text:
Where is the pseudoscalar glueball?

# Where is the pseudoscalar glueball?

M. Majewski
Department of Theoretical Physics II, University of Lodz
Pomorska 149/153, 90-236 Lodz, Poland
V. A. Meshcheryakov
Bogolyubov Laboratory of Theoretical Physics, JINR,
Dubna Moscow Region, Russia
e-mail: m.majewski@merlin.phys.uni.lodz.pl
###### Abstract

The pseudoscalar mesons , , , and are assumed to form the meson decuplet which includes the glueball as the basis state supplementing the standard nonet of light states . The decuplet is investigated by using the algebraic approach based on the hypothesis of vanishing exotic commutators (VEC) of ”charges” and their time derivatives. This leads to a system of master equations (ME) determining: (a) octet contents of the physical isoscalar mesons, (b) the mass formula relating all masses of the decuplet and (c) the mass ordering rule. The states of the physical isoscalar mesons , , are expressed as superpositions of the ”ideal” ( and ) states and the glueball one. The ”mixing matrix” realizing transformation from the unphysical states to the physical ones follows from the octet contents and is expressed totaly by the decuplet meson masses. Among four one-parameter families of the resulting mixing matrices (multitude of the solutions arising from bad quality of data on the and meson masses) there is a family attributing the glueball-dominated composition to the meson. The pseudoscalar decuplet is similar in some respects to the scalar one: both are composed of the excited states and G; the mass ordering of their N, S, G - dominated isoscalars is the same. Contrary to the Lattice QCD and other predictions, the mass of the pseudoscalar pure glueball state is smaller than the scalar one.

## 1 Introduction

The pseudoscalar glueball investigation has been initiated soon after it was realized that bound states of the gluons may play an important role in the strong interactions [1, 2]. From the very beginning the glueball state was traced within structure of the and mesons [3, 4]. At present, this is not the main purpose of the investigation, but is still continued, and not only within the meson structures [5, 6] but also within the baryon ones [7].

The discovery of the meson [8, 9] rous hopes for the existence of the glueball. The meson has been detected in the gluon rich process of the radiative decay and was immediately claimed to be a glueball. However, the glueball may exist as separate particle only if it has exotic quantum numbers; otherwise it should be mixed with the isoscalar states having the same signatures . The mass of the meson belongs to the region of a higher-lying multiplet and the states of this multiplet were (and still are) poorly known. That posed the question of how to certify such assignment. To this end several criteria have been invented which could be used in the cases of deficient multiplets. Some of them, concerning production, are pure qualitative like ”creation in the gluon-rich environment”, other ones, more regarding decay products, are semi-quantitative (big value of ”stickness” [10] and ”gluiness” [11]). At the same time, the question has been risen whether the glueball is necessary for understanding data concerning the pseudoscalar mesons known at that time [12]. This question is still alive [13].

The trend of discussion has changed since the results of the Lattice QCD (LQCD) calculations became available [14, 15, 16]. They supported the very existence of the pseudoscalar glueball, but the mass attributed was about - much above the . An attempt to lower the lattice prediction by including quark loops was not very successful [17]. Although the doubts were not dispelled (see e. g. [18]), this became a serious obstacle for to be recognized as the glueball candidate, because the results of the lattice calculations are generally accepted. On the other hand, there is no candidate having the mass predicted by lattice. Perhaps that induced several attempts to interpret the meson as the pseudoscalar glueball [19, 20, 21], although its mass might also be regarded as too low.

At the same time, starting from late 1980s there was growing conviction that the signal should be attributed to two different isoscalar mesons [22]. Much experimental effort was devoted to understanding the structure of the signal [23, 24, 25, 26, 27]. As a result, it has been split into and . Hence, since 2004 three isoscalar pseudoscalar mesons have been listed in RPP within the narrow interval of mass [28]:

 η1=η(1295),η2=η(1405),η3=η(1475). (1)

Such three isoscalar mesons with similar masses in the vicinity of the isotriplet and isodublet suggest overpopulation of a nonet and possible existence of a glueball which is hidden within the structures of three isoscalar states. The decuplet findings are very important because investigation of its properties is the most promising way of glueball search unless the glueball with exotic quantum numbers will be detected.

Information about the structures of the isoscalar mesons , , has been extracted from data on the reactions of their production and from branching ratios of their decays. Data suggest that the meson is a particle dominated by the glueball state [11, 28, 29, 31].

An unexpected objection has been risen against such a picture: the has been claimed not to be the state [13]. Even its very existence was considered uncertain. This implies the nonexistence of the decuplet and requires much more complicated spectroscopy of the pseudoscalar mesons. Therefore, we discuss this question in more detail.

The structure is put into doubt due to not occurrence of the in the reactions

 p¯p,J/ψ,γγ→, (2)

”at least not with the expected yields”. The base of such expectation is not indicated.

However, this is not the only point of view concerning these reactions. The authors of recently published, very careful analysis of the experimental data on , came to the following conclusions [29]:
(i) the charge exchange experiments definitively establish evidence of the ;
(ii) a clear signal of is seen in the radiative decay;
(iii) there is an indication for the existence in annihilation;
(iv) the LEP data on reaction are compatible with the existence of the signal.

So the may or may not be seen in the reactions (2). The three-body decays , are strongly suppressed by small phase space (cf and that may be the reason why it is difficult to observe the . It is explicitly seen in the reaction , for which high statistics is available, but the number of events observed in reactions (2) is many times smaller [28]. Obviously, more measurements are needed to elucidate the situation. But this question has no relevance to the problem of the internal structure. As in the reaction (2) the is observed throughout the products of decay, the frequency of its registration depends on width; the subsequent measurements would verify the magnitude of the width. However, the definition of the multiplet does not depend on the widths. Therefore, the widths of the particles cannot be the basis for any conclusion about the structure of the multiplet. Also the width of the cannot be the base for conclusion about its structure. An attempt to call in question this structure resembles confusion which arose after denying the structure of the meson motivated by its small width [30].

In the present paper we admit the meson to be a state and assume that the examined pseudoscalar mesons form a decuplet. We thus focus the glueball search again in the region of meson - this time being fully aware of the conflict with the lattice prediction.

The glueball assignment of the meson is also motivated on theoretical ground [32]. It is argued that meson is a natural pseudoscalar glueball candidate if the is the scalar glueball and the glueballs are described as the closed gluonic flux-tubes. Then the and the would be two parity related glueballs with equal masses. This description deserves attention in view of failure of the Lattice prediction, particularly if it can be treated not too literally.

It is thus interesting to make sure that this assignment can be confirmed by an argument based on the properties of the flavor multiplet as a whole.

We conclude the introduction with few comments concerning credibility of the approach we use in this paper. The credibility is especially important in evaluating the glueball contents of the decuplet isoscalar states.

It is currently known that broken symmetry predicts the existence of octets and nonets of light mesons. The multiplets are usually testified by the mass formula relating their masses. The Gell-Mann–Okubo (GMO) and Schwinger (S) mass formulae have been obtained by inclusion into the lagrangean the non-invariant mass term with regard to mixing of the octet isoscalar with the unitary singlet.

Our model unifies and generalizes these mass relations. The model has been introduced at the University of Lodz in the middle of ’80s [33, 34] and is based on requirement of vanishing the exotic commutators (VEC) of the ”charges” and their time derivatives. Apart from the GMO and S mass formulae it gives additional insight into the properties of the multiplet. For the S nonet the model VEC determines the mixing angle and establishes the mass ordering rule which ensures the mixing angle to be a real number. There are two possible orderings. For one of them the mixing angle is smaller than ideal , while for the other one it is bigger ().

The model also predicts the ideally mixed (ideal) nonet (I). This nonet has not been derived, as yet, from any other mixing description. In the quark model, where it is the basic object, it is postulated.

The S and I mass formulae are well obeyed by many nonets with various signatures comprising low mass mesons [35]. In general, the S nonets better describe data, although differences between I and S descriptions are small.

For the glueball quest the most important is the prediction of a decuplet [34] - a multiplet comprising three isoscalar mesons. The mass formula, mass ordering rule and the octet contents of the physical isoscalar states follow from the same constraints. The octet contents play a key role in determining mixing matrix of the isoscalar states, i.e. the contributions of glueball state to their structures. The orthogonal 3 x 3 mixing matrix can be parametrized by Euler angles. Absolute values of the trigonometric functions of these angles are expressed by the particle masses.

The VEC model does not use additional assumptions nor introduces free parameters to describe multiplets. Its predictions are definite and applicable for decuplets of any signature . If fitted with required experimental input, it offers complete description of the decuplet states. Thus, it bestows quantitative meaning to the most obvious qualitative signature of the glueball presence – overpopulation of a nonet.

The model has appeared very effective in describing the mesons. It makes possible to sort out 20 scalar mesons among multiplets and attribute the glueball dominating structure to the meson [30]. To analyze the decuplet we use essentially the same model. Our present analysis is proceeded in a different way because sample of the input data is different.

