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A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations. (English) Zbl 1185.65139
Summary: Based on the homotopy analysis method (HAM), a powerful algorithm is developed for the solution of nonlinear ordinary differential equations of fractional order. The proposed algorithm presents the procedure of constructing the set of base functions and gives the high-order deformation equation in a simple form. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter $\hslash$. The analysis is accompanied by numerical examples. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.
##### MSC:
 65L99 Numerical methods for ODE 26A33 Fractional derivatives and integrals (real functions) 34A08 Fractional differential equations


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Now showing items 546-565 of 683

• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2019)
We report on quasar outflow properties revealed by analyzing more than 60 composite outflow spectra built from ~60,000 C iv absorption troughs in the SDSS-III/BOSS DR12QBAL catalog. We assess the dependences of the equivalent ...
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2017)
Ellerman bombs (EBs) are brightenings in the Hα line wings that are believed to be caused by magnetic reconnection in the lower atmosphere. To study the response and evolution of the chromospheric line profiles, we perform ...
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2019)
During a solar flare, it is believed that reconnection takes place in the corona followed by fast energy transport to the chromosphere. The resulting intense heating strongly disturbs the chromospheric structure and induces ...
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2016)
We describe a new quiet-Sun phenomenon which we call quiet-Sun Ellerman-like brightenings (QSEB). QSEBs are similar to Ellerman bombs (EB) in some respects but differ significantly in others. EBs are transient brightenings ...
• (Master thesis / Hovedoppgave, 2005)
During inflation, the potential of the inflaton scalar field $\phi$, drives the exponential expansion of the Universe. The nature of the inflaton potential is presently unknown, but one hopes to learn more from observations ...
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2019)
Inversion codes allow the reconstruction of a model atmosphere from observations. With the inclusion of optically thick lines that form in the solar chromosphere, such modeling is computationally very expensive because a ...
• (Master thesis / Masteroppgave, 2016)
Recent observations support a significantly reduced visibility of Lyman-α (Lyα) flux from galaxies at redshifts z > 6. This is surprising considering that the visibility of Lyα flux has been observed to increase up towards ...
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2018)
We present, for the first time, numerical simulations of young tidal dwarf galaxies (TDGs), including a self-consistent treatment of the tidal arm in which they are embedded. Thereby, we do not rely on idealized initial ...
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2013)
Several extensions of general relativity and high energy physics include scalar fields as extra degrees of freedom. In the search for predictions in the nonlinear regime of cosmological evolution, the community makes use ...
• (Series / Serietittel, 1950)
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2019)
Aims. We investigate the transfer of energy between a fundamental standing kink mode and azimuthal Alfvén waves within an expanding coronal magnetic flux tube. We consider the process of resonant absorption in a loop with ...
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2015)
Transverse magnetohydrodynamic waves have been shown to be ubiquitous in the solar atmosphere and can, in principle, carry sufficient energy to generate and maintain the Sun's million-degree outer atmosphere or corona. ...
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2017)
Motivated by observational and theoretical work that suggest very small-scale (≲ 1 pc) structure in the circumgalactic medium of galaxies and in other environments, we study Lyman-α (Lyα) radiative transfer in an extremely ...
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2018)
The baryonic content and physical properties of the warm and hot (105 lesssim T lesssim 107 K) phases of the circumgalactic medium (CGM) are poorly constrained owing to the lack of observables probing the requisite range ...
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2015)
The emergence of three-dimensional magneto-hydrodynamic simulations of stellar atmospheres has sparked a need for efficient radiative transfer codes to calculate detailed synthetic spectra. We present RH 1.5D, a massively ...
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2020)
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2020)
We present a new code,SCALAR, based on the high-resolution hydrodynamics andN-body codeRAMSES, to solve the Schrödingerequation on adaptive refined meshes. The code is intended to be used to simulate axion ...
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2018)
We investigate the efficiency of screening mechanisms in the hybrid metric-Palatini gravity. The value of the field is computed around spherical bodies embedded in a background of constant density. We find a thin shell ...
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2012)
It is shown that extensions to general relativity, which introduce a strongly coupled scalar field, can be viable if the interaction has a nonconformal form. Such disformal coupling depends upon the gradients of the scalar ...
• (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2019)
We present a formalism to study screening mechanisms in modified theories of gravity through perturbative methods in different cosmological scenarios. We consider Einstein-frame posed theories that are recast as Jordan-frame ...

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# To find average speed of gas molecule when the temperature is incresed

We are given moles of gas let say $n$. Temperature is increased from $T_1$ to $T_2$. Volume is held constant at $V_0$. We need to find the average speed of gas molecule when the temperature $T_2$ is achieved.

I am confused to use $v= \sqrt{\dfrac{3k_BT}{m}}$ or to use $v= \sqrt{\dfrac{8k_BT}{\pi m}}$ ; we are given monatomic gas having atomic $m = 6.4 \times 10^{-25} \:\rm kg$.

Please explain me which formula is applicable here and why.

• Welcome to Physics Stack Exchange! Please note that Physics.StackExchange is not a homework help site. Please see this Meta post on asking homework questions. If you choose to modify your question, please tell us what "Pim" is. Apr 15 '15 at 16:29
• @user36790 Nice work, I didnt understand anything before the edit Apr 15 '15 at 18:47

The distribution of speeds in an ideal gas is given by the Maxwell-Boltzmann distribution. There are a variety of average speeds e.g. the most probable speed, the mean speed and the root mean square speed. Which one you use will depend on the application. The two equations you give are for the RMS speed:

$$\sqrt{\langle v^2 \rangle} = \sqrt{\frac{3kT}{m}}$$

and the mean speed:

$$\langle v \rangle = \sqrt{\frac{8kT}{\pi m}}$$

The question doesn't make it clear what speed is required, but I would guess that it is the mean.

• I am asked to find the average speed of a gas molecule after the final temperature is achieved. Even I am confused to take which one of these? Apr 15 '15 at 17:15
• RMS seems to me more like variation, not the average... Apr 15 '15 at 18:46
• @jaromrax: no, root mean square is a commonly used average. I wonder if you're mixing it up with the standard deviation. Apr 15 '15 at 19:24
• @JohnRennie - well, I dont want to argue with you, ($\sqrt{v^2}$ really resembles $v$), I just feel uncomfortable when I look at $root$ histogram and I see mean and RMS fields with a bit different meaning...that was my point... Apr 15 '15 at 20:48
• @jaromrax - actually the RMS will tell you something about the mean energy of the atoms, since each atom has energy proportional to $v^2$. If I have one atom with v=0 m/s and another with v=2 m/s, the RMS value is $\sqrt{2}$ m/s but the mean velocity is 1 m/s. You get the same mean when both velocities are 1 m/s, but then you would get a smaller RMS velocity (also 1 m/s) - reflecting that the energy of the 0+2 is greater than the 1+1. May 8 '15 at 18:30


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# What is the protocol or etiquette for pinging users in Ten Fold?

CV gives practically unfettered access to renowned experts in the field in a democratic setting that would have been unthinkable in the not so distant past. Incredibly, in many cases, this is even achieved anonymously, using avatars and pen names. Personally, I try to show appreciation for this incredible gift.

On the other hand, and in the frenzy to learn and participate, it is easy to lose perspective of who is who in the real world, where texting each other would be a preposterous thought.

In this context, I wonder if it is OK to ping any user at any time within Ten Fold (as the most commonly used chat room), or if the fact that it is a chat room makes it more personal than the "open forum", and considerations such as time of the day, personal rapport with the person based on prior interactions, or status of the user being pinged in the profession should be more strongly weighted than for a regular comment at the bottom of a post.

