text:
# ant-ivy-user mailing list archives

##### Site index · List index
Message view
Top
From "andy@far2gone.com" <a...@far2gone.com>
Subject Re: maven classifier on ant install task
Date Mon, 10 Feb 2014 16:22:15 GMT
```Hi,

I've been deleting my cache prior to running the build.

The only way I can seem to make this work is to have a second resolver which has
the classifier hard-coded, ie:

<chain name="chain" returnFirst="true" dual="true">

<ibiblio name="central" m2compatible="true" checkconsistency="false"

pattern="[organisation]/[module]/[revision]/[artifact]-[revision].[ext]"/>

<ibiblio name="central2" m2compatible="true" checkconsistency="false"

pattern="[organisation]/[module]/[revision]/[artifact]-[revision]-jdk15.[ext]"/>

</chain>

This seems to work (at least all the jars I expect to find are being downloaded
to my local repo).

Cheers, Andy

> On 04 February 2014 at 06:55 Marc De Boeck <mdeboec@gmail.com> wrote:
>
>  You may have to remove the json-lib entry in your cache. Check also how the
> artifacts were stored in the cache. Could you find the jdk-specific artifacts
> there ?
>
>  Marc
>
>
>  2014-02-03 andy@far2gone.com <mailto:andy@far2gone.com> <andy@far2gone.com
> <mailto:andy@far2gone.com> >:
>    > >    Hi,
> >
> >
> >    My ivysettings.xml file now looks like:
> >
> >
> >    <ivysettings>
> >
> >      <settings defaultResolver="chain" defaultConflictManager="all" /> <!--
> > get all revisions without any eviction -->
> >
> >      <caches defaultCacheDir="\${ivy.cache.dir}/no-namespace" />
> >
> >      <resolvers>
> >
> >        <filesystem name="fs1">
> >          <ivy
> >
> >             pattern="\${dest.repo.dir}/no-namespace/[organisation]/[module]/ivys/ivy-[revision](-[classifier]).xml"
> > />
> >          <artifact
> >
> >             pattern="\${dest.repo.dir}/no-namespace/[organisation]/[module]/[type]s/[artifact]-[revision](-[classifier]).[ext]"
> > />
> >
> >
> >        </filesystem>
> >
> >
> >        <chain name="chain" returnFirst="true" dual="true">
> >
> >
> >
> >
> >
> >          <ibiblio name="central" m2compatible="true"
> >
> >             pattern="[organisation]/[module]/[revision]/[artifact]-[revision](-[classifier]).[ext]"/>
> >
> >
> >
> >        </chain>
> >
> >      </resolvers>
> >
> >    </ivysettings>
> >
> >
> >    Unfortunately, when I run the ant task the result is:
> >
> >
> >
> >    [ivy:install] :: installing net.sf.json-lib#json-lib;2.4 ::
> >    [ivy:install] :: resolving dependencies ::
> >    [ivy:install] found net.sf.json-lib#json-lib;2.4 in chain
> >    [ivy:install] :: installing in fs1 ::
> >    [ivy:install] published ivy to
> > c:\tmp\ivy-test\repo\no-namespace\net.sf.json-lib\json-lib\ivys\ivy-2.4.xml
> >    [ivy:install] :: install resolution report ::
> >    [ivy:install] :: resolution report :: resolve 0ms :: artifacts dl 175ms
> >    ---------------------------------------------------------------------
> >    | | modules || artifacts |
> >    | conf | number| search|dwnlded|evicted|| number|dwnlded|
> >    ---------------------------------------------------------------------
> >    | default | 1 | 0 | 0 | 0 || 1 | 0 |
> >    ---------------------------------------------------------------------
> >    [ivy:install]
> >    [ivy:install] :: problems summary ::
> >    [ivy:install] :::: WARNINGS
> >    [ivy:install] [FAILED ] net.sf.json-lib#json-lib;2.4!json-lib.jar: (0ms)
> >    [ivy:install] ==== central: tried
> >
> >   [ivy:install]<http://repo1.maven.org/maven2/net/sf/json-lib/json-lib/2.4/json-lib-2.4.jar>
> >    [ivy:install] ::::::::::::::::::::::::::::::::::::::::::::::
> >    [ivy:install] :: ^ see resolution messages for details ^ ::
> >    [ivy:install] ::::::::::::::::::::::::::::::::::::::::::::::
> >    [ivy:install] :: net.sf.json-lib#json-lib;2.4!json-lib.jar
> >    [ivy:install] ::::::::::::::::::::::::::::::::::::::::::::::
> >    [ivy:install]
> >    [ivy:install]
> >    [ivy:install] :: USE VERBOSE OR DEBUG MESSAGE LEVEL FOR MORE DETAILS
> >
> >    BUILD FAILED
> >    D:\ar-work\tmp\ivy-classifier-test\build.xml:32: Problem happened while
> > installing modules - see output for details
> >
> >
> >
> >
> >
> >
> >    Cheers, Andy
> >
> >    > On 29 January 2014 at 21:32 Marc De Boeck < mdeboec@gmail.com
> >    > <mailto:mdeboec@gmail.com> > wrote:
> >    >
> >    >
> >    > I could reproduce the problem you are describing.
> >    > It seems that the pom-file of this json-lib is not consistent with the
> >    > published artifacts.
> >    >
> >    > Try to specify the artifact pattern in your central resolver to
> >    > something
> >    > like this:
> >    >
> >    > <ibiblio name="central" m2compatible="true">
> >    > <pattern="[organisation]/[module]/[revision]/[artifact]-[revision]
> >    > (-[classifier]).[ext]"/>
> >    > </ibiblio>
> >    >
> >    > You should also add in the same way the artifact pattern for your cache
> >    > and
> >    > for your local repo. Otherwise, all these artifacts will be mapped to
> >    > the
> >    > same artifact name, and only one of the artifacts will end up in your
> >    > local
> >    > repo.
> >    >
> >    > Regards,
> >    > Marc
> >    >
> >    >
> >    >
> >    >
> >    > 2014-01-29 andy@far2gone.com <mailto:andy@far2gone.com> <
> >    > andy@far2gone.com <mailto:andy@far2gone.com> >
> >    >
> >    > > Hi,
> >    > >
> >    > > I think my use case is straightforward, I want to use Ivy to make a
> >    > > local
> >    > > Ivy
> >    > > repository of libraries available in maven central. This repository
> >    > > is
> >    > > then used
> >    > > to find dependencies declared in a gradle build.
> >    > >
> >    > > The intent is to have a build script that will automatically trundle
> >    > > off
> >    > > and
> >    > > collect all the jars requested in "install" tasks and store them
> >    > > locally
> >    > > (whether on the laptop of a developer who is new to the project or
on
> >    > > a CI
> >    > > server etc.).
> >    > >
> >    > > I think the problem I face is that without declaring a classifier in
> >    > > the
> >    > > install
> >    > >
> >    > > I don't think I'm trying to prematurely refine the content of the
> >    > > repository, in
> >    > > fact I'd be happy if all the available variants were downloaded, the
> >    > > problem is
> >    > >
> >    > > I suspect the reason is as follows, if you take a look at the maven
> >    > > web
> >    > > site
> >    > > page for json-lib at:
> >    > >
> >    > ><http://mvnrepository.com/artifact/net.sf.json-lib/json-lib/2.4>
> >    > >
> >    > >
> >    > >
> >    > ><http://repo1.maven.org/maven2/net/sf/json-lib/json-lib/2.4/json-lib-2.4.jar>
> >    > >
> >    > >
> >    > > Looking in the directory in which the jar files are stored:
> >    > >
> >    > ><http://repo1.maven.org/maven2/net/sf/json-lib/json-lib/2.4/>
> >    > >
> >    > > It can be seen that the jar file names all include a jdk13 or a jdk15
> >    > > in
> >    > > the
> >    > > file name.
> >    > >
> >    > > So, I think I need to persuade Ivy to put a jdk15 somewhere in the
> >    > > file
> >    > > name it
> >    > > is expecting?
