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BONAMI: Resonance Self-Shielding by the Bondarenko Method

U. Mertyurek and M. L. Williams

ABSTRACT

BONAMI is a module of the SCALE code system that is used to perform Bondarenko calculations for resonance self-shielding. BONAMI obtains problem-independent cross sections and Bondarenko shielding factors from a multigroup (MG) AMPX master library, and it creates a MG AMPX working library of self-shielded, problem-dependent cross sections. Several options may be used to compute the background cross section values using the narrow resonance or intermediate resonance approximations, with and without Bondarenko iterations. A novel interpolation scheme is used that avoids many of the problems exhibited by other interpolation methods for the Bondarenko factors. BONAMI is most commonly used in automated SCALE sequences and is fully integrated within the SCALE cross section processing module, XSProc.

Acknowledgments

The authors express gratitude to B. T. Rearden and M. A. Jessee for their supervision of the SCALE project and review of the manuscript. The authors acknowledge N. M. Greene, formerly of ORNL, for his original development of and contributions to the BONAMI module and methodology. Finally, the authors wish to thank Sheila Walker for the completion and publication of this document.

Introduction

BONAMI (BONdarenko AMPX Interpolator) is a SCALE module that performs resonance self-shielding calculations based on the Bondarenko method [IlichB64]. It reads Bondarenko shielding factors (“f-factors”) and infinitely dilute microscopic cross sections from a problem-independent nuclear data library processed by the AMPX  Batson Iii committed Jan 19, 2021 69 system [WWCD15], interpolates the tabulated shielding factors to appropriate  Batson Iii committed Nov 17, 2020 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 temperatures and background cross sections for each nuclide in the system, and produces a self-shielded, problem-dependent data set.

The code performs self-shielding for an arbitrary number of mixtures using either the narrow resonance (NR) or intermediate resonance (IR) approximation [GC62]. The latter capability was introduced in SCALE 6.2. BONAMI has several options for computing background cross sections, which may include Bondarenko iterations to approximately account for the impact of resonance interference for multiple resonance absorbers. Heterogeneous effects are treated using equivalence theory based on an “escape cross section” for arrays of slabs, cylinders, or spheres. During the execution of a typical SCALE computational sequence using XSProc, Dancoff factors for uniform lattices of square- or triangular-pitched units are calculated automatically for BONAMI by numerical integration over the chord length distribution. However, for non-uniform lattices—such as those containing water holes, control rods, and so on—the SCALE module MCDancoff can be run to compute Dancoff factors using Monte Carlo for an arbitrary 3D configuration, and these values are then provided in the sequence input.

The major advantages of the Bondarenko approach are its simplicity and speed compared with SCALE’s more rigorous CENTRM/PMC self-shielding method, which performs a pointwise (PW) deterministic transport calculation “on the fly” to compute multigroup (MG) self-shielded cross sections. With the availability of IR theory in BONAMI, accurate results can be obtained for a variety of system types without the computation expense of CENTRM/PMC.

Bondarenko Self-Shielding Theory

In MG resonance self-shielding calculations, one is interested in calculating effective cross sections of the form

 Batson Iii committed Jan 19, 2021 101 (301)$\sigma^{(r)}_{X,g} = \frac{\int_{g}\sigma^{(r)}_{X}(E)\Phi(E)\text{dE}}{\int_{g}\Phi(E)\text{dE}} ,$
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where $$\sigma^{(r)}_{X,g}$$ is the shielded MG cross section for reaction type X of resonance nuclide r in group g; $$\sigma^{(r)}_{X}(E)$$ is a PW cross section; and $$\Phi(E)$$ is the PW weighting function, which approximates the flux spectrum per unit of energy for the system of interest. PW cross section values are known from processing evaluated data in ENDF/B files; therefore, resonance self‑shielding depends mainly on determining the problem-dependent flux spectrum $$\Phi(E)$$, which may exhibit significant fine structure variations as a result of resonance reactions.

The essence of the Bondarenko method is to parameterize the flux spectrum corresponding to varying degrees of self-shielding, represented by the background cross section parameter $$\sigma_0$$ (called “sigma-zero”) and the Doppler broadening temperature T. Hence,

 Batson Iii committed Jan 19, 2021 115 (302)$\Phi \text{(E)}\to \Phi \text{(E;}\,\sigma _{\text{0,g}}^{\text{(r)}}\text{,T)}\ \ \,,\,\ \text{E}\in \text{g}\ ; \text{and} \  Batson Iii committed Nov 17, 2020 116 117 118 119 120 121 122 123 124 \sigma^{(r)}_{X,g} \rightarrow \sigma^{(r)}_{X,g}(\sigma^{(r)}_{0,g},\text{T})$

With this approach, it is possible to preprocess MG data for different background cross sections representing varying degrees of resonance self-shielding. This allows the MG averaging to be performed during the original MG library processing, so that BONAMI can do a simple interpolation on the background cross section and temperature to obtain self-shielded cross sections. This procedure is much faster than the CENTRM/PMC method in SCALE, which computes a PW flux spectrum by solving the neutron transport equation on a PW energy mesh in CENTRM and then  Batson Iii committed Jan 19, 2021 125 evaluates (301). in PMC “on the fly” during a sequence execution.

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BONAMI performs two main tasks: (a) computation of background cross sections for all nuclides in each mixture in the system and (b) interpolation of shielded cross sections from the library values tabulated vs. background cross sections and temperature. The BONAMI calculation is essentially isolated from the computation of the tabulated shielded cross sections, which is performed by the AMPX processing code system—the only connection is through the definition of the background cross section used in processing the library values. Various approximations can be used to parameterize the flux spectrum in terms of a background XS, as required by the Bondarenko method. We will first consider several approaches to representing the flux in an infinite medium, which lead to different definitions of the background cross section. BONAMI’s use of equivalence theory to extend the homogeneous methods to address heterogeneous systems, such as reactor lattices, is discussed in the following section.

Parameterized Flux Spectra

Several approximations can be applied to the infinite medium transport equation to parameterize the flux spectrum in terms of a background XS, as required by the Bondarenko method. The resulting homogeneous spectra are used in AMPX to process MG cross sections which can also can be applied to heterogeneous systems (i.e., lattices) by using equivalence theory; thus the key step is determining approximations that provide parameterized solutions for homogeneous media. The neutron transport equation for a homogeneous medium at temperature T, containing a resonance nuclide r mixed with other nuclides can be expressed as

 Batson Iii committed Jan 19, 2021 153 (303)$\left( \Sigma _{\text{t}}^{\text{(r)}}\text{(E,T)}\ +\sum\limits_{j\ne r}  Batson Iii committed Nov 17, 2020 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 {\Sigma _{\text{t}}^{\text{(j)}}\text{(E,T)}} \right)\ \Phi \text{(E,T)}\ \,\,\,=\ \ \,{{\text{S}}^{\text{(r)}}}(\text{E,T})\ \,+\,\sum\limits_{j\ne r}{{{\text{S}}^{\text{(j)}}}(\text{E,T})} ,$

where $$\Sigma _{\text{t}}^{\text{(r)}}\text{(E,T)}$$ , $$\text{S}_{{}}^{\text{(r)}}\text{(E,T)}$$ are the macroscopic total XS and elastic scattering source for r, respectively; and $$\Sigma _{\text{t}}^{\text{(j)}}\text{(E,T)}$$, $$\text{S}_{{}}^{\text{(j)}}\text{(E,T)}$$ are the macroscopic total cross section and elastic source, respectively, for a nuclide j. The cross sections in all these expressions are Doppler-broadened to the temperature of the medium. The nuclides in the summations (i.e., all nuclides except r) are called background nuclides for the resonance absorber r.