## 2 The decuplet of pseudoscalar mesons

### 2.1 The model of vanishing exotic commutators (VEC)11footnotemark: 1

The following sequence of exotic commutators is assumed to vanish 22footnotetext: formerly the model was called exotic commutator model (ECM)

 [Ta,djTbdtj]=0,(j=1,2,3,...) (3)

where is generator, is the time and is an exotic combination of indices, i.e. such that the operator does not belong to the octet representation. Substituting , and using the infinite momentum approximation for one-particle hamiltonian [36], we transform eqs. (3) into the system:

 [Ta,[^m2,Tb]]=0,
 [Ta,[^m2,[^m2,Tb]]]=0,
 [Ta,[^m2,[^m2,[^m2,Tb]]]]=0, (4)
 ........................................

where is the squared-mass operator.

For the matrix elements of the commutators (4) between one-particle states (we assume one-particle initial, final and intermediate states) we obtain the sequence of equations involving expressions with different powers , where is the isoscalar state belonging to the octet. Solving these equations, we obtain the sequence of formulae for a multiplet of the light mesons. We find

 ⟨x8∣^(m2)j∣x8⟩=13aj+23bj(j=1,2,3,...). (5)

where is the mass squared of the isovector meson ; is the mass squared of the subsidiary state,

 b=2K−a, (6)

and , in turn, is the mass squared of the isospinor meson. The isoscalar octet state can be represented as the linear combination of the physical isoscalar states

 ∣x8⟩=∑li∣xi⟩. (7)

The coefficients , , ,.. determine octet contents of the physical isoscalar states , ,… Substituting (7) into (5) we obtain master equations (ME) of the multiplet.

 ∑l2ixji=13aj+23bj,(j=0,1,2,3,...) (8)

where the , , ,… are isoscalar meson masses squared. Normalization condition of the coefficients is included into (8) as equation for .

### 2.2 ME for the decuplet

The states of the decuplet belong to a reducible representation of the

 8⊕1⊕1,

where the octet and one of the singlets are considered as states while the second singlet is supposed to be a glueball .

For the decuplet we have following system of the ME [34, 33] which determines masses and mixings of the decuplet states [30]:

 l21xj1+l22xj2+l23xj3=13aj+23bj,(j=0,1,2,3) (9)

where are the masses squared of the isoscalar mesons , , .
The coefficients , , , are real, as all isoscalar mesons are neutral particles. The ME (9) are considered as a system of linear equations with respect to unknown coefficients .

The solution is given by three kinds of relations [34, 30]:
a) the octet contents (OC) of the isoscalar states

 l21=13(x2−a)(x3−a)+2(x2−b)(x3−b)(x1−x2)(x1−x3), (10a) l22=13(x1−a)(x3−a)+2(x1−b)(x3−b)(x2−x1)(x2−x3), (10b) l23=13(x1−a)(x2−a)+2(x1−b)(x2−b)(x3−x1)(x3−x2); (10c)

b) the mass formula (MF)

 f(a)+2f(b)=0, (11)

where

 f(x)=(x1−x)(x2−x)(x3−x) (12)

is the characteristic polynomial of the operator; the numbering of its eigenvalues is chosen such as to satisfy the inequality

 x1

c) the mass ordering rule (MOR)

 x1

The MF (11) is a linear equation with respect to each of the , but it is a cubic one with respect to and .

The masses and experimental status of the mesons assigned to the decuplet are quoted in tab. 1. The table shows that masses of the isoscalar mesons , , are determined with good accuracy; the meson mass has large error; the -meson is not yet established – its mass is unknown. Therefore, these two masses should be considered unknown. It is natural in this model to choose and (6) as unknown variables of the ME.

For solving the ME and constructing the mixing matrix of the decuplet the solution of the MF is needed. However, MF is a single equation and its solution cannot be unique. Yet, high precision of the data on isoscalar meson masses provides correct form of the characteristic polynomial f(x) of the operator as well as precise values of the and bands which are required by MOR (14). We hope that restrictions of the model on decuplet states will reduce ambiguity of the solution.

The restrictions imposed by MOR (14) are obvious. The unknown variables and have to satisfy MOR which requires and . That restricts also the K-meson mass. It follows from (6) and (14) that

 x1+x2<2K

or

 1353MeV

From we have

 1294MeV

Comparing the bands (17) with the range of error of the meson mass we find that the MOR cuts off lower part of the error range and that the MOR-allowed region covers the upper part of it. This is consistent with treating as an unknown quantity of the ME.

### 2.3 Families of solutions of the ME

Combining the MOR with MF we restrict the unknown masses much stronger. Moreover, as will be seen below, the allowed masses can be attributed to the solutions of ME with explicit flavor properties.

Figure 1 displays and (11) (c.f. [37]). The pair of the unknown variables () provides a solution of the MF if they are such that .

It can be seen from the figure that beside the MOR restrictions , there also appears the MF restriction forbidding . Hence, the allowed values of belong to two narrow intervals: and . To each allowed value of there correspond two values of (obeying the mass formula) placed on the opposite sides of the point . If we wish to have unique solution, we should divide the interval into two parts: and . Then, we get four domains including unique pairs of values making solutions of the MF:

 A: a ∈(x1,xP), b ∈(xR,x3), (18a) B: a ∈(x1,xP), b ∈(x2,xR), (18b) C: a ∈(xQ,x2), b ∈(xR,x3), (18c) D: a ∈(xQ,x2), b ∈(x2,xR), (18d)

where

 xP≃(1.320GeV)2,xQ≃(1.365GeV)2,xR≃(1.447GeV)2. (19)

These values correspond to

 x1=(1.294GeV)2,x2=(1.410GeV)2,x3=(1.475GeV)2. (20)

The domains A, B, C, D are shown in figure 2. We solve the ME (9) in each of them separately and express as functions of . The details of solving the MF as well as properties of the solutions are described in the appendix.

Next we calculate the octet contents of the physical isoscalar decuplet states , , (10) and construct the mixing matrix of the states , , [30]. The mixing matrix V is chosen such that

 ⎡⎢⎣η1η2η3⎤⎥⎦=V⎡⎢⎣NSG⎤⎥⎦, (21)

where

 N=1√2(u¯u+d¯d),S=s¯s,G−glueball. (22)

So V expresses the states of the physical isoscalar mesons , , in terms of the decuplet ideal states , and . The V is an orthogonal matrix. Its elements are defined by the masses.

In each of the domains A, B, C, D there is one point where the solution of the ME (9) is degenerate. The points are placed at the corners of domains A, B, C, D as is shown in Fig. 2. In the first three domains we find three different ideally mixed nonets and a detached glueball; in the domain D we obtain degenerate decuplet composed of the octet states and two singlet states detached from the octet:

 A: x1 =a,x3=b,l21=13,l22=0,l23=23, (23a) B: x1 =a,x2=b,l21=13,l22=23,l23=0, (23b) C: x2 =a,x3=b,l21=0,l22=13,l23=23, (23c) D: x2 =b=a=x8,l21=0,l22=1,l23=0. (23d)

For the wave functions one obtains:

 A: η1 =±N,η2=±G,η3=±S, (24a) B: η1 =±N,η2=±S,η3=±G, (24b) C: η1 =±G,η2=±N,η3=±S, (24c) D: η1 =  γ1,η2=±x8,η3=γ2. (24d)

Each of the degenerate solutions A, B, C points out its own candidate from among , , as a pure glueball. The solution D describes degenerate decuplet where is the octet isoscalar state and , are scalar states built as superpositions of the and G.

The intervals and of the variable allowed by MF and MOR are small. Also the intervals and of the variable are small. Therefore, the domains A, B, C, D are also small and across any domain the solutions are not much different from the degenerate ones. The solutions of ME in any given domain constitute one-parameter family. To each of the domains there corresponds such a family. The solutions belonging to the same family are dominated by the same structure (N, S, G, ) which is pure in the degenerate solutions. That can be seen from the Tab. 2. Hence, the dominant structures of the , , in the domains A, B, C, D preserve the patterns of degenerate decuplet (24):

 A: η1 ∼N,η2∼G,η3∼S, (25a) B: η1 ∼N,η2∼S,η3∼G, (25b) C: η1 ∼G,η2∼N,η3∼S, (25c) D: η1 ∼γ1,η2∼η8,η3∼γ2, (25d)

where is the octet isoscalar state and , are superpositions of the singlet and G. Their contribution to the and states can be expressed by masses of the physical isoscalar mesons and are slowly varying functions inside the domain D.

Let us give the examples of the mixing matrix of the A, B, C, D solutions near degeneracy.

In each example the value of parameter is chosen such that the deviation of the meson mass from its ideal value (i. e. from the or from the meson mass) is equal to . The choice of this number is to some extend arbitrary. We want to have a decuplet which is deviated both not too little and not too much from the degenerate one; 6 MeV is the difference between the mean RPP values of and masses.