> >    > >
> >    > > If I were using specific Ivy dependencies I think I would use a
> >    > > classifier
> >    > > in
> >    > > the dependency statement to achieve this.
> >    > >
> >    > > Where I get completely unstuck is figuring out how to persuade an Ivy
> >    > > install to
> >    > > download the files (either all of them or just the ones I ask for)...
> >    > >
> >    > > My setup is copied pretty much verbatim from the Ivy repository
> >    > > tutorial,
> >    > > ivy
> >    > > settings and ant build copied below.
> >    > >
> >    > > Cheers, Andy
> >    > >
> >    > > Ivy settings:
> >    > >
> >    > > <ivysettings>
> >    > >
> >    > > <settings defaultResolver="chain" defaultConflictManager="all" />
> >    > > <!--
> >    > > get all
> >    > > revisions without any eviction -->
> >    > >
> >    > > <caches defaultCacheDir="\${ivy.cache.dir}/no-namespace" />
> >    > >
> >    > > <resolvers>
> >    > >
> >    > > <filesystem name="fs1">
> >    > > <ivy
> >    > >
> >    > > pattern="\${dest.repo.dir}/no-namespace/[organisation]/[module]/ivys/ivy-[revision].xml"
> >    > > />
> >    > > <artifact
> >    > >
> >    > > pattern="\${dest.repo.dir}/no-namespace/[organisation]/[module]/[type]s/[artifact]-[revision].[ext]"
> >    > > />
> >    > > </filesystem>
> >    > >
> >    > >
> >    > > <chain name="chain" returnFirst="true" dual="true">
> >    > >
> >    > > <ibiblio name="central" m2compatible="true"/>
> >    > >
> >    > > </chain>
> >    > >
> >    > > <ibiblio name="libraries" m2compatible="true" />
> >    > >
> >    > > </resolvers>
> >    > >
> >    > > </ivysettings>
> >    > >
> >    > >
> >    > > Ant build:
> >    > >
> >    > > <target name="install-libs" description="--> install module from
> >    > > maven 2
> >    > > repository" depends="init-ivy">
> >    > >
> >    > > <property name="dest.repo.dir" value="\${externalBuildDir}/\${repoDir}"
> >    > > />
> >    > > <property name="ivy.cache.dir"
> >    > > value="\${externalBuildDir}/\${ivyCacheDir}"/>
> >    > >
> >    > > <ivy:settings file="\${basedir}/ivysettings.xml"/>
> >    > >
> >    > > <ivy:install organisation="log4j" module="log4j" revision="1.2.17"
> >    > > from="chain"
> >    > > to="fs1" transitive="true" overwrite="true"/>
> >    > >
> >    > > <ivy:install organisation="org.apache.solr" module="solr-solrj"
> >    > > revision="4.5.1"
> >    > > from="chain" to="fs1" transitive="true" overwrite="true"/>
> >    > >
> >    > > <!-- a load of other libraries that install ok -->
> >    > >
> >    > > <!-- the cause of my woe -->
> >    > > <ivy:install organisation="net.sf.json-lib" module="json-lib"
> >    > > revision="2.4"
> >    > > from="chain" to="fs1" transitive="false" overwrite="true" />
> >    > >
> >    > > </target>
> >    > >
> >    > >
> >    > > > On 28 January 2014 at 19:05 Kirby Files < kirby.files@masergy.com
> >    > > > <mailto:kirby.files@masergy.com> > wrote:
> >    > > >
> >    > > >
> >    > > > The install task is for copying entire modules into a repository.
> >    > > > You
> >    > > > typically don't do surgery on the artifacts, filtering by type
or
> >    > > > classifier, at this point. Once it's in your repository, you use
> >    > > > the
> >    > > > retrieve task to pull specific artifacts from dependencies.
> >    > > >
> >    > > > Perhaps you can explain your use case a little more, and
> >    > > > demonstrate
> >    > > > what you hope to accomplish with the install task.
> >    > > >
> >    > > > --Kirby
> >    > > >
> >    > > > Sent from my mobile device
> >    > > >
> >    > > > On Jan 28, 2014, at 1:30 PM, Andy Redhead < andy@far2gone.com
> >    > > > <mailto:andy@far2gone.com> > wrote:
> >    > > >
> >    > > > > Hi,
> >    > > > > Thanks for your reply and if I was using a dependency then
I
> >    > > > > think this
> >    > > > > would be sufficient.
> >    > > > >
> >    > > > > What I really need is the equivalent for the install task.
> >    > > > >
> >    > > > > Cheers, Andy
> >    > > > >
> >    > > > > On 28 Jan 2014 14:44, David Weintraub < qazwart@gmail.com
> >    > > > > <mailto:qazwart@gmail.com> > wrote:
> >    > > > >>
> >    > > > >> There are a couple of things you need:
> >    > > > >>
> >    > > > >> * You need to add the following line to the top of your
ivy.xml
> >    > > > >> file:
> >    > > > >>
> >    > > > >> <ivy-module version="1.0" xmlns:maven="<http://maven.apache.org>
> >    > > > >> ">
> >    > > > >>
> >    > > > >> Note the "xmlns" declaration. It doesn't matter where
it points
> >    > > > >> to (I
> >    > > like
> >    > > > >> using the URL of the project), you just need to declare
the
> >    > > > >> namespace.
> >    > > > >>
> >    > > > >> * Now, you can use that Maven classifier in your dependency
> >    > > declaration:
> >    > > > >>
> >    > > > >> <dependency org="net.sf.json-lib"
> >    > > > >> name="json-lib"
> >    > > > >> rev="2.4"
> >    > > > >> conf="compile->default">
> >    > > > >> <artifact name="json-lib"
> >    > > > >> type="jar"
> >    > > > >> ext="jar"
> >    > > > >> maven:classifier="jdk15"/>
> >    > > > >> </dependency>
> >    > > > >>
> >    > > > >>
> >    > > > >> --
> >    > > > >> David Weintraub
> >    > > > >> qazwart@gmail.com <mailto:qazwart@gmail.com>
> >    > > > >>
> >    > > > >> ================================================
> >    > > > >> Sure, call me crazy. They called Eddie Antar crazy too,
but
> >    > > > >> look at these low, low prices on all of these famous
name brand
> >    > > appliances.
> >    > > > >>
> >    > > > >>
> >    > > > >> On Jan 28, 2014, at 8:40 AM, andy@far2gone.com
> >    > > > >> <mailto:andy@far2gone.com> wrote:
> >    > > > >>
> >    > > > >>> Hi,
> >    > > > >>>
> >    > > > >>> I've been having a go at using the Ivy Install ant
> >    > > > >>> create a
> >    > > local
> >    > > > >>> repository based on libraries downloaded from maven
central.
> >    > > > >>>
> >    > > > >>> Having gone along with the tutorial, all seemed to
be working
> >    > > > >>> :)
> >    > > > >>>
> >    > > > >>> Until...
> >    > > > >>>
> >    > > > >>> One of the dependencies in my project is json-lib.
> >    > > > >>>
> >    > > > >>> It seems that the json-lib project uses a maven classifier
to
> >    > > discriminate
> >    > > > >>> between jars suitable for different versions of the
jvm.
> >    > > > >>>
> >    > > > >>> If I were just declaring a dependency I think I could
do this:
> >    > > > >>>
> >    > > > >>> <dependency>
> >    > > > >>> <groupId>net.sf.json-lib</groupId>
> >    > > > >>> <artifactId>json-lib</artifactId>
> >    > > > >>> <version>2.4</version>
> >    > > > >>> <classifier>jdk15</classifier>
> >    > > > >>> </dependency>
> >    > > > >>>
> >    > > > >>> However, I'm using install tasks that look like:
> >    > > > >>>
> >    > > > >>> <ivy:install organisation="log4j" module="log4j"
> >    > > > >>> revision="1.2.17"
> >    > > > >>> from="chain"
> >    > > > >>> to="fs1" transitive="true" overwrite="true"/>
> >    > > > >>>
> >    > > > >>> How should I declare the "classifier" on an "install"


text:
# Lesson 2: Linear Muscle Recruitment¶

Caution

Old tutorial: This tutorial has not yet been updated to ver. 7 of the AnyBody Modeling System. Some concepts may have changed.