The NR approximation can be used to approximate scattering sources of nuclides for which the neutron energy loss is large compared with the practical widths of resonances for the absorber materials of interest. Applying the NR approximation for the scattering source of background material j gives

 Batson Iii committed Jan 19, 2021 169 (304)$\text{S}^{(j)}(\text{E,T}) \rightarrow \Sigma^{(j)}_{p}C(E) \text{for j = a NR-scatterer nuclide}$
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where C(E) is a slowly varying function representative of the asymptotic (i.e., no absorption) flux in a homogeneous medium, which approximates the flux between resonances. In the resolved resonance range of most important resonance absorbers, the asymptotic flux per unit energy is represented as,

 Batson Iii committed Jan 19, 2021 176 (305)$C(\text{E})\ =\ \ \,\frac{{{\Phi }_{\infty }}}{E}\ \ \ ,$
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where $${{\Phi }_{\infty }}$$ is an arbitrary normalization constant that cancels from the MG cross section expression. In the thermal range a Maxwellian spectrum is used for C(E), and in the fast range a fission spectrum is used. The SCALE Cross Section Libraries section of the SCALE documentation gives analytical expressions for C(E) used in AMPX to process MG data with the NR approximation. AMPX also has an option to input numerical values for C(E), obtained for example from a PW slowing-down calculation with CENTRM. This method has been used to process MG data for some nuclides on the SCALE libraries.

Conversely, the wide resonance (WR) approximation has been used to represent elastic scattering sources of nuclides for which the neutron energy loss is small compared with the practical width of the resonance. This approximation tends to be more accurate for heavy nuclides and for lower energies. The limit of infinite mass is usually assumed, so the WR approximation is sometimes called the infinite mass (IM) approximation. Because of the assumption of IM, there is no energy loss due to collisions with WR scatterers. Applying the WR approximation for the slowing-down source of background nuclide j gives

 Batson Iii committed Jan 19, 2021 196 (306)$\text{S}^{(j)}(\text{E,T}) \rightarrow \Sigma^{(j)}_{s}(\text{E,T})\Phi(\text{E,T}) ;  Batson Iii committed Nov 17, 2020 197 198 199 200 201 202 203 204 \text{for} j = \text{a WR-scatterer nuclide}$

The IR approximation was proposed in the 1960s for scatterers with slowing-down properties intermediate between those of NR and WR scatterers [GC62]. The IR method represents the scattering source for arbitrary nuclide j by a linear combination of NR and WR expressions. This is done by introducing an IR parameter usually called lambda, such that

 Batson Iii committed Jan 19, 2021 205 206 (307)$\text{S}_{{}}^{\text{(j)}}(\text{E,T)}\,\ \to \ \,\underbrace{\lambda _{\text{g}}^{\text{(j)}}\Sigma _{\text{p}}^{\text{(j)}}\,C(E)}_{\mathbf{NR scatterer}}\ +\ \ (1-\lambda _{\text{g}}^{\text{(j)}})\,\,\underbrace{\Sigma _{\text{s}}^{\text{(j)}}(\text{E,T})\Phi (\text{E,T})}_{\mathbf{WR scatterer}}\ \,\ \,\ ;\,\,\ \ \text{E}\in \text{g}\,\text{.}$

A value of λ=1 reduces (307) to the NR expression, whereas λ=0 reduces the  Batson Iii committed Nov 17, 2020 207 208 209 210 211 212 equation to the WR expression. Fractional λ’s are for IR scatterers. Since the type of scatterer can change with the energy, the IR lambdas are functions of the energy group as well as the nuclide. The λ values represent the moderation “effectiveness” of a given nuclide, compared to hydrogen. The AMPX module LAMBDA was used to compute the IR parameters on the SCALE libraries. (See AMPX documentation distributed with SCALE)  Batson Iii committed Jan 19, 2021 213 Substituting (307) into (303) and then dividing by the absorber number  Batson Iii committed Nov 17, 2020 214 215 216 density N(r) gives the following IR approximation for the infinite medium transport equation in energy group g

 Batson Iii committed Jan 19, 2021 217 (308)$\left( \sigma _{\text{t}}^{\text{(r)}}\text{(E,T)}\ \text{+}\ \sigma _{0}^{\text{(r)}}\text{(E,T) } \right)\,{{\Phi }^{\text{(r)}}}\text{(E,T)}\ \ =\,\ \frac{\text{1}}{{{\text{N}}^{\text{(r)}}}}{{\text{S}}^{\text{(r)}}}\text{(E,T)}\ +\ \frac{\text{1}}{{{\text{N}}^{\text{(r)}}}}\sum\limits_{j\ne r}{\lambda _{\text{g}}^{\text{(j)}}\,\Sigma _{\text{p}}^{\text{(j)}}C(E)\,}$
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where the background cross section of r in the homogeneous medium is defined as

 Batson Iii committed Jan 19, 2021 221 222 (309)$\sigma _{0}^{\text{(r)}}\text{(E,T)}\ \ =\ \ \frac{1}{{{\text{N}}^{\text{(r)}}}}\,\,\sum\limits_{j\ne r}{\left( \Sigma _{\text{a}}^{\text{(j)}}(\text{E,T})+\lambda _{\text{g}}^{\text{(j)}}\,\Sigma _{\text{s}}^{\text{(j)}}(\text{E,T})\,\, \right)}$

Although (308) provides the flux spectrum as a function of the background  Batson Iii committed Nov 17, 2020 223 224 225 cross section $$\sigma \,_{0}^{(r)}(u,T)$$ it is not in a form that can be preprocessed when the MG library is generated, because the energy variation of $$\sigma \,_{0}^{(r)}(E,T)$$ must be known. If the total cross sections  Batson Iii committed Jan 19, 2021 226 of the background nuclides in (309) have different energy variations, the shape of  Batson Iii committed Nov 17, 2020 227 228 $$\sigma \,_{0}^{(r)}(E,T)$$ depends on their relative concentrations—which are not known when the MG library is processed.  Batson Iii committed Jan 19, 2021 229 However, if the cross sections in (309) are independent of energy,  Batson Iii committed Nov 17, 2020 230 so that the background cross section is constant,  Batson Iii committed Jan 19, 2021 231 (308) can be solved for any arbitrary value of $$\sigma \,_{0}^{(r)}$$  Batson Iii committed Nov 17, 2020 232 233 234 235 236 237 as a parameter. This obviously occurs for the special case in which nuclide r is the only resonance nuclide in the mixture; i.e., the background materials are nonabsorbing moderators for which the total cross section is equal to the potential cross section. In this case, $$\sigma \,_{0}^{(r)}(E,T)\quad \to \ \ \ \sigma \,_{0,g}^{(r)}$$, where

 Batson Iii committed Jan 19, 2021 238 (310)$\sigma \,_{0,g}^{(r)}\,\,=\quad \frac{1}{N_{{}}^{(r)}}\sum\limits_{j\,\ne \,i}{\ N_{{}}^{(j)}\,\lambda _{g}^{(j)}\sigma \,_{p}^{(j)}}$
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If the mixture contains multiple resonance absorbers, as is usually the case, other approximations must be made to obtain a constant background cross section.

