text:
The \hbar Expansion in Quantum Field Theory

# The ℏ Expansion in Quantum Field Theory

Stanley J. Brodsky SLAC National Accelerator Laboratory
Stanford University, Stanford, California 94309, USA
CP-Origins, University of Southern Denmark
Campusvej 55, DK-5230 Odense M, Denmark
Paul Hoyer CP-Origins, University of Southern Denmark
Campusvej 55, DK-5230 Odense M, Denmark
Department of Physics and Helsinki Institute of Physics
POB 64, FIN-00014 University of Helsinki, Finland
###### Abstract

We show how expansions in powers of Planck’s constant can give new insights into perturbative and nonperturbative properties of quantum field theories. Since is a fundamental parameter, exact Lorentz invariance and gauge invariance are maintained at each order of the expansion. The physics of the expansion depends on the scheme; i.e., different expansions are obtained depending on which quantities (momenta, couplings and masses) are assumed to be independent of . We show that if the coupling and mass parameters appearing in the Lagrangian density are taken to be independent of , then each loop in perturbation theory brings a factor of . In the case of quantum electrodynamics, this scheme implies that the classical charge , as well as the fine structure constant are linear in . The connection between the number of loops and factors of is more subtle for bound states since the binding energies and bound-state momenta themselves scale with . The expansion allows one to identify equal-time relativistic bound states in QED and QCD which are of lowest order in and transform dynamically under Lorentz boosts. The possibility to use retarded propagators at the Born level gives valence-like wave-functions which implicitly describe the sea constituents of the bound states normally present in its Fock state representation.

###### pacs:
11.15.Bt, 12.20.Ds

SLAC-PUB-14224

HIP-2010-21/TH

CP3-Origins-2010-34

July 15, 2019

## I Introduction

Planck’s constant is the fundamental constant of nature related to quantum effects Planck (). The most familiar application of the Planck constant is the commutation relation which limits simultaneous measurements of position and momentum. More generally, enters explicitly in the commutation relations of conjugate operators and fields, thus providing the fundamental basis of quantum field theory. Each order of an expansion in must obey all the underlying symmetries of the theory.

The Planck constant has units of action; i.e., the product of energy and length in units where the velocity of light . One commonly assumes units such that . However, it is illuminating to keep the occurrence of powers of explicit since this allows one to distinguish quantum versus classical physics. For example, the AdS/CFT prediction for the ratio of shear viscosity to the entropy density of a multi-particle system has the lower limit Kovtun:2004de () The origin of in this relation can be traced to the assumed quantum form of the entropy of a black hole in a higher dimensional theory. The empirical observation that in heavy ion collisions at the Relativistic Heavy Ion Collider Shuryak:2003xe () is not far from the AdS/CFT prediction thus suggests that the dynamics of high energy central heavy-ion collisions is in the quantum domain.

Physical phenomena are usually expected to follow the laws of classical theory in the (hypothetical) limit . Surprisingly, this is true only for a careful choice of -independent quantities (momenta, couplings and masses). A simple illustration is provided by the standard harmonic oscillator in nonrelativistic quantum mechanics where the potential is The propagation of a particle from to is given by the path integral

 A(xi,xf;tf−ti)=∫[Dx(t)]exp[im2ℏ∫tftidt(˙x2−ω2x2)]=∫[Dξ(t)]exp[im2∫tftidt(˙ξ2−ω2ξ2)] (1)

In the second equality we have removed the explicit dependence on by scaling the coordinates as . (The scaling of the Jacobian is irrelevant for this discussion.) Remarkably, the full quantum mechanical structure of the harmonic oscillator model persists as when one uses the scaled variables , since the propagation is independent of , as are the scaled bound-state energies . Thus there is a domain of positions and momenta where the action is proportional to and the system stays quantum mechanical even in the limit. On the other hand, the propagation between fixed (-independent) positions involves large values of , and thus the transitions between highly excited levels (with of order ) correspond to classical dynamics in the limit.

In the case of general relativity, can be eliminated Volovik:2009xs () from the equations of motion, and physical phenomena only depend on dimensionless quantities such as . Although mass cancels out of the equations of motion in classical gravity (due to the equivalence principle), it appears in the Schrödinger equation SakuraiMQM (), but always in the form . As we shall show, is the fundamental mass parameter which appears in the equations of motion for fields when one formulates quantum field theory through the action principle and functional integrals.

In quantum field theory the -matrix elements of lowest order in are usually considered to be tree diagrams (Born approximation), with each loop correction introducing one additional power of hbarloops (). However, Donoghue and Holstein and their collaborators Donoghue:2001qc () have demonstrated that classical physics (of lowest order in ) can also emerge from loop diagrams where zero mass quanta appear. As we shall see, the difference arises from the definition of the limit, i.e., the limit depends on which Lagrangian parameters are taken to be independent .

Born diagrams usually provide a good first approximation to scattering amplitudes in quantum field theory. However, the bound-state poles of a scattering amplitude are not present in tree (or any finite number of loop) diagrams, but instead are generated by the divergent perturbative expansion of the covariant Green’s function. For example, the Schrödinger and Dirac bound states, which arise from tree-level interactions of an electron in an external Coulomb potential, emerge in field theory from the infinite sum of ladder and crossed-ladder Feynman diagram contributions to the electron-muon Green’s function in the limit where the muon mass is taken to infinity Brodsky:1971sk (). The binding is caused by loop momenta which are , thus changing the relation between the number of loops and the power of .

We shall argue that the expansion can provide a systematic approximation scheme for bound states formed by interactions between particles. This expansion is equally valid for relativistic and non-relativistic dynamics. We find that the lowest order interaction kernel in indeed defines a viable “Born term for bound states”. This Born approximation is insensitive to the prescription of propagators, which allows a simple Hamiltonian equal-time development in cases where the Coulomb interaction dominates. A hidden Lorentz boost covariance provides a non-trivial test that the approximation correctly includes all contributions of lowest order in .

## Ii The ℏ expansion

There is a general understanding that each loop contribution to quantum field theory amplitudes is associated with one factor of hbarloops (). Loop integrals represent quantum fluctuations and classical physics is expected to emerge from the Born term in the limit. To fully define this limit one needs to specify the dependence of the fields, couplings and mass parameters of the action . The choice is not as obvious as it sounds since appears not only in the path integral measure but (for dimensional reasons) also in the action itself. In this section we establish the scaling with which makes the loop and expansions equivalent. In specific problems the energies and momenta may also scale with , as illustrated by the harmonic oscillator above. In the next section we show how the contribution from momenta of in the multiple loops of ladder diagrams describes the physics of scattering from a fixed potential and gives rise to Schrödinger and Dirac bound states.

We use units where the speed of light Planck’s constant then has dimension of energy times length ; i.e, . The dimensions of the fields and other parameters of the action are as usual fixed by the requirement that be dimensionless.

The gluon part of the QCD action,

 −14∫d4x(Gμν)2=−14∫d4x(∂μAν−∂νAμ+igℏ[Aμ,Aν])2 (2)

sets . The gluon coupling has dimension , ensuring that is dimensionless. This requires that appear with an inverse factor of in (2). For the same reason the classical electric charge and mass are divided by in the action of scalar QED,

 SSQED=∫d4x([(∂μ+ieℏAμ)ϕ]†[(∂μ+ieℏAμ)ϕ]−m2ℏ2ϕ†ϕ) (3)

The boson field dimensions implied by (2) and (3) agree with the dimensions of the corresponding classical fields. Fermion field dimensions are convention dependent since Grassmann numbers have no classical counterparts. While factors of are rarely displayed in quantum field theory, the convention of Ref. Sakurai () is to write the electron part of the QED action in the form

 SQED=∫d4x[¯ψ(iℏ⧸∂−e⧸A−m)ψ] (4)

with a factor in the kinetic term. This gives the electron field the dimension so that has the dimension of a probability density. With this convention the classical charge and mass appear in the action without a factor of , and the anti-commutator of the electron field is

 {ψ†(t,x),ψ(t,y)}=δ3(x−y) (5)

Insofar as is regarded as a constant of nature, the factors of in the action only serve to ensure the proper units of the parameters. However, to define an expansion we need to specify the dependence on of all quantities appearing in the action. In analogy to the harmonic oscillator (1) we take the boson fields to scale as

 ~A≡A/√ℏ,   ~ϕ≡ϕ/√ℏ (6)

with and independent of . Thus . Similarly, we take the parameters

 ~g≡gℏ,~e≡eℏand~m≡mℏ (7)

to be independent of . Thus has the dimensions of wavenumber . In the functional integral measures for the gluon and scalar fields,

 −14ℏ∫d4x(Gμν)2 = −14∫d4x(∂μ~Aν−∂ν~Aμ+i~g√ℏ[~Aμ,~Aν])2 1ℏSSQED = ∫d4x[(∂μ+i~e√ℏ~Aμ)~ϕ]†[(∂μ+i~e√ℏ~Aμ)~ϕ]−~m2~ϕ†~ϕ (8)

then appears exclusively in the combinations and . Hence loop corrections of and will be of .

With the convention of a factor in the derivative as in (4) we take the fermion field to be independent of , while the coupling and mass should scale as in (7). Then

 1ℏSQED=∫d4x[¯ψ(i⧸∂−~e√ℏ~⧸A−~m)ψ] (9)

again ensuring that each loop is of .

The present derivation is equivalent to (albeit perhaps more transparent than) the standard one of Ref. hbarloops (), which associates a factor with each propagator, an with each vertex and assumes the parameters appearing in the action to be independent of , i.e., as in (7). Thus the free gluon propagator since at Born level. Similarly the Born term of . The fermion propagator is of with the field normalization implied by (4). Each loop correction adds one power of .111Reference hbarloops () uses the normalization convention for fermion fields where there is no factor of in the kinetic part of the action (4). Then the fermion fields scale similarly to the boson fields (6), and identical results are obtained for the powers of in amplitudes. The normalization convention only affects the Jacobian of the functional integral (and the external lines of the Green function).

According to (7) the equivalence of the loop and expansions thus, remarkably, requires that the classical couplings and masses divided by are fixed in the limit. For example, the dimensionless fine structure constant,

 α≡e24πℏ=~e2ℏ4π (10)

is then . The alternative of keeping and independent of was considered in Ref. Donoghue:2001qc (), where it was found that classical physics can emerge from loops in an amplitude where pairs of massless photons appear in the -channel.

Even though variations of can not be studied experimentally, an expansion is useful as a formal tool which clarifies the structure of theory. The most appropriate expansion scheme depends on the physics at hand.

In the remaining part of this paper we shall be mostly concerned with bound-state equations which require summing an infinite number of perturbative diagrams. The expansion will allow us to identify a Born term for bound states to which loop corrections may be systematically applied. In particular, we shall make use of the fact that Born terms are insensitive to the prescription of the propagators, whereas that prescription is crucial for loops.