Let us begin this lesson with a physiological observation. We know that there is a metabolic cost involved in development of muscle force. Food is a limited resource for living organisms so it is likely that nature has programmed us to save muscle work in some sense. But exactly how?

When two muscles are sharing the load of one joint moment, the options seem to be:

• Letting muscle 1 do the work

• Letting muscle 2 do the work

• Letting the two muscles share the work as we saw in the previous lesson

So we can reduce the load on one muscle, but the cost will be more work remaining for the other muscle to do. It seems that a criterion taking both muscles into account is needed. Let us just briefly recall the formulation of the muscle recruitment problem again:

\begin{split} \begin{equation*} \begin{aligned} & {\text{minimize}} & & G\left(\mathbf{f^{(M)}}\right) \\ & \text{subject to} & & \mathbf{Cf} = \mathbf{r} \\ &&& f_i^{(M)} \geq 0, i=1\ldots n^{(M)} \end{aligned} \end{equation*} \end{split}

The simplest imaginable choice of objective function taking the effort of both muscles into account seems to be a linear combination like this:

$G = \frac{f_1}{N_1}+ \frac{f_2}{N_2}$

i.e., a linear combination of the two muscle forces, where the $$N_1$$ and $$N_2$$ are some choice of normalization factors. Thinking about the physiology we might quickly get the idea that it is reasonable if strong muscles do more work than weak muscles, so $$N_i$$ acould be measures of the strength of each muscle. We are going to stick with that idea for the remainder of this tutorial because it actually turns out to make sense also experimentally. We can get AnyBody to work with precisely this definition by making the following additions to our study:

AnyBodyStudy MyStudy = {
AnyFolder &Model = .MyModel;
Gravity = {0.0, -9.81, 0.0};
nStep = 1000; // Lots of time steps to make it run slowly enough
InverseDynamics.Criterion = {
Type = MR_Linear;
};
};

This small addition specifies that the muscles must be recruited as a linear combination. The muscles forces are automatically normalized by the strengths. Running the analysis and plotting the result as before leaves us with this somewhat surprising result:

It seems that the weaker M2 is slacking around doing nothing and leaving all the work to the stronger M1. It turns out that we always get this type of behavior when recruiting muscles with a linear objective function. In general, this method will only recruit the minimum number of muscles to balance the system, and the chosen muscles are those that by combination of their moment arms and strengths are best suited for the purpose. While this might be a reasonable strategy, it is unfortunately totally non-physiological. It is possible to measure activity in muscles electrically, and these measurements show without a doubt that several muscles spanning the same joint share the load between them.

This type of muscle recruitment is included in AnyBody for two reasons despite being non-physiological:

1. For completeness. The strategy behind AnyBody is to leave decisions about physiology to the user as much as possible.

2. This type of analysis is excellent for identifying the important muscles in a complex system in a given posture and external load. In complex systems this can be very challenging to determine manually and the result can be useful for a variety of purposes including decisions on tendon transfer surgery.

For normal purposes, however, the linear criterion is not a good choice, and we shall therefore proceed to investigate another option in

format: plain


text:
### Electronic Research Archive

2021, Issue 5: 2987-3015. doi: 10.3934/era.2021023
Special Issues

# Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron

• Received: 01 December 2020 Revised: 01 February 2021 Published: 15 March 2021
• Primary: 37G15, 92B20; Secondary: 37C29

• Post-inhibitory rebound (PIR) spike induced by the negative stimulation, which plays important roles and presents counterintuitive nonlinear phenomenon in the nervous system, is mainly related to the Hopf bifurcation and hyperpolarization-active caution ($I_h$) current. In the present paper, the emerging condition for the PIR spike is extended to the bifurcation of the big homoclinic (BHom) orbit in a model without $I_h$ current. The threshold curve for a spike evoked from a mono-stable or coexisting steady state surrounds the steady state from left, to below, and to right, because the BHom orbit is big enough to surround the steady state. The right part of the threshold curve coincides with the stable manifold of the saddle and acts the threshold for the spike induced by the positive stimulation, resembling that of the saddle-node bifurcation on an invariant cycle, and the left part acts the threshold for the PIR spike, resembling that of the Hopf bifurcation. The bifurcation curve and a codimension-2 bifurcation point related to the BHom orbit are acquired in the two-parameter plane. The results present a comprehensive viewpoint to the dynamics near the BHom orbit bifurcation, which presents a novel threshold curve and extends the conditions for the PIR spike.

Citation: Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron[J]. Electronic Research Archive, 2021, 29(5): 2987-3015. doi: 10.3934/era.2021023

### Related Papers:

• Post-inhibitory rebound (PIR) spike induced by the negative stimulation, which plays important roles and presents counterintuitive nonlinear phenomenon in the nervous system, is mainly related to the Hopf bifurcation and hyperpolarization-active caution ($I_h$) current. In the present paper, the emerging condition for the PIR spike is extended to the bifurcation of the big homoclinic (BHom) orbit in a model without $I_h$ current. The threshold curve for a spike evoked from a mono-stable or coexisting steady state surrounds the steady state from left, to below, and to right, because the BHom orbit is big enough to surround the steady state. The right part of the threshold curve coincides with the stable manifold of the saddle and acts the threshold for the spike induced by the positive stimulation, resembling that of the saddle-node bifurcation on an invariant cycle, and the left part acts the threshold for the PIR spike, resembling that of the Hopf bifurcation. The bifurcation curve and a codimension-2 bifurcation point related to the BHom orbit are acquired in the two-parameter plane. The results present a comprehensive viewpoint to the dynamics near the BHom orbit bifurcation, which presents a novel threshold curve and extends the conditions for the PIR spike.

