t n = a x r n-1 where, a is first term of GP and r is the common ratio. For example, it appears as the ratio Of a side to the base in the 720, 720 , 360 isosceles triangle. Here's what we use this for: The nth term is given by a formula. 456 and then find the 10th term. For example, if the first value of a geometric sequence is 6 and the common ratio is , we have: This sequence also converges to 0. [n = 4] Solution. Find the sum of the first ten terms of the third sequence. Find the sum to the infinity of the geometric progression. Find the first term of a geometric sequence. quantity one minus the common ratio. 4 The table below shows the balance b, in dollars, of Daryl’s savings account t years after he made an initial deposit. Find the value of the Find the value of the SPNone. How to Find Any Term of a Geometric Sequence. A sequence a 1, a 2, a 3, ,a n is said to be geometric is the ratio between consecutive terms remains. What is the first term and common ratio for this geometric sequence? Find the first term and common ratio for the geometric sequence when #a_"2"# = -15 and #a_"4"# = -375. It is often useful to find a formula for a sequence of numbers. Find the sum, if it exists, for the following series. Find the common ratio. The tribonacci numbers are similar to the Fibonacci numbers, except that each term is the sum of the three previous terms in the sequence. If we do not already have an explicit form, we must find it first before finding any term in a sequence. Find the value of a) the common ratio; b) the first term. To find the sum of the first n terms of a geometric sequence with first term a1, and common ratio r, one may use the following formula: Example: Find the sum of the first six terms of the geometric sequence with first term −3and common ratio 4. It is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. StepsIdentify the first term in the sequence, call this number a. Tell the learners that the sequence 2, 4, 8, 16, 32 is an example of a type of sequence. Guidelines to use the calculator If you select a n, n is the nth term of the sequence If you select S n, n is the first n term of the sequence For more information on how to find the common difference or sum, see this lesson Geometric sequence. Common Ratio For a geometric sequence or geometric series , the common ratio is the ratio of a term to the previous term. so, 16 = divide both sides by 2 ==> 16/3 =. series mc-TY-convergence-2009-1 In this unit we see how ﬁnite and inﬁnite series are obtained from ﬁnite and inﬁnite sequences. , the nth term of the recursively defined sequence below: 12) In an arithmetic sequence, and. Important Formula of Geometric Progression. -2 Common Ratio, r multiply each tem by 2 to arive at the next tem ordivide by al to find the common ratio, 2. Find the seventh term of the geometric sequence, given the first term and common ratio. The geometric sequence can be rewritten as where is the amount of terms, is the common ratio, and is the first term. There are few enough that you can write the series and add them up. A geometric sequence is an exponential function. The first term of an infinite geometric sequence is 3, and the common ratio is. 25, with a = 100, and 95 90. The variable a_n is equal to the value of the nth term in the given geometric sequence, while a_1 is the value of the first term in the sequence. 2: A geometric sequence has a first term of 2 and a common ratio of 1. So, a rule for the nth term is: a n a 1rn 1 Write general rule. quantity one minus the common ratio. Geometric Sequence In mathematics, a geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero. , if first term is 2, last term is 486 and the sum of these. If we do not already have an explicit form, we must find it first before finding any term in a sequence. Consecutive terms of a geometric sequence have a common ratio. as each power of r will tend to 1 as well giving: as expected!. Find the common ratio, r, in the sequence by dividing two successive terms:for every positive integer k. The common ratio is the ratio between two numbers in a geometric sequence. To find the sum of the terms of a geometric sequence, we need to know the first term, al, the common ratio, r, and the number of terms, n. Program for N-th term of Geometric Progression series Given first term (a), common ratio (r) and a integer N of the Geometric Progression series, the task is to find N th term of the series. Since the last term is 729, we can set up the equation. -2 Common Ratio, r multiply each tem by 2 to arive at the next tem ordivide by al to find the common ratio, 2. This arithmetic sequence calculator (also called the arithmetic series calculator) is a handy tool for analyzing a sequence of numbers that is created by adding a constant value each time. Trying to find the value of a certain term in a geometric sequence? Don't want to go through the terms one-by-one to find the one you want? Use the formula to find the nth term in a geometric sequence! This tutorial shows you how find that formula!. A general geometric sequence has first term a 1 and common ratio r. Find the Nth term divisible by a or b or c Program to print GP (Geometric Progression) Given first term (a), common ratio (r) and a integer n of the Geometric Progression series, the task is to print th n terms of the series. The first term is 4. 