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Regge spectra of excited mesons, harmonic confinement and QCD vacuum structure

Regge spectra of excited mesons, harmonic confinement and QCD vacuum structure

Sergei N. Nedelko111nedelko@theor.jinr.ru, Vladimir E. Voronin222voronin@theor.jinr.ru Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia
Abstract

An approach to QCD vacuum as a medium describable in terms of statistical ensemble of almost everywhere homogeneous Abelian (anti-)self-dual gluon fields is briefly reviewed. These fields play the role of the confining medium for color charged fields as well as underline the mechanism of realization of chiral and symmetries. Hadronization formalism based on this ensemble leads to manifestly defined quantum effective meson action. Strong, electromagnetic and weak interactions of mesons are represented in the action in terms of nonlocal -point interaction vertices given by the quark-gluon loops averaged over the background ensemble. New systematic results for the mass spectrum and decay constants of radially excited light, heavy-light mesons and heavy quarkonia are presented. Interrelation between the present approach, models based on ideas of soft wall AdS/QCD, light front holographic QCD, and the picture of harmonic confinement is outlined.

pacs:
12.38.Aw, 12.38.Lg, 12.38.Mh, 11.15.Tk

I Introduction

Almost forty five years ago Feynman, Kislinger, and Ravndal noticed Feynman () that the Regge spectrum of meson and baryon masses could be universally described by assuming the four-dimensional harmonic oscillator potential acting between quarks and antiquarks. During subsequent years the idea of four-dimensional harmonic oscillator re-entered the discussion about quark confinement several times in various ways. Leutwyler and Stern developed the formalism devoted to the covariant description of bilocal meson-like fields combined with the idea of harmonic confinement Leutwyler:1977vz (); Leutwyler:1977vy (); Leutwyler:1977pv (); Leutwyler:1977cs (); Leutwyler:1978uk (). Considerations of paper Feynman () and Leutwyler-Stern formalism Leutwyler:1977vz (); Leutwyler:1977vy (); Leutwyler:1977pv (); Leutwyler:1977cs (); Leutwyler:1978uk () can be seen as the forerunners to at present time very popular soft wall AdS/QCD models Karch:2006pv () and the light front holographic QCD deTeramond:2005su (); Brodsky:2006uqa (); deTeramond:2008ht (). In recent years, the approaches to confinement based on the ideas of soft wall AdS/QCD model and light front holography demonstrated an impressive phenomenological success Karch:2006pv (); deTeramond:2005su (); Brodsky:2006uqa (); deTeramond:2008ht (); Swarnkar (); Gutsche:2015xva (). The crucial for phenomenology features of these approaches are the particular dilaton profile and the harmonic oscillator form of the confining potential as the function of fifth coordinate . All these approaches begin with different motivation but finally come to the Schrödinger type differential equation with the harmonic potential in defining the wave functions and mass spectrum of mesons and baryons.

The physical origin of the above-mentioned particular form of dilaton profile in AdS/QCD and light front holography as well as the harmonic potential in the Stern-Leutwyler studies and, hence, the Laguerre polynomial form of the meson wave functions, could not be identified within these approaches themselves. The preferable form of the dilaton profile and/or the potential are determined by the phenomenological requirement of Regge character of the excited meson mass spectrum Karch:2006pv (); Feynman (); Leutwyler:1977vz (); Leutwyler:1977vy (); Leutwyler:1977pv (); Leutwyler:1977cs (); Leutwyler:1978uk ().

The approach presented in this paper has been developed in essence twenty years ago EN1 (). It clearly incorporates the idea of harmonic confinement both in terms of elementary color charged fields and the composite colorless hadron field. The distinctive feature of the present approach is that it basically links the concept of harmonic confinement and Regge character of hadron mass spectrum to the specific class of nonperturbative gluon configurations – almost everywhere homogeneous Abelian (anti-)self-dual gluon fields. A posteriori a close interrelation of the Abelian (anti-)self-dual fields and the hadronization based on harmonic confinement can be read off the papers Leutwyler1 (); Leutwyler2 (); Minkowski (); Leutwyler:1977vz (); Leutwyler:1977vy (); Leutwyler:1977pv (); Leutwyler:1977cs (); Leutwyler:1978uk (). In brief, the line of arguments is as follows (for more detailed exposition see NV ()).

An important benchmark has been the observation of Pagels and Tomboulis Pagels () that Abelian self-dual fields describe a medium infinitely stiff to small gauge field fluctuations, i.e. the wave solutions for the effective quantum equations of motion are absent. This feature was interpreted as suggestive of confinement of color. Strong argumentation in favour of the Abelian (anti-)self-dual homogeneous field as a candidate for the global nontrivial minimum of the effective action originates from the papers Minkowski (); Leutwyler2 (); NG2011 (); Pawlowski (); George:2012sb (). In particular, Leutwyler has shown that the constant gauge field is stable (tachyon free) against small quantum fluctuations only if it is Abelian (anti-)self-dual covariantly constant field Minkowski (); Leutwyler2 (). Nonperturbative calculation of the effective potential within the functional renormalization group Pawlowski () supported the earlier one-loop results on existence of the nontrivial minimum of the effective action for the Abelian (anti-)self-dual field.

The eigenvalues of the Dirac and Klein-Gordon operators in the presence of Abelian self-dual field are purely discrete, and the corresponding eigenfunctions of quarks and gluons are of the bound state type. This is a consequence of the fact that these operators contain the four-dimensional harmonic oscillator, acting as a confining harmonic potential. Eigenmodes of the color charged fields have no (quasi-)particle interpretation but describe field fluctuations decaying in space and time. The consequence of this property is that the momentum representation of the translation invariant part of the propagator of the color charged field in the background of (anti-)self-dual Abelian gauge field is entire analytical function. The absence of pole in the propagator was treated as the absence of the particle interpretation of the charged field Leutwyler1 (). However just the absence of a single quark or anti-quark in the spectrum can not be considered as sufficient condition for confinement. One has to explain the most peculiar feature of QCD – the Regge character of the physical spectrum of colorless hadrons. Usually Regge spectrum is related to the string picture of confinement, justified in two complementary ways and limits: classical relativistic rotating string connecting massless quark and antiquark, and the linear potential between nonrelativistic heavy quark and antiquark with the area law for the temporal Wilson loop as a relevant criterion for static quark confinement. Neither the homogeneous Abelian (anti-)self-dual field itself nor the form of gluon propagator in the presence of this background had the clue to linear quark-antiquark potential. Nevertheless, the analytic structure of the gluon and quark propagators and assumption about the randomness of the background field ensemble led both to the area law for static quarks and the Regge spectrum for light hadrons.

Randomness of the ensemble of almost everywhere homogeneous Abelian (anti-)self-dual gluon fields has been taken into account implicitly in the model of hadronization developed in EN1 (); EN () via averaging of the quark loops over the parameters of the random fields. The nonlocal quark-meson vertices with the complete set of meson quantum numbers were determined in this model by the form of the color charged gluon propagator. The spectrum of mesons displayed the Regge character both with respect to total angular momentum and radial quantum number of the meson. The reason for confinement of a single quark and Regge spectrum of mesons turned out to be the same – the analytic properties of quark and gluon propagators.

This result has almost completed the quark confinement picture based on the random almost everywhere homogeneous Abelian (anti-)self-dual fields. Self-duality of the fields plays the crucial role in this picture. This random field ensemble represents a medium where the color charged elementary excitations exist as quickly decaying in space and time field fluctuations but the collective colorless excitations (mesons) can propagate as plain waves (particles). It should be stressed that in this formalism any meson looks much more like a complicated collective excitation of a medium (QCD vacuum) involving quark, antiquark and gluon fields than a nonrelativistic quantum mechanical bound state of charged particles (quark and anti-quark) due to some potential interaction between them. Within this relativistic quantum field description the Regge spectrum of color neutral collective modes appeared as a ”medium effect” as well as the suppression (confinement) of a color charged elementary modes.

However, besides this dynamical color charge confinement, a correct complete picture must include the limit of static quark-antiquark pair with the area law for the temporal Wilson loop. In order to explore this aspect an explicit construction of the random domain ensemble was suggested in paper NK1 (), and the area law for the Wilson loop was demonstrated by the explicit calculation. Randomness of the ensemble (in line with Olesen7 ()) and (anti-)self-duality of the fields are crucial for this result.

In this paper we briefly review the approach to confinement, chiral symmetry realization and bosonization based on the representation of QCD vacuum in terms of the statistical ensemble of almost everywhere homogeneous Abelian (anti-)self-dual gluon fields, systematically calculate the spectrum of radial meson excitations and their decay constants and outline the possible relation between the formalism of soft wall AdS/QCD and light-front holography, and this, at first sight, different approach.

The paper is organized as follows. Section II is devoted to motivation of the approach. Derivation of the effective meson action is considered in section III. Results for the masses, transition and decay constants of various mesons are presented in section IV. In the section V we outline possible relation between the present hadronization approach and the formalism of the soft wall AdS/QCD model, light front holographic QCD, compare the quark and gluon propagators of the present approach with the results of functional renormalization group (FRG) and Dyson-Schwinger equations (DSE). Important technical details are given in the appendices.

Ii Domain wall networks as QCD vacuum

The primary phenomenological basis of the present approach is the existence of nonzero condensates in QCD, first of all – the scalar gluon condensate . In order to incorporate this condensate into the functional integral approach to quantization of QCD one has to choose appropriate conditions for the functional space of gluon fields to be integrated over (see, e.g., Ref.faddeev ()). Besides the formal mathematical content, these conditions play the role of substantial physical input which, together with the classical action of QCD, complements the statement of the quantization problem. In other words, starting with the very basic representation of the Euclidean functional integral for QCD,

 Z=N∫FBDA∫ΨDψD¯ψexp{−S[A,ψ,¯ψ]}, (1)

one has to specify integration spaces for gluon and for quark fields. Bearing in mind a nontrivial QCD vacuum structure encoded in various condensates, one have to define permitting gluon fields with nonzero classical action density,

 FB={A:limV→∞1V∫Vd4xg2Faμν(x)Faμν(x)=B2}.

It is assumed that the constant may have a nonzero value. The gauge fields that satisfy this condition have a potential to provide the vacuum with the whole variety of condensates.