The approximation of “no resonance interference” assumes that resonances of background nuclides do not overlap with those of nuclide r, so their total cross sections can be approximated by the potential values within resonances of r where self-shielding occurs. In this  Batson Iii committed Jan 19, 2021 246 approximation, the expression in (310) is also used for the background  Batson Iii committed Nov 17, 2020 247 248 249 250 251 cross section.

Another approximation is to represent the energy-dependent cross sections of the background nuclides by their group-averaged (i.e., self-shielded cross) values; thus

 Batson Iii committed Jan 19, 2021 252 253 (311)$\sigma \,_{a}^{(j)}(E,T)\quad \to \ \ \ \sigma \,_{a,g}^{(j)}\ \quad ;\quad \ \ \quad \sigma \,_{s}^{(j)}(E,T)\quad \to \ \ \ \sigma \,_{s,g}^{(j)}\text{ for }E\in g$

In this case, the background cross section in (309) for nuclide r is the  Batson Iii committed Nov 17, 2020 254 255 group-dependent expression,

 Batson Iii committed Jan 19, 2021 256 257 (312)$\sigma _{0,g}^{\text{(r)}}\ \ =\ \ \frac{1}{{{\text{N}}^{\text{(r)}}}}\,\,\sum\limits_{j\ne r}{\left( \Sigma _{\text{a,g}}^{\text{(j)}}+\lambda _{\text{g}}^{\text{(j)}}\,\Sigma _{\text{s,g}}^{\text{(j)}}\, \right)}$

An equation similar to (312) is used for the background cross sections of  Batson Iii committed Nov 17, 2020 258 259 260 261 262 263 264 265 266 267 all resonance nuclides; thus the self-shielded cross sections of each resonance absorber depend on the shielded cross sections of all other resonance absorbers in the mixture. When self-shielding operations are performed with BONAMI for this approximation, “Bondarenko” iterations are performed to account for the inter-dependence of the shielded cross sections.

Assuming that $$\sigma \,_{0}^{(r)}$$ is represented as a groupwise-constant based on one of the previous approximations, several methods can be used to obtain a parameterized flux spectrum for preprocessing Bondarenko data in the MG libraries. In the simpliest approach, the scattering source of the resonance  Batson Iii committed Jan 19, 2021 268 nuclide r in (308) is represented by the NR approximation,  Batson Iii committed Nov 17, 2020 269 $${{\text{S}}^{\text{(r)}}}(\text{E,T})$$ to $$\Sigma _{\text{p}}^{\text{(r)}}C(E)$$.  Batson Iii committed Jan 19, 2021 270 In this case, (308) can be solved analytically to obtain the following  Batson Iii committed Nov 17, 2020 271 272 expression for the flux spectrum used to process MG data as a function of $$\sigma \,_{0}^{(r)}$$:

 Batson Iii committed Jan 19, 2021 273 (313)${{\Phi }^{\text{(r)}}}\text{(E;}\,\sigma _{0}^{\text{(r)}}\text{,T)}\ \ =\,\ \frac{\sigma _{\text{p}}^{\text{(r)}}\ +\ \,\frac{\text{1}}{{{\text{N}}^{\text{(r)}}}}\sum\limits_{j\ne r}{\,\Sigma _{\text{p}}^{\text{(j)}}\,}\ }{\sigma _{\text{t}}^{\text{(r)}}\text{(E,T)}\ \text{+}\ \sigma _{0}^{\text{(r)}}}C(E)\ \ \,\ \to \ \ \,\frac{C(E)\ }{\sigma _{\text{t}}^{\text{(r)}}\text{(E,T)}\ \text{+}\ \sigma _{0}^{\text{(r)}}}$
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where C(E) includes is an arbitrary constant multiplier that cancels  Batson Iii committed Jan 19, 2021 275 from (301).

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A more accurate approach that does not require using the NR approximation is to directly solve the IR form of the neutron transport equation using PW cross sections, with the assumption of no interference between mixed absorber resonances. The IRFfactor module of AMPX uses XSProc to calculate the self-shielded flux spectrum for MG data processing using one of two options:

1. A homogeneous model corresponding to an infinite medium of the resonance nuclide mixed with hydrogen, in which the ratio of the absorber to hydrogen number densities is varied in CENTRM to obtain the desired background cross section values;

2. A heterogeneous model corresponding to a 2D unit cell from an infinite lattice, in which the cell geometry (e.g., pitch) as well as the absorber number density is varied in CENTRM to obtain the desired background cross section values.

Both of these models provide a numerical solution for the flux spectrum. Details on these approaches are given in reference 2.

Self-Shielded Cross Section Data in SCALE Libraries

The AMPX code system processes self-shielded cross sections using the flux expressions described in the preceding section. For MG libraries in  Batson Iii committed Jan 19, 2021 299 SCALE-6.2 and later versions, the NR approximation in (313) is used to  Batson Iii committed Nov 17, 2020 300 301 302 303 304 305 306 307 308 309 310 311 312 313 represent the flux spectrum for nuclides with masses below A=40, since the NR approximation is generally accurate for low-mass nuclides and/or high energies. The standard AMPX weight functions are used to represent C(E) over the entire energy range for all nuclides with A<40, except for hydrogen and oxygen which use a calculated C(E) from CENTRM. The NR approximation with a calculated C(E) function is also used to represent the spectrum above the resolved resonance range for nuclides with A>40; but in the resolved resonance range of these nuclides, AMPX processes shielded cross sections with flux spectra obtained from CENTRM calculations using either a homogeneous or heterogeneous model. Regardless of the method used to obtain the flux spectrum, the parameterized shielded cross sections for absorber nuclide “r” are computed from the expression,

 Batson Iii committed Jan 19, 2021 314 (314)$\sigma _{\text{X,g}}^{\text{(r)}}(\sigma \,_{0}^{(r)}\,,T)\quad =\quad \,\frac{\int_{g}{\ \ \,\sigma _{X}^{(r)}(E,T)\ \,\Phi (E;\,\,\sigma \,_{0}^{(r)}\,,T)\ dE}}{\int_{g}{\ \,\Phi (E;\,\,\sigma \,_{0}^{(r)}\,,T)\ \,dE}}\quad ,$
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where $$\Phi (E;\,\,\sigma \,_{0}^{(r)}\,,T)$$ is the flux for a given value of $$\sigma \,_{0}^{(r)}$$ and T.

Rather than storing self-shielded cross sections in the master library, AMPX converts them to Bondarenko shielding factors, also called f-factors, defined as the ratio of the shielded cross section to the infinitely dilute cross section. Thus the MG libraries in SCALE contain Bondarenko data consisting of f‑factors defined as

 Batson Iii committed Jan 19, 2021 323 (315)$f_{\text{X,g}}^{\text{(r)}}(\sigma \,_{0}^{{}}\,,T)\quad =\quad \,\frac{\sigma _{\text{X,g}}^{\text{(r)}}(\sigma \,_{0}^{{}},T)}{\sigma _{\text{X,g}}^{\text{(r)}}(\infty )}\quad ,$
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and infinitely dilute cross sections defined as,

 Batson Iii committed Jan 19, 2021 326 (316)$\sigma _{\text{X,g}}^{\text{(r)}}(\infty )\quad =\quad \,\sigma _{\text{X,g}}^{\text{(r)}}(\sigma \,_{0}^{{}}=\infty ,T={{T}_{ref}}) \to \ \ \,\frac{\int_{g}{\ \sigma _{X}^{(r)}(E,{{T}_{ref}})\ C(E)\ \,dE}}{\int_{g}{\ \,C(E)\ \,dE}}\quad .$
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In AMPX, the reference temperature for the infinitely dilute cross section is normally taken to be 293 K. Bondarenko data on SCALE libraries are provided for all energy groups and for five reaction types: total, radiative capture, fission, within-group scattering, and elastic scatter. Recent SCALE libraries include f-factors at ~10–30 background cross section values (depending on nuclide) ranging from ~10−3 to ~1010 barns, which span the range of self-shielding conditions. Typically the f-factor data are tabulated at five temperature values. Background cross sections and temperatures available for each nuclide in the SCALE MG libraries are given in the SCALE Cross Section Libraries chapter.