A. , , ,
,  ,  ,

 VA=⎛⎜⎝0.9754880.0218500.2189660.217116−0.257600−0.9415430.0358330.966004−0.256030⎞⎟⎠; (26)

B. , , ,
,  ,  ,

 VB=⎛⎜⎝0.9918720.0334350.1227710.059285−0.975166−0.2133910.1125880.218935−0.969222⎞⎟⎠; (27)

C. , , ,
,  ,  ,

 VC=⎛⎜⎝0.2338520.0984450.9672760.971521−0.062752−0.2284910.0382040.993162−0.110316⎞⎟⎠; (28)

D. , , ,
,  ,  ,

 VD=⎛⎜⎝0.5307360.3453340.7739920.595077−0.802101−0.0501760.6034920.487215−0.631205⎞⎟⎠ (29)

Table 2 also exhibits intervals of the admissible K meson mass corresponding to these solutions. Its changes under variations of are in all domains relatively small and the ranges of admissible values in different domains are strictly separated.

It follows that the states of the pseudoscalar mesons , , , , may constitute solution of the ME. The price to pay for ignorance concerning both and meson masses is the ambiguity of the solution: instead of unique solution we have four qualitatively different one-parameter families of solutions. These families are defined within four separated domains A, B, C, D of the plane and can be distinguished due to the fact that the isoscalar physical states are dominated by one of the , , or component. Hence the domination pattern enables us to distinguish between the families of the solutions A, B, C, D of the ME. With the present data on masses of the and mesons we can only distinguish between the families. If one of these masses was known then there would be only two solutions (not two families of solutions !) as can be seen from the Fig. 1.

The data on flavor properties of the isoscalar mesons indicate the family A as the one which points out the meson as a particle dominated by the glueball state.

## 3 Comments on solutions of the ME

A decuplet of mesons is a multiplet such that the octet isoscalar state contributes to three isoscalar physical states . The contributions of the isoscalar octet state to the physical states , given by s (10), constitute a solution of the ME. The coefficients are real numbers, therefore, the following conditions should be satisfied

 l2i>0/i=1,2,3/. (30)

This property is not guaranteed by solution (10). Requiring it, we put constraints on the masses of the decuplet.

The knowledge of the octet contents s provides very convenient way for constructing the mixing matrix.

This 3 x 3 orthogonal matrix can be parametrized by Euler angles. The absolute values of trigonometric functions of two of the angles can be expressed explicitly by s, i.e. by the masses. The requirement of the glueball flavor independence

 ==, (31)

relates them to the trigonometric functions of the third angle. However, some signs of the trigonometric functions cannot be determined if only masses are known. To find them we use available information on domination of the states by one of the , , , states. As a result, all the Euler angles, and consequently, all elements of the mixing matrix are uniquely determined [30].

The VEC description of the decuplet depends on the number of ME (9) which are taken into consideration. There are two kinds of decuplets [30].

i) A decuplet which is based on the assumption that three exotic commutators vanish. Then four ME arise. If they were applied to the nonet, they would define the ideal (I) one. We may imagine the isoscalar states as superpositions of a glueball and the I nonet states. We say that this decuplet is of the kind I. It complies with one mass formula (11) which, together with conditions (30), defines explicit MOR restrictions for the masses (14). To also admit the degenerate solutions of the ME one must allow for some and in the MOR instead of the . The solutions of the ME do not include free parameters – all predicted quantities and relations are expressed by physical masses.

ii) A decuplet arising under assumption that two exotic commutators vanish. Then there are three ME. If applied to the nonet, they give the S one. The decuplet is of the kind S if it is formed as superposition of the glueball and the S nonet. In this case the mass formula and ordering rule do not arise. The restrictions on the masses are not so strict and follow from the conditions (30).

If the masses of ten mesons with proper quantum numbers are known and satisfy (30), but do not satisfy the MF, the decuplet is of the kind S. If one of the masses is unknown then we can determine it from the MF and the decuplet becomes of the kind I, provided the masses satisfy (30). However, the states constituting these two distinct decuplets may be not very different.

The solutions A, B, C, D described above concern the decuplet I. We now summarize the main features of the solution.

The analysis of the pseudoscalar decuplet presented here does not give the unique result due to the fact that for determining two unknown masses we have only one MF. Moreover, the MF is represented by polynomial of the third degree with respect to each of these variables.

The masses of and mesons are unknown. So the values and which are natural variables in the VEC model are also unknown, but they are bounded by MOR from the below and above. These restrictions are helpful in choosing the proper solution of the nonlinear equation (11). Further restriction is provided by the mass formula which cuts out the central part of the MOR-allowed interval of and thus reduce it to two narrow disconnected subintervals (see figure 1). To each belonging to them there correspond two values of . We divide the interval of the values to two parts. As a result, the whole domain of the values of the and is reduced to four small domains A, B, C, D which are shown in figure 2. In these domains the solution is unique if one of the variables, or is known.

Hence, due to the restrictions of the ME on and , the solution is split into four one-parameter, qualitatively different families. The partition allows us to look for a solution in each domain separately. Still we have two unknown masses and only one MF equation relating them, but the domains are small and the solutions are only slightly changing across them. The ranges of the masses of the and mesons over the domains can be found out from the table 2.

The partition of the whole domain of variables is especially helpful for describing properties of mixing matrix. To each of the domains A, B, C, D there is attributed a separate one-parameter family of solutions of the MF determining the decuplet – among them the degenerate one. A family of the MF solutions, in turn, induce one-parameter family of mixing matrices. All the matrices of the family preserve a common dominance pattern. This pattern is determined by the dominance of one of the N, S, G, amplitudes in the , , states and can be read out from the degenerate solution. In each domain the degenerate decuplet corresponds to a point at the outer corner of the domain (see figure 2). Across the domain the pure state of the degenerate decuplet is transformed into dominating one and all isoscalar states become mixed. Within the domains A, B, C the glueball dominates , , states, respectively.

In the domain D the dominance pattern is different. The degenerate decuplet consists of the octet of the exact symmetry and two separate singlets being mixed states of the and . is the pure octet , while and are pure singlets. The rates of the and states in the structures of the and mesons are comparable, slowly changing functions of the parameter within the domain. In spite of the identical flavor properties of the constituents, the properties of the and mesons should be different and the difference is changing across the domain. This is mainly due to the fact that they are opposite superpositions of the and amplitudes. In this family of solutions the glueball state is not apparent.

On account of so different properties of the families A, B, C, D the qualitative information on the isoscalar mesons is sufficient to make the choice. The proper family can be chosen on the basis of the flavor properties of the isoscalar mesons , , . As it has been pointed out, even if all masses are known and satisfy MF and MOR, such an extra information is necessary for constructing the mixing matrix. An exact solution of the ME corresponding to definite values of the and K(1460) masses would be determined by suitable value of .

The restrictions following from figure 1 do not hold for the decuplet of the type S. There is no mass formula in this case; therefore, there is no connection between the variables and and there is no mass gap () of the meson. Also the MOR (14) does not exist. Such a situation can arise for the decuplet we discuss if the measured masses of the and K(1460) mesons will not satisfy MF. However, to define a decuplet of any type we always need mesons having such masses that conditions (30) are satisfied. These conditions give weaker constraints on and . We find

 x1

where is given by the GMO mass formula,

 x8=13a+23b. (33)

The glueball contents of the isoscalar mesons , , can be always calculated from (10) if the masses and satisfy /i=1,2,3/.

Having known the s we can construct the mixing matrix. If the state of is predicted to be dominated by the solution of the ME should be similar to the one describing the states of the family A.

## 4 Pseudoscalar versus scalar meson multiplets

### 4.1 Parity related spectra of the spin 0 mesons

Having described the multiplets of pseudoscalar mesons we get the opportunity to confront its properties with the properties of the corresponding multiplets of scalar mesons [30]. The comparison may reveal some new features of the meson spectroscopy.

Let us compare the and multiplets.
The ground states form the nonets:

 π,K,η,η′, (34)
 a0(980),K0(1430),f0(980),f0(1710). (35)

which are followed by the decuplets:

 π(1300),K(1460),η(1295),η(1405),η(1475), (36)
 a0(1450),K0(1950),f0(1370),f0(1500),f0(2200)/f0(2330). (37)

In both cases we have the same sequence of the multiplets. Some of the masses are not exactly known, but this does not spoil the general picture.

Let us observe that not only the sequences of the multiplets are similar, but also the inner structures of the decuplets are; namely:
- the physical mesons and which are dominated by glueball states are settled just between the remaining isoscalars which are expected to be mostly the and quark states,
- both decuplets involve excited states; hence both glueballs mix with the excited states.

The latter property suggests affinity of the glueball with the excited states. This is especially prompted by the mixing of the glueball. Its mass belongs to the region where the nonet ground states and the decuplet excited states are overlapping, but the glueball prefers mixing just with the excited states – there is no trace of mixing with the ground states [30].

The and mesons form the parity related spectra of multiplets (nonets and decuplets). The sequences of these multiplets differ only due to existence of the scalar meson which has no adequate pseudoscalar partner. But the nature of this meson is still a matter of discussion. Several authors suggest that its nature is different from the nature of other mesons [38, 39, 40]. By ignoring the we find that and mesons form parity related spectra of multiplets.