 [1] Local control of postinhibitory rebound spiking in CA1 pyramidal neuron dendrites. J. Neurosci. (2010) 30: 6434-6442. [2] Dynamics and bifurcations in a silicon neuron. IEEE Trans. Biomed. Circuits Syst. (2010) 4: 320-328. [3] Postinhibitory rebound during locomotor-like activity in neonatal rat motoneurons in vitro. J. Neurophysiol. (1998) 79: 342-351. [4] P. Channell, G. Cymbalyuk and A. Shilnikov, Origin of bursting through homoclinic spike adding in a neuron model, Phys. Rev. Lett., 98 (2007), 134101. doi: 10.1103/PhysRevLett.98.134101 [5] Dynamics of neurons in the pre-Bötzinger complex under magnetic flow effect. Nonlinear Dyn. (2018) 94: 1961-1971. [6] Bursting and two-parameter bifurcation in the Chay neuronal model. Discrete Contin. Dyn. Syst. Ser. B (2011) 16: 445-456. [7] B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems. A Guide to XPPAUT for Researchers and Students, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898718195 [8] Improving control effects of absence seizures using single-pulse alternately resetting stimulation (SARS) of corticothalamic circuit. Appl. Math. Mech-Engl. (2020) 41: 1287-1302. [9] Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophys. (1955) 17: 257-278. [10] Synchronization and rhythmic processes in physiology. Nature (2001) 410: 277-284. [11] Slow and persistent postinhibitory rebound acts as an intrinsic short-term memory mechanism. J. Neurosci. (2010) 30: 4687-4692. [12] L. Guan, B. Jia and H. Gu, A novel threshold across which negative stimulation evokes action potential near a saddle-node bifurcation in a neuronal model with $I_h$ current, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950198, 26 pp. doi: 10.1142/S0218127419501980 [13] Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete Contin. Dyn. Syst. Ser. S (2009) 2: 851-872. [14] Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete Contin. Dyn. Syst. Ser. B (2011) 16: 457-474. [15] The local electric changes associated with repetitive action in a non-medullated axon. J. Physiol. (1948) 107: 165-181. [16] Neural excitability, spiking and bursting. Internat. J. Bifur. Chaos Appl. Sci. Engrg. (2000) 10: 1171-1266. [17] Which model to use for cortical spiking neurons?. IEEE T. Neural Network (2004) 15: 1063-1070. [18] (2007) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting.MIT press. [19] Dynamics analysis of the hippocampal neuronal model subjected to cholinergic action related with alzheimer's disease. Cogn. Neurodyn. (2020) 14: 483-500. [20] T. Malashchenko, A. Shilnikov and G. Cymbalyuk, Bistability of bursting and silence regimes in a model of a leech heart interneuron, Phys. Rev. E, 84 (2011), 041910. doi: 10.1103/PhysRevE.84.041910 [21] Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain. Discrete Contin. Dyn. Syst. Ser. B (2018) 23: 3935-3947. [22] J. Mitry, M. McCarthy, N. Kopell and M. Wechselberger, Excitable neurons, firing threshold manifolds and canards, J. Math. Neurosci., 3 (2013), Art. 12, 32 pp. doi: 10.1186/2190-8567-3-12 [23] J. Rinzel and G. Ermentrout, Analysis of neural excitability and oscillations, in Methods in Neuronal Modeling (eds. C. Koch and I. Segev), MIT Press, Cambridge, (1998), Chap. 7,251–291. [24] The potassium A-current, low firing rates and rebound excitation in Hodgkin-Huxley models. Bull. Math. Biol. (1995) 57: 899-929. [25] Codimension-two bursting analysis in the delayed neural system with external stimulations. Nonlinear Dyn. (2012) 67: 309-328. [26] Extrinsic modulation and motor pattern generation in a feeding network: A cellular study. J. Neurosci. (2001) 21: 1767-1778. [27] Synchronization transitions induced by partial time delay in an excitatory-inhibitory coupled neuronal network. Nonlinear Dyn. (2017) 89: 2509-2520. [28] A. Tonnelier, Threshold curve for the excitability of bidimensional spiking neurons, Phys. Rev. E, 90(2014), 022701. doi: 10.1103/PhysRevE.90.022701 [29] F. Q. Wu and H. Gu, Bifurcations of negative responses to positive feedback current mediated by memristor in neuron model with bursting patterns, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2030009, 22 pp. doi: 10.1142/S0218127420300098 [30] F. Wu, H. Gu and Y. Li, Inhibitory electromagnetic induction current induced enhancement instead of reduction of neural bursting activities, Commun. Nonlinear Sci. Numer. Simul., 79 (2019), 104924, 15 pp. doi: 10.1016/j.cnsns.2019.104924 [31] Stochastic dynamics of conduction failure of action potential along nerve fiber with hopf bifurcation. Sci. China Technol. Sci. (2019) 62: 1502-1511. [32] A feasible neuron for estimating the magnetic field effect. Nonlinear Dyn. (2020) 102: 1849-1867. [33] Bifurcation analysis of bursting solutions of two Hindmarsh-Rose neurons with joint electrical and synaptic coupling. Discrete Contin. Dyn. Syst. Ser. B (2011) 16: 637-651. [34] Z. Zhao and H. Gu, Transitions between classes of neuronal excitability and bifurcations induced by autapse, Sci. Rep., 7 (2017), 6760. doi: 10.1038/s41598-017-07051-9 [35] Z. Zhao, L. Li and H. Gu, Dynamical mechanism of hyperpolarization-activated non-specific cation current induced resonance and spike-timing precision in a neuronal model, Front. Cell. Neurosci., 12 (2018), 62. doi: 10.3389/fncel.2018.00062
• ##### Reader Comments
• © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

## Metrics

Article views(907) PDF downloads(242) Cited by(1)

Article outline

Figures(17)

## Other Articles By Authors

• On This Site
• On Google Scholar

/

DownLoad:  Full-Size Img  PowerPoint


text:
# Dark Matter Detection

### Problem 1

Estimate the WIMPs' flow onto the Earth's surface.

### Problem 2

What are the main processes due to which the WIMPs can be detected?

### Problem 3

Show that WIMPs of mass $\sim100GeV$ being elastically scattered on Xenon nuclei with mass $\sim130GeV$ lead to energy recoil $\le40keV$.

### Problem 4

Show that the WIMPs' elastic scattering experiments will be most efficient if the target nuclei's mass is comparable with the WIMPs' mass.

### Problem 5

Determine the minimum velocity of WIMPs that can transmit energy $Q$ to a nucleus with mass $m_N$.

### Problem 6

Estimate the count rate for the detector registering elastic events of WIMPs.

### Problem 7

Obtain the expression for the count rate of the detector registering elastic events of WIMPs.

### Problem 8

Reconstruct the one-dimensional WIMPs' velocities distribution $f_1(v)$ (see the previous problem) basing on the given count rate of the elastic events.

### Problem 9

Construct a model-independent scheme for WIMPs' mass determination using the results of WIMPs' elastic scattering events detection for two or more sets of experimental data with detectors of different composition.

### Problem 10

Show that if a WIMP has the mass of the order $100GeV$ and velocity of the order of $300km/s$, then it coherently interacts with nucleons of a detector nuclei.

### Problem 11

Find the total WIMP-nucleus cross-section determining the elastic events' count rate.

### Problem 12

How will the number of counts for the elastic events detector be affected by transition to heavier target nuclei at fixed detector mass?

### Problem 13

Show that if the WIMPs' mass is of the order of $100GeV$ then the elastic spin-independent cross-section for WIMPs on a nucleus with $A\sim100$ is eight orders of magnitude larger then the corresponding cross-section on a nucleon

### Problem 14

Estimate the annual modulations' amplitude of the WIMPs elastic scattering cross-section.

### Problem 15

Estimate the diurnal modulations' amplitude of the WIMPs elastic scattering cross-section.


text:
<meta http-equiv="refresh" content="1; url=/nojavascript/"> Review of VCV Shortening Rules | CK-12 Foundation
You are reading an older version of this FlexBook® textbook: Basic Speller Student Materials Go to the latest version.

# 16.23: Review of VCV Shortening Rules

Difficulty Level: At Grade Created by: CK-12

## Review of VCV Shortening Rules

1. The Suffix -ity Rule. The vowel right in front of the suffix -ity will always be ______________, even if it is the first vowel in a ______________ string.

2. The Third Vowel Rule. The third vowel sound from the end of a word will often be ______ if it is ______________, even if it is the first vowel in a ______ string.

3. French Lemon Rule. Words that have ____________ vowel sounds and were borrowed from ______________ will have a ______________ first vowel, even in a _________ string.

4. Some of the words below are instances of the three rules above. Sort them into the table:

$&\text {electricity}&& \text {moment}&& \text {society}&& \text {rationality}\\&\text {equality}&& \text {chocolate}&& \text {ferocity}&& \text {hesitate}\\&\text {educate}&& \text {gravel}&& \text {citizen}&& \text {analysis}\\&\text {assiduous}&& \text {recent}&& \text {positive}&& \text {definite}\\&\text {physics}&& \text {balance}&& \text {stomach}&& \text {personality}\\&\text {anxiety}&& \text {susceptibility}&& \text {agent}&& \text {legend}\\&\text {precious}&& \text {simplicity}&& \text {dozen}&& \text {focus}$

5. Be ready to discuss this question: What were your reasons for excluding each of the six words that you did not write into the table?

## Subjects:

1 , 2 , 3 , 4 , 5

Feb 23, 2012

Jan 16, 2015


text:
# Newton's Third law of Motion, who can tell me how to deduct below?

Consider a system of N mutually interacting point objects.

