text:
# thermo.lennard_jones module¶

thermo.lennard_jones.Stockmayer(Tm=None, Tb=None, Tc=None, Zc=None, omega=None, CASRN='', AvailableMethods=False, Method=None)[source]

This function handles the retrieval or calculation a chemical’s Stockmayer parameter. Values are available from one source with lookup based on CASRNs, or can be estimated from 7 CSP methods. Will automatically select a data source to use if no Method is provided; returns None if the data is not available.

Prefered sources are ‘Magalhães, Lito, Da Silva, and Silva (2013)’ for common chemicals which had valies listed in that source, and the CSP method Tee, Gotoh, and Stewart CSP with Tc, omega (1966) for chemicals which don’t.

Parameters: Returns: Tm : float, optional Melting temperature of fluid [K] Tb : float, optional Boiling temperature of fluid [K] Tc : float, optional Critical temperature, [K] Zc : float, optional Critical compressibility, [-] omega : float, optional Acentric factor of compound, [-] CASRN : string, optional CASRN [-] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K] methods : list, only returned if AvailableMethods == True List of methods which can be used to obtain epsilon with the given inputs Method : string, optional A string for the method name to use, as defined by constants in Stockmayer_methods AvailableMethods : bool, optional If True, function will determine which methods can be used to obtain epsilon for the desired chemical, and will return methods instead of epsilon

Notes

These values are somewhat rough, as they attempt to pigeonhole a chemical into L-J behavior.

The tabulated data is from [R423], for 322 chemicals.

References

 [R422] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006
 [R423] (1, 2) Magalhães, Ana L., Patrícia F. Lito, Francisco A. Da Silva, and Carlos M. Silva. “Simple and Accurate Correlations for Diffusion Coefficients of Solutes in Liquids and Supercritical Fluids over Wide Ranges of Temperature and Density.” The Journal of Supercritical Fluids 76 (April 2013): 94-114. doi:10.1016/j.supflu.2013.02.002.

Examples

>>> Stockmayer(CASRN='64-17-5')
1291.41

thermo.lennard_jones.molecular_diameter(Tc=None, Pc=None, Vc=None, Zc=None, omega=None, Vm=None, Vb=None, CASRN='', AvailableMethods=False, Method=None)[source]

This function handles the retrieval or calculation a chemical’s L-J molecular diameter. Values are available from one source with lookup based on CASRNs, or can be estimated from 9 CSP methods. Will automatically select a data source to use if no Method is provided; returns None if the data is not available.

Prefered sources are ‘Magalhães, Lito, Da Silva, and Silva (2013)’ for common chemicals which had valies listed in that source, and the CSP method Tee, Gotoh, and Stewart CSP with Tc, Pc, omega (1966) for chemicals which don’t.

Parameters: Returns: Tc : float, optional Critical temperature, [K] Pc : float, optional Critical pressure, [Pa] Vc : float, optional Critical volume, [m^3/mol] Zc : float, optional Critical compressibility, [-] omega : float, optional Acentric factor of compound, [-] Vm : float, optional Molar volume of liquid at the melting point of the fluid [K] Vb : float, optional Molar volume of liquid at the boiling point of the fluid [K] CASRN : string, optional CASRN [-] sigma : float Lennard-Jones molecular diameter, [Angstrom] methods : list, only returned if AvailableMethods == True List of methods which can be used to obtain epsilon with the given inputs Method : string, optional A string for the method name to use, as defined by constants in molecular_diameter_methods AvailableMethods : bool, optional If True, function will determine which methods can be used to obtain sigma for the desired chemical, and will return methods instead of sigma

Notes

These values are somewhat rough, as they attempt to pigeonhole a chemical into L-J behavior.

The tabulated data is from [R425], for 322 chemicals.

References

 [R424] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006
 [R425] (1, 2) Magalhães, Ana L., Patrícia F. Lito, Francisco A. Da Silva, and Carlos M. Silva. “Simple and Accurate Correlations for Diffusion Coefficients of Solutes in Liquids and Supercritical Fluids over Wide Ranges of Temperature and Density.” The Journal of Supercritical Fluids 76 (April 2013): 94-114. doi:10.1016/j.supflu.2013.02.002.

Examples

>>> molecular_diameter(CASRN='64-17-5')
4.23738

thermo.lennard_jones.sigma_Flynn(Vc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical volume. CSP method by [R426] as reported in [R427].

$\sigma = 0.561(V_c^{1/3})^{5/4}$
Parameters: Vc : float Critical volume of fluid [m^3/mol] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Vc is originally in units of mL/mol.

References

 [R426] (1, 2) Flynn, L.W., M.S. thesis, Northwestern Univ., Evanston, Ill. (1960).
 [R427] (1, 2) Stiel, L. I., and George Thodos. “Lennard-Jones Force Constants Predicted from Critical Properties.” Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023

Examples

>>> sigma_Flynn(0.000268)
5.2506948422196285

thermo.lennard_jones.sigma_Bird_Stewart_Lightfoot_critical_2(Tc, Pc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical temperature and pressure. CSP method by [R428].

$\sigma = 2.44(T_c/P_c)^{1/3}$
Parameters: Tc : float Critical temperature of fluid [K] Pc : float Critical pressure of fluid [Pa] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Original units of critical pressure are atmospheres.

References

 [R428] (1, 2) Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> sigma_Bird_Stewart_Lightfoot_critical_2(560.1, 4550000)
5.658657684653222

thermo.lennard_jones.sigma_Bird_Stewart_Lightfoot_critical_1(Vc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical volume. CSP method by [R429].

$\sigma = 0.841 V_c^{1/3}$
Parameters: Vc : float Critical volume of fluid [m^3/mol] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Original units of Vc are mL/mol.

References

 [R429] (1, 2) Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> sigma_Bird_Stewart_Lightfoot_critical_1(0.000268)
5.422184116631474

thermo.lennard_jones.sigma_Bird_Stewart_Lightfoot_boiling(Vb)[source]

Calculates Lennard-Jones molecular diameter. Uses molar volume of liquid at boiling. CSP method by [R430].

$\sigma = 1.166V_{b,liq}^{1/3}$
Parameters: Vb : float Boiling molar volume of liquid [m^3/mol] sigma : float Lennard-Jones collision integral, [Angstrom]

Notes

Original units of Vb are mL/mol.

References

 [R430] (1, 2) Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> sigma_Bird_Stewart_Lightfoot_boiling(0.0001015)
5.439018856944655

thermo.lennard_jones.sigma_Bird_Stewart_Lightfoot_melting(Vm)[source]

Calculates Lennard-Jones molecular diameter. Uses molar volume of a liquid at its melting point. CSP method by [R431].

$\sigma = 1.222 V_{m,sol}^{1/3}$
Parameters: Vm : float Melting molar volume of a liquid at its melting point [m^3/mol] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Original units of Vm are mL/mol.

References

 [R431] (1, 2) Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> sigma_Bird_Stewart_Lightfoot_melting(8.8e-05)
5.435407341351406

thermo.lennard_jones.sigma_Stiel_Thodos(Vc, Zc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical volume and compressibility. CSP method by [R432].

$\sigma = 0.1866 V_c^{1/3} Z_c^{-6/5}$
Parameters: Vc : float Critical volume of fluid [m^3/mol] Zc : float Critical compressibility of fluid, [-] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Vc is originally in units of mL/mol.

References

 [R432] (1, 2) Stiel, L. I., and George Thodos. “Lennard-Jones Force Constants Predicted from Critical Properties.” Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023

Examples

Monofluorobenzene

>>> sigma_Stiel_Thodos(0.000271, 0.265)
5.94300853971033

thermo.lennard_jones.sigma_Tee_Gotoh_Steward_1(Tc, Pc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical temperature and pressure. CSP method by [R433].

$\sigma = 2.3647 \left(\frac{T_c}{P_c}\right)^{1/3}$
Parameters: Tc : float Critical temperature of fluid [K] Pc : float Critical pressure of fluid [Pa] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Original units of Pc are atm. Further regressions with other parameters were performed in [R433] but are not included here, except for sigma_Tee_Gotoh_Steward_2.

References

 [R433] (1, 2, 3) Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. “Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential.” Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011

Examples

>>> sigma_Tee_Gotoh_Steward_1(560.1, 4550000)
5.48402779790962

thermo.lennard_jones.sigma_Tee_Gotoh_Steward_2(Tc, Pc, omega)[source]

Calculates Lennard-Jones molecular diameter. Uses critical temperature, pressure, and acentric factor. CSP method by [R434].

$\sigma = (2.3551 - 0.0874\omega)\left(\frac{T_c}{P_c}\right)^{1/3}$
Parameters: Tc : float Critical temperature of fluid [K] Pc : float Critical pressure of fluid [Pa] omega : float Acentric factor for fluid, [-] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Original units of Pc are atm. Further regressions with other parameters were performed in [R434] but are not included here, except for sigma_Tee_Gotoh_Steward_1.

References

 [R434] (1, 2, 3) Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. “Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential.” Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011

Examples

>>> sigma_Tee_Gotoh_Steward_2(560.1, 4550000, 0.245)
5.412104867264477

thermo.lennard_jones.sigma_Silva_Liu_Macedo(Tc, Pc)[source]

Calculates Lennard-Jones molecular diameter. Uses critical temperature and pressure. CSP method by [R435].

$\sigma_{LJ}^3 = 0.17791 + 11.779 \left( \frac{T_c}{P_c}\right) - 0.049029\left( \frac{T_c}{P_c}\right)^2$
Parameters: Tc : float Critical temperature of fluid [K] Pc : float Critical pressure of fluid [Pa] sigma : float Lennard-Jones molecular diameter, [Angstrom]

Notes

Pc is originally in bar. An excellent paper. None is returned if the polynomial returns a negative number, as in the case of 1029.13 K and 3.83 bar.

References

 [R435] (1, 2) Silva, Carlos M., Hongqin Liu, and Eugenia A. Macedo. “Models for Self-Diffusion Coefficients of Dense Fluids, Including Hydrogen-Bonding Substances.” Chemical Engineering Science 53, no. 13 (July 1, 1998): 2423-29. doi:10.1016/S0009-2509(98)00037-2

Examples

>>> sigma_Silva_Liu_Macedo(560.1, 4550000)
5.164483998730177

thermo.lennard_jones.epsilon_Flynn(Tc)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses critical temperature. CSP method by [R436] as reported in [R437].