The transparency of this picture confirms not only the opinion about the distinct nature of the , but also supports correctness of sorting out the scalar mesons between the overlapping multiplets [30].

However, there is also a difference between these spectra. The mass spread of the multiplets is shrinking for consecutive multiplets (nonet , decuplet , perhaps degenerate octet at 1800 MeV). The tendency of shrinking the mass spread of the higher lying multiplets is even more clearly expressed in the spectrum of mesons where all known multiplets above 1400 MeV (at 1400 MeV and 1800 MeV) are degenerate octets [35]. But this tendency is not seen in the spectrum of the multiplets - at least below 2300 MeV.

### 4.2 The masses of the spin 0 glueballs

The LQCD calculations predict for the lower bound of the lightest glueball the mass [14, 15, 16]

 mG++≈1.500GeV. (38)

Such mass allows to attribute the glueball nature to several mesons – among them to favored .

For the lightest glueball these calculations predict the lower bound at

 mG−+≈2.300GeV. (39)

With this value no isoscalar meson discussed here can be assigned to be the glueball. Attempts to diminish this bound were unsuccessful.

The LQCD calculations also predict the lower bounds for the masses of many other glueballs with different . The result is that the mass (38) marks the minimum of these lower bounds. However, this result can be obtained also in other approaches (see, e.g. West’s theorem [41]). Therefore, it is considered more general and independent of particular approach.

The LQCD predicts masses of pure glueball states. Also the flux tube (FT) [32] concerns such states.

The FT approach predicts, however, that masses of the and glueballs should be equal. Since is the favored glueball candidate, we should expect the glueball mass at about . Hence, the LQCD and FT predictions on glueball mass are contradictory.

The VEC prediction of the glueball mass has different source. It refers to broken unitary symmetry which collects the mesons in multiplets – the octets and the nonets. Several of them are well established in the low mass region at various . We assume that at higher masses the mesons are collected in multiplets as well. In the case where the glueball appears we expect the decuplet. The three isoscalar components of the decuplet are superpositions of the and states. There is no pure glueball state but such state may dominate one of the isoscalars.

The mass formula for the decuplet relates physical masses. Also the mixing matrix is explicitly determined by physical masses. There is no ambiguity and only physical masses enter. Therefore, predictions are definite and, in favorable case, may help to perceive something new.

Using the mixing matrix we can calculate the mass of the pure glueball state. We can do this for the decuplets and separately (not assuming any relation between them).

From the decuplet , assuming solution A and the mass input appropriate to the mixing matrix (26) we find

 mG−+=1.369GeV. (40)

From the decuplet for solution 1 [30] we get

 mG++=1.497GeV. (41)

The difference between these predictions is approximately equal to the meson mass.

 mG++−mG−+=mπ. (42)

Also observe that the inequality

 mG−+

holds for all families A, B, C, D despite of LQCD calculations and West’s Theorem predictions.

Let us comment.

The VEC search of the glueball is carried on within the isoscalar sector of the decuplet. The prediction of the glueball mass consists in setting all masses of the decuplet and fitting the mixings of the isoscalar components to their flavor properties. Hence, the mixings play an important role in glueball determination.

The FT prediction of the equality of the and glueball masses is approximately obeyed. This may follow from the fact that contributions to their structure are high and almost equal:
for decuplet (solution A (26)),

 VA=⎡⎢⎣xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx−0.94154xxxxxxxxxxxxxxxxxx⎤⎥⎦. (44)

for the decuplet [30],

 V1=⎡⎢⎣xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx−0.88466xxxxxxxxxxxxxxxxxx⎤⎥⎦. (45)

In fact, the masses of the and glueballs are not identical. Difference between them, although small in the scale of the mass of these glueballs, is not negligible. This difference is a result of the parallel but independent examination of the relations between physical masses of the decuplet particles. Perhaps, different masses of these glueballs suggest that different decuplets affect the glueball component nonidentically. That may concern not only the cases of different but also decuplets of the same in different mass regions (if such ones exist).

The West’s theorem is not fulfilled: the observed mass difference (left part of the (42)) has opposite sign. The absolute value of the difference is not predicted. Also the difference between the masses predicted by LQCD has wrong sign. Beside, the value of this difference is probably too large to be explained as the mixing effect.

## 5 Conclusions

1. Owing to the unknown masses of the and mesons the solution of the ME for decuplet , , , , is not unique. In spite of that due to the restrictions of the VEC model all solutions can be classified into four separate families; one of the families points out as the particle dominated by the glueball state.
2. The scalar and pseudoscalar glueballs belong to the decuplets formed by mixing with excited isoscalar states.
3. The spectra of the known multiplets of the and mesons are parity related provided the is ignored.
4. The mass of the pure pseudoscalar glueball state is smaller than the mass of the scalar glueball one.

## 6 Acknowledgments

The authors thank Professors P. Maslanka, J. Rembielinski, W. Tybor and management of BLTF JINR for promoting our cooperation as well as Dr K. Smolinski for help in computer operations. Especially we thank Professor S. B. Gerasimov for many valuable discussions in early stage of this work and Professor P. Kosinski for many interesting comments. This work was financially supported by JINR B-I Fond and by grants nos690 and 795 of University of Lodz.

## 7 Appendix. Solving the MF

Introduce in each of the domains A, B, C, D shown in figure 2 the small variables and . They can be chosen to be nonnegative . Therefore, in the individual domains A, B, C, D, we have:

 A: a =x1+Δa, b =x3−Δb, (46a) B: a =x1+Δa, b =x2+Δb, (46b) C: a =x2−Δa, b =x3−Δb, (46c) D: a =x2−Δa, b =x2+Δb. (46d)

Further procedure is the following. Substituting and into (10), we express the coefficients as functions of the and . Putting and into MF (46) we get the relation between the and . This relation is a cubic equation with respect to any of these variables. For our purposes it is sufficient to find the approximate solution. We have

 Δb=Δb(Δa,x1,x2,x3). (47)

Substituting this function into (10) we obtain the functions . If all these functions are positive, we may consider the corresponding values of and , together with the functions , as an approximate solution of the ME (9) in the appropriate domain. This procedure is to be performed for all the domains A, B, C, D.

In the domains A and B we may neglect all terms of (11) containing higher degrees of or and restrict ourselves to the linear dependence between them. The approximation is plausible for covering all the interval . We obtain

 A: Δb =Δa2x2−x1x3−x1, (48a) B: Δb =Δa2x3−x1x3−x2. (48b)

In the domains C and D we also take into account the term quadratic in and all powers of . This is to avoid in the domain C and to extend applicability of this approximation towards the largest values of in the domain D. In these cases the expressions for are the solutions of the quadratic equation, so they are simple but long and we do not write them out. Two solutions of the quadratic equation for do not cause confusion, as only one of them complies with the condition for all i=1,2,3. In both regions the approximation is plausible everywhere, except the small surroundings of the point .

In all the solutions A, B, C, D the value implies . The degeneracy of the decuplet is destroyed if . An isoscalar state having pure or pure structure becomes mixed. However, it is still dominated by the same state provided is sufficiently small. The mixing is intensified and the dominance is getting weaker as is increasing. By examining the mixing matrix we can check whether the dominance is kept inside all domains.

Table 2 shows the range of change of the squared matrix elements , , expressing contribution of the glueball to the , , respectively and the octet content under change of in the solutions A, B, C, D. It can be seen that dominance of the and states is kept over the whole domain of these solutions. The and dominance of the other states belonging to the same solution (not shown in the table) is preserved across the domains A, B, C as well; however, there is no dominance of by in the case of the solution D. We thus find that within each domain the solution has specific dominance pattern which does not change under variation of . (Obviously, the degree of the dominance does depend on the ). The dominance patterns of the solutions A, B, etc., are identical with the patterns of ideal structures of the degenerate decuplet (24). These structures correspond to the points at the outer corners of the appropriate domain.

To conclude, all solutions of the ME are split into four separate one-parameter families. The solutions belonging to the same family are slightly different. The solutions belonging to different families have different dominance patterns.

## References

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# Standard direct materials cost per unit from variance data The following data relating to direct materials cost for October of the current year are taken from the records of Good Clean Fun Inc., a manufacturer of organic toys: Quantity of direct materials used 3,000 lb. Actual unit price of direct materials $5.50 per lb. Units of finished product manufactured 1,400 units Standard direct materials per unit of finished product 2 lb. Direct materials quantity variance—unfavorable$1,000 Direct materials price variance —unfavorable $1,500 Determine the standard direct materials cost per unit of finished product, assuming that there was no inventory of work in process at either the beginning or end of the month. BuyFind ### Accounting 27th Edition WARREN + 5 others Publisher: Cengage Learning, ISBN: 9781337272094 BuyFind ### Accounting 27th Edition WARREN + 5 others Publisher: Cengage Learning, ISBN: 9781337272094 #### Solutions Chapter Section Chapter 23, Problem 23.6EX Textbook Problem ## Standard direct materials cost per unit from variance data The following data relating to direct materials cost for October of the current year are taken from the records of Good Clean Fun Inc., a manufacturer of organic toys: Quantity of direct materials used 3,000 lb. Actual unit price of direct materials$5.50 per lb. Units of finished product manufactured 1,400 units Standard direct materials per unit of finished product 2 lb. Direct materials quantity variance—unfavorable $1,000 Direct materials price variance —unfavorable$1,500  Determine the standard direct materials cost per unit of finished product, assuming that there was no inventory of work in process at either the beginning or end of the month.