## Iii Application of the ℏ expansion to bound states

Bound states in quantum field theory can be identified from the divergence of the perturbative series of the -matrix at each bound-state energy. In the familiar case of non-relativistic QED atoms, the divergence arises from ladder diagrams with loop momenta which scale with . The binding energy of hydrogen thus scales as . Hence an expansion at bound-state poles requires that the external momenta in the -matrix scale with . In such a limit the usual relation between the number of loops and power of need not hold. The ladder diagrams are in fact sensitive to infrared loop momenta that contribute inverse powers of . The lowest-order contribution in to the hydrogen atom is also of lowest order in , i.e., it is a Born term for the bound state. The loop momenta do not scale with in the higher-order corrections to propagators and vertices, implying the usual relation between the number of such loops and the power of .

The bound-state poles of non-relativistic QED atoms are obtained by summing ladder diagrams, the first two of which are shown in Fig. 1. The tree diagram in (a) generally dominates the loop diagram (b) at small coupling strength . However, for external momenta within the range of the atomic wave function the loop diagram is unsuppressed even in the limit. In the center-of-mass system this requires to be of ( being the mass of the constituents) and to be of . The restricted range of the loop momentum, in the bound-state domain, as well as the additional factor of in the loop diagram, is balanced (relative to the tree diagram) by its three additional propagators, each of which is of . The situation repeats for all of the higher-order, multi-loop ladder diagrams, allowing the ladder series to diverge and bound-state poles to occur at any value of .

Since the previous argument demonstrates that the loop diagram in Fig. 1(b) is, in the bound-state domain, of the same order also in as the tree diagram of Fig. 1(a). The loop does not bring an extra factor of since we consider a limit where the (external and internal) momenta depend on , similarly to the case of the harmonic oscillator (1). Repeating the argument for ladder diagrams with more rungs, we may conclude that the Schrödinger equation defines the Born term for non-relativistic atoms. The standard higher order corrections in to the binding energies and wave functions are also of higher order in .

The energy exchanged in the ladder loops is of higher order in than the 3-momentum . Thus in the Born approximation we may set in the photon propagators. This makes the non-relativistic bound-state dynamics equivalent to tree-level scattering from an external, instantaneous potential.

The relativistic Dirac-Coulomb electron bound-states are obtained by summing scattering diagrams which include all ladder and crossed-ladder photon exchanges Brodsky:1971sk (). In the limit where the muon mass is large, the sum of crossed and uncrossed photon exchanges gives a which suppresses energy exchange and reduces the dynamics to tree-level scattering. In this sense the relativistic Dirac-Coulomb bound states are also of lowest order in .

## Iv Bound states in a static external potential

The prescription in propagators is related to the boundary condition in time and is irrelevant at lowest order in . Heuristically, this is seen from the damping factors which give energy denominators where . In perturbative expansions of the S-matrix the tree diagrams are in fact independent of . Conversely, the prescription determines the discontinuities of diagrams which are of higher order in through the pinching of loop integrals. For Born level bound states this prescription independence allows alternative but equivalent wave functions as we shall next discuss.

Consider first the Dirac-Coulomb bound states in a static external potential. In four-momentum space the interactions with the Coulomb potential do not change the energy component of the particle’s momentum. Denoting a single Coulomb photon interaction by the Green’s function of the particle can be expanded as

 G(p0,p)=S+SKS+SKSKS+…=S+SKG=R(E,p)p0−E+… (11)

In the last equality we displayed the pole contribution of a bound state with energy , whose residue is easily seen to satisfy the Dirac equation. Since preserves the Green’s function is independent of the prescription at the negative-energy pole of the Dirac propagator . In other words, for we have , where . In particular, the bound-state energy is the same whether we use a Feynman or retarded propagator,

 SF/R(p0,p)≡i⧸p+m(p0−Ep+iε)(p0+Ep∓iε) (12)

If we time-order the interactions through a Fourier transform the bound state gives a stationary contribution However, the Fock state decomposition of its equal-time wave function depends on the choice of propagator (12). In the Feynman propagator the negative energy components move backward in time. This gives rise to “”-diagrams describing particle-antiparticle pair fluctuations in the time-ordered interactions with the static potential. The equal-time wave function of a Dirac-Coulomb bound state thus has Fock components with any number of pairs, and its explicit expression is, to the best of our knowledge, not known even in simple cases such as a potential.

In the case of the retarded propagator (12), the negative-energy components move forward in time,

 SR(t,p)=θ(t)2Ep[(Epγ0−p⋅γ+m)e−iEpt+(Epγ0+p⋅γ−m)eiEpt] (13)

The corresponding time-ordered interactions have no -contributions, thus only the single bound particle is present at any intermediate time. As a consequence of this, the retarded propagator is local in space at infinitesimal times,

 limt→0+SR(t,x)=γ0δ3(x) (14)

By contrast, the right-hand side would be non-local in for the Feynman propagator . This may be understood by invoking a completeness sum over states at an earlier time: the particle first moves backward in time and then returns to at .

The absence of pair production in retarded propagation as well as the locality property (14) allows a one-particle Hamiltonian description as in relativistic quantum mechanics Hoyer:2009ep (). The bound-state wave functions are then given by the Dirac equation. These single-particle wave functions describe the same bound states, which using Feynman propagators, contain an indefinite number of particle pairs arising from -diagrams. As we have seen from (11), the bound-state energies of both pictures agree at lowest order in .

## V Relativistic bound states in field theory

The Hamiltonian description of relativistic bound states in an external Coulomb potential can in certain cases be extended to field theory Hoyer:2009ep (). According to the above arguments the binding energies will at Born level be independent of the prescription used in the bound particle propagators. This allows the use of the retarded propagator (13) which suppresses -diagrams and makes the time evolution local as in (14). Coulomb exchange gives an instantaneous potential due to the absence of terms in the Lagrangian density. Transversely polarized photons propagate in time and thus they generate higher Fock states. Here we only consider cases where Coulomb exchange is dominant.

In dimensions only Coulomb exchange contributes in a gauge where . For an state where the electron is at and the muon at

 |E,t=0⟩=∫dx1dx2ψ†e(t=0,x1)χ(x1,x2)ψμ(t=0,x2)|0⟩ (15)

the equation of motion (Gauss’ law),

 −∂2xA0(x;x1,x2)=e[δ(x−x1)−δ(x−x2)] (16)

determines the instantaneous Coulomb field,

 −A0(x;x1,x2)=12e[|x−x1|−|x−x2|] (17)

It is then straightforward Hoyer:2009ep () to determine the condition on the bound-state wave function which ensures a stationary time development,

 γ0(−i→∂x1γ1+me)χ(x1,x2)−χ(x1,x2)γ0(i←∂x2γ1+mμ)=[E−V(x1−x2)]χ(x1,x2) (18)

Here the kinetic term of the electron operates on from the left, and that of the muon from the right. The potential following from (17) is

 V(x)=12e2|x| (19)

In dimensions the Dirac matrices as well as the wave function may be taken to be matrices.

Not surprisingly, the bound-state equation (18) has a “double Dirac” form, and as such was proposed already by Breit Breit:1929zz (). Now this equation is seen to provide an approximation of lowest order in to relativistic bound states, with the potential (19) in 1+1 dimensions fixed by QED.

A stringent test that (18) actually represents the exact Born level result is that it is consistent with the Poincaré invariance of QED. Translational invariance is explicit, but the boost invariance is hidden (dynamic), since the constituents are at equal time in all frames. It turns out Hoyer:1985tz () that for the linear potential (19) (and for no other form of the potential) the bound-state energy indeed has the correct dependence on the CM momentum of the bound state,

 E=√P2+M2 (20)

The -dependence of the wave function is non-trivial. It resembles an ordinary Lorentz transformation, but with (the canonical energy) in the boost parameter. Thus the wave function contracts at a rate which depends on the separation between the constituents.

The properties of the relativistic bound states defined by (18) merit further study. At large separations between the constituents, where , the wave function has a constant, non-vanishing density in . This may reflect the particle pairs which are polarized from the vacuum in a formulation using Feynman propagators.

The two very different but equivalent pictures of bound states that we find here using retarded vs. Feynman time development may be related to the well-known puzzle of the quark model vs. parton model views of real hadrons: Hadron excitation spectra reflect mainly their valence quark degrees of freedom, even though sea quarks contribute importantly to deep inelastic scattering at low and low cooper ().

A bound-state equation analogous to (18) may be derived in dimensions by assuming a non-trivial boundary condition in Gauss’ law (16). The homogenous solution for then gives rise to a linear potential. Poincaré invariance is respected similarly to the 1+1-dimensional case. It is also possible to derive the analogous Born level meson and baryon bound states in QCD Hoyer:2009ep (). The potential is gauge covariant and confines the three quarks in a symmetric way. In some respects this resembles the soft-wall AdS/QCD models deTeramond:2008ht (); Karch:2006pv () which utilize a linear potential in an effective Dirac equation in AdS space.

## Vi Conclusions

The Planck constant is a fundamental constant of nature which gives a measure of quantum effects. In this paper we have shown how expansions in powers of Planck’s constant can give new insights into perturbative and nonperturbative properties of quantum field theories. Since is a fundamental parameter, exact Lorentz invariance and gauge invariance are maintained at each order of the expansion.

It is common to set since there is a general belief that the power of is given by the number of loops in the perturbative expansion hbarloops (). In the functional integral appears in the measure , and the action is usually assumed to be independent of . It is then argued that classical physics emerges in the limit since the rapidly varying phase selects field configurations for which the action is stationary. This argument is, however, somewhat oversimplified since there may be field configurations which make . We have illustrated this with the harmonic oscillator, whose full quantum mechanical bound-state spectrum persists in the limit.

In fact, the physics of the expansion depends on the scheme; i.e., different expansions are obtained depending on which quantities (momenta, couplings and masses) are assumed to be independent of . We have shown that if the coupling and mass parameters appearing in the Lagrangian density are taken to be independent of , then each loop in perturbation theory brings a factor of . In the case of quantum electrodynamics, this scheme implies that the classical charge , as well as the fine structure constant are linear in .

We have also noted that for dimensional reasons the QCD coupling is divided by in the action (2). Similarly, the classical charge and mass are divided by in the QED action (3).

It may seem natural to fix the classical quantities , and as , as is done in Donoghue:2001qc (). However, this scheme introduces a non-trivial dependence on into the action . We also note that the fine structure constant diverges in the limit if the classical coupling is held constant. Furthermore, it has been demonstrated through explicit examples in this scheme that loops containing massless quanta, such as the gravitational form factors of the electron, can contribute to classical physics in the limit Donoghue:2001qc (). The underlying reason is that an expansion is not uniquely defined without specifying which quantities (momenta, couplings and masses) are to be regarded as independent of .

On the other hand, if the parameters of the action, , and are taken to be independent of , the fine structure constant and the expansion is equivalent to the perturbative expansion. For example, in the limit, the loops which define the function vanish, so that the running coupling is constant as in a conformal theory.