$\epsilon/k = 1.77 T_c^{5/6}$
Parameters: Tc : float Critical temperature of fluid [K] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K]

References

 [R436] (1, 2) Flynn, L.W., M.S. thesis, Northwestern Univ., Evanston, Ill. (1960).
 [R437] (1, 2) Stiel, L. I., and George Thodos. “Lennard-Jones Force Constants Predicted from Critical Properties.” Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023

Examples

>>> epsilon_Flynn(560.1)
345.2984087011443

thermo.lennard_jones.epsilon_Bird_Stewart_Lightfoot_critical(Tc)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses critical temperature. CSP method by [R438].

$\epsilon/k = 0.77T_c$
Parameters: Tc : float Critical temperature of fluid [K] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K]

References

 [R438] (1, 2) Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> epsilon_Bird_Stewart_Lightfoot_critical(560.1)
431.27700000000004

thermo.lennard_jones.epsilon_Bird_Stewart_Lightfoot_boiling(Tb)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses boiling temperature. CSP method by [R439].

$\epsilon/k = 1.15 T_b$
Parameters: Tb : float Boiling temperature [K] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K]

References

 [R439] (1, 2) Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> epsilon_Bird_Stewart_Lightfoot_boiling(357.85)
411.5275

thermo.lennard_jones.epsilon_Bird_Stewart_Lightfoot_melting(Tm)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses melting temperature. CSP method by [R440].

$\epsilon/k = 1.92T_m$
Parameters: Tm : float Melting temperature [K] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K]

References

 [R440] (1, 2) Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> epsilon_Bird_Stewart_Lightfoot_melting(231.15)
443.808

thermo.lennard_jones.epsilon_Stiel_Thodos(Tc, Zc)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses Critical temperature and critical compressibility. CSP method by [R441].

$\epsilon/k = 65.3 T_c Z_c^{3.6}$
Parameters: Tc : float Critical temperature of fluid [K] Zc : float Critical compressibility of fluid, [-] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K]

References

 [R441] (1, 2) Stiel, L. I., and George Thodos. “Lennard-Jones Force Constants Predicted from Critical Properties.” Journal of Chemical & Engineering Data 7, no. 2 (April 1, 1962): 234-36. doi:10.1021/je60013a023

Examples

Fluorobenzene

>>> epsilon_Stiel_Thodos(358.5, 0.265)
196.3755830305783

thermo.lennard_jones.epsilon_Tee_Gotoh_Steward_1(Tc)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses Critical temperature. CSP method by [R442].

$\epsilon/k = 0.7740T_c$
Parameters: Tc : float Critical temperature of fluid [K] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K]

Notes

Further regressions with other parameters were performed in [R442] but are not included here, except for epsilon_Tee_Gotoh_Steward_2.

References

 [R442] (1, 2, 3) Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. “Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential.” Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011

Examples

>>> epsilon_Tee_Gotoh_Steward_1(560.1)
433.5174

thermo.lennard_jones.epsilon_Tee_Gotoh_Steward_2(Tc, omega)[source]

Calculates Lennard-Jones depth of potential-energy minimum. Uses critical temperature and acentric factor. CSP method by [R443].

$\epsilon/k = (0.7915 + 0.1693 \omega)T_c$
Parameters: Tc : float Critical temperature of fluid [K] omega : float Acentric factor for fluid, [-] epsilon_k : float Lennard-Jones depth of potential-energy minimum over k, [K]

Notes

Further regressions with other parameters were performed in [R443] but are not included here, except for epsilon_Tee_Gotoh_Steward_1.

References

 [R443] (1, 2, 3) Tee, L. S., Sukehiro Gotoh, and W. E. Stewart. “Molecular Parameters for Normal Fluids. Lennard-Jones 12-6 Potential.” Industrial & Engineering Chemistry Fundamentals 5, no. 3 (August 1, 1966): 356-63. doi:10.1021/i160019a011

Examples

>>> epsilon_Tee_Gotoh_Steward_2(560.1, 0.245)
466.55125785

thermo.lennard_jones.collision_integral_Neufeld_Janzen_Aziz(Tstar, l=1, s=1)[source]

Calculates Lennard-Jones collision integral for any of 16 values of (l,j) for the wide range of 0.3 < Tstar < 100. Values are accurate to 0.1 % of actual values, but the calculation of actual values is computationally intensive and so these simplifications are used, developed in [R444].

$\Omega_D = \frac{A}{T^{*B}} + \frac{C}{\exp(DT^*)} + \frac{E}{\exp(FT^{*})} + \frac{G}{\exp(HT^*)} + RT^{*B}\sin(ST^{*W}-P)$
Parameters: Tstar : float Reduced temperature of the fluid [-] l : int term s : int term Omega : float Collision integral of A and B

Notes

Acceptable pairs of (l,s) are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), and (4, 4).

$T^* = \frac{k_b T}{\epsilon}$

Results are very similar to those of the more modern formulation, collision_integral_Kim_Monroe.

Calculations begin to yield overflow errors in some values of (l, 2) after Tstar = 75, beginning with (1, 7). Also susceptible are (1, 5) and (1, 6).

References

 [R444] (1, 2) Neufeld, Philip D., A. R. Janzen, and R. A. Aziz. “Empirical Equations to Calculate 16 of the Transport Collision Integrals Omega(l, S)* for the Lennard-Jones (12-6) Potential.” The Journal of Chemical Physics 57, no. 3 (August 1, 1972): 1100-1102. doi:10.1063/1.1678363

Examples

>>> collision_integral_Neufeld_Janzen_Aziz(100, 1, 1)
0.516717697672334

thermo.lennard_jones.collision_integral_Kim_Monroe(Tstar, l=1, s=1)[source]

Calculates Lennard-Jones collision integral for any of 16 values of (l,j) for the wide range of 0.3 < Tstar < 400. Values are accurate to 0.007 % of actual values, but the calculation of actual values is computationally intensive and so these simplifications are used, developed in [R446].

$\Omega^{(l,s)*} = A^{(l,s)} + \sum_{k=1}^6 \left[ \frac{B_k^{(l,s)}} {(T^*)^k} + C_k^{(l,s)} (\ln T^*)^k \right]$
Parameters: Tstar : float Reduced temperature of the fluid [-] l : int term s : int term Omega : float Collision integral of A and B

Notes

Acceptable pairs of (l,s) are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), and (4, 4).

$T^* = \frac{k_b T}{\epsilon}$

References

 [R446] (1, 2) Kim, Sun Ung, and Charles W. Monroe. “High-Accuracy Calculations of Sixteen Collision Integrals for Lennard-Jones (12-6) Gases and Their Interpolation to Parameterize Neon, Argon, and Krypton.” Journal of Computational Physics 273 (September 15, 2014): 358-73. doi:10.1016/j.jcp.2014.05.018.

Examples

>>> collision_integral_Kim_Monroe(400, 1, 1)
0.4141818082392228

thermo.lennard_jones.Tstar(T, epsilon_k=None, epsilon=None)[source]

This function calculates the parameter Tstar as needed in performing collision integral calculations.

$T^* = \frac{kT}{\epsilon}$
Parameters: epsilon_k : float, optional Lennard-Jones depth of potential-energy minimum over k, [K] epsilon : float, optional Lennard-Jones depth of potential-energy minimum [J] Tstar : float Dimentionless temperature for calculating collision integral, [-]

Notes

Tabulated values are normally listed as epsilon/k. k is the Boltzman constant, with units of J/K.

References

 [R448] Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, Revised 2nd Edition. New York: John Wiley & Sons, Inc., 2006

Examples

>>> Tstar(T=318.2, epsilon_k=308.43)
1.0316765554582887


text:
# Designing GANs: A Likelihood Ratio Approach

We are interested in the design of generative adversarial networks. The training of these mathematical structures requires the definition of proper min-max optimization problems. We propose a simple methodology for constructing such problems assuring, at the same time, that they provide the correct answer. We give characteristic examples developed by our method, some of which can be recognized from other applications and some introduced for the first time. We compare various possibilities by applying them to well known datasets using neural networks of different configurations and sizes.

## Authors

• 6 publications
• 13 publications
• ### Training generative networks using random discriminators

In recent years, Generative Adversarial Networks (GANs) have drawn a lot...
04/22/2019 ∙ by Babak Barazandeh, et al. ∙ 0

• ### Kernel-Based Training of Generative Networks

Generative adversarial networks (GANs) are designed with the help of min...
11/23/2018 ∙ by Kalliopi Basioti, et al. ∙ 4

• ### Optimizing Shallow Networks for Binary Classification

Data driven classification that relies on neural networks is based on op...
05/24/2019 ∙ by Kalliopi Basioti, et al. ∙ 0

• ### LEAD: Least-Action Dynamics for Min-Max Optimization

Adversarial formulations in machine learning have rekindled interest in ...
10/26/2020 ∙ by Reyhane Askari Hemmat, et al. ∙ 22

• ### Interior Point Methods with Adversarial Networks

We present a new methodology, called IPMAN, that combines interior point...
05/23/2018 ∙ by Rafid Mahmood, et al. ∙ 0

• ### Training Neural Networks for Likelihood/Density Ratio Estimation

Various problems in Engineering and Statistics require the computation o...
11/01/2019 ∙ by George V. Moustakides, et al. ∙ 0

##### This week in AI

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## 1 Introduction

The problem we are interested in can be summarized as follows: We are given two collections of training data and . In the first set the samples follow the originprobability density and in the second the target density . The target density is considered unknown while can either be known with the possibility to produce samples every time it is necessary or unknown in which case we have a second fixed training set . Our goal is to design a deterministic transformation so that the data produced by applying the transformation onto follow the target density .

Of course one may wonder whether the proposed problem enjoys any solution, namely, whether there indeed exists a transformation capable of transforming into with the former following the origin density and the latter the target density

. The problem of transforming random vectors has been analyzed by

(Box & Cox, 1964) where existence is shown under general conditions. Computing, however, the actual transformation is a completely different challenge with one of the possible solutions relying on adversarial approaches applied to neural networks.

The most well known usage of this result is, clearly, the possibility to generate synthetic data that follow the unknown target density . In this case is selected to be simple (e.g. i.i.d. standard Gaussian or i.i.d. uniform) so that generating realizations from is straightforward. As mentioned, the adversarial approach can be applied even if the origin density is unknown provided that we have a dataset with data following the origin density. When, however, is known and we can generate any number of realizations , it is expected, the adversarial approach, to identify the transformation with higher accuracy due to the possibility of generating more training data.

It was (Goodfellow et al., 2016) that first introduced the idea of adversarial (min-max) optimization and demonstrated that it results in the determination of the desired transformation . Alternative adversarial approaches by (Arjovsky et al., 2017; Binkowski et al., 2018) were subsequently suggested and shown to also deliver the correct transformation . Finally, we must mention the work by (Nowozin et al., 2016) which is closely related to our results and for which, at the end of Section 2, we give details, emphasizing differences and similarities with our method. As in (Nowozin et al., 2016), we will show that our methods provides an abundance of adversarial problems that are capable of identifying the appropriate transformation . Furthermore, we will also provide a simple recipe as to how we can successfully construct such problems.