Expert Solution
To determine

Standard cost:

In the accounting records, the term standard cost refers to the practice of replacement of an expected cost for an actual cost. Then the difference between the expected costs and actual costs showing the variance are also recorded periodically. A standard costs is also known as target cost or predetermined cost.

Direct material variances:

The difference between the actual material cost per unit, and the standard material cost per unit for the direct material purchased is known as direct material cost variance. The direct material variance can be classified as follows:

• Direct materials price variance.
• Direct materials quantity variance.

To determine: The standard direct materials cost per unit of finished products.

### Explanation of Solution

The standard direct materials cost per unit of finished products is ($5.00 × 2lb.)$10.

Working notes:

Determine the direct materials price variance as follows:

Direct materials price variance = [(Actual priceStandard price)× Actual quantity]$1,500(Unfavorable) =[($5.50Standard price)×3,000 lb

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## Nature reflections on policing

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , on June 24, 2020 by xi'an

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , on April 30, 2019 by xi'an

Ziheng Yang and Tianqui Zhu published a paper in PNAS last year that criticises Bayesian posterior probabilities used in the comparison of models under misspecification as “overconfident”. The paper is written from a phylogeneticist point of view, rather than from a statistician’s perspective, as shown by the Editor in charge of the paper [although I thought that, after Steve Fienberg‘s intervention!, a statistician had to be involved in a submission relying on statistics!] a paper , but the analysis is rather problematic, at least seen through my own lenses… With no statistical novelty, apart from looking at the distribution of posterior probabilities in toy examples. The starting argument is that Bayesian model comparison is often reporting posterior probabilities in favour of a particular model that are close or even equal to 1.

“The Bayesian method is widely used to estimate species phylogenies using molecular sequence data. While it has long been noted to produce spuriously high posterior probabilities for trees or clades, the precise reasons for this over confidence are unknown. Here we characterize the behavior of Bayesian model selection when the compared models are misspecified and demonstrate that when the models are nearly equally wrong, the method exhibits unpleasant polarized behaviors,supporting one model with high confidence while rejecting others. This provides an explanation for the empirical observation of spuriously high posterior probabilities in molecular phylogenetics.”

The paper focus on the behaviour of posterior probabilities to strongly support a model against others when the sample size is large enough, “even when” all models are wrong, the argument being apparently that the correct output should be one of equal probability between models, or maybe a uniform distribution of these model probabilities over the probability simplex. Why should it be so?! The construction of the posterior probabilities is based on a meta-model that assumes the generating model to be part of a list of mutually exclusive models. It does not account for cases where “all models are wrong” or cases where “all models are right”. The reported probability is furthermore epistemic, in that it is relative to the measure defined by the prior modelling, not to a promise of a frequentist stabilisation in a ill-defined asymptotia. By which I mean that a 99.3% probability of model M¹ being “true”does not have a universal and objective meaning. (Moderation note: the high polarisation of posterior probabilities was instrumental in our investigation of model choice with ABC tools and in proposing instead error rates in ABC random forests.)

The notion that two models are equally wrong because they are both exactly at the same Kullback-Leibler distance from the generating process (when optimised over the parameter) is such a formal [or cartoonesque] notion that it does not make much sense. There is always one model that is slightly closer and eventually takes over. It is also bizarre that the argument does not account for the complexity of each model and the resulting (Occam’s razor) penalty. Even two models with a single parameter are not necessarily of intrinsic dimension one, as shown by DIC. And thus it is not a surprise if the posterior probability mostly favours one versus the other. In any case, an healthily sceptic approach to Bayesian model choice means looking at the behaviour of the procedure (Bayes factor, posterior probability, posterior predictive, mixture weight, &tc.) under various assumptions (model M¹, M², &tc.) to calibrate the numerical value, rather than taking it at face value. By which I do not mean a frequentist evaluation of this procedure. Actually, it is rather surprising that the authors of the PNAS paper do not jump on the case when the posterior probability of model M¹ say is uniformly distributed, since this would be a perfect setting when the posterior probability is a p-value. (This is also what happens to the bootstrapped version, see the last paragraph of the paper on p.1859, the year Darwin published his Origin of Species.)

## model misspecification in ABC

Posted in Statistics with tags , , , , , , , , on August 21, 2017 by xi'an

With David Frazier and Judith Rousseau, we just arXived a paper studying the impact of a misspecified model on the outcome of an ABC run. This is a question that naturally arises when using ABC, but that has been not directly covered in the literature apart from a recently arXived paper by James Ridgway [that was earlier this month commented on the ‘Og]. On the one hand, ABC can be seen as a robust method in that it focus on the aspects of the assumed model that are translated by the [insufficient] summary statistics and their expectation. And nothing else. It is thus tolerant of departures from the hypothetical model that [almost] preserve those moments. On the other hand, ABC involves a degree of non-parametric estimation of the intractable likelihood, which may sound even more robust, except that the likelihood is estimated from pseudo-data simulated from the “wrong” model in case of misspecification.

In the paper, we examine how the pseudo-true value of the parameter [that is, the value of the parameter of the misspecified model that comes closest to the generating model in terms of Kullback-Leibler divergence] is asymptotically reached by some ABC algorithms like the ABC accept/reject approach and not by others like the popular linear regression [post-simulation] adjustment. Which suprisingly concentrates posterior mass on a completely different pseudo-true value. Exploiting our recent assessment of ABC convergence for well-specified models, we show the above convergence result for a tolerance sequence that decreases to the minimum possible distance [between the true expectation and the misspecified expectation] at a slow enough rate. Or that the sequence of acceptance probabilities goes to zero at the proper speed. In the case of the regression correction, the pseudo-true value is shifted by a quantity that does not converge to zero, because of the misspecification in the expectation of the summary statistics. This is not immensely surprising but we hence get a very different picture when compared with the well-specified case, when regression corrections bring improvement to the asymptotic behaviour of the ABC estimators. This discrepancy between two versions of ABC can be exploited to seek misspecification diagnoses, e.g. through the acceptance rate versus the tolerance level, or via a comparison of the ABC approximations to the posterior expectations of quantities of interest which should diverge at rate Vn. In both cases, ABC reference tables/learning bases can be exploited to draw and calibrate a comparison with the well-specified case.

## La déraisonnable efficacité des mathématiques

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , on May 11, 2017 by xi'an

Although it went completely out of my mind, thanks to a rather heavy travel schedule, I gave last week a short interview about the notion of mathematical models, which got broadcast this week on France Culture, one of the French public radio channels. Within the daily La Méthode Scientifique show, which is a one-hour emission on scientific issues, always a [rare] pleasure to listen to. (Including the day they invited Claire Voisin.) The theme of the show that day was about the unreasonable effectiveness of mathematics, with the [classical] questioning of whether it is an efficient tool towards solving scientific (and inference?) problems because the mathematical objects pre-existed their use or we are (pre-)conditioned to use mathematics to solve problems. I somewhat sounded like a dog in a game of skittles, but it was interesting to listen to the philosopher discussing my relativistic perspective [provided you understand French!]. And I appreciated very much the way Céline Loozen the journalist who interviewed me sorted the chaff from the wheat in the original interview to make me sound mostly coherent! (A coincidence: Jean-Michel Marin got interviewed this morning on France Inter, the major public radio, about the Grothendieck papers.)

## Why should I be Bayesian when my model is wrong?

Posted in Books, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , on May 9, 2017 by xi'an

Guillaume Dehaene posted the above question on X validated last Friday. Here is an except from it:

However, as everybody knows, assuming that my model is correct is fairly arrogant: why should Nature fall neatly inside the box of the models which I have considered? It is much more realistic to assume that the real model of the data p(x) differs from p(x|θ) for all values of θ. This is usually called a “misspecified” model.

My problem is that, in this more realistic misspecified case, I don’t have any good arguments for being Bayesian (i.e: computing the posterior distribution) versus simply computing the Maximum Likelihood Estimator.

Indeed, according to Kleijn, v.d Vaart (2012), in the misspecified case, the posterior distribution converges as nto a Dirac distribution centred at the MLE but does not have the correct variance (unless two values just happen to be same) in order to ensure that credible intervals of the posterior match confidence intervals for θ.

Which is a very interesting question…that may not have an answer (but that does not make it less interesting!)