The expansion is particularly non-trivial and illuminating in the case of bound states. We have considered whether one can define a Born term for bound states, which is equivalent to a tree (no loop) approximation when , and are fixed. This would allow an unambiguous and physically motivated approximation scheme for relativistic bound states, maintaining all symmetries of the theory at each order in . In the familiar case of non-relativistic QED atoms the binding energy depends on . In order to stay on the bound state pole of a Green’s function in the limit we must allow the momenta to scale with , which introduces additional sources of . This is why the sum of multi-loop ladder diagrams in perturbation theory reduces to a Born approximation for atomic states.

Born terms are insensitive to the prescription of perturbative propagators. This is explicitly seen for tree diagrams of the S-matrix, and we have shown that it holds also for bound states in a Coulomb potential. Using Feynman or retarded propagators does, however, make a consequential difference for the wave functions of the equal-time bound states. Relativistic bound states given by the Dirac equation have an indefinite number of particle pair constituents in their Fock expansion which arise from -diagrams due to Feynman propagators. In contrast, if one chooses retarded propagators, the bound state appears to contain just a single particle (of positive or negative energy) whose distribution is given by the standard Dirac wave function.

Our understanding of bound states formed by the mutual interactions of relativistic particles is still very limited. This may be due to the lack of a physically well motivated and manageable first approximation akin to the tree diagrams of scattering amplitudes. In this paper we have shown that there is a well-defined Born approximation for relativistic bound states. With retarded propagation one obtains simple and specific bound-state equations. The two-body equations in QED and QCD have a hidden boost invariance which strongly suggests that they include all effects of lowest order in . Just as for the Dirac equation, the resulting valence wave functions implicitly describe the multiple pair constituents generated by Feynman propagation.

###### Acknowledgements.
We thank J. F. Donoghue and B. R. Holstein for a critical reading of our manuscript and a helpful discussion concerning the definition of expansions. We have also benefitted from discussions with C. Carlson and S. Peigné.

## References

• (1) Max Planck, Ann. Phys. (Leipzig) 309, 553 (1901)
• (2) P. Kovtun, D. T. Son, A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005) [hep-th/0405231].
• (3) E. Shuryak, Prog. Part. Nucl. Phys. 53, 273 (2004) [hep-ph/0312227].
• (4) G. E. Volovik, JETP Lett. 90, 697 (2010) [arXiv:0904.1965 [gr-qc]].
• (5) J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading, MA, 1994) p. 126.
• (6) J. Iliopoulos, C. Itzykson and A. Martin, Rev. Mod. Phys. 47 165 (1975);
C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980), Sec. 6-2-1.
• (7) J. F. Donoghue, B. R. Holstein, B. Garbrecht and T. Konstandin, Phys. Lett. B 529, 132 (2002) [arXiv:hep-th/0112237];
N. E. J. Bjerrum-Bohr, J. F. Donoghue and B. R. Holstein, Phys. Rev. D 68, 084005 (2003) [Erratum-ibid. D 71, 069904 (2005)] [arXiv:hep-th/0211071];
N. E. J. Bjerrum-Bohr, J. F. Donoghue and B. R. Holstein, Phys. Rev. D 67, 084033 (2003) [Erratum-ibid. D 71, 069903 (2005)] [arXiv:hep-th/0211072];
B. R. Holstein and J. F. Donoghue, Phys. Rev. Lett. 93, 201602 (2004) [hep-th/0405239].
• (8) D. R. Yennie, Brandeis Summer Institute Lectures on Strong and Electromagnetic Interactions (Brandeis University, Waltham, MA, 1963), Vol. I.;
S. J. Brodsky, in Brandeis University Summer Institute in Theoretical Physics, 1969: Atomic Physics and Astrophysics edited by M. Chrétien and E. Lipworth (Gordon and Breach, New York, 1971) Vol. 1; Report No. SLAC-PUB-1010 (1971);
A. R. Neghabian and W. Glöckle, Can. J. Phys. 61, 85 (1983).
• (10) P. Hoyer, arXiv:0909.3045 [hep-ph];
P. Hoyer, PoS EPS-HEP2009, 073 (2009);
P. Hoyer, Acta Phys. Polon. B41, 2701 (2010) [arXiv:1010.5431 [hep-ph]].
• (11) G. Breit, Phys. Rev. 34, 553 (1929).
• (12) P. Hoyer, Phys. Lett. B 172, 101 (1986) and Report No. HU-TFT-85-49 (1985).
• (13) A. M. Cooper-Sarkar, 38th International Symposium On Multiparticle Dynamics ISMD08, Hamburg, Germany, 2008, (DESY, Hamburg, 2009), http://www-library.desy.de/preparch/desy/proc/proc09-01.html [arXiv:0901.4001 [hep-ph]].
• (14) A. Karch, E. Katz, D. T. Son et al., Phys. Rev. D74, 015005 (2006) [hep-ph/0602229].
• (15) G. F. de Teramond, S. J. Brodsky, Phys. Rev. Lett. 102, 081601 (2009) [arXiv:0809.4899 [hep-ph]].
You are adding the first comment!
How to quickly get a good reply:
• Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
• Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
• Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters


text:
# 9-23 STOCK VALUATION Robert Balik and Carol Kiefer are senior vice presidents of the Mutual of...

9-23         STOCK VALUATION Robert Balik and Carol Kiefer are senior vice presidents of the Mutual of Chicago Insur- ance Company. They are codirectors of the company’s pension fund management division, with Balik having re- sponsibility for fixed-income securities (primarily bonds) and Kiefer being responsible for equity investments. A major new client, the California League of Cities, has requested that Mutual of Chicago present an investment seminar to the mayors of the represented cities; and Balik and Kiefer, who will make the actual presentation, have asked you to help them.

To illustrate the common stock valuation process, Balik and Kiefer have asked you to analyze the Bon Temps Company, an employment agency that supplies word processor operators and computer programmers to busi- nesses with temporarily heavy workloads. You are to answer the following questions:

a.        Describe briefly the legal rights and privileges of common stockholders.

b.       (1) Write a formula that can be used to value any stock, regardless of its dividend pattern.

(2)     What is a constant growth stock? How are constant growth stocks valued?

(3)     What are the implications if a company forecasts a constant g that exceeds its rs? Will many stocks have expected g > rs in the short run (that is, for the next few years)? in the long run (that is, forever)?

c.        Assume that Bon Temps has a beta coefficient of 1.2, that the risk-free rate (the yield on T-bonds) is 7%, and that the required rate of return on the market is 12%. What is Bon Temps’s required rate of return?

d.       Assume that Bon Temps is a constant growth company whose last dividend (D0, which was paid yesterday) was $2.00 and whose dividend is expected to grow indefinitely at a 6% rate. (1) What is the firm’s expected dividend stream over the next 3 years? (2) What is its current stock price? (3) What is the stock’s expected value 1 year from now? (4) What are the expected dividend yield, capital gains yield, and total return during the first year? e. Now assume that the stock is currently selling at$30.29. What is its expected rate of return?

f.        What would the stock price be if its dividends were expected to have zero growth?

g.       Now assume that Bon Temps is expected to experience nonconstant growth of 30% for the next 3 years, then return to its long-run constant growth rate of 6%. What is the stock’s value under these conditions? What are its expected dividend and capital gains yields in Year 1? Year 4?

h.       Suppose Bon Temps is expected to experience zero growth during the first 3 years and then resume its steady- state growth of 6% in the fourth year. What would be its value then? What would be its expected dividend and capital gains yields in Year 1? in Year 4?

i.         Finally, assume that Bon Temps’s earnings and dividends are expected to decline at a constant rate of 6% per year, that is, g = -6%. Why would anyone be willing to buy such a stock, and at what price should it sell? What would be its dividend and capital gains yields in each year?

j.         Suppose Bon Temps embarked on an aggressive expansion that requires additional capital. Management decided to finance the expansion by borrowing $40 million and by halting dividend payments to increase retained earnings. Its WACC is now 10%, and the projected free cash flows for the next 3 years are -$5 mil- lion, $10 million, and$20 million. After Year 3, free cash flow is projected to grow at a constant 6%. What is Bon Temps’s total value? If it has 10 million shares of stock and $40 million of debt and preferred stock com- bined, what is the price per share? k. Suppose Bon Temps decided to issue preferred stock that would pay an annual dividend of$5.00 and that the issue price was \$50.00 per share. What would be the stock’s expected return? Would the expected rate of return be the same if the preferred was a perpetual issue or if it had a 20-year maturity?

format: plain


text:
# Solve the Diophantine Equation $x^2 + 1 = 2y^4$ over $\mathbb{Z}$.

Solve the Diophantine Equation $x^2 + 1 = 2y^4$ over $\mathbb{Z}$.

I have found few elementary solutions like $(1,1)$.

I have tried it with variable replacements. After solving it a bit it becomes clear that both $x$ and $y$ are odd. Evaluating it for $x = 2x_1+1$ and $y=2y_1+1$ I have reached:

$$2x_1^2 + 2x_1 + 1 = (2y_1 + 1)^4$$

or

$$x_1^2 + (x_1 + 1)^2 = (2y_1 + 1)^4$$

I don't know how to solve this further. Do you have any ideas, hints or techniques to solve this?

Thanks.

• @Jean Corrected it. Mar 28, 2016 at 11:14
• I know the Pythagoras Theorem but what do I do with it. Also, the hypotenuse must be a perfect square. Mar 28, 2016 at 11:19
• So $(\pm 239,\pm 13),(\pm 1, \pm 1)$ appears to be the only solutions. Mar 28, 2016 at 11:35
• @K.K.McDonald Given the reputation of Smarandache, the article has to be carefully checked I guess :) Mar 28, 2016 at 11:38
• xD ok I googled him and now I know what you mean! Mar 28, 2016 at 11:39

The only solutions to this equation are with $(x,y)=(1,1)$ and $(239,13)$. This was first proved by Ljunggren via an elaborate version of Skolem's $p$-adic method. Other proofs all rely upon pretty sophisticated techniques (linear forms in logarithms, the hypergeometric method, Frey curves, etc). Mordell asked many years ago (as reported in Guy's UPINT) for an elementary proof. As of now, there is none known.

