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Connecting The Non-Singular Origin of the Universe, The Vacuum Structure and The Cosmological Constant Problem

# Connecting The Non-Singular Origin of the Universe, The Vacuum Structure and The Cosmological Constant Problem

Eduardo I. Guendelman Physics Department, Ben Gurion University of the Negev, Beer Sheva 84105, Israel    Pedro Labraña Departamento de Física, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile and
Departament d’Estructura i Constituents de la Matèria, Institut de Ciències del Cosmos, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain.
###### Abstract

We consider a non-singular origin for the Universe starting from an Einstein static Universe, the so called “emergent universe” scenario, in the framework of a theory which uses two volume elements and , where is a metric independent density, used as an additional measure of integration. Also curvature, curvature square terms and for scale invariance a dilaton field are considered in the action. The first order formalism is applied. The integration of the equations of motion associated with the new measure gives rise to the spontaneous symmetry breaking (S.S.B) of scale invariance (S.I.). After S.S.B. of S.I., it is found that a non trivial potential for the dilaton is generated. In the Einstein frame we also add a cosmological term that parametrizes the zero point fluctuations. The resulting effective potential for the dilaton contains two flat regions, for relevant for the non-singular origin of the Universe, followed by an inflationary phase and , describing our present Universe. The dynamics of the scalar field becomes non linear and these non linearities produce a non trivial vacuum structure for the theory and are responsible for the stability of some of the emergent universe solutions, which exists for a parameter range of values of the vacuum energy in , which must be positive but not very big, avoiding the extreme fine tuning required to keep the vacuum energy density of the present universe small. The non trivial vacuum structure is crucial to ensure the smooth transition from the emerging phase, to an inflationary phase and finally to the slowly accelerated universe now. Zero vacuum energy density for the present universe defines the threshold for the creation of the universe.

###### pacs:
98.80.Cq, 04.20.Cv, 95.36.+x

## I Introduction

One of the most important and intriging issues of modern physics is the so called “Cosmological Constant Problem” CCP1 (); CCP2 (); CCP3 (); CCP4 (); CCP5 (), (CCP), most easily seen by studying the apparently uncontrolled behaviour of the zero point energies, which would lead to a corresponding equally uncontrolled vacuum energy or cosmological constant term. Even staying at the classical level, the observed very small cosmological term in the present universe is still very puzzling.

Furthermore, the Cosmological Constant Problem has evolved from the “Old Cosmological Constant Problem”, where physicist were concerned with explaining why the observed vacuum energy density of the universe is exactly zero, to different type of CCP since the evidence for the accelerating universe became evident, for reviews see AU1 (); AU2 (). We have therefore since the discovery of the accelerated universe a “New Cosmological Constant Problem” W2 (), the problem is now not to explain zero, but to explain a very small vacuum energy density.

This new situation posed by the discovery of a very small vacuum energy density of the universe means that getting a zero vacuum energy density for the present universe is definitely not the full solution of the problem, although it may be a step towards its solution.

One point of view to the CCP that has been popular has been to provide a bound based on the “anthropic principle” Anthropic (). In this approach, a too large Cosmological Constant will not provide the necessary conditions required for the existence of life, the anthropic principle provides then an upper bound on the cosmological constant.

One problem with this approach is for example that it relies on our knowledge of life as we know it and ignores the possibility that other life forms could be possible, for which other (unknown) bounds would be relevant, therefore the reasoning appears by its very nature subjective, since of course if the observed cosmological constant will be different, our universe will be different and this could include different kind of life that may be could have adjusted itself to a higher cosmological constant of the universe. But even accepting the validity of anthropic considerations, we still do not understand why the observed vacuum energy density must be positive instead of possibly a very small negative quantity. Accepting the anthropic explanation means may be also giving up on discovering important physics related to the CCP and this may be the biggest objection.

Nevertheless, the idea of associating somehow restrictions on the origin of the universe with the cosmological constant problem seems interesting. We will take on this point of view, but leave out the not understood concept of life out from our considerations. Instead, we will require, in a very specific framework, the non-singular origin of the universe. The advantage of this point of view is that it is formulated in terms of ideas of physics alone, without reference to biology, which unlike physics, has not reached the level of an exact science. Another interesting consequence is that we can learn that of a non-singularly created universe may not have a too big cosmological constant, an effect that points to a certain type of gravitational suppression of UV divergences in quantum field theory.

In this respect, one should point out that even in the context of the inflationary scenario Inflation1 (); Inflation2 (); Inflation3 (); Inflation4 () which solves many cosmological problems, one still encounters the initial singularity problem which remains unsolved, showing that the universe necessarily had a singular beginning for generic inflationary cosmologies singularities1 (); singularities2 (); singularities3 (); singularities4 (); singularities5 ().

Here we will adopt the very attractive “Emergent Universe” scenario, where those conclusions concerning singularities can be avoided emerging1 (); emerging2 (); emerging3 (); emerging4 (); emerging5 (); emerging6 (); emerging7 (); emerging8 (); emerging9 (). The way to escape the singularity in these models is to violate the geometrical assumptions of these theorems, which assume i) that the universe has open space sections ii) the Hubble expansion is always greater than zero in the past. In emerging1 (); emerging2 () the open space section condition is violated since closed Robertson Walker universes with are considered and the Hubble expansion can become zero, so that both i) and ii) are avoided.

In emerging1 (); emerging2 () even models based on standard General Relativity, ordinary matter and minimally coupled scalar fields were considered and can provide indeed a non-singular (geodesically complete) inflationary universe, with a past eternal Einstein static Universe that eventually evolves into an inflationary Universe.

Those most simple models suffer however from instabilities, associated with the instability of the Einstein static universe. The instability is possible to cure by going away from GR, considering non perturbative corrections to the Einstein’s field equations in the context of the loop quantum gravity emerging3 (), a brane world cosmology with a time like extra dimension emerging4 (); emerging5 () considering the Starobinski model for radiative corrections (which cannot be derived from an effective action) emerging6 () or exotic matter emerging7 (). In addition to this, the consideration of a Jordan Brans Dicke model also can provide a stable initial state for the emerging universe scenario emerging8 (); emerging9 ().

In this review we study a different theoretical framework where such emerging universe scenario is realized in a natural way, where instabilities are avoided and a succesfull inflationary phase with a graceful exit can be achieved. The model we will use was studied first in SIchile (), however, in the context of this model, a few scenarios are possible. For example in the first paper on this model SIchile () a special choice of state to describe the present state of our universe was made. Then in SICCP () a different candidate for the vacuum that represents our present universe was made. The way in which we best represents the present state of the universe is crucial, since as it should be obvious, the discussion of the CCP depends on what vacuum we take. In SICCP () we expressed the stability and existence conditions for the non-singular universe in terms of the energy of the vacuum of our candidate for the present Universe. In SICCP () a few typos in SIchile () were corrected and also the discussion of some notions discussed was improved in SICCP () and more deeper studies will be done in this review.

Indeed in this review, all those topics will be further clarified, in particular the vacuum structure of this model will be extended. A very important new feature that will be presented in this review is the existence of a “kinetic vacuum”, that produces a vacuum energy state which is degenerate with the vacuum choice made in SICCP (), this degeneracy is analyzed and the dynamical role of this kinetic vacuum in the evolution of the universe and the CCP is analyzed.

We work in the context of a theory built along the lines of the two measures theory (TMT). Basic idea is developed in TMT1a (), TMT1b ()-TMT1r () TMT2 (), TMT3a ()-TMT3d (), TMT4a ()-TMT4e (), TMT5 () and more specifically in the context of the scale invariant realization of such theories TMT2 (), TMT3a ()-TMT3e (), TMT4a ()-TMT4e (), TMT5 (). These theories can provide a new approach to the cosmological constant problem and can be generalized to obtain also a theory with a dynamical spacetime dyn (), furthermore, string and brane theories, as well as brane world scenarios can be constructed using Two Measure Theories ideas G16a ()-G20 (). We should also point out that the Hodge Dual construction of TMT1d () for supergravity constitutes in fact an example of a TMT. The construction by Comelli TMT1e () where no square root of the determinant of the metric is used and instead a total divergence appears is also a very much related approach.

The two measure theories have many points of similarity with “Lagrange Multiplier Gravity (LMG)” Lim2010 (); Capozziello2010 (). In LMG there is a Lagrange multiplier field which enforces the condition that a certain function is zero. For a comparison of one of these lagrange multiplier gravity models with observations see LabranaMultiplier (). In the two measure theory this is equivalent to the constraint which requires some lagrangian to be constant. The two measure model presented here, as opposed to the LMG models of Lim2010 (); Capozziello2010 () provide us with an arbitrary constant of integration. The introduction of constraints can cause Dirac fields to contribute to dark energy FDM () or scalar fields to behave like dust like in Lim2010 () and this dust behaviour can be caused by the stabilization of a tachyonic field due to the constraint, accompanied by a floating dark energy component GSY (); AG (). TMT models naturally avoid the 5th force problem 5th ().

We will consider a slight generalization of the TMT case, where, we consider also the possible effects of zero point energy densities, thus “softly breaking” the basic structure of TMT for this purpose. We will show how the stated goals of a stable emergent universe can be achieved in the framework of the model and also how the stability of the emerging universe imposes interesting constraints on the energy density of the ground state of the theory as defined in this paper: it must be positive but not very large, thus the vacuum energy and therefore the term that softly breaks the TMT structure appears to be naturally controlled. An important ingredient of the model considered here is its softly broken conformal invariance, meaning that we allow conformal breaking terms only though potentials of the dilaton, which nevertheless preserve global scale invariance. In another models for emergent universe we have studied SICh2 (), that rule of softly broken conformal invariance was taken into account. It is also a perfectly consistent, but different approach.

The review will be organized as follows: First we review the principles of the TMT and in particular the model studied in TMT2 (), which has global scale invariance and how this can be the basis for the emerging universe. Such model gives rise, in the effective Einstein frame, to an effective potential for a dilaton field (needed to implement an interesting model with global scale invariance) which has a flat region. Following this, we look at the generalization of this model TMT5 () by adding a curvature square or simply “ term” and show that the resulting model contains now two flat regions. The existence of two flat regions for the potential is shown to be consequence of the s.s.b. of the scale symmetry. We then consider the incorporation in the model of the zero point fluctuations, parametrized by a cosmological constant in the Einstein frame. In this resulting model, there are two possible types of emerging universe solutions, for one of those, the initial Einstein Universe can be stabilized due to the non linearities of the model, provided the vacuum energy density of the ground state is positive but not very large. This is a very satisfactory results, since it means that the stability of the emerging universe prevents the vacuum energy in the present universe from being very large!. The transition from the emergent universe to the ground state goes through an intermediate inflationary phase, therefore reproducing the basic standard cosmological model as well. We end with a discussion section and present the point of view that the creation of the universe can be considered as a “threshold event” for zero present vacuum energy density, which naturally gives a positive but small vacuum energy density.