A few thoughts about that meme that all models are wrong: (resonating from last week discussion):

1. While the hypothetical model is indeed almost invariably and irremediably wrong, it still makes sense to act in an efficient or coherent manner with respect to this model if this is the best one can do. The resulting inference produces an evaluation of the formal model that is the “closest” to the actual data generating model (if any);
2. There exist Bayesian approaches that can do without the model, a most recent example being the papers by Bissiri et al. (with my comments) and by Watson and Holmes (which I discussed with Judith Rousseau);
3. In a connected way, there exists a whole branch of Bayesian statistics dealing with M-open inference;
4. And yet another direction I like a lot is the SafeBayes approach of Peter Grünwald, who takes into account model misspecification to replace the likelihood with a down-graded version expressed as a power of the original likelihood.
5. The very recent Read Paper by Gelman and Hennig addresses this issue, albeit in a circumvoluted manner (and I added some comments on my blog).
6. In a sense, Bayesians should be the least concerned among statisticians and modellers about this aspect since the sampling model is to be taken as one of several prior assumptions and the outcome is conditional or relative to all those prior assumptions.


text:
Remarks on relativistic Schrödinger operators and their extensions
Annales de l'I.H.P. Physique théorique, Volume 67 (1997) no. 1, p. 29-39
@article{AIHPA_1997__67_1_29_0,
author = {Ben-Artzi, Matania and Nemirovsky, Jonathan},
title = {Remarks on relativistic Schr\"odinger operators and their extensions},
journal = {Annales de l'I.H.P. Physique th\'eorique},
publisher = {Gauthier-Villars},
volume = {67},
number = {1},
year = {1997},
pages = {29-39},
zbl = {0885.35135},
mrnumber = {1463003},
language = {en},
url = {http://www.numdam.org/item/AIHPA_1997__67_1_29_0}
}

Ben-Artzi, Matania; Nemirovsky, Jonathan. Remarks on relativistic Schrödinger operators and their extensions. Annales de l'I.H.P. Physique théorique, Volume 67 (1997) no. 1, pp. 29-39. http://www.numdam.org/item/AIHPA_1997__67_1_29_0/

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[5] M. Ben-Artzi and A. Devinatz, Local Smoothing and Convergence Properties for Schrödinger-Type Equations, J. Func. Anal., Vol. 101, 1991, pp. 231-254. | MR 1136935 | Zbl 0762.35022

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text:
A simple question about the center of a finite group

Let $G$ be a finite group. $N$ a normal subgroup of $G$ and $K$ a characteristic subgroup of $N$: $$K \text{ char } N \triangleleft G.$$ If $Z(G/N)=1$ and $Z(G)=1$, does it follow that $Z(G/K)=1$?

-
What's the relation between $N$ and $K$? –  Chris Godsil Oct 25 '12 at 2:31
Yes, when G is the trivial group. –  Mustafa Gokhan Benli Oct 25 '12 at 2:42
Sorry,I have corrected it.$K$char$N$. –  Tom Oct 25 '12 at 4:03
As you state the question, no. What it $G$ is any finite group with $Z(G) = 1$ and we take $N = G, K =[G,G]$? Then $Z(G) = Z(G/N) = 1$ and $G/K$ is Abelian (here, $[G,G]$ means the derived group). –  Geoff Robinson Oct 25 '12 at 4:45
This is not a research level question, so is not suitable for MO. With the question as it is worded now, a counterexample is $S_3 \times A_5$, with $N=S_3$, $|K|=3$. –  Derek Holt Oct 25 '12 at 9:12

As it is not entirely clear where the containments are meant to be, I will assume you are asking about the case $N\triangleleft K\triangleleft G$; in this case $Z(G) = Z(G/N) = 1$ is not sufficient in general to guarantee $Z(G/K)$ is also trivial. As an example, take $N$ to be the Klein-4 subgroup of $S_4$ and $K = A_4$. Then $G/N\cong S_3$ so $Z(G) = Z(G/N)$ is trivial, but $G/K$ is abelian (isomorphic to $\mathbb{Z}/2\mathbb{Z}$) and so $Z(G/K)$ is not trivial.
I am sorry.It is not the case.It is $K$char$N\triangleleft G$. Thank you all the same. –  Tom Oct 25 '12 at 4:35


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1. polynomial

23984024294.png

2. Re: polynomial

Hints:

a) Factor Theorem

b) Factor Theorem

c) Differentiation and Factor Theorem

3. Re: polynomial

This is where you consider the product of roots, which is 1.

$Since \ \alpha \ is a multiple root, then it must AT LEAST have a multiplicity of 2, we're also given that \frac{1}{\alpha} \ is a root, so if \ \alpha \ \ has a multiplicity of 3, then the roots would be \alpha , \alpha , \alpha , \frac{1}{\alpha} which would imply that \ \alpha =+-1 , \ \ but we know that can't happen from part a). Therefore \alpha only has a multiplicity of 2. Therefore we know that 3 of the roots are \alpha , \alpha,\frac{1}{\alpha}. The fourth root must also be \frac{1}{\alpha} to satisfy the product of roots.$

4. Re: polynomial

$Now we consider the sum of roots and the sum of roots 2 at a time. \\ \\ -A = 2 \alpha + \frac{2}{\alpha} \\ \\ B = \alpha^2 + \frac{1}{\alpha^2} + 4 \\ \\ Wtih proper subsitution , you should get the given result in the question.$

5. Re: polynomial

If a and b are roots of the equation x^2 + 8x -5=0, find the quadratic equation whose roots are a/b and b/a

6. Re: polynomial

Originally Posted by 19KANguy
If a and b are roots of the equation x^2 + 8x -5=0, find the quadratic equation whose roots are a/b and b/a
Some hints.

$\frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab} = \frac{74}{-5}$

$\frac{a}{b} \cdot \frac{b}{a} = 1$

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# A triangle has corners at (3 ,8 ), (5 ,9 ), and (8 ,5 ). What is the area of the triangle's circumscribed circle?

Jun 1, 2018

Area of the triangle's circumscribed circle$= \frac{53 \pi}{9}$ units.

#### Explanation:

First of all, we have to find out circumcenter (G) of the triangle.

$G = \frac{3 + 5 + 8}{3} , \frac{8 + 9 + 5}{3}$

$G = \frac{16}{3} , \frac{22}{3}$

For area of the triangle's circumscribed circle, we have to calculate radius of the circumscribed circle. And, it is equal to the distance between G and any of the vertices of the triangle.

R= sqrt((16/3 -3)^2+(22/3 -8)^2

R= sqrt(((16-9)/3)^2+((22-24)/3)^2

R= sqrt(((7)/3)^2+((-2)/3)^2

R= sqrt(49/9+4/9

R= sqrt(53/9

$R = \frac{\sqrt{53}}{3}$ units

Area of the triangle's circumscribed circle$= \pi {R}^{2}$

Area of the triangle's circumscribed circle$= \pi \times \frac{53}{9}$

Area of the triangle's circumscribed circle$= \frac{53 \pi}{9}$ units.

Jun 5, 2018

{ 2125 pi }/242

#### Explanation:

Archimedes' Theorem says for a triangle with sides $a , b , c :$

$16 \setminus {\textrm{a r e a}}^{2} = 4 {a}^{2} {b}^{2} - {\left({c}^{2} - {a}^{2} - {b}^{2}\right)}^{2}$

The radius of the circumcircle equals the product of the sides of the triangle divided by four times the area of the triangle. This is more useful squared:

${r}^{2} = \frac{{a}^{2} {b}^{2} {c}^{2}}{16 \setminus {\textrm{a r e a}}^{2}} = \frac{{a}^{2} {b}^{2} {c}^{2}}{4 {a}^{2} {b}^{2} - {\left({c}^{2} - {a}^{2} - {b}^{2}\right)}^{2}}$

${a}^{2} , {b}^{2} \mathmr{and} {c}^{2}$ are available directly from $\left(3 , 8\right) , \left(5 , 9\right) , \left(8 , 5\right)$

${a}^{2} = {\left(3 - 5\right)}^{2} + {\left(8 - 9\right)}^{2} = 5$

${b}^{2} = {\left(5 - 8\right)}^{2} + {\left(9 - 5\right)}^{2} = 25$

${c}^{2} = {\left(8 - 3\right)}^{2} + {\left(5 - 8\right)}^{2} = 34$

 text{circumcircle area} = pi r^2 = pi {(5)(25)(34) }/{ 4(5)(25) - (34-5-25)^2 } ={2125 pi }/242

It appears me or the other answer is wrong.

Check: Alpha $\quad$ I'm right quad sqrt


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# Math Insight

### Mean path length definition

In a network, the mean path length is the average shortest path between two nodes.

Let $d_{ij}$ be the length of the shortest path between nodes $i$ and $j$. In a network, the length of a path is the number of edges that the path contains. The shortest path between two points is called geodesic. There will typically be many geodesics between a pair of nodes, but, by definition, they will all have the same length $d_{ij}$.

The mean path length is the average of the shortest path length, averaged over all pairs of nodes. For an undirected graph of $N$ nodes, the mean path length is \begin{gather*} \ell = \frac{1}{N(N-1)} \sum_{i \ne j} d_{ij}, \end{gather*} where the sum is over all pairs of distinct nodes.