Arguing along the same lines of the existing min-max formulations: We would like to optimally specify a vector transformation , the generator, and a scalar function , the discriminator. To achieve this, for each combination we define the cost function

 J(G,D)=Ef[ϕ(D(X))]+Eh[ψ(D(G(Z)))] (1)

where are two scalar functions of the scalar and denote expectation with respect to the density respectively. The optimum combination generator/discriminator is then identified by solving the following min-max problem

 (2)

We must point out that our goal is not to solve (2), but rather find a class of functions so that the transformation that will come out of the solution of (2) is such that follows the target density when follows the origin density .

If is random following then is also random and we denote with its corresponding probability density. Clearly, there exists a correspondence between transformations and densities when the density of is fixed. Since we can write

this allows us to argue that the min-max problem in (2) is equivalent to

 (3)

It is now possible to combine the two expectations by applying a change of measure and a change of variables and equivalently write (3) as follows

 (4)

where denotes the corresponding likelihood ratio. Since is also fixed, there is again a correspondence between and , hence the previous min-max problem becomes equivalent to

 minr(X)∈RfmaxD(X)Ef[ϕ(D(X))+r(X)ψ(D(X))]. (5)

Here denotes the class of all likelihood ratios with respect to the density , namely, all the functions that satisfy

 Rf={r(X): r(X)≥0, ∫r(X)f(X)dX=1}. (6)

Using these definitions, let us define the cost

 J(r,D)=Ef[ϕ(D(X))+r(X)ψ(D(X))] (7)

and, according to (5), we are interested in the following min-max problem

 minr(X)∈RfmaxD(X)J(r,D). (8)

As mentioned, our actual goal is not to solve the adversarial problem. Instead, we would like to properly identify pairs of functions so that (8) accepts as solution the function . Indeed, if is the solution to (8), this means that is the solution to (3) and, finally, that the optimum obtained from (1) is such that follows which, of course, is our original objective. Even though the min-max problem in (1) is what we attempt to solve, it is through (8) that we understand what its solution entails. In the next section we focus on (7), (8) and propose a simple design method (recipe) for the two functions that assures that the solution of (8) is indeed .

## 2 A Class of Functions ϕ(z),ψ(z)

Suppose that is a strictly increasing and (left and right) differentiable scalar function of the nonnegative scalar , i.e. . Denote with the range of values of and let be the inverse function of which is defined for . Let be a positive scalar function also defined for then, using and , we propose the following pair

 ϕ′(z)=−ω−1(z)ρ(z),  ψ′(z)=ρ(z), (9)

where “” denotes derivative. Since and are arbitrary (provided they satisfy the strict increase and positivity constraint respecitively), the class of pairs defined by (9) is very rich allowing for a multitude of choices. We show next that any such pair gives rise to a min-max problem, as in (8), that accepts as its unique solution. We prove this claim in two steps. The first, involves a theorem where we consider a simplified version of the min-max problem.

###### Theorem 1

Let and be defined as above with the additional constraint . Fix and consider as a function of the scalar . Then, for any , we have that

 ϕ(D)+rψ(D)≤ϕ(ω(r))+rψ(ω(r)), (10)

with equality if and only if .

Consider next the minimization with respect to of the maximal value in (10). It is true that

 minr≥0{ϕ(ω(r))+rψ(ω(r))}=ϕ(ω(1)), (11)

with equality if and only if .

Proof  We note that the constraint does not affect the generality of our class of functions since from (9) we have that , after integration, is defined up to an arbitrary additive constant. We can always select this constant so that the constraint is satisfied. We would also like to emphasize that this constraint is needed only for the proof of this theorem and it is not necessary for the corresponding min-max problem defined in (8).

For fixed , to find the maximum of we consider the derivative with respect to which, using (9), takes the form

 ϕ′(D)+rψ′(D)=(r−ω−1(D))ρ(D).

The strict increase of is inherited by its inverse function which, combined with the positivity of , implies that the previous expression has the same sign as or . Consequently is the only critical point of which is a global maximum. Of course there are possibilities for extrema at the two end points of but they can only be (local) minima.

Let us now focus on the resulting function . Taking its derivative with respect to yields

 ddr{ϕ(ω(r))+rψ(ω(r))}=\allowdisplaybreaks{ϕ′(ω(r))+rψ′(ω(r))}ω′(r)+ψ(ω(r))=ψ(ω(r)),

where the last equality is due to the specific definition of the two functions in (9). Since , this implies that is strictly increasing, being also the integral of it is continuous in . If we combine this property with the strict increase and continuity (as a result of left and right differentiability) of we conclude that is also strictly increasing and continuous in . We recall that is selected to satisfy , consequently for the function has a unique minimum which is global and no other critical points. Of course it can still exhibit extrema at and/or but they can only be (local) maxima.

A consequence of Theorem 1 is the next corollary, which constitutes the second and final step in proving that the adversarial problem defined in (8) has as unique solution the function .

###### Corollary 1

If the functions satisfy (9) and is strictly increasing and left and right differentiable, then in the adversarial problem defined in (8) the maximizer is and the minimizer is , while the resulting min-max value is equal to

 minr(X)∈RfmaxD(X)Ef[ϕ(D(X))+r(X)ψ(D(X))]=ϕ(ω(1))+ψ(ω(1)). (12)

Proof  The proof is simple. First we observe that

 Ef[ϕ(D(X))+r(X)ψ(D(X))]=Ef[ϕ(D(X))+r(X)~ψ(D(X))]+ψ(ω(1)) (13)

with the last equality being true since and where . We start with the maximization problem. Since is a function of we have

 maxD(X)Ef[ϕ(D(X))+r(X)~ψ(D(X))]\allowdisplaybreaks=\allowdisplaybreaksEf[maxD(X){ϕ(D(X))+r(X)~ψ(D(X))}]. (14)

The maximization under the expectation can be performed for each fixed . However, when we fix then becomes a constant and the result of the maximization depends only on the actual value of . This suggests that we can limit ourselves to functions of the form . After this observation we can drop the dependence on and perform, equivalently, the maximization

 maxD{ϕ(D(r))+r~ψ(D(r))}

for each fixed . The pair satisfies the assumptions of Theorem 1, therefore maximization is achieved for . This implies that

We can now continue in a similar way for the minimization problem. Specifically

with the last inequality being true since the minimization that follows is unconstrained and the last equality being a consequence of Theorem 1. The final lower bound is clearly attained by , which is also a legitimate solution of the constrained minimization, since belongs to the class of likelihood ratios. Consequently is the solution to the min-max problem. Returning to the original min-max setup with replacing , we can clearly see that it satisfies (12). This completes the proof.

###### Remark 1

The adversarial problem is defined with the help of the two functions which, according to (9), can be obtained by integrating the corresponding derivatives. However, this integration might not always be possible, analytically. As we will have the chance to confirm in Section 4, in an actual optimization algorithm (e.g. of gradient type) that solves (2), the exact form of is not necessary. Instead, what is required is their derivatives which are analytically available from (9).

We must emphasize that there already exists the significant work by (Nowozin et al., 2016) that addresses a similar problem as our current work, namely the definition of a class of min-max optimizations that can be used to design the generator/discriminator pair. The class in (Nowozin et al., 2016) is defined in terms of a convex function which can be shown to correspond to the outcome of our maximization, namely the function . This establishes a one-to-one correspondence between the two methods under the ideal (non data-driven) setup. However, we believe that, our approach enjoys certain significant advantages:

First, the definition of the two functions in Equ. (9) is straightforward while in (Nowozin et al., 2016) requires the solution of an optimization problem.

Second, in our case we have complete control over the result of the maximization problem that defines the discriminator. In other words we can decide what transformation of the likelihood ratio

, the discriminator must estimate. In

(Nowozin et al., 2016) such flexibility does not exist.

Controlling the function we estimate with the discriminator plays a significant role in the implementation of our method. Indeed when we use a neural network to approximate the optimum discriminator, this affects the overall quality of the resulting generator/discriminator pair. We should also note that there are important applications in Statistics where one is interested in estimating only the transformation of the likelihood ratio, with the most common cases being the likelihood ratio itself, its logarithm (log-likelihood ratio), or the ratio

which plays the role of the posterior probability between two densities. In other words, there are applications where one is interested only in the “max” part of the min-max problem. In fact, in the next section we give examples of various choices of

and mention problems where the discriminator function becomes the actual target and not the generator.

## 3 Examples

Let us now present characteristic cases for the function and give pairs that satisfy (9). As we proved, this implies that the corresponding adversarial problem in (8) accepts the desired solution .

### 3.1 Case ω(r)=rα

For we have that and . According to (9), for we must define

 ϕ′(z)=−z1αρ(z),  ψ′(z)=ρ(z). (15)

The following examples can be shown to satisfy (15).

A1) If we select , with , this yields and . For , , , . For , , , .

A2) If we select , then, and .

A3) If we select , , this yields and . This example corresponds to functions that are not both available in closed form. However they can still be used to define an optimization problem whose solution is numerically computable.

For the particular selection (corresponding to ) we can show that the resulting cost is equivalent to the Bregman cost (Bregman, 1967). In fact there is a one-to-one correspondence between our function and the function that defines the Bregman cost. This correspondence however is lost once we switch to a different or a different function, suggesting that the proposed class of pairs , is far richer than the class induced by the Bregman cost.

We should mention that in A1) the selection is known as the mean square error criterion and if we apply only the maximization problem then this corresponds to a likelihood ratio estimation technique proposed in the literature (Sugiyama et al., 2010, 2013). Under the adversarial approach the cost takes the following interesting form

 J(r,D)=Ef[−0.5D2(X)+r(X)D(X)]\allowdisplaybreaks=12Ef[−(D(X)−r(X))2+(r(X)−1)2]+12

where the equality is a consequence of being a likelihood ratio with respect to . As we can see, the maximization problem indeed yields while the minimization that must follow, captures the desired solution .

### 3.2 Case ω(r)=α−1logr

For we have and . As before must be strictly positive and, according to (9), for all real we must define

 ϕ′(z)=−eαzρ(z), ψ′(z)=ρ(z). (16)

We have the following examples that satisfy these equations.

B1) If with , this produces If then , , . If then , and .

B2) If , then, and .

B3) If , this yields and . The two functions can be written in terms of the Exponential integral or with the help of a power series expansion, but they do not enjoy any closed form expressions. On the other hand, their derivatives are simple and can be clearly used in a gradient type algorithm to numerically compute the solution of the corresponding optimization.