If $$x$$ and $$y$$ are integers such that $$x^2 + 1 = 2y^4$$ then in $$\Bbb{Z}[i]$$ we have $$2y^4=(x+i)(x-i).$$ Of course $$x$$ is odd and so the greatest common divisor of the two factors on the right hand side is $$1+i$$. Because $$\Bbb{Z}[i]$$ is a unique factorization domain it follows that $$x+i=i^k(1+i)(u+vi)^4,$$ for some integers $$k$$, $$u$$ and $$v$$. Expanding the right hand side shows that $$x+i=i^k((u^4-4u^3v-6u^2v^2+4uv^3+v^4)+(u^4+4u^3v-6u^2v^2-4uv^3+v^4)i),$$ and so comparing imaginary parts and changing the sign of $$u$$ if necessary, we see that $$u^4+4u^3v-6u^2v^2-4uv^3+v^4=\pm1.$$ This is a Thue equation, which has only finitely many integral solutions and there exist effective methods to find them all. With the help of a computer I've found that they are, up to sign, $$(0, 1),\quad(1, 0),\quad (2, -3),\quad (3, 2).$$ These correspond to $$x=\pm1$$ and $$x=\pm239$$, with $$y=\pm1$$ and $$y=\pm13$$ respectively.


text:
# IBPS PO Quantitative Aptitude Questions with Answers Practice online test 15

Description: free IBPS PO Quantitative Aptitude Questions with Answers Practice test 15 for IBPS PO Preliminary and Main online test Prepare bank PO banking mock exams adda

1 . What approximate value should come in place of question mark (?) in the following questions?
23$\times$15 - 60 + ? $\div$ 31 = 292
 218 186 217 201 None of these
2 . What approximate value should come in place of question mark (?) in the following questions?
 1 3 4 5 none of these
3 . What approximate value should come in place of question mark (?) in the following questions?
 4 $1\over40$ 5 $1\over40$ 6 $1\over40$ 5 $3\over40$ none of these
4 . What approximate value should come in place of question mark (?) in the following questions?
 231.4 233.6 231.8 231.6 None of these
5 . What approximate value should come in place of question mark (?) in the following questions?

 1 2 3 4 5
6 .
The circumferences of two circles are 132 metres and 176 metres respectively. What is the difference between the area of the larger circle and that of the smaller circle?
 1048 sq m 1076 sq m 1078 sq m 1090 sq m None of these
7 .
Arun invested a sum of money at a certain rate of simple interest for a period of four years. Had he invested the same sum for a period of six years, the total interest earned by him would have been fifty percent more than the earlier interest amount. What was the rate of interest percent per annum?
 4 8 5 Cannot be determined None of these
8 .
180-metre long train crosses another 270- metre long train running in the opposite direction in 10.8 seconds. If the speed of the first train in 60 kmph, what is the speed of the second train in kmph?
 80 90 150 Cannot be determined None of these
If the numerator of a fraction is increased by 150% and the denominator of the fraction is increased by 350%, the resultant fraction is $25\over51$ . What is the original fraction?
 $11\over17$ $11\over15$ $15\over17$ $13\over15$ none of these


text:
The Vertices of a Triangle Are (6, 0), (0, 6) and (6, 6). the Distance Between Its Circumcentre and Centroid is - Mathematics

MCQ

The vertices of a triangle are (6, 0), (0, 6) and (6, 6). The distance between its circumcentre and centroid is

Options

• $2\sqrt{2}$

• 2

• $\sqrt{2}$

• 1

Solution

$\sqrt{2}$ Let A(0, 6), B(6, 0) and C(6, 6) be the vertices of the given triangle.

$\text { Centroid of } \bigtriangleup \text { ABC } = \left( \frac{0 + 6 + 6}{3}, \frac{6 + 0 + 6}{3} \right)$

$= \left( 4, 4 \right)$

$\text { Coordinates of N } = \left( \frac{6 + 6}{2}, \frac{6 + 0}{2} \right)$

$= \left( 6, 3 \right)$

$\text { Coordinates of P } = \left( \frac{0 + 6}{2}, \frac{6 + 6}{2} \right)$

$= \left( 3, 6 \right)$

Equation of MN is y = 3

Equation of MP is x = 3

As , we know that circumcentre of a triangle is the intersection of the perpendicular

bisectors of any two sides .

Therefore, coordinates of circumcentre is (3, 3)

Thus, the coordinates of the circumcentre are (3, 3) and the centroid of the triangle is (4,4).
Let d be the distance between the circumcentre and the centroid.

$\therefore d = \sqrt{\left( 4 - 3 \right)^2 + \left( 4 - 3 \right)^2} = \sqrt{2}$

Is there an error in this question or solution?

APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 23 The straight lines
Q 31 | Page 135


text:
## Results (displaying all 13 matches)

Label Base field Conductor norm Conductor label Isogeny class Weierstrass coefficients
75.1-a2 $$\Q(\sqrt{-3})$$ 75 75.1 75.1-a $$\bigl[1$$ , $$1$$ , $$1$$ , $$0$$ , $$0\bigr]$$
1875.1-b2 $$\Q(\sqrt{-3})$$ 1875 1875.1 1875.1-b $$\bigl[1$$ , $$0$$ , $$1$$ , $$-1$$ , $$23\bigr]$$
11025.1-c2 $$\Q(\sqrt{-3})$$ 11025 11025.1 11025.1-c $$\bigl[a + 1$$ , $$-a - 1$$ , $$1$$ , $$0$$ , $$11 a - 21\bigr]$$
11025.3-c2 $$\Q(\sqrt{-3})$$ 11025 11025.3 11025.3-c $$\bigl[a + 1$$ , $$a + 1$$ , $$0$$ , $$2 a - 1$$ , $$-11 a - 10\bigr]$$
12675.1-a2 $$\Q(\sqrt{-3})$$ 12675 12675.1 12675.1-a $$\bigl[1$$ , $$a + 1$$ , $$1$$ , $$a$$ , $$7 a + 3\bigr]$$
12675.3-a2 $$\Q(\sqrt{-3})$$ 12675 12675.3 12675.3-a $$\bigl[a$$ , $$-a - 1$$ , $$a$$ , $$1$$ , $$-7 a + 10\bigr]$$
19200.1-g2 $$\Q(\sqrt{-3})$$ 19200 19200.1 19200.1-g $$\bigl[0$$ , $$1$$ , $$0$$ , $$0$$ , $$-12\bigr]$$
57600.1-j2 $$\Q(\sqrt{-3})$$ 57600 57600.1 57600.1-j $$\bigl[0$$ , $$-a - 1$$ , $$0$$ , $$0$$ , $$72 a - 36\bigr]$$
57600.1-k2 $$\Q(\sqrt{-3})$$ 57600 57600.1 57600.1-k $$\bigl[0$$ , $$a + 1$$ , $$0$$ , $$2 a - 1$$ , $$-71 a + 35\bigr]$$
81225.1-a2 $$\Q(\sqrt{-3})$$ 81225 81225.1 81225.1-a $$\bigl[a$$ , $$a$$ , $$a + 1$$ , $$-a - 1$$ , $$30 a + 60\bigr]$$
81225.3-a2 $$\Q(\sqrt{-3})$$ 81225 81225.3 81225.3-a $$\bigl[1$$ , $$a$$ , $$a$$ , $$a$$ , $$-31 a + 91\bigr]$$
102675.1-a2 $$\Q(\sqrt{-3})$$ 102675 102675.1 102675.1-a $$\bigl[a + 1$$ , $$-a + 1$$ , $$0$$ , $$1$$ , $$-47 a + 34\bigr]$$
102675.3-a2 $$\Q(\sqrt{-3})$$ 102675 102675.3 102675.3-a $$\bigl[1$$ , $$-a$$ , $$0$$ , $$a - 1$$ , $$47 a - 13\bigr]$$


text:
# a basic definite integration and its result used in evaluating limit

Question :

$$\mathbf\Omega(n)=\displaystyle\int _{0}^{2\pi}\log(n^2-2n\cos t+1)dt\ ,\ n\geq1$$

then , find :

$$\displaystyle \lim_{n \to \infty} \left(1+\dfrac{\mathbf \Omega(n)}{4\pi}\right)^{\log(n+1)}$$

my attempt:

$$\mathbf\Omega(n)=\displaystyle\int _{0}^{2\pi}\log(n-e^{it})dt+\displaystyle\int _{0}^{2\pi}\log(n-e^{-it})dt\$$

$$\mathbf\Omega(n)=2\displaystyle\int _{0}^{2\pi} \log\ n \ dt+\displaystyle \int_{0}^{2\pi}\log\left(1-\dfrac{e^{it}}{n}\right)dt+\displaystyle \int_{0}^{2\pi}\log\left(1-\dfrac{e^{-it}}{n}\right)dt$$

$$\mathbf\Omega(n)=2\pi \log\ n-\displaystyle \int_{0}^{2\pi}\displaystyle \sum_{k=1}^{\infty}\ \dfrac{e^{ikt}}{k\ n^k}dt-\displaystyle \int_{0}^{2\pi}\displaystyle \sum_{k=1}^{\infty}\ \dfrac{e^{-ikt}}{k\ n^k}dt$$

changing order of summation and integration

$$\mathbf\Omega(n)=2\pi \log\ n -\displaystyle \sum_{k=1}^{\infty}\dfrac{1}{kn^k}\displaystyle \int_{0}^{2\pi}e^{ikt}dt\ - \displaystyle \sum_{k=1}^{\infty}\dfrac{1}{kn^k}\displaystyle \int_{0}^{2\pi}e^{-ikt}dt$$

both integrals becomes zero becuase they are integration over a period of sinusoid

$$\mathbf\Omega(n)=2\pi \log\ n$$

therefore,

$$\displaystyle \lim_{n \to \infty} \left(1+\dfrac{\mathbf \Omega(n)}{4\pi}\right)^{\log(n+1)}=\displaystyle \lim_{n \to \infty} \left(1+\dfrac{\log n}{2}\right)^{\log(n+1)}$$

after this step, i don't know how to proceed to find final answer. my integral might be wrong so, please help me re-evaluate if possible and also help me find limit

You don't really need to calculate $$\Omega(n)$$.
Note that $$\Omega(n)\geq c\log n$$ for some $$c>0$$ and all sufficiently large $$n$$ (e.g., from bounding the integrand $$\log(n^2-2n\log t+1)$$ by $$2\log(n-1)$$), so $$(1+\Omega(n)/(4\pi))^{\log(n+1)}$$ is of the form $$\infty^\infty$$ so must $$\to\infty$$ as $$n\to\infty$$.


text:
Home > Standard Error > Why Is Standard Error Divided By Square Root Of N

# Why Is Standard Error Divided By Square Root Of N

## Contents

With n = 2 the underestimate is about 25%, but for n = 6 the underestimate is only 5%. As the sample size increases, the sampling distribution become more narrow, and the standard error decreases. The sample mean will very rarely be equal to the population mean. If people are interested in managing an existing finite population that will not change over time, then it is necessary to adjust for the population size; this is called an enumerative check my blog

And we've seen from the last video that, one, if-- let's say we were to do it again. Thanks! Let me try this... Your cache administrator is webmaster. https://en.wikipedia.org/wiki/Standard_error

## Square Root N Law

Thanks, David Harper CFA FRM, Nov 5, 2013 #5 cdbsmith Member David, Thanks for clarifying that standard error calculation for me. Eventually, you do this a gazillion times-- in theory, infinite number of times-- and you're going to approach the sampling distribution of the sample mean. Let's see if I can remember it here.

Watch QueueQueueWatch QueueQueue Remove allDisconnect Loading... Consider the following scenarios. The SD/sqrt(N) trick is handy if you don't own a computer, but in this day and age there is absolutely no reason to ever use it. Standard Error Regression So this is the mean of our means.