• FYI I believe it only works when the user pinged has visited the chat room not too long ago. – Scortchi - Reinstate Monica Oct 4 '16 at 13:48
• @Scortchi I was not aware of the "not too long ago" part. – amoeba says Reinstate Monica Oct 4 '16 at 14:00
• I don't think we need rules for everything. If someone wants to communicate that way, they'll reply; if not, they'll ignore you or ask you to ask a question directly or communicate otherwise. More importantly to me, I don't mind too much one-to-one conversations, but the purpose of chat in my view is to supplement the main site, not act as a medium for personal exchanges better carried out in some other way (e.g. email). A little related, but I am not keen on people who I've answered on the forum contacting me privately for further help (yes, I know, I should make myself less identifiable). – Nick Cox Oct 4 '16 at 14:37
• Otherwise put, a positive principle for chat is that you should be able to imagine that whatever you say could be of interest to others, even if it is only as a joke or a grumble. That principle makes marginal some exchanges of the form: Dear X: I would value your advice on Y. – Nick Cox Oct 4 '16 at 14:39
• @NickCox Your views are always insightful, and I couldn't agree more with the danger of stifling spontaneity. Please note my emphasis on etiquette. The question is more about using the resources on the site at large with propriety... I missed initially the second part of your comment. What do you mean by 'makes marginal'? – Antoni Parellada Oct 4 '16 at 14:42
• @AntoniParellada Thanks. Here is a blunter (or perhaps sharper) version. I don't want to see others' essentially private conversations as part of CV chat. – Nick Cox Oct 4 '16 at 14:57
• @amoeba: I recollected hearing of some recency criterion: here it's said to be 48 hours. – Scortchi - Reinstate Monica Oct 5 '16 at 8:29

You may be worrying more than you need.

I think it's within the rules of the site to ping absolutely anyone -- but not everyone would see the ping.

However what's within the rules is not always the same as what people might see as reasonable. I think it would generally be reasonable to:

a) ping a user that you have a specific reason to contact (like an issue with a question or answer you want to raise with them beyond what would be suitable for comments -- such as extended discussion). Given your judgement of what the question was about, your ping should arguably fall under this category.

b) ping a user you have interacted with regularly, for any reasonable purpose consistent with your personal relationship and the (fairly lax) chat rules (this is really a personal judgement about individual interactions so it's hard to say much ... and this is what I think you're asking about. We can't really establish a policy on individual judgement about the social suitability of a ping)

c) ping any user who is a regular in chat, for any purpose reasonably related to what the site is for, unless they have indicated otherwise. (If you're a chat regular I think it's reasonable to expect to be pinged there)

• Because $T^-$ (meaning "test negative") was read by me as $TN$ ("true negative"), I felt a sense of urgency to correct what (under the light of this misunderstanding) seemed like an outrageous ("accepted") mistake, reflecting poorly on CV, precisely because of its basic nature. And I couldn't think of a more authoritative voice to correct an issue on epidemiology than Professor Harrell. – Antoni Parellada Oct 4 '16 at 22:24
• @Antoni Yes, I understand this in retrospect (I was talking about it in my answer). In that light it made some sense to do what you did, but I didn't interpret the situation that way at the time. That's the different understanding of the circumstances I was discussing above. – Glen_b -Reinstate Monica Oct 4 '16 at 22:28
• If you want to delete this question, I have no problem. If you can't, because my answer has an upvote I can delete my answer. (If the upvote is yours you may be able to remove it and delete on your own). Or it can stay, if you prefer, I don't mind either way. – Glen_b -Reinstate Monica Oct 4 '16 at 22:30


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Confining solitons in the Higgs phase of {\mathbb{C}}P^{N-1} model: Self-consistent exact solutions in large-N limit

Confining solitons in the Higgs phase of CPN−1 model: Self-consistent exact solutions in large-N limit

Abstract

The quantum model is in the confining (or unbroken) phase with a full mass gap in an infinite space, while it is in the Higgs (broken or deconfinement) phase accompanied with Nambu-Goldstone modes in a finite space such as a ring or finite interval smaller than a certain critical size. We find a new self-consistent exact solution describing a soliton in the Higgs phase of the model in the large- limit on a ring. We call it a confining soliton. We show that all eigenmodes have real and positive energy and thus it is stable.

1 Introduction

The two-dimensional model Eichenherr:1978qa (); Golo:1978dd (); Cremmer:1978bh () shares many common non-perturbative phenomena with four dimensional Yang-Mills theory, such as confinement, asymptotic freedom, dynamical mass generation, and instantons Polyakov:1975rr (); Polyakov:1975yp (); Bardeen:1976zh (); Brezin:1976qa (); DAdda:1978vbw (); DAdda:1978etr (); Witten:1978bc (); Novikov:1984ac (). The two-dimensional Gross-Neveu (GN) Gross:1974jv () or Nambu Jona-Lasino (NJL) model Nambu:1961tp () exhibits dynamical mass gap and dynamical chiral symmetry breaking, with sharing these properties with four dimensional QCD. These two-dimensional theories, one consisting of bosons with nonlinear interaction and the other describing fermions with a four Fermi interaction, have been well studied in the large- or mean field approximation (where is the number of flavors for the GN model), as toy models to study non-perturbative effects in four-dimensional gauge theories.

The model is in the confining (or unbroken) phase with a mass gap when it is defined in an infinite space, in agreement with the Coleman-Mermin-Wagner (CMW) theorem prohibiting a spontaneous symmetry breaking or a long-range order in 1+1D systems Coleman:1973ci (); Mermin:1966fe (). While this can be well shown in the large- or mean field approximation in general, the mass gap can be exactly shown by using the mirror symmetry Hori:2000kt () in the case of the supersymmetric model Witten:1977xn (); DiVecchia:1977nxl (). On the other hand, the Higgs (broken or deconfinement) phase accompanied with Nambu-Goldstone (NG) modes is possible when the is defined in a finite system; Recently phase transitions between the confining and Higgs phases in finite systems have been discovered in the model on a ring Monin:2015xwa (); Monin:2016vah () and on a finite interval Milekhin:2012ca (); Milekhin:2016fai (); Pavshinkin:2017kwz (). In these cases, the Higgs phase is favored when the system size is smaller than some critical size. It is consistent with the CMW theorem because, in a finite size, this phase does not have a long range order correlation inhibited by the theorem.

The GN model is in the broken phase in the large- or mean field approximation, in which the order parameter is constant and the symmetry is dynamically broken. The chiral GN (or NJL) model, having a complex order parameter, is in the broken phase (or Bardeen-Cooper-Schrieffer (BCS) phase in the context of superconductivity) in the ground state in the infinite system in the large- limit. Though one might think this phase to be incompatible with the CMW theorem because of symmetry breaking, it was shown to be compatible with the CMW theorem in Ref. Witten:1978qu ().

The above arguments only consider homogeneous configurations. Although it is usually the case that a constant condensation is the lowest energy state in a homogeneous system without any external field, inhomogeneous condensations may appear as the lowest energy states in certain situations. For instance, a twisted boundary condition unavoidably make a configuration inhomogeneous, e.g., the 1D Ising model with the anti-parallel spin orientation on the opposite boundary. Many examples of inhomogeneous configurations have been known for long time in the (chiral) GN model, corresponding to conducting polymers or superconductors in physical systems. For instance, a phase twist on a superconductor results in the Fulde-Ferrell (FF) state Fulde:1964zz (), in which the order parameter is plane-wave like. Another possibility is the presence of the external fields; In the case of superconductors, an applied magnetic field induces the spin excess by the Zeeman effect and thus the order parameter becomes nodal Machida:1984zz (), which is known as the Larkin-Ovchinnikov (LO) state Larkin:1964zz () or a real kink crystal. A self-consistent exact solution for it was obtained in Ref. Brazovskii (). Those self-consistent solutions were generalized to a system with a spin imbalance Yoshii:2011yt (); Takahashi:2012aw (). A self-consistent analytic solution of the FFLO state or a twisted kink crystal was found in Ref. Basar:2008im (), in which both the amplitude and phase of the order parameter modulate. A superconducting ring with a Zeeman magnetic field and magnetic flux penetrating the ring exhibits similar configurations Yoshii:2014fwa (). Inhomogeneous configurations also appear in a finite interval with the Dirichlet boundary condition Flachi:2017cdo (). Apart from inhomogeneous ground states, solitons as excited states have been constructed as self-consistent analytic solutions, such as a kink Dashen:1975xh (); Takayama:1980zz (), kink-anti-kink Dashen:1975xh (); Campbell:1981dc (), kink-anti-kink-kink Okuno:1983 (); Feinberg:1996gz (); Feinberg:2002nq () for the real GN model, and twisted kink Shei:1976mn () and multiple twisted kinks with arbitrary separations Takahashi:2012pk () in the chiral GN model.