We would like to point out that the previous examples are presented for the first time and can be used either under a min-max setting for the determination of the generator/discriminator pair or under a purely maximization setting for the direct estimation of the log-likelihood ratio function .

### 3.3 Case ω(r)=rr+1

When we have and . For we must define the functions according to (9)

 ϕ′(z)=−z1−zρ(z),  ψ′(z)=ρ(z). (17)

The next set of examples can be seen to satisfy (17).

C1) If we select , this yields and .

C2) Selecting , with , yields and . For , we have and , , while for we have and , .

In C1) we recognize the functions used in the original article by (Goodfellow et al., 2016). C2) appears for the first time.

### 3.4 Case ω(r)=sign(logr)

This is a special case of with the corresponding function not being strictly increasing. It turns out that we can still come up with optimization problems, two of which are known and used in practice, by considering as a limit of a sequence of strictly increasing functions.

Monotone Loss: As a first approximation we propose where a parameter. We note that . Using this approximation we can write

 (18)

As we mentioned, we have exact equality for . Let us perform our analysis by assuming that is finite. We note that and . Consequently, if for , we must define

 ϕ′(z)=−(1+z1−z)1cρ(z), ψ′(z)=ρ(z). (19)

D1) In (19) if we let in order to converge to the desired sign function, this yields and . This suggests that is decreasing and is increasing. In fact any strictly increasing function can be adopted provided we select .

There is a popular combination that falls under Case D1). In particular, the selection known as Wasserstein GAN is proposed in (Arjovsky et al., 2017). We recall that in this case .

Hinge Loss: As a second approximation we use the expression , which is strictly increasing, continuous and converges to as . This suggests that

 sign(logr)≈sign(logr)|logr|1c=ω(r), (20)

and . Since can assume any real value we conclude that which, clearly, differs from the previous approximation where we had . If then, according to (9) we must define

 ϕ′(z)=−ezcρ(z), ψ′(z)=ρ(z). (21)

We present the following case that leads to a very well known pair from a completely different application.

D2) Following (21), if we select then . If we now let , we obtain the limiting form for the derivatives which become and . By integrating we arrive at and . The cost based on this particular pair is called the hinge loss (Tang, 2013) and it is very popular in binary classification where one is interested only in the maximization problem. The corresponding method is known to exhibit an overall performance which in practice is considered among the best (Rosasco et al., 2004; Janocha & Czarnecki, 2017). Here, as in (Zhao et al., 2017), we propose the hinge loss as a means to perform adversarial optimization for the design of the generator .

This completes our presentation of examples. However, we must emphasize, that these are only a few illustrations of possible pairs one can construct. Indeed combining, as dictated by (9), any strictly increasing function with any positive function generates a legitimate pair and a corresponding min-max problem (8) that enjoys the desired solution .

## 4 Data-Driven Setup and Neural Networks

Let us now consider the data-driven version of the problem. As mentioned, the target density is unknown. Instead we are given a collection of realizations that follow and a second collection that follows the origin density . These data constitute our training set. Regarding the second set it can either become available “on the fly” when is known by generating realizations every time they are needed, or it can be considered fixed from the start exactly as , if is also unknown.

As we pointed out in Section 1, we are interested in designing a generator so that when we apply it onto the data , that is, the resulting will follow a density that matches the target density .

Since we are now considering the data-driven version of the problem, we are going to limit to be the outputs of corresponding neural networks. Therefore the generator is replaced by while the discriminator by where summarize the parameters of the two neural networks. Of course instead of neural networks one could use any other parametric family, as SVMs, capable of efficiently approximating any nonlinear function.

Once we have selected our favorite and functions we can compute from (9) the functions that enter into the min-max problem defined in (2). This problem, after limiting the generator and discriminator to neural networks, can be rewritten as follows

 minθmaxϑJ(θ,ϑ)=\allowdisplaybreaksminθmaxϑ{Ef[ϕ(D(X,ϑ))]+Eh[ψ(D(G(Z,θ),ϑ))]}. (22)

If are the corresponding optimum parameter values, and the structure of the two networks is sufficiently rich, we expect that will approximate the optimum functions of the ideal problem in (2) respectively. In particular for , the generator , whenever applied onto any that follows , it will result in a that follows a density which is expected to be close to the target density .

A simple stochastic gradient algorithm that can solve the min-max optimization in (22) is the following

 ϑt=ϑt−1+μ{ϕ′(D(Xt,ϑt−1))∇ϑD(Xt,ϑt−1)+ψ′(D(Yt,ϑt−1))∇ϑD(Yt,ϑt−1)}, (23)

corresponding to the maximization problem and

 (24)

for the minimization. Here denotes the Jacobian of with respect to , and is the learning rate of the two updates. With we denote a training sample from the collection while with either a sample from when is unknown or a new realization following if the latter is known. If the collection of training data is exhausted after applying the iterations several times, then we simply reuse them.

With (23), (24) we confirm Remark 1, namely that in an optimization algorithm we do not necessarily need the functions explicitly, but only their derivatives.

It has also been observed (Goodfellow et al., 2016; Arjovsky et al., 2017) that in order for the optimization algorithm to converge properly, for each iteration of the minimization problem we must perform several iterations of the maximization problem (common practice suggests at least five iterations of the maximization problem for each iteration of the minimization).

###### Remark 2

When replacing with neural networks we must take special care of the corresponding outputs. Basically, we must guarantee that they are of the correct form. This is particularly important in the case of the scalar output of the discriminator. We recall that the optimum discriminator is . This implies that we need to assure that takes values in (the range of ). Consequently, we must apply the proper nonlinearity in the output of the discriminator that will guarantee this fact.

## 5 Experiments

We implemented most of the examples mentioned in Section 3 using the datasets MNIST, CelebA and CIFAR-10. Before presenting our results, we would like to give details about the following pairs that exhibited the best overall performance in our experiments:

Exponential: From Example B1), , yields , , while . No nonlinearity is needed in the discriminator output.

B1b: From Example B1), for we obtain and with . No nonlinearity is needed in the discriminator output.

B2: Example B2), with and and . No nonlinearity is needed in the discriminator output.

Cross entropy: This is the classical method proposed in (Goodfellow et al., 2016) corresponding to Example C1) with , and

. To the discriminator output we apply the sigmoid function.

Wasserstein: We are in Example D1) with and . To limit the output of the discriminator we use the function .

Hinge: From Example D2), , and . No nonlinearity is needed in the discriminator output.

For each dataset we present the best five methods in terms of convergence rate and quality of synthetic results produced by the generator.

We recall that GANs are notorious for their nonrobust behavior (Bengio, 2012; Creswell et al., 2018; Mescheder et al., 2017). For the stabilization of the training process, we used the gradient-penalty methodology described in (Gulrajani et al., 2017) which was generalized to a class of Lipschitz GANs in (Zhou et al., 2019).

For the generator, we used a four-layer neural network where the first layer is linear and the remaining deconvolutional; with ReLU activation functions between the layers except the final layer where we used a sigmoid function since the output is an image with pixel values in the range

. The generator input is a standard i.i.d. normal vector with dimension 64 for MNIST and 128 for CelebA and CIFAR-10.

For the discriminator, we used a four-layer neural network with three convolutional layers followed by a linear layer. We applied Leaky ReLUs between the layers except for the final layer where we adopted proper functions based on the range . For the training of the two neural networks we applied the Adam algorithm (Kingma et al., 2015) with , , learning rate and batch size 50 for MNIST and 64 for CelebA and CIFAR-10. For all datasets the training lasted iterations.

The first set of experiments involves training with MNIST. Table 1 summarizes the

top five attained Frechet Inception Distances (FID) (Heusel et al., 2017) and Kernel Inception Distances (KID) (Binkowski et al., 2018) by the various methods.

In Figure 1 we present examples of generated synthetic numerals by the corresponding methods. We observe that for this particular dataset the designed GANs have comparable performance with the Wasserstein and Cross Entropy exhibiting the smallest and B1b and Exponential the highest scores.

The second set of experiments involves the CelebA dataset. Table 2 reports the best observed FID, KID

scores of the competing methods, while Figures 33 depict the evolution of the corresponding scores with the number of iterations during training. From these figures we observe that both scores of the Hinge method exhibit a high variability while B2, Cross entropy, Exponential and Wasserstein have comparable behavior. In Figure 4 we have examples of synthetic faces generated by each method.

The third and last set of experiments involves the far more challenging CIFAR-10 dataset. Table 3 summarizes the best FID, KID scores, while Figures 6, 6 capture their evolution during training; finally, Figure 7 hosts examples of corresponding synthetic images. Interestingly, from Figure 6, we distinguish for this dataset, two performance groups. We can see that B1b and Wasserstein have a visibly better performance than the second group which includes the Cross Entropy, Exponential and B2.

## Acknowledgement

This work was supported by the US National Science Foundation, Grant CIF 1513373, through Rutgers University.

## References

• Arjovsky et al. (2017) Arjovsky, M., Chintala, S., and Bottou, L. Wasserstein generative adversarial networks.

Proceedings of Machine Learning Research (PMLR 2017)

, pp. 214–223, 2017.
• Bengio (2012) Bengio, Y. Practical recommendations for gradient-based training of deep architectures. arXiv:1206.5533, 2012.
• Binkowski et al. (2018) Binkowski, M., Sutherland, D. J., Arbel, M., and Gretton, A. Demystifying MMD GANs. Proceedings Interational Conference on Learning Representations, 2018.
• Bregman (1967) Bregman, L. M. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics, 7:200–217, 1967.
• Box & Cox (1964) Box, G. E. P., and Cox, D. R. An analysis of transformations. Journal of the Royal Statistical Society. Series B, 26(2):211–252, 1964.
• Creswell et al. (2018) Creswell, A., White, T., Dumoulin, V., Arulkumaran, K., Sengupta, B., and Bharath, A. A. Generative adversarial networks: An overview IEEE Signal Processing Magazine, 35(1):53–65, January 2018.
• Goodfellow et al. (2016) Goodfellow, I. J., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., and Bengio, Y. Generative adversarial nets. arXiv: 1406.2661, 2014.
• Gulrajani et al. (2017) Gulrajani, I., Ahmed, F., Arjovsky, M., Dumoulin, V., and Courville, A. Improved training of Wasserstein GANs. arXiv: 1704.00028, 2017.
• Heusel et al. (2017) Heusel, M., Ramsauer, H., Unterthiner, T., and Nessler, B. GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium. Proceedings Advances Neural Information Processing Systems Conference, 2017.
• Janocha & Czarnecki (2017) Janocha, K., and Czarnecki, W. M. On loss functions for deep neural networks in classification. arXiv:1702.05659, 2017.
• Mescheder et al. (2017) Mescheder, L. M., Nowozin, S. and Geiger A. The numerics of GANs. Proceedings Advances Neural Information Processing Systems Conference, 2017.
• Moulin & Veeravalli (2019) Moulin, P., and Veeravalli, V. V. Statistical Inference for Engineers and Data Scientists. Cambridge, New York, 2019.
• Kingma et al. (2015) Kingma, D. P., and Ba J. L. Adam: A method for Stochastic Optimization. Proceedings International Conference on Learning Representations, 2015.
• Nowozin et al. (2016) Nowozin, S., Cseke, B., and Tomioka, R. f-GAN: Training generative neural samplers using variational divergence minimization. Proceedings of the 30th International Conference on Neural Information Processing Systems, pp. 271–279, 2016.
• Rosasco et al. (2004) Rosasco, L., De Vito, E., Caponnetto, A., Piana, M., and Verri, A.