Background Cross Section Options in BONAMI

To compute self-shielded cross sections for nuclide r, BONAMI first computes the appropriate background cross section for the system of interest and then interpolates the library Bondarenko data to obtain the f-factor corresponding to this σ0 and nuclide temperature. Several options are available in BONAMI to compute the background cross  Batson Iii committed Jan 19, 2021 346 section, based on (310) and (312) in the preceding section. The options are  Batson Iii committed Nov 17, 2020 347 348 349 350 351 specified by input parameter “iropt” and have the following definitions:

1. iropt = 0 => NR approximation with Bondarenko iterations:

 Batson Iii committed Jan 19, 2021 352

Background cross sections for all nuclides are computed using (312) with  Batson Iii committed Nov 17, 2020 353 354 λ=1; therefore,

 Batson Iii committed Jan 19, 2021 355 (317)$\sigma _{0}^{\text{(r)}}\ =\ \frac{1}{{{\text{N}}^{\text{(r)}}}}\,\,\sum\limits_{j\ne r}{\Sigma _{\text{t,g}}^{\text{(j)}}} .$
 Batson Iii committed Nov 17, 2020 356 357 358 359 360

Since the background cross section for each nuclide depends on the shielded total cross sections of all other nuclides in the mixture, “Bondarenko iterations” are performed in BONAMI to obtain a consistent set of shielded cross sections. Bondarenko iterations provide a crude method of accounting for resonance interference effects that are ignored by the  Batson Iii committed Jan 19, 2021 361 approximation for $$\sigma \,_{0}^{(r)}$$ in (310). The BONAMI  Batson Iii committed Nov 17, 2020 362 363 364 365 366 367 iterative algorithm generally converges in a few iterations. Prior to SCALE-6.2, this option was the only one available in BONAMI, and it is still the default for XSProc.

1. iropt = 1 => IR approximation with no resonance interference (potential cross sections):

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Background cross sections for all nuclides are computed using (310). No  Batson Iii committed Nov 17, 2020 369 370 371 372 373 Bondarenko iterations are needed.

1. iropt t = 2 => IR approximation with Bondarenko iterations, but no resonance scattering:

 Batson Iii committed Jan 19, 2021 374

Background cross sections for all nuclides are computed using (312) with  Batson Iii committed Nov 17, 2020 375 376 377 the scattering cross section approximated by the potential value; therefore,

 Batson Iii committed Jan 19, 2021 378 (318)$\sigma _{0}^{\text{(r)}}\ \ =\ \ \frac{1}{{{\text{N}}^{\text{(r)}}}}\,\,\sum\limits_{j\ne r}{\left( \Sigma _{\text{a,g}}^{\text{(j)}}+\lambda _{\text{g}}^{\text{(j)}}\,\Sigma _{\text{p}}^{\text{(j)}}\, \right)}$
 Batson Iii committed Nov 17, 2020 379 380 381 382 383 384 385

Since the background cross section for each resonance nuclide includes the shielded absorption cross sections of all other nuclides, Bondarenko interactions are performed.

1. iropt = 3 => IR approximation with Bondarenko iterations:

Background cross sections for all nuclides are computed using the full  Batson Iii committed Jan 19, 2021 386 IR expression in (312). Bondarenko interactions are performed.

 Batson Iii committed Nov 17, 2020 387 388 389 390 391 392 393 394 395 396 397 398 399

Computation of the background cross sections in BONAMI generally requires group-dependent values for the IR parameter λ. These are calculated by a module in AMPX during the library process and are stored in the MG libraries under the reaction identifier (MT number), MT=2000.

Self-Shielded Cross Sections for Heterogeneous Media

Equivalence theory can be used to obtain shielded cross sections for heterogeneous systems containing one or more “lumps” of resonance absorber mixtures separated by moderators, such as reactor lattices. It can be shown that if the fuel escape probability is represented by the Wigner rational approximation, the collision probability formulation of the neutron transport equation for an absorber body in a heterogeneous  Batson Iii committed Jan 19, 2021 400 medium can be reduced to a form identical to (303). This can be done for  Batson Iii committed Nov 17, 2020 401 402 403 404 405 406 407 408 409 410 411 412 an “equivalent” infinite homogeneous medium consisting of the same absorber body mixture plus an additional NR scatterer with a constant cross section called the “escape cross section” [Lam66]. Equivalence theory states that the self-shielded cross section for resonance absorber r in the heterogeneous medium is equal to the self-shielded cross section of r in the equivalent infinite homogeneous medium; therefore the f-factors that were calculated for homogenous mixtures can also be used to compute self-shielded cross sections for heterogeneous media by simply interpolating the tabulated f-factors in the library to the modified sigma-zero value of

 Batson Iii committed Jan 19, 2021 413 (319)$\hat{\sigma }_{0}^{(r)}\quad =\quad \sigma _{0}^{(r)}\ +\ \ \,\sigma _{esc}^{(r)}$
 Batson Iii committed Nov 17, 2020 414 415 416 417 418 419 420 421

where,

$$\hat{\sigma }_{0}^{(r)}$$ = background cross section of r in the absorber lump of the heterogeneous system;

$$\sigma \,_{0}^{(r)}$$ = background cross section defined in Parameterized Flux Spectra for an infinite homogeneous medium of the absorber body mixture;

$$\sigma _{esc}^{(r)}$$ = microscopic escape cross section for nuclide r, defined as

 Batson Iii committed Jan 19, 2021 422 (320)$\sigma _{esc}^{(r)}\quad =\quad \frac{{{\Sigma }_{esc}}}{{{N}^{(r)}}}$
 Batson Iii committed Nov 17, 2020 423 424 425 426

$${{\Sigma }_{esc}}$$ = macroscopic escape cross section for the absorber lump defined in BONAMI as

 Batson Iii committed Jan 19, 2021 427 (321)${{\Sigma }_{esc}}\quad =\quad \,\frac{(1\quad -\quad c)A}{\bar{\ell }\ \,\ \left[ 1\quad +\quad \left( A\quad -\quad 1 \right)c \right]}$
 Batson Iii committed Nov 17, 2020 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459

where

$$\bar{\ell }$$ = average chord length of the absorber body = $$4\ \ \,\times \ \frac{volume}{surface\ \ area}$$;

A = Bell factor, used to improve the accuracy of the Wigner rational approximation;

c = lattice Dancoff factor, which is equal to the probability that a neutron escaping from one absorber body will reach another absorber body before colliding in the intervening moderator.