On the other hand, inhomogeneous or soliton solutions for the quantum model had not been discussed until recently, while various classical solitons in the model have been known for long time, such as kinks (domain walls) Abraham:1992vb (); Gauntlett:2000ib (); Isozumi:2004jc (), lumps (instantons) Polyakov:1975yp (), and their composites Gauntlett:2000de (); Isozumi:2004vg (). In the case of a finite interval with the Dirichlet boundary condition, it is inevitable that the Higgs field and mass gap function are inhomogeneous Bolognesi:2016zjp (); Betti:2017zcm (); Flachi:2017xat (); Nitta:2018lnn (); Bolognesi:2018njt (). Self-consistent analytic solutions for this case was obtained in Refs. Flachi:2017xat (); Nitta:2018lnn (). A similar problem in a two-dimensional disk was discussed in Refs. Gorsky:2013rpa (); Pikalov:2017lrb ().

The hidden connection between the and GN models has been recently revealed; the self-consistent equations obtained in the leading order of the large- limit for the both theories coincide Nitta:2017uog (). This correspondence provides a map from solutions in the GN model to those in the model, and by using it, we have constructed several self-consistent inhomogeneous solutions of the model as summarized in Table 1. One is a self-consistent soliton solution in the confining phase of the model in an infinite space or on a ring Nitta:2017uog (). This soliton is stable in the linear order because all eigenmodes have real and non-negative energy. The Higgs phase with NG modes is localized inside the soliton, thereby called the Higgs soliton. Multiple Higgs solitons with arbitrary separation have also been obtained. The twisted boundary condition results in the flavor rotating solutions in the model Nitta:2018lnn () (see also Bolognesi:2018njt ()).

A natural question arising immediately is whether any soliton can exist in the Higgs phase of the model. In this paper, we construct self-consistent analytical soliton solutions of a kink-shape or kink-anti kink crystal shape in the Higgs phase of the model. The solution is LO-like in the sense that the amplitude of the Higgs field modulates. We call them confining solitons since the confining phase is localized inside the solitons. Confining solitons are stable at least in the linear order; all eigenmodes are real and non-negative. While classical solutions such as kinks (domain walls), lumps (instantons) and their composites were known in the Higgs phase in the model, our new solitons in this paper are the first examples of quantum solitons in the Higgs phase of the model.

Since the 1+1D model appears on a non-Abelian vortex string in 3+1D (supersymmetric) gauge theory Hanany:2003hp (); Auzzi:2003fs (); Eto:2005yh (); Tong:2005un (); Eto:2006pg (); Shifman:2007ce (); Tong:2008qd (); Hanany:2004ea (); Shifman:2004dr () or dense QCD (for ) Nakano:2007dr (); Eto:2013hoa (), the above studies of quantum phases of the model describe quantum vortex states. The model on a ring and interval describe a closed string and open string stretched between monopoles or domain walls, respectively.

The sigma model which is equivalent to the model also appears in the condensed matter physics. It is the continuous model of the Heisenberg anti-ferromagnetic spin chain Haldane:1982rj (); Affleck (), and the quantum phase transition, so-called deconfined criticality, is proposed in the anti-ferromagnetic system Senthil (); Nogueira:2013oza (). Our solution may be relevant to this case.

This paper is organized as follows. In Sec. 2, we briefly review our method to obtain self-consistent solutions. In Sec. 3, a kink solution is introduced and the existence of the solution in finite systems is discussed. In Sec. 4, the solution in Sec. 3 is generalized to a kink crystal solution. Finally we summarize the results of our paper and discuss future problems in Sec. 5.

2 Model and method

The action of the model is given by

 S=∫dtdx[(Dμni)∗(Dμni)−λ(n∗ini−r)], (1)

where the first term describes the kinetic term of complex scalar fields () with the covariant derivative , and the second term gives the constraint by the Lagrange multiplier .

When this model is considered as the effective theory of a non-Abelian vortex in gauge theory, the “radius” can be written as , where is a coupling constant in the Yang-Mills theory.

Here we note that the gauge field in this model is auxiliary (no kinetic term) and thus we set in the following. One can obtain the effective action for by integrating out the rests ():

 Seff=∫dtdx[(N−1)Trln(−∂μ∂μ+λ)+∂μσ∂μσ−λ(σ2−r)]. (2)

The total energy functional can be written as

 E=(N−1)∑nωn+∫dx[(∂xσ)2+λ(σ2−r)]. (3)

The stationary conditions for and , respectively, yield Bolognesi:2016zjp ()

 N−12∑n|fn|2ωn+σ2−r=0, (4) (−∂2x+λ)σ=0, (5)

where and are given as orthonormal eigenstates and eigenvalues of the following equation

 (−∂2x+λ)fn(x)=ω2nfn(x). (6)

Thus we end up with three Eqs. (4)–(6). We here note that Eqs. (5) is nothing but the zero mode equation of Eq. (6).

In the previous paper Nitta:2017uog (), we have found a map from above self-consistent equations to the gap equation and eigenvalue equation for the GN model. We have introduced the auxiliary field as

 Δ2+∂xΔ=λ(x), (7)

and find the solution for Eq. (5),

 σ=Aexp[∫xdyΔ(y)], (8)

with is the integral constant which must be determined from Eq. (4).

The energy functional Eq. (3) becomes

 Etot=N∑nωn−r∫∞−∞dxλ(x)+σ∂xσ|∞−∞. (9)

The important point is that the real solution (order parameter) for the GN model automatically satisfies the self-consistent equations in the model. Thanks to this map we have found the various new solutions which are inhomogeneous in the space. In the present paper, we find a new solution with kink crystal Higgs profiles and also discuss a possible setup in which the new solution can exist.

3 Confining soliton in the Higgs phase

As we already mentioned, the homogeneous confining and Higgs phases were known in the model. Those solutions are mapped from (BCS phase) and (normal phase), respectively in the GN model. In addition, we have found the following inhomogeneous solutions (the corresponding solution in the GN model are denoted in the brackets in the following, see also Table 1); localized Higgs soliton (kink solution) and the localized Higgs soliton lattice solution (kink crystal solution), two Higgs soliton solution with the arbitrary separation (kink-anti-kink-kink solution), and the “confining” solution (BCS-like solution) and the “Higgs” solution (normal-like solution) in a finite interval.

We start from a brief review for the homogeneous Higgs phase in a periodic system with the length . This phase corresponds to and thus

 λ=0, σ=A. (10)

In this case, the eigenfunctions and eigenvalues are, respectively, given as

 fk(x)=√1Leikx, k=2πnL, (11)

where is an integer. By substituting Eqs. (10) and (11) into the gap equation (4), we obtain

 N−12L∑n=1L2πn+A2−r=0. (12)

This equation allows the normalized zero mode with renormalized coupling constant . After the continuous approximation, the situation becomes more clearer. By using the replacement , one obtains

 N−14π∫ΛUV2π/Ldk1k+A2−r=N−14πlnΛUV2π/L+A2−r=0. (13)

This shows that the size introduces a low-energy cutoff and the solution is forbidden in the infinite size limit since must diverges as in the limit.

Now we move to inhomogeneous solutions. A candidate of a new solution would be the following

 σ=Atanhmx2,λ=−m22cosh2mx/2. (14)

In Fig. 1, we plot the Higgs field configuration and the corresponding mass gap function of the configuration in Eq. (14).

The Higgs field exhibits a kink profile for this solution, whereas the mass gap function has a dip at the node of the kink. This configuration is obtained through the map in the last section from a singular solution

 Δ=m/sinhmx (15)

in the GN model. While the singular point is present at in Eq. (15) in the GN model, the configuration in Eq. (14) in the model is regular everywhere. When we restrict Eq. (15) to the region, it is a solution on a half infinite line with the Dirichlet boundary condition, which reduces to the normal phase in the limit of the boundary sent to the infinity .

Though the mass gap function in Eq. (14) takes negative values everywhere and seems to be problematic, unstable modes are absent. In fact, the above mass gap function is called the Pöschl-Teller potential and all the energy spectra for this potential are known to be positive PoschlTeller (); Lekner (). The eigenfunctions are explicitly given as

 fk=1√24√k2+m2eikmx/2(ik−mtanhmx/2), (16)

where the eigenvalues are given as

 ωk=k/2,k>0. (17)

A set of configurations in Eqs. (14) and (16) satisfies Eqs. (5) and (6).