1 Arithmetic and Geometric Sequences Definitions: (yes, that's right, this is important, know these!) A sequence is a set of numbers, called terms, arranged in some particular order. To use it, we find our values, plug them in, and evaluate. Example 1: Find the 35 th term in the arithmetic sequence 3, 9, 15, 21, … There are three things needed in order to find the 35 th term using the formula: the first term ( {a_1}) the common difference between consecutive terms (d) and the term position (n ) From the given sequence, we can easily read off the first term and common difference. At some point, your pre-calculus teacher will ask you to find the general formula for the nth term of an arithmetic sequence without knowing the first term or the common difference. Given that the first term of a geometric sequence is -2 and the common ratio is -1/4. Solution: First term, a = 1/3, common ratio, r = 3. [3 marks] Determine the set of values of n for which. Trying to find the value of a certain term in a geometric sequence? Don't want to go through the terms one-by-one to find the one you want? Use the formula to find the nth term in a geometric sequence! This tutorial shows you how find that formula!. The rst term of a geometric series is 20 and the common ratio is 7 8. This arithmetic sequence calculator (also called the arithmetic series calculator) is a handy tool for analyzing a sequence of numbers that is created by adding a constant value each time. Geometric sequences are a little more complicated, depending on the ratio $$\rho$$ (recall Theorem 2. We can also use formula (n/2) (a + l) to find sum, where a is the first term and l is the last term of given arithmetic progression. List of important geometric progression formulas are given below: The Formula for nth term in the geometric progression (G. For example in the arithmetic sequence - 10, 20, 30, 40 É. find the first term and common ratio [Solved!] Alicia 28 Nov 2015, 06:03. Write the first 5 terms of a geometric sequence with first term 3 and common ratio 2. Find the first term, common ratio, and an explict rule for the nth term. Since we only know the second term, let’s first use the Finite Geometric Sequences formula. Normally we call the number in a geometric sequence that each term is multiplied by r, which is really short for the term 'the common ratio'. , the first example is denoted by s n = 1-2 n. Geometric Progression, Series & Sums Introduction. Given that the fourth term is 4/3 and the seventh term is 32/81 in a geometric sequence, find the common ratio, r. The name “Fibonacci sequence” was first used by the 19th-century number theorist Edouard Lucas. to generate the next term. You can also solve for the nth term, first term, number of terms, or common ratio of a geometric sequence. " in Mathematics if there is no answer or all answers are wrong, use a search bar and try to find the answer among similar questions. So, a rule for the nth term is: a n = a 1 + (n º 1)d Write general. Series Consider the arithmetic sequence 7, 13, 19, 25, 31, … What if we wanted to find the sum of the first 100 terms? We can find the 100th term using the formula from. A geometric series has the form "a*r^k", where "a" is the first term of the series, "r" is the common ratio and "k" is a variable. \) with the specific property that the ratio between two consecutive terms of the sequence is ALWAYS constant, equal to a certain value $$r$$. series mc-TY-convergence-2009-1 In this unit we see how ﬁnite and inﬁnite series are obtained from ﬁnite and inﬁnite sequences. (1) 2nd term: -24, 5th term: 81 [Sol] Let a be the 1st term and r be the common ratio. Use the explicit formula to write a geometric sequence whose common ratio is a decimal number between 0 and 1. To use it, we find our values, plug them in, and evaluate. This shows indeed that this sequence is not created by adding or subtracting a common term. Find the 6th term of the sequence? a n = a 1 rn-1 Write the formula. Write the first 5 terms of an arithmetic sequence with first term of 3 and common difference 2. Repeated multiplication by a number whose magnitude is larger than 1 makes the resulting magnitude increase without bound. Varsitytutors. A recursive definition, since each term is found by multiplying the previous term by the common ratio, a k+1 =a k * r. This is an example of an arithmetic progression (AP) and the constant value that defines the difference between any two consecutive terms is called the common difference. , if first term is 2, last term is 486 and the sum of these. P series, which is optional. Multiply each term by the common ratio to find the next three terms. Here, we used For Loop to display the G. The geometric sequence has first term and common ratio r = 1. The ratio between any two adjacent numbers will give the factor. Objective A Identify a geometric sequence (or geometric progression) and find its common ratio. 