An analytical approach to definition and calculation of the functional integral can be based on separation of modes responsible for nonzero condensates from the small perturbations . This separation must be supplemented with gauge fixing. Background gauge fixing condition is the most natural choice. To perform separation, one inserts identity

 1=∫BDBΦ[A,B]∫QDQ∫ΩDωδ[Aω−Qω−Bω]δ[D(Bω)Qω]

in the functional integral and arrives at

 Z = N′∫BDB∫ΨDψD¯ψ∫QDQdet[D(B)D(B+Q)]δ[D(B)Q]e−SQCD[B+Q,ψ,¯ψ] = ∫BDBexp{−Seff[B]}.

Thus defined quantum effective action has a physical meaning of the free energy of the quantum field system in the presence of the background gluon field . In the limit global minima of determine the class of gauge field configurations representing the equilibrium state (vacuum) of the system.

Quite reliable argumentation in favour of (almost everywhere) homogeneous Abelian (anti-)self-dual fields as dominating vacuum configurations was put forward by many authors Pagels (); Leutwyler2 (). As it has already been mentioned in Introduction, nonperturbative calculation of QCD quantum effective action within the functional renormalization group approach Pawlowski () supported the one-loop result Pagels (); Minkowski (); Leutwyler2 () and indicated the existence of a minimum of the effective potential for nonzero value of Abelian (anti-)self-dual homogeneous gluon field.

Ginzburg-Landau (GL) approach to the quantum effective action indicated a possibility of the domain wall network formation in QCD vacuum resulting in the dominating vacuum gluon configuration seen as an ensemble of densely packed lumps of covariantly constant Abelian (anti-)self-dual field NK1 (); NG2011 (); NV (); George:2012sb (). Nonzero scalar gluon condensate postulated by the effective potential

 Ueff = Λ412Tr(C1˘f2+43C2˘f4−169C3˘f6), (2)

with being a scale of QCD and , leads to the existence of twelve discrete degenerate global minima of the effective action (see Fig.1),

 ˘Aμ∈{˘B(kl)μ| k=0,1,…,5; l=0,1},  ˘B(kl)μ=−12˘nkB(l)μνxν, ~B(l)μν=12εμναβB(l)αβ=(−1)lB(l)μν, ˘nk=T3 cos(ξk)+T8 sin(ξk),  ξk=2k+16π, (3)

where and correspond to the self-dual and anti-self-dual field respectively, matrix belongs to Cartan subalgebra of with six values of the angle corresponding to the boundaries of the Weyl chambers in the root space of .

The minima are connected by the parity and Weyl group reflections. Their existence indicates that the system is prone to the domain wall formation. To demonstrate the simplest example of domain wall interpolating between the self-dual and anti-self-dual Abelian configurations, one allows the angle between chromomagnetic and chromoelectric fields to vary from point to point in and restricts other degrees of freedom of gluon field to their vacuum values. In this case Ginsburg-Landau Lagrangian leads to the sine-Gordon equation for with the standard kink solution (for details see Ref. NG2011 (); NV ())

 ω(xν)=2 arctan(exp(μxν)).

Away from the kink location vacuum field is almost self-dual () or anti-self-dual (). Exactly at the wall it becomes purely chromomagnetic (). Domain wall network is constructed by means of the kink superposition. In general kink can be parametrized as

 ζ(μi,ηiνxν−qi)=2πarctanexp(μi(ηiνxν−qi)),

where is inverse width of the kink, is a normal to the wall and are coordinates of the wall. A single lump in two, three and four dimensions is given by

 ω(x)=πk∏i=1ζ(μi,ηiνxν−qi).

for , respectively. The general kink network is then given by the additive superposition of lumps

 ω=π∞∑j=1k∏i=1ζ(μij,ηijνxν−qij).

Topological charge density distribution for a network of domain walls with different width is illustrated in Fig.2.

Based on this construction, the measure of integration over the background field can be constructively represented as the infinite dimensional (in the infinite volume) integral over the parameters of domain walls in the network: their positions, orientations and widths, with the weight determined by the effective action. It should be noted that chronologically the explicit construction of the domain wall network is the most recent development of the formalism that have been studied in the series of papers EN (); EN1 (); NK1 (); NK4 (); NK6 (), in which the domain wall defects in the homogeneous Abelian (anti-)self-dual field were taken into account either implicitly or in an explicit but simplified form with the spherical domains. The practical calculations in the next sections will be done within combined implementation of domain model given in paper NK4 (): propagators in the quark loops are taken in the approximation of the homogeneous background field and the quark loops are averaged over the background field, the correlators of the background field are calculated in the spherical domain approximation.

Iii Hadronization within the domain model of QCD vacuum

The haronization formalism based on domain model of QCD vacuum was elaborated in the series of papers EN1 (); EN (); NK1 (); NK4 (). We refer to these papers for most of the technical details omitted in this brief presentation. It has been shown that the model embraces static (area law) and dynamical quark confinement (propagators in momentum representation are entire analytical functions) as well as spontaneous breaking of chiral symmetry by the background domain structured field itself. problem was resolved without introducing the strong CP-violation NK6 (). Estimation of masses of light, heavy-light mesons and heavy quarkonia along with their orbital excitations EN1 (); EN (); NK4 () demonstrated promising phenomenological performance. However, calculations in Refs. EN (); NK4 () have been done neglecting a mixing between radially excited meson fields. Below we present results of calculation refined in this respect.

In the spherical domain approximation, the background gluon fields are represented by the ensemble of domain-structured fields with the strength tensor NK1 (); NK4 ()

 Faμν(x)=N∑k=1n(k)aB(k)μνθ(1−(x−zk)2/R2),B(k)μνB(k)μρ=B2δνρ,B=2√3Λ2, ~B(k)μν=±B(k)μν,^n(k)=t3cosξk+t8sinξk,ξk∈{π6(2k+1), k=0,…,5},

where is the space-time coordinate of the -th domain center, scale and mean domain radius are parameters of the model related to the scalar gluon condensate and topological susceptibility of pure Yang-Mills vacuum, respectively NK1 ().

The measure of integration over ensemble of background fields is defined as NK1 (); NK4 ()

 ∫BdB… = ∏k124π2limV→∞1V∫Vd4zk2π∫0dφk∫π0dθksinθk × ∫2π0dξk3,4,5∑l=0,1,2δ(ξk−(2l+1)π6)∫π0dωk∑n=0,1δ(ωk−πn)…

Once the measure is specified, one can return to the functional integral (1) and integrate out fluctuation part of the gluon fields :

 Z = ∫dB∫ΨDψD¯ψ∫QDQδ[D(B)Q]ΔFP[B,Q]e−SQCD[Q+B,ψ,¯ψ] = ∫dB∫ΨDψD¯ψexp{∫dx¯ψ(i⧸∂+g⧸B−m)ψ}W[j],

where is the local quark current. Recalling the definition of Green functions,

 Ga1…anμ1…μn(x1,…,xn|B)=1gnδnlnW[j]δja1μ1(x1)…δjanμn(xn)∣∣ ∣∣j=0,

we arrive at the representation

 W[j|B] = exp{∑ngnn!∫d4x1…∫d4xnja1μ1(x1)…janμn(xn)Ga1…anμ1…μn(x1,…,xn|B)},

where by construction the gauge coupling constant and the exact renormalized -point gluon Green functions of pure gauge theory in the presence of the background field appear to be renormalized within appropriate renormalization scheme. It is needless to say that the functional form of these Green functions, gluon propagator in particular, has been a subject of many investigations carried out over decades. Quite reliable information about two- and three-point Euclidean Green functions was obtained within the functional renormalization group, Lattice QCD as well as calculations based on Dyson-Schwinger equations.

At this step one has to set up the approximation scheme. We truncate the exponent in up to the four-fermion interaction term. Interaction between standard local color charged quark currents is described by the product of the renormalized coupling constant squared and exact gluon propagator which will be approximated by the gluon propagator in the presence of homogeneous Abelian (anti-)self-dual field. Radiative corrections due to the gluon and ghost field fluctuations are neglected (for more details see Refs. EN1 (); EN (); NK4 ()). It should be noted that omitted radiative corrections can be represented in terms of the standard for pure gluodynamics set of Feynman graphs for gluon polarization function but the internal lines in the graphs correspond to the gluon and ghost propagators in the background field . In other words, the approximation in use corresponds to the lowest (tree level) order with respect to perturbative fluctuations , but the background field (vacuum field ) itself is taken into account exactly.

The randomness of domain ensemble is taken into account implicitly by means of averaging the nonlocal meson-meson interaction vertices over all possible configurations of the homogeneous background field at the final stage of derivation of the effective meson action EN (); NK4 ().

Relevant truncated part of QCD functional integral reads

 Z=∫BdB∫ΨDψD¯ψexp{∫d4x¯ψ(i⧸∂+g⧸B−m)ψ+L}, (4) L=g22∫d4x∫d4y Gabμν(x,y|B)jaμ(x)jbν(y),

where is a diagonal quark mass matrix. By means of the standard Fierz transformation of color, Dirac and flavour matrices the four-quark interaction can be rewritten as

 L=g22∑J,cCJ∫d4x∫d4yG(x−y)JJc(x,y|B)JJc(y,x|B),

where numerical coefficients are different for different spin-parity . Here bilocal color neutral quark currents,

 JJc(x,y|B)=¯ψ(x)λcΓJexp{i2xμ^Bμνyν}ψ(y),

are singlets with respect to the local background gauge transformations. In the center of quark mass coordinate system bilocal currents take the form

 JJc(x,y|B)→JJc(x,z|B)=¯ψf(x)λcΓJexp(izμ\lx@stackrel↔Dμff′(x))ψf′(x), (5) \lx@stackrel↔Dff′μ=ξf\lx@stackrel←Dμ−ξf′\lx@stackrel→Dμ,  \lx@stackrel←Dμ(x)=\lx@stackrel←∂μ+i^Bμ(x),  \lx@stackrel→Dμ(x)=\lx@stackrel→∂μ−i^Bμ(x), ξf=mf′mf+mf′, ξf′=mfmf+mf′,

and their interaction is described by the action EN1 ()

 S=g22∑J,cCJ∫d4x∫d4zG(z)J†Jc(x,z|B)JJc(x,z|B), (6) G(z)=14π2z2exp{−14Λ2z2}, (7)

where - center of quark mass coordinates, and - relative coordinates of quark and antiquark. It has to be noted here that quark fields are seen as pure fluctuations describable in terms of four-dimentional harmonic oscillator eigenmodes of the bound state type NV (); NK1 (); NK4 () in . Interpretation of the quark field in terms of point-like particle is simply does not exist in the confining background under consideration. Function originates from the gluon propagator in the presence of the homogeneous Abelian (anti-)self-dual gluon field EN1 (). It differs from the free massless scalar propagator by the Gaussian exponent, which completely changes the IR properties of the propagator but leaves its UV asymptotic behaviour unchanged. In momentum representation it takes the form

 ~G(p)=1p2(1−e−p2/Λ2). (8)

It is important that nonzero gluon condensates and represented by the Abelian (anti-)self-dual vacuum field remove the pole from the propagator which can be treated as dynamical confinement of the color charged fields Leutwyler1 ().

