text:
Computing derivative-based global sensitivity measures using polynomial chaos expansions

# Computing derivative-based global sensitivity measures using polynomial chaos expansions

<NAME>, Institute of Structural Engineering, Chair of Risk, Safety & Uncertainty Quantification, Stefano-Franscini-Platz 5, CH-8093 Zürich, Switzerland
###### Abstract

In the field of computer experiments sensitivity analysis aims at quantifying the relative importance of each input parameter (or combinations thereof) of a computational model with respect to the model output uncertainty. Variance decomposition methods leading to the well-known Sobol’ indices are recognized as accurate techniques, at a rather high computational cost though. The use of polynomial chaos expansions (PCE) to compute Sobol’ indices has allowed to alleviate the computational burden though. However, when dealing with large dimensional input vectors, it is good practice to first use screening methods in order to discard unimportant variables. The derivative-based global sensitivity measures (DGSM) have been developed recently in this respect. In this paper we show how polynomial chaos expansions may be used to compute analytically DGSMs as a mere post-processing. This requires the analytical derivation of derivatives of the orthonormal polynomials which enter PC expansions. The efficiency of the approach is illustrated on two well-known benchmark problems in sensitivity analysis.

###### keywords:
global sensitivity analysis, derivative-based global sensitivity measures (DGSM), polynomial chaos expansions, Morris method
journal: Reliability Engineering & System Safety

## 1 Introduction

Nowadays, the increasing computing power allows one to use numerical models to simulate or predict the behavior of physical systems in various fields, e.g. mechanical engineering (Reuter and Liebscher, 2008), civil engineering (Hasofer, 2009), chemistry (Campolongo et al., 2007), etc. The considered systems usually lead to highly complex models with numerous input factors (possibly tens to hundreds (Sudret et al., 2013; Patelli and Pradlwarter, 2010)) that are required to represent all the parameters driving the system’s behaviour, e.g. boundary and initial conditions, material properties, external excitations, etc. On the one hand, this increases dramatically the computational cost. On the other hand, the input factors that are not perfectly known may introduce uncertainties into the predictions of the system response. In order to take into account the uncertainty, probabilistic approaches have been developed in the last two decades, in which the model input parameters are represented by random variables. Then the input uncertainties are propagated through the computational model and the distribution, moments or probability of exceeding prescribed thresholds may be computed (Sudret, 2007; de Rocquigny, 2012).

In this context, sensitivity analysis (SA) examines the sensitivity of the model output with respect to the input parameters, i.e. how the output variability is affected by the uncertain input factors (Saltelli et al., 2000, 2004, 2008). The use of SA is common in various fields: engineering (Hasofer, 2009; Pappenberger et al., 2008; Reuter and Liebscher, 2008; Kala, 2011), chemistry Campolongo et al. (2007), nuclear safety Fassò (2013), economy Borgonovo and Peccati (2006), biology Marino et al. (2008), and medicine Abraham et al. (2007), among others. One can traditionally classify SA into local and global sensitivity analyses. The former aims at assessing the output sensitivity to small input perturbations around the nominal values of input parameters. The latter aims at assessing the overall or average influence of input parameters onto the output. Local SA has the disadvantages of being related to a fixed nominal point in the input space, and the interaction between the inputs is not accounted for Kucherenko et al. (2009). On the other hand, global SA techniques take into account the input interaction and are not based on the choice of a nominal point but account for the whole input space.

The most common sensitivity analysis methods found in the literature are the method of Morris (1991), FAST Cukier et al. (1973, 1978); Mara (2009) and variance decomposition methods originally investigated in Sobol’ (1993); Sobolâ² (2001); Sobol’ and Kucherenko (2005); Archer et al. (1997); Saltelli (2002); Saltelli et al. (2010). Usually standard Monte Carlo simulation (MCS) is employed for estimating the sensitivity indices in all these techniques. This approach requires a large number of model evaluations though and therefore is usually computationally expensive. When complex systems are of interest, the standard MCS becomes unaffordable. To overcome this problem, metamodels (also called surrogate models or emulators) are usually used in order to carry out the Monte Carlo simulation Sathyanarayanamurthy and Chinnam (2009); Zhang and Pandey (2014). In particular, polynomial chaos expansions (PCE) have been recognized as a versatile tool for building surrogate models and for conducting reliability and sensitivity analyses, as originally shown in Sudret (2006, 2008); Blatman and Sudret (2010a). Using PCE, variance-based sensitivity analysis becomes a mere post-processing of the polynomial coefficients once they have been computed.

More recently, a new gradient-based technique has been proposed for screening unimportant factors. The so-called derivative-based global sensitivity measures (DGSM) are shown to be upper bounds of the total Sobol’ indices while being less computationally demanding Sobol’ and Kucherenko (2009, 2010); Kucherenko et al. (2009); Lamboni et al. (2013). Although the computational cost of this technique is reduced compared to the variance-based technique Kucherenko et al. (2009), its practical computation still relies on the Monte Carlo simulation approach.

In this paper we investigate the potential of polynomial chaos expansions for computing derivative-based sensitivity indices and allow for an efficient screening procedure. The paper is organized as follows: the classical derivation of Sobol’ indices and their link to derivative-based sensitivity indices is summarized in Section 2. The machinery of polynomial chaos expansions and the link with sensitivity analysis is developed in Section 3. The computation of the DGSM based on PC expansions is then presented in Section 4, in which an original method for computing the derivatives of orthogonal polynomials is presented. Finally two numerical tests are carried out in Section 5.

## 2 Derivative-based global sensitivity measures

### 2.1 Variance-based sensitivity measures

Global sensitivity analysis (SA) aims at quantifying the impact of input parameters onto the output quantities of interest. One input factor is considered insignificant (unessential) when it has little or no effect on the output variability. In practice, screening out the insignificant factors allows one to reduce the dimension of the problem, e.g. by fixing the unessential parameters.

Variance-based SA relies upon the decomposition of the output variance into contributions of different components, i.e. marginal effects and interactions of input factors. Consider a numerical model where the input vector contains independent input variables with uniform distribution over the unit-hypercube and is the scalar output. The Sobol’ decomposition reads Sobol’ (1993):

 Y=M(X)=M0+M∑i=1Mi(Xi)+∑1≤i

in which is a constant term and each summand is a function of the variables . For the sake of conciseness we introduce the following notation for the subset of indices:

 u\lx@stackrel\tiny def={i1,…,is} (2)

and denote by the subvector of that consists of the variables indexed by . Using this set notation, Eq. (1) rewrites:

 (3)

in which is the summand including the subset of parameters . According to Sobol’ (1993), a unique decomposition requires the orthogonality of the summands, i.e.:

 E[Mu(Xu)Mv(Xv)]=∫HMMu(xu)Mv(xv)dx=0,u≠v (4)

In particular each summand shall be of zero mean value. Accordingly the variance of the response reads:

 D\lx@stackrel\tiny def=Var[Y]=∑u⊂{1,…,M}u≠0Var[Mu(Xu)]. (5)

In this expansion is the contribution of summand to the output variance.

The Sobol’ sensitivity index for the subset of variables is defined as follows Sobolâ² (2001):

 Su\lx@stackrel\tiny def=DuD=Var[Mu(Xu)]D (6)

The total sensitivity index for subset is given by Sobolâ² (2001):

 STu\lx@stackrel\tiny def=DTuD=∑v⊃uVar[Mv(Xv)]D (7)

where the sum is extended over all sets which contains . It represents the total amount of uncertainty apportioned to the subset of variables . For instance, for a single variable the Sobol’ sensitivity index reads:

 Si=Var[Mi(Xi)]D, (8)

and the total Sobol’ sensitivity index reads:

 STi=∑v∋iVar[Mv(Xv)]D. (9)

and respectively represent the sole and total effect of the factor on the system’s output variability. The smaller is, the less important the factor is. In the case when , is considered as unimportant (unessential or insignificant) and may be replaced in the analysis by a deterministic value.

In the literature one can find different approaches for computing the total Sobol’ indices, such as the Monte Carlo simulation (MCS) and the spectral approach. Sobol’ (1993); Sobolâ² (2001) proposed direct estimation of the sensitivity indices for subsets of variables using only the model evaluations at specially selected points. The approach relies on computing analytically the integral representations of and respectively defined in Eq. (6) and Eq. (7).

Let us denote by the set that is complementary to , i.e. . Let and be vectors of independent uniform variables defined on the unit hypercube and define . The partial variance is represented as follows Sobol’ and Kucherenko (2005):

 (10)

The total variance is given by Sobol’ and Kucherenko (2005):

 DTu=12∬[M(x)−M(x′u,x¯¯¯u)]2dxdx¯¯¯u (11)

A Monte Carlo algorithm is used to estimate the above integrals. For each sample point, one generates two -dimensional samples and . The function is evaluated at three points , and . Using independent sample points, one computes the quantities of interest , and by means of the following crude Monte Carlo estimators:

 M0=1NN∑i=1M(x(i)) (12)
 D+M02=1NN∑i=1[M(x(i))]2 (13)
 Du+M02=1NN∑i=1M(x(i))M(x(i)u,x(i)′¯¯¯u) (14)
 DTu=1NN∑i=112[M(x(i))−M(x(i)′u,x(i)¯¯¯u)]2 (15)

The computation of the sensitivity indices by MCS may exhibit reduced accuracy when the mean value is large. In addition, the computational cost is prohibitive: in order to compute the sensitivity indices for parameters, MCS requires model runs, in which is the number of sample points typically chosen equal to to reach an acceptable accuracy. Saltelli (2002) suggested a procedure that is more efficient for computing the first and total sensitivity indices. Sobol’ et al. (2007) modified the MCS procedure in order to reduce the lack of accuracy. Some other estimators for the sensitivity indices by MCS may be found in Monod et al. (2006); Janon et al. (2013).

### 2.2 Derivative-based sensitivity indices

The total Sobol’ indices may be used for screening purposes. Indeeed a negligible total Sobol’ index means that variable does not contribute to the output variance, neither directly nor in interaction with orther variables. In order to avoid the computational burden associated with estimating all total Sobol’ indices, a new technique based on derivatives has been recently proposed by Kucherenko et al. (2009).