81 4 3 n 1 Substitute for a 1 and r. Geometric sequence worksheets are prepared for determining the geometric sequence, finding first term and common ratio, finding the n th term of a geometric sequence, finding next three terms of the sequence and much more. For example, in the series 2, 4, 8, 16 the factor is 16/8 or 8/4 = 2. Finite Geometric Series A series that ends and whose successive terms have a common ratio. It is often useful to find a formula for a sequence of numbers. 580 Chapter 11 Sequences and Series Find Arithmetic Means Find the four arithmetic means between 16 and 91. VA-Algebra II Honors Scope and Sequence Unit Lesson Lesson Objectives The Quadratic Formula Find real and complex solutions of quadratic equations using the quadratic formula. In the case of the flea jumps, the original sequence is geometric with first term and common ratio l. The goal of this activity is for students to use the formula for the sum of the first $$n$$ terms of a geometric sequence in a non-geometric context. Example Find the nth term of the geometric sequence: 2, 2. In the above sequence, the second term (a 2) is obtained by multiplying a fixed quantity, the common ratio, to the first term (a 1). The Mathematics Standards of Learning Enhanced Scope and Sequence is a resource intended to help teachers align their classroom instruction with the Mathematics Standards of Learning that were adopted by the Board of Education in October 2001. So then, the first element is $$a_1$$, the next one is $$a_1 r$$, the next one is $$a_1 r^2$$, and so on. - the common difference would be 10. [Addition of fractions formula] 02:34. Each number in a sequence is called a term. a 6 = 5(2)6-1 Substitute 5 for a 1,6 for n, and 2 for r. From Ramanujan to calculus co-creator Gottfried Leibniz, many of the world's best and brightest mathematical minds have belonged to autodidacts. Here, we used For Loop to display the G. Objective A Identify a geometric sequence (or geometric progression) and find its common ratio. Find the common ratio of an infinite geometric series with the given sum and first term. n must be a positive integer. Finding the nth term of a geometric sequence. The first term is a, and the constant multiplier, or common ratio of successive terms, is x. Therefore, 2 nd term is ar, 3 rd term is ar 2, 4 th term is ar 3, Example: 1. r can be found by taking any term except the first and dividing it by the preceding term. Given that the sum of the first four terms of a geometric sequence is 10 and the fourth term is half the third term. Get an answer to your question "Find the first 5 terms of the sequence: a1 = 500, an = (an-1) / 5. Alternatively, the sequence can be defined recursively by where. The first term is 4. Since the geometric sum equation is a(1-r^n)/(1-r), a being the first term and r being the common ratio, we can find the sum of the first 8 by: 3(1 - 5^8)/(-4) = 292968. Common Ratio Next Term N-th Term Value given Index Index. graphics calculator instructions quadratics functions exponentials logarithms transforming functions sequences and series the binomial expansion the unit circle and radian measure non-right angled triangle trigonometry trigonometric functions. We can write this as an algebraic expression. I hope you can understand this. Find a and b. 2 Consider the geometricsequence with u5 =18and 8 = 486. The formula of common ratio is dividing second term with the first one. the sum to infinity is 27. The common ratio of the terms in a geometric series is 2 x a) State the set values of x for which the sum to infinity of the series exists. If the absolute value of the common ratio is less than , , the sum of terms always approaches a definite limit as increases without bounds. Here I'm multiplying it by a different amount. The sequence <1,2,4,8,16,… = is a geometric sequence with common ratio 2, since each term is obtained from the preceding one by doubling. For this series, find (a) the common ratio,(2) (b) the first term,(2) (c) the sum of the first 20 terms, giving your answer to the nearest whole number. I hope you can understand this. Now, the other terms are: Second term. The first term is a, and the constant multiplier, or common ratio of successive terms, is x. Give an example and explain WHY is constitutes. The nth term of a geometric sequence is , where is the first term and is the common ratio. 