Newton's second law of motion applied to the $$i$$ th object yields: $$m_i \frac {d^2 \vec {r_i}}{dt^2}=\sum_{j=1,N}^{j\neq i} \vec {f_{ij}}$$ Let us now take the above equation and sum it over all objects. We obtain $$\sum_{i=1,N} m_i \frac {d^2 \vec {r_i}}{dt^2}= \sum_{i,j=1,N}^{j\neq i} \vec {f_{ij}}$$ because of newton's third law of motion , the right side of equation is equal to 0,but the question is that i can't understand how the left side of equation turn to below? $$M\frac {d^2\vec {r_{cm}}}{dt^2}=\vec 0$$ where $$M=\sum_{i=1}^{N}m_i$$ is the total mass. The quantity $$\vec {r_{cm}}$$is the vector displacement of the center of mass of the system, which is an imaginary point whose coordinates are the mass weighted averages of the coordinates of the objects that constitute the system: i.e., $$\vec {r_{cm}}=\frac{\sum_{i=1}^{N}m_i \vec {r_i}}{\sum_{i=1}^{N}m_i}$$

The position vector of the centre of mass is defined as: $$\mathbf {r_{cm}}=\frac{\sum_{i=1}^{N}m_i \mathbf {r_i}}{M}$$ i.e., $$M \mathbf {r_{cm}}=m_1 \mathbf r_1+m_2 \mathbf r_2+ m_3 \mathbf r_3+...+m_N \mathbf r_N$$ $$\Rightarrow M \frac {d^2 \mathbf {r_{cm}}}{dt^2} = m_1 \frac {d^2 \mathbf {r_{1}}}{dt^2}+ m_2 \frac {d^2 \mathbf {r_{2}}}{dt^2}+...+m_N \frac {d^2 \mathbf {r_{N}}}{dt^2}$$ (Double Differentiating both sides) Here $$M=\sum_{i=1}^{N}m_i$$. Since single differentiation of the position vector gives velocity $$\mathbf v$$ and double differentiation gives acceleration $$\mathbf a$$. Therefore $$\frac {d^2 \mathbf {r_{cm}}}{dt^2} = \mathbf a$$ This means that $$\sum_{i=1}^{N} m_i \frac {d^2 \mathbf {r_i}}{dt^2} = M \frac {d^2 \mathbf {r_{cm}}}{dt^2} = M \mathbf a = \mathbf {F_{net}}$$

Now if the net force acting on the object is $$\mathbf 0$$ then $$M \mathbf a= M\frac {d^2 \mathbf {r_{cm}}}{dt^2}=\mathbf 0$$

Note that internal forces cannot cause acceleration as they always come in action-reaction pair.

Try this: take the definition

$$\vec {r_{cm}}=\frac{\sum_{i=1}^{N}m_i \vec {r_i}}{\sum_{i=1}^{N}m_i},$$

multiply both sides by $$M$$, and differentiate the equation twice with respect to time.

Clear answers has been given before I have written this comment. Probably the easiest way to derive the equation in question is by definition of center of mass. These answers have been given so I will not do it here. I will however try to arrive to the question from forces acting on a system of particles, as I hope it will give you some insight.

Consider system of n particles. I will represent total force acting on a i'th particle as sum of external force and forces experienced from other particles.

Therefore: $$\text{Force on }i^{th}\text{partilcle:}\qquad\vec F_i=\vec F_i^{ext}+\sum_{j=1,j\neq i}^n \vec F_{ij}$$ $$\text{By Newton's }3^{rd}\,\text {law:}\qquad\vec F_{ij}=-\vec F_{ji},\text{where i, j represent force on }i^{th}\,\text{particle due to}j^{th}\,\text{particle.}$$ Thus total force on n particles is given by: $$\sum_{i=1}^n\vec F_i=\vec F=\sum_{i=1}^n\vec F_i^{ext}+\sum_{i=1}^n\ \Biggl(\sum_{j=1,j\neq i}^n \vec F_{ij}\Biggr)$$ $$\sum_{i=1}^n\ \Biggl(\sum_{j=1,j\neq i}^n \vec F_{ij}\Biggr)=\sum_{j=1}^n\ \Biggl(\sum_{i=1,i\neq j}^n \vec F_{ji}\Biggr)=-\sum_{j=1}^n\ \Biggl(\sum_{i=1,i\neq j}^n \vec F_{ij}\Biggr)=-\sum_{i=1}^n\ \Biggl(\sum_{j=1,j\neq i}^n \vec F_{ij}\Biggr)$$ It implies that $$\sum_{i=1}^n\ \Biggl(\sum_{j=1,j\neq i}^n \vec F_{ij}\Biggr)=0$$

Therefore total force on n particles is sum of all external forces, as internal forces between particle do cancel out.

$$\text{Finally getting back your answer, we know that:}\quad\vec F=M\vec a_c\quad$$ Thus: $$\vec F=\sum_{i=1}^n\vec F^{ext}_i=\sum_{i=1}^nm_i\vec a_i\implies \frac{\sum_{i=1}^nm_i\vec a_i}{M}=\vec a_c\quad \text{where M}=\sum_{i=1}^nm_i$$

The equation tells us that sum of each external force on i'th can be expressed simply as total mass M and acceleration of center of mass. Trivially, if external forces sum out to zero, Newton's First Law applies.


text:
# Total spin angular momentum meaning

1. Aug 9, 2013

### galvin452

Two questions

1) The total orbital spin angular momentum is given as L2=l(l+1). What is the source of or meaning of the (l+1).

2) Similarly for the electron the total spin angular momentum is given as S2-1/2(1/2+1) hbar2. Is the total angular momentum precessing to give sz 1/2 hbar object?

2. Aug 9, 2013

### Simon Bridge

Last edited: Aug 9, 2013
3. Aug 10, 2013

### galvin452

Or is it that the electron has two intrinsic spin values?

4. Aug 10, 2013

### Staff: Mentor

There's no such thing as "total orbital spin angular momentum."

Orbital angular momentum has magnitude $L = \sqrt{l(l+1)} \hbar$, and its component along any direction (usually we use the z-direction) is $L_z = m_l \hbar$ where $m_l = -l \cdots +l$ in steps of 1.

Spin angular momentum has magnitude $S = \sqrt{s(s+1)} \hbar$, and its component along any direction (usually we use the z-direction) is $S_z = m_s \hbar$ where $m_s = -s \cdots +s$ in steps of 1. For e.g. an electron, s = 1/2, so ms = -1/2 or +1/2.

Total angular momentum has magnitude $J = \sqrt{j(j+1)} \hbar$, and its component along any direction (usually we use the z-direction) is $J_z = m_j \hbar$ where $m_j = -j \cdots +j$ in steps of 1.

(Some books use different notation.)

For orbital angular momentum, one way to get the $l(l+1)$ is to apply the (orbital) angular momentum operator to the solutions to the Schrödinger equation for e.g. the hydrogen atom. There's probably a "deeper" way to get it which applies to all three kinds of angular momentum, but someone else will have to provide it.

5. Aug 10, 2013

### Simon Bridge

No that's pretty much it - in QM classes (here anyway) we force students to do that calculation.
The quantum number basically comes from counting the states. There are three dimensions and the surd comes from the vector sum. You actually have to crunch through the equations to see it.

One may expect that the lth L state would have L=l\hbar ... but that neglects stuff like that there are three dimensions. The x(x+1) pattern is kind-of a symmetry in angular momentum.

Last edited: Aug 10, 2013


text:
Template:Expert

In physics, in terms of a metric theory of gravitation, a gravitational wave is a fluctuation in the curvature of space-time which propagates as a wave. Gravitational radiation results when gravitational waves are emitted from some object or system of gravitating objects.

(Gravitational waves are sometimes called gravity waves, but this term should be reserved for a completely different kind of wave encountered in hydrodynamics.)