Derivative-based sensitivity analysis originates from the Morris method introduced in Morris (1991). The idea is to measure the average of the elementary effects over the input space. Considering variable one first samples an experimental design (ED) in the input space and then varies this sample in the direction. The elementary effect () is defined as:

 EE(j)i=M(x(j)r)−M(x(j))Δ (16)

in which is the sample point and is the perturbed sample point in the -th direction. The Morris importance measure (Morris factor) is defined as the average of the ’s:

 μi=1NN∑j=1EE(j)i (17)

By definition, the variance of the s is calculated from:

 σ2i=1N−1N∑j=1(EE(j)i−μi)2 (18)

Kucherenko et al.(Kucherenko et al., 2009) generalized these quantities as follows:

 μi\lx@stackrel\tiny def=E[∂M∂xi(X)]=∫HM∂M∂xi(x)dx (19)
 σ2i=∫HM[∂M∂xi(x)]2dx−μ2i (20)

provided that is square-integrable. Any input parameter with and is considered as unimportant. It has been shown that the Morris factor has a higher convergence rate compared to variance-based methods, which makes it attractive from a computational viewpoint. Because the elementary effects may be positive or negative, they can cancel each other, which might lead to a misinterpretation of the importance of . To avoid this, Campolongo et al. (2007) modified the Morris factor as follows:

 μ∗i=E[∣∣∣∂M∂xi(X)∣∣∣] (21)

Recently, Sobol’ and Kucherenko (2009) introduced a new sensitivity measure (SM) which is the mean-squared derivative of the model with respect to :

 νi=E[(∂M∂xi(X))2] (22)

Sobol’ and Kucherenko (2009) and Lamboni et al. (2013) could establish a link between the in Eq. (22) and the total Sobol’ indices in Eq. (9). In case is a uniform random variable over , one gets:

 STi≤SDGSMi\lx@stackrel\tiny def=νiπ2D (23)

where is the upper-bound to the total sensitivity index and is the model output variance. In case of a uniform variable this upper bound scales to:

 STi≤SDGSMi=(bi−ai)2π2νiD (24)

Finally the above results can be extended to other types of distributions. If is a Gaussian random variable with mean and variance and respectively, one gets:

 STi≤SDGSMi=bi2νiD (25)

In the general case, Lamboni et al. (2013) define the upper bound of the total Sobol’ index of as:

 SDGSMi=4Ci2νiD (26)

in which is the Cheeger constant, is the cumulative distribution function of and is the probability density function of .

## 3 Polynomial chaos expansions

Let us consider a numerical model where the input vector is composed of independent random variables and is the output quantity of interest. Assuming that has a finite variance, it can be represented as follows Ghanem and Spanos (2003); Soize and Ghanem (2004):

 Y=M(X)=∞∑j=0yjϕj(X) (27)

in which form a basis on the space of second order random variables and ’s are the coordinates of onto this basis. In case the basis terms are multivariate orthonormal polynomials of the input variables , Eq. (27) is called polynomial chaos expansion.

Assuming that the input vector has independent components with prescribed probability distribution functions , one obtains the joint probability density function:

 fX(x)=M∏i=1fXi(xi) (28)

For each , one can construct a family of orthogonal univariate polynomials with respect to the probability measure satifying:

 ⟨P(i)j,P(i)k⟩\lx@stackrel\tiny def% =E[P(i)j(Xi)P(i)k(Xi)]=∫P(i)j(xi)P(i)k(xi)fXi(xi)dxi=c(i)jδjk (29)

where is the inner product defined on the space associated with the probability measure , is the Kronecker symbol with if , otherwise and is a constant. The univariate polynomial belongs to a specific class according to the distribution of . For instance, if is standard uniform (resp. Gaussian) random variable, are orthogonal Legendre (resp. Hermite) polynomials. Then the orthonormal univariate polynomials are obtained by normalization:

 Ψ(i)j=P(i)j/√c(i)j (30)

Introducing the multi-indices , a multivariate polynomial can be defined by tensor product as:

 Ψα(x)\lx@stackrel% \tiny def=M∏i=1Ψ(i)αi(xi) (31)

Soize and Ghanem (2004) prove that the set of all multivariate polynomials in the input random vector forms a basis of the Hilbert space of second order random variables:

 Y=∑α∈NMaαΨα(X) (32)

where ’s are the deterministic coefficients of the representation.

In practice, the input random variables are usually not standardized, therefore it is necessary to transform the input vector into a set of standard variables. We define the isoprobabilistic transform which is a unique mapping from the original random space of ’s onto a standard space of basic independent random variables ’s. As an example may be a standard normal random variable or a uniform variable over .

In engineering applications, only a finite number of terms can be computed in Eq.(32). Accordingly, the truncated polynomial chaos expansion of can be represented as follows Sudret (2007):

 Y=M(X)=M(T(Z))=∑α∈AaαΨα(Z) (33)

in which is the set of multi-indices ’s retained by the truncation scheme.

The application of PCE consists in choosing a suitable polynomial basis and then computing the appropriate coefficients ’s. To this end, there exist several techniques including spectral projection Le Maître et al. (2002); Matthies and Keese (2005), stochastic collocation method Xiu (2009) or least square analysis (also called regression, see (Berveiller et al., 2004, 2006)). A review of these so-called non-intrusive techniques is given in (Blatman, 2009). Recently, the least-square approach has been extended to obtain sparse expansions (Blatman and Sudret, 2010b, 2011). This technique has been applied to global sensitivity analysis in Blatman and Sudret (2010a). The main results are now summarized.

First note that the orthonormality of the polynomial basis leads to the following properties:

 E[Ψα(Z)]=0andE[Ψα(Z)Ψβ(Z)]=δαβ (34)

As a consequence the mean value of the model output is whereas the variance is the sum of the square of the other coefficients:

 D=Var[Y]=Var[∑α∈AaαΨα(Z)]=∑α∈Aα≠0aα2Var[Ψα(Z)]=∑α∈Aα≠0aα2 (35)

Making use of the unique orthonormality properties of the basis, Sudret (2006, 2008) proposed an original post-processing of the PCE for performing global sensitivity analysis. For any subset variables , one defines the set of multivariate polynomials which depends only on :

 (36)

The ’s form a partition of , thus the Sobol’ decomposition of the truncated PCE in Eq. (33) may be written as follows:

 (37)

where:

 (38)

In other words the Sobol’ decomposition is directly read from the PC expansion. Consequently, due to the orthonormality of PC basis, the partial variance reads:

 Du=Var[Mu(Zu)]=∑α∈Aua2α (39)

As a cosequence the Sobol’ indices at any order may be computed by a mere combination of the squares of the coefficients. As an illustration, the first order PC-based Sobol’ indices read:

 Si=∑α∈Aia2α/D,Ai={α∈A:αi>0,αj≠i=0} (40)

whereas the total PC-based Sobol’ indices are:

 STi=∑α∈ATia2α/D,ATi={α∈A:αi>0} (41)

## 4 Derivative of polynomial chaos expansions

In this paper, we consider the combination of polynomial chaos expansions with derivative-based global sensitivity analysis. On the one hand, PCE are already known to provide accurate metamodels at reasonable cost. On the other hand, the derivative-based sensitivity measure (DGSM) is effective for screening unimportant input factors. The combination of PCE and DGSM appears as a promising approach for effective low-cost SA. In fact, once the PCE metamodel is built, the DGSM can be computed as a mere post-processing of the metamodel which simply consists of polynomial functions.

As seen in Section 2, a DGSM is related to the expectation of the square of the model derivative, which was denoted by . We will express the model derivative in a way such that the expectation operator can be easily computed, more precisely by projecting the components of the gradient onto a PC expansion.

### 4.1 Hermite polynomial chaos expansions

In this section we consider a numerical model where is the scalar output and is the input vector composed of independent Gaussian variables . The isoprobabilistic transform reads:

 X=T(Z):Xi=μi+σiZi (42)

where are standard normal random variables. The truncated PCE of reads:

 Y=M(X)=M(T(Z))=∑α∈AaαΨα(Z) (43)

in which is a multi-index, is the set of indices in the truncated expansion, is the multivariate polynomial basis obtained as the tensor product of univariate orthonormal Hermite polynomials (see A) and is the deterministic coefficient associated with .

Since is a one-to-one mapping with , the derivative-based sensitivity index reads:

 νi=E[(∂M∂xi(X))2]=E[(∂M∘T∂zi∂zi∂xi)2]=1σi2E[(∂M∘T∂zi(Z))2] (44)

The DGSM of , in other words the corresponding upper bound to the total Sobol’ index , is computed according to Eq. (25):

 SDGSMi=σ2iνiD=1DE[(∂M∘T∂zi(Z))2]=1DE⎡⎣(∂∂zi∑α∈AaαΨα(Z))2⎤⎦ (45)

in which . This requires computing the partial derivatives of polynomial functions of the form . One can prove that the derivatives read (see A):

 ~He′n(z)=√n~Hen−1(z) (46)

Therefore the derivative of the multivariate orthonormal Hermite polynomial with respect to reads:

 ∂Ψα∂zi(z)=M∏j=1j≠i~Heαj(zj)√αi~Heαi−1(zi) (47)

provided that , and otherwise. Then the derivative of a Hermite PCE with respect to is given the following expression:

 ∂MA∂zi(z)=∑α∈A(i)√αiaαΨα′i (48)

in which is the set of multi-indices having a non-zero component and is the index vector derived from by subtracting 1 from . The expectation of the squared derivative in Eq. (45) is reformulated as:

 E[(∂MA∂zi(Z))2]=E⎡⎢⎣∑α∈A(i)∑β∈A(i)√αiβiaαaβΨαi′Ψβ′i⎤⎥⎦ (49)

Due to the linearity of the expectation operator, the above equation requires computing . Note that the orthonormality of the polynomial basis leads to where is the Kronecker symbol. Thus one has:

 E[(∂MA∂zi(Z))2]=∑α∈A(i)αia2α (50)

As a consequence, in case of a Hermite PCE the DGSM can be given the following analytical expression:

 ^SDGSMi=1D∑α∈A(i)αia2α=∑α∈A(i)αia2α∑α∈A,α≠0a2α (51)

Note that the total Sobol’ indices can be obtained directly from the PCE by as shown in Eq. (41). With integer indices , it is clear that the inequality is always true by construction.