5 and 25 is also 1/2, and so on down the whole sequence. Geometric Sequence. To find the sum of the first S n terms of a geometric sequence use the formula S n = a 1 (1 − r n) 1 − r, r ≠ 1, where n is the number of terms, a 1 is the first term and r is the common ratio. Are you trying to find a particular term in the geometric sequence? For example, in the geometric sequence 2,4,8,16 to find the 15th term we would use the formula a sub n = a sub 1 x r ^(n-1), where a sub n is the nth term of the geometric sequence a sub 1 is the 1st term of the geometric sequence r is common ratio between successive terms. Relates any term in a sequence to only the first term and the common ratio. 5, ), is a geometric sequence since the ratio between 25 and 50 is 1/2, the ration between 12. So let's look at some geometric sequences. How many terms are in the geometric sequence having a first term 2, a last term 32, and a common ratio −2? 11. There is a common ratio of 4. Find the first term and a recursive rule for the n-th term. Also every following term of the sequence has certain relation with the first term. commutative: The property of a binary operation such that its operands can always be swapped around without affect its value. Substituting our pizza values into the formula, we see that the sum will equal one half divided. Where a is the first term, 'n' is the number of terms and 'r' is the common ratio. A geometric sequence can be defined recursively by the formulas a 1 = c, a n+1 = ra n, where c is a constant and r is the common ratio. The Fibonacci sequence obeys the recursion relation P(n) = P(n-1) + P(n-2). The first term of a geometric sequence is 200 and the sum of the first four terms is 324. In other words, a sequence is a list of numbers generated by some mathematical rule and typically expressed in terms of n. I need a formula for looking the common ratio of a geometric series. Finding the Terms of a Geometric Sequence:. A sequence a 1, a 2, a 3, ,a n is said to be geometric is the ratio between consecutive terms remains. The (n+1) th term of GP can be calculated as (n+1) th = n th x R where R is the common ratio (n+1) th /n th The formula to calculate N th term of GP : t n = a x r n-1 where, a is first term of GP and r is the common ratio. To define an arithmetic or geometric sequence, we have to know not just the common difference or ratio, but also the initial value (called a). StepsIdentify the first term in the sequence, call this number a. A Sequence is a set of things (usually numbers) that are in order. a our beginning term of our sequence, and r our common ratio. geometric progression,sequence,arithmetic progression,sum of n terms. That means that the first term of your sequence is 90. The power in 'r' is one less than the nth term of a geometric sequence. 5, ), is a geometric sequence since the ratio between 25 and 50 is 1/2, the ration between 12. Geometric Sequences and Series Part III The sequence is an example of a Geometric sequence A sequence is geometric if where r is a constant called the common ratio In the above sequence, r = 2 A geometric sequence or geometric progression (G. Rule of an Arithmetic Sequence: The nth term of an arithmetic sequence with first term a1. For example, in the sequence above, you would not be expect-ed to know that the 6th term is 33 without being given. The (n+1) th term of GP can be calculated as (n+1) th = n th x R where R is the common ratio (n+1) th /n th The formula to calculate N th term of GP : t n = a x r n-1 where, a is first term of GP and r is the common ratio. Byju's Geometric Sequence Calculator is a tool which makes calculations very simple and interesting. The sum of a finite geometric sequence (the value of a geometric series) can be found according to a simple formula. If you can find a simple form for this function g, then bully. Alternatively, the sequence can be defined recursively by where. Work Together • Using the sequence given at the right. jpg mc017-4. Check for a common ratio by dividing each term by the term that precedes it. What was the first term? 15. ) is of the form The nth term of an G. Find the sum of the first 7 terms. C Program to find Sum of Geometric Progression Series Example. The general term for a geometric sequence with a common ratio of 1 is $\large a_n = a r^{n-1}= a \cdot 1^{n-1} = a$ So, a sequence with common ratio of 1 is a rather boring geometric sequence, with all the terms equal to the first term. Find the first term and a recursive rule for the n-th term. For example, it appears as the ratio Of a side to the base in the 720, 720 , 360 isosceles triangle. In geometric sequences there is a case of repeated multiplication Look down for more a,ar,ar^2,ar^(n-1 ----n---- So the sum of the first n terms of sequence is ; S_n =( a(1-r^n))/(1-r) Now given you know