## Ii Introducing a new measure

The general structure of general coordinate invariant theories is taken usually as

 S1=∫L1√−gd4x (1)

where . The introduction of is required since by itself is not a scalar but the product is a scalar. Inserting , which has the transformation properties of a density, produces a scalar action , as defined by Eq.(1), provided is a scalar.

In principle nothing prevents us from considering other densities instead of . One construction of such alternative “measure of integration”, is obtained as follows: given 4-scalars (a = 1,2,3,4), one can construct the density

 Φ=εμναβεabcd∂μφa∂νφb∂αφc∂βφd (2)

and consider in addition to the action , as defined by Eq.(1), , defined as

 S2=∫L2Φd4x (3)

is again some scalar, which may contain the curvature (i.e. the gravitational contribution) and a matter contribution, as it can be the case for , as defined by Eq.(1). For an approach that uses four-vectors instead of four-scalars see (four-vector, ).

In the action defined by Eq.(3) the measure carries degrees of freedom independent of that of the metric and that of the matter fields. The most natural and successful formulation of the theory is achieved when the connection is also treated as an independent degree of freedom. This is what is usually referred to as the first order formalism.

One can consider both contributions, and allowing therefore both geometrical objects to enter the theory and take as our action

 S=∫L1√−gd4x+∫L2Φd4x (4)

Here and are independent.

We will study now the dynamics of a scalar field interacting with gravity as given by the following action, where except for the potential terms and we have conformal invariance, the potential terms and break down this to global scale invariance.

 SL=∫L1√−gd4x+∫L2Φd4x (5)
 L1=U(ϕ) (6)
 L2=−1κR(Γ,g)+12gμν∂μϕ∂νϕ−V(ϕ) (7)
 R(Γ,g)=gμνRμν(Γ),Rμν(Γ)=Rλμνλ (8)
 Rλμνσ(Γ)=Γλμν,σ−Γλμσ,ν+ΓλασΓαμν−ΓλανΓαμσ. (9)

The suffix in is to emphasize that here the curvature appears only linearly. Here, except for the potential terms and we have conformal invariance, the potential terms and break down this to global scale invariance. Since the breaking of local conformal invariance is only through potential terms, we call this a “soft breaking”.

In the variational principle , the measure fields scalars and the “matter” - scalar field are all to be treated as independent variables although the variational principle may result in equations that allow us to solve some of these variables in terms of others.

For the case the potential terms we have local conformal invariance

 gμν→Ω(x)gμν (10)

and is transformed according to

 φa→φ′a=φ′a(φb) (11)
 Φ→Φ′=J(x)Φ (12)

where is the Jacobian of the transformation of the fields.

This will be a symmetry in the case if

 Ω=J (13)

Notice that can be a local function of space time, this can be arranged by performing for the fields one of the (infinite) possible diffeomorphims in the internal space.

We can still retain a global scale invariance in model for very special exponential form for the and potentials. Indeed, if we perform the global scale transformation ( = constant)

 gμν→eθgμν (14)

then (9) is invariant provided and are of the form TMT2 ()

 V(ϕ)=f1eαϕ,U(ϕ)=f2e2αϕ (15)

and is transformed according to

 φa→λabφb (16)

which means

 Φ→det(λab)Φ≡λΦ (17)

such that

 λ=eθ (18)

and

 ϕ→ϕ−θα. (19)

We will now work out the equations of motion after introducing and and see how the integration of the equations of motion allows the spontaneous breaking of the scale invariance.

Let us begin by considering the equations which are obtained from the variation of the fields that appear in the measure, i.e. the fields. We obtain then

 Aμa∂μL2=0 (20)

where . Since it is easy to check that , it follows that det if . Therefore if we obtain that , or that

 L2=−1κR(Γ,g)+12gμν∂μϕ∂νϕ−V=M (21)

where M is constant. Notice that this equation breaks spontaneously the global scale invariance of the theory, since the left hand side has a non trivial transformation under the scale transformations, while the right hand side is equal to , a constant that after we integrate the equations is fixed, cannot be changed and therefore for any we have obtained indeed, spontaneous breaking of scale invariance.

We will see what is the connection now. As we will see, the connection appears in the original frame as a non Riemannian object. However, we will see that by a simple conformal tranformation of the metric we can recover the Riemannian structure. The interpretation of the equations in the frame gives then an interesting physical picture, as we will see.

Let us begin by studying the equations obtained from the variation of the connections . We obtain then

 −Γλμν−Γαβμgβλgαν+δλνΓαμα+δλμgαβΓγαβgγν−gαν∂μgαλ+δλμgαν∂βgαβ−δλνΦ,μΦ+δλμΦ,νΦ=0 (22)

If we define as where is the Christoffel symbol, we obtain for the equation

 −σ,λgμν+σ,μgνλ−gναΣαλμ−gμαΣανλ+gμνΣαλα+gνλgαμgβγΣαβγ=0 (23)

where .

The general solution of Eq.(24) is

 Σαμν=δαμλ,ν+12(σ,μδαν−σ,βgμνgαβ) (24)

where is an arbitrary function due to the - symmetry of the curvature Lambda () ,

 Γαμν→Γ′αμν=Γαμν+δαμZ,ν (25)

Z being any scalar (which means ).

If we choose the gauge , we obtain

 Σαμν(σ)=12(δαμσ,ν+δανσ,μ−σ,βgμνgαβ). (26)

Considering now the variation with respect to , we obtain

 Φ(−1κRμν(Γ)+12ϕ,μϕ,ν)−12√−gU(ϕ)gμν=0 (27)

solving for from Eq.(27) and introducing in Eq.21, we obtain

 M+V(ϕ)−2U(ϕ)χ=0 (28)

a constraint that allows us to solve for ,

 χ=2U(ϕ)M+V(ϕ). (29)

To get the physical content of the theory, it is best consider variables that have well defined dynamical interpretation. The original metric does not has a non zero canonical momenta. The fundamental variable of the theory in the first order formalism is the connection and its canonical momenta is a function of , given by,

 ¯¯¯gμν=χgμν (30)

and given by Eq.(29). Interestingly enough, working with is the same as going to the “Einstein Conformal Frame”. In terms of the non Riemannian contribution dissappears from the equations. This is because the connection can be written as the Christoffel symbol of the metric . In terms of the equations of motion for the metric can be written then in the Einstein form (we define usual Ricci tensor in terms of the bar metric and )

 ¯¯¯¯Rμν(¯¯¯gαβ)−12¯¯¯gμν¯¯¯¯R(¯¯¯gαβ)=κ2Teffμν(ϕ) (31)

where

 Teffμν(ϕ)=ϕ,μϕ,ν−12¯¯¯gμνϕ,αϕ,β¯¯¯gαβ+¯¯¯gμνVeff(ϕ) (32)

and

 Veff(ϕ)=14U(ϕ)(V+M)2. (33)

In terms of the metric , the equation of motion of the Scalar field takes the standard General - Relativity form

 1√−¯¯¯g∂μ(¯¯¯gμν√−¯¯¯g∂νϕ)+V′eff(ϕ)=0. (34)

Notice that if and also, provided is finite and there. This means the zero cosmological constant state is achieved without any sort of fine tuning. That is, independently of whether we add to a constant piece, or whether we change the value of , as long as there is still a point where , then still and ( still provided is finite and there). This is the basic feature that characterizes the TMT and allows it to solve the “old” cosmological constant problem, at least at the classical level.

In what follows we will study the effective potential (33) for the special case of global scale invariance, which as we will see displays additional very special features which makes it attractive in the context of cosmology.

Notice that in terms of the variables , , the “scale” transformation becomes only a shift in the scalar field , since is invariant (since and )

 ¯¯¯gμν→¯¯¯gμν,ϕ→ϕ−θα. (35)

If and as required by scale invariance Eqs. (14, 16, 17, 18, 19), we obtain from the expression (33)

 Veff=14f2(f1+Me−αϕ)2 (36)

Since we can always perform the transformation we can choose by convention . We then see that as const. providing an infinite flat region as depicted in Fig. 1. Also a minimum is achieved at zero cosmological constant for the case at the point

 ϕmin=−1αln∣f1M∣. (37)

In conclusion, the scale invariance of the original theory is responsible for the non appearance (in the physics) of a certain scale, that associated to M. However, masses do appear, since the coupling to two different measures of and allow us to introduce two independent couplings and , a situation which is unlike the standard formulation of globally scale invariant theories, where usually no stable vacuum state exists.

The constant of integration plays a very important role indeed: any non vanishing value for this constant implements, already at the classical level S.S.B. of scale invariance.

## Iii Generation of two flat regions after the introduction of a R2 term

As we have seen, it is possible to obtain a model that through a spontaneous breaking of scale invariace can give us a flat region. We want to obtain now two flat regions in our effective potential. A simple generalization of the action will fix this. The basic new feature we add is the presence is higher curvature terms in the action barrow1 ()-mijic3 (), which have been shown to be very relevant in cosmology. In particular he first inflationary model from a model with higher terms in the curvature was proposed in mijic3 ().

What one needs to do is simply consider the addition of a scale invariant term of the form

 SR2=ϵ∫(gμνRμν(Γ))2√−gd4x (38)

The total action being then . In the first order formalism is not only globally scale invariant but also locally scale invariant, that is conformally invariant (recall that in the first order formalism the connection is an independent degree of freedom and it does not transform under a conformal transformation of the metric).

Let us see what the equations of motion tell us, now with the addition of to the action. First of all, since the addition has been only to the part of the action that couples to , the equations of motion derived from the variation of the measure fields remains unchanged. That is Eq.(21) remains valid.

The variation of the action with respect to gives now

 Rμν(Γ)(−Φκ+2ϵR√−g)+Φ12ϕ,μϕ,ν−12(ϵR2+U(ϕ))√−ggμν=0 (39)

It is interesting to notice that if we contract this equation with , the terms do not contribute. This means that the same value for the scalar curvature is obtained as in section 2, if we express our result in terms of , its derivatives and . Solving the scalar curvature from this and inserting in the other - independent equation we get still the same solution for the ratio of the measures which was found in the case where the terms were absent, i.e. .