### 4.2 Legendre polynomial chaos expansions

Consider now a computational model where the input vector contains independent uniform random variables . We first use an isoprobabilistic transform to convert the input factors into normalized variables :

 X=T(Z):Xi=bi+ai2+bi−ai2Zi (52)

where are uniform random variables. The Legendre PCE has the form of the expansion in Eq. (43), except that is now the multivariate polynomial basis made of univariate orthonormal Legendre polynomials (see B). Again, since is a one-to-one linear mapping with the derivative-based sensitivity index reads:

 νi=E[(∂M∂xi(X))2]=4(bi−ai)2E[(∂M∘T∂zi(Z))2] (53)

Similarly to Eq. (45), the upper bound DGSM to the total Sobol’ index is computed from Eq. (24) as:

 SDGSMi=(bi−ai)2π2νiD=4π2DE[(∂M∘T∂zi(Z))2]=4π2DE⎡⎣(∂∂zi∑α∈AaαΨα(Z))2⎤⎦ (54)

Thus the derivative of univariate and multivariate Legendre polynomials are required. Denoting by , one shows in B that:

 {~Le′1(z),…,~Le′n(z)}T=CLe⋅{~Le0(z),…,~Len−1(z)}T (55)

in which is a constant matrix whose row contains the coordinates of the derivative of onto a basis made of lower-degree polynomials . In other words, . Using this notation, the derivative of the multivariate orthonormal Legendre polynomials with respect to reads:

 ∂Ψα∂zi(z)=M∏j=1j≠i~Leαj(zj)(αi∑l=1CLeαil~Lel−1(zi)) (56)

For a given let us define by the index vector having the component equal to :

 αri=⎧⎪⎨⎪⎩α1,…,ithpositionr,…,αM⎫⎪⎬⎪⎭ (57)

Using this notation Eq. (56) rewrites as follows:

 ∂Ψα∂zi(z)=αi∑l=1CLeαilΨαl−1i (58)

Denote by the set of having a non-zero index , i.e. . The derivative of a Legendre PCE with respect to then reads:

 ∂MA∂zi(z)=∑α∈A(i)aα∂Ψα∂zi(z)=∑α∈A(i)αi∑l=1aαCLαilΨαl−1i(z) (59)

Denote by the set of multi-indices representing the ensemble of multivariate polynomials generated by differentiating the linear combination of polynomials . is obtained as:

 B(i)={β=α+(k−αi)⋅ei,α∈A(i),k=0,…,αi−1} (60)

where:

 ei=(0,…,0,ithpos.1,0…,0) (61)

The derivative of Legendre PCE rewrites:

 ∂MA∂zi(z)=∑β∈B(i)bβΨβ(z) (62)

in which the coefficient is obtained from Eq.(59). Since the polynomials are also orthonormal, one obtains:

 E[(∂MA∂zi(Z))2]=∑β∈B(i)b2β (63)

Finally, the DGSMs read:

 ^SDGSMi=4π2∑β∈B(i)b2β∑α∈A,α≠0a2α (64)

### 4.3 General case

Consider now the general case where the input vector contains independent random variables with different prescribed probability distribution functions, i.e. Gaussian, uniform or others. Such a problem can be addressed using generalized polynomial chaos expansions Xiu and Karniadakis (2002). As the above derivations for Hermite and Legendre polynomials are valid componentwise, they remain identical when dealing with generalized expansions. Only the proper matrix yielding the derivative of the univariate polynomials in the same univariate orthonormal basis is needed, see Appendix A for Hermite polynomials and Appendix B for Legendre polynomials. The derivation for Laguerre polynomials is also given in Appendix C for the sake of completeness.

## 5 Application examples

### 5.1 Morris function

We first consider the Morris function that is widely used in the literature for sensitivity analysis Morris (1991); Lamboni et al. (2013). This function reads:

 y=βo+20∑i=1βiωi+20∑i

in which:

• except for where ,

• the input vector contains 20 uniform random variables ,

• for ,

• for ,

• for ,

• ,

• the remaining first and second order coefficients are defined by , and ,

• and the remaining third order coefficients are set to 0.

First, a PCE is built using the Least Angle Regression technique based on a Latin Hypercube experimental design of size . Then the PCE is post-processed to obtain the total Sobol’ indices and the upper-bound derivative-based sensitivity measures (DGSMs) using Eq. (41) and Eq. (64), respectively. The procedure is replicated 100 times in order to provide the 95% confidence interval of the resulting sensitivity indices.

As a reference, the total Sobol’ indices are computed by Monte Carlo simulation as described in Section 2.1 using the sensitivity package in R Pujol et al. (2013). One samples two experimental designs of size denoted respectively by and then computes the corresponding output vectors and . To estimate the total sensitivity index with respect to random variable , one replaces the entire column in sample (which contains the samples of ) by the column in sample to obtain a new experimental design denoted by . Then the output is computed from the input . The variance-based is obtained by means of , and using the sobol2007 function Pujol et al. (2013); Saltelli et al. (2010). The total number of model evaluations required by the MCS approach is . After 100 replications we also obtain the 95% confidence interval of the sensitivity indices.

For each input variable, Figure 1 depicts the total sensitivity indices computed by MCS and PCE approaches, and the DGSM derived from PCE as well as their corresponding 95% confidence intervals (the bounds of the latter being obtained from the 100 replications). Figure 1 shows that PCE derivative-based sensitivity measures and total Sobol’ indices for parameters are both close to zero, i.e. are unimportant factors. This is consistent with the MCS-based sensitivity measures. Using a sample set of size , the PCE-based approach can detect the non-significant input parameters with acceptable accuracy compared to MCS which requires model runs. In addition, the PCE-based total Sobol’ indices for the remaining parameters vary in a significantly narrower confidence intervals, i.e. are more reliable, than the MCS-based sensitivity indices. Finally, the obtained total Sobol’ indices are always smaller than the DGSMs. The less significant the parameter, the closer DGSM gets to the total Sobol’ index.

### 5.2 Oakley & O’Hagan function

The second numerical example is the Oakley & O’Hagan function Sobol’ and Kucherenko (2009); Oakley and O’Hagan (2004) which reads:

 f(X)=aT1X+aT2cos(X)+aT3sin(X)+XTMX (66)

in which the input vector consists of 15 independent standard normal random variables . The vectors and the matrix are provided at www.sheffield.ac.uk/st1jeo.

Given the complexity of the function, the PCE-based approach is run with a Latin Hypercube experimental design of size . The size of a single sample set for the MCS approach is resulting in model runs. The procedure is similar as in Section 5.1.

Figure 2 shows that the PCE-based approach using only model evaluations can estimate the least significant parameters


text:
# How was Bohr's third postulate later found to be wrong?

Why does the Bohr's third postulate was later found to be wrong? I read it in a note but don't know why is it?

The third postulate is:

The orbits of electronic motion are circular and well defined and are such that angular momentum $mvr$ of the electron is quantized in units of $h/2\pi$.

Around 1927, there was a transition from the "old" quantum mechanics to the "new" quantum mechanics. There were lots of problems with the old quantum mechanics, and the shortcomings of the Bohr model were just one such problem. In the new quantum mechanics, the idea of classical trajectories for particles is completely abandoned. They propagate as waves instead.

Ultimately, the reason for preferring the new quantum mechanics is that it is in better agreement with experiment. One decisive experiment was the 1926 Bothe-Geiger experiment, which disproved Bohr's idea that light could still be described classically, and only the atom needed to be quantized. The old quantum mechanics also got various things wrong in its description of the atom, e.g., it predicted the hydrogen atom to have an orbital angular momentum of $1\hbar$ in its ground state, rather than the correct value of zero.

So in summary, Bohr's third postulate was not one isolated problem with the old quantum mechanics that needed to be fixed for one specific reason.

Bohr's stipulation that the orbits are circular and quantized in units of $\hbar$ was expanded by Sommerfeld to elliptical orbits and the condition $$\oint pdq=n\hbar$$ The (relativistic) semi-classical Bohr-Sommerfeld theory of the hydrogen atom could explain the observed spectrum of hydrogen as well as the solutions of the Dirac equation but failed in describing higher order perturbation effects of the Stark effect correctly and the Lamb shift.

• Some good points here but the Lamb shift was measured in 1947, well after Bohr’s theory had been abandoned. – ZeroTheHero May 12 '18 at 14:44
• @ZeroTheHero- You are, of course, right, neither the Bohr-Sommerfeld nor the Dirac theory explains the Lamb- shift. Still, the agreement of this "wrong" semi-classical Bohr-Sommerfeld theory with advanced QM theory in describing the H-spectrum is rather astounding. – freecharly May 12 '18 at 17:53

The reason why this is wrong is because :

1. If the orbit of an electron is well defined, then it must mean that it is rotating around the nucleus ( like a planetary system ).

$\implies$ They are accelerating ( since rotation changes the velocity vector at every instant ) and from electrodynamics, we have that an accelerating charge should radiate.

$\implies$ All electrons would gradually lose energy to radiation, spiral into the nucleus and atoms would be impossible.

1. With the advent of the probabilistic nature of physics ( Born's rule and the Uncertainty Principle ), orbits just went out of the picture because they would represent definite positions of the electron, which is impossible via the Uncertainty Principle. So, the postulate utterly fails at this point too...
• I don't think this is a very good explanation. Radiation is a secondary issue. The main issue is just how quantum mechanics really is found to work. Bohr was fully aware of the radiation issue, and he simply postulated that there was no radiation except in a transition between allowed states. – user4552 May 12 '18 at 13:19
• Maybe it's not complete, we can't say it's wrong. Even if Bohr postulated that quantised orbits won't radiate, it goes straight against electrodynamics. – Yuzuriha Inori May 12 '18 at 13:57
• A body emitting synchrotron radiation loses energy continuously, but the energy of an electron in a Bohr orbit is forbidden to change its energy continuously, any energy changes must be discrete. Thus the model explains the discrete spectra of atoms and the non-emission of synchrotron radiation; pity about the other empirical properties of atoms which it fails abysmally with. ;) – PM 2Ring May 12 '18 at 20:39

format: plain


text:
Boost.Hana  1.4.0 Your standard library for metaprogramming
cartesian_product.hpp File Reference

Forward declares boost::hana::cartesian_product. More...

## Namespaces

boost::hana
Namespace containing everything in the library.

## Variables

constexpr auto boost::hana::cartesian_product
Computes the cartesian product of a sequence of sequences.Given a sequence of sequences, cartesian_product returns a new sequence of sequences containing the cartesian product of the original sequences. For this method to finish, a finite number of finite sequences must be provided. More...

## Detailed Description

Forward declares boost::hana::cartesian_product.


text:
Chapter 5 - Linear Functions - Chapter Review - 5-8 Graphing Absolute Value Functions: 42

$y=|x|+13$

Work Step by Step

The graph of $y=|x|+k$, where $k$ is a positive number, translates the graph $y=|x|$ up $k$ units. When we substitute $13$ for $k$, the equation represents $y=|x|$, translated $13$ units up. This gives us the equation $y=|x|+13$.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# Write the trigonometric expression as an algebraic expression

1. Mar 12, 2012

### jkristia

1. The problem statement, all variables and given/known data

Not long ago I had a similar problem, which I was able to solve after reading this thread, but for this question I'm stuck and I could use a small hint.