Math Meeting 351,386 views 8:26 11 5 Square Roots of Variable Expressions Pt 1 - Duration: 6:32. Sigma Divided By Square Root Of N So the lower tail is insignificant and it’s a right tailed test. It doesn't matter what our n is. http://www.statisticshowto.com/sigma-sqrt-n-used/ I'm going to remember these.

Difference Between a Statistic and a Parameter 3. Standard Error Of The Mean Definition Second, the standard deviation given in the question is the population standard deviation. National Center for Health Statistics typically does not report an estimated mean if its relative standard error exceeds 30%. (NCHS also typically requires at least 30 observations – if not more I had found in a previous part of the problem that $\hat \sigma=.33853$ and the sample consists of $16$ measurements.

## Sigma Divided By Square Root Of N

But you can also compute the SE of any statistic, not just the mean. https://www.reddit.com/r/askscience/comments/2l57ff/can_someone_explain_to_me_why_the_standard_error/ These assumptions may be approximately met when the population from which samples are taken is normally distributed, or when the sample size is sufficiently large to rely on the Central Limit Square Root N Law Of course, this would be prohibitively time consuming, so instead you can use a trick called bootstrapping. Standard Error Formula And it actually turns out it's about as simple as possible.

For the purpose of this example, the 9,732 runners who completed the 2012 run are the entire population of interest. http://3cq.org/standard-error/which-is-larger-standard-error-or-standard-deviation.php How's that? It is true you can argue that you still can define a standard deviation by associating the group with the process of "picking randomly one element of that group". 3) So Two data sets will be helpful to illustrate the concept of a sampling distribution and its use to calculate the standard error. Standard Error Vs Standard Deviation

In regression analysis, the term "standard error" is also used in the phrase standard error of the regression to mean the ordinary least squares estimate of the standard deviation of the The sample standard deviation s = 10.23 is greater than the true population standard deviation σ = 9.27 years. Thank you. news Relative standard error See also: Relative standard deviation The relative standard error of a sample mean is the standard error divided by the mean and expressed as a percentage.

All of these things I just mentioned, these all just mean the standard deviation of the sampling distribution of the sample mean. Standard Error Of Proportion Sign in to make your opinion count. The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample used to compute the estimate.

## In order to figure out which formula to use, and why we use sigma / sqrt (n), I tried to really understand what I was doing by asking questions: Q.

permalinkembedsaveparentgive gold[–]rlee89 0 points1 point2 points 1 year ago(0 children) There are two things I'm confused about, the first is why we are converting the SD from the sample population back to variance Loading... I understand the aim is to find how the samples vary on average. Difference Between Standard Error And Standard Deviation Blackwell Publishing. 81 (1): 75–81.

For example finding the standard error when the standard deviation is 18, sample mean is 45 and the sample is 100. Quantitative Methods (20%)' started by monsieuruzairo3, Nov 5, 2013. The confidence interval of 18 to 22 is a quantitative measure of the uncertainty – the possible difference between the true average effect of the drug and the estimate of 20mg/dL. http://3cq.org/standard-error/when-to-use-standard-error-versus-standard-deviation.php Yes, a stats 101 question, but my brain can't seem to go that far back.

The mean age was 33.88 years. You just have to recompute your statistic over and over, and see how it varies. Ecology 76(2): 628 – 639. ^ Klein, RJ. "Healthy People 2010 criteria for data suppression" (PDF). Hopefully that makes sense permalinkembedsaveparentgive gold[–]rlee89 0 points1 point2 points 1 year ago(0 children) Also how does the sample size (N) relate to the No.

Popular Articles 1. Khan Academy 718,269 views 9:49 Statistics: Standard deviation | Descriptive statistics | Probability and Statistics | Khan Academy - Duration: 13:07. doi:10.2307/2340569. The true standard error of the mean, using σ = 9.27, is σ x ¯   = σ n = 9.27 16 = 2.32 {\displaystyle \sigma _{\bar {x}}\ ={\frac {\sigma }{\sqrt

You don't use it as a number, you use it as a criterion. So 9.3 divided by 4. Statistics Blog > sigma / sqrt (n) You'll come across a couple of different formulas when calculating z-scores: When I first started learning statistics, the different formulas completely confused me. Correction for finite population The formula given above for the standard error assumes that the sample size is much smaller than the population size, so that the population can be considered

Ecology 76(2): 628 – 639. ^ Klein, RJ. "Healthy People 2010 criteria for data suppression" (PDF). And of course, the mean-- so this has a mean. For a value that is sampled with an unbiased normally distributed error, the above depicts the proportion of samples that would fall between 0, 1, 2, and 3 standard deviations above Numberphile 1,799,169 views 8:49 Standard deviation - Statistics - Duration: 8:26.

With statistics, I'm always struggling whether I should be formal in giving you rigorous proofs, but I've come to the conclusion that it's more important to get the working knowledge first permalinkembedsavegive gold[–]pythagorian 2 points3 points4 points 1 year ago(2 children)Although this is a correct technical definition, I think it will not help someone understand the overall meaning of standard error. Tweet Welcome to Talk Stats! As your sample tends towards infinity, the SD of the sampling distribution will tend towards zero.

Discrete vs. Then the mean here is also going to be 5. So I'm taking 16 samples, plot it there. Study Planner Features & Pricing Forum FAQs Blog Bionic Turtle Home Forums > Financial Risk Manager (FRM).


text:
# E2T How To: Install a Wind Turbine

If you read our write-up about four wee wind startups making turbines for the small-scale wind power market and are looking for a weekend project, there are a few things you should check out before climbing onto the roof with a pinwheel.

Do you have enough wind?

Before you can exploit your wind resource, you need to make sure you’ve got enough of a breeze to make it worthwhile. The DOE’s Department for Energy Efficiency and Renewable Energy recommends an average wind speed of 9 mph for a stand-alone system and 10 mph for a grid-connected turbine. If you can’t get your hands on Second Wind‘s super-sweet Triton Sonic Wind Profiler, which measures wind speeds using sound, check out the wind resource map from the National Renewable Energy Lab to make sure you’ve got the gusts.

What will the neighbors say?

Before you get too excited about your very own windmill, you should check the local zoning ordinances lest you upset the Joneses. It’s best to talk to your neighbors before you start unless you want to defend your breezy hopes at a public hearing called in your name. In most areas there is a height limit on the structures you can have on your property and most turbines need to up at about 80 feet. Having at least half an acre of land is a good minimum to make sure you’re not treading too close to the picket fences.

A wind energy system can run anywhere from $6,000 to$22,000. As for a return on your investment, the American Wind Energy Association estimates that a residential turbine can make your money back in as little at six years but more likely it will take around 15 years to see your investment blow into the black.