Are loss functions all the same?

Journal Neural Computation, 16(5):1063–1076, 2004.
• Sugiyama et al. (2010) Sugiyama, M., Suzuki, T., and Kanamori, T. Density ratio estimation: A comprehensive review. RIMS Kokyuroku, 1703: 10–31, 2010.
• Sugiyama et al. (2013) Sugiyama, M., Suzuki, T., and Kanamori, T. Density Ratio Estimation in Machine Learning. Cambridge, New York, 2013.
• Tang (2013) Tang, Y. Deep learning using linear support vector machines. arXiv:1306.0239, 2013.
• Zhao et al. (2017) Zhao, J., Mathieu, M., and LeCun Y. Energy-Based Generative Adversarial Networks. Proceedings International Conference on Learning Representations, 2015.
• Zhou et al. (2019) Zhou, Z., Liang, J., Song, Y., Yu, L., Wang, H., Zhang, W., Yu, Y., and Zhang, Z. Lipschitz generative adversarial nets. Proceedings of the 36th International Conference on Machine Learning, pp. 7584–7593, 2019.

format: plain


text:
### SLA_DVN

Normalize Vector

ACTION:
Normalize a 3-vector, also giving the modulus (double precision).
CALL:
CALL sla_DVN (V, UV, VM)
##### GIVEN:
 V D(3) vector

##### RETURNED:
 UV D(3) unit vector in direction of V VM D modulus of V

NOTE:
If the modulus of V is zero, UV is set to zero as well.


text:
Main Article
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This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.
Relations among units

For the quantitative measurement of different aspects of radioactivity, there are a substantial number of units of radioactivity. The number is relatively high because the aspects include the energy of a radiation source, as well as the effect of ionizing radiation in air, absorbed in arbitrary materials, and specific effects on biological systems. Doubling the number of units is the reality that they are defined both for the International System of Units (SI) and in "traditional" or "customary" units.

Properly interpreting the units can be quite complex. A given quantum of ionizing radiation will have different effects not only due to the type of radiation, which is considered in the computation of the units of biological effect, sieverts and rems, but also due to the energy level within a radiation type. Fast, medium-speed and slow neutrons, for example, have different biological effects.

## Activity of a source of radiation

 Case Study On arriving at an accident site, the Incident Commander tells the radiation survey specialist that a package of medical isotopes is on a crashed truck. It is labeled to contain 0.2 Ci or 7.4 x 109 Bq of Cs-137. She wants to know the risks it presents, and get advice on how to handle it.'[1] This is a practical example how single units do not fully characterize a hazard. The basic quantitative measurements define the amount of potential radioactivity in the container: two-tenths of a curie, or 200 millicuries, of cesium (Cs). This is equivalent to 7,400,000,000 Bq, or 7.4 GBq. To assess hazard, you must consider the isotope involved. Cesium-137, widely used in medicine and industry, and also a likely contaminant from a nuclear reactor accident, is a beta and gamma emitter with a half-life of 30 years. This half-life means it will decay to 0.1 curie in 30 years. Risk assessment requires a knowledge of the ionizing radiation emitted, and also the physical form of the isotope. Assume the package is intact, although some of the outer shielding may have torn off; there is no dispersal hazard. Beta and gamma radiation have different biological effects. They also have very different penetrating power. Beta particles are principally an internal hazard and a slight hazard to the outer skin. As long as the package contents are not dispersed, the beta aspect is relatively safe. Beta particles cannot travel, in air, for long distances. Even though the biological effect of gamma rays is less, the radiation can travel long distances, penetrate light shielding, and, as an external source, is a hazard to living things. "Practical steps that can be taken to reduce your internal risk to Cs-137 would include wearing anti-contamination clothing complete with face mask or respirator (if the responder is trained and respirator fitted.) Your exposure to the gamma emitter in Cs-137 can be reduced by relying on the exposure control methods of: ${exposure} = {time} * {distance} * {shielding}$ "Time spent in the radiation field may be lessened by rotating the crew. Unless you have a designated function, stay out of the radiation field. Put as much shielding between you and the radiation source as possible. The denser the material the better the shielding. For example, a fire truck may provide better shielding than a concrete block wall."

As opposed to most of the other units, the SI unit becquerel[2] (Si) and the common unit curie (Ci) deal with the activity of the source, not the effects on radioactivity reaching its destination. Linked with area or volume measurements, these units are useful in giving a quantitative measurement of contamination of areas of the ground or volumes of water. When the source contains multiple isotopes, as with a reactor accident, it is most useful to state activity of each isotope.

### Basic units

1 Bq = 1 event of radiation emission per second. Since this is a very small unit, common measurements are:

• 1 kBq = 103 Bq
• 1 MBq = 106 Bq
• 1 GBq = 109 Bq

The older unit, the curie (Ci), is equal to 37 GBq. Curies are large units, so common representations are

• 1 mCi = 10-3Ci
• 1 uCi = 10-6Ci
• 1 nCi = 10-9Ci
• 1 pCi = 10-12Ci

Linking the two sets of units,

$1 Bq = 27 pCi$

### Measuring contamination

Contamination of surfaces, such as the ground near a radiation release, is stated in Bq or Ci units per square metre. Contamination of water is expressed in Bg or Ci per liter.

### Refining activity

Neither the Bq nor the Ci measure the energy of release from the source, only its rate. The energy is specified in electronvolts.

## Ionization of air

Roentgens measure the ionization of air from a radioactive source, but knowing that is less important than knowing other measurements. They measure exposure to radioactivity, but, since absorption and biological effects are more operationally important, R, and its SI approximation in Gy/Hr, are much less widely used in safety work.

## Biologic effects

Radiation damages cells by two classes of mechanism: direct and indirect. Direct damage results from chemical bonds directly absorbing ionizing radiation and being disrupted. Indirect damage comes from irradiation producing free radicals from water and other substances, and having these products chemically poison cells.[3]

The greatest direct damage comes from the most efficient linear energy transfer to complex molecules such as DNA.

Radiation type Energy level Weighting factor (WR))
Gamma rays and x rays All 1
Beta particles All 1
Neutrons < 10 KeV 5
> 10 keV to 100 keV 10
> 100 keV to 2 MeV 20
> 2 MeV to 20 MeV 10
> 20 MeV 5
Alpha particles All 20

### Linear energy transfer

Linear Energy Transfer (LET) provides one of the reasons that some types of radiation have more or less biological effect than others. Measured in keV/µm-kilo (electron volt per micrometer), it is the energy transferred per unit length of the track of radiation through tissue. [5]

$L= \frac{dE}{dl}$
where:
$dE$ energy transferred by a charged particle of specified energy
in traversing a distance of $dl$

## References

1. Radiological Emergency Response Independent Study, Emergency Management Institute, Federal Emergency Management Agency, January 1998, IS 301, pp. 2-1 to 2-2
2. While the abbreviations of units are capitalized, the spelled-out names are not, to avoid confusion with the person they honor.
3. FEMA, pp. 3-3 to 3-5
5. Mansoor Ahmed, Lecture 1: Interaction of Radiation with Biological Systems, Mammalian Radiation Biology Course, Sylvester Comprehensive Cancer Center, University of Miami Health System, p. 33
Summary of units
Property measured SI unit Other unit Notes
Rate of emission from a source
• becquerel (Bq)
• curie (Ci)
• Counts (or disintegrations) per minute


text:
# Qualitative description of unbounded ocean system

## Homework Statement

[/B]
a) Suppose that you have the unbounded ocean show in the picture, where $h = 2000$ m. At $t = 0$, the system is allowed to relax. Make a qualitative description of its evolution and final state if i) $L = 100$ m, and ii) $L = 10,000$ m.

b) At what speed should an observer move in order to observe the same wavelength of a wave packet traveling over a basin of constant depth?

## Homework Equations

None, because this is a qualitative question? I guess $c_g = \frac{\partial \omega}{\partial k}$ may be useful for part b.

## The Attempt at a Solution

For part a.i, because $L$ is small, is seems as though what will happen is the higher sides of that rectangular space in the ocean will come crashing towards each other, then hit, then they'll sort of rebound back out , and eventually the ocean will settle back into a still surface. For part a.ii., because $L$ is bigger, the water from each higher side will take longer to reach each other, but I'm not sure how else that would change the situation, and I have no idea what $h$ has to do with this whole situation at all, really.

For part b), I am confused on the wording of this part. When it says in order to observe the same wavelength of a wave packet, what is the "same" referring to? The observer observes the same wavelength as...?

I'm sorry for how messy my answers here are. I guess I really do have a hard time with qualitative questions like this. I would really appreciate any help!

## Answers and Replies

Bystander
Science Advisor
Homework Helper
Gold Member
"Wavelength" is the length of the displaced surface.


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Limiting Reagent Protein Nutrition - ChemPRIME

# Limiting Reagent Protein Nutrition

## Vegetarian Diets

Although a vegetarian diet has many health and environmental advantages, some people worry that they will get insufficient protein in their diet if they exclude meat. But to produce 1 pound of protein in the form of beef or veal, we must feed 21 pounds of vegetable protein to the animal (not to mention gallons of water). It may be that the amount of protein lost to humans in this way is equivalent to 90% of the annual world protein deficit. In the contemporary US it is virtually impossible to suffer protein malnutrition no matter what our diet (we generally eat a huge excess of protein), but in some poorly developed nations certain vegetarian diets may not supply enough protein, even though they provide enough calories, and diseases like kwashiorkor are common. There is animal evidence for increase in disease when single amino acids are deficient [1].