2. Relevant equations

3. The attempt at a solution

Thanks

#### Attached Files:

• ###### trig-question.png
File size:
4.5 KB
Views:
599
2. Mar 12, 2012

### scurty

I would first solve for $arctan(v)$ in terms of $\alpha$. See if you can do the rest!

3. Mar 12, 2012

### SammyS

Staff Emeritus
Label the triangle differently. Label it so that the tangent of α is v .

$\displaystyle \tan(\alpha)=\frac{v}{1}\,.$

4. Mar 12, 2012

### jkristia

>>I would first solve for arctan(v) in terms of α

>>Label the triangle differently. Label it so that the tangent of α is v .

hmm, I think that is what I have, tan (α) = v = sqrt..., I'm still missing something.

5. Mar 12, 2012

### scurty

Okay, so you have $tan(\alpha) = v$. What is $arctan(v)$? Then, working from the inside out, what is $sin(arctan(v))$?

6. Mar 13, 2012

### jkristia

That is the part I'm missing, how do I find the exact value of atan(v). I know what the result is, but I dont know how to get to it.

7. Mar 13, 2012

### scurty

Look in your original picture. v is equal to four quantities, which one would give you a nice value for $arctan(v)$?

8. Mar 13, 2012

### jkristia

I appreciate the help and the hints, but at the moment I'm blank. I'm sure the answer is staring right at me, and it is pretty straight forward - but I just can't see it. I will think more about this during the day and hopefully tonight when I get back to it, the answer jumps right out of the picture.

9. Mar 13, 2012

### jkristia

ah - I think I got it, will try later. I need to find the inverse of v = sqrt()/x.

10. Mar 13, 2012

### jkristia

A little closer, but not there yet.

#### Attached Files:

• ###### trig2.png
File size:
2.1 KB
Views:
295
11. Mar 13, 2012

### scurty

Sorry for the late reply, I took a nap!

What is $arctan(tan(x))$ equal to?

Can you see which value to plug in now?

Then you need to calculate sine of that number you get, which is also listed in the picture as a number not involving trig functions or trig inverses.

12. Mar 13, 2012

### jkristia

I have not been able to figure this out by myself, I kept going in circles. Finally when I was about to give up I Bing'ed it and found this explanation http://mathforum.org/library/drmath/view/53946.html - and now I can see how simple it is, and I can see my main problem was the labeling as was pointed out in one of the answers. Had I labeled the figure correct, then I might have been able to figure it out on my own.

13. Mar 13, 2012

### scurty

Yeah, I've had plenty of those moments, and then it just clicks. Just so we all know you understand it correctly, what answer did you get?

14. Mar 14, 2012

### jkristia

This is how I solved it. It is exactly the same steps as on the link. I read the explanation, closed the browser and solved it myself by first drawing a new trigangle with the 'correct' labels, and then it pretty much fell in place.

I think the trick is to label the triangle with the sides as v/1 = v for the inner trig function, and that is where I made the mistake.... I think

Again, thank you for your help.

Edit: I just noticed that the tanθ does not provide any additional information.

#### Attached Files:

• ###### trig3.png
File size:
1.8 KB
Views:
277
Last edited: Mar 14, 2012
15. Mar 14, 2012

### scurty

That looks good to me based off of the labels of the diagram! Just wanted to make sure you figured out the problem correctly!


text:
# How to apply glm(generalized linear model) in this simple example?

We are given
1) Y = $(Y_1,Y_2,...,Y_n)^T$ ~ Exponential
2) E[Y] = $\mu$ = X$\beta$, where X $\in R^{nxr}$ and $\beta \in R^r$

My question is can we apply the glm in this case? The case where the canonical link does not relate $\mu$ and $X\beta$. If the answer is Yes, then how? if no, then what is the suitable modelling technique instead?

In addition, if we are given Y = $(Y_1,Y_2,...,Y_n)^T$ ~ Poisson or Inverse Gaussian instead, how can we apply glm?

Since you have $\mu = X\beta$ this is the identity link.

To fit an exponential GLM, you fit a gamma GLM but instead of estimating the dispersion parameter $\phi$ from the data, you specify that it's 1.

This doesn't affect the parameter estimation, only the standard errors.

[Consequently in R you simply specify the dispersion parameter when calling summary (summary(myExpGLMfitobject,dispersion=1)) . See ?summary.glm.]

For the Poisson and Inverse Gaussian cases you simply specify the relevant family. [In R, see ?family.]

• Yes this is the identical link, which is why precisely it is NOT the canonical link, since in the case where y's are exponentially distributed, the canonical link should be inverse. – mathguy Jul 22 '16 at 13:46
• Btw, in r, is there anyway to specify the link function no matter how ridiculous the link function is? Like what if somebody suspects the link function in the data to be arcsin(E(y)) = Xb? – mathguy Jul 22 '16 at 13:48
• @mathguy you can write your own link function (you also supply some other things along with it, such as the inverse link). The help on family contains an example for the binomial. Beware, however that as you move away from the canonical link (in some sense), the easier it becomes to encounter difficulties in optimizing the likelihood. (Good starting points will often help, but some of the potential problems are inherent). – Glen_b Jul 23 '16 at 2:20


text:
Last updated

A threaded rod, also known as a stud, is a relatively long rod that is threaded on both ends; the thread may extend along the complete length of the rod. [1] They are designed to be used in tension. [2] Threaded rod in bar stock form is often called all-thread.

## Studs

Studs may have a drive slot in one end to facilitate installing the stud.

### Types

Undercut studs (rolled thread) are also stronger because the metal is "rolled" up to the major diameter, not removed. This preserves the grain of the steel, and in some cases even enhances it. Full bodied studs (cut thread) are weaker because metal is removed to create the thread, disturbing the grain of the steel. [4] [5]

Undercut studs are only required in applications where the stud is exposed to fatigue. Cut threads are entirely suitable for many applications, even when rolled threads might be slightly stronger. Mass-produced fasteners (standard bolts and studs) are usually rolled, but jobbed parts with custom features and small lot sizes are likely to be cut.

## Strength

The allowable tensile force for a non-undercut threaded rod is defined by the American Institute of Steel Construction (AISC) as: [6]

${\displaystyle P={\frac {0.33F_{u}\pi d^{2}}{4}}}$

where d is the nominal diameter and Fu is the ultimate tensile strength of the material. For undercut threaded rods the allowable tensile stress is defined as: [6]

${\displaystyle P={\frac {0.6F_{y}\pi d_{s}^{2}}{4}}}$

where ds is the diameter of the shank and Fy is the yield strength of the material.

Both equations give units of force for the result, i.e. pounds force or newtons.

## Class

Metric threaded rods are marked on the end with a color code to define the ISO strength class. The color codes are: [7]

• Unmarked — 4.6 class (tensile strength = 400 N/mm2, yield strength 240 N/mm2)
• Yellow — 8.8 class (800 N/mm2, 640 N/mm2)
• Green — A2 stainless steel (304)
• Red — A4 stainless steel (316)
• White — 10.9 class (1000 N/mm2, 900 N/mm2)

## Related Research Articles

A rivet is a permanent mechanical fastener. Before being installed, a rivet consists of a smooth cylindrical shaft with a head on one end. The end opposite to the head is called the tail. On installation, the rivet is placed in a punched or drilled hole, and the tail is upset, or bucked, so that it expands to about 1.5 times the original shaft diameter, holding the rivet in place. In other words, the pounding or pulling creates a new "head" on the other end by smashing the "tail" material flatter, resulting in a rivet that is roughly a dumbbell shape. To distinguish between the two ends of the rivet, the original head is called the factory head and the deformed end is called the shop head or buck-tail.

American National Standard Pipe Thread standards, often called national pipe thread (NPT) standards for short, are U.S. national technical standards for screw threads used on threaded pipes and pipe fittings. They include both tapered and straight thread series for various purposes including rigidity, pressure-tight sealing, or both. The various types are each named with a symbol and a full name; examples of the symbols include NPT, NPS, NPTF, NPSC, and others.

A fastener or fastening is a hardware device that mechanically joins or affixes two or more objects together. In general, fasteners are used to create non-permanent joints; that is, joints that can be removed or dismantled without damaging the joining components. Welding is an example of creating permanent joints. Steel fasteners are usually made of stainless steel, carbon steel, or alloy steel.

Bolted joints are one of the most common elements in construction and machine design. They consist of fasteners that capture and join other parts, and are secured with the mating of screw threads.

A screw thread, often shortened to thread, is a helical structure used to convert between rotational and linear movement or force. A screw thread is a ridge wrapped around a cylinder or cone in the form of a helix, with the former being called a straight thread and the latter called a tapered thread. A screw thread is the essential feature of the screw as a simple machine and also as a threaded fastener.

A dowel is a cylindrical rod, usually made from wood, plastic, or metal. In its original manufactured form, a dowel is called a dowel rod. Dowel rods are often cut into short lengths called dowel pins. Dowels are commonly used as structural reinforcements in cabinet making and in numerous other applications, including:

Anchor bolts are used to connect structural and non-structural elements to concrete. The connection can be made by a variety of different components: anchor bolts, steel plates, or stiffeners. Anchor bolts transfer different types of load: tension forces and shear forces.

A punch is a hard metal rod with a narrow tip at one end and a broad flat "butt" at the other. When used, the narrower end is pointed against a target surface and the broad end is then struck with a hammer or mallet, causing the blunt force of the blow to be transmitted down the rod body and focused more sharply onto a small area. Typically, woodworkers use a ball-peen hammer to strike a punch.

Shearing forces are unaligned forces pushing one part of a body in one specific direction, and another part of the body in the opposite direction. When the forces are aligned into each other, they are called compression forces. An example is a deck of cards being pushed one way on the top, and the other at the bottom, causing the cards to slide. Another example is when wind blows at the side of a peaked roof of a house - the side walls experience a force at their top pushing in the direction of the wind, and their bottom in the opposite direction, from the ground or foundation. <NAME> defines shear force in terms of planes: "If a plane is passed through a body, a force acting along this plane is called a shear force or shearing force."

Multi-jackbolt tensioners (MJT) are an alternative to traditional bolted joints. Rather than needing to tighten one large bolt, MJTs use several smaller jackbolts to significantly reduce the torque required to attain a certain preload. MJTs range in thread sizes from 34 in (19 mm) to 32 in (810 mm) and can achieve 20 million pounds-force or more. MJTs only require hand-held tools, such as torque wrenches or air/electric impacts, for loading and unloading bolted joints.