If two nodes are disconnected, meaning there is no path between them, then the path length between them in infinite. As a consequence, if a network contains disconnected components (collections of nodes that have no paths between them), then the mean path length $\ell$ also diverges to infinity. One way to avoid this problem is to calculate $\ell$ only from nodes in the largest connected component.


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### Home > ACC6 > Chapter 5 Unit 12 > Lesson CC2: 5.2.6 > Problem5-86

5-86.

Darnell is designing a new game. He will have $110$ different-colored blocks in a bag. While a person is blindfolded, he or she will reach in and pull out a block. The color of the block determines the prize according to Darnell’s sign at right.

 blue → small toy  purple → hat   green → large stuffed animal
1. If he wants players to have a $60\%$ probability of winning a small toy, how many blue blocks should he have?

Since picking a blue block gives a small toy prize, Darnell can consider the probability of picking a blue block as the probability of winning a small toy.

If he wants players to have a $60\%$ probability of winning a small toy, $60\%$ of the $110$ blocks in the bag should be blue.

$66$ blue blocks

2. If he wants players to have a $10\%$ probability of winning a large stuffed animal, how many green blocks should he have?

See part (a).


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# Abelian Group or Commutative Group

If the commutative law holds in a group, then such a group is called an Abelian group or Commutative group. Thus the group $\left( {G, * } \right)$ is said to be an Abelian group or commutative group if $a * b = b * a,\forall a,b \in G$.

A group which is not Abelian is called a non-Abelian group. The group $\left( {G, + } \right)$ is called the group under addition while the group $\left( {G, \times } \right)$ is known as group under multiplication.

Examples:

• The structure $\left( {\mathbb{Z}, + } \right)$ is a group, i.e., the set of integers with the addition composition is a group. This is so because addition in numbers is associative, the additive identity $0$ belongs to $\mathbb{Z}$, and the inverse of every element $a$, viz. $- a$ belongs to $\mathbb{Z}$. This is known as additive Abelian group of integers.
• The structures $\left( {\mathbb{Z}, + } \right),\left( {\mathbb{R}, + } \right),\left( {\mathbb{C}, + } \right)$ are all groups, i.e., the sets of rational numbers, real numbers, complex numbers, each with the additive composition, form an Abelian group. But the same sets with the multiplication composition do not form a group, for the multiplicative inverse of the number zero does not exist in any of them.
• The structure $\left( {{\mathbb{Q}_o}, \times } \right)$ is an Abelian group, where ${\mathbb{Q}_o}$ is the set of non-zero rational numbers. This is so because the operation is associative, the multiplicative identity $1$ belongs to ${\mathbb{Q}_o}$, and the multiplicative inverse of every element $a$ in the set is $1/a$, which also belongs to ${\mathbb{Q}_o}$. This is known as the multiplicative Abelian group of non-zero rational.
Obviously $\left( {{\mathbb{R}_o}, \times } \right)$ and $\left( {{\mathbb{C}_o}, \times } \right)$ are groups, where ${\mathbb{R}_o}$ and ${\mathbb{C}_o}$ are respectively the sets of non-zero real numbers and non-zero complex numbers.


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# Adaptive Dynamic Programming: An Introduction

@article{Wang2009AdaptiveDP,
author={F. Wang and Huaguang Zhang and Derong Liu},
journal={IEEE Computational Intelligence Magazine},
year={2009},
volume={4}
}
• Published 2009
• Computer Science
• IEEE Computational Intelligence Magazine
In this article, we introduce some recent research trends within the field of adaptive/approximate dynamic programming (ADP), including the variations on the structure of ADP schemes, the development of ADP algorithms and applications of ADP schemes. For ADP algorithms, the point of focus is that iterative algorithms of ADP can be sorted into two classes: one class is the iterative algorithm with initial stable policy; the other is the one without the requirement of initial stable policy. It is… Expand
659 Citations

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It is proved that any of iterative control obtained in the proposed algorithm can stabilize the nonlinear system which overcomes the disadvantage of traditional value iteration algorithms. Expand
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It is proved that all the Iterative controls obtained in the iterative θ-ADP algorithm can stabilize the nonlinear system which means that the iteratives θ, which is feasible for implementations both online and offline, is feasible. Expand

#### References

SHOWING 1-10 OF 144 REFERENCES
A performance gradient perspective on approximate dynamic programming and its application to partially observable Markov decision processes
• Computer Science
• 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control
• 2006
This paper shows an approach to integrating common approximate dynamic programming (ADP) algorithms into a theoretical framework to address both analytical characteristics and algorithmic features.Expand
Model-free Approximate Dynamic Programming Schemes for Linear Systems
• Computer Science, Mathematics
• 2007 International Joint Conference on Neural Networks
• 2007
Online model-free adaptive critic schemes based on approximate dynamic programming (ADP) to solve optimal control problems in both discrete-time and continuous-time domains for linear systems with unknown dynamics are presented. Expand
Direct Neural Dynamic Programming
• Computer Science
• 2004
This chapter discusses the relationships, results, and challenges of various approaches under the theme of ADP, and introduces the fundamental principles of the direct neural dynamic programming (NDP), which is demonstrated for a continuous state control problem using an industrial scale Apache helicopter model. Expand
• Mathematics, Computer Science
• IEEE Trans. Syst. Man Cybern. Part C
• 2002
An adaptive dynamic programming algorithm (ADPA) is described which fuses soft computing techniques to learn the optimal cost functional for a stabilizable nonlinear system with unknown dynamics and hard Computing techniques to verify the stability and convergence of the algorithm. Expand
A Novel Infinite-Time Optimal Tracking Control Scheme for a Class of Discrete-Time Nonlinear Systems via the Greedy HDP Iteration Algorithm
• Computer Science, Medicine
• IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics)
• 2008
In this paper, we aim to solve the infinite-time optimal tracking control problem for a class of discrete-time nonlinear systems using the greedy heuristic dynamic programming (HDP) iterationExpand
Continuous-Time ADP for Linear Systems with Partially Unknown Dynamics
• Mathematics
• 2007 IEEE International Symposium on Approximate Dynamic Programming and Reinforcement Learning
• 2007
Approximate dynamic programming has been formulated and applied mainly to discrete-time systems. Expressing the ADP concept for continuous-time systems raises difficult issues related to samplingExpand
Discrete-Time Nonlinear HJB Solution Using Approximate Dynamic Programming: Convergence Proof
• Mathematics, Computer Science
• IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics)
• 2008
It is shown that HDP converges to the optimal control and the optimal value function that solves the Hamilton-Jacobi-Bellman equation appearing in infinite-horizon discrete-time (DT) nonlinear optimal control. Expand
Neural-Network-Based Near-Optimal Control for a Class of Discrete-Time Affine Nonlinear Systems With Control Constraints
• Mathematics, Computer Science
• IEEE Transactions on Neural Networks
• 2009
In this paper, the near-optimal control problem for a class of nonlinear discrete-time systems with control constraints is solved by iterative adaptive dynamic programming algorithm. First, a novelExpand
Handbook of Learning and Approximate Dynamic Programming
• Computer Science
• IEEE Transactions on Automatic Control
• 2006
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Model-free multiobjective approximate dynamic programming for discrete-time nonlinear systems with general performance index functions
• Mathematics, Computer Science
• Neurocomputing
• 2009
In this paper, a forward-in-time optimal control method for a class of discrete-time nonlinear systems with general multiobjective performance indices is proposed with unknown system dynamics. TheExpand


text:
Department of Mathematics Search | Help | Login | |

Math @ Duke

 ....................... ....................... Webpage

## Publications [#357478] of Hau-Tieng Wu

Papers Published

1. Steinerberger, S; Wu, H-T, On Zeroes of Random Polynomials and an Application to Unwinding, International Mathematics Research Notices, vol. 2021 no. 13 (June, 2021), pp. 10100-10117, Oxford University Press (OUP) [doi]
(last updated on 2021/10/16)

Abstract:
Abstract Let $\mu$ be a probability measure in $\mathbb{C}$ with a continuous and compactly supported density function, let $z_1, \dots , z_n$ be independent random variables, $z_i \sim \mu$, and consider the random polynomial $p_n(z) = \prod _{k=1}^{n}{(z - z_k)}.$ We determine the asymptotic distribution of $\left \{z \in \mathbb{C}: p_n(z) = p_n(0)\right \}$. In particular, if $\mu$ is radial around the origin, then those solutions are also distributed according to $\mu$ as $n \rightarrow \infty$. Generally, the distribution of the solutions will reproduce parts of $\mu$ and condense another part on curves. We use these insights to study the behavior of the Blaschke unwinding series on random data.

dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821

Mathematics Department
Duke University, Box 90320
Durham, NC 27708-0320


text:
## Categories and orbispaces. (arXiv:1810.06632v1 [math.AT])