text: For example, the mean of the numbers 5, 7, 9 is 4 since 5 + 7 + 9 = 21 and 21 divided by 3 [there are three numbers] is 7. a_n = a_1 + (n - 1) d Answer 2. Problem 2. arithmetic series word problems with answers Question 1 : A man repays a loan of 65,000 by paying 400 in the first month and then increasing the payment by 300 every month. Basic Arithmetic Lesson 1 – Whole Numbers 1 LESSON 1 – WHOLE NUMBERS INTRODUCTION We will begin our study of Basic Arithmetic by learning about whole numbers. We have the formula that gives the sum of the first n terms of an arithmetic sequence knowing the first and last term of the sequence and the number of terms (see formula above). Make sure you answer the question asked. Arithmetic Series We can use what we know of arithmetic sequences to understand arithmetic series. By using this website, you agree to our Cookie Policy. Problem: Scott took 7 math tests in one marking period. Tell whether if the sequence is arithmetic or not. Example Question #1 : How To Solve Arithmetic Word Problems Bob gets paid an hour for the regular hours he works and an hour for any overtime hours he works. 1) Find the general term of the A.P. Scroll down the page for examples and solutions on how to use the formulas. { x } ^ { 2 } - 4 x - 5 = 0. x2 − 4x − 5 = 0. What is the common difference of the arithmetic progression 10, 5, 0, -5? Do the numbers 2, 6, 10, 12, 16... form an arithmetic progression? Types of Arithmetic Problems with Solutions . Learn and practice on fully solved and frequently asked questions covering all the topics on quantitative and arithmetic aptitude. Arithmetic Reasoning questions are math problems expressed in a story format. Calculator Tips. Trigonometry. So, here I am sharing arithmetical reasoning tricks for your better preparation. = 3 n + 3 Arithmetic mean formula. 4sinθcosθ = 2sinθ. ... Arithmetic Problems. Dividing the sum by the number of test scores we get: Answer: The mean test score is 85. Questions include rate, percent, and measurement problems, geometry problems, and distribution of a quantity into its fractional parts. . Basic Math Solver offers you solving online fraction problems, metric conversions, power and radical problems. The sum sn of the first n terms of an arithmetic sequence is defined by s_n = \dfrac{n (a_1 + a_n)}{2} After Centuries, a Simple Math Problem Gets an Exact Solution Mathematicians have long pondered a deceptively easy puzzle about the reach of a goat … of terms. NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions give the step by step solutions to these optional problems also. Solution: 25 + … We will discuss some arithmetic Progression problems with solutions in which students are facing problems while solving it. Find its 100 th term.Solution to Problem 4: We use the n th term formula for the 5 th and 15 th terms to write a 5 = a 1 + (5 - 1 ) d = 22a 15 = a 1 + (15 - 1 ) d = 62We obtain a system of 2 linear equations where the unknown are a1 and d. Subtract the right and left term of the two equations to obtain 62 - 22 = 14 d - 4 dSolve for d. d = 4Now use the value of d in one of the equations to find a1. . Mathematically, Arithmetic Mean= average = Sum of terms/ No. Since $\frac{1}{b + c}, \frac{1}{c + a}, \frac{1}{a + b}$ are in A.P. Arithmetic Calculator with step by step solutions Long Arithmetic, Rational Numbers, Operations with Fractions, Ratios, Proportions, and Percents, Measurement, Area, and Volume, Factors, Fractions, and Exponents, Unit Conversions, Data Measurement and Statistics. You will need to get assistance from your school if you are having problems entering the answers into your online assignment. \] Properties of average. $89, 73, 84, 91, 87, 77, 94 Solution: The sum of these numbers is 595. a_n = a_1 + (n - 1) d \\ Question 1: The sum of two numbers is 50, and their difference is 30. ... Let $${a_n}$$ be an arithmetic progression. Sequence A : - 1,{\rm{ }} - … Find a formula for the n th term and the value of the 50 th termeval(ez_write_tag([[728,90],'analyzemath_com-medrectangle-4','ezslot_1',340,'0','0']));Solution to Problem 1: Use the value of the common difference d = 3 and the first term a1 = 6 in the formula for the n th term given above \( Arithmetic sequences are used throughout mathematics and applied to engineering, sciences, computer sciences, biology and finance problems.A set of problems and exercises involving arithmetic sequences, along with detailed solutions and answers, are presented. Arithmetical reasoning is an important topic in verbal reasoning. y = 3x + 4. You may speak with a member of our customer support team by calling 1-800-876-1799. Basic and general quantitative aptitude problems or questions and answers with simple explanation of numerical arithmetic ability for all competitive examinations like bank, CAT, SSC, interviews, quiz tests. Find the value of the 20 th termSolution to Problem 2: Use the value of the common difference d = -10 and the first term a1 = 200 in the formula for the n th term given above and then apply it to the 20 th term a 20 = 200 + (-10) (20 - 1 ) = 10Problem 3 An arithmetic sequence has a common difference equal to 10 and its 6 th term is equal to 52. Arithmetic Mean Formula Sum of all of the numbers of a group, when divided by the number of items in that list is known as the Arithmetic Mean or Mean of the group. Quadratic equation. The average of the first and last term would also be the average of all the terms of the sequence. All … Solution : Here a = x + b , d = x + 3b - (x + b) d = x + 3b - x - b. Get help with your Math Word Problems homework. Type a math problem. Covid-19 has affected physical interactions between people. Arithmetic Series Online Calculator. . It’s not just you. Find second geometric mean. What is the mean test score? Arithmetic Progressions: Problems with Solutions. Also, the answer key and explanations are given for the same. An arithmetic series is a series or summation that sums the terms of an arithmetic sequence. QuickMath allows students to get instant solutions to all kinds of math problems, from algebra and equation solving right through to calculus and matrices. Arithmetic Progressions Practice Problems: Level 01 Solve the given practice questions based on arithmetic progression. You can find area and volume of rectangles, circles, triangles, trapezoids, boxes, cylinders, cones, pyramids, spheres. Problem 1 The first term of an arithmetic sequence is equal to 6 and the common difference is equal to 3. Review the latest information on permitted and prohibited calculators. For algebra problems like this one, you need to understand the laws of exponents. Arithmetic Progression problems with solutions. REVIEW OF ARITHMETIC SEQUENCESThe formula for the n th term aeval(ez_write_tag([[300,250],'analyzemath_com-medrectangle-3','ezslot_3',320,'0','0']));n of an arithmetic sequence with a common difference d and a first term a1 is given by Now take a 5 = 48 ⇒ a r 5-1 = 48 ⇒ a 2 4 = 48 ⇒ a = 3. https://artofproblemsolving.com/wiki/index.php/Modular_arithmetic/Introduction We will discuss some arithmetic Progression problems with solutions in which students are facing problems while solving it. . As in our initial clock example, let's work in modulus 12. 5c5d6f3da65850013a992201 5a673ccd63b436e8b899e064. Basic Arithmetic - Problem Solving on Brilliant, the largest community of math and science problem solvers. a_{50} = 3 (50) + 3 = 153 Arithmetic Word Problems - Sample Math Practice Problems The math problems below can be generated by MathScore.com, a math practice program for schools and individual families. Slader teaches you how to learn with step-by-step textbook solutions written by subject matter experts. Math Problems – Answers 1 to 5 Answer 1. Simple solutions to hard problems. All the problems can be solved without a calculator. . This category includes addition, subtraction, multiplication, and division as well as percent problems, decimal recognition, fraction and percent equivalences, and estimation problems. Solution: Formula – The ‘n’ numbers G 1, G 2, G 3, . References to complexity and mode refer to the overall difficulty of the problems as they appear in the main program. Solution: Here a = 1000 and d = 110 and nth term is 160. nth term = a + ( n-1)d. So charge for borewell work ( i.e 160 feet digging) = 1000 + ( 161-1)250 = ₹41,000. © and ™ ask-math.com. You are not required to use a calculator. BASIC&ARITHMETIC&2&CUMULATIVE&REVIEW&2&Answers&.....&237& vi ! Locate your solution among the answer choices. Solve the problem. All Rights Reserved. We use the n th term formula to find the 50 th term a 50 = 2 + 2 (50 - 1) = 100We now the first term and last term and the number of terms in the sequence, we now find the sum of the first 50 terms s 50 = 50 (2 + 100) / 2 = 2550. given that the first few terms of an arithmetic sequence are given by 6,12,18,... Geometric Sequences Problems with Solutions, Math Problems, Questions and Online Self Tests, Math Word Problems with Solutions and Answers for Grade 10, High School Math (Grades 10, 11 and 12) - Free Questions and Problems With Answers. Your goal is to determine what the question is asking by picking out the relevant factors needed to solve the problem, set up mathematical equations as needed, and arrive at the correct solution. Example -11: Four geometric means are inserted between 1/8 and 128. Basic Arithmetic Lesson 1 – Whole Numbers 2 LESSON CHECKLIST G n are said to be Geometric means in between ‘a’ and ‘b’. Here a_{4} + a_{8} = 24 and a_{6} + a_{10} = 34, From arithmetic progression problems with solutions to Home.$ We use the n th term formula for the 6 th term, which is known, to write. a6= 52 = a1+ 10 (6 - 1 ) The … 2010 - 2013. Stay Home , Stay Safe and keep learning!!! Access the answers to hundreds of Math Word Problems questions that are explained in a way that's easy for you to understand. Applications and problem-solving. Phone support is available Monday-Friday, 9:00AM-10:00PM ET. a 15 = 2 + 10 (15 - 1) = 142Problem 4 An arithmetic sequence has a its 5 th term equal to 22 and its 15 th term equal to 62. ... • Disjointed Solution • Problem Solving Process . Symbolab: equation search and math solver - solves algebra, trigonometry and calculus problems step by step This website uses cookies to ensure you get the best experience. ... Let [ tex ] { a_n } [ /tex ] be an arithmetic progression problems with in. 4 \sin \theta find your textbook online math Solver with free step by step solutions to algebra,,. – 4: find the arithmetic progression whether if the sequence is equal to 3 conceptual about. 6 th term formula for the 6 th term formula for the same 1 the and. Problems as they appear in the main program optional problems also REVIEW & 2 & CUMULATIVE & REVIEW 2. Your conceptual understanding about the arithmetic progression problems with solutions in which students are problems! We can use what we know of arithmetic sequences to understand the laws exponents. First term of an arithmetic series with a = 3 polynomials, combine expressions Word problems questions that are in. The latest information on permitted and prohibited calculators marking period, cones, pyramids, spheres { a_n } /tex! A_N } [ /tex ] be an arithmetic sequence is arithmetic or not how learn. As they appear in the main program, 94 solution: formula the! 0, -5 to find the general term of an arithmetic sequence distribution of a into. On permitted and prohibited calculators would also be the average of the arithmetic progression 10, 12, 16 form. The topics on quantitative and arithmetic problems with solutions aptitude: Four geometric means in between ‘ a ’ ‘... 73, 84, 91, 87, 77, 94 solution: sum. 48 ⇒ a 2 4 = 22Solve for a1 to obtain 2 4 = 3... A member of our customer support team by calling 1-800-876-1799: Four geometric means are inserted 1/8. By calling 1-800-876-1799 understanding about the arithmetic progression problems with solutions in which students are facing problems solving... Be geometric means are inserted between 1/8 and 128 problems entering the answers to hundreds math! + 4 = 48 ⇒ a = -1.5 and d = -0.5 has. The n th term, which is known, to write to the difficulty! Subject matter experts questions based on arithmetic progression, percent, and distribution of a quantity into its fractional.... 1 – Whole numbers 2 Lesson CHECKLIST arithmetic mean formula topic in verbal reasoning practice. Your conceptual understanding about the arithmetic progression Class 10 Maths Chapter 5 Progressions!, -5 include rate, percent, and other math problems – answers 1 5!, factor/multiply polynomials, combine expressions conceptual understanding about the arithmetic progression online calculator calculate... Answer: the mean test score is 85 step by step solutions algebra! 3, inserted between 1/8 and 128 rectangles, circles, triangles, trapezoids, boxes, cylinders,,. This one, you need to get assistance from your school if you having! D = -0.5, 10, 5, 0, -5 10, 5, 0, -5 example Let... Answers to hundreds of math Word problems questions that are explained in a way that 's easy for to! Problems can be solved without a calculator here I am sharing arithmetical reasoning tricks for your better preparation ’... 48 ⇒ a 2 4 = a 3 + 4 = 22Solve for a1 obtain! Summation that sums the terms of an arithmetic series member of our support., a+3d can find area and volume of rectangles, circles,,... Understand arithmetic series work in modulus 12 school if you are having problems entering the to. 46 to 50, a+d, a+2d, a+3d: Answer: arithmetic problems with solutions. G 1, G 3, solutions for Class 10 by solving the problems as they in. Find the general term of an arithmetic sequence is equal to 3 of math Word problems questions that explained. & CUMULATIVE & REVIEW & 2 & answers &..... & 237 & vi can use what we of., metric conversions, power and radical problems your school if you are having problems entering the answers for 46... Be geometric means are inserted arithmetic problems with solutions 1/8 and 128 having problems entering the answers your! Access the answers to hundreds of math Word problems questions that are explained in a way 's. 73, 84, 91, 87, 77, 94 solution: 25 + … to... On how to use the rules of modular arithmetic to Solve the following problems..... & 237 & vi the. That are explained in a story format can be solved without a calculator arithmetic formula. Difference is equal to 1 \cos \theta = 2 \sin \theta of arithmetic to... &..... & 237 & vi solutions in which students are facing problems solving... A 1 + ( 5 - 1 ) 4 = 48 ⇒ a r 5-1 = 48 ⇒ r. A calculator in the main program and practice on fully solved and frequently asked covering... Of arithmetic sequences to understand the laws of exponents your better preparation arithmetic problems with solutions! And the common difference of the arithmetic progression 10, 5, 0, -5 a 1 + 5!... form an arithmetic series is a series or summation that sums terms!, combine expressions of arithmetic sequences to understand arithmetic series that are explained a. You how to use the arithmetic problems with solutions of modular arithmetic to Solve the given questions. As they appear in the main program also, the Answer key and explanations are given the... Covering all the topics on quantitative and arithmetic aptitude 's work in 12., a+d, a+2d arithmetic problems with solutions a+3d score is 85 key and explanations given. Topics on quantitative and arithmetic aptitude = 48 ⇒ a 2 4 = 48 ⇒ a r =. Practice on fully solved and frequently asked questions covering all the problems can be without! Geometry problems, and other math problems expressed in a story format + 4 = 22Solve for to! 94 solution: formula – the ‘ n ’ numbers G 1, G 3,, power and problems. } [ /tex ] be an arithmetic progression problems with solutions in which students are facing while! Can simplify and evaluate expressions, factor/multiply polynomials, combine expressions a 3 × a 4 = 48 ⇒ =. Know of arithmetic sequences to understand the laws of exponents online calculator to calculate the sum these! One marking period to calculate the sum of these numbers is 50, and other math –... May speak with a = -1.5 and d = -0.5 prohibited calculators 84 91! Entering the answers to hundreds of math Word problems questions that are explained in a story format a and! Means are inserted between 1/8 and 128: 25 + … Go the... Is 50, and their difference is 30 the following diagrams give two formulas to the. Some arithmetic progression 10, 5, 0, -5 to 5 Answer 1, to.... Important topic in verbal reasoning & REVIEW & 2 & CUMULATIVE & REVIEW & &. Last term is 1000 and the last term would also be the average of the A.P with a -1.5. Students are facing problems while solving it entering the answers for questions 46 to 50 step step..., Let 's work in modulus 12 Home, stay Safe and keep learning!!. Algebra problems like this one, you agree to our Cookie Policy CUMULATIVE & &... On fully solved and frequently asked questions covering all the terms of an arithmetic progression the step by solutions... And volume of rectangles, circles, triangles, trapezoids, boxes,,! Initial clock example, Let 's work in modulus 12 237 & vi 0. x2 − 4x − 5 48! 4X − 5 = 0 complexity and mode refer to the answers into your assignment. Sharing arithmetical reasoning is an important topic in verbal reasoning some arithmetic progression to understand the laws of.... Means are inserted between 1/8 and 128 they appear in the main program period. And prohibited calculators key and explanations are given for the same laws of exponents n are to., 12, 16... form an arithmetic series we can use what know... Our customer support team by calling 1-800-876-1799, a+d, a+2d, a+3d ‘! A quantity into its fractional parts expressions, factor/multiply polynomials, combine expressions you can and... Problems with solutions in which students are facing problems while solving it work in modulus 12 a calculator progression 10. 73, 84, 91, 87, 77, 94 solution: 25 + … Go the. Get: Answer: the sum of the terms of an arithmetic progression to use formulas... 1 and the common difference is equal to 6 and the common difference is equal to 6 the. Sum of the first and last arithmetic problems with solutions would also be the average of the A.P [ tex ] { }. } ^ { 2 } - 4 x - 5 = 0. x2 4x... Following diagrams give two formulas to find the general term of an arithmetic progression with. And their difference is equal to 1 down the page for examples and solutions on how to use n! Of modular arithmetic to Solve the given practice questions based on arithmetic progression the page for examples and on. That are explained in a story format solving it: Four geometric means in ‘! = -0.5 of arithmetic sequences to understand arithmetic series & CUMULATIVE & REVIEW & &! And volume of rectangles, circles, triangles, trapezoids, boxes, cylinders,,... Combine arithmetic problems with solutions to these optional problems also between 1/8 and 128: the sum of terms/ No Level 01 the...: formula – the ‘ n ’ numbers G 1, G 2, G 2 6...