A swage nut or self-clinching nut is a type of nut or threaded insert that is used on sheet metal.

A screw and a bolt are similar types of fastener typically made of metal, and characterized by a helical ridge, known as a male thread. Screws and bolts are used to fasten materials by the engagement of the screw thread with a similar female thread in the matching part.

Threading is the process of creating a screw thread. More screw threads are produced each year than any other machine element. There are many methods of generating threads, including subtractive methods ; deformative or transformative methods ; additive methods ; or combinations thereof.

ISO 898 is an international standard that defines mechanical and physical properties for metric fasteners. This standard is the origin for other standards that define properties for similar metric fasteners, such as SAE J1199 and ASTM F568M. It is divided into five (nonconsecutive) parts:

ASTM A325 is an ASTM International standard for heavy hex structural bolts, titled Standard Specification for Structural Bolts, Steel, Heat Treated, 120/105 ksi Minimum Tensile Strength. It defines mechanical properties for bolts that range from 12 to 1 12 inches in diameter.

ASTM F568M is an ASTM International standard for metric bolts, screws and studs that are used in general engineering applications. It is titled: Standard Specification for Carbon and Alloy Steel Externally Threaded Metric Fasteners. It defines mechanical properties for fasteners that range from M1.6 to 100 in diameter. The standard was withdrawn in 2012. and has been replaced by ISO 898-1

ASTM A354 is an ASTM International standard that defines chemical and mechanical properties for alloy steel bolts, screws, studs, and other externally threaded fasteners. It is officially titled: Standard Specification for Quenched and Tempered Alloy Steel Bolts, Studs, and Other Externally Threaded Fasteners.

ASTM A490 and ASTM A490M are ASTM International standards for heavy hex structural bolts made from alloy steel. The imperial standard is officially titled Standard Specification for Structural Bolts, Alloy Steel, Heat Treated, 150 ksi Minimum Tensile Strength, while the metric standard (M) is titled Standard Specification for High-Strength Steel Bolts, Classes 10.9 and 10.9.3, for Structural Steel Joints.

A bolt is a form of threaded fastener with an external male thread. Bolts are very closely related to screws.

A mechanical joint is a section of a machine which is used to connect one or more mechanical part to another. Mechanical joints may be temporary or permanent, most types are designed to be disassembled. Most mechanical joints are designed to allow relative movement of these mechanical parts of the machine in one degree of freedom, and restrict movement in one or more others.

## References

1. <NAME> (1957), Fasteners handbook, Reinhold Publishing, p. 90.
2. <NAME>. (1997), Mechanical fastening, joining, and assembly, CRC Press, p. 54, ISBN   978-0-8247-9835-2.
3. <NAME> (2008), High Performance Fasteners & Plumbing, Penguin Group, p. 49, ISBN   978-1-55788-523-4.
4. Machinery's Handbook (1996), pp. 1828–1830.
5. Machinery's Handbook (1996), pp. 1842.
6. <NAME> (2001), Structural Steel Design, 1, Kaplan AEC Engineering, p. 151, ISBN   978-1-58001-055-9.


text:
## A Modified Collatz sequence

### Problem 277

A modified Collatz sequence of integers is obtained from a starting value $a_1$ in the following way:

$a_{n+1} = \, \,\, \frac {a_n} 3 \quad$ if $a_n$ is divisible by $3$. We shall denote this as a large downward step, "D".

$a_{n+1} = \frac {4 a_n+2} 3 \, \,$ if $a_n$ divided by $3$ gives a remainder of $1$. We shall denote this as an upward step, "U".

$a_{n+1} = \frac {2 a_n -1} 3 \, \,$ if $a_n$ divided by $3$ gives a remainder of $2$. We shall denote this as a small downward step, "d".

The sequence terminates when some $a_n = 1$.

Given any integer, we can list out the sequence of steps.
For instance if $a_1=231$, then the sequence $\{a_n\}=\{231,77,51,17,11,7,10,14,9,3,1\}$ corresponds to the steps "DdDddUUdDD".

Of course, there are other sequences that begin with that same sequence "DdDddUUdDD....".
For instance, if $a_1=1004064$, then the sequence is DdDddUUdDDDdUDUUUdDdUUDDDUdDD.
In fact, $1004064$ is the smallest possible $a_1 > 10^6$ that begins with the sequence DdDddUUdDD.

What is the smallest $a_1 > 10^{15}$ that begins with the sequence "UDDDUdddDDUDDddDdDddDDUDDdUUDd"?


text:
# The typical range of Prandtl number for water is

1. 0.004-0.300
2. 1.7-13.7
3. 50.500
4. 2000-1000

Option 2 : 1.7-13.7

## Detailed Solution

Explanation:

Prandtl member is the ratio of momentum diffusivity to thermal diffusivity.

$$Pr = \frac{\nu }{\alpha } = \frac{\mu }{{\frac{{\rho k}}{{\rho {C_p}}}}} = \frac{{\mu {C_p}}}{k}$$

Typical ranges of Prandtl member is listed below

 Fluid Pr Liquid metals 0.004 – 0.030 Gases 0.7 – 1.0 Water 1.7 – 13.7

# The insulating ability of an insulator with the presence of moisture would

1. increase
2. decrease
3. remain unaffected
4. Non of the above

Option 2 : decrease

## Detailed Solution

Explanation:

Insulation

• It is defined as a process of preventing the flow of heat from the body by applying insulator materials to the surface which controls the rate of heat transfer.
• The insulating ability of an insulator depends on various factors:
• thickness of insulator
• material of insulator
• surrounding conditions
• temperature difference
• Generally, air packets are present in porous insulating materials.
• Since water which is a more conductive material is replacing air which is a less conductive material, so the overall insulating ability of the insulator will decrease. Most insulators are porous in nature.
• If it has been about Non-porous insulators then the insulating ability will remain unaffected.

# $$Nu = CR_e^mP_r^n$$ represents heat transfer under

1. Free convection
2. Forced convection
3. Combined convection
4. None of the above

Option 2 : Forced convection

## Detailed Solution

For forced convection

Nu = C Rem Prn

where C, m and n are constants to be determined from the experiment data.

Re = Reynolds Number

Pr = Prandtl Number

Nu = C (Gr Pr) m

Where Gr is Grashof Number and Pr is Prandtl Number

The Product of Gr and Pr is also called Rayleigh Number.

# As the temperature increases, the thermal conductivity of a gas

1. increases
2. decreases
3. remains constant
4. increases up to a certain temperature and then decreases

Option 1 : increases

## Detailed Solution

Explanation:

Gases transfer heat by the collision of molecules.

As the temperature increases, the kinetic energy of molecules of gases also increases and eventually collision between molecules also increases which increases the thermal conductivity of gases.

∴ As temperature increases the thermal conductivity of gases increases.

For liquid and solids, generally as the temperature increases, the thermal conductivity decreases.

# Boiling point of water at sea level is ______.

1. 210° F
2. 212° F
3. 208° F
4. 214° F

Option 2 : 212° F

## Detailed Solution

The correct answer is 212° F.

• At 1 atmosphere of pressure (sea level), water boils at 100° C (212° F).
• When a liquid is heated, it eventually reaches a temperature at which the vapor pressure is large enough that bubbles form inside the body of the liquid. This temperature is called the boiling point.
• Once the liquid starts to boil, the temperature remains constant until all of the liquid has been converted to a gas.

Important Points

• The boiling point of water depends on the atmospheric pressure, which changes according to elevation.
• Water boils at a lower temperature as you gain altitude (e.g., going higher on a mountain).
• Water boils at a higher temperature if you increase atmospheric pressure (coming back down to sea level or going below it).
• The boiling point of water also depends on the purity of the water.
• Water that contains impurities (such as salted water) boils at a higher temperature than pure water. This phenomenon is called boiling point elevation.
• It is one of the colligative properties of matter.

Key Points

• Liquids have a characteristic temperature at which they turn into solids, known as their freezing point.
• Water freezes at 32° F or 0° C or 273.15 Kelvin.
• Pure, crystalline solids have a characteristic melting point, the temperature at which the solid melts to become a liquid.
• In theory, the melting point of a solid should be the same as the freezing point of the liquid.

# Solar radiation of 1000 W/m2 is incident on a grey opaque surface with emissivity of 0.4 and emissive power of 400 W/m2. The radiosity of the surface will be:

1. 940 W/m2
2. 850 W/m2
3. 760 W/m2
4. 670 W/m2

Option 3 : 760 W/m2

## Detailed Solution

Concept:

Irradiation (G): Total radiation incident upon a surface per unit time per unit area.

Radiosity (J): Total radiation leaving a surface per unit time per unit area.

Radiosity comprises the original emittance from the surface plus the reflected portion of any radiation incident upon it.

J = E + ρG

J = εEb + ρG

Eb = Emissive power of a perfect black body

α + ρ + τ = 1

For opaque body: τ = 0 ⇒ α + ρ = 1 ⇒ ρ = 1 - α = 1 - ε

J = εEb + (1 - ε)G = E + (1 - ε)G

Calculation:

Given:

G = 1000 W/m2, Eb = 400 W/m2, ϵ = 0.4

J = 400 × 0.4 + (1 - 0.4)1000 = 760 W/m2

# If one of the two fluids flowing through a heat exchanger of NTU = 2 remains at constant temperature throughout the exchanger length, the effectiveness of the heat exchanger will be

1. 1 – e-4
2. 1 – e-2
3. $$\dfrac{{1 - {e^{ - 2}}}}{2}$$
4. $$\dfrac{{1 - {e^2}}}{2}$$

Option 2 : 1 – e-2

## Detailed Solution

Explanation:

Heat Exchanger:

It is the steady flow adiabatic open system in which two flowing fluids exchange or transfer heat between them, without losing or gaining any heat from the ambient.

In the heat exchanger, the rate of enthalpy decrease of hot fluid = The rate of enthalpy increase of cold fluid.

Using the energy balance equation$$\dot{m}_h\times c_{ph}\times{(T_{hi}\;-\;T_{he})}=\dot{m}_c\times c_{pc}\times{(T_{ce}\;-\;T_{ci})}$$

h, c = Mass flow rate of hot and cold fluid.

cph, cpc = Specific heats of hot and cold fluid.

Thi, Tci = Hot and cold fluid inlet temperature, The,Tce = Hot and cold fluid exit temperature.