Gravitational waves are very weak. The strongest gravitational waves we can expect to observe on Earth would be generated by very distant and ancient events in which a great deal of energy moved very violently (examples include the collision of two neutron stars, or the collision of two super massive black holes). Such a wave should cause relative changes in distance everywhere on Earth, but these changes should be on the order of at most one part in 1021. In the case of the arms of the LIGO gravitational wave detector, this is less than one thousandth of the "diameter" of a proton. This should give some indication of why it has proven very difficult to detect even the strongest gravitational waves!

The existence and indeed ubiquity of gravitational waves is an unambiguous prediction of Einstein's theory of General relativity. All competing gravitation theories currently thought to be viable (apparently in agreement to the level of accuracy with all available evidence) feature predictions about the nature of gravitational radiation. In principle, these predictions are sometimes significantly different from those of general relativity, but unfortunately, at present it seems to be sufficiently challenging simply to directly confirm the existence of gravitational radiation, much less study its detailed properties.

Although gravitational radiation has not yet been unambiguously and directly detected, there is already significant indirect evidence for its existence. Most notably, observations of the binary pulsar PSR1913+16, which is thought to consist of two neutron stars orbiting rather tightly and rapidly around each other, have revealed a gradual in spiral at exactly the rate which would be predicted by general relativity. The simplest (and almost universally accepted) explanation for these observations is that general relativity must give an accurate account of gravitational radiation in such systems. Joseph H. Taylor Jr. and Russell A. Hulse shared the Nobel Prize in Physics in 1993 for this work.

## Overview

In Einstein's theory of General Relativity, gravitation is, essentially, identified with spacetime curvature. In the famous slogan promulgated by John Archibald Wheeler, matter tells spacetime how to curve, and spacetime tells matter how to move. For example, humans feel the ground pressing against their feet (or behind, according to stance). From the viewpoint of general relativity, this means that contact with the ground is preventing them from falling freely, thereby accelerating them. Since acceleration is identified with bending of world lines, this means that the world line of a human who is not in freefall is not a geodesic. On the other hand, far from any mass-energy, spacetime is almost perfectly flat, so geodesics behave much like straight lines in familiar solid geometry, so small objects can exhibit rectilinear inertial motion.

General relativity (and similar theories of gravitation) are expressed by writing down a field equation (and possibly an equation of motion, if this is not implicit from the field equation, as it is, remarkably, in the case of general relativity). That is, these are classical relativistic field theories in which the gravitational field is at least partially identified with spacetime curvature. As a more or less inevitable consequence, in these theories, the rapid motion of mass-energy in some region will generate ripples in spacetime itself which radiate outward as gravitational waves. Indeed, this is in a sense how "field updating information" is typically conveyed from place to place.

Like electromagnetic radiation, in general relativity (and many other theories), gravitational radiation travels at the speed of light and is transverse (meaning that the major effects of a gravitational wave on the motion of test particles occurs in a plane orthogonal to the direction of propagation). However (roughly speaking):

• gravitational waves represent perturbations in a second rank tensor field (and, borrowing a term from quantum field theory, are said to spin-two),
• electromagnetic waves result from perturbations in a vector field (and are said to be spin-one).
• various other waves treated in physics result from perturbations in a scalar field (and are said to be spin-zero).

In electromagnetism, certain motions of charged particles, like electrons, will radiate electromagnetic waves. Analogously in gravity, certain motions of mass or energy will radiate gravitational waves. In the quantum field theory which arises from classical electromagnetism, called quantum electrodynamics, there is a massless particle associated with electromagnetic radiation, called the photon. Attempts to create an analogous quantum field theory for classical gravity (general relativity) led to an analogous concept, a massless particle called the graviton. However, it turns out that this route to quantizing general relativity ultimately fails, which renders the possible role of the graviton somewhat problematic in gravitational physics. Probably it is best to think of this notion as one which arises in an approximation and which has some virtues, but which is probably not as the notion of the photon.

## The Nature of Gravitational Waves

Gravitational waves represent fluctatations in the metric of space-time. That is, they alter the relative distance between test particles. It follows that to directly detect a gravitational wave, you should in essence look for tiny relative motions between two objects. In the case of the LIGO detectors, this is essentially relative motion between two suspended mirrors, and as we saw above the motion to be detected is far smaller than the size of an atom, in fact smaller than the "size" of an atomic nucleus. Since thermal motion in each mirror is far larger than this, understanding why anyone would expect LIGO to work takes some explaining! (See LIGO.)

Imagine a perfect flat region of spacetime, with a bunch of mutually motionless test particles. Along comes a monochromatic linearly polarized gravitational wave. What happens to the test particles? Roughly speaking, they will oscillate in a cruciform manner, orthogonal to the direction of motion:

• first, East/West separated particles draw together while North/South separated particles draw apart,
• next, East/West separated particles draw apart while North/South separated particles draw together,

and so forth. (Diagonally separated particles exhibit a relative motion which is more difficult to describe verbally, but which is more or less implied by this description.) The cross-sectional of a small box of test particles is invariant under these changes, and there is neglible motion in the direction of propagation (at least, neglecting gravitomagnetic effects; that is, we are tacitly assuming that the relative motion of our test particles is not very rapid).

A monochromatic circularly polarized induces similar cruciform oscillation, except that the crucifix rotates with the same frequency as the frequency of cruciform oscillation.

Interestingly enough, after the wave has passed, there may be some residual "secular" relative motion of the test particles. There are also some interesting optical effects. If, before the wave arrives, we look through the oncoming wavefronts at objects behind these wavefronts, we can see no optical distortion (if we could, of course, we would have advance notice of its impending arrival, in violation of the principle of causality). But if, after the wave has passed by, we turn and look through the departing wavefronts at objects which the wave has not yet reached, we will see optical distortions in the images of small shapes such as galaxies. Unfortunately, this is an utterly impractical method of detecting the very weak waves we can expect to occur in the vicinity of the solar system.

## Sources of Gravitational Waves

Gravitational waves are caused by certain motions of mass or energy. The type of motion required is different from electromagnetism where any accelerating charge will radiate electromagnetic radiation. [1]

According to general relativity, the quadrupole moment (or some higher moment) of an isolated system must be time-varying in order for it to emit gravitational radiation. Here are some examples which illustrate when we should (assuming general relativity gives accurate predictions) expect a system to emit gravitational radiation:

• An isolated object in approximately "rectilinear" motion will not radiate. (Needless to say, this motion is wrt some observer and can be only approximately rectilinear. Technically, this entails defining a weakly gravitating system possessing a time varying dipole moment but stationary quadrupole moment, with all moments being taken with respect to the origin.) This can be regarded as a consequence of the principle of conservation of linear momentum. (Caveat: this example is trickier than it looks, and in the case of a small object falling toward a large one, say, it leads to one of the most vexed questions in general relativity, the problem of treating radiation reaction).
• A spherically pulsating spherical star (nonzero and non-stationary monopole moment or mass, but vanishing and hence stationary quadrupole moment) will not radiate, in agreement with Birkhoff's theorem.
• A spinning disk (nonzero but stationary monopole and quadrupole moments) will not radiate. This can be regarded as a consequence of the principle of conservation of angular momentum. (Caveat: in general relativity, unlike Newtonian gravitation, a spinning disk will not generate an external field identical to the field of an equivalent but non-spinning disk, due to gravitomagnetic effects, but this does not contradict the absence of radiation. Roughly speaking, the field is generated as we concentrate matter, and if that matter has some angular momentum, but we end up with a stationary external gravitational field, that field will exhibit gravitomagnetism but not radiation.)
• Two objects mounted on the endpoints of an isolated extensible curtain rod, which is provided with some kind of engine and which oscillates long/short/long with frequency ω, gives a system with time-varying quadrupole moment, so this system will radiate. Observers far from the rod and in the equatorial plane of the rod will observe linearly polarized radiation (aligned with the rod) with frequency ω. Observers lying on the axis of symmetry of the rod will observe no radiation, however.