The quark propagator in the homogeneous as well as domain structured NK1 () Abelian (anti-)self-dual gluon field also demonstrates confinement. Momentum representation of the translation invariant part of the quark propagator in the presence of the homogeneous field,

 S(x,y)=exp(−i2xμBμνyν)H(x−y),

is an entire analytical function of momentum:

 ~Hf(p)=12vΛ2∫10dse(−p2/2vΛ2)s(1−s1+s)m2f/4vΛ2[s1−s2pαγα±isγ5γαfαβpβ (9) fαβ=^n2vΛ2Bαβ, v=diag(16,16,13),  ^Bρμ^Bρν=4v2Λ4δμν.

The propagator has a rich Dirac structure including not only the vector and scalar parts but also the pseudoscalar, axial vector and tensor terms (flavour index is omitted for the sake of brevity)

 ~H(p)=m2vΛ2HS(p2)∓γ5m2vΛ2HP(p2)+γαpα2vΛ2HV(p2)±iγ5γαfαβpβ2vΛ2HA(p2)+σαβmfαβ4vΛ2HT(p2). (10)

Here ”” corresponds to self-dual and anti-self-dual background field configurations. One can easily reconstruct explicit form of functions from Eq.(III). More detailed description of different form factors, particularly the scalar one, and their role in the chiral symmetry realization will be given in section V. This structure of the quark propagator plays important role for successful description of the meson spectrum, especially for the ground state light mesons.

There are two equivalent ways to derive effective meson action based on the functional integral (4) with the interaction term taken in the form (6). The first one is bosonization of the functional integral in terms of bilocal meson-like fields (see for example Ref. roberts ()). We shall return to this option in the discussion section. Another, more elucidative way is to decompose the bilocal currents (5) over complete set of functions orthogonal with the weight determined by function originating from the gluon propagator (7) in Eq.(6)

 JaJ(x,z)=∞∑n,l=0(z2)l/2fnlμ1…μl(z)JaJlnμ1…μl(x).

Here is the radial quantum number and is the orbital momentum. Coefficient quark currents have to describe intrinsic structure of the collective meson-like excitations with complete set of quantum numbers. The form of interaction (6) and natural requirement of diagonality (with respect to and ) of the four-quark interaction, expressed in terms of the currents , indicate the choice of

 fnlμ1…μl=Lnl(z2)T(l)μ1…μl(nz),nz=z/√z2. (11)

Here are irreducible tensors of four-dimensional rotational group, and generalized Laguerre polynomials obey relation

 ∫∞0duρl(u)Lnl(u)Ln′l(u)=δnn′,ρl(u)=ule−u.

The weight comes from the gluon propagator (7). Nonlocal quark currents with complete set of meson quantum numbers can be explicitly calculated and depend only on the center of mass coordinate  EN (); NK4 (),

 JaJlnμ1…μl(x)=¯q(x)VaJlnμ1…μl(\lx@stackrel↔D(x)Λ)q(x), VaJlnμ1…μl(x)=CJlnMaΓJFnl(\lx@stackrel↔D2(x)Λ2)T(l)μ1…μl(1i\lx@stackrel↔D(x)Λ), (12) Fnl(s)=sn∫10dttn+lexp(st), C2Jln=CJl+12ln!(n+l)!,CS/P=2CV/A=19,

where and are flavour and Dirac matrices respectively. The four-fermion interaction takes the form of an infinite sum of the current-current interactions diagonal with respect to all quantum numbers

 L=g22∑aJln∫d4xJ†aJln(x)JaJln(x).

It has to be stressed that the nonlocal quark currents are invariant with respect to the local gauge transformations of the background gauge field as the vertices (12) depend on the covariant derivatives.

The truncated QCD functional integral can be rewritten in terms of the composite colorless meson fields by means of the standard bosonization procedure: introduce the auxiliary meson fields, integrate out the quark fields, perform the orthogonal transformation of the auxiliary fields that diagonalizes the quadratic part of the action and, finally, rescale the meson fields to provide the correct residue of the meson propagator at the pole corresponding to its physical mass (if any). More details can be found in Ref. EN (); EN1 (); NK4 (). The result can be written in the following compact form

 Z=N∫DϕQexp{−Λ22h2Qg2∫d4xϕ2Q(x)−∞∑k=21kWk[ϕ]}, (13) Wk[ϕ]=∑Q1…QkhQ1…hQk∫d4x1…∫d4xkΦQ1(x1)…ΦQk(xk)Γ(k)Q1…Qk(x1,…,xk), (14) ΦQ(x)=∫d4p(2π)4eipxOQQ′(p)~ϕQ′(p), (15)

where condensed index denotes all relevant meson quantum numbers and indices. Integration variables in the functional integral (13) correspond to the physical meson fields that diagonalize the quadratic part of the effective meson action (14) in momentum representation, which is achieved by means of orthogonal transformation .

Interactions between physical meson fields are described by -point nonlocal vertices ,

 Γ(2)Q1Q2=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(2)Q1Q2(x1,x2)−Ξ2(x1−x2)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(1)Q1G(1)Q2, Γ(3)Q1Q2Q3=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(3)Q1Q2Q3(x1,x2,x3)−32Ξ2(x1−x3)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(2)Q1Q2(x1,x2)G(1)Q3(x3) +12Ξ3(x1,x2,x3)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(1)Q1(x1)G(1)Q2(x2)G(1)Q3(x3), Γ(4)Q1Q2Q3Q4=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(4)Q1Q2Q3Q4(x1,x2,x3,x4)−43Ξ2(x1−x2)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(1)Q1(x1)G(3)Q2Q3Q4(x2,x3,x4) −12Ξ2(x1−x3)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(2)Q1Q2(x1,x2)G(2)Q3Q4(x3,x4) +Ξ3(x1,x2,x3)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(1)Q1(x1)G(1)Q2(x2)G(2)Q3Q4(x3,x4) −16Ξ4(x1,x2,x3,x4)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(1)Q1(x1)G(1)Q2(x2)G(1)Q3(x3)G(1)Q4(x4),

subsequently tuned to the physical meson representation by means of corresponding orthogonal transformations . Vertices are expressed via 1-loop diagrams which include nonlocal quark-meson vertices (12) and quark propagators (III) :

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(k)Q1…Qk(x1,…,xk)=∫dBTrVQ1(x1|B)S(x1,x2)…VQk(xk|B)S(xk,x1),

Bar denotes integration over all configurations of the background fields. As it is illustrated in Fig.3, vertex functions include, in general, several one-loop diagrams correlated via the background field. In the simplified model of spherical domains, the -point correlator is given by a volume of overlap of four-dimensional hyperspheres NK1 (); NK4 ().

It has to be noted that though all Dirac structures, besides the vector and scalar ones, are nullified by the integration of the propagator (10) over the background field, all of them give highly nontrivial contribution to the quark loops (products of several propagators). For example, the two-point correlators responsible for the mass spectrum contain not only the one gluon (fluctuation !) exchange interaction hidden in the vertex but also additional , , , and interactions effectively generated by the background gluon field .

It has to be stressed that the terms linear in meson fields are absent in (13). The linear terms naturally vanish for all mesons besides the scalar ones, and their elimination for the scalar fields requires solution of an infinite system of equations

 Λ2Φ(0)Q1=∞∑k=1gkk∑Q1…QkΦ(0)Q2…Φ(0)QkΓ(k)Q1…Qk, (16)

where and can be treated as an infinite set of scalar quark condensates labelled by the radial quantum number . As we shall discuss in section V, solution of this system of equations leads to the interesting details of the chiral symmetry realization in the presence of the background field under consideration. Actual calculations further below will be done with constant mass which from now on will be treated as the infrared limit of the running nonperturbative quark masses considered as parameters of the model.

The mass spectrum of mesons and quark-meson coupling constants are determined by the quadratic part of the effective meson action via equations

 1=g2Λ2~ΠQ(−M2Q|B), (17) h−2Q=ddp2~ΠQ(p2)|p2=−M2Q, (18)

where is the diagonalized two-point correlator put on mass shell:

 ~ϕ†Q(−p)[OT(p)~Γ(2)(p)O(p)]QQ′~ϕQ′(p)|p2=−M2Q=~ΠQ(−M2Q)~ϕ†Q(−p)~ϕQ(p)|p2=−M2Q.

Explicit construction of will be discussed in the next section. Solution of Eq. (17) identifies the position of the pole in the propagator of the meson with quantum numbers . Definition (18) of the meson-quark coupling constant provides correct residue at the pole.

The free parameters of the model are the IR limits of the running renormalized strong coupling constant , quark masses , , , , and the scales and . By construction, the coupling constant and the quark masses correspond to the background Feynman gauge condition and momentum subtraction (MOM) renormalization scheme at subtraction point . The scale and mean domain size are related to the scalar gluon condensate and topological susceptibility of pure gluodynamics respectively,

 ⟨αsF2⟩=23Λ4π,χYM=172Λ8R4128π2.

It should be noted that decomposition (11) and (12) attributes the same radial form factor to the mesons with different spin-parity . Moreover, the form of appears to be the same for all quarkonium-like collective excitations with different quark content and spin-parity such as and mesons. On the contrary, the physical meson states correspond to the momentum dependent transformed basis and respectively transformed quark current

 faJlnp(z)=∞∑n′=0Onn′aJl(p)fln′(z),  OaJlOTaJl=I, ~JaJ(p,z)=∞∑nl(z2)l/2faJnlp(z)Onn′aJl~JaJln′(p),

where is an orthogonal transformation of the initial basis taking into account two-point function . All this means that though ab initio the basic property of quark-meson interaction form factor is set up by gluon propagator, it is the quark loop that defines its final physical form which is different for different mesons.