Kwashiorkor victim[2]

## Limiting Reactants in Protein Biosynthesis

Nutritionists look to chemists to explain protein deficiency in terms of amino acid composition of foods and "Limiting Reactants". When a limiting reactant is used up, the other reactants, no matter how plentiful they are, have nothing to react with, so the reaction ceases and the left over reactants are wasted.

Human proteins are complex polymers ("polypeptides") made by joining about 8 essential amino acids that our bodies can't synthesize (and aother 12 that our bodies synthesize) in particular orders that define the activity of a protein (which may be an enzyme, a structural body component, or serve many other functions). Molecular structures of two essential amino acids, lysine and tryptophan, are shown here.

 The lysine molecule [3] The tryptophan molecule[4]

The amino acids are joined by a very simple "condensation" reaction where two molecules join by eliminating water, as shown in the figure below. The portion of the molecule labeled "R" stands for a variable part of the molecule that distinguishes different amino acids; the rest of the molecule is common to all amino acids.

This process can continue with different amino acids adding to either end of the protein chain. The H-N of an amino acid adds to the C-OH end of the protein, eliminating water (H-OH) and forming a C-N bond where the water was eliminated (or the C-OH end of an amino acid adds to the N-H end of the protein). This process continues until the protein has hundreds of different amino acids.

Formation of a peptide bond [5]

## Example 1

Show two ways that one lysine molecule and one tryptophan molecule can combine via a peptide bond.

Solution In the figure showing the formation of a peptide bond, we see that two amino acids are different only because they have different "R" groups. The "R" group of lysine is -CH2CH2CH2NH2, and that of tryptophan is -CH2C8H6N (with the fused 5 and 6 atom rings). It's always the terminal acid (-COOH) and alpha amino acid (-NH2 on the adjacent carbon atom) that react to form the peptide bond. If we replace "R" on the left amino acid with -CH2CH2CH2NH2 it becomes lysine. Replacing "R" on the right amino acid with -CH2C8H6N we get tryptophan, and the amine (-NH2) group of tryptophan bonds to the acid group (-COOH) of lysine. If we switch the "R" groups, then the amine group of the lysine on the right will bond with the acid group of the tryptophan on the left to make a different dipeptide.

We can understand why a vegetarian diet with plenty of calories could cause protein malnutrition by imagining simple vegetable proteins that contained only two amino acids, lysine (abbreviated LYS) and tryptophan (TRP), which are consumed by a person and must be used to make human protein. This example is highly simplified to demonstrate a principle; the actual synthesis of human protein from vegetable is much more complex.

## Example 2

A legume food provides 100.0 g of and 100.0 g of tryptophan, but the Human polypeptide requires equal amounts of each. Which is the limiting reagent?

Solution The balanced equation

LYS + TRP → (LYS1TRP1)n + n H2O

tells us that according to the atomic theory, 1 mol LYS is required for each mole of TRP to make a long chain polypeptide with a large number "n" of repeating LYS-TRP units: ~LYS-TRP-LYS-....TRP~ . That is, the stoichiometric ratio S(LYS/TRP) = 1 mol LYS / 1 mol TRP. Let us see how many moles of each we actually have in the legume food. By inspecting the molecular structure, we see that LYS is C6H14N2O2 and has a molar mass M = 146.19 g mol-1 and similarly TRP is C11H12N2O2 has M = 204.23 g mol-1.

\begin{align} & n_{\text{TRP}}=\text{100}\text{.0 g}\times \frac{\text{1 mol TRP}}{\text{204}\text{.23 g}}=\text{0}\text{.4896 mol TRP} \\ & n_{\text{LYS}}=\text{100}\text{.0 g}\times \frac{\text{1 mol LYS}}{\text{146}\text{.19 g}}=\text{0}\text{.684 mol LYS} \\ \end{align}

When the reaction ends, 0.4896 mol TRP will have reacted with 0.4896 mol LYS and there will be

(0.6840 – 0.4896) mol LYS = 0.195 mol LYS

left over. TRP is therefore the limiting reagent, and 0.1273 mol (about 19 g or 20% of the total) is entirely wasted.

But suppose at the same time that the legume was consumed, a serving of wheat was also consumed that provided 140 g of TRP and only 71.5 g of LYS (wheat is notoriously low in LYS, just as legumes are high in LYS). Now

\begin{align} & n_{\text{TRP}}=\text{140}\text{.0 g}\times \frac{\text{1 mol TRP}}{\text{204}\text{.23 g}}=\text{0}\text{.685 mol TRP} \\ & n_{\text{LYS}}=\text{72.9}\text{.0 g}\times \frac{\text{1 mol LYS}}{\text{146}\text{.19 g}}=\text{0}\text{.489 mol LYS} \\ \end{align}

Now there is 0.196 mol of TRP left over from the wheat protein, exactly enough to react with the excess LYS from the legumes, and no amino acid is wasted! Thus wheat and beans are complementary sources of protein, and if eaten together, provide optimal nutrition, as shown in the diagram below, where the deficiencies in each cause neither to be fully utilized, but when eaten together, about equal amounts of the four amino acids are provided, which leads to essentially perfect stoichiometric ratios for synthesis of human protein.

Complementary Amino Acids

In order to design a proper diet, complementary protein foods would have to be chosen from tables of amino acid contents of foods. Some anthropologists notice how often diets of complementary foods show up in different parts of the world (i.e. tacos and beans), and attribute the society's survival value to their choice of food.

From the legume example you can begin to see what needs to be done to determine which of two reagents, X or Y, is limiting. We must compare the stoichiometric ratio S(X/Y) with the actual ratio of amounts of X and Y which were initially mixed together. In Example 1 this ratio of initial amounts

$\frac{n_{\text{TRP}}\text{(initial)}}{n_{\text{LYS}\text{(initial)}}}=\frac{\text{0}\text{.4986 mol TRP}}{\text{0}\text{.6840 mol LYS}}=\frac{\text{0}\text{.716 mol TRP}}{\text{1 mol LYS}}$

was greater than the stoichiometric ratio

$\text{S}\left( \frac{\text{TRP}}{\text{LYS}} \right)=\frac{\text{1 mol TRP}}{\text{1 mol LYS}}$

This indicated that there was not enough TRP to react with all the LYS and TRP was the limiting reagent. The corresponding general rule, for any reagents X and Y, is

\begin{align} & \text{If}\frac{n_{\text{X}}\text{(initial)}}{n_{\text{Y}}\text{(initial)}}\text{is less than S}\left( \frac{\text{X}}{\text{Y}} \right)\text{, then X is limiting}\text{.} \\ & \\ & \text{If}\frac{n_{\text{X}}\text{(initial)}}{n_{\text{Y}}\text{(initial)}}\text{is greater than S}\left( \frac{\text{X}}{\text{Y}} \right)\text{, then Y is limiting}\text{.} \\ \end{align}

These calculations can be organized as a table, with entries below the respective reactants and products in the chemical equation. Calculations are shown for each possible case, assuming that one reactant is completely consumed and determining if enough of the other reactants is present to consume it. If not, that scenario is discarded.

LYS + TRP → (LYS1TRP1)n + n H2O
m (g) 100 100
M (g/mol) 146.19 204.23
n (mol) 0.684 0.490
if all LYS reacts -0.684 -0.684
if all TRP reacts -0.490 -0.490
Actual Reaction
Amounts
-0.490 -0.490
Actual Reaction
Masses
-71.6 -100

(Of course, when the amounts of X and Y are in exactly the stoichiometric ratio, both reagents will be completely consumed at the same time, and neither is in excess.). This general rule for determining the limiting reagent is applied in the next example.

## Example 3

To detect an amino acid (even in a fingerprint in forensic chemistry), the ninhydrin test is often used.

Ninhydrin

In the ninhydrin test, two ninhydrin molecules (C9H6O4, shown to the left) become linked by the N attached to the first carbon of the amino acid chain, producing the blue/purple ion shown at right.

Blue/Purple Product[6]

The balanced chemical equation is:

2 C9H6O4 + C11H12N2O2 → (C9H5O2)-N=(C9H4O2) + C10H9NO + CO2 + 3 H2O

If 2.00 mg of ninhydrin (Nin) is used to detect 2 mg of TRP, has enough ninhydrin been added to react with all the TRP? Which is the limiting reagent, and what mass of H2O will be formed?

Solution

The stoichiometric ratio connecting Nin and TRP is

$\text{S}\left( \frac{\text{Nin}}{\text{TRP}} \right)=\frac{\text{2 mol Nin}}{\text{1 mol TRP}}$

The initial amounts of Nin and TRP are calculated using appropriate molar masses(160.13 g mol−1 for Nin and 204.23 g mol−1 for TRP: \begin{align} & \text{ }n_{\text{Nin}}\text{(initial)}=\text{2}\text{.00}\times \text{10}^{\text{-3}}\text{g}\times \frac{\text{1 mol Nin}}{\text{178}\text{.14 g}}=\text{1}\text{.12}\times \text{10}^{\text{-5}}\text{mol Nin} \\ & \\ & n_{\text{TRP}}\text{(initial)}=\text{2}\text{.00}\times \text{10}^{\text{-3}}\text{g}\times \frac{\text{1 mol TRP}}{\text{204}\text{.23 g}}=\text{9}\text{.79}\times \text{10}^{\text{-6}}\text{mol TRP} \\ \end{align}

Their ratio is

$\frac{n_{\text{Nin}}\text{(initial)}}{n_{\text{TRP}}\text{(initial)}}=\frac{\text{1}\text{.123}\times \text{10}^{\text{-5}}\text{mol Nin}}{\text{9}\text{.79}\times \text{10}^{\text{-6}}\text{mol TRP}}=\frac{\text{1}\text{.15 mol Nin}}{\text{1 mol TRP}}$

Since this ratio is smaller than the 2:1 stoichiometric ratio, you have insufficient Nin to react with all the TRP, so Nin is the limiting reagent. To ensure detection, in would be better to add excess Ninhydrin.

b) The amount of water product formed in a reaction may be calculated via an appropriate stoichiometric ratio from the amount of a reactant which was consumed. Some of the excess reactant TRP will be left over, but all the initial amount of Nin will be consumed. Therefore we use nNin (initial) to calculate how much H2O can be obtained

$n_{\text{Nin}}\text{ }\xrightarrow{S\text{(water/Nin}\text{)}}\text{ }n_{\text{water}}\xrightarrow{M_{\text{water}}}\text{ }m_{\text{water}}$

$m_{\text{water}}=\text{1}\text{.12 }\times \text{ 10}^{\text{-5}}\text{ mol Nin }\text{ }\times \text{ }$ $\frac{\text{3 mol water}}{\text{2 mol Nin}}\text{ }\times \text{ }\frac{\text{18}\text{.016 g}}{\text{mol water}}=\text{3}\text{.02 }\times \text{ 10}^{\text{-4}}\text{ g water}$

This is 0.302 mg water.