For the map between the and GN models, we use Eq. (4) differentiated with respect to instead of Eq. (4) itself. Thus the left hand side of Eq. (4) can be shown to be constant, but the map of the solution in the GN model does not necessary yield a solution of the model.1 For the present case, the first term in the left hand side of the gap Eq. (4) becomes

 const×∫ΛUVΛIR1kk2+m2tanh2mx/2k2+m2, (18)

where and are the ultraviolet and infrared cutoff, respectively. In the case of the infinite system, and the gap equation cannot be satisfied, which is consistent with the CMW theorem. However if we consider a finite system with the length the infrared cutoff because the quantization of the energy levels are introduced, and thus a solution can exist. For instance, a ring of the length with anti-periodic boundary condition in which the Higgs field changes the sign ( rotation) allows a solution, as discussed in the next section.

For this configuration all eigenmodes have real and non-negative energy, and thus the solution is energetically stable in the linear order. The minimum energy state is homogeneous solution as follows. The energy difference between the homogeneous Higgs solution with the confining soliton and the homogeneous Higgs phase can be calculated as

 Ekink−EHiggs=−r∫dx(−m22cosh2mx/2)=rm2. (19)

Here we have used the fact that the energy spectra for both solutions are the same and thus the energy difference only comes from the second term in Eq. (9).

Here we note a similarity and difference from the Higgs soliton solution in the confining phase, which we obtained before. In the Higgs soliton solution Nitta:2017uog ()

 σ=Acoshmx, λ=m2(1−21cosh2mx), (20)

the localized Higgs field appears in the confining background. The corresponding mass gap function has a dip at the peak of the Higgs field in the background constant mass gap. The energy spectra and the mass gap parameter coincide with those of the homogeneous confining phase.

On the other hand, in the present solution, the localized confining part () or localized confining soliton appears in the Higgs phase. Thus we call this solution as a confining soliton. The mass gap function has a dip around the confining soliton and it becomes zero far away from the soliton. The energy spectra of the confining soliton solution coincide with those of the homogeneous Higgs phase. The confining soliton solution contains the mass parameter , although the homogeneous Higgs phase does not have it.

It may be interesting to note that this solution is similar to the kink solution in the GN model Dashen:1975xh (); Takayama:1980zz (); the order parameter for the kink solution is also given by the hyperbolic tangent . However, this coincidence is no longer valid in the soliton lattice case as we will see in the next section.

4 Confining soliton lattice solution

In the last section, we have considered the single confining soliton. This configuration is a special case of the following solution

 σ=ACSL−dn(mx,ν)+1sn(mx,ν), λ=cn2(mx,ν)−dn(mx,ν)sn2(mx,ν), (21)

where are Jacobi’s elliptic functions, is the elliptic parameter, and is a normalization constant. Here the is determined by the gap equation (4) as

 A2CSL=s(x)−2(r−N−12∑n|fCSLn|2ωn), (22)

where and is the eigenfunctions obtained from Eq. (5) for the current solution. The mapping between the GN and models ensures that the left hand side of Eq. (22) becomes a constant. Though it can be fixed by a numerical summation, it is difficult to give an explicit form of the normalization factor for the present solution. In Fig. 2, we plot the Higgs fields and the mass gap functions for various parameters.

In this configuration, the Higgs field exhibits a kink crystal structure and the mass gap function does a soliton lattice structure. This configuration describes a confining soliton lattice in the Higgs phase. The periodicity of the solution is given by the complete elliptic integral of the first kind,

 l=2K(ν). (23)

The single soliton configuration presented in the last section is corresponding to the case of . For this solution, we need to be careful since the energy spectra are not always positive in the infinite system. For instance, if we consider the limit, the mass gap function becomes (Fig. 3). This means that the energy spectra start from which does not cause unstable modes when the size of the system is smaller than , since the system with the length has the lowest kinetic energy . This means that the system with the size cannot have larger than . The condition for the existence of the stable solution is the same with one for the existence of the homogeneous Higgs solution in the cases of a ring Monin:2015xwa (); Monin:2016vah () and a finite interval Milekhin:2012ca (); Milekhin:2016fai (); Pavshinkin:2017kwz ().

Let us make a comment on the similarity between solutions in the and GN models. The solution has an amplitude modulation of the Higgs field and thus it can be considered as a counterpart of the LO solution in the GN model, although the solution for the soliton lattice solution is not identical to the LO solution Brazovskii () in the GN model. As we saw in the last section, only in the single kink limit, those functions coincide.

Using the configuration in Eq. (21), we can construct a self-consistent analytic solutions on a ring. Here we consider (a) periodic or (b) anti-periodic boundary conditions.

(a) For the case of the periodic boundary condition, one can satisfy the condition by choosing such that the () to be the size of the ring. In this case, the homogeneous Higgs solution also satisfies the same boundary condition.

(b) For the case of the anti-periodic boundary condition, one needs to choose to satisfy with . In this case the homogeneous Higgs solution with the flavor rotation Nitta:2018lnn () can also be the solution of the same boundary condition.

5 Summary

In this paper, we have constructed the self-consistent localized confining soliton solution in the Higgs phase of the model. While this solution is inhibited in the infinite system as the homogeneous Higgs phase is inhibited in the same limit, it is allowed in finite systems due to the existence of the infrared cutoff . Our solutions are stable since the all eigenmodes have real and positive energy. We have generalized the kink solution to the case of the soliton lattice.

Our present calculation relies on the large- limit, however the resulting self-consistent equations are the same with the equations obtained by the mean field approximation. Thus we expect that our solution is qualitatively reliable even for the case of .

Throughout the present paper, we have discussed the model, but the same analysis is applicable to the model. Thus the present solutions are expected to appear in the spin system.

Although our solutions are stable at linear order, the minimum energy state should be the homogeneous Higgs phase. It remains an interesting future problem to look for conditions that the confining soliton lattice becomes the minimum energy state. For instance, if we consider the anti-periodic boundary condition, solutions satisfying the same boundary condition known so far are the confining soliton in this paper and the flavor rotating solutions obtained in the previous paper. Suppose we have an anisotropy which makes Higgs field tend to align in a specific direction, we expect that the confining kink solution is favored since the flavor rotating solution costs more energy to have the components not aligned to the selected direction. Another possibility could be the presence of a defect. For example, if the Higgs fields are obliged to vanish at the defect, the kink solution might be chosen energetically. We leave this for the future problem.

Our solution may be relevant for a gauge field localization in the brane world scenario, see, e.g., Refs. Dvali:1996xe (); Ohta:2010fu (). This is because gauge field is absent outside the soliton in the Higgs phase while it is present as a composite gauge boson inside the soliton. In order to make this more realistic we have to introduce more dimensions orthogonal to the soliton to yield it the world-volume directions.

The inhomogeneous solutions obtained so far are given by the /GN correspondence, in which we can use only real condensates of the real GN model. Thus, a possible generalization should be the case of the chiral GN model which has the complex order parameters. We also leave this for the future problem.

Acknowledgement

The support of the Ministry of Education, Culture, Sports, Science (MEXT)-Supported Program for the Strategic Research Foundation at Private Universities ‘Topological Science’ (Grant No. S1511006) is gratefully acknowledged. The work of M. N. is supported in part by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (KAKENHI Grant No. 16H03984 and 18H01217) and by a Grant-in-Aid for Scientific Research on Innovative Areas “Topological Materials Science” (KAKENHI Grant No. 15H05855) from the MEXT of Japan.

Footnotes

1. In fact the homogeneous Higgs solution is known to be inappropriate as the solution in the infinite system, though the corresponding solution so-called normal phase exists in the GN model.