• A spinning non-axisymmetric planetoid (say with a large bump or dimple on the equator) will define a system with a time-varying quadrupole moment, so this system will radiate. As an idealization, one can study an isolated uniform mass curtain rod which is spinning with angular frequency ω about a rotation axis orthogonal to the rod, but passing through some point other than the centroid of the rod. This gives a system with time varying quadrupole moment, so the system will radiate. Observers far from the system and lying in the plane of rotation will observe linearly polarized radiation with frequency . Observers far from the system and near its axis of symmetry will observe circularly polarized radiation with frequency ω.
• Two objects orbiting each other with angular frequency ω in a quasi-Keplerian planar orbit, gives a system with time-varying quadrupole moment, so this system will radiate. Observers far from the system and in its equatorial plane will observe linearly polarized radiation (aligned with the rod) with frequency . Observers far from the system and lying on its axis of symmetry will observe circularly polarized radiation with frequency ω.

The last three examples illustrate a general rule-of-thumb: far from a radiatiating system, projection of the system on the "viewing plane" affords a rough and ready indication of what kind of radiation will be observed.

These examples (and others) are most commonly studied using a simplified version of general relativity, sometimes called linearized general relativity, which gives indistinguishable results in the case of weak gravitational fields. (The external field of our Sun would be considered "weak" in this terminology.) Similar conclusions hold for the fully nonlinear theory, but it is much more difficult to obtain analytic results outside the domain of the linearized theory. This is one reason why so much work on phenomena such as the collision and merger of two black holes currently requires numerical analysis.

Gravitational radiation carries energy away from a radiating system. Consequently, in the case of the quasi-Keplerian system discussed above, the two objects will gradually spiral in towards one another, becoming more tightly bound to compensate for this loss of energy. The predicted rate of this inspiral can also be computed, using the linearized approximation, and the result gives excellent agreement for observed binary pulsars (this is the theoretical basis for the Nobel Prize awarded to Hulse and Taylor). In the late stages of the inspiral of two neutron stars or black holes, however, the linearized theory is no longer adequate, so one must result to more complicated approximations, and eventually to numerical simulations.

Similarly, in the case of the eccentric rotating rod, the frequency will decrease as the radiation gradually carries off energy from the system.

We stress that some theories of gravitation give significantly different predictions concerning the nature and generation of gravitational radiation, while others give predictions which are almost identical to those of general relativity. All currently known theories other than general relativity are either in disagreement with observation, or in some sense more complicated than general relativity (see for example Brans-Dicke theory for an example illustrating the latter possibility).

### Examples

• Our solar system can not radiate very much gravitational radiation because the planets are mostly on the same plane of orbit with spins aligned to that plane. And even if this were not the case, the gravitational forces in our solar system are very weak with the exception of Mercury and the Sun.
• If two spinning black holes were to collide, they could emit an enormous amount of gravitational radiation and lose energy in the process.

## Detection

As mentioned above, observations of the binary pulsar PSR J0737-3039 appeared to confirm predictions of general relativity with respect to energy emitted by gravitational waves, with the system's orbit observed to shrink 7 mm per day. This further confirms predictions made by Russell Alan Hulse and Joseph Hooton Taylor Jr. observing binary pulsar PSR B1913+16, for the discovery and analysis of which they were awarded the Nobel Prize in Physics in 1993.

But to directly detect gravitational waves you would have to look for any motion they cause. Typically you would look for the expansion and contraction oscillations caused by the gravitational wave. A simple version of this setup is called a Weber bar -- a large, solid piece of metal with electronics attached to detect any vibrations. Unfortunately, Weber bars are not likely to be sensitive enough to detect anything but very powerful gravitational waves. A more sensitive version is the Interferometer, with test masses placed as many as four kilometers apart. Ground-based interferometers such as LIGO are now coming on line. The motion to be detected would be very slight -- a small fraction of the width of an atom, over a distance of four kilometers. Space-based interferometers, such as LISA are also being developed.

One reason for the lack of direct detection so far is that the gravitational waves that we expect to be produced in nature are very weak, so that the signals for gravitational waves, if they exist, are buried under noise generated from other sources. Reportedly, ordinary terrestrial sources would be undetectable, despite their closeness, because of the great relative weakness of the gravitational force.

Gravitational radiation has not been directly observed, although there are a number of existing and proposed experiments such as LIGO that intend to do so. A number of teams are working on making more sensitive and selective gravitational wave detectors and analysing their results. A commonly used technique to reduce the effects of noise is to use coincidence detection to filter out events that do not register on both detectors. There are two common types of detectors used in these experiments:

There are other prospects such as MiniGRAIL, a spherical gravitational wave antenna based at Leiden University. Some scientists even want to use the moon as a giant gravitational wave detector. The moon should be somewhat pliable to the contortions caused by gravitational waves.

In November 2002, a team of Italian researchers at the Instituto Nazionale di Fisica Nucleare and the University of Rome La Sapienza produced an analysis of their experimental results that may be further indirect evidence of the existence of gravitational waves. Their paper, entitled "Study of the coincidences between the gravitational wave detectors EXPLORER and NAUTILUS in 2001", is based on a statistical analysis of the results from their detectors which shows that the number of coincident detections is greatest when both of their detectors are pointing into the center of our Milky Way galaxy.

### Einstein@Home

Bruce Allen of UWM's LIGO Scientific Collaboration (LSC) group is leading the development of the Einstein@Home project, developed to search data for signals coming from selected, extremely dense, rapidly rotating stars observed from LIGO in the US and the GEO 600 gravitational wave observatory in Germany . Such sources are believed to be either quark stars or neutron stars; a subclass of these stars are already observed by conventional means and are known as pulsars, electromagnetic wave-emitting celestial bodies. If some of these stars are not quite near-perfectly spherical, they should emit gravitational waves, which LIGO and GEO 600 may begin to detect.

Einstein@Home is a small part of the LSC scientific program. It has been set up and released as a distributed computing project similar to SETI@home. That is, it relies on computer time donated by private computer users to process data generated by LIGO's and GEO 600's search for gravity waves.

## Prospects

Scientists are eager to directly measure gravitational waves from astronomical sources, as they can probe phenomena that are difficult or impossible to study with electromagnetic radiation. For instance, although a black hole emits no visible radiation in the way that a regular star does, gravitational waves can be emitted when an object falls into a black hole, or when two black holes collide. If the inspiraling mass is significantly smaller than the central black hole, the emitted gravitational waves may, at least in some circumstances, allow physicists to directly probe the spacetime geometry around the event horizon (such observations are a primary goal of the LISA mission). Also, because gravitational waves are so weak (and thus difficult to detect), objects opaque to light are often transparent to gravitational radiation. In particular, gravitational waves could propagate while the universe was still opaque to light (i.e., at times before recombination). In this way, gravitational waves could help reveal information about the very structure of the universe.

In contrast to electromagnetic radiation, it is not fully understood what difference the presence of gravitational radiation would make for the workings of the universe. A sufficiently strong sea of primordial gravitational radiation, with an energy density exceeding that of the big bang electromagnetic radiation by a few orders of magnitude, would shorten the life of the universe, violating existing data that show it is at least 13 billion years old. More promising is the hope to detect waves emitted by sources on astronomic size scales, such as:

## Derivation

### Perturbation off Flat Space-time

Consider that the full metric g is nearly the flat metric η plus some small perturbation h.

gμν = ημν + hμν

The Einstein equation in vacuum is

$R_{\mu \nu} = \mathbf{0}$

Where R is the Ricci curvature. We will expand R in perturbatively in powers of h.

$R_{\mu \nu} = \mathbf{0} + \delta R_{\mu \nu} + \delta^2 R_{\mu \nu} + \cdots$

The zeroth order term can only be a function of the flat metric and therefore is identically zero. As the perturbation is to be small, we will solve only for the first order term and ignore all higher orders.

$R_{\mu \nu} = \delta R_{\mu \nu} + \mathbf{0}$

Where δRμν is the deviation from the flat (and thus zero) Ricci curvature that depends linearly on the perturbation h.