Values for the mean chord length $$\bar{\ell }$$ are computed in BONAMI for slab, sphere, and cylinder absorber bodies. In the most common mode of operation where BONAMI is executed through the XSProc module in SCALE, Dancoff factors for uniform lattices are computed automatically and provided as input to BONAMI. For nonuniform lattices—such as those containing water holes, control rods, etc.—it may be desirable for the user to run the SCALE module MCDancoff to compute Dancoff factors using Monte Carlo for an arbitrary 3D configuration. In this case the values are provided in the MORE DATA input block of XSProc. The Bell factor “A” is a correction factor to account for errors caused by use of the Wigner rational approximation to represent the escape probability from a lump. Two optional Bell factor corrections are included in BONAMI. The first uses expressions developed by Otter that essentially force the Wigner escape probability for an isolated absorber lump to agree with the exact escape probability for the particular geometry by determining a value of A as a function of $${{\Sigma }_{T}}\bar{\ell }$$ for slab, cylindrical, or spherical geometries. Since the Otter expression was developed for isolated bodies, it does not account for errors in the Wigner rational approximation due to lattice effects. BONAMI also includes a Bell factor correction based on a modified formulation developed by Leslie [LHJ65] that is a function of the Dancoff factor.

Interpolation Scheme

After the background cross section for a system has been computed, BONAMI interpolates f-factors at the appropriate σ0 and  Batson Iii committed Jan 19, 2021 460 temperature from the tabulated values in the library. Fig. 202 shows  Batson Iii committed Nov 17, 2020 461 462 463 464 465 a typical variation of the f-factor vs. background cross sections for the capture cross section of 238U in the SCALE 252 group library.

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Fig. 202 Plot of f-factor variation for 238U capture reaction.

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Interpolation of the f-factors can be problematic, and several different schemes have been developed for this purpose. Some of the interpolation methods that have been used in other codes are constrained Lagrangian, [DYB77] arc-tangent fitting, [Kid74] and an approach developed by Segev [Seg81]. All of these were tested and found to be inadequate for use with the SCALE libraries, which may have multiple energy groups within a single resonance. BONAMI uses a unique interpolation method developed by Greene, which is described in [Gre82]. Greene’s interpolation method is essentially a polynomial approach in which the powers of the  Batson Iii committed Jan 19, 2021 477 polynomial terms can vary within a panel, as shown in (325):

 Batson Iii committed Nov 17, 2020 478
 Batson Iii committed Jan 19, 2021 479 (322)$f\left( \sigma \right)\quad =\quad f\left( \sigma {{\,}_{1}} \right)\quad +\quad \frac{\sigma {{\,}^{q(\sigma )}}\quad -\quad \sigma \,_{1}^{q(\sigma )}}{\sigma \,_{2}^{q(\sigma )}\quad -\quad \sigma \,_{1}^{q(\sigma )}}\quad \left( f\left( {{\sigma }_{2}} \right)\quad -\quad f\left( {{\sigma }_{1}} \right) \right)\quad ,$
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where

 Batson Iii committed Jan 19, 2021 482 483 (323)$q\left( \sigma \right)\quad =\quad q\left( \sigma {{\,}_{1}} \right)\quad +\quad \frac{\sigma \quad -\quad \sigma \,_{1}^{{}}}{\sigma \,_{2}^{{}}\quad -\quad \sigma \,_{1}^{{}}}\quad \left( q\left( {{\sigma }_{2}} \right)\quad -\quad q\left( {{\sigma }_{1}} \right) \right)\quad .$

Fig. 203 illustrates the expected behavior of (322) caused by varying  Batson Iii committed Nov 17, 2020 484 485 486 487 488 489 490 491 492 493 the powers in a panel.

By allowing the power q to vary as a function of independent variable σ, we can move between the various monotonic curves on the graph in a monotonic fashion. Note that when p crosses the p = 1 curve, the shape changes from concave to convex, or vice versa. This shape change means that we can use the scheme to introduce an inflection point, which is exactly the situation needed for interpolating f-factors.

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Fig. 203 Illustration of the effects of varying “powers” in the Greene interpolation method.

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Fig. 204 and Fig. 204 show typical “fits” of the f-factors using  Batson Iii committed Nov 17, 2020 497 498 499 500 501 502 503 504 505 the Greene interpolation scheme for two example cases. Note, in particular, that since this scheme has guaranteed monotonicity, it easily accommodates the end panels that have the smooth asymptotic variation. Even considering the extra task of having to determine the powers for temperature and σ0 interpolations, the method is not significantly more time-consuming than the alternative schemes for most applications.

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Fig. 204 Use of Greene’s method to fit the σ0 variation of Bondarenko factors for case 1.

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Fig. 205 Use of Greene’s method to fit the σ0 variation of Bondarenko factors for case 2.

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Input Instructions

BONAMI is most commonly used as an integral component of XSProc through SCALE automated analysis sequences. XSProc automatically prepares all the input data for BONAMI and links it with the other self-shielding modules. During a SCALE sequence execution, the data are provided directly to BONAMI in memory through XSProc. Some of the input parameters can be modified in the MOREDATA block in XSProc.

However, the legacy interface to execute stand-alone BONAMI calculations has been preserved for expert users. The legacy input to BONAMI uses the FIDO schemes described in the FIDO chapter of the SCALE manual. The BONAMI input for standalone execution is given below, where the MOREDATA input keywords are marked in bold.

Data Block 1

0$Logical Unit Assignments [4] 1. masterlib— input master library (Default = 23) 2. mwt—not used 3. msc—not used 4. newlib—output master library (Default = 22) 1$ Case Description [6]

1. cellgeometry—geometry description

0 homogeneous

1 slab

2 cylinder

3 sphere

2. numzones—number of zones or material regions

3. mixlength—mixing table length. This is the total number of entries needed to describe the concentrations of all constituents in all mixtures in the problem.

1. ib—not used

2. crossedt—output edit option

0 no output (Default)

1 input echo

2 iteration list, timing

3 background cross section calculation details

4 shielded cross sections, Bondarenko factors

3. issopt—not used

4. iropt—resonance approximation option

0 NR (Default) (Bondarenko iterations)

1 IR with potential scattering

2 IR with absorption and potential scattering (Bondarenko iterations)

3 IR with absorption and elastic scattering (Bondarenko iterations)

5. bellopt—Bell factor calculation option

0 Otter 1 Leslie (Default)

6. escxsopt—escape cross section calculation option

0 consistent

1 inconsistent (Default)

2* Floating-Point Constants [2]

1. bonamieps—convergence criteria for the Bondarenko iteration (Default = 0.001)

2. bellfact—geometrical escape probability adjustment factor. See notes below on this parameter (Default = 0.0).

T Terminate Data Block 1.