In the presence of the term in the action, Eq. (22) gets modified so that instead of , = appears. This in turn implies that Eq.(23) keeps its form but where is replaced by , where once again, .

Following then the same steps as in the model without the curvature square terms, we can then verify that the connection is the Christoffel symbol of the metric given by

 ¯¯¯gμν=(Ω√−g)gμν=(χ−2κϵR)gμν (40)

defines now the “Einstein frame”. Equations (39) can now be expressed in the “Einstein form”

 ¯¯¯¯Rμν−12¯¯¯gμ ν¯¯¯¯R=κ2Teffμν (41)

where

 Teffμν=χχ−2κϵR(ϕ,μϕ,ν−12¯¯¯gμνϕ,αϕ,β¯¯¯gαβ)+¯¯¯gμνVeff (42)

where

 Veff=ϵR2+U(χ−2κϵR)2 (43)

Here it is satisfied that , equation that expressed in terms of becomes

. This allows us to solve for and we get,

 R=−κ(V+M)+κ2¯¯¯gμν∂μϕ∂νϕχ1+κ2ϵ¯¯¯gμν∂μϕ∂νϕ (44)

Notice that if we express in terms of , its derivatives and , the result is the same as in the model without the curvature squared term, this is not true anymore once we express in terms of , its derivatives and .

In any case, once we insert (44) into (43), we see that the effective potential (43) will depend on the derivatives of the scalar field now. It acts as a normal scalar field potential under the conditions of slow rolling or low gradients and in the case the scalar field is near the region .

Notice that since , then if , then, as in the simpler model without the curvature squared terms, we obtain that at that point without fine tuning (here by we mean the derivative of with respect to the scalar field , as usual).

In the case of the scale invariant case, where and are given by equation (15), it is interesting to study the shape of as a function of in the case of a constant , in which case can be regarded as a real scalar field potential. Then from (44) we get , which inserted in (43) gives,

 Veff=(f1eαϕ+M)24(ϵκ2(f1eαϕ+M)2+f2e2αϕ) (45)

The limiting values of are:

First, for asymptotically large positive values, ie. as , we have .

Second, for asymptotically large but negative values of the scalar field, that is as , we have: .

In these two asymptotic regions ( and ) an examination of the scalar field equation reveals that a constant scalar field configuration is a solution of the equations, as is of course expected from the flatness of the effective potential in these regions.

Notice that in all the above discussion it is fundamental that . If the potential becomes just a flat one, everywhere (not only at high values of ). All the non trivial features necessary for a graceful exit, the other flat region associated to the Planck scale and the minimum at zero if are all lost . As we discussed in the model without a curvature squared term, implies the we are considering a situation with S.S.B. of scale invariance.

These kind of models with potentials giving rise to two flat potentials have been applied to produce models for bags and confinement in a very natural way bags and confinement ().

## Iv A Note on the the “Einstein” metric as a canonical variable of the Theory

One could question the use of the Einstein frame metric in contrast to the original metric . In this respect, it is interesting to see the role of both the original metric and that of the Einstein frame metric in a canonical approach to the first order formalism. Here we see that the original metric does not have a canonically conjugated momentum (this turns out to be zero), in contrast, the canonically conjugated momentum to the connection turns out to be a function exclusively of , this Einstein metric is therefore a genuine dynamical canonical variable, as opposed to the original metric. There is also a lagrangian formulation of the theory which uses , as we will see in the next section, what we can call the action in the Einstein frame. In this frame we can quantize the theory for example and consider contributions without reference to the original frame, thus possibly considering breaking the TMT structure of the theory through quantum effects, but such breaking will be done “softly” through the introduction of a cosmological term only. Surprisingly, the remaining structure of the theory, reminiscent from the original TMT structure will be enough to control the strength of this additional cosmological term once we demand that the universe originated from a non-singular and stable emergent state.

## V Generalizing the model to include effects of zero point fluctuations

The effective energy-momentum tensor can be represented in a form like that of a perfect fluid

 Teffμν=(ρ+p)uμuν−p~gμν,whereuμ=ϕ,μ(2X)1/2 (46)

here . This defines a pressure functional and an energy density functional. The system of equations obtained after solving for , working in the Einstein frame with the metric can be obtained from a “k-essence” type effective action, as it is standard in treatments of theories with non linear kinetic terms or k-essence modelsk-essence1 ()-k-essence4 (). The action from which the classical equations follow is,

 Seff=∫√−¯¯¯gd4x[−1κ¯¯¯¯R(¯¯¯g)+p(ϕ,R)] (47)
 p=χχ−2κϵRX−Veff (48)
 Veff=ϵR2+U(χ−2κϵR)2 (49)

where it is understood that,

 χ=2U(ϕ)M+V(ϕ). (50)

We have two possible formulations concerning : Notice first that and are different objects, the is the Riemannian curvature scalar in the Einstein frame, while is a different object. This will be treated in two different ways:

1. First order formalism for . Here is a lagrangian variable, determined as follows, that appear in the expression above for can be obtained from the variation of the pressure functional action above with respect to , this gives exactly the expression for that has been solved already in terms of , etc, see Eq. (44).

2. Second order formalism for . that appear in the action above is exactly the expression for that has been solved already in terms of , etc. The second order formalism can be obtained from the first order formalism by solving algebraically R from the Eq. (44) obtained by variation of , and inserting back into the action.

One may also use the method outlined in (R2Bulg, ) to find the effective action in the Einstein frame, in (R2Bulg, ) the problem of a curvature squared theory with standard measure was studied. The methods outlined there can be also applied in the modified measure case (mahary, ), thus providing another derivation of the effective action explained above.

The problem that we have to solve to find the effective lagrangian is basically finding that lagrangian tat will produce the effective energy momentum tensor in the Einstein frame by the variation of the metric

 Teffμν=¯¯¯gμνLeff(h)−2∂Leff∂¯¯¯gμν (51)

In contrast to the simplified models studied in literaturek-essence1 (); k-essence2 (); k-essence3 (); k-essence4 (), it is impossible here to represent in a factorizable form like . The scalar field effective Lagrangian can be taken as a starting point for many considerations.

In particular, the quantization of the model can proceed from (47) and additional terms could be generated by radiative corrections. We will focus only on a possible cosmological term in the Einstein frame added (due to zero point fluctuations) to (47), which leads then to the new action

 Seff,Λ=∫√−¯¯¯gd4x[−1κ¯R(¯¯¯g)+p(ϕ,R)−Λ] (52)

This addition to the effective action leaves the equations of motion of the scalar field unaffected, but the gravitational equations aquire a cosmological constant. Adding the term can be regarded as a redefinition of

 Veff(ϕ,R)→Veff(ϕ,R)+Λ (53)

As we will see the stability of the emerging Universe imposes interesting constraints on

After introducing the term, we get from the variation of the same value of , unaffected by the new term, but as one can easily see then does not have the interpretation of a curvature scalar in the original frame since it is unaffected by the new source of energy density (the term), this is why the term theory does not have a formulation in the original frame, but is a perfectly legitimate generalization of the theory, probably obtained by considering zero point fluctuations, notice that quantum theory is possible only in the Einstein frame. Notice that even in the original frame the bar metric (not the original metric) appears automatically in the canonically conjugate momenta to the connection, so we can expect from this that the bar metric and not the original metric be the relevant one for the quantum theory.

In Figure 1 and 2 we have plotted the effective potential as a function of the scalar field, for and respectively. We consider unit where , , and different values for .

## Vi Analysis of the Emergent Universe solutions

We now want to consider the detailed analysis of The Emerging Universe solutions and in the next section their stability in the TMT scale invariant theory. We start considering the cosmological solutions of the form

 ds2=dt2−a(t)2(dr21−r2+r2(dθ2+sin2θdϕ2)),ϕ=ϕ(t) (54)

in this case, we obtain for the energy density and the pressure, the following expressions. We will consider a scenario where the scalar field is moving in the extreme right region , in this case the expressions for the energy density and pressure are given by,

 ρ=A2˙ϕ2+3B˙ϕ4+C (55)

and

 p=A2˙ϕ2+B˙ϕ4−C (56)

It is interesting to notice that all terms proportional to behave like “radiation”, since is satisfied. here the constants and are given by,

 A = f2f2+κ2ϵf21, (57) B = ϵκ24(1+κ2ϵf21/f2)=ϵκ24A, (58) C = f214f2(1+κ2ϵf21/f2)+Λ=f214f2A+Λ. (59)

It will be convenient to “decompose” the constant into two pieces,

 Λ=−14κ2ϵ+Δλ (60)

since as , . Therefore has the interesting interpretation of the vacuum energy density in the vacuum. As we will see, it is remarkable that the stability and existence of non-singular emergent universe implies that , and it is bounded from above as well.

The equation that determines such static universe , , gives rise to a restriction for that have to satisfy the following equation in order to guarantee that the universe be static, because is proportional to , we must require that , which leads to

 3B˙ϕ40+A˙ϕ20−C=0, (61)

This equation leads to two roots, the first being

 ˙ϕ21=√A2+12BC−A6B. (62)

The second root is:

 ˙ϕ22=−√A2+12BC−A6B. (63)

It is also interesting to see that if the discriminant is positive, the first solution has automatically positive energy density, if we only consider cases where , which is required if we want the emerging solution to be able to turn into an inflationary solution eventually. One can see that the condition for the first solution reduces to the inequality , where , since we must have , otherwise we get a negative kinetic term during the inflationary period, and as we will see in the next section, we must have that from the stability of the solution, and as long as , it is always true that this inequality is satisfied.

Before going into the subject of the small perturbations and stability of these solutions, we would like to notice the “entropy like” conservation laws that may be useful in a non perturbative analysis of the theory.

In fact in the region, we have the exact symmetry . and considering that the effective matter action here is , we have the conserved quantity

 πϕ=a3(A˙ϕ+4B˙ϕ3) (64)

It is very interesting to notice that

 πϕ=S=a3s (65)

where assumes the “entropy density” form

 s=(ρ+p)/T (66)

provided we identify the “Temperature” T with .