Constructing and manipulating homotopy types from categorical input data has been an important theme in algebraic topology for decades. Every category gives rise to a `classifying space', the geometric realization of the nerve. Up to weak homotopy equivalence, every space is the classifying space of a small category. More is true: the entire homotopy theory of topological spaces and continuous maps can be modeled by categories and functors. We establish a vast generalization of the equivalence of the homotopy theories of categories and spaces: small categories represent refined homotopy types of orbispaces whose underlying coarse moduli space is the traditional homotopy type hitherto considered. A global equivalence is a functor between small categories that induces weak equivalences of nerves of the categories of \$G\$-objects, for all finite groups \$G\$. We show that the global equivalences are part of a model structure on the category of small categories, which is moreover Quillen equi查看全文

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 Constructing and manipulating homotopy types from categorical input data has been an important theme in algebraic topology for decades. Every category gives rise to a `classifying space', the geometric realization of the nerve. Up to weak homotopy equivalence, every space is the classifying space of a small category. More is true: the entire homotopy theory of topological spaces and continuous maps can be modeled by categories and functors. We establish a vast generalization of the equivalence of the homotopy theories of categories and spaces: small categories represent refined homotopy types of orbispaces whose underlying coarse moduli space is the traditional homotopy type hitherto considered. A global equivalence is a functor between small categories that induces weak equivalences of nerves of the categories of \$G\$-objects, for all finite groups \$G\$. We show that the global equivalences are part of a model structure on the category of small categories, which is moreover Quillen equi


text:
Rebus puzzle “cancelled”

I have a rebus puzzle I can't figure out.
The word "Cancelled" is written with the "CAN" angled downwards and the "CELLED" angled upwards :

                D
E
L
L
C       E
A   C
N

What does that mean ?

• Could you maybe add a visual so we can see what you are talking about? – Micha Sprengers Jun 16 '16 at 9:48
• if this is really a rebus puzzle please add the rebus tag. Unless this isn't a rebus and more of a riddle?? – Ben Jun 16 '16 at 10:04
• I'va added a visual representation, remove/edit it if this is not your puzzle – Fabich Jun 16 '16 at 13:07
• This can be an answer of interest. You might be interested to accept it. – EKons Jun 16 '16 at 14:28

1 Answer

I think that this rebus was published in America and is

cancelled check
But hang on -- if published in America, shouldn't it be CANCELED with one L?

• Out of curiosity, what makes you think this was published in America? – Gordon Allocman Jun 16 '16 at 13:16
• @GordonAllocman I could be wrong about that, of course. But the mark that the OP described (and which Lord of Dark has now shown in text art, though that text art wasn't in the OP when I answered it) -- the mark such as the green one Puzzling.SE displays by an accepted answer -- is in Britain called a tick, not a check. And the word "check" for that slip of paper ordering a sum of money to be paid from someone's account to a named person -- is spelt "cheque" in Britain. – Rosie F Jun 16 '16 at 13:33
• @RosieF Credit Mr Webster for that particular confusion. They are both used, almost equally, in AE. Canceled just looks wrong to me. – Kevin Jun 16 '16 at 14:19
• Also "cheque" is the correct spelling in countries that aren't the US. – Ian MacDonald Jun 16 '16 at 20:22
• I spell cancelled with 2 Ls and I am in the US. – Joshua Jun 16 '16 at 20:38


text:
### Still have math questions?

Algebra
Question

Find the value of a that makes the following function differentiable for all values of $$x$$ .

$$g ( x ) = \left\{ \begin{array} { l l } { a x } & { ifx< 0}\\ { x ^ { 2 } - 21 x } & { ifx\ge 0}\end{array} \right.$$

The function $$g ( x )$$ is differentiable for all values of $$x$$ when a $$= \square$$


text:
# Frary Walkway

Location.  Walkway spanning the distance from Frary Dining Hall to College Way, midway between Walker and Clark V student dormitories. Pomona College, Claremont, CA.

Description of the space.  The walkway is wide, with a few trees, picnic tables, and small lawn spaces flanking both sides (in front of the dormitories). Small, unpaved pathways branch off the main walkway and lead to the entrances of each dormitory. The area is usually occupied by any number of students; however, because the recording was taken at a later time of night, the area and its soundscape have a fairly subdued quality. Beyond the stretch between the two dormitories, the area surrounding the walkway is generally open.

Recording setup.  The microphone and balloon, approximately 3 feet apart, were held at 4-5 feet from the ground. The microphone (a TASCAM DR-40 recorder) was supported by a tripod, and the balloon was held manually.

Reverberation time.

30 dB decay: 0.4s

60 dB decay: Not measured

decay to background level: approximately 0.8 sec, although hard to tell with the background noise.

minimum: 0.8 sec

Acoustic description.  The resonance in this space is barely perceptible, due to the relatively open area in which it was recorded. However, likely a result of the buildings that hem the walkway on both sides, a faint echo is audible at approximately 0.8,1.4, and 1.6 seconds. These reverberances are visualized as (faint) vertical bands on the spectrograph.

Posted by Yannai Kashtan and Jeanne Rasmussen


text:
# A Poset Classifying Non-Commutative Term Orders

Abstract : We study a poset $\Re$ on the free monoid (X*) on a countable alphabet X.This poset is determined by the fact that its total extensions are precisely the standard term orders on X*. We also investigate the poset classifying degree-compatible standard term orders, and the poset classifying sorted term orders. For the latter poset, we give a Galois coconnection with the Young lattice.
Keywords :
Document type :
Conference papers
Domain :

Cited literature [12 references]

https://hal.inria.fr/hal-01182966
Contributor : Coordination Episciences Iam Connect in order to contact the contributor
Submitted on : Thursday, August 6, 2015 - 11:28:03 AM
Last modification on : Tuesday, March 7, 2017 - 3:00:10 PM
Long-term archiving on: : Friday, May 5, 2017 - 11:23:26 AM

### File

dmAA0122.pdf
Publisher files allowed on an open archive

### Citation

Jan Snellman. A Poset Classifying Non-Commutative Term Orders. Discrete Models: Combinatorics, Computation, and Geometry, DM-CCG 2001, 2001, Paris, France. pp.301-314, ⟨10.46298/dmtcs.2286⟩. ⟨hal-01182966⟩

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text:
1. 离散情形

1.2. 一维离散型随机变量函数的分布

$X$ $x_1$ $x_2$ $\cdots$ $x_k$ $\cdots$
$Y$ $g(x_1)$ $g(x_2)$ $\cdots$ $g(x_k)$ $\cdots$
$P\lbrace X=x_k\rbrace$ $p_1$ $p_2$ $\cdots$ $p_k$ $\cdots$

1.3. 二维离散型随机变量函数的分布

$(X,Y)$ $(x_1,y_1)$ $(x_1,y_2)$ $\cdots$ $(x_1,y_n)$ $(x_2,y_1)$ $\cdots$
$Z$ $g(x_1,y_1)$ $g(x_1,y_2)$ $\cdots$ $g(x_1,y_n)$ $g(x_2,y_1)$ $\cdots$
$P\lbrace X=x_i,Y=y_j\rbrace$ $p_{11}$ $p_{12}$ $\cdots$ $p_{1n}$ $p_{21}$ $\cdots$

2. 连续情形

2.1. 一维连续随机变量函数的分布

1. 确定$Y$的值域$R(Y)$。
2. 对任意的$y\in R(Y)$，求出$Y$的分布函数：
$$F_{Y}(y)=P\lbrace Y \leq y\rbrace = P\lbrace g(X) \leq y\rbrace =P\lbrace X \in G(y)\rbrace = \int_{G(N)} f_{X}(x) d x$$ 其中$G(Y)$由不等式$g(X)\leq y$解出，积分部分一般不积出。
3. 利用变上限函数求导$F_{Y}^{\prime}(y)=f_{Y}(y) \quad y \in R(Y)$

$$f_{Y}(y)=\frac{d}{d y} \int_{G(y)} f_{X}(x) d x ， y \in R(Y) \ f_{Y}(y)=0, y \notin R(Y)$$

2.2. 二维连续随机变量函数的分布

1. 确定$Z$的值域$R(Z)$。
2. 对任意的$Z\in R(Z)$，求出$Z$的分布函数：
$$F_{Z}(z)=P{Z \leq z}=P{g(X, Y) \leq z}=P{(X, Y) \in G(z)}=\iint_{G(z)} f(x, y) d \sigma$$
3. 求导$f_{z}(z)=F_{z}^{\prime}(z) z \in R(Z)$

$$f_{z}(z)=F_{z}^{\prime}(z),z \in R(Z) \ f_{z}(z)=0 ,z \notin R(Z)$$

2.3. 和的分布

$$f_{z}(z)=\int_{-\infty}^{+\infty} f(x, z-x) d x=\int_{-\infty}^{+\infty} f(z-y, y) d y$$特别地，当$X,Y$独立时:
$$f_{z}(z)=\int_{-\infty}^{+\infty} f_{X}(x) f_{Y}(z-x) d x=\int_{-\infty}^{+\infty} f_{X}(z-y) f_{Y}(y) d y$$

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( ,,´･ω･)ﾉ"(´っω･｀｡)
╮(╯▽╰)╭
o(*////▽////*)q
＞﹏＜
( ๑´•ω•) "(ㆆᴗㆆ)

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