Data Block 2

3$Mixture numbers in the mixing table [mixlength] 4$ Component (nuclide) identifiers in the mixing table [mixlength] 5* Concentrations (atoms/b-cm) in the mixing table [mixlength] 6$Mixtures by zone [numzones] 7* Outer radii (cm) by zone [numzones] 8* Temperature (k) by zone [numzones] 9* Escape cross section (cm-1) by zone [numzones] 10$ Not used 11$Not used 12* Temperature (K) of the nuclide in a one-to-one correspondence with the mixing table arrays. 13* Dancoff factors by zone [numzones] 14* Lbar ($$\bar{\ell }$$) factors by zone [numzones] T Terminate Data Block 2. This concludes the input data required by BONAMI. Notes on input In the 1$ array, cellgeometry specifies the geometry. The geometry information is used in conjunction with the 7* array to calculate mean chord length Lbar if it is not provided by the user in the 14* array.

numzones, the number of zones, may or may not model a real situation. It may, for example, be used to specify numzones independent media to perform a cell calculation in parallel with one or more infinite medium calculations. The geometry description in 1$array applies only to mean chord length calculations unless it is provided in 14*. In the 2* array, bonamieps is used to specify the convergence expected on all macroscopic total values by zone, that is, each $${{\Sigma }_{t}}(g,j)$$ in group g and zone j is converged so that  Batson Iii committed Jan 19, 2021 623 (324)$\frac{\left| \,\Sigma \,_{t}^{i}(g,j)\quad -\quad \Sigma \,_{t}^{i-1}(g,j)\, \right|}{\Sigma \,_{t}^{i}(g,j)}\quad \le \quad bonamieps\quad .$  Batson Iii committed Nov 17, 2020 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673  The “Bell” factor in the 2* array is the parameter used to adjust the Wigner rational approximation for the escape probability to a more correct value. It has been suggested that if one wishes to use one constant value, the Bell factor should be 1.0 for slabs and 1.35 otherwise. In the ordinary case, BONAMI defaults the Bell factor to zero and uses a prescription by Otter [Ott64] to determine a cross-section geometry-dependent value of the Bell factor for isolated absorber bodies. It uses a prescription by Leslie6 to determine the Dancoff factor–dependent values of the Bell factor for lattices, which are much more accurate than the single value. The user who wishes to determine the constant value can, however, use it by inputting a value other than zero. The 3$, 4$, and 5* arrays are used to specify the concentrations of the constituents of all mixtures in the problem as follows: Entry 3$ (Mixture Number) 4$(Nuclide ID) 5* (Concentrations) 1 2 . . . . mixlength . Because of the manner in which BONAMI references the nuclides in a calculation, each nuclide in the problem must have a unique entry in the mixing table. Thus one cannot specify a mixture and subsequently load it into more than one zone, as can be the case with many modules requiring this type of data. The 12* array is used to allow varying the temperatures by nuclide within a zone. In the event this array is omitted, the 12* array will default by nuclide to the temperature of the zone containing the nuclide. The mixture numbers in each zone are specified in the 6$ array. Mixture numbers are arbitrary and need only match up with those used in the 3\$ array.

The radii in the 7* array are referenced to a zero value at the left boundary of the system.

In the event the temperatures in the 8* array are not bounded by temperature values in the Bondarenko tables, BONAMI will extrapolate using the three temperature points closest to the value. For example, a request for 273 K for a nuclide with Bondarenko sets at 300, 900, and 2,100 K would use the polynomial fit from those three temperature points to extrapolate the 273 K value.

The escape cross sections in the 9* array allow a macro escape cross section ($$\Sigma _{e}^{input}$$) to be specified by zone. (This array can be ignored if Dancoff factors are provided.) If the Dancoff factor for a zone is specified as −1 in the input, then the user-specified escape cross section is used in calculating the background cross sections σ0 as follows:

 Batson Iii committed Jan 19, 2021 674 (325)${{\sigma }_{0}}\quad =\quad \frac{\sum\limits_{n\ne i}{{{N}_{n}}\ \sigma \,_{t}^{n}\quad +\quad \Sigma _{e}^{input}}}{{{N}_{i}}}\quad$
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Sample Problem

In most cases, the input data to BONAMI are simple and obvious because the complicated parameters are determined internally based on the options selected. The user describes his geometry, the materials contained therein, the temperatures, and a few options.

This problem is for a system of iron-clad uranium (U238 – U235 ) fuel pins arranged in a square lattice in a water pool.

Our number densities are

Fuel:

$${{N}_{{}^{235}U}}$$ = 1.4987 × 10−4

$${{N}_{{}^{238}U}}$$ = 2.0664× 10−-2

$${{N}_{{}^{56}Fe}}$$ = 9.5642× 10−5

Water:

NH = 6.6662 × 10−2

NO = 3.3331 × 10−2

For the problem, we choose iropt = 1 (IR approximation with scattering approximated by λΣp) and crossedt = 4 for the most detailed output edits. An 8-group test library is used for fast execution and a short output file.

The XSProc/CSAS1X SCALE sequence input file, the corresponding i_bonami FIDO input file created by the sequence under the temporary working directory, and an abbreviated copy of the output from this case follows.

=csas1x Assembly pin test-8grp read comp ' fuel u-235  1 0 1.4987e-4 297.15 end u-238  1 0 2.0664e-2 297.15 end ' clad fe-56  2  0 9.5642e-5 297.15 end ' coolant h      3 0 6.6662e-2 297.15 end o      3 0 3.3331e-2 297.15 end end comp ' ==================================================================== read celldata latticecell squarepitch  pitch=1.26 3  fuelr=0.405765 1                                        cladr=0.47498  2 end  moredata iropt=1 crossedt=4 end moredata end celldata ' ==================================================================== end