## Vii Stability of the static solution

We will now consider the perturbation equations. Considering small deviations of the from the static emerging solution value and also considering the perturbations of the scale factor , we obtain, from Eq. (55)

 δρ=A˙ϕ0δ˙ϕ+12B˙ϕ30δ˙ϕ (67)

at the same time can be obtained from the perturbation of the Friedmann equation

 3(1a2+H2)=κρ (68)

and since we are perturbing a solution which is static, i.e., has , we obtain then

 −6a30δa=κδρ (69)

we also have the second order Friedmann equation

 1+˙a2+2a¨aa2=−κp (70)

For the static emerging solution, we have , , so

 2a20=−2κp0=23κρ0=Ω0κρ0 (71)

where we have chosen to express our result in terms of , defined by , which for the emerging solution has the value . Using this in 69, we obtain

 δρ=−3Ω0ρ0a0δa (72)

and equating the values of as given by 67 and 72 we obtain a linear relation between and , which is,

 δ˙ϕ=D0δa (73)

where

 D0=−3Ω0ρ0a0˙ϕ0(A+12B˙ϕ20) (74)

we now consider the perturbation of the eq. (70). In the right hand side of this equation we consider that , with

 Ω=2(1−Ueffρ), (75)

where,

 Ueff=C+B˙ϕ4 (76)

and therefore, the perturbation of the Eq. (70) leads to,

 −2δaa30+2δ¨aa0=−κδp=−κδ((Ω−1)ρ) (77)

to evaluate this, we use 75, 76 and the expressions that relate the variations in and (73). Defining the “small” variable as

 a(t)=a0(1+β) (78)

we obtain,

 2¨β(t)+W20β(t)=0, (79)

where,

 W20=Ω0ρ0⎡⎣24B˙ϕ20A+12˙ϕ20B−6(C+B˙ϕ40)ρ0−3κΩ0+2κ⎤⎦, (80)

notice that the sum of the last two terms in the expression for , that is vanish since , for the same reason, we have that , which brings us to the simplified expression

 W20=Ω0ρ0⎡⎣24B˙ϕ20A+12˙ϕ20B−4⎤⎦, (81)

For the stability of the static solution, we need that , where is defined either by E. (62) () or by E. (63) (). If we take E. (63) () and use this in the above expression for , we obtain,

 W20=Ω0ρ0[4√A2+12BC−2√A2+12BC−A], (82)

to avoid negative kinetic terms during the slow roll phase that takes place following the emergent phase, we must consider , so, we see that the second solution is unstable and will not be considered further.

Now in the case of the first solution, E. (62) (), then becomes

 W20=Ω0ρ0[−4√A2+12BC2√A2+12BC−A], (83)

so the condition of stability becomes , or , squaring both sides and since , we get , which means , and therefore , multiplying by , we obtain, , replacing the values of , given by 57 we obtain the condition

 Δλ>0, (84)

Now there is the condition that the discriminant be positive

 Δλ<112(−ϵ)κ2[f2f2+κ2ϵf21], (85)

since , , meaning that , we see that we obtain a positive upper bound for the energy density of the vacuum as , which must be positive, but not very big.

## Viii Inflation and its Graceful Exit

The emerging phase owes its existence to a strictly constant vacuum energy (which here is represented by the value of ) at very large values of the field . In fact, while for the effective potential of the scalar field is perfectly flat, for any the effective potential acquires a non trivial shape. This causes the transition from the emergent phase to a slow roll inflationary phase which will be the subject of this section.

Following SIchile (), we consider now then the relevant equations for the model in the slow roll regime, i.e. for small and when the scalar field is large, but finite and we consider the first corrections to the flatness to the effective potential. Dropping higher powers of in the contributions for the kinetic energy and in the scalar curvature , we obtain

 ρ=12γ˙ϕ2+Veff, (86) γ=χχ−2κϵR, (87) R=−κ(V+M). (88)

Here, as usual . In the slow roll approximation, we can drop the second derivative term of and the second power of in the equation for and we get

 3Hγ˙ϕ=−V′eff, (89) 3H2=κVeff, (90)

where . The relevant expression for will be that given by (45), i.e., where all higher derivatives are ignored in the potential, consistent with the slow roll approximation.

We now display the relevant expressions for the region of very large, but not infinite , these are:

 Veff=C+C1exp(−αϕ), (91) χ=2f2f1exp(αϕ)−2Mf2f21, (92)

and

 γ=γ0+γ1exp(−αϕ). (93)

The relevant constants that will affect our results are, , as given by (59) and and given by

 C1=−8ϵκ2f31M(4f2+4κ2ϵf21)2+2f1M4f2+4κ2ϵf21, (94)

and

 γ0=f2f2+κ2ϵf21, (95)

respectively.

Using Eq. (91) we can calculates the key landmarks of the inflationary history: first, the value of the scalar field where inflation ends, and a value for the scalar field bigger than this () and which happens earlier, which represents the “horizon crossing point”. We must demand then that a typical number of e-foldings, like , takes place between , until the end of inflation at .

To determine the end of inflation, we consider the quantity and consider the point in the evolution of the Universe where , only when , we have an accelerating Universe, so the point represents indeed the end of inflation. Calculating the derivative with respect to cosmic time of the Hubble expansion using (90) and (89), we obtain that the condition gives

 δ=12

format: plain


text:
### Recursive Proof Composition without a Trusted Setup

Sean Bowe, Jack Grigg, and Daira Hopwood

##### Abstract

Non-interactive arguments of knowledge are powerful cryptographic tools that can be used to demonstrate the faithful execution of arbitrary computations with publicly verifiable proofs. Increasingly efficient protocols have been described in recent years, with verification time and/or communication complexity that is sublinear in the size of the computation being described. These efficiencies can be exploited to realize recursive proof composition: the concept of proofs that attest to the correctness of other instances of themselves, thereby allowing large computational effort to be incrementally verified. All previously known realizations of recursive proof composition have required a trusted setup and cycles of expensive pairing-friendly elliptic curves. We obtain and implement Halo, the first practical example of recursive proof composition without a trusted setup, using the discrete log assumption over normal cycles of elliptic curves. In the process we develop several novel techniques that may be of independent interest.

Available format(s)
Category
Cryptographic protocols
Publication info
Preprint. MINOR revision.
Keywords
recursive proofsincrementally verifiable computationzero knowledge
Contact author(s)
sean @ electriccoin co
History
2020-02-18: last of 2 revisions
See all versions
Short URL
https://ia.cr/2019/1021

CC BY

BibTeX

@misc{cryptoeprint:2019/1021,
author = {Sean Bowe and Jack Grigg and Daira Hopwood},
title = {Recursive Proof Composition without a Trusted Setup},
howpublished = {Cryptology ePrint Archive, Paper 2019/1021},
year = {2019},
note = {\url{https://eprint.iacr.org/2019/1021}},
url = {https://eprint.iacr.org/2019/1021}
}

Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.


text:
J. Korean Ceram. Soc. > Volume 40(12); 2003 > Article
 Journal of the Korean Ceramic Society 2003;40(12): 1183. doi: https://doi.org/10.4191/kcers.2003.40.12.1183
 LCP(Liquid Crystal Polymer)와 BaTiO3복합체의 제조 및 유전특성 윤기현, 이승주, 양병덕, 김종희1 연세대학교 세라믹공학과1삼성전기 주식회사 Preparation and Dielectric Properties of LCP and BaTiO3 Composites ABSTRACT LCP (Liquid Crystal Polymer)/BT(BaTiO$_3$) composite was prepared by mixing LCP varnish and BT slurry. And the effect of the composition of LCP/BT composites and the size of BT powder on dielectric properties of LCP/BT composite were investigated. The dispersion properties of BT slurries were also studied as a function of volume fraction of dispersant, GLYMO (3-glycidyloxypropyltrimethoxysilane) added to various sized BT powders. The quantity of optimum dispersant in BT slurry decreased with an increase of BT powder size because of the decrease of specific surface area of BT powder. LCP/BT composite sheet was prepared by tape-casting method with mixing BT slurry and LCP varnish. The dielectric constant of the composites increased from 34.3 to 44.1 and their dielectric loss increased from 0.05 to 0.063 as the size of BT powder increased from 100 nm to 500 nm. Key words: Liquid crystal polymer, $BaTiO_3$, Composite, Dielectric, Dispersion
TOOLS
Full text via DOI


text:
# Properties

 Label 27T19 Degree $27$ Order $81$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_3\wr C_3$

## Group action invariants

 Degree $n$: $27$ Transitive number $t$: $19$ Group: $C_3\wr C_3$ Parity: $1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $9$ Generators: (1,4,25)(2,5,26)(3,6,27)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,10,21)(2,11,19)(3,12,20)(4,13,24)(5,14,22)(6,15,23)(7,18,25)(8,16,26)(9,17,27)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$27$:  $C_3^2:C_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$ x 4

Degree 9: $C_3^2$, $C_3 \wr C_3$ x 3

## Low degree siblings

9T17 x 3, 27T21, 27T27 x 3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $3$ $( 7,12,14)( 8,10,15)( 9,11,13)(16,23,21)(17,24,19)(18,22,20)$ $3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $3$ $( 7,14,12)( 8,15,10)( 9,13,11)(16,21,23)(17,19,24)(18,20,22)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,19,22)(17,20,23) (18,21,24)(25,26,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7,15,11)( 8,13,12)( 9,14,10)(16,24,20)(17,22,21) (18,23,19)(25,26,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,22,19)(17,23,20) (18,24,21)(25,27,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7,11,15)( 8,12,13)( 9,10,14)(16,20,24)(17,21,22) (18,19,23)(25,27,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,10,13)( 8,11,14)( 9,12,15)(16,24,20) (17,22,21)(18,23,19)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1, 5,27)( 2, 6,25)( 3, 4,26)( 7,13,10)( 8,14,11)( 9,15,12)(16,22,19) (17,23,20)(18,24,21)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 6,26)( 2, 4,27)( 3, 5,25)( 7,12,14)( 8,10,15)( 9,11,13)(16,21,23) (17,19,24)(18,20,22)$ $9, 9, 9$ $9$ $9$ $( 1, 7,17,26,14,24, 6,12,19)( 2, 8,18,27,15,22, 4,10,20)( 3, 9,16,25,13,23, 5, 11,21)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $9$ $3$ $( 1, 8,16)( 2, 9,17)( 3, 7,18)( 4,11,19)( 5,12,20)( 6,10,21)(13,24,27) (14,22,25)(15,23,26)$ $9, 9, 9$ $9$ $9$ $( 1, 9,18, 6,11,20,26,13,22)( 2, 7,16, 4,12,21,27,14,23)( 3, 8,17, 5,10,19,25, 15,24)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $9$ $3$ $( 1,16, 8)( 2,17, 9)( 3,18, 7)( 4,19,11)( 5,20,12)( 6,21,10)(13,27,24) (14,25,22)(15,26,23)$ $9, 9, 9$ $9$ $9$ $( 1,17,12, 6,19,14,26,24, 7)( 2,18,10, 4,20,15,27,22, 8)( 3,16,11, 5,21,13,25, 23, 9)$ $9, 9, 9$ $9$ $9$ $( 1,18,13,26,22,11, 6,20, 9)( 2,16,14,27,23,12, 4,21, 7)( 3,17,15,25,24,10, 5, 19, 8)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1,26, 6)( 2,27, 4)( 3,25, 5)( 7,14,12)( 8,15,10)( 9,13,11)(16,23,21) (17,24,19)(18,22,20)$