References

1. H. Eichenherr, “SU(N) Invariant Nonlinear Sigma Models,” Nucl. Phys. B 146, 215 (1978).
2. V. L. Golo and A. M. Perelomov, “Few Remarks On Chiral Theories With Sophisticated Topology,” Lett. Math. Phys. 2, 477 (1978); “Solution of the Duality Equations for the Two-Dimensional SU(N) Invariant Chiral Model,” Phys. Lett. 79B, 112 (1978).
3. E. Cremmer and J. Scherk, “The Supersymmetric Nonlinear Sigma Model in Four-Dimensions and Its Coupling to Supergravity,” Phys. Lett. 74B, 341 (1978).
4. A. M. Polyakov, “Interaction of Goldstone Particles in Two-Dimensions. Applications to Ferromagnets and Massive Yang-Mills Fields,” Phys. Lett. 59B, 79 (1975).
5. A. M. Polyakov and A. A. Belavin, “Metastable States of Two-Dimensional Isotropic Ferromagnets,” JETP Lett. 22, 245 (1975) [Pisma Zh. Eksp. Teor. Fiz. 22, 503 (1975)].
6. W. A. Bardeen, B. W. Lee and R. E. Shrock, “Phase Transition in the Nonlinear Sigma Model in Two + Epsilon Dimensional Continuum,” Phys. Rev. D 14, 985 (1976).
7. E. Brezin and J. Zinn-Justin, “Spontaneous Breakdown of Continuous Symmetries Near Two-Dimensions,” Phys. Rev. B 14, 3110 (1976).
8. A. D’Adda, M. Luscher and P. Di Vecchia, “A 1/n Expandable Series of Nonlinear Sigma Models with Instantons,” Nucl. Phys. B 146, 63 (1978).
9. A. D’Adda, M. Luscher and P. Di Vecchia, “Topology and Higher Symmetries of the Two-dimensional Nonlinear Model,” Phys. Rept. 49, 239 (1979).
10. E. Witten, “Instantons, the Quark Model, and the 1/n Expansion,” Nucl. Phys. B 149, 285 (1979).
11. V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, “Two-Dimensional Sigma Models: Modeling Nonperturbative Effects of Quantum Chromodynamics,” Phys. Rept. 116, 103 (1984) [Sov. J. Part. Nucl. 17, 204 (1986)] [Fiz. Elem. Chast. Atom. Yadra 17, 472 (1986)].
12. D. J. Gross and A. Neveu, “Dynamical Symmetry Breaking in Asymptotically Free Field Theories,” Phys. Rev. D 10, 3235 (1974).
13. Y. Nambu and G. Jona-Lasinio, “Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I.,” Phys. Rev. 122, 345 (1961); “Dynamical Model Of Elementary Particles Based On An Analogy With Superconductivity. II,” Phys. Rev. 124, 246 (1961).
14. S. R. Coleman, “There are no Goldstone bosons in two-dimensions,” Commun. Math. Phys. 31, 259 (1973).
15. N. D. Mermin and H. Wagner, “Absence of ferromagnetism or antiferromagnetism in one-dimensional or two-dimensional isotropic Heisenberg models,” Phys. Rev. Lett. 17, 1133 (1966).
16. K. Hori and C. Vafa, “Mirror symmetry,” hep-th/0002222.
17. E. Witten, “A Supersymmetric Form of the Nonlinear Sigma Model in Two-Dimensions,” Phys. Rev. D 16, 2991 (1977).
18. P. Di Vecchia and S. Ferrara, “Classical Solutions in Two-Dimensional Supersymmetric Field Theories,” Nucl. Phys. B 130, 93 (1977).
19. S. Monin, M. Shifman and A. Yung, “Non-Abelian String of a Finite Length,” Phys. Rev. D 92, no. 2, 025011 (2015) [arXiv:1505.07797 [hep-th]].
20. S. Monin, M. Shifman and A. Yung, “Heterotic Non-Abelian String of a Finite Length,” Phys. Rev. D 93, no. 12, 125020 (2016) [arXiv:1604.01003 [hep-th]].
21. A. Milekhin, “CP(N-1) model on finite interval in the large N limit,” Phys. Rev. D 86, 105002 (2012) [arXiv:1207.0417 [hep-th]];
22. A. Milekhin, “CP(N) sigma model on a finite interval revisited,” Phys. Rev. D 95, no. 8, 085021 (2017) [arXiv:1612.02075 [hep-th]].
23. D. Pavshinkin, “Grassmannian sigma model on a finite interval,” Phys. Rev. D 97, no. 2, 025001 (2018), [arXiv:1708.06399 [hep-th]].
24. E. Witten, “Chiral Symmetry, the 1/n Expansion, and the SU(N) Thirring Model,” Nucl. Phys. B 145, 110 (1978).
25. P. Fulde and R. A. Ferrell, “Superconductivity in a Strong Spin-Exchange Field,” Phys. Rev. 135, A550 (1964).
26. K. Machida and H. Nakanishi, “Superconductivity under a ferromagnetic molecular field,” Phys. Rev. B 30, 122 (1984).
27. A. I. Larkin and Y. N. Ovchinnikov, “Nonuniform state of superconductors,” Zh. Eksp. Teor. Fiz. 47, 1136 (1964) [Sov. Phys. JETP 20, 762 (1965)].
28. S. A. Brazovskii, S. A. Gordynin, and N. N. Kirova, “Exact solution of the Peierls model with an arbitrary number of electrons in the unit cell,” JETP Lett. 31, 456 (1980); S. A. Brazovskii and N. N. Kirova, “Excitons, polarons, and bipolarons in conducting polymers,” JETP Lett. 33, 4 (1981).
29. G. Basar and G. V. Dunne, “Self-consistent crystalline condensate in chiral Gross-Neveu and Bogoliubov-de Gennes systems,” Phys. Rev. Lett. 100, 200404 (2008) [arXiv:0803.1501 [hep-th]]; “A Twisted Kink Crystal in the Chiral Gross-Neveu model,” Phys. Rev. D 78, 065022 (2008) [arXiv:0806.2659 [hep-th]].
30. R. Yoshii, S. Tsuchiya, G. Marmorini and M. Nitta, “Spin imbalance effect on Larkin-Ovchinnikov-Fulde-Ferrel state,” Phys. Rev. B 84, 024503 (2011) [arXiv:1101.1578 [cond-mat.supr-con]]; R. Yoshii, G. Marmorini and M. Nitta, “Spin Imbalance Effect on Josephson Junction and Grey Soliton,” J. Phys. Soc. Jpn, 81, 094704 (2012).
31. D. A. Takahashi, S. Tsuchiya, R. Yoshii and M. Nitta, “Fermionic solutions of chiral Gross-Neveu and Bogoliubov-de Gennes systems in nonlinear Schródinger hierarchy,” Phys. Lett. B 718, 632 (2012) [arXiv:1205.3299 [cond-mat.supr-con]].
32. R. Yoshii, S. Takada, S. Tsuchiya, G. Marmorini, H. Hayakawa and M. Nitta, “Fulde-Ferrell-Larkin-Ovchinnikov states in a superconducting ring with magnetic fields: Phase diagram and the first-order phase transitions,” Phys. Rev. B 92, no. 22, 224512 (2015) [arXiv:1404.3519 [cond-mat.supr-con]].
33. A. Flachi, M. Nitta, S. Takada and R. Yoshii, “Sign Flip in the Casimir Force for Interacting Fermion Systems,” Phys. Rev. Lett. 119, no. 3, 031601 (2017) [arXiv:1704.04918 [hep-th]].
34. R. F. Dashen, B. Hasslacher and A. Neveu, “Semiclassical Bound States in an Asymptotically Free Theory,” Phys. Rev. D 12, 2443 (1975).
35. H. Takayama, Y. R. Lin-Liu and K. Maki, “Continuum model for solitons in polyacetylene,” Phys. Rev. B 21, 2388 (1980).
36. D. K. Campbell and A. R. Bishop, “Solitons in polyacetylene and relativistic-field-theory models,” Phys. Rev. B 24, 4859(R) (1981); “Soliton Excitations in Polyacetylene and Relativistic Field Theory Models,” Nucl. Phys. B 200, 297 (1982).
37. S. Okuno and Y. Onodera, “Coexistence of a Soliton and a Polaron in Trans-Polyacetylene” J. Phys. Soc. Jap. 52,3495 (1983).
38. J. Feinberg, “Marginally stable topologically nontrivial solitons in the Gross-Neveu model,” Phys. Lett. B 569, 204 (2003) [hep-th/0209108].
39. J. Feinberg and A. Zee, “Dynamical generation of extended objects in a (1+1)-dimensional chiral field theory: Nonperturbative Dirac operator resolvent analysis,” Phys. Rev. D 56, 5050 (1997) [cond-mat/9603173].
40. S. S. Shei, “Semiclassical Bound States in a Model with Chiral Symmetry,” Phys. Rev. D 14, 535 (1976).