Capacity rate ratio (C):

$$C = \frac{{{{\left( {\dot m{c_p}} \right)}_{smaller}}}}{{{{\left( {\dot m{c_p}} \right)}_{larger}}}}$$,  (0 ≤ C ≤ 1)

C = 0, when one of the fluids undergoing phase change like in steam condenser or evaporator or boiler.

Number of transfer unit (NTU)

$$NTU = \frac{{UA}}{{{{\left( {\dot m{c_p}} \right)}_{smaller}}}}$$, U = overall heat transfer coefficient, A = heat transfer area

The formula of effectiveness for:

 Capacity rate ratio (C) Parallel Flow Heat Exchanger Counter Flow Heat Exchanger 0 < C < 1 $${\epsilon_{parallel}} = \frac{{1 - {e^{ - \left( {1 + c} \right)NTU}}}}{{1 + c}}$$ $${\epsilon_{counter}} = \frac{{1 - {e^{ - \left( {1 - c} \right)NTU}}}}{{1 - c{e^{ - \left( {1 - c} \right)NTU}}}}$$ C = 1 $${\epsilon_{parallel}} = \frac{{1 - {e^{ - 2NTU}}}}{2}$$ $$\epsilon_{counter} = \frac{{NTU}}{{\;1 + NTU}}$$ C = 0 $${\epsilon_{parallel}} = 1 - {e^{ - NTU}}$$ $${\epsilon_{counter}} = 1 - {e^{ - NTU}}$$

Calculation:

Given:

NTU = 2 and C = 0 (since phase change)

$${\epsilon_{parallel}} = {\epsilon_{counter}} =1 - {e^{ - NTU}}= 1 - {e^{ - 2}}$$

# Consider one-dimensional steady-state heat conduction, without heat generation, in a plane wall; with boundary conditions as shown in the figure below. The conductivity of the wall is given by k = k0 + bT; where k0 and b are positive constants and T is temperature.As x increases, the temperature gradient (dT/dx) will

1. Remain constant
2. Be zero
3. Increase
4. Decrease

Option 4 : Decrease

## Detailed Solution

Explanation

The conductivity of the wall, k = k0 + bT

And T2 > T1

Since conductivity is a function of temperature, As we move from left to right temperature increases. Thus thermal conductivity of the wall also increases.

Using Fourier's law of heat conduction.

$$Q = kA\left( {\frac{{dT}}{{dx}}} \right)$$

As thermal conductivity, k is increasing so to keep the heat transfer constant the temperature gradient $$\left( {\frac{{dT}}{{dx}}} \right)$$  must decrease.

# In a heat exchanger, it is observed that ΔT1 = ΔT2, where ΔT1 is the temperature difference between the two single phase fluid streams at one end and ΔT2 is the temperature difference at the other end. This heat exchanger is

1. a condenser
2. an evaporator
3. a counter flow heat exchanger
4. a parallel flow heat exchanger

Option 3 : a counter flow heat exchanger

## Detailed Solution

Explanation:

In case of the counter-flow heat exchanger when the heat capacities of both the fluids are the same.

i.e. ṁhch = ṁccc

Q = ṁhch(Th1 – Th2) = ṁccc(Tc2 – Tc1)

⇒ (Th1 – Th2) = (Tc2 – Tc1)

⇒ (Th1 – Tc2) = (Th2 – Tc1)

⇒ ΔT1 = ΔT2

For parallel flow heat exchanger,  ΔT1 will always be greater than ΔT2.

# In which process, the rate of transfer of heat is maximum:

1. conduction
2. convection
4. In all, heat is transferred with the same speed

## Detailed Solution

Concept:

 Three modes of Transmission of heat/heat flow Sl. No Conduction Convection Radiation 1 Heat dissipates from one place to another by molecular vibration Heat is transfer from one place to another by the transfer of molecules It transfers heat in the form of electromagnetic wave 2 Conduction is relevant to solid only. Convection happens in liquid or gases It can heat any form of material. 3 Need medium to transfer heat Need medium to transfer heat No need for medium 4 Good Conductor- The objects which transfer heat easily. Ex- metals, human body etc Bad Conductor- The objects which do not transfer heat easily. Ex- Wood, Air, etc. Thermal Insulator- No heat is transferred by any means. Ex- Abonite, asbestos etc. When molecules are heated they headed upward and upper molecules go downward and this cyclic process continues.   Boiling of fluid Heat travels in terms of energy packets or waves The heat absorbed by the body gain energy   Radiations of Sun

Explanation:

• As we know the radiation travels with the speed of light, thus the rate of heat transfer is maximum in radiation in form of electromagnetic radiations

# For a hydrodynamically and thermally fully developed laminar flow through a circular pipe of constant cross-section, the Nusselt number at constant wall heat flux (Nuq) and that at constant wall temperature (NuT) are related as

1. Nu> NuT
2. Nuq = NuT
3. Nuq < NuT
4. Nuq = (NuT)2

Option 1 : Nu> NuT

## Detailed Solution

Concept:

Always remember standard results mentioned below;

Part – I For constant surface heat flux (qs = constant);

Hydrodynamically and thermally fully developed laminar flow through a circular pipe of constant cross section

Nuq = 4.36      …1)

Part – II For constant wall temperature (Tw = constant)

For this case

NuT = 3.66     …2)

Calculation:

Comparing 1) and 2)

Nuq > NuT ⇒ Option A is correct.

Key Points

Go through both derivations (i.e. qs = constant and T = constant) and remember graphs of both cases.

# In a double-pipe heat exchanger, the cold fluid is water with an inlet temperature 20°C and mass flow rate 20 kg/s and the hot fluid water inlet temperature 80°C and mass flow rate 10 kg/s. Assume that for water Cp = 4.2 kJ/kg°C, independent of temperature. What is the maximum temperature to which the cold fluid can be heated in a parallel flow and in a counter flow heat exchanger?

1. 80°C in both parallel flow and counter flow
2. 50°C in both parallel flow and counter flow
3. 40°C in parallel flow and 50°C in counter flow
4. 40°C in parallel flow and 80°C in counter flow

Option 3 : 40°C in parallel flow and 50°C in counter flow

## Detailed Solution

For parallel flow Heat exchanger:

For maximum temperature the outlet temperature of cold fluid must be equal to the outlet temperature of hot water.

Heat lost by hot water = Heat gain by cold water

$${m_n}{C_{ph}}\left( {{f_{{n_1}}} - {f_{{n_2}}}} \right) = {M_c}{C_{pc}}\left( {{t_{{c_1}}} - {t_{{c_2}}}} \right)$$

$$\because {t_{{n_2}}} = {t_{{c_2}}}\;\& \;{C_{ph}} = {C_{pc}}$$

$$\therefore \;Let\;{t_{{n_2}}} = {t_{{c_2}}} = t$$

$$\therefore 10 \times {C_{ph}} \cdot \left( {80 - t} \right) = 20 \times {C_{pc}} \times \left( {t - 20} \right)$$

80 - t = 2t - 40

⇒ 3t = 120

t = 40°C

For counter flow Heat exchanger:

In Counter flow heat exchanger for maximum temperature the two outlet temperature of hot water in equal to the

Inlet temperature of cold water

$$\therefore {t_{{n_2}}} = {t_{{c_1}}} = 20^\circ C$$

∴ Heat lost by hot water = Heat gain by cold water

$${m_h}{C_{ph}} \times \left( {{t_{{n_1}}} - {t_{{n_2}}}} \right) = {m_c} \times {C_{pc}} \times \left( {{t_{{c_2}}} - {t_{{c_1}}}} \right)$$

$$10\;\left( {80 - 20} \right) = 20 \times \left( {{t_{{c_2}}} - 20^\circ } \right)$$

$$\frac{{60}}{2} = {t_{{c_2}}} - 20$$

$$\therefore {t_{{c_2}}} = 50^\circ C$$

# In a furnace, the inner and outer sides of the brick wall (k1 = 2.5 W/m.K) are maintained at 1100°C and 700°C, respectively as shown in figureThe brick wall is covered by an insulating material of thermal conductivity k2. The thickness of the insulation is 1/4th of the thickness of the brick wall. The outer surface of the insulation is at 200°C. The heat flux through the composite walls is 2500 W/m2. The value of k2 is _____W/m.K (round off to one decimal place).

## Detailed Solution

Concept:

Heat flux through the wall is given by

$$Q = \; - kA\frac{{dT}}{{dx}}$$

Where k is the thermal conductivity, A is the area perpendicular to the direction of heat flow, dT is the temperature difference and dx is the thickness of the wall.
Heat flux flowing through wall remains same.

∴ Heat flux through Brick wall = Heat flux through insulation

Calculation:

Given,

For brick wall, k1 = 2.5 W/m.k, dT1 = 700 – 1100 = -400 K

For Insulation, dT2 = 200 – 700 = -500 K and L2 = L1/4

Since the heat flux through the wall remains same,

$${k_1}\frac{{d{T_1}}}{{d{x_1}}} = {k_2}\frac{{d{T_2}}}{{d{x_2}}}\;$$

$$- 2.5 \times \frac{{\left( { - 400} \right)}}{{{L_1}}} = - \frac{{{k_2}\left( { - 500} \right)}}{{{L_1}/4}}$$

$$\Rightarrow {k_2} = \;\frac{{2.5 \times 400}}{{4 \times 500}} = 0.5\;W/m.K$$

# If $$\sqrt{\frac{hA}{kP}}~$$ is _____, then addition of fin to the surface increases the heat transfer. (Where: h: Convective heat transfer co-efficient, A: Area, P: Perimeter and k: Thermal conductivity of material)

1. Equal to one
2. Greater than one
3. Less than one
4. Greater than one but less than two

Option 3 : Less than one

## Detailed Solution

Explanation:

Fins are the projections protruding from a hot surface into ambient fluid and they are meant for increasing the heat transfer rate by increasing the area of heat transfer convection area.

The effectiveness of fin decides whether adding the fin to the hot surface will necessarily increase the heat transfer rate.

Effectiveness of a fin is defined as the ratio of heat transfer with fin and without fin.

$${{\epsilon }_{fin}}=\frac{{{q}_{with~fin}}}{{{q}_{without~fin}}}$$

If effectiveness is greater than one then only added fin will increase the heat transfer rate otherwise it will have no meaning of adding it to the surface.