## Group invariants

 Order: $81=3^{4}$ Cyclic: no Abelian: no Solvable: yes GAP id: [81, 7]
 Character table:  3 4 3 3 3 3 3 3 3 3 4 2 2 2 2 2 2 4 1a 3a 3b 3c 3d 3e 3f 3g 3h 3i 9a 3j 9b 3k 9c 9d 3l 2P 1a 3b 3a 3e 3f 3c 3d 3h 3g 3l 9c 3k 9d 3j 9a 9b 3i 3P 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 3l 1a 3i 1a 3i 3l 1a 5P 1a 3b 3a 3e 3f 3c 3d 3h 3g 3l 9c 3k 9d 3j 9a 9b 3i 7P 1a 3a 3b 3c 3d 3e 3f 3g 3h 3i 9a 3j 9b 3k 9c 9d 3l X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 1 1 1 1 1 B B B /B /B /B 1 X.3 1 1 1 1 1 1 1 1 1 1 /B /B /B B B B 1 X.4 1 1 1 B B /B /B B /B 1 1 B /B /B 1 B 1 X.5 1 1 1 /B /B B B /B B 1 1 /B B B 1 /B 1 X.6 1 1 1 B B /B /B B /B 1 B /B 1 B /B 1 1 X.7 1 1 1 /B /B B B /B B 1 /B B 1 /B B 1 1 X.8 1 1 1 B B /B /B B /B 1 /B 1 B 1 B /B 1 X.9 1 1 1 /B /B B B /B B 1 B 1 /B 1 /B B 1 X.10 3 A /A . . . . . . 3 . . . . . . 3 X.11 3 /A A . . . . . . 3 . . . . . . 3 X.12 3 . . C D /C -D -/C -C /A . . . . . . A X.13 3 . . /C -D C D -C -/C A . . . . . . /A X.14 3 . . -/C C -C /C D -D /A . . . . . . A X.15 3 . . -C /C -/C C -D D A . . . . . . /A X.16 3 . . D -/C -D -C C /C /A . . . . . . A X.17 3 . . -D -C D -/C /C C A . . . . . . /A A = 3*E(3) = (-3+3*Sqrt(-3))/2 = 3b3 B = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 C = -2*E(3)-E(3)^2 = (3-Sqrt(-3))/2 = 1-b3 D = E(3)-E(3)^2 = Sqrt(-3) = i3


text:
Nano alternative for windows powershell

I am looking for software similar to nano for linux bash but for windows powershell. Is there any built in so I do not have to install something?

EDIT Nano is a text editor that runs within the bash. You can open a text like document (.txt, .c etc) in the bash to edit it on the fly or just view it and close it again.

• I don't know if this works but maybe it's possible to get edit.com off an XP machine and use it in powershell on windows 7. I don't know if Win7 32bit has edit.com but win7 64bit doesn't have edit.com – barlop Jul 8 '15 at 22:14
• Assume someone knows everything about PowerShell and could help you, but doesn't know much about Linux or what Nano may be. Maybe you should describe what you want to do. – Peter Hahndorf Jul 9 '15 at 2:45
• @PeterHahndorf You were right, I edited it – John Demetriou Jul 9 '15 at 6:25

There is now a way to use nano and vim with powershell by installing "Bash on Windows". More information on Scott Hanselman blog

From command line you can run

bash -c "vi filename.txt"
bash -c "nano filename.txt"

function vi ($File){ bash -c "vi$File"
}

function nano ($File){ bash -c "nano$File"
}

The blog source where I got the information from

• Those smart quotes in your post will become stupid to the shells and cause problems. I've fixed them for you this time – phuclv Oct 17 '17 at 3:25
• Yes. Since powershell has been updated. Thank you for the updated answer, I swapped the accepted one to yours – John Demetriou Oct 17 '17 at 8:25

Nano is available for powershell. If you have the Chocolatey package manager installed in your system you can install nano with:

choco install nano

You can install Chocolatey through the command line with:

Set-ExecutionPolicy Bypass -Scope Process -Force; iex ((New-Object System.Net.WebClient).DownloadString('https://chocolatey.org/install.ps1'))

My personal experience is that it nano performs great in Windows 10 but it's really slow to start up the first time in Windows 7.

The only built-in editor in Windows is Notepad. It should already be in your path, so you can just type notepad something.txt in the PowerShell console.

If you want console-based editors, there are some here: https://stackoverflow.com/questions/11045077/edit-a-text-file-on-the-console-in-64-bit-windows

A useful thing to do is to make an alias called "edit" (for example) for your favorite text editor. Put something like this in your profile:

set-alias edit "\${env:ProgramFiles}\Sublime Text 3\sublime_text.exe"

• Newer versions of Windows also have powershell_ise.exe built-in and in the path. A pretty good editor for PowerShell scripts. – Peter Hahndorf Jul 9 '15 at 7:24
• so no built in powershell editor inside the shell? only external programs that run outside of powershell? – John Demetriou Jul 9 '15 at 7:30
• @John Demetriou, I'm not sure what you mean by "built in" now. There are two stock text editors that come with Windows: Notepad and PowerShell ISE (thanks Peter). If you want a console-based editor, then the link above has some. There is no stock, console-based editor in recent versions of Windows. – dangph Jul 9 '15 at 7:43
• that's what I meant, stock console based editor. ok thanks. I will look in the link you provide and choose. thanks – John Demetriou Jul 9 '15 at 7:58
• To use the ISE editor: psEdit \path\to\file.txt ... To switch back and forth between the editor and powershell ctrl + s and ctrl + d – Kolob Canyon Jan 17 '17 at 23:54

Just install Windows Subsystem for Linux (WSL). Then, type.

wsl nano

or

wsl nano textfilenametoedit.txt

Quotes are not needed.

• This may be obvious to some people, but it tripped me up for a bit: if you're supplying a full path to the file, make sure you pass the path that you would use in WSL, not in Windows. For example, if you want to edit C:\text.txt, instead of using wsl nano C:\text.txt, use wsl nano /mnt/c/text.txt – Matt Apr 12 '19 at 18:30

To add to the answers you've already received, you can have a shell editor in Windows, by installing Vim for windows, from Vim's official page.


text:
Precalculus (6th Edition) Blitzer

The solutions of the equation are $\theta =\frac{3\pi }{2}+2n\pi$, where n is any integer.
\begin{align} & 3sin\theta +5=-2\sin \theta \\ & 3sin\theta +2sin\theta =-5 \\ & 5\sin \theta =-5 \\ & \sin \theta =-1 \end{align} $sin\frac{\pi }{2}=1$, and the solutions of $\sin \theta =-1$ in [0,2 $\pi$ ) are: \begin{align} & \theta =\pi +\frac{\pi }{2} \\ & =\frac{2\pi }{2}+\frac{\pi }{2} \\ & =\frac{3\pi }{2} \end{align} \begin{align} & \theta =2\pi -\frac{\pi }{2} \\ & =\frac{4\pi }{2}-\frac{\pi }{2} \\ & =\frac{3\pi }{2} \end{align} The period of the sine function is $2\pi$, and the solutions are given by: $\theta =\frac{3\pi }{2}+2n\pi$, where n is any integer.


text:
# Search Jobs Europass - Europa EU

Översättning Engelska-Spanska :: tilde :: ordlista

The consonant ñ gives the Spanish alphabet one more letter than the English alphabet. When you see a wiggly line — called a tilde (teel -deh) — on top of the letter n that looks like ñ, use the ny sound that you use for the English word canyon. In pronunciation brackets, this sound is shown as ny. Following are some examples: The ~ accent, called a tilde, on the Spanish letter Ñ indicates that the word used to be spelled with two Ns but now the tilde stands in for the second one. Ñ is pronounced like the ni in onion. It’s very important to include the tilde when writing or typing, because N and Ñ are two different letters. á é í ó ú.

Unless you have a Spanish keyboard, you may find it difficult to include special characters or letters such as ¿ ¡ ñ or é in your text. And writing the correct letter can make a big difference. 2019-11-13 · A row of characters including the Spanish characters will pop up (along with characters using other types of diacritical marks such as those of French). Simply slide your finger over to the character you want, such as the é, and release. Similarly, the ñ can be selected by pressing and holding the virtual "n" key. The Tilde in Spanish (ñ) This accent mark is only ever found above an “n”.

The accent mark is called a tilde (and informally referred to as a "squiggly").

## Att göra spanska substantiv och adjektiv plural

It’s very important to include the tilde when writing or typing, because N and Ñ are two different letters. That funny little squiggle is called a tilde, and it tells us to place more emphasis on the letter “n” in a word.

### BT Bättre Svenska, Bosse Thorén

Focus on one accent: mixing multiple accents can get really confusing Meaning and examples for 'hoarse' in Spanish-English dictionary. More for horse sound, knowing what ' s not do n't know how the Dictionary is giving an sound! 06:52  spanish.begagnade smartphones stockholm.iphone 7 plus australia.kГ¶pa hГҐrfГ¶n munstycke fГ¶r lockigt hГҐr.depend nagellampa.playstation 4 byta telefonnummer.hvad kan man fГҐ for en iphone 7.bosch accent line  Spanish in Cadiz (SIC) grundades 2003 och är en liten och modern skola som ligger i centrala Cádiz med bara ett par minuters promenad till stadens historiska  Quick help to Type Spanish n with Tilde.

The accent mark is called a tilde (and informally referred to as a "squiggly"). Since this character is not used in the English alphabet, most United States keyboards do not have a key specifically for it. The use of accent marks – the small diagonal lines, which appear over certain letters when written down – is one of the most instantly notable features of the Spanish language. However, understanding when and where to use them can seem like a daunting task, especially if you are a novice. After inserting accent marks in your email message, you can save these accent marks as AutoText entries with Kutools for Outlook’s Auto Text utility, and then reuse these accent marks with only double clicking in Outlook message. Kutools for Outlook: Add more than 100 handy tools for Outlook, free to try with no limitation in 60 days.
Vindex ab allabolag

From a doubled letter N in Spanish words of Latin origin.