r n and the sum you find a by re arranging Additionally If you re given the nth term then' a_n = ar^(n-1) You may often has to use both these equation to get to the answer. Since the last term is 729, we can set up the equation. asked • 10/08/18 How to find the common ratio of a geometric sequence, if given only the first term as 200 and the sum of the first four terms as 324. Find the intercepts of 7xy – 15 = 4x + 12y – 1. A sequence a 1, a 2, a 3, ,a n is said to be geometric is the ratio between consecutive terms remains. NEWTON, ISAAC (b. c Find the sum of the first 10 terms of the corresponding geometric series. Could you please explain to me how to do geometric sequences and how to find the different terms and sums. Explicit rule appears exponential. The above sequence has a first term equal to 2 and a common difference d = 2. ſa, = ar = -24 D Tag = ard= 0 From 2:0, r = , Ledam Substituting into a= Therefore, the 1st term is common ratio is (2) 6th term: 25, gth term: 100. The common ratio of a geometric sequence is 3 and the sum of the first five terms is 968. Given the first term and the common ratio of a geometric sequence find the first five terms and the explicit formula. Menu Algebra 2 / Sequences and series / Geometric sequences and series A geometric sequence is a sequence of numbers that follows a pattern were the next term is found by multiplying by a constant called the common ratio, r. Finding a Term of a Geometric Sequence Find the 12th term of the geometric sequence Solution The common ratio of this sequence is Because the first term is you can determine the 12th term to be Formula for geometric sequence Substitute 5 for 3 for and 12 for Use a calculator. Alternatively, the sequence can be defined recursively by where. P are 3 and 1875 respectis€y. P) is a sequence of numbers such that the difference of any two successive numbers of the sequence is a constant. In order for an infinite geometric series to have a sum, the common ratio r must be between − 1 and 1. [a = 2, r = 3] Solution Example 3. Common Ratio  The ratio of a number in a geometric sequence to the preceding number in the sequence. Geometric Sequences and Series Part III The sequence is an example of a Geometric sequence A sequence is geometric if where r is a constant called the common ratio In the above sequence, r = 2 A geometric sequence or geometric progression (G. the previous term by a constant ratio Example: 42 1 0. Solution: Assume a Geometric progression with first term=a and ratio=r The given terms can be written as follows, $a_4= ar^3$ $a_7= ar^6$. From Ramanujan to calculus co-creator Gottfried Leibniz, many of the world's best and brightest mathematical minds have belonged to autodidacts. That is, in this question we have: 4, as the first number. The sum of the first n terms of a series is (3)n – 1. A sequence is a special case of a function. The nth Term of a Geometric Sequence Formula: Example 2: Find the 15 th term of the geometric sequence whose first term is 20 and whose common ratio is 1. note that the first, and largest, term in this series is. Geometric Sequence. Writing a Rule for the nth Term Write a rule for the nth term of the sequence 50, 44, 38, 32,. List of important geometric progression formulas are given below: The Formula for nth term in the geometric progression (G. The first term of an infinite geometric sequence is 3, and the common ratio is. In a number sequence, order of the sequence is important, and depending on the sequence, it is possible for the same terms to appear multiple times. The general term of. For example, if I know that the 10 th term of a geometric sequence is 24, and the 9 th term of the sequence is 6, I can find the common ratio by dividing the 10 th term by the 9 th term: 24 / 6 = 4.


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Enter the focal length (mm) and the aperture (mm^2) into the Focal Ratio Calculator. The calculator will evaluate and display the Focal Ratio.

## Focal Ratio Formula

The following formula is used to calculate the Focal Ratio.

FR = FL / A

• Where FR is the Focal Ratio
• FL is the focal length (mm)
• A is the aperture (mm^2)

## How to Calculate Focal Ratio?

The following example problems outline how to calculate Focal Ratio.

Example Problem #1:

1. First, determine the focal length (mm).
• The focal length (mm) is given as: 5.
2. Next, determine the aperture (mm^2).
• The aperture (mm^2) is provided as: 50.
3. Finally, calculate the Focal Ratio using the equation above:

FR = FL / A

The values given above are inserted into the equation below and the solution is calculated:

FR = 5 / 50 = .10

Example Problem #2:

For this problem, the variables required are provided below:

focal length (mm) = 6

aperture (mm^2) = 70

Test your knowledge using the equation and check your answer with the calculator above.FR = FL / A = ?