These calculations can be organized as a table, with entries below the respective reactants and products in the chemical equation. Calculations are shown for each possible case, assuming that one reactant is completely consumed and determining if enough of the other reactants is present to consume it. If not, that scenario is discarded.

2 C9H6O4 + C11H12N2O2 (C9H5O2)-N=(C9H4O2)) + C10H9NO + CO2 + 3 H2O
m (g) 0.0020 0.0020
M (g/mol) 178.1 204.23 303.3 159.2 44.0 18.0
n (mol) 1.12 x 10-5 9.79 x 10-6 -- -- -- --
if all C9H6O4 reacts -1.12 x 10-5 -5.62 x 10-6
if all C11H12N2O2 reacts -1.96 x 10-5 -9.79 x 10-6
Actual Reaction
Amounts
-1.12 x 10-5 -5.62 x 10-6 5.62 x 10-6 5.62 x 10-6 5.62 x 10-6 1.68 x 10-5
Actual Reaction
Masses
-0.0020 -0.00115 1.71 x 10-3 8.93 x 10-4 2.47 x 10-4 3.02 x 10-4

As you can see from the example, in a case where there is a limiting reagent, the initial amount of the limiting reagent must be used to calculate the amount of product formed. Using the initial amount of a reagent present in excess would be incorrect, because such a reagent is not entirely consumed.

The concept of a limiting reagent was used by the nineteenth century German chemist Justus von Liebig (1807 to 1873) to derive an important biological and ecological law. Liebig’s law of the minimum states that the essential substance available in the smallest amount relative to some critical minimum will control growth and reproduction of any species of plant or animal life. When a group of organisms runs out of that essential limiting reagent, the chemical reactions needed for growth and reproduction must stop. Vitamins, protein, and other nutrients are essential for growth of the human body and of human populations. Similarly, the growth of algae in natural bodies of water such as Lake Erie can be inhibited by reducing the supply of nutrients such as phosphorus in the form of phosphates. It is for this reason that many states have regulated or banned the use of phosphates in detergents and are constructing treatment plants which can remove phosphates from municipal sewage before they enter lakes or streams.


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### Home > AC > Chapter 6 > Lesson 6.1.3 > Problem6-29

6-29.

As Sachiko solved the equation $(x+2)+3=9$, she showed her work in the table below. Copy the table and provide justification for each step.

Statement

1. $(x+2)+3=9$

2. $x+(2+3)=9$

3. $x+5=9$

4. $x+5−5=9−5$

5. $x=4$

Reason

Given

Decide what rule goes with each step being made.

Statement

1. $(x+2)+3=9$

2. $x+(2+3)=9$

3. $x+5=9$

4. $x+5−5=9−5$

5. $x=4$

Reason

Given

Associative Property


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## Persistent pushd and popd

 1  2022-11-29T12:57:50-0700

## Overview

I use tmux on Linux and ConEMU on Windows. Both tools are some kind of terminal multiplexer. I often create a new terminal window, but the window just sends me back to my home directory. There are some ways around this, but I find this to be so annoying that I just created my own set of tools around it.

## Method

To solve this problem, I write my “working directory” to a file. This makes a lot of sense for me. When I want to work in a directory, I often wish to do this across reboots. Then when any new shell is created, it changes directory to that working directory. I wrote other tools that allow me to push and pop directories to a text file, allowing me to treat this directory store not just as a slot, but as a stack.

The tools are as follows:

• ppushd <dir>: Push directory <dir> to the persistent directory store, and change directories to it.
• ppopd: Pop a directory off of the directory store, and change directories to it.
• pchanged: Change directories to the current working directory.
• plistd: List the contents of the persistent directory stack.

Finally, I put a pchanged call at the end of my init shell script (\$profile for PowerShell, ~/.zshrc for Mac and Linux).

Currently, I only have this implemented for windows, but plan on using the tools on Linux as well in the future.

## Conclusion

The last idea – putting pchanged at the end of the init script – feels dirty, but it’s changed my workflow for the better. I context switch a lot less now.

Hopefully this will prove useful to someone else out there as well :)


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# Preference for tin or iron in the reduction of nitrobenzene

My NCERT book(Class 12) said the following for reduction of nitrobenzene using $$\ce{Sn/HCl}$$ and $$\ce{Fe/HCl}$$ : Now, I have been following this mechanism for this particular reduction, and it seems to me that the metal is getting oxidized to +2 state wherever it is shown that 2 electrons are being transferred. If that is the case, then:

•Why is $$\ce{Fe + HCl}$$ being preferred?

•Because I looked it up on Wikipedia and it says that $$\ce{SnCl2}$$ also gets hydrolysed to liberate $$\ce{HCl}$$ as follows: $$\ce{SnCl2 (aq) + H2O (l) ⇌ Sn(OH)Cl (s) + HCl (aq)}$$

Or have I interpreted the mechanism incorrectly?

• Probably preferred for cost reasons, iron being cheaper than tin – Waylander Mar 8 '19 at 10:14
• @Waylander But the way the book has stated it,it seems that this regeneration of HCl takes place only with Fe. So is there a problem with the yield in case of Sn+HCl? – Yusuf Hasan Mar 8 '19 at 10:37
• I can see no problem with your. My guess is that the preference the authors express is for Fe/HCl over catalytic hydrogenation. – Waylander Mar 8 '19 at 10:39

The book's reasoning is correct. Fe/HCl gives $$\ce{FeCl2}$$ on reaction and it gets hydrolyzed by steam vapors to produces more hydrochloric acid and hydrogen to push the reaction forward thus making the reaction self-sustaining to produce more aniline from nitrobenzene.
$$\ce{Fe + 2HCl ->[160°C,CH3OH] FeCl2 + H2}$$
$$\ce{3FeCl2 + 4H2O(steam) ->[Hydrolysis] Fe3O4 + 6HCl + H2 }$$
whereas $$\ce{SnCl2}$$ gives $$\ce{Sn(OH)Cl}$$ and 1 mole of hydrogen chloride which is insufficient and it gets exhausted after sometime.


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# What is a single word that describes the idea of the second time derivative of energy?

I think about position, its time derivative speed, and its second time derivative, acceleration. I would like to identify a single word that can be used as a handle for the second time derivative of energy (i.e., the time derivative of power). If there is a widely used term, I'd prefer to use it. If not, I'd like to get your suggestions as to what a term might be. Any ideas?

-
"Power-up rate"? :) –  Lagerbaer Dec 15 '11 at 22:57
A heuristical argument why this is not really a physically relevant quantity: the nontrivial physical solutions to second-order differential equations are essentially oscillatory, but if you study nontrivial energy changes these are typically due to thermodynamic processes involving entropy increase and therefore not reversible. –  leftaroundabout Dec 16 '11 at 2:16
Within the full systems studied by proper physics, the single word you're looking for is "zero". The reason is called the energy conservation law. ;-) Well, a more accurate term has several words: "a silly awkward way to write zero as a time derivative of yet another zero". :-) –  Luboš Motl Dec 16 '11 at 17:03

Within power systems such as regional or national electricity grids, $\frac{\mathrm{d}^2E}{\mathrm{d}t^2}$ is called the slew rate: it's used to denote the rate of change of power demanded from, or supplied to, electricity grids. It's typically either expressed as MW/s or GW/h, being two time periods of interest in balancing electricity grids.

Inconveniently, you might find that (in some contexts) slew rate is also used to refer to rate of change of voltage, or of current, with respect to time (in units of V/s or A/s respectively).

-

The quantity itself, $\frac{\mathrm{d}^2E}{\mathrm{d}t^2}$, is not widely used (as a matter of fact I can't think of any equation in which it appears off the top of my head), so correspondingly there is no widely used term for it. You can just say "second time derivative of energy" or "rate of change of power" or some such thing and it will get the point across.

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Thanks David, this explains why searching Google turned up nothing. If you are curious, we are considering the second time derivative of energy as it applies to the rate of change of metabolic rate incurred by different activities. –  Sipp Dec 16 '11 at 14:21


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Using the Quadratic Equation, solve the following for X: 4X{eq}^2 {/eq} + 5X + 6 = 0

Question:

Using the Quadratic Equation, solve the following for X: 4X{eq}^2 {/eq} + 5X + 6 = 0

The quadratic equation has three options for its solution:

• Two different real roots
• Two equal real roots
• Two conjugate complex roots

This equation has a quadratic formula that defines the solutions using the coefficients of the equation:

\begin{align} x_{1,2} &= \frac { -b \pm \sqrt{b^2-4ac}}{2a} && \left[ \textrm {Second grade solutions formula} \quad ax^2+bx+c=0 \right] \\[0.3cm] \end{align}

Become a Study.com member to unlock this answer!

$$4x^2+5x+6=0$$

Solve the equation using quadratic formula:

\begin{align} 4x^2+5x+6 &=0 \\[0.3cm] x_{1,2} &= \frac...


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# Conservation Law from Metric

• I
Consider the following metric for a 2D spacetime:
##g_{tt} = -x ##
##g_{tx} = g_{xt} = 3##
##g_{xx} = 0##

i.e.
$$g_{\mu \nu} = \left( \begin{array}{cc} -x & 3\\ 3 & 0 \end{array} \right)$$

Now, since the metric is independent of time (t), there is supposedly a conservation law containing ##\frac{dx(\tau)}{d\tau}## and ##\frac{dt(\tau)}{d\tau}##. What is this conservation law, and why does the time independence of the metric imply it?

haushofer
To find the isometries of such a spacetime, you have to solve for the Killing equations. The tt-component of this equation should answer your question.

A note: GR in two dimensions is a bit problematic; the Einstein-Hilbert action is a topological term, meaning that the Einstein equations are an identity.

vanhees71
pervect
Staff Emeritus
Consider the following metric for a 2D spacetime:
##g_{tt} = -x ##
##g_{tx} = g_{xt} = 3##
##g_{xx} = 0##

i.e.
$$g_{\mu \nu} = \left( \begin{array}{cc} -x & 3\\ 3 & 0 \end{array} \right)$$

Now, since the metric is independent of time (t), there is supposedly a conservation law containing ##\frac{dx(\tau)}{d\tau}## and ##\frac{dt(\tau)}{d\tau}##. What is this conservation law, and why does the time independence of the metric imply it?

Consider a curve ##u^0(\tau) = t(\tau), u^1(\tau) = x(\tau)##. The tangent to this curve is a vector, i.e.

$$\left( \frac{dt}{d\tau}, \frac{dx}{d\tau} \right)$$

is a vector.