Se hela listan på lawlessspanish.com It chooses the right encoding by some glyphs from your language correspondingly to the encoding of the source file. \documentclass [11pt,spanish] {article} \usepackage [T1] {fontenc} \usepackage {selinput} \SelectInputMappings {% aacute= {á}, ntilde= {ñ}, Euro= {€} } \usepackage {babel} \begin {document} Hola! The "Spanish N," as it is frequently called, is an "N" with a small accent mark above it. The accent mark is called a tilde (and informally referred to as a "squiggly").
1630 usd to sek

deweys decimalsystem
aaron antonovsky river of life
tandhygienister malmö

### BT Bättre Svenska, Bosse Thorén

\n\nSERIAL NUMBER 90026 \n\nONLY TWO OWNERS, ALWAYS n\nVega Sicilia is undoubtedly a unique wine and one of the great Spanish classics. The Vega Sicilia enjoy a certain antique but stately accent, like the  Subtitling, transcription and translation in both Spanish and English.

Grav accent (),  följt av bokstav. Tilde (~), Skift Tilde (ñ), AltGr + n. N n versal | Spanska Alphabet | M(A)L MasterAnyLanguage.com (Svenska) o versal accent Spanish for Beginners Lesson 1 - Spanish Alphabet 1  i små bokstäver utan accent. variationer. Lär dig Pause.


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## G = D4×Dic6order 192 = 26·3

### Direct product of D4 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — D4×Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — D4×Dic3 — D4×Dic6
 Lower central C3 — C2×C6 — D4×Dic6
 Upper central C1 — C22 — C4×D4

Generators and relations for D4×Dic6
G = < a,b,c,d | a4=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 632 in 280 conjugacy classes, 123 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C22×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C6×D4, D4×Q8, C4×Dic6, C122Q8, Dic3.D4, C12⋊Q8, C12.48D4, D4×Dic3, D4×C12, C22×Dic6, D4×Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C24, Dic6, C22×S3, C22×D4, C22×Q8, 2- 1+4, C2×Dic6, S3×D4, S3×C23, D4×Q8, C22×Dic6, C2×S3×D4, Q8○D12, D4×Dic6

Smallest permutation representation of D4×Dic6
On 96 points
Generators in S96
(1 65 89 28)(2 66 90 29)(3 67 91 30)(4 68 92 31)(5 69 93 32)(6 70 94 33)(7 71 95 34)(8 72 96 35)(9 61 85 36)(10 62 86 25)(11 63 87 26)(12 64 88 27)(13 60 77 37)(14 49 78 38)(15 50 79 39)(16 51 80 40)(17 52 81 41)(18 53 82 42)(19 54 83 43)(20 55 84 44)(21 56 73 45)(22 57 74 46)(23 58 75 47)(24 59 76 48)
(1 34)(2 35)(3 36)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(49 84)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 81)(59 82)(60 83)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 22 7 16)(2 21 8 15)(3 20 9 14)(4 19 10 13)(5 18 11 24)(6 17 12 23)(25 37 31 43)(26 48 32 42)(27 47 33 41)(28 46 34 40)(29 45 35 39)(30 44 36 38)(49 67 55 61)(50 66 56 72)(51 65 57 71)(52 64 58 70)(53 63 59 69)(54 62 60 68)(73 96 79 90)(74 95 80 89)(75 94 81 88)(76 93 82 87)(77 92 83 86)(78 91 84 85)

G:=sub<Sym(96)| (1,65,89,28)(2,66,90,29)(3,67,91,30)(4,68,92,31)(5,69,93,32)(6,70,94,33)(7,71,95,34)(8,72,96,35)(9,61,85,36)(10,62,86,25)(11,63,87,26)(12,64,88,27)(13,60,77,37)(14,49,78,38)(15,50,79,39)(16,51,80,40)(17,52,81,41)(18,53,82,42)(19,54,83,43)(20,55,84,44)(21,56,73,45)(22,57,74,46)(23,58,75,47)(24,59,76,48), (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(49,84)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,81)(59,82)(60,83)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,22,7,16)(2,21,8,15)(3,20,9,14)(4,19,10,13)(5,18,11,24)(6,17,12,23)(25,37,31,43)(26,48,32,42)(27,47,33,41)(28,46,34,40)(29,45,35,39)(30,44,36,38)(49,67,55,61)(50,66,56,72)(51,65,57,71)(52,64,58,70)(53,63,59,69)(54,62,60,68)(73,96,79,90)(74,95,80,89)(75,94,81,88)(76,93,82,87)(77,92,83,86)(78,91,84,85)>;

G:=Group( (1,65,89,28)(2,66,90,29)(3,67,91,30)(4,68,92,31)(5,69,93,32)(6,70,94,33)(7,71,95,34)(8,72,96,35)(9,61,85,36)(10,62,86,25)(11,63,87,26)(12,64,88,27)(13,60,77,37)(14,49,78,38)(15,50,79,39)(16,51,80,40)(17,52,81,41)(18,53,82,42)(19,54,83,43)(20,55,84,44)(21,56,73,45)(22,57,74,46)(23,58,75,47)(24,59,76,48), (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(49,84)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,81)(59,82)(60,83)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,22,7,16)(2,21,8,15)(3,20,9,14)(4,19,10,13)(5,18,11,24)(6,17,12,23)(25,37,31,43)(26,48,32,42)(27,47,33,41)(28,46,34,40)(29,45,35,39)(30,44,36,38)(49,67,55,61)(50,66,56,72)(51,65,57,71)(52,64,58,70)(53,63,59,69)(54,62,60,68)(73,96,79,90)(74,95,80,89)(75,94,81,88)(76,93,82,87)(77,92,83,86)(78,91,84,85) );

G=PermutationGroup([[(1,65,89,28),(2,66,90,29),(3,67,91,30),(4,68,92,31),(5,69,93,32),(6,70,94,33),(7,71,95,34),(8,72,96,35),(9,61,85,36),(10,62,86,25),(11,63,87,26),(12,64,88,27),(13,60,77,37),(14,49,78,38),(15,50,79,39),(16,51,80,40),(17,52,81,41),(18,53,82,42),(19,54,83,43),(20,55,84,44),(21,56,73,45),(22,57,74,46),(23,58,75,47),(24,59,76,48)], [(1,34),(2,35),(3,36),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(49,84),(50,73),(51,74),(52,75),(53,76),(54,77),(55,78),(56,79),(57,80),(58,81),(59,82),(60,83),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,85),(68,86),(69,87),(70,88),(71,89),(72,90)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,22,7,16),(2,21,8,15),(3,20,9,14),(4,19,10,13),(5,18,11,24),(6,17,12,23),(25,37,31,43),(26,48,32,42),(27,47,33,41),(28,46,34,40),(29,45,35,39),(30,44,36,38),(49,67,55,61),(50,66,56,72),(51,65,57,71),(52,64,58,70),(53,63,59,69),(54,62,60,68),(73,96,79,90),(74,95,80,89),(75,94,81,88),(76,93,82,87),(77,92,83,86),(78,91,84,85)]])

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L ··· 4Q 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E ··· 12L order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 ··· 4 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 6 6 6 6 12 ··· 12 2 2 2 4 4 4 4 2 2 2 2 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + - + + + + + - - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 D6 D6 D6 Dic6 2- 1+4 S3×D4 Q8○D12 kernel D4×Dic6 C4×Dic6 C12⋊2Q8 Dic3.D4 C12⋊Q8 C12.48D4 D4×Dic3 D4×C12 C22×Dic6 C4×D4 Dic6 C3×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C6 C4 C2 # reps 1 1 1 4 2 2 2 1 2 1 4 4 1 2 1 2 1 8 1 2 2

Matrix representation of D4×Dic6 in GL6(𝔽13)

 1 3 0 0 0 0 8 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 3 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 11 0 0 0 0 1 1 0 0 0 0 0 0 1 12 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 6 0 0 0 0 4 1 0 0 0 0 0 0 6 10 0 0 0 0 3 7

G:=sub<GL(6,GF(13))| [1,8,0,0,0,0,3,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,3,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,11,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,4,0,0,0,0,6,1,0,0,0,0,0,0,6,3,0,0,0,0,10,7] >;

D4×Dic6 in GAP, Magma, Sage, TeX

D_4\times {\rm Dic}_6
% in TeX

G:=Group("D4xDic6");
// GroupNames label

G:=SmallGroup(192,1096);
// by ID

G=gap.SmallGroup(192,1096);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,387,675,80,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^12=1,d^2=c^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽


text:
# Math Help - Writing an equation of a line in standard form

1. ## Writing an equation of a line in standard form

Recall how to write an equation of a line in standard form given 2 points
• no fractions
• no decimals
• x is positive
**********
1) (1/2,-2); (1/2,5)

2) (3,-6); (10,-6)
i planned to use m=y2-y1/x2-x1, then y=mx+b, and then put it into Ax+By=C. but after using m=y2-y1/x2-x1, i kept getting a fraction: 7/0 for #1 and 0/7 for #2. i can't remember which one is indefinite, but i'm stuck on the first step because the teacher said no fractions... thanks in advance

2. Line #1 is simply: $2x=1$.
Some textbooks insist that standard form is: $Ax+By+C=0,\;A>0$.
Be sure to see what your text says.

3. Originally Posted by Plato
Line #1 is simply: $2x=1$.
Some textbooks insist that standard form is: $Ax+By+C=0,\;A>0$.
Be sure to see what your text says.
2x=1? the thing is that the publisher hasn't sent our textbooks yet, we are working blindly w/problems she throws at us. she hasn't explained it either because it's, "review." may i ask the steps you would use or how you got 2x=1?

4. Originally Posted by trancefanatic
2x=1? the thing is that the publisher hasn't sent our textbooks yet, we are working blindly w/problems she throws at us. she hasn't explained it either because it's, "review." may i ask the steps you would use or how you got 2x=1?
Bless you my friend, you are the casualty of fraud.
If you want to take legal action, I will take you side.
The whole “online teaching of mathematic is a scam”!

5. Originally Posted by Plato
Bless you my friend, you are the casualty of fraud.
If you want to take legal action, I will take you side.
The whole “online teaching of mathematic is a scam”!
..............wait, what???!

6. Originally Posted by trancefanatic
..............wait, what???!
Here is 47 years of teaching experience speaking.
If you or the institution offering an online course in mathematics thinks that such a course will teach you the equivalent of an on campus course, then I have a big bridge in NY to sell you both for \$1.

Any online course in mathematics is a joke!
It cannot be done. You can pay your money. You can even get a passing grade. But you will not understand the material that is in 70% of the cases a prerequisite to the rest of your life. WHY TAKE THE CHANCE

7. Well, this might make you laugh. It's not an online course! The institution just opened a few weeks ago and haven't received text books yet. Why? I don't know, but all I know is that this teacher makes no sense to me whatsoever because of her extreme methods that are "outside of the box." Although these problems are due tomorrow... I think it'd be a risk I'd have to take (F).