Iv Masses and decay constants of mesons

iv.1 Mass spectrum of radial excitations of light, heavy-light mesons and heavy quarkonia

Meson masses are defined by the algebraic equation (17). This equation emerges as follows. In the momentum representation, the quadratic part of the effective action pseudoscalar and vector meson fields with zero orbital momentum has the form

 S2=−12∫d4p(2π)4~ΦaV0nμ(−p)[Λ2δaa′δμμ′δnn′−g2~Γ(2)μμ′aV0n,a′V0n′(p)]Φa′V0n′μ′(p) −12∫d4p(2π)4~ΦaP0n(−p)[Λ2δaa′δnn′−g2~Γ(2)aP0n,a′P0n′(p2)]~Φa′P0n′(p),

where vector two point correlator has the structure

 ~Γ(2)μμ′aV0n,a′V0n′(p)=~Γ(2)aV0n,a′V0n′(p2)δμμ′+~LaV0n,a′V0n′(p2)pμpμ′. (19)

Vector fields (see Eq.(15)) are subject to the on-shell condition

 pμϕaV0nμ=0, p2=−M2aV0n,

while the mass () is determined by (17) with

 ~ΠQ(p)

format: plain


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Documentation

## Maximize Long-Term Investments Using Linear Programming: Problem-Based

This example shows how to use the problem-based approach to solve an investment problem with deterministic returns over a fixed number of years `T`. The problem is to allocate your money over available investments to maximize your final wealth. For the solver-based approach, see Maximize Long-Term Investments Using Linear Programming: Solver-Based.

### Problem Formulation

Suppose that you have an initial amount of money `Capital_0` to invest over a time period of `T` years in `N` zero-coupon bonds. Each bond pays a fixed interest rate, compounds the investment each year, and pays the principal plus compounded interest at the end of a maturity period. The objective is to maximize the total amount of money after `T` years.

You can include a constraint that no single investment is more than a certain fraction of your total capital at the time of the investment.

This example shows the problem setup on a small case first, and then formulates the general case.

You can model this as a linear programming problem. Therefore, to optimize your wealth, formulate the problem using the optimization problem approach.

### Introductory Example

• The starting amount to invest `Capital_0` is \$1000.

• The time period `T` is 5 years.

• The number of bonds `N` is 4.

• To model uninvested money, have one option B0 available every year that has a maturity period of 1 year and a interest rate of 0%.

• Bond 1, denoted by B1, can be purchased in year 1, has a maturity period of 4 years, and interest rate of 2%.

• Bond 2, denoted by B2, can be purchased in year 5, has a maturity period of 1 year, and interest rate of 4%.

• Bond 3, denoted by B3, can be purchased in year 2, has a maturity period of 4 years, and interest rate of 6%.

• Bond 4, denoted by B4, can be purchased in year 2, has a maturity period of 3 years, and interest rate of 6%.

By splitting up the first option B0 into 5 bonds with maturity period of 1 year and interest rate of 0%, this problem can be equivalently modeled as having a total of 9 available bonds, such that for `k=1..9`

• Entry `k` of vector `PurchaseYears` represents the beginning of the year that bond `k` is available for purchase.

• Entry `k` of vector `Maturity` represents the maturity period ${m}_{k}$ of bond `k`.

• Entry `k` of vector `MaturityYears` represents the end of the year that bond `k` is available for sale.

• Entry `k` of vector `InterestRates` represents the percentage interest rate ${\rho }_{k}$ of bond `k`.

Visualize this problem by horizontal bars that represent the available purchase times and durations for each bond.

```% Time period in years T = 5; % Number of bonds N = 4; % Initial amount of money Capital_0 = 1000; % Total number of buying oportunities nPtotal = N+T; % Purchase times PurchaseYears = [1;2;3;4;5;1;5;2;2]; % Bond durations Maturity = [1;1;1;1;1;4;1;4;3]; % Bond sale times MaturityYears = PurchaseYears + Maturity - 1; % Interest rates in percent InterestRates = [0;0;0;0;0;2;4;6;6]; % Return after one year of interest rt = 1 + InterestRates/100; plotInvestments(N,PurchaseYears,Maturity,InterestRates)```

### Decision Variables

Represent your decision variables by a vector `x`, where `x(k)` is the dollar amount of investment in bond `k`, for `k = 1,...,9`. Upon maturity, the payout for investment `x(k)` is

`$x\left(k\right)\left(1+{\rho }_{k}/100{\right)}^{{m}_{k}}.$`

Define ${\beta }_{k}=1+{\rho }_{k}/100$ and define ${r}_{k}$ as the total return of bond `k`:

`${r}_{k}=\left(1+{\rho }_{k}/100{\right)}^{{m}_{k}}={\beta }_{k}^{{m}_{k}}.$`

```x = optimvar('x',nPtotal,'LowerBound',0); % Total returns r = rt.^Maturity;```

### Objective Function

The goal is to choose investments to maximize the amount of money collected at the end of year `T`. From the plot, you see that investments are collected at various intermediate years and reinvested. At the end of year `T`, the money returned from investments 5, 7, and 8 can be collected and represents your final wealth:

`$\underset{x}{\mathrm{max}}{x}_{5}{r}_{5}+{x}_{7}{r}_{7}+{x}_{8}{r}_{8}$`

Create an optimization problem for maximization, and include the objective function.

```interestprob = optimproblem('ObjectiveSense','maximize'); interestprob.Objective = x(5)*r(5) + x(7)*r(7) + x(8)*r(8);```

### Linear Constraints: Invest No More Than You Have

Every year, you have a certain amount of money available to purchase bonds. Starting with year 1, you can invest the initial capital in the purchase options ${x}_{1}$ and ${x}_{6}$, so:

`${x}_{1}+{x}_{6}={Capital}_{0}$`

Then for the following years, you collect the returns from maturing bonds, and reinvest them in new available bonds to obtain the system of equations:

`$\begin{array}{ll}{x}_{2}+{x}_{8}+{x}_{9}& ={r}_{1}{x}_{1}\\ {x}_{3}& ={r}_{2}{x}_{2}\\ {x}_{4}& ={r}_{3}{x}_{3}\\ {x}_{5}+{x}_{7}& ={r}_{4}{x}_{4}+{r}_{6}{x}_{6}+{r}_{9}{x}_{9}\end{array}$`

```investconstr = optimconstr(T,1); investconstr(1) = x(1) + x(6) == Capital_0; investconstr(2) = x(2) + x(8) + x(9) == r(1)*x(1); investconstr(3) = x(3) == r(2)*x(2); investconstr(4) = x(4) == r(3)*x(3); investconstr(5) = x(5) + x(7) == r(4)*x(4) + r(6)*x(6) + r(9)*x(9); interestprob.Constraints.investconstr = investconstr;```

### Bound Constraints: No Borrowing

Because each amount invested must be positive, each entry in the solution vector $x$ must be positive. Include this constraint by setting a lower bound on the solution vector $x$. There is no explicit upper bound on the solution vector.

`x.LowerBound = 0;`

### Solve the Problem

Solve this problem with no constraints on the amount you can invest in a bond. The interior-point algorithm can be used to solve this type of linear programming problem.

```options = optimoptions('linprog','Algorithm','interior-point'); [sol,fval,exitflag] = solve(interestprob,'options',options)```
```Solving problem using linprog. Solution found during presolve. ```
```sol = struct with fields: x: [9x1 double] ```
```fval = 1.2625e+03 ```
```exitflag = OptimalSolution ```

### Visualize the Solution

The exit flag indicates that the solver found an optimal solution. The value `fval`, returned as the second output argument, corresponds to the final wealth. Look at the final sum of investments, and the investment allocation over time.

```fprintf('After %d years, the return for the initial \$%g is \$%g \n',... T,Capital_0,fval);```
```After 5 years, the return for the initial \$1000 is \$1262.48 ```
`plotInvestments(N,PurchaseYears,Maturity,InterestRates,sol.x)`

### Optimal Investment with Limited Holdings

To diversify your investments, you can choose to limit the amount invested in any one bond to a certain percentage `Pmax` of the total capital that year (including the returns for bonds that are currently in their maturity period). You obtain the following system of inequalities:

`$\begin{array}{ll}{x}_{1}& \le Pmax×{Capital}_{0}\\ {x}_{2}& \le Pmax×\left({\beta }_{1}{x}_{1}+{\beta }_{6}{x}_{6}\right)\\ {x}_{3}& \le Pmax×\left({\beta }_{2}{x}_{2}+{\beta }_{6}^{2}{x}_{6}+{\beta }_{8}{x}_{8}+{\beta }_{9}{x}_{9}\right)\\ {x}_{4}& \le Pmax×\left({\beta }_{3}{x}_{3}+{\beta }_{6}^{3}{x}_{6}+{\beta }_{8}^{2}{x}_{8}+{\beta }_{9}^{2}{x}_{9}\right)\\ {x}_{5}& \le Pmax×\left({\beta }_{4}{x}_{4}+{\beta }_{6}^{4}{x}_{4}+{\beta }_{8}^{3}{x}_{8}+{\beta }_{9}^{3}{x}_{9}\right)\\ {x}_{6}& \le Pmax×{Capital}_{0}\\ {x}_{7}& \le Pmax×\left({\beta }_{4}{x}_{4}+{\beta }_{6}^{4}{x}_{4}+{\beta }_{8}^{3}{x}_{8}+{\beta }_{9}^{3}{x}_{9}\right)\\ {x}_{8}& \le Pmax×\left({\beta }_{1}{x}_{1}+{\beta }_{6}{x}_{6}\right)\\ {x}_{9}& \le Pmax×\left({\beta }_{1}{x}_{1}+{\beta }_{6}{x}_{6}\right)\end{array}$`

```% Maximum percentage to invest in any bond Pmax = 0.6; constrlimit = optimconstr(nPtotal,1); constrlimit(1) = x(1) <= Pmax*Capital_0; constrlimit(2) = x(2) <= Pmax*(rt(1)*x(1) + rt(6)*x(6)); constrlimit(3) = x(3) <= Pmax*(rt(2)*x(2) + rt(6)^2*x(6) + rt(8)*x(8) + rt(9)*x(9)); constrlimit(4) = x(4) <= Pmax*(rt(3)*x(3) + rt(6)^3*x(6) + rt(8)^2*x(8) + rt(9)^2*x(9)); constrlimit(5) = x(5) <= Pmax*(rt(4)*x(4) + rt(6)^4*x(6) + rt(8)^3*x(8) + rt(9)^3*x(9)); constrlimit(6) = x(6) <= Pmax*Capital_0; constrlimit(7) = x(7) <= Pmax*(rt(4)*x(4) + rt(6)^4*x(6) + rt(8)^3*x(8) + rt(9)^3*x(9)); constrlimit(8) = x(8) <= Pmax*(rt(1)*x(1) + rt(6)*x(6)); constrlimit(9) = x(9) <= Pmax*(rt(1)*x(1) + rt(6)*x(6)); interestprob.Constraints.constrlimit = constrlimit;```

Solve the problem by investing no more than 60% in any one asset. Plot the resulting purchases. Notice that your final wealth is less than the investment without this constraint.