Because none of the components of the metric is a function of time, the vector (in the above sense) ##\xi^a = (1,0) = dt/d\tau## is a Killing vector.

The dot product of a tangent vector of a geodesic curve and a Killing vector is a conserved quantity - it is the same everywhere along the geodesic curve. If we let ##u^a = (u^0(\tau), u^1(\tau) ) = (t(\tau), x(\tau))## be a geodesic curve then we can say that the dot product is given by ##g_{ab} \xi^a u^b = \xi_b u^b## is constant. Because ##\xi^a = (1,0)## and ##\xi_a = (-x,3)## we can say that ##-x (dt/d\tau) + 3 (dx/d\tau)## is constant along the curve, i.e. is independent of ##\tau##.

As a check, we can consider ## (d/d\tau) (-x (dt/d\tau) + 3 (dx/d\tau)) = 0##, we know ##d/d\tau## of a constant is zero. Expanding this out using the chain rule, we get:
$$-\frac{dx}{d\tau} \frac{dt}{d\tau} + -x \frac{dt^2}{d\tau^2} + 3 \frac{dx^2}{d\tau^2} = 0$$

We can derive this from the geodesic equation. We compute the christoffel symbols

$$\Gamma^t{}_{tt} = \frac{1}{6} \quad \Gamma^x{}_{tt} = \frac{x}{18} \quad \Gamma^x{}_{tx} = -\frac{1}{6}$$

and confirm that we can derive the above from the geodesic equations, which in tensor notation are:

$$\frac{d^2 u^i}{d\tau^2} + \Gamma^i{}_{jk} \frac{du^j}{d\tau}\frac{du^k}{d\tau} = 0$$

Tensor notaton is very compact - the non-repeated index i generates two equations, one where we substitute i=0, the other where we substitute i=1. For each of these equations, j and k are repeated indices, so we sum over the repeated indicies, i.e we sum , for all four combinations of j,k, (0,0), (0,1), (1,0), (1,1).


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### Souvenirs from Polania

I recently returned from the 33rd International Workshop on Operator Theory and its Applications (IWOTA). IWOTA is a series of large conferences, that has become so central to the field that it attracts hundreds of participants and also some big names. Now IWOTA has its own wikipedia page and even a youtube channel (this year’s plenary talks are not yet uploaded). I like going to IWOTA because I get to hear talks in a very large range of topics, on the one hand, while also having the opportunity to go to very specialized small parallel sessions.

Every time IWOTA is held at a different location – this year it was held in Krakow, Poland. This was the first time I traveled abroad since January 2020 and it was also the first conference in a couple of years for many of the participants and that was maybe the main (unofficial) theme of the conference: so nice to get back together, so nice get back to normal (honestly, I feel somewhat corrupt to speak of flying to conferences in a different continent, sleeping in hotels and eating out every day for a week as “normal” … about this we shall need to talk another day). It was nice to meet and shake hands with people whose papers I have read and admired, and exchange a few words. There was also a scientific program – check it here (clicking the speaker name leads to a pdf file with title and abstract).

#### My talks

I was invited to speak in two special sessions: the first one on “Functional Calculus and Spectral Constraints” and in it I spoke about “A von Neumann type inequality for commuting row contractions”. I spoke about my joint work with Hartz and Richter, which has been recently published in Math Z. In the comments and questions part of the talk I learned about a very interesting conjecture or Matsaev’s, which has been apparently solved in the negative by Drury a little more than a decade ago. The conjecture is that for every $p \in (1,\infty)$ every polynomial $p$ and every contraction $T \in B(\ell^p)$, it holds that

$\|p(T)\| \leq \|p(S)\|$

where $S$ is the unilateral shift on $\ell^p$. It seemed to me that some of the participants in the session who knew about this paper didn’t completely believe the counter example because it requires numerical verification. Drury’s paper has a link to code that one can download and check. This problem and its solution is definitely something I’d like to learn more about, and maybe this will be next undergraduate project that I will offer. In any case, a friend told me that it is still an open question whether $\|p(T)\| \leq C\|p(S)\|$ with some constant $C$.

The second session I spoke in was “Operator Space Techniques in Operator Algebras”, and in it I spoke about “Dilations of unitary tuples and their surprising applications”, highlighting my recent work with Malte Gerhold. I also wanted to advertise my survey on dilation theory (because I recently recalled how good it is) but I ran out of time so I forgot to do the little advertisement in the end.

#### My favorite talks

I usually actively look for talks that would excite and inspire me, to note to myself directions that I should follow up on after the conference. Let me tell you about the three talks that did it for me this time.

The first one was also the first talk in the conference, Nikhil Srivastava’s talk on numerical and algorithmic aspects of diagonalizing a matrix. I find it quite comforting that supposedly completely solved problems like the diagonalization of a matrix are actually not completely settled once you look at them in practice. There are incredible software packages that one can use that actually usually work excellently; however, the speaker pointed out that that he is after algorithm algorithms that provably always work with provable bounds on running time and precision. Building on decades of remarkable work in numerical algebra, they obtain some striking improvements to what was known. If I understood correctly, the algorithms they use are have been around in one form or another for decades, and the main new ingredient that one can tell of without going into details, is the analysis of what happens to the matrix $A + G$during the algorithms, where $A$ is the matrix you want to diagonalize, and $G$ is a random perturbation. This is good because if you diagonalize $A + G$ and $G$ is small, then you have found a diagonal $D$ and an invertible $V$ such that $A - U^{-1}DV$ is small, which is one reasonable way to state the problem of numerically diagonalizing a matrix $A$.

The second talk that I want to mention was the talk by Ion Nechita, which highlighted connections between stuff I work on (free spectrahedra and matrix convexity) and stuff I don’t work on: quantum information. I was already aware of Nechita’s work but this talk really inspired me to look more into the interesting connections and applications of the mathematics I do to quantum information.

The last talk I want to highlight is the talk by Mirte van der Eyden, whose talk title was “Halos and the undecidability of tensor stable positive maps”, where she presented this paper with the same title. The talk was captivating both because of the very interesting content as well as for that fact that it was also the best talk.

The problem treated is that of existence of essential tensor stable positive maps. A map $P: M_n \to M_k$ is said to be tensor stable if the maps $P$, $P \otimes P$, $P \otimes P \otimes P$, etc. are all positive. If a map $P$ is completely positive, then it is tensor stable. Also, if $P$ is a completely positive map followed by a transpose, then it is also tensor stable. A map that is of the previous two kinds is said to be a trivial tensor stable map, a map that is tensor stable but not of the previous two kinds is said to be an essential tensor stable map. The question is: do there exist any essential tensor stable maps?

The authors treat this problem in two interesting and non-standard ways. The first attack is literally based on nonstandard analysis. They work over the hypercomplex numbers (warning: on wikipedia these are called surcomplex numbers) and show that in that setting there do exist essential tensor stable maps (they can actually construct such a map). The “halo” in the title refers to the image in the hypercomplex world, where the tensor stable maps are surrounded by a halo consisting of the hypercomplex tensor stable maps.

The second approach to the problem is the following: they tried to show that the problem of deciding whether a given map is tensor stable is undecidable. Why would this be helpful? Well, deciding whether a map is completely positive, or a composition of a completely positive map and a transposition, is decidable. Thus, if they prove that the problem of deciding whether a map is tensor stable is an undecidable problem then there has to be an essential one! That’s quite clever. Unfortunately they can only show that a related problem is undecidable, but this is still very interesting.

#### Auschwitz-Birkenau

Krakow is not far from the small town Staszow in Poland (which my grandmother would call Polania) which is where my father’s mother was born and raised. My grandmother immigrated to Israel before the war, but her family stayed behind and her parents and some of her siblings were murdered in the holocaust (two brothers fled the Nazis, and two of sisters survived the death camp, I think Treblinka). I have a friend who is a mathematician that works and lives in Krakow, and (knowing about my family’s origins) on the last day of the conference he very kindly suggested to drive me around the area, and since another friends of ours (a German) was already planning to go to Auschwitz, we decided to go there. And so we went to Auschwitz, a Pole, a German and a Jew. This was a very special experience that I will not forget.

What is there to say? By the end of the day I forgot that I was at a conference that ended that same morning. I don’t know what is the lesson to be learned and what I am supposed to do now. Just one thing is very clear: we should always remember and continue to remind.

### Bounded perturbations of the Heisenberg commutation relation via dilation theory (preprint)

Malte Gerhold and I recently uploaded our new preprint, “Bounded perturbations of the Heisenberg commutation relation via dilation theory“, to the arxiv. In the paper we modify the notion of dilation distance that was introduced in a previous paper of ours (joint with Satish Pandey and Baruch Solel) and we apply it to one-parameter unitary groups. Our main result is that a small dilation distance between groups implies that their generators are a bounded perturbation one of the other, and our main application is a new (and easy, if you ask me) proof of the theorem of Haagerup and Rordam that the infinite ampliation of the canonical position and momentum operators satisfying the Heisenberg commutation relation $[P,Q]=iI$ is a bounded perturbation of a pair of strongly commuting selfadjoint operators. This result is striking, since it can be shown that the finite ampliations of the position and momentum operators are not a bounded perturbation of strongly commuting operators.

It was a nice surprise to discover that our dilation techniques have applications also in the realm of unbounded operators. It was also a nice surprise to find myself writing a 9 page paper, I forgot that this is even possible!

I will be speaking about this result in the upcoming IWOTA and then in the IMU meeting.

### Dilation and classification in operator algebra theory (workshop)

I am really looking forward to the following workshop “Dilation and classification in operator algebra theory” to take place in the University of Copenhagen this October. I will be speaking about… dilation theory and applications, what else.

### Noncommutative analysis at the Technion 2022

The workshop NCAT 2022 that was held in honour of Paul Muhly at the Technion two weeks ago ended, to my great relief as organizer, with no disasters, thank god 🙂 In fact it was a delightful event, with some excellent talks, some new collaborations starting up, a lot of words of kindness and friendship, old friendships and partnerships rekindled, I am very happy to have taken a part in the organization, together with my friends Baruch Solel and Adam Dor-On. Needless to say, all of the actual physical organization was done by the current and former CMS coordinators, the excellent Karla Konik and Yael Stern.

Check out the website to see the program, abstracts, and the slides of some of the talks. As we say in Hebrew: it was good, and it is good that it was.

### Workshop in honour of Paul Muhly: Noncommutative Analysis at the Technion, June 2022

Registration is open until june 19th

https://cms-math.net.technion.ac.il/noncommutative-analysis-at-the-technion-2/