41. D. A. Takahashi and M. Nitta, “Self-Consistent Multiple Complex-Kink Solutions in Bogoliubov-de Gennes and Chiral Gross-Neveu Systems,” Phys. Rev. Lett. 110, no. 13, 131601 (2013) [arXiv:1209.6206 [cond-mat.supr-con]]; “On reflectionless nature of self-consistent multi-soliton solutions in Bogoliubov-de Gennes and chiral Gross-Neveu models,” J. Low. Temp. Phys. 175, 250 (2014) [arXiv:1307.3897 [cond-mat.supr-con]].
42. E. R. C. Abraham and P. K. Townsend, “Q kinks,” Phys. Lett. B 291, 85 (1992); E. R. C. Abraham and P. K. Townsend, “More on Q kinks: A (1+1)-dimensional analog of dyons,” Phys. Lett. B 295, 225 (1992); M. Arai, M. Naganuma, M. Nitta and N. Sakai, “Manifest supersymmetry for BPS walls in N=2 nonlinear sigma models,” Nucl. Phys. B 652, 35 (2003) [hep-th/0211103].
43. J. P. Gauntlett, D. Tong and P. K. Townsend, “Multidomain walls in massive supersymmetric sigma models,” Phys. Rev. D 64, 025010 (2001) [hep-th/0012178]; D. Tong, “The Moduli space of BPS domain walls,” Phys. Rev. D 66, 025013 (2002) [hep-th/0202012].
44. Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, “Construction of non-Abelian walls and their complete moduli space,” Phys. Rev. Lett. 93, 161601 (2004) [hep-th/0404198]; Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, “Non-Abelian walls in supersymmetric gauge theories,” Phys. Rev. D 70, 125014 (2004) [hep-th/0405194]; M. Eto, Y. Isozumi, M. Nitta, K. Ohashi, K. Ohta and N. Sakai, “D-brane construction for non-Abelian walls,” Phys. Rev. D 71, 125006 (2005) [hep-th/0412024].
45. J. P. Gauntlett, R. Portugues, D. Tong and P. K. Townsend, “D-brane solitons in supersymmetric sigma models,” Phys. Rev. D 63, 085002 (2001) [hep-th/0008221].
46. Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, “All exact solutions of a 1/4 Bogomol’nyi-Prasad-Sommerfield equation,” Phys. Rev. D 71, 065018 (2005) [hep-th/0405129].
47. S. Bolognesi, K. Konishi and K. Ohashi, “Large- sigma model on a finite interval,” JHEP 1610, 073 (2016) [arXiv:1604.05630 [hep-th]].
48. A. Betti, S. Bolognesi, S. B. Gudnason, K. Konishi and K. Ohashi, “Large-N sigma model on a finite interval and the renormalized string energy,” JHEP 1801, 106 (2018) [arXiv:1708.08805 [hep-th]].
49. A. Flachi, M. Nitta, S. Takada and R. Yoshii, “Casimir Force for the Model,” arXiv:1708.08807 [hep-th].
50. M. Nitta and R. Yoshii, “Self-consistent Analytic Solutions in Twisted Model in the Large- Limit,” arXiv:1801.09861 [hep-th].
51. S. Bolognesi, S. B. Gudnason, K. Konishi and K. Ohashi, “Large- sigma model on a finite interval: general Dirichlet boundary conditions,” arXiv:1802.08543 [hep-th].
52. A. Gorsky and A. Milekhin, “CP(N-1) model on a disk and decay of a non-Abelian string,” Phys. Rev. D 88, no. 8, 085017 (2013) [arXiv:1306.3565 [hep-th]].
53. A. Pikalov, “CP(N) model on regions with boundary,” arXiv:1710.00699 [hep-th].
54. M. Nitta and R. Yoshii, “Self-Consistent Large- Analytical Solutions of Inhomogneous Condensates in Quantum Model,” JHEP 1712, 145 (2017) [arXiv:1707.03207 [hep-th]].
55. A. Hanany and D. Tong, “Vortices, instantons and branes,” JHEP 0307, 037 (2003) [hep-th/0306150].
56. R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, “NonAbelian superconductors: Vortices and confinement in N=2 SQCD,” Nucl. Phys. B 673, 187 (2003) [hep-th/0307287].
57. M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, “Moduli space of non-Abelian vortices,” Phys. Rev. Lett. 96, 161601 (2006) [hep-th/0511088]; M. Eto, K. Konishi, G. Marmorini, M. Nitta, K. Ohashi, W. Vinci and N. Yokoi, “Non-Abelian Vortices of Higher Winding Numbers,” Phys. Rev. D 74, 065021 (2006) [hep-th/0607070]; M. Eto, K. Hashimoto, G. Marmorini, M. Nitta, K. Ohashi and W. Vinci, “Universal Reconnection of Non-Abelian Cosmic Strings,” Phys. Rev. Lett. 98, 091602 (2007) [hep-th/0609214].
58. D. Tong, “TASI lectures on solitons: Instantons, monopoles, vortices and kinks,” hep-th/0509216.
59. M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, “Solitons in the Higgs phase: The Moduli matrix approach,” J. Phys. A 39, R315 (2006) [hep-th/0602170].
60. M. Shifman and A. Yung, “Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories,” Rev. Mod. Phys. 79, 1139 (2007) [hep-th/0703267].
61. D. Tong, “Quantum Vortex Strings: A Review,” Annals Phys. 324, 30 (2009) [arXiv:0809.5060 [hep-th]].
62. A. Hanany and D. Tong, “Vortex strings and four-dimensional gauge dynamics,” JHEP 0404, 066 (2004) [hep-th/0403158].
63. M. Shifman and A. Yung, “NonAbelian string junctions as confined monopoles,” Phys. Rev. D 70, 045004 (2004) [hep-th/0403149].
64. E. Nakano, M. Nitta and T. Matsuura, “Non-Abelian strings in high density QCD: Zero modes and interactions,” Phys. Rev. D 78, 045002 (2008) [arXiv:0708.4096 [hep-ph]]; M. Eto and M. Nitta, “Color Magnetic Flux Tubes in Dense QCD,” Phys. Rev. D 80, 125007 (2009) [arXiv:0907.1278 [hep-ph]]; M. Eto, E. Nakano and M. Nitta, “Effective world-sheet theory of color magnetic flux tubes in dense QCD,” Phys. Rev. D 80, 125011 (2009) [arXiv:0908.4470 [hep-ph]]; M. Eto, M. Nitta and N. Yamamoto, “Instabilities of Non-Abelian Vortices in Dense QCD,” Phys. Rev. Lett. 104, 161601 (2010) [arXiv:0912.1352 [hep-ph]];
65. M. Eto, Y. Hirono, M. Nitta and S. Yasui, “Vortices and Other Topological Solitons in Dense Quark Matter,” PTEP 2014, no. 1, 012D01 (2014) [arXiv:1308.1535 [hep-ph]].
66. F. D. M. Haldane, “Continuum dynamics of the 1-D Heisenberg antiferromagnetic identification with the O(3) nonlinear sigma model,” Phys. Lett. A 93, 464 (1983).
67. I. Affleck, “Quantum spin chains and the Haldane gap” J. Phys.: Condens. Matter 1, 3047 (1989).
68. T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, “Deconfined quantum critical points,” Science, 303, 1490 (2004) [cond-mat.str-el/0311326]; T. Senthil, L. Balents, A. Vishwanath, S. Sachdev, and M. P. A. Fisher, “Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm,” Phys. Rev. B, 70, 144407 (2004) [cond-mat.str-el/0312617].
69. F. S. Nogueira and A. Sudbo, “Deconfined Quantum Criticality and Conformal Phase Transition in Two-Dimensional Antiferromagnets,” EPL 104, no. 5, 56004 (2013) [arXiv:1304.4938 [cond-mat.str-el]].
70. G. Pöschl and E. Teller, “Bemerkungen zur Quantenmechanik des Anharmonischen Oszillators,” Z. Phys. 83, 143 (1933).
71. J. Lekner, “Reflectionless eigenstates of the sech potential,” Am. J. Phys. 875 1151(2007).
72. G.R.Dvali and M.A.Shifman, “Domain walls in strongly coupled theories,” Phys. Lett. B 396, 64 (1997), Erratum: [Phys. Lett. B 407, 452 (1997)], [hep-th/9612128].
73. K.Ohta and N.Sakai, “Non-Abelian Gauge Field Localized on Walls with Four-Dimensional World Volume,” Prog. Theor. Phys. 124, 71 (2010), Erratum: [Prog. Theor. Phys. 127, 1133 (2012)], [arXiv:1004.4078 [hep-th]].
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# What is the standard notation for a set of equivalence classes?