`[sol,fval] = solve(interestprob,'options',options);`
```Solving problem using linprog. Minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the selected value of the function tolerance, and constraints are satisfied to within the selected value of the constraint tolerance. ```
```fprintf('After %d years, the return for the initial \$%g is \$%g \n',... T,Capital_0,fval);```
```After 5 years, the return for the initial \$1000 is \$1207.78 ```
`plotInvestments(N,PurchaseYears,Maturity,InterestRates,sol.x)`

### Model of Arbitrary Size

Create a model for a general version of the problem. Illustrate it using `T` = 30 years and 400 randomly generated bonds with interest rates from 1 to 6%. This setup results in a linear programming problem with 430 decision variables.

```% For reproducibility rng default % Initial amount of money Capital_0 = 1000; % Time period in years T = 30; % Number of bonds N = 400; % Total number of buying oportunities nPtotal = N + T; % Generate random maturity durations Maturity = randi([1 T-1],nPtotal,1); % Bond 1 has a maturity period of 1 year Maturity(1:T) = 1; % Generate random yearly interest rate for each bond InterestRates = randi(6,nPtotal,1); % Bond 1 has an interest rate of 0 (not invested) InterestRates(1:T) = 0; % Return after one year of interest rt = 1 + InterestRates/100; % Compute the return at the end of the maturity period for each bond: r = rt.^Maturity; % Generate random purchase years for each option PurchaseYears = zeros(nPtotal,1); % Bond 1 is available for purchase every year PurchaseYears(1:T)=1:T; for i=1:N % Generate a random year for the bond to mature before the end of % the T year period PurchaseYears(i+T) = randi([1 T-Maturity(i+T)+1]); end % Compute the years where each bond reaches maturity at the end of the year MaturityYears = PurchaseYears + Maturity - 1;```

Compute the times when bonds can be bought or sold. The `buyindex` matrix holds the potential purchase times, and the `sellindex` matrix holds the potential sales times for each bond.

```buyindex = false(nPtotal,T); % allocate nPtotal-by-T matrix for ii = 1:T buyindex(:,ii) = PurchaseYears == ii; end sellindex = false(nPtotal,T); for ii = 1:T sellindex(:,ii) = MaturityYears == ii; end```

Set up the optimization variables corresponding to the bonds.

`x = optimvar('x',nPtotal,1,'LowerBound',0);`

Create the optimization problem and objective function.

```interestprob = optimproblem('ObjectiveSense','maximize'); interestprob.Objective = sum(x(sellindex(:,T)).*r(sellindex(:,T)));```

For convenience, create a temporary array xBuy, whose columns represent the bonds we can buy at each time period.

`xBuy = repmat(x,1,T).*double(buyindex);`

Similarly, create a temporary array xSell, whose columns represent the bonds we can sell at each time period.

`xSell = repmat(x,1,T).*double(sellindex);`

The return generated for selling these bounds is

`xReturnFromSell = xSell.*repmat(r,1,T);`

Create the constraint that the amount you invest in each time period is the amount that you sold in the previous time period.

```interestprob.Constraints.InitialInvest = sum(xBuy(:,1)) == Capital_0; interestprob.Constraints.InvestConstraint = sum(xBuy(:,2:T),1) == sum(xReturnFromSell(:,1:T-1),1);```

### Solution with No Holding Limit

Solve the problem.

```tic [sol,fval,exitflag] = solve(interestprob,'options',options);```
```Solving problem using linprog. Minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the selected value of the function tolerance, and constraints are satisfied to within the selected value of the constraint tolerance. ```
`toc`
```Elapsed time is 0.218624 seconds. ```

How well did the investments do?

```fprintf('After %d years, the return for the initial \$%g is \$%g \n',... T,Capital_0,fval);```
```After 30 years, the return for the initial \$1000 is \$5167.58 ```

### Solution with Limited Holdings

To create constraints that limit the fraction of investments in each asset, set up a matrix that keeps track of the active bonds at each time. To express the constraint that each investment must be less than `Pmax` times the total value, set up a matrix that keeps track of the value of each investment at each time. For this larger problem, set the maximum fraction that can be held to 0.4.

`Pmax = 0.4;`

Create an `active` matrix corresponding to times when a bond can be held, and a `cactive` matrix that holds the cumulative duration of each active bond. So the value of bond `j` at time `t` is `x(j)*(rt^cactive)`.

```active = double(buyindex | sellindex); for ii = 1:T active(:,ii) = double((ii >= PurchaseYears) & (ii <= MaturityYears)); end cactive = cumsum(active,2); cactive = cactive.*active;```

Create the matrix whose entry (j,p) represents the value of bond j at time period p:

`bondValue = repmat(x, 1, T).*active.*(rt.^(cactive));`

Determine the total value of the investments at each time interval so you can impose the constraint on limited holdings. `mvalue` is the money invested in all the bonds at the end of each time period, an `nPtotal`-by-`T` matrix.moneyavailable is the sum over the bonds of the money invested at the beginning of the time period, meaning the value of the portfolio at each time.

```constrlimit = optimconstr(nPtotal,T); constrlimit(:,1) = xBuy(:,1) <= Pmax*Capital_0; constrlimit(:,2:T) = xBuy(:,2:T) <= repmat(Pmax*sum(bondValue(:,1:T-1),1), nPtotal, 1).*double(buyindex(:,2:T)); interestprob.Constraints.constrlimit = constrlimit;```

Solve the problem with limited holdings.

```tic [sol,fval,exitflag] = solve(interestprob,'options',options);```
```Solving problem using linprog. Minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the selected value of the function tolerance, and constraints are satisfied to within the selected value of the constraint tolerance. ```
`toc`
```Elapsed time is 1.561167 seconds. ```
```fprintf('After %d years, the return for the initial \$%g is \$%g \n',... T,Capital_0,fval);```
```After 30 years, the return for the initial \$1000 is \$5095.26 ```

To speed up the solver, try the dual-simplex algorithm.

```options = optimoptions('linprog','Algorithm','dual-simplex'); tic [sol,fval,exitflag] = solve(interestprob,'options',options);```
```Solving problem using linprog. Optimal solution found. ```
`toc`
```Elapsed time is 0.659623 seconds. ```
```fprintf('After %d years, the return for the initial \$%g is \$%g \n',... T,Capital_0,fval);```
```After 30 years, the return for the initial \$1000 is \$5095.26 ```

In this case, the dual-simplex algorithm took less time to obtain the same solution.

### Qualitative Result Analysis

To get a feel for the solution, compare it to the amount `fmax` that you would get if you could invest all of your starting money in one bond with a 6% interest rate (the maximum interest rate) over the full 30 year period. You can also compute the equivalent interest rate corresponding to your final wealth.

```% Maximum amount fmax = Capital_0*(1+6/100)^T; % Ratio (in percent) rat = fval/fmax*100; % Equivalent interest rate (in percent) rsol = ((fval/Capital_0)^(1/T)-1)*100; fprintf(['The amount collected is %g%% of the maximum amount \$%g '... 'that you would obtain from investing in one bond.\n'... 'Your final wealth corresponds to a %g%% interest rate over the %d year '... 'period.\n'], rat, fmax, rsol, T)```
```The amount collected is 88.7137% of the maximum amount \$5743.49 that you would obtain from investing in one bond. Your final wealth corresponds to a 5.57771% interest rate over the 30 year period. ```
`plotInvestments(N,PurchaseYears,Maturity,InterestRates,sol.x,false)`


text:
# Where do the coefficients in the Catmull-Clark subdivision algorithm come from?

I'm learning about subdivision surface algorithms. The "Catmull-Clark" algorithm seems to be one of the most widely-used classical algorithms.

The introduction of new face points and edges is straightforward, but I'm confused by the following formula for modifying the locations of original points

$$\frac{F + 2R + (n-3)P}{n}$$

where:

• $P$ is the original point
• $R$ is the average of all $n$ edges midpoints touching $P$
• $F$ is the average of all $n$ face points touching $P$

I understand the result is a weighted average of three points $F$, $R$ and $P$, but where do the coefficients $1$, $2$, and $(n-3)$ come from? Why $2R$? Why not $2.01R$? Or $1R$ and $2F$?

All the introductions I've seen just present the formula without presenting a justification.

The derivation is presented in the original paper that introduced CC subdivisions as a generalisation of B-Spline patches: https://people.eecs.berkeley.edu/~sequin/CS284/PAPERS/CatmullClark_SDSurf.pdf

• Thanks. I'm struggling to follow the derivation somewhat. If I'm not wrong, every entry of the $H_1$ matrix should be multiplied by a factor of $\frac{1}{8}$. Feb 15 '17 at 5:08
• Thanks for the link, but in general link-only answers are discouraged on StackExchange. I'm not suggesting you paste the entire derivation in, but perhaps you could summarize the key points (e.g. assumptions that lead to the specific coefficients) in your answer? Feb 20 '17 at 16:00

I'm just going to add a few more details to the paper that Stefan linked to.

First of all, the matrix $H_1$ displayed in the paper is incorrect; every element should be multiplied by a factor of $\frac{1}{8}$.

$$H_1 = \frac{1}{8} \begin{bmatrix} 4 & 4 & 0 & 0 \\ 1 & 6 & 1 & 0 \\ 0 & 4 & 4 & 0 \\ 0 & 1 & 6 & 1 \end{bmatrix}$$

Second, the derivation of the new face point formula for quads in the first half of the paper is confusing, so I will fill in some of the details.

The matrix $G_1 = HGH^T$ contains the 16 control points for one sub-quad of the original quad with control points $G$. The entries of $G_1$ are referred to with $q_{i,j}$. Of most interest are the elements $q_{1,1}$, $q_{1,2}$ and $q_{2,2}$, which represent control points for a corner, edge, and interior of the $4\times 4$ control point grid for the sub-quad. All the other elements have similar formulas due to the symmetry of the subdivision process.