The most common fin used is the adiabatic tip (fin is finite in length)

$${{q}_{with~fin}}=\sqrt{hpkA}~\left( {{T}_{o}}-{{T}_{\infty }} \right)\tanh ml$$

$${{q}_{without~fin}}=hA~\left( {{T}_{o}}-{{T}_{\infty }} \right)$$

Where, h: Convective heat transfer co-efficient, A: Area, P: Perimeter and k: Thermal conductivity of the material

$${{\epsilon }_{fin}}=\frac{\sqrt{hpkA}~\left( {{T}_{o}}-{{T}_{\infty }} \right)\tanh ml}{hA~\left( {{T}_{o}}-{{T}_{\infty }} \right)}$$

$${{\epsilon }_{fin}}=\frac{~\tanh ml}{\sqrt{\frac{hA}{kp}}}$$

Now if the value of $$\sqrt{\frac{hA}{kp}}$$ < 1 then the effectiveness will be greater than one, which ultimately will increase the heat transfer.

# The ratio of the thickness of the thermal boundary layer to the thickness of the hydrodynamic boundary layer is equal to (Prandtl number)n, where n is______.

1. - 1/3
2. - 2/3
3. 1
4. 1/3

Option 1 : - 1/3

## Detailed Solution

δ = the thickness of the hydrodynamic boundary layer; the region of flow where the velocity is less than 99% of the far-field velocity.

δT = the thickness of the thermal boundary layer; the region of flow where the local temperature nearly reaches the value (99%) of the bulk flow temperature

$$\begin{array}{l} \frac{\delta }{{{\delta _T}}} = {\left( {{\rm{Pr}}} \right)^{\frac{1}{3}}}\;\\ \frac{{{\delta _T}}}{\delta } = {\left( {{\rm{Pr}}} \right)^{ - \frac{1}{3}}} \Rightarrow n = - \frac{1}{3} \end{array}$$

# Heat is transferred by all three modes of transfer, viz, conduction, convection and radiation in

1. Electric heater
2. Steam condenser
3. Melting of ice
4. Boiler

Option 4 : Boiler

## Detailed Solution

Explanation:

• There are three methods of heat transfer between the two systems. They are conduction, convection, and radiation.
• Conduction is a method of heat transfer in solids and heat transfer takes place without the movement of particles.
• Convection is a method of heat transfer in fluids (gases and liquids) and heat transfer takes place due to the movement of particles.
• Radiation is a method of heat transfer where heat is transferred from one place to another without affecting the medium of heat transfer.

Now let's see what happens in a steam boiler:

• A steam boiler is designed to absorb the maximum amount of heat released from the process of combustion.
• Heat transfer within the steam boiler is accomplished by three methods: radiation, convection, and conduction. The heating surface in the furnace area receives heat primarily by radiation.
• The remaining heating surface in the steam boiler receives heat by convection from the hot flue gases. Heat received by the heating surface travels through the metal by conduction
• Heat is then transferred from the metal to the water by convection.

# A composite slab has two layers of different materials having internal conductivities, k1 and k2. If each layer has the same thickness, then what is the equivalent thermal conductivity of the slab?

1. $$\frac {k_1 k_2}{k_1 + k_2}$$
2. $$\frac {k_1 k_2}{2(k_1 + k_2)}$$
3. $$\frac {2k_1 }{k_1 + k_2}$$
4. $$\frac {2k_1 k_2}{k_1 + k_2}$$

Option 4 : $$\frac {2k_1 k_2}{k_1 + k_2}$$

## Detailed Solution

Concept:

Thermal resistance:

• Thermal resistance is defined as the ratio of the temperature difference between the two faces of a material to the rate of flow per unit area.

$${\bf{i}}.{\bf{e}}.\;{{\bf{R}}_{{\bf{thermal}}}} = \frac{{{\bf{\Delta T}}}}{{\bf{Q}}}$$

• Its unit is K/W
• The concept of thermal resistance is used to solve composite layer problems.

For a solid plate, thermal resistance is given by:

$${{\bf{R}}_{{\bf{thermal}}}} = \frac{{\bf{L}}}{{{\bf{kA}}}}$$

Since the slabs are in contact with each other side by side. Therefore, the thermal resistances will be in series.

The equivalent thermal resistance to the flow of heat is given by,

$${R_{eq}} = {R_1} + {R_2}$$

$$\frac{{{L_{eq}}}}{{{k_{eq}}A}} = \frac{{{L_1}}}{{{k_1}A}} + \frac{{{L_2}}}{{{k_2}A}}$$

Since $${L_1} = {L_2} = L$$

$$\frac{{2L}}{{{k_{eq}}}} = \frac{L}{{{k_1}}} + \frac{L}{{{k_2}}}$$

$$\frac{{{k_{eq}}}}{2} = \frac{{{k_1}{k_2}}}{{{k_1} + {k_2}}}\\$$

$${k_{eq}} = \frac{{2{k_1}{k_2}}}{{{k_1} + {k_2}}}$$

Important Points

Thermal resistance for various shapes:

 Heat conduction through the plane wall $$Q = \frac{{{T_1} - {T_2}}}{{\frac{L}{{kA}}}}$$ Heat conduction through a hollow cylinder $$Q = \frac{{{T_1} - {T_2}}}{{\frac{{\ln \left( {\frac{{{r_o}}}{{{r_i}}}} \right)}}{{2\pi kL}}}}$$ Heat conduction through the hollow sphere $$Q = \frac{{{T_1} - {T_2}}}{{\frac{{{r_o} - {r_i}}}{{4\pi k{r_o}{r_i}}}}}$$

# In the figure given below, curve A will be applicable when thermal conductivity of the material.

1. Increases with increase in temperature
2. Decreases with increase in temperature
3. Is very large
4. Is constant at all the temperatures

Option 1 : Increases with increase in temperature

## Detailed Solution

Explanation:

• The variation in thermal conductivity of a material with temperature in the temperature range of interest is given by:
• k(T) = k0 (1 + βT) where β is called the temperature coefficient of thermal conductivity.
• The variation of temperature in a plane wall during steady one - dimensional heat conduction for the cases of constant and variable thermal conductivity is

# For a heat exchanger, ΔTmax is the maximum temperature difference and ΔTmin is the minimum temperature difference between the two fluids. LMTD is the log mean temperature difference. Cmin and Cmax are the minimum and the maximum heat capacity rates. The maximum possible heat transfer (Qmax) between the two fluids is

1. Cmin LMTD
2. Cmin ΔTmax
3. Cmax ΔTmax
4. Cmax ΔTmin

Option 2 : Cmin ΔTmax

## Detailed Solution

Concept:

1. When heat capacity rate is minimum the heat will transfer quickly hence for maximum heat transfer we will take Cmin .

2. And more the temperature difference greater will be the value of heat transfer.

Cmin = Minimum heat capacity rate

Cmax = Maximum heat capacity rate

∴ For maximum heat transfer:

$$\Rightarrow {\dot q_{max}} = {C_{min}}{\rm{\Delta }}{T_{max}}$$

# Grashof number signifies the ratio of

1. inertia force to viscous force
2. buoyancy force to viscous force
3. buoyancy force to inertia force
4. inertia force to surface tension force

Option 2 : buoyancy force to viscous force

## Detailed Solution

The dimensionless parameter which represents the natural convection effects is called the Grashof number.

Grashof number, Gr, is the ratio between the buoyancy force and the viscous force:

$$Gr = \frac{{g\beta \left( {{T_s} - {T_\infty }} \right)L_c^3}}{{{\nu ^2}}}$$

Nusselt number is a function of the Grashof number and the Prandtl number alone. Nu = f (Gr, Pr)

Important Non-dimensional numbers:

• Biot number  → Ratio of internal thermal resistance to boundary layer thermal resistance
• Grashof number  → Ratio of buoyancy to viscous force
• Prandtl number  → Ratio of momentum to thermal diffusivities
• Reynolds number  → Ratio of inertia force to viscous force


text:
Lie-algebra valued forms

1. Sep 4, 2010

camel_jockey

In trying to understand connection one-forms, I have to learn what a Lie-algebra valued form is.

I already understand what a vector-valued form is. I also understand why

\nabla (e) = e \otimes \omega

where \omega is a one-form, \nabla is an affine connection /covariant differentiation and e is some basis vector. Here \omega (vector) = number

But in the case of a connection one-form matrix, I am trying to understand why, when supplied with a "vector", it produces an element of the Lie-algebra. So all of a sudden, it would appear, \omega (vector) = lie-algebra element = element of a vector space!

Can someone explain this to me?

Many thanks!


text:
• Browse all
Dependence of inclusive jet production on the anti-$k_\mathrm{T}$ distance parameter in pp collisions at $\sqrt{s} =$ 13 TeV

The collaboration
JHEP 12 (2020) 082, 2020.

Abstract (data abstract)
No description provided.

• Table 1

Data from Figure 4

10.17182/hepdata.95241.v1/t1

Ratio of differential cross section of AK1 jets with respect to AK4 jets a function of jet PT in the...

• Table 2

Not in paper

10.17182/hepdata.95241.v1/t2

Ratio of differential cross section of AK1 jets with respect to AK4 jets a function of jet PT in the...

• Table 3

Not in paper

10.17182/hepdata.95241.v1/t3

Ratio of differential cross section of AK1 jets with respect to AK4 jets a function of jet PT in the...

• Table 4

Not in paper

10.17182/hepdata.95241.v1/t4

Ratio of differential cross section of AK1 jets with respect to AK4 jets a function of jet PT in the...

• Table 5

Data from Figure 4

10.17182/hepdata.95241.v1/t5

Ratio of differential cross section of AK2 jets with respect to AK4 jets a function of jet PT in the...

• Table 6

Not in paper

10.17182/hepdata.95241.v1/t6

Ratio of differential cross section of AK2 jets with respect to AK4 jets a function of jet PT in the...

• Table 7

Not in paper

10.17182/hepdata.95241.v1/t7

Ratio of differential cross section of AK2 jets with respect to AK4 jets a function of jet PT in the...

• Table 8

Not in paper

10.17182/hepdata.95241.v1/t8

Ratio of differential cross section of AK2 jets with respect to AK4 jets a function of jet PT in the...

• Table 9

Data from Figure 4

10.17182/hepdata.95241.v1/t9

Ratio of differential cross section of AK3 jets with respect to AK4 jets a function of jet PT in the...

• Table 10

Not in paper

10.17182/hepdata.95241.v1/t10

Ratio of differential cross section of AK3 jets with respect to AK4 jets a function of jet PT in the...

• Table 11

Not in paper

10.17182/hepdata.95241.v1/t11

Ratio of differential cross section of AK3 jets with respect to AK4 jets a function of jet PT in the...