Oh well.

8. Originally Posted by trancefanatic
Recall how to write an equation of a line in standard form given 2 points
• no fractions
• no decimals
• x is positive
**********
1) (1/2,-2); (1/2,5)

2) (3,-6); (10,-6)
i planned to use m=y2-y1/x2-x1, then y=mx+b, and then put it into Ax+By=C. but after using m=y2-y1/x2-x1, i kept getting a fraction: 7/0 for #1 and 0/7 for #2. i can't remember which one is indefinite, but i'm stuck on the first step because the teacher said no fractions... thanks in advance
1) Note that any points on a line passing though these points is going to have the same x value. Thus $x = \frac{1}{2}$. Now put that in standard form without any fractions.

2) Note that any points on a line passing through these points is going to have the same y value. etc.

-Dan


text:
# Christian Muslim Relations Online I

## 600-1500

Christian-Muslim Relations: A Bibliographical History Online (CMR Online) is a general online history of relations between the faiths. It covers the period from 600 to 1500 CE, when encounters took place through the extended Mediterranean basin and are recorded in Syriac, Arabic, Greek, Latin and other languages. It comprises introductory essays on the treatment of Christians in the Qur’an, Qur’an commentaries, biographies of the Prophet, Hadith and Sunni law, and of Muslims in canon law, and the main body of more than two hundred detailed entries on all the works recorded, whether surviving or lost. The result of collaboration between leading scholars, CMR Online is intended as an invaluable tool for research in Christian-Muslim relations.

Christian Muslim Relations: A Bibliographical History is also available in print, visit https://brill.com/view/serial/HCMR.

Features and Benefits
- International expert authorship
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text:
# Do Trig Ever Age?

Geometry Level 4

$2\sin{2^\circ}, 4\sin{4^\circ}, 6\sin{6^\circ}, \cdots, 180\sin{180^\circ}$

Find the average of the above sequence of numbers.

×


text:
# Tag Info

3

Joshphysics has already given a nice answer showing that in 2+1 dimensional Einstein gravity any metric is locally equivalent to a metric of constant curvature. As dilaton mentioned in a comment this in particular means that there are no local excitations. The updated question also asks about 1+1 dimensions. In this case the answer is even simpler: the 1+1 ...

6

There is nothing "wrong" with the Einstein field equations in $2+1$ as indicated by the comments, but they do have interesting, significantly restricted behavior in $2+1$ dimensions. For example, the Wikipedia page referred to by Olof in the comments says that in $2+1$, every vacuum solution is locally either $\mathbb R^{2,1}$, $\mathrm{AdS_3}$, or ...

0

Solitons are just nonlinear waves. They appear in almost any nonlinear system, similar to usual (linear) waves that characterize excitations in different systems (deformation waves, acoustic waves, electromagnetic waves). A distinguishing feature of a soliton is that it is localized in space. Usually, a soliton has a bell-shaped form (sometimes, this type is ...

0

First, what is a difference between linear and nonlinear physical processes? If a deviation of a system from an equilibrium is small, then the system is said to be linear. Formally, in this case, the system is described by a linear equation. A simple example of a linear system is a pendulum that performs small oscillations near the equilibrium (vertical) ...

Top 50 recent answers are included


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# Changes between Initial Version and Version 1 of TracRepositoryAdmin

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Timestamp:
Sep 2, 2019, 11:30:15 AM (3 years ago)
Comment:

--

### Legend:

Unmodified
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The version control systems can be enabled by adding the following to the [components] section of your [TracIni#components-section trac.ini], or enabling the components through the //Plugins// admin page. {{{#!ini tracopt.versioncontrol.svn.* = enabled }}} {{{#!ini tracopt.versioncontrol.git.* = enabled }}} == Specifying repositories #Repositories Trac supports multiple repositories per environment, and the repositories may be for different version control systems. Each repository must be defined in a repository configuration provider. Repository providers included with Trac are the [#ReposDatabase database store], the [#ReposTracIni trac.ini configuration file] and the [#ReposGitWeb GitWeb configuration file]. Additional providers are available as plugins. You can define your repositories through a mix of providers, but each repository should only be defined in a single provider. The repository names must be unique across all providers and duplicate names are discarded. It is possible to define aliases of repositories that act as "pointers" to real repositories. This can be useful when renaming a repository, to avoid breaking links to the old name. === Default Repository Trac's repositories are listed in the Repository Index when navigating to //Browse Source//. The default repository is displayed first, followed by the Repository Index. TracLinks without a repository name in the path specification (e.g. [1] rather than [1/repos1]) refer to the default repository. !TracLinks for repositories other than the default must include the repository name in the path specification. From the Repository Admin page, the default repository is specified by leaving the //Name// attribute empty. From the command line, the default repository is specified using the string (default) or "". 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Defaults to true for version control systems that support caching. || ||description ||\ ||The text specified in the description attribute is displayed below the top-level entry for the repository in the source browser. It supports WikiFormatting. || ||dir ||\ ||The dir attribute specifies the location of the repository in the filesystem. The alias and dir attributes are mutually exclusive. || ||hidden ||\ || When set to true, the repository is hidden from the repository index page in the source browser. Browsing the repository is still possible, and links referencing the repository remain valid. || ||sync_per_request||\ || When set to true the repository will be synchronized on every request (implicit synchronization). This is generally not recommended. See [#Synchronization repository synchronization] for a comparison of explicit and implicit synchronization. The attribute defaults to false. || ||type ||\ || The type attribute specifies the version control system used by the repository. Trac provides support for Subversion and Git, and plugins add support for several other systems. If type is not specified, it defaults to the value of the [versioncontrol] [wiki:TracIni#versioncontrol-default_repository_type-option default_repository_type] option. || ||url ||\ || The url attribute specifies the root URL to be used for checking out from the repository. When specified, a "Repository URL" link is added to the context navigation links in the source browser, that can be copied into the tool used for creating the working copy. || === Scoped Repository For some version control systems, it is possible to specify not only the path to the repository in the dir attribute, but also a ''scope'' within the repository. Trac will then only show information related to the files and changesets below that scope. The scope is specified by appending a path that is relative to the repository root. The Subversion backend for Trac supports this. For example, assume a repository at filesystem path /var/svn/repos1 with several directories at the root of the repository: /proj1, /proj2, etc. The following configuration would scope the repository to /proj1: {{{#!ini proj1.dir = /var/svn/repos1/proj1 proj1.type = svn }}} For other repository types, check the corresponding plugin's documentation. === In the database #ReposDatabase Repositories can also be specified in the database, using either the //Repositories// admin page under //Version Control//, or the trac-admin $ENV repository commands. The admin panel shows the list of all repositories defined in the Trac environment. It allows adding repositories and aliases, editing repository attributes and removing repositories. Note that repositories defined in trac.ini are displayed but cannot be edited. The following [wiki:TracAdmin trac-admin] commands can be used to perform repository operations from the command line. repository add [type]:: Add a repository  located at , and optionally specify its type. repository alias :: Create an alias  for the repository . repository remove :: Remove the repository . repository set :: Set the attribute  to  for the repository . Note that the default repository has an empty name, so it will need to be quoted when running trac-admin from a shell. Alternatively, the name (default) can be used instead, for example when running trac-admin in interactive mode. === In trac.ini #ReposTracIni Repositories and repository attributes can be specified in the [repositories] section of [wiki:TracIni#repositories-section trac.ini]. Every attribute consists of a key structured as {name}.{attribute} and the corresponding value separated with an equal sign (=). The name of the default repository is empty. The main advantage of specifying repositories in trac.ini is that they can be inherited from a [wiki:TracIni#GlobalConfiguration global configuration]. Cached repositories defined in trac.ini at the time of [TracEnvironment#SourceCodeRepository environment initialization] will be automatically synchronized if the repository's connector is enabled. One drawback is that due to limitations in the ConfigParser class used to parse trac.ini, the repository name is always lowercase. The following example defines two Subversion repositories named project and lib, and an alias to project as the default repository. This is a typical use case where a Trac environment previously had a single repository (the project repository), and was converted to multiple repositories. The alias ensures that links predating the change continue to resolve to the project repository. {{{#!ini [repositories] project.dir = /var/repos/project project.description = This is the ''main'' project repository. project.type = svn project.url = http://example.com/svn/project project.hidden = true lib.dir = /var/repos/lib lib.description = This is the secondary library code. lib.type = svn lib.url = http://example.com/svn/lib .alias = project }}} Note that name.alias = target makes name an alias for the target repo, not the other way around. === In !GitWeb #ReposGitWeb [https://git-scm.com/book/en/v2/Git-on-the-Server-GitWeb GitWeb] is a CGI script that comes with Git for web-based visualization of repositories. Trac can read the gitweb-formatted project.lists file. The configuration is done through the [gitweb-repositories] section of [[TracIni#gitweb-repositories-section|trac.ini]]. == Repository caching Caching improves the performance browsing the repository, viewing logs and viewing changesets. Cached repositories must be [#Synchronization synchronized], using either explicit or implicit synchronization. When searching changesets, only cached repositories are searched. Repositories that support caching are cached by default. The Subversion and Git backends support caching. The [trac:TracMercurial Mercurial plugin] does not yet support caching ([trac:#8417]). To disable caching, set the cached attribute to false. After adding a cached repository, the cache must be populated with the trac-admin$ENV repository resync command. repository resync :: Re-synchronize Trac with a repository. == Repository synchronization #Synchronization Either explicit or implicit synchronization can be used. Implicit synchronization is easier to configure, but may result in noticeably worse performance. The changeset added and modified events can't be triggered with implicit synchronization, so the [#Automaticchangesetreferencesintickets commit ticket updater] won't be available. === Explicit synchronization #ExplicitSync This is the preferred method of repository synchronization. It requires adding a call to trac-admin in the post-commit hook of each repository. Additionally, if a repository allows changing revision metadata, a call to trac-admin must be added to the post-revprop-change hook as well. changeset added [...]:: Notify Trac that one or more changesets have been added to a repository. changeset modified [...]:: Notify Trac that metadata on one or more changesets in a repository has been modified. The  argument can be either a repository name (use "(default)" for the default repository) or the path to the repository. Note that you may have to set the environment variable PYTHON_EGG_CACHE to the same value as was used for the web server configuration before calling trac-admin, if you changed it from its default location. See [wiki:TracPlugins#Settinguptheplugincache TracPlugins] for more information. ==== Subversion ===== Using trac-svn-hook In a Unix environment, the simplest way to configure explicit synchronization is by using the [trac:source:branches/1.4-stable/contrib/trac-svn-hook contrib/trac-svn-hook] script. trac-svn-hook starts trac-admin asynchronously to avoid slowing the commit and log editing operations. The script comes with a number of safety checks and usage advice. Output is written to a log file with prefix svn-hooks- in the environment log directory, which can make configuration issues easier to debug. There's no equivalent trac-svn-hook.bat for Windows yet, but the script can be run by Cygwin's bash. The documentation header of trac-svn-hook contains a Cygwin configuration example. Follow the help in the documentation header of the script to configure trac-svn-hook. You'll need to minimally set the TRAC_ENV variable, and may also need to set TRAC_PATH and TRAC_LD_LIBRARY_PATH for a non-standard installation or a virtual environment. Configuring the hook environment variables is even easier in Subversion 1.8 and later using the [http://svnbook.red-bean.com/en/1.8/svn.reposadmin.create.html#svn.reposadmin.hooks.configuration hook script environment] configuration. Rather than directly editing trac-svn-hook to set the environment variables, or exporting them from the hook that invokes trac-svn-hook, they can be configured through the repository conf/hooks-env file. Here is an example, using a Python virtual environment at /usr/local/venv: {{{#!ini [default] TRAC_ENV=/var/trac/project-1 TRAC_PATH=/usr/local/venv/bin }}} ===== Writing Your Own Hook Script The following examples are complete post-commit and post-revprop-change scripts for Subversion. They should be edited for the specific environment, marked executable (where applicable) and placed in the hooks directory of each repository. On Unix (post-commit): {{{#!sh #!/bin/sh export PYTHON_EGG_CACHE="/path/to/dir" /usr/bin/trac-admin /path/to/env changeset added "$1" "$2" }}} Adapt the path to the actual location of trac-admin. On Windows (post-commit.cmd): {{{#!bat @C:\Python26\Scripts\trac-admin.exe C:\path\to\env changeset added "%1" "%2" }}} The post-revprop-change hook for Subversion is very similar. On Unix (post-revprop-change): {{{#!sh #!/bin/sh export PYTHON_EGG_CACHE="/path/to/dir" /usr/bin/trac-admin /path/to/env changeset modified "$1" "$2" }}} On Windows (post-revprop-change.cmd): {{{#!bat @C:\Python26\Scripts\trac-admin.exe C:\path\to\env changeset modified "%1" "%2" }}} The Unix variants above assume that the user running the Subversion commit has write access to the Trac environment, which is the case in the standard configuration where both the repository and Trac are served by the web server. If you access the repository through another means, for example svn+ssh://, you may have to run trac-admin with different privileges, for example by using sudo. See the [http://svnbook.red-bean.com/en/1.7/svn.reposadmin.create.html#svn.reposadmin.create.hooks section about hooks] in the Subversion book for more information. Other repository types will require different hook configuration. ==== Git [https://git-scm.com/book/en/v2/Customizing-Git-Git-Hooks Git hooks] can be used in the same way for explicit syncing of Git repositories. If your repository is one that only gets pushed to, add the following to the hooks/post-receive file in the repo: {{{#!sh #!/bin/sh tracenv=/path/to/env  # set to your Trac environment's path repos=                # set to your repository's name while read oldrev newrev refname; do if [ "$oldrev" = 0000000000000000000000000000000000000000 ]; then git rev-list --reverse "$newrev" -- else git rev-list --reverse "$newrev" "^$oldrev" -- fi | xargs trac-admin "$tracenv" changeset added "$repos" done }}} The repos variable is the repository name (use "(default)" for the default repository). Alternatively, if your git repository is one that gets committed to directly on the machine that hosts Trac, add the following to the hooks/post-commit file in your Git repository: {{{#!sh #!/bin/sh tracenv=/path/to/env  # set to your Trac environment's path repos=                # set to your repository's name REV=$(git rev-parse HEAD) trac-admin "$tracenv" changeset added "$repos"$REV }}} The post-commit hook will do nothing if you only update the repository by pushing to it. Be sure to set the hook scripts as executable. ==== Mercurial For Mercurial, add the following entries to the .hgrc file of each repository accessed by Trac (if [trac:TracMercurial] is installed in a Trac plugins directory, download [trac:source:mercurial-plugin/tracext/hg/hooks.py hooks.py] and place it somewhere accessible): {{{#!ini [hooks] ; If mercurial-plugin is installed globally commit = python:tracext.hg.hooks.add_changesets changegroup = python:tracext.hg.hooks.add_changesets ; If mercurial-plugin is installed in a Trac plugins directory commit = python:/path/to/hooks.py:add_changesets changegroup = python:/path/to/hooks.py:add_changesets [trac] env = /path/to/env trac-admin = /path/to/trac-admin }}} === Per-request synchronization #PerRequestSync If the post-commit hooks are not available, the environment can be set up for per-request synchronization. The sync_per_request attribute for each repository in the database and in [wiki:TracIni#trac-section trac.ini] must be set to true. Note that in this case, the changeset listener extension point is not called, and therefore plugins that depend on the changeset added and modified events won't work correctly. For example, automatic changeset references cannot be used with implicit synchronization. == Automatic changeset references in tickets #CommitTicketUpdater You can automatically add a reference to the changeset as a ticket comment whenever changes are committed to the repository. The description of the commit needs to contain one of the following patterns: * '''Refs #123''' - to reference this changeset in #123 ticket * '''Fixes #123''' - to reference this changeset and close #123 ticket with the default status ''fixed'' This functionality requires installing a post-commit hook as described in [#ExplicitSync explicit synchronization], and enabling the optional commit updater components through the //Plugins// admin panel or by adding the following line to the [components] section of your [wiki:TracIni#components-section trac.ini]: {{{#!ini tracopt.ticket.commit_updater.* = enabled }}} For more information, see the documentation of the CommitTicketUpdater component in the //Plugins// admin panel and the [trac:CommitTicketUpdater] page. == Troubleshooting === My trac-post-commit-hook doesn't work anymore #trac-post-commit-hook You must now use the optional components from tracopt.ticket.commit_updater.*, which you can activate through the //Plugins// admin page, or by directly modifying the [TracIni#components-section "[components]"] section in the trac.ini. Be sure to use [#ExplicitSync explicit synchronization]. See [trac:CommitTicketUpdater#Troubleshooting] for more troubleshooting tips. === Git control files missing If your repository is not browseable and you find a message in the log that looks like: {{{ 2017-08-08 10:49:17,339 Trac[PyGIT] ERROR: GIT control files missing in '/path/to/git-repository' 2017-08-08 10:49:17,339 Trac[git_fs] ERROR: GitError: GIT control files not found, maybe wrong directory? }}} First check that the path to your repository is correct. If the path is correct, you may have a permission problem whereby the web server cannot access the repository. You can use Git to verify the repository. On a Debian-like Linux OS, the following command should help: {{{#!sh \$ sudo -u www-data git --git-dir=/path/to/git-repository fsck }}} On other platforms you may need to modify the command to use the user under which the webserver runs.


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## concept

$\frac{d[R]}{dt}=-k[R]; \ln [R]=-kt + \ln [R]_{0}; t_{\frac{1}{2}}=\frac{0.693}{k}$

CaminaB_1D
Posts: 63
Joined: Fri Sep 28, 2018 12:16 am

### concept

Is there a conceptual meaning behind a second order reaction?

Reva Kakaria 1J
Posts: 62
Joined: Fri Sep 28, 2018 12:23 am
Been upvoted: 1 time

### Re: concept

A second order reaction means that the reaction rate is proportional to either the square of the concentration of one reactant, or to the concentrations of two reactants multiplied together; so two molecules have to collide for the reaction.

mcredi
Posts: 63
Joined: Fri Sep 28, 2018 12:16 am

### Re: concept

A second order reaction is a reaction where x + y = 2. This can happen if one reactant is consumed at a rate proportional to the square of the reactant's concentration (rate = k[A]2) or both reactants are consumed linearly over time (rate = k[A][B]).


text:
Note

These slides are also available in PDF format: 4:3 PDF, 16:9 PDF, 16:10 PDF.

## Learning Group Activity¶

Spend 10 minutes comparing your code from the LGA with your learning groups’ code. How did your implementations differ? Share anything cool you did too!

Haskell uses inferred static typing. However, you may specify the types of a function if you wish.

To declare the type of a function in Haskell, use the following notation:

double :: Int -> Int
double x = x * 2

double is a function which takes an Int and returns an Int.

## Review: Currying¶

• Haskell takes advantage of currying to support functions with multiple arguments. That is, functions take a single argument and return a function ready to take the next argument.
• We call the function ready to take the next argument a partially applied function.

## Multiple Arguments¶

Declaring types of a function gets a little more tricky once we involve currying:

gcd :: Int -> Int -> Int
gcd a b = ...

gcd is a function which takes an Int, and returns a function that takes an Int and returns an Int.

## Lists in Type Specifiers¶

To specify a list of some type, surround the type in square brackets. For example:

frobnicate :: Int -> [Char] -> [Char]
frobnicate n st = ...

However, this might be more cleanly written as:

frobnicate :: Int -> String -> String
frobnicate n st = ...

as String is just a synonym for [Char].

## Type Variables¶

Haskell has a cool little trick: type variables. Consider this type specifier for the head function:

head :: [a] -> a

head takes a list of some type a, and returns something of type a.

This makes sense. head can be preformed on lists of any type, and we can guarantee it will give us something of the type the list is of.

## Typeclasses¶

Suppose that we wanted to specify restrictions on what our type variables can be. Haskell features typeclasses for this. For example, what if we wanted to declare the type of a sort function:

sort :: (Ord a) => [a] -> [a]

sort takes a list of type a, and returns a list of type a, so as long as the type a is something that can be ordered.

## More Typeclasses¶

• Num a: a is numeric
• Eq a: a can be tested for equality
• Integral a: a is an integer (either bounded or unbounded)
• More examples in LYH.

## Querying GHC for Inferred Type¶

Wondering what GHC guessed? Use :t in GHCi to check the type of a function:

GHCi> :t fst
fst :: (a, b) -> a

## Quiz Study Time¶

1. As a learning group, create a quiz that you think I could have potentially written for you today.
2. When done, trade quizzes with another learning group and complete their quiz.

I am also happy to answer any questions or go over old slides during this time. Plan is to start the real quiz in about 15 to 20 minutes.