Now we need the formula for the Ricci curvature.

$R_{\mu \nu} = \partial_{\alpha} \Gamma_{\mu \nu}^\alpha - \partial_{\nu} \Gamma_{\mu \alpha}^\alpha + \Gamma_{\mu \nu}^\alpha \Gamma_{\alpha \beta}^\beta - \Gamma_{\mu \beta}^\alpha \Gamma_{\nu \alpha}^\beta$

Where Γ are the Christoffel symbols and $\partial_{\alpha}$ is shorthand for $\frac{\partial}{\partial x^{\alpha}}$. Only first two terms which are linear in Γ will contribute to the first order correction.

$\delta R_{\mu \nu} = \partial_{\alpha} \delta \Gamma_{\mu \nu}^\alpha - \partial_{\nu} \delta \Gamma_{\mu \alpha}^\alpha$

Next we need the formula for the Christoffel symbols.

$\Gamma^\alpha_{\mu \nu} = \frac{1}{2} g^{\alpha \gamma} \left( \partial_{\nu} g_{\gamma \mu} + \partial_{\mu} g_{\gamma \nu} - \partial_{\gamma} g_{\mu \nu} \right)$

Seeing as the flat metric is constant, the only first order terms will involve derivatives of the perturbation.

$\delta \Gamma^\alpha_{\mu \nu} = \frac{1}{2} \eta^{\alpha \gamma} \left( \partial_{\nu} h_{\gamma \mu} + \partial_{\mu} h_{\gamma \nu} - \partial_{\gamma} h_{\mu \nu} \right)$

The linearized Einstien equation now becomes

$\delta R_{\mu \nu} = \frac{1}{2} \left( \Box^2 h_{\mu \nu} + \partial_\alpha V_\beta + \partial_\beta V_\alpha \right)$

Where Vα substitutes the expression $\partial_\beta h_\alpha^\beta - \frac{1}{2} \partial_\alpha h_\beta^\beta$ and $\Box^2 = \partial_t^2 - \nabla^2$ is the d'Alembertian or 4-Laplacian. Raising and lowering indices can be tricky. To first order you only use the flat metric. Also note the inverse metric has a negative perturbation plus higher order terms.

Next we choose a particular coordinate system where Vα is identically zero. Some proof is necessary to make sure this is possible, but it is. We are left with a wave equation and our gauge condition.

$\Box^2 h_{\mu \nu} = \mathbf{0}$
$\partial_\beta h_\alpha^\beta = \frac{1}{2} \partial_\alpha h_\beta^\beta$

From experience with simpler wave equations we can guess the general form of the solution.

$h_{\mu \nu} = A_{\mu \nu} e^{\imath k \cdot x}$

Where $k \cdot k = 0$ is a null vector. The wave equation is now satisfied, but what choices of A will satisfy the gauge condition we used.

$A_\alpha^\beta \partial_\beta e^{\imath k \cdot x} = A_\beta^\beta \partial_\alpha e^{\imath k \cdot x}$
$A_\alpha^\beta k_\beta = A_\beta^\beta k_\alpha$

If we don't want transformations to disturb our choice of gauge, then we better make the wave traceless, $A_\beta^\beta = 0$, and transverse, $A_\alpha^\beta k_\beta = 0$.

For a wave traveling in the z direction, k = (1,0,0,1), the perturbation will take the following form.

$h_{\mu \nu} = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & a & b & 0\\ 0 & b & -a & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} e^{\imath k \cdot x}$

Thus the oscillations are transverse spacial distortions. The wave is called spin-2 because there are 2 different polarizations a,b. Light only has one! a is called the plus polarization and b is called the cross polarization.

## Perturbative versus Exact

Gravitational waves differ markedly from electromagnetic waves in that electromagnetic waves can be derived exactly from Maxwell's equations. However gravitational waves, as a linear, spin-2 wave, as they are often thought of are only perturbations to certain space-time geometries. In other words, classically there are always linear, spin-1 E&M waves, but there are never linear, spin-2 gravitational waves. There are still wave-like fluctuations, but in general things are nonlinear, as is always the case in General Relativity. This is one of the reasons there may be no graviton.

The thing that is analogous to electromagnetic radiation is the Weyl curvature, not the linear, spin-2 wave.

## Gravitational waves transmit energy

Within parts of the scientific community there was initially some confusion as to if gravitational waves could transmit energy like electromagnetic waves can. This confusion came from the fact that gravitational waves have no local energy density - no contribution to the stress-energy tensor. Unlike Newtonian gravity, Einstein gravity is not a force theory. Gravity is not a force in General Relativity, it is geometry. Therefore the field was thought to not contain energy, like a gravitational potential would. But the field can most certainly carry energy as it can do mechanical work at a distance. And this has been proven using stress-energy pseudo tensors that transport energy as well as seeing how radiation can carry energy out to infinity.


text:
# SOLVING UNKNOWNS IN PROPORTION

Proportion, as I told you before, states that two ratios are the same.

So, the statement $\frac{7}{4}=\frac{28}{16}$ because the ratios are the same when simplify.

A simpler way of determining if two ratios are proportional is to use cross-multiplication. This is because two ratios are proportional if their cross-products are the same.

$$\frac{7}{4}=\frac{28}{16}$$

$$7 \times 16 = 28 \times 4$$

$$112=112$$

Because their cross-products are the same, the two ratios are the same

But, what if we replace 7 with x so that it becomes:

$$\frac{x}{4}=\frac{28}{14}$$

Since we know that cross product of a proportion is the same.

$$x \times 14= 28 \times 4$$

$$14x=112$$

$$x=7$$

Using this fact, let's solve few examples

Example 1

Solve for x in $\frac{145}{232}=\frac{10}{2y}$

Solution:

By cross multiplication

$$145 \times 2y=232 \times 10$$

$$290y=2320$$

$$\frac{290y}{290}=\frac{2320}{290}$$

$$y=8$$

Example 2

Solve $\frac{8}{10z}=\frac{0.4}{1.4}$

Solution:

By cross multiplication

$$8 \times 1.4= 10z \times 0.4$$

$$11.2=4z$$

$$z=2.8$$

Example 3

Solve for x in $\frac{5x}{150}=\frac{\frac{2}{3}}{5}$

Solution:

First, we evaluate the right-hand side

$$\frac{5x}{150}=\frac{2}{3}\times\frac{1}{5}$$

$$\frac{5x}{150}=\frac{2}{15}$$

$$5x ×15=150 × 2$$

$$75x=300$$

$$x=4$$

Example 4

Solve for x in $\frac{32}{100}=\frac{40}{4y}$

Solution:

By cross multiplication

$$128y=4000$$

$$\frac{128y}{128}=\frac{4000}{128}$$

$$y=31.25$$

That will be all for now. If you got questions relating to this, do well to ask our telegram community

Help us grow our readership by sharing this post


text:
# The square of a sum of two expressions is equal to the square of the first, plus twice the product of the first and second, plus the square of the second.

## Important!

The square of the sum formula shall be used for finding squares of large numbers.

## Derivation of the formula of the square of the sum. Method 1

Let us prove the formula from left to right, i.e. prove that:

## Step 1

The square of the expression is the expression mutiplied by itself:

## Step 2

Expand the brackets:

## Step 3

As the product remains the same by reordering its factors, then:

## Step 4

Collect like terms and multiply the elements:

## Step 5

As a result, we can derive:

Consider:

## Step 2

Let us write in the form:

## Step 3

Apply this to the expression under consideration:

Simplify:

Factorize:

## Step 6

Factorize the expression (a+b):

## Step 7

As a result of transformations we have a product of the expression by itself, and it is the square of this expression:

Hence, we have proved that:

The formula of the square of the sum has been derived.