text:
Card-matching game

I updated my code according to a few suggestions from my last post, so I thought I'd repost it to see what other feedback I could get. I also commented it heavily and fixed some of the functionality.

One question I do have is what is the difference between defining a function by doing

var funcName = function(){};

rather than

function funcName() {};

Which one is better or is neither better and it's just a convention to do one over the other?

You can check out the game at mattgowie.com/project-2/. And here is the source:

    // Author: Matt Gowie
// Created on: 10/01/12
// Project for Web Dev 2400
$(document).ready(function(){ "use strict"; var numberOfPairs = 30; var cardsToFlip = []; var prevCardClasses = []; var matches = 0; var score = 0; var multiplier = 10; var that = this; var$board = $('.board'),$winMessage = $('.win-message'); // Check if the player won the game by matching all of the card pairs var checkForWin = function() { // Are there no more cards on the board? if($('.card-container').length === 0) {
// Then the player must have won! Hide the board and show the win message
$board.hide();$winMessage.show();
}
};

$('.win-message a').click(function() { // How are the previous cards being reset here?$('.matches').html('0');
$('.score').html('0');$('.mult').html('10X');
matches    = 0;
score      = 0;
multiplier = 10;
buildGameBoard();
});

// Build a deck of cards
//
// numberOfPairs - integer for how many pairs we need, and is
//   currently restricted to <= 30
//
// Returns an array of pairs of integers. ex: [1, 1, 2, 2, 3, 3, ... ]
function buildDeck(numberOfPairs) {
var deck = [];
for(var i = 0; i < numberOfPairs; i++){
deck.push(i,i);
}
return deck;
};

// Shuffle the given ordered deck
// deck - An ordered array of pairs of integers
// Returns an array of pairs of integers which are no longer ordered
function shuffleDeck(deck) {
var rand, shuffled = [];
// Randomly access each pair in the given deck and add it to the resulting
// shuffled deck
while(deck.length > 0){
rand = Math.random() * deck.length;
shuffled.push(deck.splice(rand, 1)[0]);
}
return shuffled;
};

// Build the cards for the given deck and append them to the board
// deck - a shuffled deck
// Returns void
function buildCards(deck) {
var row = 1, col = 1;
// Loop through each card in the deck, initializing it with it's relevent info
for( var i = 1; i <= deck.length ; i++ ) {
buildCard(deck[i - 1], row, col, i);
col += 1;
// Increment the row and reset the cols for every 10 cards
if(i % 10 === 0) { row += 1; col = 1; }
}
};

// Build the html for a card and append it to the board
// cardClass - integer which corresponds to a specific card in the deck
// row       - integer corresponding to this cards row number
// col       - integer corresponding to this cards col number
// id        - integer for identifing the card for click events
// Returns void
function buildCard(cardClass, row, col, id) {
var $card =$('

');
var $container =$('
');
var $cardContainer =$('
');

// Associate this card with the given class, so we know what it is when flipped
$card.data('cardClass', 'card' + cardClass); if(cardClass === 's-ace'){$card.addClass('debug'); } // For debugging!

// Associate the cards container with the current side of the card, which
// when the game starts is always on its back
$cardContainer.data('cardSide', 'back'); // Add the id, row, and col to the card container for positioning and // card identification when clicking$container.attr('id', "card" + id).addClass("row" + row).addClass("col" + col);

// Put the card and containers inside one another, and then append them to the board
$container.html($cardContainer.html($card));$board.append($container); } // Reset and build the game board by creating a deck, shuffling, and dealing function buildGameBoard() { // Reset the GameBoard area$winMessage.hide();
$board.show().html(''); // Initialize a new deck, shuffle, and deal. var deck = buildDeck(numberOfPairs); deck = shuffleDeck(deck); buildCards(deck); // Bind the click function for every card on the new board$('.container').click(function() { clickCard(this); });
};
buildGameBoard();

// Flip the clicked card and if > 1 card is flip check for a match
// container - the container of the card that was clicked
var clickCard = function(container) {
var cardId = $(container).attr('id'); // Do we currently have two cards flipped? Which implies we are checking for a match if(cardsToFlip.length < 2) { // Flip the card if we don't have two cards flipped and the card clicked // is not the same card that is already flipped if(cardsToFlip.length <= 1 && cardsToFlip[0] !== cardId){ cardsToFlip.push(cardId); flipCard(container); } // If there are two cards flipped over check for a match if(cardsToFlip.length === 2){ // checkMatch is delayed 2 seconds to allow player to look at cards setTimeout(function(){ checkMatch(); }, 2000); } } } function incrementScoreBoard() { matches += 1;$('.matches').html(matches);
score += 10 * multiplier;
$('.score').html(score);$('.mult').html('10X');
multiplier = 10;
}

// Check to see if the two cards which are flipped over are a match
var checkMatch = function() {
console.log('checkMatch called!');
var cardOneClass = $('#' + cardsToFlip[0]).find('.card').data('cardClass'); var cardTwoClass =$('#' + cardsToFlip[1]).find('.card').data('cardClass');

if(cardOneClass === cardTwoClass){  // Do these cards match?
// Clear the cards from the board, increment the score board, and
//   check to see if the user won.
$(cardsToFlip).each(function(i, e){$("#" + e).html(''); });
incrementScoreBoard();
checkForWin();
} else {
// No match so lower the multiplyer
if(multiplier !== 1){ multiplier -= 1; }
$('.mult').html(multiplier + 'X'); // Flip the cards onto their back side$(cardsToFlip).each(function(i, e){ flipCard($("#" + e)); }); } // Add the cards which were just flipped to the previous card area$('.prev-card1').attr('class', 'card back prev-card1 ' + prevCardClasses[0]);
$('.prev-card2').attr('class', 'card back prev-card2 ' + prevCardClasses[1]); prevCardClasses = []; cardsToFlip = []; }; // Flip the card container in the given container. var flipCard = function(container) { var classToRemove, classToAdd, flipDir; var$cardContainer = $(container).children('.card-container'); var$card = $cardContainer.children('.card'); var cardSide =$cardContainer.data('cardSide');
var cardClass = $card.data('cardClass'); if(cardSide === 'back'){ // Is this card face down? classToRemove = 'back'; classToAdd = cardClass; flipDir = 'rl';$cardContainer.data('cardSide', 'front');
} else { // This card is face up.
classToRemove = 'front ' + cardClass;
flipDir = 'lr';
$cardContainer.data('cardSide', 'back'); prevCardClasses.push(cardClass); } // Use jquery.flip plugin to flip the card up or down depending on vars // set previously$card.css("background-color", "#AB0000" );
$card.flip({ direction: flipDir, onBefore: function() {$card.removeClass(classToRemove).addClass(classToAdd);
},
onEnd: function() {
\$card.css({ 'background-color': 'rgba(0, 0, 0, 0.0)' });
},
speed: '200',
color: '#AB0000'
});
};

});

• You can look at this post to see the [difference between a function expression vs declaration in Javascript][1] [1]: stackoverflow.com/questions/1013385/… – Kinjal Oct 15 '12 at 1:10
• Pretty nice update. Perhaps the checkMatch function is still doing a little too much (i.e. not just checking for a match, but also calculating the multiplier, flipping cards, updating the prev-cards stuff). That's the kind of stuff that could be broken into separate functions. But I'll let someone else take a closer look this time :) – Flambino Oct 15 '12 at 10:25
• The link you provided is dead, can you provide a link to the previous review ? – konijn Dec 23 '13 at 16:07

I like your code

• "Use Strict" within a IFFE
• Good level of comments
• Correct casing and decent naming
• Nice size of methods

Some very minor stuff

• JsHint says you never use that ( line 12 ) nor i declard on line 163 and line 171
• function(){} should not have a semicolon after the closing curly brace (buildCards)
• var f = function(){} should have a semicolon after the closing curly brace (clickCard)
• You have a magical constant in buildCards which could be extracted and named.
• You have a magical constant in incrementScoreBoard which could be extracted and named.
• You should take out all debugging related code in your final version

As for var funcName = function(){}; versus function funcName() {};, I always thought the second form was silly. Until I read this:

// Don't do this:
function getData() { }

// Do this instead:
var getData = function() { };

Whenever JR says something about JavaScript, one should pay attention. Since 2011 though we have come to the practical conclusion that anonymous functions in the callstack are a major PITA. which means that if we stick to 'Do this instead' we ought to write.

var getData = function getData(){};

And that just looks wrong ( not DRY ), and should be avoided in my book.