FIDO input i_bonami

-1$$a0001 500000 e 0$$  a0001         11           0          18           1 e  1$$a0001 1 3 5 0 4 1010 1 -1 -1 e 2** a0001 1.00000E-03 0.00000E+00 e t 3$$  a0001          1           1           2           3           3 e  4$$a0001 92235 92238 26056 1001 8016 e 5** a0001 1.49870E-04 2.06640E-02 9.56420E-05 6.66620E-02 3.33310E-02 e 6$$  a0001          1           2           3 e 7**  a0001  4.05765E-01   4.74980E-01   7.10879E-01  e 8**  a0001  2.97150E+02   2.97150E+02   2.97150E+02  e 9**  a0001  1.11870E+00   4.15813E+00   1.78119E-01  e 10$$a0001 92235 92238 26056 1001 8016 e 11$$  a0001          0           0           0 e 13**  a0001  2.71260E-01   5.20852E-01   9.24912E-01  e 14**  a0001  8.11530E-01   1.38430E-01   4.71798E-01  e 15**  a0001  0.00000E+00   0.00000E+00   0.00000E+00   0.00000E+00  e 16$$a0001 2 2 2 e 17$$  a0001          0           0           0           0 e t
program verification information                                 code system:  SCALE    version:  6.2               program:  bonami      creation date:   unknown            library:  /home02/u2m/Workfolder/sampletmp           test code:  bonami            version:  6.2.0            jobname:  u2m       machine name:  node22.ornl.gov  date of execution:  04_dec_2013  time of execution:  21:43:54.38
1                 BONAMI CELL PARAMETERS --------------------------------------------- Bonami Print Option          : 4 BellFactor                   : 0 Bondarenko Iteration eps     : 0.001 Resonance Option             : 1 Bell Factor  Option          : LESLIE Escape CrossSection  Option  : INCONSISTENT CellGeometry                 : 2 MasterLibrary                : Number oF Neutron Groups     : 8 First Thermal Neutron Group  : 5 __________________________________________ Processing Zone               : 1 Mixture Number                : 1 Number Of Nuclides            : 2 Dancoff Factor                : 0.27126 Lbar                          : 0.81153 Escape Cross Section Input    : 1.1187 Material Temeprature          : 297.15  Processing Nuclide :  92235  Number Density : 0.00014987 Processing Nuclide :  92238  Number Density : 0.020664  Bondarenko Iterations iteration    Nuclide Group   MaxChange   Selfsig0     Effsig0      1        92235     0             0          0   0      1        92238     0             0          0   0  Total number of Bondarenko Iterations  : 1 Max Change in Group                    : 0  Group  Eff Macro Sig0      Escape Xsec    1      0.2351032         0.9075513    2      0.2351032         0.9075513    3      0.2351032         0.9075513    4      0.2351032         0.9075513    5      0.2351032         0.9075513    6      0.2351032         0.9075513    7      0.2351032         0.9075513    8      0.2351032         0.9075513  ---------------------------------------------------
Shielding Nuclide 92235  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec    1     1    7612.71875         7.19131      0.999998   7.19129    1     2    7612.71875         10.2521      0.999616   10.2481    1     3    7612.71875         24.9361       1.00241   24.9963    1     4    7612.71875         75.1109       1.05902   79.5436    1     5    7612.71875         56.0286       1.00205   56.1434    1     6    7612.71875         198.645        1.0008   198.805    1     7    7612.71875         347.945       1.00024   348.028    1     8    7612.71875         761.257        1.0066   766.282  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec    2     1    7612.71875         3.71448      0.999999   3.71448    2     2    7612.71875         7.63235       0.99935   7.62739    2     3    7612.71875          11.841      0.999444   11.8345    2     4    7612.71875         11.5408       1.00561   11.6055    2     5    7612.71875         12.5449       1.00001   12.545    2     6    7612.71875         14.2501       1.00007   14.2511    2     7    7612.71875         14.8125       1.00003   14.8128    2     8    7612.71875         15.1274       1.00015   15.1297  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec   18     1    7612.71875         1.21846      0.999996   1.21846   18     2    7612.71875         1.40834        1.0002   1.40862   18     3    7612.71875         8.92885       1.00132   8.94062   18     4    7612.71875         39.2086       1.06274   41.6686   18     5    7612.71875         32.7026       1.00105   32.737   18     6    7612.71875         153.511       1.00089   153.647   18     7    7612.71875         285.775       1.00026   285.848   18     8    7612.71875         636.445       1.00655   640.611  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec  102     1    7612.71875        0.060296             1   0.0602962  102     2    7612.71875        0.317627       1.00352   0.318746  102     3    7612.71875         4.16593       1.01325   4.22113  102     4    7612.71875         24.3615       1.07832   26.2695  102     5    7612.71875          10.781       1.00749   10.8618  102     6    7612.71875         30.8844       1.00074   30.9073  102     7    7612.71875         47.3579        1.0002   47.3671  102     8    7612.71875         109.685       1.00781   110.542  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec 1007     1    7612.71875               0             0   0 1007     2    7612.71875               0             0   0 1007     3    7612.71875               0             0   0 1007     4    7612.71875               0             0   0 1007     5    7612.71875         12.5448       1.00001   12.5449 1007     6    7612.71875         14.2501       1.00007   14.2511 1007     7    7612.71875         14.8125       1.00003   14.8129 1007     8    7612.71875         15.1278       1.00015   15.13  ---------------------------------------------------
Shielding Nuclide 92238  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec    1     1    44.0034676         7.33815      0.999983   7.33803    1     2    44.0034676         10.3566       1.00418   10.3999    1     3    44.0034676         15.0517      0.976844   14.7032    1     4    44.0034676          15.951      0.983793   15.6925    1     5    44.0034676         9.43867       1.00002   9.43887    1     6    44.0034676         10.1008       1.00008   10.1015    1     7    44.0034676         10.7744       1.00004   10.7748    1     8    44.0034676         12.2124       1.00145   12.2301  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec    2     1    44.0034676          4.0228      0.999974   4.0227    2     2    44.0034676         9.05886       1.00575   9.11093    2     3    44.0034676         14.0213      0.979923   13.7398    2     4    44.0034676         11.9032       0.98795   11.7598    2     5    44.0034676         8.86555      0.999984   8.86541    2     6    44.0034676         9.24452       1.00002   9.24471    2     7    44.0034676          9.2797       1.00002   9.27987    2     8    44.0034676          9.3077       1.00009   9.30853  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec   18     1    44.0034676        0.376356       1.00001   0.376361   18     2    44.0034676     0.000528746       1.00019   0.000528845   18     3    44.0034676     0.000308061      0.966052   0.000297603   18     4    44.0034676     4.75014e-06      0.967842   4.59738e-06   18     5    44.0034676     2.60878e-06       1.00006   2.60893e-06   18     6    44.0034676     5.27139e-06       1.00071   5.27512e-06   18     7    44.0034676      9.3235e-06       1.00018   9.32514e-06   18     8    44.0034676     1.81868e-05       1.00588   1.82937e-05  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec  102     1    44.0034676       0.0554327       1.00006   0.0554359  102     2    44.0034676         0.17972      0.978628   0.175879  102     3    44.0034676         1.03011      0.934934   0.963087  102     4    44.0034676         4.04777      0.971568   3.93268  102     5    44.0034676        0.573119        1.0006   0.573462  102     6    44.0034676        0.856257       1.00068   0.856839  102     7    44.0034676         1.49471       1.00017   1.49497  102     8    44.0034676         2.90465       1.00586   2.92168  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec 1007     1    44.0034676               0             0   0 1007     2    44.0034676               0             0   0 1007     3    44.0034676               0             0   0 1007     4    44.0034676               0             0   0 1007     5    44.0034676         8.86549      0.999984   8.86535 1007     6    44.0034676         9.24445       1.00002   9.24463 1007     7    44.0034676         9.27974       1.00002   9.27992 1007     8    44.0034676         9.30769       1.00009   9.30852 Zone Calculation is completed in 0 seconds                 BONAMI CELL PARAMETERS --------------------------------------------- Bonami Print Option          : 4 BellFactor                   : 0 Bondarenko Iteration eps     : 0.001 Resonance Option             : 1 Bell Factor  Option          : LESLIE Escape CrossSection  Option  : INCONSISTENT CellGeometry                 : 2 MasterLibrary                : Number oF Neutron Groups     : 8 First Thermal Neutron Group  : 5 __________________________________________ Processing Zone               : 2 Mixture Number                : 2 Number Of Nuclides            : 1 Dancoff Factor                : 0.520852 Lbar                          : 0.13843 Escape Cross Section Input    : 4.15813 Material Temeprature          : 297.15  Processing Nuclide :  26056  Number Density : 9.5642e-05  Bondarenko Iterations iteration    Nuclide Group   MaxChange   Selfsig0     Effsig0      1        26056     0             0          0   0  Total number of Bondarenko Iterations  : 1 Max Change in Group                    : 0  Group  Eff Macro Sig0      Escape Xsec    1     0.0003553244          3.487286    2     0.0003553244          3.487286    3     0.0003553244          3.487286    4     0.0003553244          3.487286    5     0.0003553244          3.487286    6     0.0003553244          3.487286    7     0.0003553244          3.487286    8     0.0003553244          3.487286  ---------------------------------------------------
Shielding Nuclide 26056  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec    1     1    36461.8672         3.07957       1.00005   3.07972    1     2    36461.8672         4.68958       1.00091   4.69382    1     3    36461.8672         7.85712      0.999843   7.85589    1     4    36461.8672         12.0029             1   12.0029    1     5    36461.8672         12.3689       1.00001   12.369    1     6    36461.8672         12.8598       1.00003   12.8602    1     7    36461.8672         13.5237      0.999906   13.5224    1     8    36461.8672         15.0714       0.99949   15.0637  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec    2     1    36461.8672         2.26476       1.00047   2.26583    2     2    36461.8672          4.6817        1.0009   4.68592    2     3    36461.8672         7.81457      0.999813   7.81311    2     4    36461.8672         11.9143             1   11.9143    2     5    36461.8672         12.0468       1.00001   12.0469    2     6    36461.8672          12.065       1.00002   12.0653    2     7    36461.8672         12.0887       1.00005   12.0893    2     8    36461.8672         12.2042       1.00013   12.2057  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec  102     1    36461.8672      0.00206393        1.0015   0.00206702  102     2    36461.8672      0.00787763        1.0035   0.00790524  102     3    36461.8672       0.0425504       1.00623   0.0428155  102     4    36461.8672       0.0885525             1   0.0885529  102     5    36461.8672        0.322101       1.00002   0.322109  102     6    36461.8672        0.794804        1.0002   0.79496  102     7    36461.8672         1.43496      0.998734   1.43314  102     8    36461.8672         2.86723      0.996792   2.85803  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec 1007     1    36461.8672               0             0   0 1007     2    36461.8672               0             0   0 1007     3    36461.8672               0             0   0 1007     4    36461.8672               0             0   0 1007     5    36461.8672         12.0468       1.00001   12.0469 1007     6    36461.8672          12.065       1.00002   12.0653 1007     7    36461.8672         12.0887       1.00005   12.0893 1007     8    36461.8672         12.2042       1.00013   12.2057 Zone Calculation is completed in 0 seconds
BONAMI CELL PARAMETERS --------------------------------------------- Bonami Print Option          : 4 BellFactor                   : 0 Bondarenko Iteration eps     : 0.001 Resonance Option             : 1 Bell Factor  Option          : LESLIE Escape CrossSection  Option  : INCONSISTENT CellGeometry                 : 2 MasterLibrary                : Number oF Neutron Groups     : 8 First Thermal Neutron Group  : 5 __________________________________________ Processing Zone               : 3 Mixture Number                : 3 Number Of Nuclides            : 2 Dancoff Factor                : 0.924912 Lbar                          : 0.471798 Escape Cross Section Input    : 0.178119 Material Temeprature          : 297.15  Processing Nuclide :   1001  Number Density : 0.066662 Processing Nuclide :   8016  Number Density : 0.033331  Bondarenko Iterations iteration    Nuclide Group   MaxChange   Selfsig0     Effsig0      1         1001     0             0          0   0      1         8016     0             0          0   0  Total number of Bondarenko Iterations  : 1 Max Change in Group                    : 0  Group  Eff Macro Sig0      Escape Xsec    1       1.494705         0.1593803    2       1.494705         0.1593803    3       1.494705         0.1593803    4       1.494705         0.1593803    5       1.494705         0.1593803    6       1.494705         0.1593803    7       1.494705         0.1593803    8       1.494705         0.1593803  ---------------------------------------------------
Shielding Nuclide 1001  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec    1     1    4.33502197         2.98905      0.999485   2.98751    1     2    4.33502197         9.87269      0.999169   9.86448    1     3    4.33502197         19.9332      0.999972   19.9326    1     4    4.33502197         20.4672      0.998926   20.4453    1     5    4.33502197         21.1735       1.00001   21.1736    1     6    4.33502197         26.1886       0.99995   26.1873    1     7    4.33502197         35.0621      0.999821   35.0558    1     8    4.33502197         54.9507      0.997361   54.8057  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec    2     1    4.33502197         2.98901      0.999485   2.98747    2     2    4.33502197          9.8726      0.999168   9.86439    2     3    4.33502197         19.9315      0.999972   19.9309    2     4    4.33502197         20.4556      0.998926   20.4336    2     5    4.33502197         21.1321       1.00001   21.1322    2     6    4.33502197         26.0865       0.99995   26.0852    2     7    4.33502197         34.8778      0.999829   34.8718    2     8    4.33502197         54.5786      0.997396   54.4365  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec  102     1    4.33502197     3.56422e-05       1.00003   3.56433e-05  102     2    4.33502197     8.96827e-05      0.998165   8.9518e-05  102     3    4.33502197      0.00171679      0.999794   0.00171643  102     4    4.33502197       0.0116042       1.00002   0.0116044  102     5    4.33502197       0.0413709       1.00001   0.0413714  102     6    4.33502197        0.102043       1.00007   0.10205  102     7    4.33502197        0.184322      0.998389   0.184025  102     8    4.33502197        0.372079      0.992163   0.369163  mt   Group      sig0      infDiluted Xsec    f-factor  shielded Xsec 1007     1    4.33502197               0             0   0 1007     2    4.33502197