By carrying out the matrix multiplication, we get:

$q_{2,2} = \frac{P_{1,1} +P_{3,1} +P_{3,1} +P_{3,3}}{64} + \frac{6(P_{1,2} +P_{2,1} +P_{3,2} +P_{2,3})}{64} + \frac{36P_{2,2}}{64}$

We can see from this formula that, when relocating a control point, its new location is determined by the $3\times 3$ grid of points surrounding it. Here we see that in this grid, corner points have $\frac{1}{64}$ influence, edge points have $\frac{6}{64} = \frac{3}{32}$ influence, and the center ("old") point has $\frac{36}{64} = \frac{9}{16}$ influence. These weights are consistent with the ones given at the bottom of this blog post.

With a bit of rearranging, we get:

$q_{2,2} = \frac{1}{4} \big[ \frac{1}{4}(\frac{P_{1,1} + P_{1,2} + P_{2,1} + P_{2,2}}{4} + \frac{P_{1,2} + P_{2,2} + P_{1,3} + P_{2,3}}{4} + \frac{P_{2,1} + P_{2,2} + P_{3,2} + P_{3,1}}{4} + \frac{P_{2,2} + P_{2,3} + P_{3,2} + P_{3,3}}{4}) + 2\frac{1}{4}(\frac{P_{1,2} + P_{2,2}}{2} + \frac{P_{2,1} + P_{2,2}}{2} + \frac{P_{2,3} + P_{2,2}}{2} + \frac{P_{3,2} + P_{2,2}}{2}) + P_{2,2}\big]$

The first term is merely the average of the four surrounding face points; in other words, it is $F$. The second term is the average of the four surrounding edge midpoints multiplied by $2$; in other words, it is $2R$. Finally, $P_{2,2}$ is simply the original control point $P$.

And so we see this is the same as the original formula for quads: $\frac{1}{4}(F + 2R + P)$.

The paper does not provide a derivation of the general case for an $n$-sided polygon, but it seems like a reasonable extension based on intuition.

As an aside, here is an explanation for the other points, and their meanings as seen on wikipedia:

$q_{1,1} = \frac{P_{1,1} +P_{1,2} +P_{2,1} +P_{2,2}}{4}$

This is simply the formula for adding a new control point at the center of an existing quad.

$q_{1,2} = \frac{P_{1,1} +P_{1,2} +P_{2,1} +P_{2,3}}{16} + \frac{6(P_{1,2} +P_{2,1})}{16}$

Rearranging this, we see it is just the average of the two neighbouring face points and the edge endpoints:

$q_{1,2} = \frac{1}{4}(\frac{P_{1,1} +P_{2,1} +P_{1,2} +P_{2,2}}{4} + \frac{P_{1,2} +P_{2,2} +P_{1,3} +P_{2,3}}{4} + P_{1,2} + P_{2,1})$


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# Math Help - a sequence of real numbers

1. ## a sequence of real numbers

Let
{xn} be a sequence of real numbers that is bounded above by M and such that

xn ® x . Prove that x £ M .

we know that
A
sequence {xn} is bounded if there exists an x ÎX and a real number

M > 0 such that d (xn , x) £ M for all n . also it convergent to x.

how can I use this to prove. any help will be apprciated.

2. Originally Posted by koko2009
Let

{xn} be a sequence of real numbers that is bounded above by M and such that

xn ® x . Prove that x £ M .

we know that

A
sequence {xn} is bounded if there exists an x ÎX and a real number
M > 0 such that d (xn , x) £ M for all n . also it convergent to x.

how can I use this to prove. any help will be apprciated.
$x>M \Longrightarrow\,\, choose\,\, e=\frac{x-M}{2}\Longrightarrow |x_n-x|N\,,\,\,for\,\,some\,\,N \in \mathbb{N}$ $\Longrightarrow \frac{M-x}{2}= -e \leq x_n-x \leq e=\frac{x-M}{2}$

Now get your contradiction that $\forall n \in \mathbb{N}\,,\,|x_n|

Tonio


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# Tag Info

46

A specific parameter might correspond to a specific (SI) unit, but not all units correspond to a specific parameter! Kinetic energy is \begin{align} K&=\frac{1}{2} mv^2 \\ \text{Joules}&=\frac{1}{2} \text{kilograms}\times\text{meters}^2/\text{seconds}^2 \end{align} We also have gravitational potential energy: \begin{align}U&=mgh \\ \... 9 They are technically units for incommensurate quantities, but in practice this is often just a technicality. The magnetic field that makes sense (B) is measured in teslas (SI) or gauss (CGS), and the magnetic field that people spoke about 100 years ago (H) is measured in amps per meter (SI, also equivalent to a number of other things) or oersteds (CGS). ... 7 For one, the laws of physics are the same whether you're working in SI or Imperial. F = ma, regardless if your m is in kg, pounds mass, or solar masses. Actually, in college we had a bonus challenge to give the answer to a problem in craziest units of energy that we could come up with that actually worked. "Slug lightyears" was a pretty good one, but the ... 7 Writing equations using only units would not work at all for dimensionless equations. For example the Snell's lawn_1 \sin\theta_1=n_2 \sin\theta_2.$$You would also lose many of the dimensionless (but usefull) parameters in physics such as the Lorentz factor$$\gamma=\frac{1}{\sqrt{1-v^2/c^2}}. Also consider equations whose all variables have the same ...

7

For simple equations, the two might be equivalent. Certainly, dimensionally an equation must always be correct. But there are plenty of situations where the units may not obviously reflect a particular quantity; and clarity of communication improves understanding. Take electrostatics. If I say "1 Volt" you know what I mean; an electric field is "Volts per ...

7

Light years and parsecs have been used since long before SI existed, so a lot of it is tradition. But using light years also makes it very obvious how long the light has traveled to get here, and thus which era of the universe we are seeing the object in. Something that is 11 billion light years away dates from the era of early galaxies, for example. If you ...

6

This is a relatively tricky one, because it involves the differences between the $\mathbf B$ field and the $\mathbf H$ field in the SI and CGS systems, and those relationships change in the different systems. In short: Oersteds are used to measure the $\mathbf H$ field in CGS units. Teslas are used to measure the $\mathbf B$ field in SI units. In the SI ...

6

W.u stands for Weisskopf unit: [ref 1, ref 2]. Despite being called a 'unit', it does not have a universal value; the value of the Weisskopf unit depends on the mass number of the nucleus in question and which transition the nucleus is undergoing ($E\lambda$ or $M\lambda$). The references contain expressions for the value of a Weisskopf unit in terms of $A$ ...

5

In nuclear Physics estimates can be made using the shell model of the nucleus of the gamma ray transition rates in excited nucleii and such estimates are named after Victor Weisskopf. A measured rate is compared with the Weisskopf estimate and the ratio is said to be in Weisskopf units (W.u.).

5

Disclaimer: I have no idea what question everyone else is answering; none of the other answers seem to address the question as I understand it. We don't use units in formula because not everything we want a variable to represent (like $a$ for acceleration) has its own nice unit. Acceleration is a perfect example: it's just too long to say "meters per ...

4

If a physics equation is to be valid, it is necessary for the units to work out, but it is not sufficient. The handling of dimensionality only provides part of the information. Units and dimensionality are good checks to make sure you did the equation right, but the mere fact that the units were right does not automatically mean the equation was right. ...

4

The purpose of units is to assign numbers to measurements. They are necessary but of secondary importance to the thing being measured. Scientists want to describe the real world with their equations, not just their measurement tools.

4

From a quick google search, it seems that Oersteds are used for defining magnetic field strength and Teslas are used for defining magnetic field strength in terms of flux density. They seem to not really be meant to be converted between, though you technically can (as evidenced by the other answers here). This website and this website might be helpful to ...

3

It is not incorrect, it is incomplete. A physical value consists of: Magnitude - this is "the number" Unit - "type" of the value Physical values are not only magnitudes, they also have a unit. It is inseparable. Unit defines a physical meaning to a value. You could formally treat it as a pair (magnitude, unit). All calculations in physics are done this ...

3

The basic fact is units are a demarcation of the quantity of something, so they can never be used as a fundamental equation. As you have mentioned F=ma can also be written as N=kg ms^-2 , then how can you say that the equation is of Newton's Law only? The equation can be used to define force dimensionally which is [M L T^-2].

2

Why does Light-years or Parsecs seem to be the standard rather than SI? In the solar system astronomy, the astronomical unit is much more widely used rather than meters for distance, days (86400 seconds), Julian years (365.25 days), or Julian centuries (36525 days) are used rather than seconds for time, and the solar mass is used rather than kilograms for ...

1

SI units are created for everyday life and therefore convenient to use in everyday life. The following sentences makes sense in modern human mind; "I live 300 meters away from here" or "My new boat is 35 meters long" or "I am bust now please call in 5 minutes". Moreover the calculations with these numbers are easy. If you hear someone saying "I drove 50 km ...

1

I'm not up to the task of rebuilding geometry from the ground up, but my intuition is that this is sensible and it can be done consistently. It seems to be true that logarithms (or inverse trig functions) take you into a different (transcendental) numerical realm, and exponentials (or trig functions) take you back, and adding quantities from different realms ...

Only top voted, non community-wiki answers of a minimum length are eligible


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# Why does thrust-to-weight ratio depend from max Mach number?

Why, for jet airplane design, does maximum Mach (or maximum speed) affect thrust-to-weight ratio so strictly? (as you can see for example in Daniel P. Raymer "Aircraft Design: A conceptual Approach" Table 5.3 page 80, where you can find $\frac{T}{W_o}=a\cdot M_{max}^C$, where “$a$” and “$C$” are empirical factors, for jet transport $a=0.267$ and $C=0.363$)

Just now I did notice to detail that Raymer say : "The equations were developed by the author using data from Ref 1" Ref1: i.e. the famous annual book : "Jane's All the world Aircrafts" . But anyway "maximum speed" (or maximum Mach number) in this case the maximum speed is ...the speed that the aircraft is not able to exceed or.... the speed that is forbidden for the pilot to try to exceed because it can be dangerous? Because, if it's the first case it's more understandable; in fact, if the other parameters are fixed, with the same fixed distance, if T/W is low the acceleration is lower and so before the end of the distance, you can arrive to low speed. At the opposite if T/W is high the acceleration is higher and so before the end of the distance, you can arrive to high speed. What do you think? So, in Janes book, what is the definition of Maximum Mach number?