What is the standard notation for a set of equivalence relations? Specifically, I have a pair of objects, call them $x$ and $y$ and I denote the ordered pair as $\left(x,y\right)$. I have a set of equivalence relations such as:

$\left(x,y\right) \sim \left(x^{-1},y\right) \sim \left(x,y^{-1}\right) \sim \left(y^{-1},x^{-1}\right)$

I would like to write this compactly, but I'm unsure of what the standard notation would be. Is the following appropriate?

$\left\{\left(x,y\right),\left(x^{-1},y\right),\left(x,y^{-1}\right),\left(y^{-1},x^{-1}\right)\right\}$

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I think the usual way is to write just what you have written (perhaps followed by "$\forall x, y \in G$"), and say the quotient is $G/\sim$. –  Kundor Apr 23 '13 at 17:20

You can represent an equivalence class by using a representative from the class, and denoting the entire class by, say, $[(a, b)]$: this represents the set of all ordered pairs $(x, y)$ such that $(x, y) \sim (a, b)$.

So, for a given equivalence relation denoted by $\sim$: one of its equivalence classes can be denoted: $$[(a, b)] = \{(x, y)\mid (x, y) \sim (a, b)\}$$

If there are many equivalence classes determined by an equivalence relation, and you want to denote the set of equivalence classes, you can list the equivalence classes as elements of a set:

• $\{[(a, b)], [(c,d)], [(e, f)], \cdots [(y, z)]\}$ if there is a finite set of them.
• For example, the set of equivalence classes determined by the equivalence relation of congruence modulo $4$ on the set of integers, you could write $\{[0], [1], [2], [3]\}$,
• or, in the case of an infinite number of equivalence classes, like those corresponding to equality/identity on the natural numbers, one can write $\{[1], [2], [3], \cdots \}$.

If you are asking how to denote a set of different equivalence relations, where the elements of the set are relations, I'm not aware of the standard notation. (That's not to say it doesn't exist.)

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What I want to do is have a way to denote the entire set of equivalence relations, like $P=\left\{\left(x,y\right),\left(x^{-1},y\right),\left(x,y^{-1}\right),\left(y^{-1},x^{-1}\right)\right\}$, so that I can later define the quotient set as $G/P$, so I need a way to denote the entire set of equivlanece relations not just one of them. –  okj Apr 22 '13 at 19:27
Try using square brackets to denote that the elements of $P$ are equivalence classes:$$P=\left\{\left[\left(x,y\right)\right],\left[\left(x^{-1},y\right)\right],\left[\left(x,y^{-1}\right)\right],\left[\left(y^{-1},x^{-1}\right)\right]\right\}$$ –  amWhy Apr 22 '13 at 21:38
@amWhy: Very nice answer +1 –  Amzoti Apr 23 '13 at 0:17

If $\sim$ is an equivalence class over $A$, then in many places we write $A/{\sim}$ as the set of equivalence classes.

This notation is similar, and on purpose, to the notation from algebra when writing $V/W$ for the quotient subspace, or $G/H$ for the quotient group, or $R/I$ when taking a quotient of a ring by an ideal.

The reason is that all those quotients actually induce an equivalence relation, and we have a natural structure on the set of equivalence classes.

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I honestly don't know what could be wrong in this answer, and I would be very glad if someone would tell me. Perhaps something is wrong with the user posting it instead... –  Asaf Karagila Dec 16 '13 at 21:30
–  Daniel Fischer Dec 17 '13 at 10:43


text:
Illinois Journal of Mathematics

Angular derivatives and semigroups of holomorphic functions

Nikolaos Karamanlis

Abstract

A simply connected domain $\Omega \subset \mathbb{C}$ is convex in the positive direction if for every $z\in \Omega$, the half-line $\{z+t:t\geq 0\}$ is contained in $\Omega$. We provide necessary and sufficient conditions for the existence of an angular derivative at $\infty$ for domains convex in the positive direction which are contained either in a horizontal half-plane or in a horizontal strip. This class of domains arises naturally in the theory of semigroups of holomorphic functions, and the existence of an angular derivative has interesting consequences for the semigroup.

Article information

Source
Illinois J. Math., Volume 63, Number 3 (2019), 403-424.

Dates
Revised: 19 June 2019
First available in Project Euclid: 19 September 2019

https://projecteuclid.org/euclid.ijm/1568858865

Digital Object Identifier
doi:10.1215/00192082-7897499

Mathematical Reviews number (MathSciNet)
MR4012349

Zentralblatt MATH identifier
07110747

Citation

Karamanlis, Nikolaos. Angular derivatives and semigroups of holomorphic functions. Illinois J. Math. 63 (2019), no. 3, 403--424. doi:10.1215/00192082-7897499. https://projecteuclid.org/euclid.ijm/1568858865

References

• [1] M. Abate, Iteration Theory of Holomorphic Maps on Taut Manifolds, Res. Lecture Notes Math. Complex Anal. Geom., Mediterranean Press, Rende, 1989.
• [2] L. V. Ahlfors, Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen, Acta Soc. Sci. Fenn. N. S. A. 1 (1930), no. 9, 1–40.
• [3] L. Arosio and F. Bracci, Canonical models for holomorphic iteration, Trans. Amer. Math. Soc. 368 (2016), no. 5, 3305–3339.
• [4] A. Baernstein, Ahlfors and conformal invariants, Ann. Sci. Fenn. Ser. A.I. Mathematica 13 (1988), 289–312.
• [5] D. Betsakos, Geometric description of the classification of holomorphic semigroups, Proc. Amer. Math. Soc. 144 (2016), no. 4, 1595–1604, 2016.
• [6] D. Betsakos, M. D. Contreras, and S. Díaz-Madrigal, On the rate of convergence of semigroups of holomorphic functions at the Denjoy–Wolff point, to appear in Rev. Mat. Iberoam. (2019).
• [7] F. Bracci, M. D. Contreras, S. Díaz-Madrigal, and H. Gaussier, Backward orbits and petals of semigroups of holomorphic self-maps of the unit disk, Ann. Math. Pura Appl. 198 (2019), 411–441.
• [8] K. Burdzy, Brownian excursions and minimal thinness Part III: Applications to the angular derivative problem, Math Z. 192 (1986), 89–107.
• [9] M. D. Contreras and S. Díaz-Madrigal, Analytic flows on the unit disk: Angular derivatives and boundary fixed points, Pac. J. Math. 222 (2005), no. 2, 253–286.
• [10] M. D. Contreras, S. Díaz-Madrigal, and Ch. Pommerenke, Second angular derivatives and parabolic iteration in the unit disk, Trans. Amer. Math. Soc. 362 (2010), 357–388.
• [11] C. C. Cowen, Iteration and the solution of functional equations for functions analytic in the unit disk, Trans. Amer. Math. Soc. 265 (1981), no. 1, 69–95.
• [12] M. Elin and D. Shoikhet, Linearization Models for Complex Dynamical Systems: Topics in Univalent Functions, Functional Equations and Semigroup Theory, Oper. Theory Adv. Appl. 208, Birkhäuser, Basel, 2010.
• [13] C. H. FitzGerald, B. Rodin, and S. E. Warschawski, Estimates of the harmonic measure of a continuum in the unit disk, Trans. Amer. Math. Soc. 287 (1985), 681–685.
• [14] J. B. Garnett and D. E. Marshall, Harmonic Measure, Cambridge Univ. Press, Cambridge, 2005.
• [15] J. A. Jenkins, On comb domains, Proc. Amer. Math. Soc. 124 (1996), no. 1, 187–191.
• [16] J. A. Jenkins and K. Oikawa, Conformality and semi-conformality at the boundary, J. Reine Angew. Math. 291 (1977), 92–117.
• [17] N. Karamanlis, On the angular derivative of comb domains, Comput. Methods Funct. Theory, published online July 11, 2019.
• [18] D. E. Marshall, Angular derivatives and Lipschitz majorants, preprint, 1995, http://math.washington.edu/~marshall/preprints/preprints.html.
• [19] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, New York, 1992.
• [20] B. Rodin and S. E. Warschawski, Extremal length and the boundary behavior of conformal mappings, Ann. Acad. Sci. Fenn. Ser. A. I. Mathematica 2 (1976), 467–500, 1976.
• [21] B. Rodin and S. E. Warschawski, Extremal length and univalent functions I. The angular derivative, Math. Z. 153 (1977), 1–17.
• [22] A. Yu. Solynin, Harmonic measure of continua with fixed diameter (in Russian), Zap. Nauch. Semin. LOMI 144 (1985), 146–149; English translation in J. Soviet Math. 38 (1987), 2140–2142.
• [23] S. E. Warschawski, On conformal mapping of infinite strips, Trans. Amer. Math. Soc. 51 (1942), 280–335.