• Table 12

Not in paper

10.17182/hepdata.95241.v1/t12

Ratio of differential cross section of AK3 jets with respect to AK4 jets a function of jet PT in the...

• Table 13

Data from Figure 4

10.17182/hepdata.95241.v1/t13

Ratio of differential cross section of AK4 jets with respect to AK4 jets a function of jet PT in the...

• Table 14

Not in paper

10.17182/hepdata.95241.v1/t14

Ratio of differential cross section of AK4 jets with respect to AK4 jets a function of jet PT in the...

• Table 15

Not in paper

10.17182/hepdata.95241.v1/t15

Ratio of differential cross section of AK4 jets with respect to AK4 jets a function of jet PT in the...

• Table 16

Not in paper

10.17182/hepdata.95241.v1/t16

Ratio of differential cross section of AK4 jets with respect to AK4 jets a function of jet PT in the...

• Table 17

Data from Figure 4

10.17182/hepdata.95241.v1/t17

Ratio of differential cross section of AK5 jets with respect to AK4 jets a function of jet PT in the...

• Table 18

Not in paper

10.17182/hepdata.95241.v1/t18

Ratio of differential cross section of AK5 jets with respect to AK4 jets a function of jet PT in the...

• Table 19

Not in paper

10.17182/hepdata.95241.v1/t19

Ratio of differential cross section of AK5 jets with respect to AK4 jets a function of jet PT in the...

• Table 20

Not in paper

10.17182/hepdata.95241.v1/t20

Ratio of differential cross section of AK5 jets with respect to AK4 jets a function of jet PT in the...

• Table 21

Data from Figure 4

10.17182/hepdata.95241.v1/t21

Ratio of differential cross section of AK6 jets with respect to AK4 jets a function of jet PT in the...

• Table 22

Not in paper

10.17182/hepdata.95241.v1/t22

Ratio of differential cross section of AK6 jets with respect to AK4 jets a function of jet PT in the...

• Table 23

Not in paper

10.17182/hepdata.95241.v1/t23

Ratio of differential cross section of AK6 jets with respect to AK4 jets a function of jet PT in the...

• Table 24

Not in paper

10.17182/hepdata.95241.v1/t24

Ratio of differential cross section of AK6 jets with respect to AK4 jets a function of jet PT in the...

• Table 25

Data from Figure 4

10.17182/hepdata.95241.v1/t25

Ratio of differential cross section of AK7 jets with respect to AK4 jets a function of jet PT in the...

• Table 26

Not in paper

10.17182/hepdata.95241.v1/t26

Ratio of differential cross section of AK7 jets with respect to AK4 jets a function of jet PT in the...

• Table 27

Not in paper

10.17182/hepdata.95241.v1/t27

Ratio of differential cross section of AK7 jets with respect to AK4 jets a function of jet PT in the...

• Table 28

Not in paper

10.17182/hepdata.95241.v1/t28

Ratio of differential cross section of AK7 jets with respect to AK4 jets a function of jet PT in the...

• Table 29

Data from Figure 4

10.17182/hepdata.95241.v1/t29

Ratio of differential cross section of AK8 jets with respect to AK4 jets a function of jet PT in the...

• Table 30

Not in paper

10.17182/hepdata.95241.v1/t30

Ratio of differential cross section of AK8 jets with respect to AK4 jets a function of jet PT in the...

• Table 31

Not in paper

10.17182/hepdata.95241.v1/t31

Ratio of differential cross section of AK8 jets with respect to AK4 jets a function of jet PT in the...

• Table 32

Not in paper

10.17182/hepdata.95241.v1/t32

Ratio of differential cross section of AK8 jets with respect to AK4 jets a function of jet PT in the...

• Table 33

Data from Figure 4

10.17182/hepdata.95241.v1/t33

Ratio of differential cross section of AK9 jets with respect to AK4 jets a function of jet PT in the...

• Table 34

Not in paper

10.17182/hepdata.95241.v1/t34

Ratio of differential cross section of AK9 jets with respect to AK4 jets a function of jet PT in the...

• Table 35

Not in paper

10.17182/hepdata.95241.v1/t35

Ratio of differential cross section of AK9 jets with respect to AK4 jets a function of jet PT in the...

• Table 36

Not in paper

10.17182/hepdata.95241.v1/t36

Ratio of differential cross section of AK9 jets with respect to AK4 jets a function of jet PT in the...

• Table 37

Data from Figure 4

10.17182/hepdata.95241.v1/t37

Ratio of differential cross section of AK10 jets with respect to AK4 jets a function of jet PT in the...

• Table 38

Not in paper

10.17182/hepdata.95241.v1/t38

Ratio of differential cross section of AK10 jets with respect to AK4 jets a function of jet PT in the...

• Table 39

Not in paper

10.17182/hepdata.95241.v1/t39

Ratio of differential cross section of AK10 jets with respect to AK4 jets a function of jet PT in the...

• Table 40

Not in paper

10.17182/hepdata.95241.v1/t40

Ratio of differential cross section of AK10 jets with respect to AK4 jets a function of jet PT in the...

• Table 41

Data from Figure 4

10.17182/hepdata.95241.v1/t41

Ratio of differential cross section of AK11 jets with respect to AK4 jets a function of jet PT in the...

• Table 42

Not in paper

10.17182/hepdata.95241.v1/t42

Ratio of differential cross section of AK11 jets with respect to AK4 jets a function of jet PT in the...

• Table 43

Not in paper

10.17182/hepdata.95241.v1/t43

Ratio of differential cross section of AK11 jets with respect to AK4 jets a function of jet PT in the...

• Table 44

Not in paper

10.17182/hepdata.95241.v1/t44

Ratio of differential cross section of AK11 jets with respect to AK4 jets a function of jet PT in the...

• Table 45

Data from Figure 4

10.17182/hepdata.95241.v1/t45

Ratio of differential cross section of AK12 jets with respect to AK4 jets a function of jet PT in the...

• Table 46

Not in paper

10.17182/hepdata.95241.v1/t46

Ratio of differential cross section of AK12 jets with respect to AK4 jets a function of jet PT in the...

• Table 47

Not in paper

10.17182/hepdata.95241.v1/t47

Ratio of differential cross section of AK12 jets with respect to AK4 jets a function of jet PT in the...

• Table 48

Not in paper

10.17182/hepdata.95241.v1/t48

Ratio of differential cross section of AK12 jets with respect to AK4 jets a function of jet PT in the...

• Table 49

Not in paper

10.17182/hepdata.95241.v1/t49

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 50

Not in paper

10.17182/hepdata.95241.v1/t50

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 51

Not in paper

10.17182/hepdata.95241.v1/t51

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 52

Data from Figure 6

10.17182/hepdata.95241.v1/t52

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 53

Not in paper

10.17182/hepdata.95241.v1/t53

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 54

Not in paper

10.17182/hepdata.95241.v1/t54

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 55

Data from Figure 6

10.17182/hepdata.95241.v1/t55

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 56

Not in paper

10.17182/hepdata.95241.v1/t56

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 57

Data from Figure 6

10.17182/hepdata.95241.v1/t57

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 58

Not in paper

10.17182/hepdata.95241.v1/t58

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 59

Not in paper

10.17182/hepdata.95241.v1/t59

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 60

Not in paper

10.17182/hepdata.95241.v1/t60

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 61

Not in paper

10.17182/hepdata.95241.v1/t61

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 62

Not in paper

10.17182/hepdata.95241.v1/t62

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 63

Not in paper

10.17182/hepdata.95241.v1/t63

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 64

Not in paper

10.17182/hepdata.95241.v1/t64

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 65

Not in paper

10.17182/hepdata.95241.v1/t65

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 66

Not in paper

10.17182/hepdata.95241.v1/t66

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 67

Not in paper

10.17182/hepdata.95241.v1/t67

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 68

Not in paper

10.17182/hepdata.95241.v1/t68

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 69

Not in paper

10.17182/hepdata.95241.v1/t69

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 70

Not in paper

10.17182/hepdata.95241.v1/t70

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 71

Not in paper

10.17182/hepdata.95241.v1/t71

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 72

Data from Figure 6

10.17182/hepdata.95241.v1/t72

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 73

Not in paper

10.17182/hepdata.95241.v1/t73

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 74

Not in paper

10.17182/hepdata.95241.v1/t74

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 75

Data from Figure 6

10.17182/hepdata.95241.v1/t75

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 76

Not in paper

10.17182/hepdata.95241.v1/t76

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 77

Data from Figure 6

10.17182/hepdata.95241.v1/t77

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 78

Not in paper

10.17182/hepdata.95241.v1/t78

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 79

Not in paper

10.17182/hepdata.95241.v1/t79

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 80

Not in paper

10.17182/hepdata.95241.v1/t80

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 81

Not in paper

10.17182/hepdata.95241.v1/t81

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 82

Not in paper

10.17182/hepdata.95241.v1/t82

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 83

Not in paper

10.17182/hepdata.95241.v1/t83

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 84

Not in paper

10.17182/hepdata.95241.v1/t84

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 85

Not in paper

10.17182/hepdata.95241.v1/t85

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 86

Not in paper

10.17182/hepdata.95241.v1/t86

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 87

Not in paper

10.17182/hepdata.95241.v1/t87

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...

• Table 88

Not in paper

10.17182/hepdata.95241.v1/t88

Ratio of cross section of inclusive anti-KT jets with respect to AK4 jets as a function of jet size in...


text:
In mathematics, a number q is called a quadratic residue modulo p if there exists an integer x such that:

${x^2}\equiv{q}\ (mod\ p)$

Otherwise, q is called a quadratic non-residue.

In effect, a quadratic residue modulo p is a number that has a square root in modular arithmetic when the modulus is p. The law of quadratic reciprocity says something about quadratic residues and primes.


text:
# GROUP ACTIONS ON KAC ALGEBRAS

For a group action ${\alpha}$ on a Kac algebra $\mathbb{K}$ with the crossed product Kac algebra $\mathbb{K}{\rtimes}_{\alpha}G$, we will show that ${\pi}_{\alpha}(\mathbb{K})$ is a sub-Kac algebra of $\mathbb{K}{\rtimes}_{\alpha}G$. We will also investigate the intrinsic group $G(\mathbb{K})$ of $\mathbb{K}$ and get a group action ${\beta}$ on a symmetric Kac algebra $\mathbb{K}_s(G(\mathbb{K})$ with the crossed product sub-Kac algebra $\mathbb{K}_s(G(\mathbb{K}){\rtimes}_{\beta}G$ of $\mathbb{K}{\rtimes}_{\alpha}G$.