• Sounds backwards... T:W would drive the max achievable speed. Even that assumes, though, that your Max achievable speed isn't limited by an MMO or VMO. An F-15 probably doesn't care; many airliners can reach MMO in level flight even at max weight. – Ralph J Mar 8 '16 at 13:46
• @RalphJ it just depends on which way you look at it. When designing an aircraft, max Mach would be a typical requirement, and the given equation will allow you to estimate the T:W ratio needed to achieve it. – fooot Mar 8 '16 at 15:24
• but, why does each affect other? – d.pensopositivo Mar 9 '16 at 9:49
• How can we demonstrate the dependence each other (theoretically, with formula not only with empirical factors)? – d.pensopositivo Apr 21 '16 at 15:59

Disclaimer: I have no copy of Raymer's book, so I have to guess what this table looks like.

There should be no direct dependency, except that all aircraft need to operate in the real world. If you look at the wing loading of regular jet transports, you will notice that they bunch together around 500 - 600 kg/m². This is the maximum wing loading which still allows them to operate from normal airports - go to higher values an you need a dry lakebed for take-offs and especially landings.

To push the resulting wing to higher flight speeds needs more thrust - hence the dependency. If you start with a higher wing loading (say, 800 kg/m²), the thrust/weight ratio for reaching the same Mach number is lower. The approach and landing speeds grow with wing loading as well, which in turn affects the resulting landing distance. I guess there is a maximum landing distance requirement for all aircraft in the table which is not explicitly mentioned.

• Peter you say: " If you start with a higher wing loading (say, 800 kg/m²), the thrust/weight ratio for reaching the same Mach number is lower. " Is not the opposite? – d.pensopositivo May 9 '16 at 14:29
• @d.pensopositivo: No, the smaller wing creates less drag and needs less thrust. Once the aircraft flies fast enough, a high wing loading will make it more efficient. – Peter Kämpf May 9 '16 at 20:59
• @d.pensopositivo Peter Kämpf is correct. At high speeds parasitic drag dominates, and this is proportional to the wing area. If thrust and weight are held constant, higher wing loading -> less wing area -> less drag -> higher speed. See the formula and derivation in my answer for exact details. – Cody P May 12 '16 at 22:24

In a short answer: In practice Mmax and Thrust to weight ratio are not correlated because other factors are more important, but in theory there is some dependency. The answer depends if you're trying to compare planes, in which case you're looking at correlations among dozens of performance parameters and trying to fit estimates into simple equations. Or you could be designing a plane, in which case you'd run equations to make sure the plane has enough thrust to meet the target Mmo. The case presented by Raymer seems to be a mix of the two. He's using industry data to predict necessary T/W early in the design process.

## Design Theory

The most confusing part here is that normally we think of thrust working against drag to provide acceleration, yet here we're talking about steady-state speed, not acceleration. The catch here is that drag is not constant as speed increases. As you accelerate you reach a point where drag becomes strong enough to counteract the maximum engine thrust, and you can't go any faster in level flight (see the first graph [source])).

Assuming thrust is constant with respect to velocity, then from the formulas for drag as a function of speed we can derive the maximum velocity. This maximum velocity (in a thrust-limited level flight) can be written as a function of thrust-to-weight ratio and wing loading: (see derivation here (pdf), compare with here (pdf)) $$V_{max}=\sqrt{\frac{\frac{T_{A,max}}{W}\frac{W}{S}+\frac{W}{S}\sqrt{(\frac{T_{A,max}}{W})^2-4C_{D_{0}}K}}{\rho _{\infty }C_{D_{0}}}}$$ Where $T_a$ is available thrust, $W$ is the weight of the aircraft, $\frac{W}{S}$ is wing loading, $K$ is the correction factor for induced drag, and $C_{D_0}$ is the drag coefficient of aircraft when lift equals zero.

However, you'll notice that there are other parameters in this formula besides thrust to weight ratio like wing loading and drag coefficient. In practice these are not trivial considerations, especially when comparing different planes. Raymer himself notes "Later in the design process, aerodynamics calculation of drag at the design maximum speeds will be used, with other criteria, to establish the required T/W." There are also factors this formula doesn't consider like the variation in thrust as speed changes. These are the reasons thrust to weight ratio and maximum speed are not correlated when comparing different planes.

## Comparing Planes

Let's see if this relationship actually holds out when comparing different planes. What real-world perfomance parameter should we reference for Vmax? Mmo, high speed cruise TAS, or something else? Technically speaking, Mmo is the certified max operational speed and is a safety limit, not just a performance limit. During descents you can pick up energy very quickly and exceed the maximum level flight speed and Mmo. However, in practice (for business jets at least) the Mmo, high speed cruise TAS, and maximum level flight speeds are correlated. After all, you don't design many planes for their diving performance and many jets are designed to cruise close to their Mmo.

The truth is that none of these speeds is closely correlated with thrust/weight when comparing jets. In my data set of 31 business jets from "Business Jets 2012", the correlation between thrust to weight and Mmo has a pitiful R-squared value of 0.20 (see the second graph). I'm not sure what data Raymer was using and if anyone has better data for showing a trend let me know. I should note that once we account for the effects of wing-loading, there is a weak relationship between thrust to weight ratio and maximum speed. So the effect of thrust/weight on Mmax in the above equation is very weak but present (see the third graph).

There are a lot of parameters that loosely correlate with Mmo besides thrust, including aircraft type, cruise altitude, wing loading, and range. These parameters need to be engineered in conjunction to make sure the airplane meets all the performance expectations. Mmo may be reduced or increased so that one of these parameters reaches its target and vice versa.

In theory, there is a correlation, but I have to say my stats don't agree with Raymer's on the variables being "closely related" when comparing different jets. Other variables like drag and wing loading are too important to be neglected.

• Talking about "maximum Mach number" I didn't understand if Raymer means Mach number coming from V(Emax) (for levelling unaccelerate flight) or someone among limit speeds. I suppose, because it recalls Referes from Jane's, that we can find in Jane's, but till now.... – d.pensopositivo May 15 '16 at 15:19
• @d.pensopositivo I believe Raymer is referring to the maximum speed you want the engines to be capable of achieving in level flight during the design process. I'm not expert on how to choose that number and what it is in relation to Mmo, but I do know that for existing planes these top speeds and high speed cruises tend to be pretty closely related. – Cody P May 16 '16 at 7:05


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# Create Mars tensor¶

You can create mars tensor from Python array like object just like Numpy, or create from Numpy array directly. More details on array creation routine and random sampling.

 mars.tensor.tensor mars.tensor.array Create a tensor.

## Create tensor on GPU¶

Mars tensor can run on GPU, for tensor creation, just add a gpu parameter, and set it to True.

import mars.tensor as mt

a = mt.random.rand(1000, 2000, gpu=True)  # allocate the tensor on GPU

## Create sparse tensor¶

Mars tensor can be sparse, unfortunately, only 2-D sparse tensors are supported for now, multi-dimensional tensor will be supported later soon.

import mars.tensor as mt

a = mt.eye(1000, sparse=True)  # create a sparse 2-D tensor with ones on the diagonal and zeros elsewhere

## Chunks¶

In mars tensor, we tile a tensor into small chunks. Argument chunk_size is not always required, a chunk’s bytes occupation will be 128M for the default setting. However, user can specify each chunk’s size in a more flexible way which may be adaptive to the data scale. The fact is that chunk’s size may effect heavily on the performance of execution.

The options or arguments which will effect the chunk’s size are listed below:

• Change options.tensor.chunk_size_limit which is 128*1024*1024(128M) by default.
• Specify chunk_size as integer, like 5000, means chunk’s size is 5000 at most for all dimensions
• Specify chunk_size as tuple, like (5000, 3000)
• Explicitly define sizes of all chunks along all dimensions, like ((5000, 5000, 2000), (2000, 1000))

### Chunks Examples¶

Assume we have such a tensor with the data shown below.

0 9 6 7 6 6
5 7 5 6 9 0
1 6 7 8 6 1
8 0 9 9 9 3
5 4 3 5 8 2
6 2 2 6 9 3
4 2 4 6 2 0
6 8 2 6 5 4

We will show how different chunk_size arguments will tile the tensor.

chunk_size=3:

0 9 6  7 6 6
5 7 5  6 9 0
1 6 7  8 6 1

8 0 9  9 9 3
5 4 3  5 8 2
6 2 2  6 9 3

4 2 4  6 2 0
6 8 2  6 5 4

chunk_size=2:

0 9  6 7  6 6
5 7  5 6  9 0

1 6  7 8  6 1
8 0  9 9  9 3

5 4  3 5  8 2
6 2  2 6  9 3

4 2  4 6  2 0
6 8  2 6  5 4

chunk_size=(3, 2):

0 9  6 7  6 6
5 7  5 6  9 0
1 6  7 8  6 1

8 0  9 9  9 3
5 4  3 5  8 2
6 2  2 6  9 3

4 2  4 6  2 0
6 8  2 6  5 4

chunk_size=((3, 1, 2, 2), (3, 2, 1)):

0 9 6  7 6  6
5 7 5  6 9  0
1 6 7  8 6  1

8 0 9  9 9  3

5 4 3  5 8  2
6 2 2  6 9  3

4 2 4  6 2  0
6 8 2  6 5  4


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# Reliability of Complete Cubic Networks under the Condition of g-Good-Neighbor

Jinqiang Li

The Computer Journal, pp. 625-635, 2017.

EI WOS
Keywords:
complete cubic networksPMC modelMM* modelfault-tolerance analysisdiagnosabilityMore(1+)

Abstract:

Fault tolerance is the ability such that a multiprocessor system operates properly in the event of the failure of some of its components. Latifi et al. [IEEE Trans. Comput. 43 (2) (1994) 218–222] proposed the notion of Rg-connectivity $(\kappa ^g)$ of a network modeled by graph G such that at least $\kappa ^g$ vertices in G should be dele...More

Code:

Data:


text:
# What was the second oldest college in america?

###### Question:

What was the second oldest college in america?

### Will give medal !! Please someone help me ? The choices are a) 2 b) 4 c) 6 d) 8

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### On a sunny day, if a 36-inch yardstick casts a 21-inch shadow, how tall is a tree whose shadow is 168 in.? A.143 in. B.196 in. C.288 in.

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### Which equation represents the hyperbola that has its Foci farthest from its Center

which equation represents the hyperbola that has its Foci farthest from its Center...

### Where would i be standing if im at 45 degrees North, 135 degrees west?

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### The plot of "Fine?" centers on Bailey's fear of what her illness is. Go back through the story and make a list of important events. Label each event to identify what happens at each stage of the plot-exposition, rising action, climax, falling action, and resolution. Explain how each event fits its plot stage.