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# Difference between revisions of "2012 AMC 12B Problems/Problem 19"

## Problem 19

A unit cube has vertices $P_1,P_2,P_3,P_4,P_1',P_2',P_3',$ and $P_4'$. Vertices $P_2$, $P_3$, and $P_4$ are adjacent to $P_1$, and for $1\le i\le 4,$ vertices $P_i$ and $P_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $P_1P_2$, $P_1P_3$, $P_1P_4$, $P_1'P_2'$, $P_1'P_3'$, and $P_1'P_4'$. What is the octahedron's side length?

$[asy] import three; size(7.5cm); triple eye = (-4, -8, 3); currentprojection = perspective(eye); triple[] P = {(1, -1, -1), (-1, -1, -1), (-1, 1, -1), (-1, -1, 1), (1, -1, -1)}; // P[0] = P[4] for convenience triple[] Pp = {-P[0], -P[1], -P[2], -P[3], -P[4]}; // draw octahedron triple pt(int k){ return (3*P[k] + P[1])/4; } triple ptp(int k){ return (3*Pp[k] + Pp[1])/4; } draw(pt(2)--pt(3)--pt(4)--cycle, gray(0.6)); draw(ptp(2)--pt(3)--ptp(4)--cycle, gray(0.6)); draw(ptp(2)--pt(4), gray(0.6)); draw(pt(2)--ptp(4), gray(0.6)); draw(pt(4)--ptp(3)--pt(2), gray(0.6) + linetype("4 4")); draw(ptp(4)--ptp(3)--ptp(2), gray(0.6) + linetype("4 4")); // draw cube for(int i = 0; i < 4; ++i){ draw(P[1]--P[i]); draw(Pp[1]--Pp[i]); for(int j = 0; j < 4; ++j){ if(i == 1 || j == 1 || i == j) continue; draw(P[i]--Pp[j]); draw(Pp[i]--P[j]); } dot(P[i]); dot(Pp[i]); dot(pt(i)); dot(ptp(i)); } label("P_1", P[1], dir(P[1])); label("P_2", P[2], dir(P[2])); label("P_3", P[3], dir(-45)); label("P_4", P[4], dir(P[4])); label("P'_1", Pp[1], dir(Pp[1])); label("P'_2", Pp[2], dir(Pp[2])); label("P'_3", Pp[3], dir(-100)); label("P'_4", Pp[4], dir(Pp[4])); [/asy]$

$\textbf{(A)}\ \frac{3\sqrt{2}}{4}\qquad\textbf{(B)}\ \frac{7\sqrt{6}}{16}\qquad\textbf{(C)}\ \frac{\sqrt{5}}{2}\qquad\textbf{(D)}\ \frac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \frac{\sqrt{6}}{2}$

## Solution

Observe the diagram above. Each dot represents one of the six vertices of the regular octahedron. Three dots have been placed exactly x units from the (0,0,0) corner of the unit cube. The other three dots have been placed exactly x units from the (1,1,1) corner of the unit cube. A red square has been drawn connecting four of the dots to provide perspective regarding the shape of the octahedron. Observe that the three dots that are near (0,0,0) are each (x)($\sqrt{2}$) from each other. The same is true for the three dots that are near (1,1,1). There is a unique x for which the rectangle drawn in red becomes a square. This will occur when the distance from (x,0,0) to (1,1-x, 1) is (x)($\sqrt{2}$).

Using the distance formula we find the distance between the two points to be: $\sqrt{{(1-x)^2} + {(1-x)^2} + 1}$ = $\sqrt{2x^2 - 4x +3}$. Equating this to (x)($\sqrt{2}$) and squaring both sides, we have the equation:

$2{x^2}$ - $4x + 3$ = $2{x^2}$

$-4x + 3 = 0$

$x$ = $\frac{3} {4}$.

Since the length of each side is (x)($\sqrt{2}$), we have a final result of $\frac{3 \sqrt{2}}{4}$. Thus, Answer choice A is correct.

(If someone can draw a better diagram with the points labeled P1,P2, etc., I would appreciate it).

--Jm314 14:55, 26 February 2012 (EST)


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## Notes on Notation – Variables

Variables have often been seen as the big bugaboo of the middle school math curriculum and beyond.  They have become inextricably identified with that 7-letter curse word “algebra”.  I submit most teachers have encountered some variation on student Jesse who says in utter frustration: “How can a possibly equal b?  a is already equal to a!”  Yet before we dismiss Jesse as confused, it would be useful to note that Jesse clearly got what was important in school up till now: that you keep all your letters straight.

The introduction of letters for numbers in school marks the end of the transition of kids from arithmetic to algebra, not the beginning.  This transition, from the doing of arithmetic to the thinking about arithmetic (both numbers and operations) as a domain with patterns and regularities, involves many mathematical ideas, and that doesn’t wait till middle school, and it doesn’t wait for formal algebra with formal variables.

And yet the introduction of letters as variables, letters for numbers, does mark a big hurdle for many students.  The last shred of pretense that what we’re doing is arithmetic, that we’re out to produce a number, is now out the window.

If we look at the notational conventions for variables, it seems that the most obvious aspect of it is also the most important: variables look different from numbers.  For the old Greeks, this was never obvious, for their way of writing numbers re-used their letters!  (They used nine letters to indicate 1-9, nine different letters to indicate 10-90, and so on.)  For them, letter variables wouldn’t have been clearly distinct from numbers.  To conclude, as some have done, that this was single-handedly what kept the Greeks from inventing algebra strikes me as rather silly – they could have  found any of another number of small variations that would have dealt with the difficulty.  But for us, letters aren’t confused with numbers, so hence letters for variables.

However, there is an additional convention that variables in mathematics are always single letters.  You often see this in physics, too, so that you see $d = \frac{1}{2}a t ^ 2$ instead of $distance = \frac{1}{2} acceleration \times time^2$.  The latter version favors ease of reading over compactness of writing: sometimes I get the impression that the entirety of mathematics still carries around the traces of old economic trade-offs: the Greeks wrote their mathematics with a stick in the sand; much of school work over the last several centuries was done with a marker on a small slate: a board literally made out of slate – easily written on, and easily erased.  In those centuries, paper was very expensive, and not wasted on small matters such as calculating sums.

If you look at modern programming languages for computers and websites, every single one allows variables to be more than a single letter.  Variables tend to be chosen so that they are meaningful for the reader, so ‘distance’ rather than ‘d’.  Of course, in movies it would be much less imposing to have a scientist with wild hair say “energy is mass times the square of the speed of light” instead of $E = mc^2$, but I’d settle gladly for a mathematics that didn’t sound like magical incantations that have to be repeated just right.

Einstein on variables

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### 3 Responses to Notes on Notation – Variables

1. José says:

You say that:
“And yet the introduction of letters as variables, letters for numbers, does mark a big hurdle for many students. The last shred of pretense that what we’re doing is arithmetic, that we’re out to produce a number, is now out the window.”

How is the transition to variables a hurdle? Maybe a very, very small hurdle, but nothing of great difficulty. If anything, the use of variables makes arithmetic steps much easier. The variables are like a name for a number when the value is not immediately needed (until later). Variables make many things in arithmetic easier to understand.

2. Bert Speelpenning says:

Thanks for your comment. You raise an important issue.

There is a distinction between what we think ought to be hard or easy, on the one hand, and what children can be observed to have difficulty with.
Those are not the same. Many things we think ought to be easy are consistently stumbled over, and many things we think ought to be hard are taken in stride by generation after generation of kids.

As you read more of this blog you’ll see me drag in examples of both kinds. The examples come from fairly extensive personal observations as well as reports from many teachers.

The connection between algebra and the ability to deal with numbers whose value is not immediately needed – that’s a great point. I’m referring to that in the blog as “deferred computation” and I’ve got many entries on just that.