• Hey, I posted this a long time ago and it's long since been forgotten, but thanks for the feedback. One interesting thing you brought up that I've just recently started changing about my code is the var funcName = function(){}; versus function funcName() {}; convention. After this project I start doing the former, but I recently switched to the latter. I switched because of the function.name field which isn't populated when using the anon function route. Using your last example ('var getData..') would fix this issue, but I agree with you on the ugliness. What do you do exactly? – Gowie47 Jan 7 '14 at 0:31


text:
Home > How To > How To Reduce Sampling Frame Error

# How To Reduce Sampling Frame Error

## Contents

Zia ABBASI Regional Director Australian Bureau of Statistics The purpose of this paper is to discuss some of the causes of measurement error and suggest possible ways to Whenever the sampling frame includes units that do not exist anymore (e.g., because the sample frames are incorrect and outdated) it will be impossible to obtain any samples from such non The implications of this sampling problem and ways to take it into account are discussed in (Shoham et al. 2006). Logic edits involve specifying checks in advance to data collection. have a peek here

NON-RESPONSE—Non-response errors occur when respondents are different than those who do not respond. The limited sampling bias gives problems in the determination of the causal relation between sensory stimuli and certain features of the neuronal population responses, because it may artificially increase the mutual References Cochran W.G. (1977), “Sampling Techniques”, Wiley Groves R. Let us assume that these are in reality completely independent of one another, and therefore observing values of $$x$$ can not help predict likely values of $$y\ .$$ However, an experimentalist http://fluidsurveys.com/university/tips-for-overcoming-researcher-bias

## Types Of Sampling Errors

Stay in the loop: You might also like: Market Research How to Label Response Scale Points in Your Survey to Avoid Misdirecting Respondents Shares Market Research Two More Tips for Explain the types of Survey's? If most speedometers are tuned to overestimate the speed, and to overestimate it more at higher speed, then the resulting sampled distribution will be biased toward high velocities. A classic frame error occurred in the 1936 presidential election between Roosevelt and Landon.

It is recognized that this procedure may bias the sampling toward larger neurons (emitting signals that are easier to detect) and toward most active neurons (Shoham et al. 2006). It happens when the majority of those contacted in a survey have a special interest or strong views in the survey topic. Explain Information Systems and Quality? How To Reduce Bias In Qualitative Research That way, you can make sure you’re not forgetting a critical segment and also that your sampling method is correct.

For instance, in a typical household survey a sample of households is selected in two stages: in a first stage there is a selection of villages or parts of cities (cluster) Ensure that this is the case by allowing all respondents the ability to gain knowledge and take part in your survey. A way to quantify the amount of statistical dependence between the random variables $$X$$ and $$Y$$ is to measure mutual information $$I(X;Y)$$ $$\tag{1} I(X;Y) = \sum_{x,y} P(x,y) \, log_2 \frac{P(x,y)}{P(x) \cdot imp source For example, with \(N$$= 200 samples; (100 samples for each value of $$x\ ;$$ Figure 2D-F), the limited sampling bias of mutual information is 0.033 bits.

This inevitably involves forcing a statistical technique on the data. Researcher Bias Example Stefano Panzeri, The University of Manchester, UK and IIT, Genoa, Italy Mr. What is Sampling? However, due to limited sampling, the estimated probabilities (red line of Figure 2A and Figure 2B) differ markedly from 0.1 and from one another, and the mutual information estimate obtained by

## How To Reduce Bias In Research

Trick to deal with sample frame error: take some serious time to study the industry or area of interest of your survey before sampling. http://mospi.nic.in/informal_paper_17.htm Post a comment Name Email Website Cancel reply Share Information MBA Tutor GD Topics MBA Colleges MBA Colleges India Assam Delhi Haryana Jammu and Kashmir Jharkhand Punjab MBA Courses Business Environment Types Of Sampling Errors SELECTION ERROR—This occurs when respondents self select their participation in the study – only those that are interested respond. Non Sampling Error But the common mistakes are rarely about the big picture.

This involves checking that all documents for a record are together and correctly labelled. When adopting such complex sample designs it is essential to ensure that the sample frame information is used properly and that the probability and random selection are implemented and documented at Researchers should carefully investigate the methodology used in any given survey. The cards would be shuffled and then the interviewer would ask a series of questions out of sequence, record each answer and then reassemble the questions in the right sequence to How To Reduce Sampling Error

and Cooper M. (1998) “Nonresponse in Household Interview Surveys”, New York: John Wiley and Sons Panzeri S., Senatore R., Montemuro M.A., Petersen R.S. (2007), "Correcting for the Sampling Bias Problem in It is therefore highly probable that estimates produced from the sample would not accurately reflect the entire community. This article has been written by KJ Singh a MBA Graduate from a prestigious Business School In India Article Published:August 14, 2010 More Entries : What are the methods of Sampling http://joomlamoro.com/how-to/how-to-reduce-systematic-error-in-measurement.php We can use information from other sources to create a more accurate description of the population.

Call backs for those not available and follow-ups can increase response rates for those who, initially, were unable to reply. How Can Bias Be Avoided However, the problem may not be overcome by just increasing the sample size, particularly if the non-responding units have different characteristics to the responding units. It is essentially a questionnaire design problem that can be surprisingly difficult to detect, because it’s just difficult to know when you have mishandled the questions if the respondents have returned

## Figure 1: Possible sources of bias occurring in the selection of a sample from a population.

unbiased). Combatting Researcher Bias The following is a list of the common types of researcher bias and some tips on how to avoid each. Sampling bias can lead to a bias of a probability functional. How To Avoid Sampling Error It is best for the researcher and the organization with the problem (sometimes the same person) to discuss and agree on the population definition and how it will be targeted.

Any sampling method that excludes people that fall under the researcher’s target population, in a non-random way, will create sampling frame error. Inappropriate edit checks and inaccurate weights in the estimation procedure can also introduce errors to the data at the editing and estimation stage. The best way to avoid surrogate information error is through exploratory research. Non-response Error this is a response problem that can at first seem inconsequential.

This bias can also be due to a lack of a clearly defined target group for the study. What are its Characteristics, Advantages and Disadvantages?


text:
Surface tension and capillarity

The cause of surface tension is said to be asymmetry in the forces experienced by the molecules at the surface due to different interactions with air and liquid, but then the same argument also applies for all other surfaces, where the fluid is in contact with the container then why isn't all strange surface tension phenomenon seen at those surfaces? And how can surface tension be described as perpendicular forces to an imaginary line at the surface, isn't it supposed to act in all direction??

Regarding capillarity, in deriving the equation $\gamma= \frac{r h \rho g}{2 \cos \theta_Y}$, we use the relation of excess(over the pressure on convex side) pressure on the concave side which is derived only for spherical bubbles or drops. then how can we assume the meniscus formed will be spherical and not defined by some other function?

-

Your question consists of three parts so let's answer them one by one.

1) Surface tension between liquid-solid and air-solid

why isn't all strange surface tension phenomenon seen at those surfaces?

Just a short note upfront: I don't know what you mean by 'strange surface tension phenomenon' so I will explain in general how surface tension is also present for the liquid-solid and the air-solid interface.

You are absolutely correct in saying that these interactions also exist at other interfaces, so also at the interface of the liquid and the container (solid) and at the interface of the air and the container. The main reason why this is not easy to see is that the solid is extremely rigid as compared to the liquid and gas and therefore doesn't significantly deform because of surface tension acting on it.

You can, however, see a signature of the surface tensions of gas-solid and liquid-solid in the case of a sessile drop. What happens there is that the droplet while adjust it's shape to a certain contact angle ($\theta$). According to the balance of there 3 surface tensions. This can be visualized as follows:

If you balance the surface tensions you get an equation which is called Young's equation: $\gamma_{sg}=\gamma_{sl}+\gamma_{lg}\cos\theta$. If, for example, the surface tension between the liquid and the solid is extremely high, this will result in a contact angle that is close to 0.

2) Direction of surface tension

how can surface tension be described as perpendicular forces to an imaginary line at the surface, isn't it supposed to act in all direction?

This is actually not true. Surface tension is a force parallel to the surface. The reason comes from the molecular origin of surface tension. Let's explain it in terms of the liquid-air interface. Liquid molecules prefer to have a certain amount of interaction with their surrounding liquid molecules. If a molecule is located at the air-liquid interface, it doesn't have sufficient interaction with the air molecules. To 'solve' this, the molecules at the surface will pull more strongly on the neighbors that are also on the surface, thus exerting a force parallel to the surface.

3) Capillarity

how can we assume the meniscus formed will be spherical and not defined by some other function?

I assume you meant to write the equation for capillary action: $h = \frac{2\gamma \cos \theta}{\rho g r}$. The assumption there is indeed that the meniscus is spherical. This is necessary to simplify the mathematics and be able to solve the liquid height in a capillary tube analytically.

You can estimate when it is still a reasonable assumption by calculating the capillary length: $\lambda_c=\sqrt{\frac{\gamma}{\rho g}}$ and comparing this to the radius of the tube. If the radius of the tube is below this length then surface tension is 'sufficiently dominant' to make sure that the meniscus is a spherical cap. You can see this capillary length in the equation for capillary action: $$h = \frac{2\gamma \cos \theta}{\rho g r}= \frac{2 \lambda_c^2 \cos \theta}{r}$$

-
quite relieved by the answer. But just a minor spinoff of my question, is the tendency to minimize surface area a consequence of the tendency to minimize surface potetial energy or is it a consequence of the invard direction of the force on the surface? –  Satwik Pasani Mar 30 '13 at 8:29
Those two things (minimize surface energy and inward force) are fully equivalent. Remember that force is nothing more then the derivative of an energy with respect to position ($F=\frac{dE}{dx}$). For minimization of $E$ you will look for the derivative to be 0, for a force balance you will look for $F=0$ which is thus exactly the same thing. (check out the second paragraph of this page: en.wikipedia.org/wiki/Surface_tension) –  Michiel Mar 30 '13 at 8:49


text:
# Centering Math Formulas

I am using LaTeXila in Ubuntu and I want to center my math equation so I have this written:

``````\$\[ 2^{10} \]\$
``````

But when I try to compile into PDF, it outputs the error:

``````Bad math environment delimeter
``````

How do I get it to center math text properly? Thanks

-
This is not valid latex syntax. You have to omit both dollar-signs to achieve what you want. –  user1146332 Jan 30 '13 at 8:07
@user1146332 wow thanks, such an easy fix. –  Richard Jan 30 '13 at 8:09
`\$[itex]\$` is for inline mathematical expressions. –  Svend Tveskæg Jan 30 '13 at 8:16
For basics and not only you can see here ctan.org/tex-archive/info/lshort/english ... :) –  karathan Jan 30 '13 at 8:21

## 1 Answer

``````\$2^{10}\$
``````

will give you inline math, i.e., a formula in the middle of the text.

``````\[ 2^{10} \]
``````

will give you display math, i.e., a formula separated from the text, and centered.

-