The plot of "Fine?" centers on Bailey's fear of what her illness is. Go back through the story and make a list of important events. Label each event to identify what happens at each stage of the plot-exposition, rising action, climax, falling action, and resolution. Explain how each event fits its p...

### Which sentence could best be revised to active voice? A. Plays were written and acted by the children. B. Artifacts were found in the archaeological dig. C. Sugar is produced in the fermentation process. D. Success is achieved through work and dedication.

Which sentence could best be revised to active voice? A. Plays were written and acted by the children. B. Artifacts were found in the archaeological dig. C. Sugar is produced in the fermentation process. D. Success is achieved through work and dedication....

### W See oiner posts. assical, Medieval, or Thread 1 - Which do you think is more beautiful- Classical Art, Medieval Art, or Romantic Art? 0 0 0

w See oiner posts. assical, Medieval, or Thread 1 - Which do you think is more beautiful- Classical Art, Medieval Art, or Romantic Art? 0 0 0...

### Plz help me with this I will mark as brainlest

plz help me with this I will mark as brainlest...

### In Miami the largest Spanish speaking group is Guatemalan True or false

In Miami the largest Spanish speaking group is Guatemalan True or false...

### What is a homogeneous and heterogeneous mixture

What is a homogeneous and heterogeneous mixture...

### Can you find and tell me all the prepositional phrases on 9 through 16 .......thanks

Can you find and tell me all the prepositional phrases on 9 through 16 .......thanks...

### ___ divided by 3x3 =12

___ divided by 3x3 =12...

### Thelma classifies the following system of equations. Y= -1/3x + 3. Y=1/3x + 3 Select Correct or Not correct for each statement. Correct Not correct Statement The system is consistent because the slopes are the same. The system is independent because the y-intercepts are different. 1b

Thelma classifies the following system of equations. Y= -1/3x + 3. Y=1/3x + 3 Select Correct or Not correct for each statement. Correct Not correct Statement The system is consistent because the slopes are the same. The system is independent because the y-intercepts are different. 1b...

### The Dance Ensemble at a local high school is selling tickets to their holiday show. On Monday, they sold 24 adult tickets and 2 child tickets totaling $150. The following day they took in$144 by selling 20 adult tickets and 8 child tickets.

The Dance Ensemble at a local high school is selling tickets to their holiday show. On Monday, they sold 24 adult tickets and 2 child tickets totaling $150. The following day they took in$144 by selling 20 adult tickets and 8 child tickets....


text:
NIPS 2017
Mon Dec 4th through Sat the 9th, 2017 at Long Beach Convention Center
Paper ID: 2950 Learning Populations of Parameters

### Reviewer 1

The paper considers a statistical framework where many observations are available but only a few concern each individual parameter of interests. Instead of estimating each parameter individually, following an idea of Stein, the idea is to estimate population parameters", that is to provide an estimate of the empirical distribution of these parameters. Theoretically, an estimation of the empirical distribution based on its first moments is proposed and a control of the risk of this estimator with respect to the Wasserstein distance is proved. Then, an algorithm for practical implementation of this estimator is proposed and illustrates the performances of the estimator on both synthetic and real data-sets. The interest of the statistical problem is shown on political tendency and sports performances at the end of the paper. I think that the statistical problem, the way to analyse it and the moment solution are interesting and I strongly recommand the paper for publication. Nevertheless, I think that many references and related works could be mentionned. For example, it seems that the estimator would provide an interesting relevant prior for an empirical Bayes procedure, the connection with latent variables in mixed effects models should be discussed, as well as the connection of this procedure with hidden" variables model like HMM. Finally, a moment procedure was proposed recently to estimate the unobserved random environment" of a Markov Chain. Comparisons with these related works could improved the paper.

### Reviewer 2

This paper establishes an interesting new problem setting, that of estimating the distribution of parameters for a group of binomial observations, gives an algorithm significantly better (in earth-mover distance) than the obvious one, and shows both a bound on its performance and an asymptotically-matching lower bounds. It also gives empirical examples on several datasets, showing the algorithm's practical advantages. I did not closely verify the proofs, but they seem reasonable. Presentation-wise, they are somewhat disorganized: it would be helpful to label the sections e.g. Proof of Theorem 3, and provide a formal statement of the proposition being proved in the noise-free case, etc. Unfortunately, the algorithm does involve solving a fairly large linear program. Explicitly characterizing the computational complexity and accuracy effects of m would be helpful. It would also be much nicer to not need a hard cutoff on the moment estimators s. Perhaps you could weight the loss function by the standard error of their estimators, or something similar? The estimators should be asymptotically normal, and you could e.g. maximize the likelihood under that distribution (though their asymptotic correlations might be difficult to characterize). Though it's an interesting problem setting, it also doesn't seem like one likely to take the world by storm; this paper is probably of primarily theoretical interest. Some interesting potential applications are suggested, however. A question about the proof: in lines 400-401, I think some more detail is needed in the proof of the variance bound. Choosing the t/k independent sets would give the stated bound, but of course there are \binom{t}{k} - t/k additional sets correlated to those t/k. It seems likely that averaging over these extra sets decreases the variance, but this is not immediate: for example, if all of the other sets were identical, the variance would be increased. I think some work is needed here to prove this, though maybe there's an obvious reason that I'm missing. Minor notes: Line 47: should be "Stein's phenomenon." Line 143, "there exists only one unbiased estimator": If you add N(0, 1) noise to any unbiased estimator, it's still unbiased. Or you could weight different samples differently, or any of various other schemes. Your proof in Appendix E (which should be referenced by name in the main text, to make it easier to find...) rather shows that there is a unique unbiased estimator which is a deterministic function of the total number of 1s seen. There is of course probably no reason to use any of those other estimators, and yours is the MVUE, but the statement should be corrected. Lines 320-322: (6) would be clearer if you established t to be scaled by c explicitly. y axis limits for CDF figures (2, 3, 4) should be clamped to [0, 1]. Probably the y lower bound in Figure 1a should be 0 as well.


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# Which of the following is an Extreme Value of the function f(c,y) = 2'+y4 4ry - + 1Lutfen birini secin: (0,1)b. (-1,0)c (1,-1)d. (-1,-1)e. (-~1,1)f. (1,0)g. (0...

## Question

###### Which of the following is an Extreme Value of the function f(c,y) = 2'+y4 4ry - + 1Lutfen birini secin: (0,1)b. (-1,0)c (1,-1)d. (-1,-1)e. (-~1,1)f. (1,0)g. (0,_1)

Which of the following is an Extreme Value of the function f(c,y) = 2'+y4 4ry - + 1 Lutfen birini secin: (0,1) b. (-1,0) c (1,-1) d. (-1,-1) e. (-~1,1) f. (1,0) g. (0,_1)

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# Download E-books Combinatorial Methods in Topology and Algebraic Geometry (Contemporary Mathematics) PDF

This assortment marks the hot resurgence of curiosity in combinatorial equipment, due to their deep and numerous purposes either in topology and algebraic geometry. approximately thirty mathematicians met on the college of Rochester in 1982 to survey numerous of the components the place combinatorial equipment are proving specially fruitful: topology and combinatorial staff concept, knot idea, 3-manifolds, homotopy idea and endless dimensional topology, and 4 manifolds and algebraic surfaces. This fabric is obtainable to complicated graduate scholars with a normal direction in algebraic topology in addition to a few paintings in combinatorial staff concept and geometric topology, in addition to to validated mathematicians with pursuits in those areas.For either pupil mathematicians, the ebook offers useful feedback for learn instructions nonetheless to be explored, in addition to the cultured pleasures of seeing the interaction among algebra and topology that is attribute of this box. in different components the booklet includes the 1st basic exposition released at the topic. In topology, for instance, the editors have incorporated M. Cohen, W. Metzler and ok. Sauerman's article on 'Collapses of $K\times I$ and workforce displays' and Metzler's 'On the Andrews-Curtis-Conjecture and comparable problems'. additionally, J. M. Montesino has supplied precis articles on either three and 4-manifolds.

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## A minimax inequality and its applications to ordinary differential equations.(English)Zbl 1009.49004

The aim of this paper is twofold. First, the author establishes several abstract results related to the minimax inequality which is involved in the three critical points theorem of Ricceri. Equivalent formulations are shown, and characterization is argued for a special class of functionals. In the second part of the paper the author applies these results in the study of the nonlinear eigenvalue differential equation $$u''+\lambda f(u)=0$$, subject to the Dirichlet conditions $$u(0)=u(1)=0$$, where $$f:\mathbb{R}\rightarrow \mathbb{R}$$ is a continuous function.

### MSC:

 49J35 Existence of solutions for minimax problems 34A40 Differential inequalities involving functions of a single real variable 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 47E05 General theory of ordinary differential operators

### Keywords:

minimax inequality; critical point; multiplicity result
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### References:

 [1] Ricceri, B., On a three critical points theorem, Arch. math. (basel), 75, 220-226, (2000) · Zbl 0979.35040 [2] Cordaro, G., On a minimax problem of ricceri, J. inequal. app., 6, 261-285, (2001) · Zbl 0986.49003 [3] Avery, R.I.; Henderson, J., Three symmetric positive solutions for a second-order boundary value problem, Appl. math. lett., 13, 1-7, (2000) · Zbl 0961.34014 [4] Bonanno, G., Existence of three solutions for a two point boundary value problem, Appl. math. lett., 13, 53-57, (2000) · Zbl 1009.34019 [5] Henderson, J.; Thompson, H.B., Existence of multiple solutions for second order boundary value problems, J. differential equations, 166, 443-454, (2000) · Zbl 1013.34017 [6] Korman, P.; Ouyang, T., Exact multiplicity results for a class of boundary value problems with cubic nonlinearities, J. math. anal. appl., 194, 328-341, (1995) · Zbl 0837.34033 [7] Korman, P.; Ouyang, T., Exact multiplicity results for two classes of boundary value problem, Diff. integral equations, 6, 1507-1517, (1993) · Zbl 0780.34013 [8] B. Ricceri, Existence of three solutions for a class of elliptic eigenvalue problems, Math. Comput. Modelling 32 1485-1494 · Zbl 0970.35089 [9] Gelfand, I.M., Some problems in the theory of quasilinear equations, (), 295-381 · Zbl 0127.04901 [10] Candito, P., Existence of three solutions for a nonautonomous two point boundary value problem, J. math. anal. appl., 252, 532-537, (2000) · Zbl 0980.34015 [11] R. Livrea, Existence of three solutions for a quasilinear two point boundary value problem, Arch. Math. (Basel), to appear · Zbl 1015.34012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.