text:
Finite quasisimple implies every endomorphism is trivial or an automorphism

Statement

Verbal statement

In a finite quasisimple group (i.e., a finite group that is also quasisimple), every endomorphism is either the trivial map, or an automorphism.

Definitions used

Quasisimple group

A group $G$ is termed quasisimple if $G/Z(G)$ is a simple group, and $G$ is a perfect group.

Note that $G/Z(G)$ is forced to be a simple non-Abelian group (if it were Abelian, $G$ would be solvable and hence not perfect). Thus, $G/Z(G)$ is centerless.

Facts used

1. Proper and normal in quasisimple implies central: In a quasisimple group, any proper normal subgroup is contained in the center.
2. Product formula
3. Cocentral implies Abelian-quotient: If a subgroup, along with the center, generates the whole group, then it must contain the commutator subgroup.

Proof

Given: A finite quasisimple group $G$ with center $Z(G)$. An endomorphism of $G$ with kernel $N \triangleleft G$ and with image $H \le G$

To prove: $N$ is either trivial or the whole group $G$

Proof: Assume that $N$ is not the whole of $G$. Then, by fact (1), $N \le Z(G)$.

Clearly, $Z(G)/N$ is a proper subgroup of $G/N$ contained in the center of $G/N$. Moreover, the quotient $(G/N)/(Z(G)/N)$ is isomorphic to $G/Z(G)$, hence is simple.

Now the image of $Z(G/N)$ under the quotient map by $Z(G)/N$ is a proper normal subgroup of $G/Z(G)$, hence is trivial. The upshot: $Z(G/N) = Z(G)/N$.

Since $H \cong G/N$, we get:

$\left |Z(H) \right | = \frac{\left|Z(G)\right|}{\left|N\right|}$

Further, we have:

$\left|H \cap Z(G) \right| \le \left| Z(H) \right|$

Thus:

$\left| H \cap Z(G) \right| \le \frac{\left|Z(G)\right|}{\left|N\right|} = \frac{\left|Z(G)\right|\left|H \right|}{\left| G \right|}$

Rearranging, we get:

$\left| H \cap Z(G) \right|\left| G \right| \le \left|Z(G)\right|\left|H \right|$

Using the product formula (fact (2)), we get:

$|G| \le |HZ(G)|$

Since $HZ(G) \le G$, this forces:

$HZ(G) = G$

By fact (3), $H$ contains the commutator subgroup of $G$. But since $G$ is assumed to be perfect, $H = G$, forcing $N$ be be trivial, and completing the proof.

format: plain


text:
# Chapter 30 - Induction and Inductance - Problems - Page 895: 4b

$\varepsilon=0.0V$

#### Work Step by Step

Remember that the induced emf is equal to $$\varepsilon=\frac{d \phi}{dt}=\frac{d}{dt} \int \vec{B} d\vec{A}$$ There is no change in area or magnetic field between $t=2.0s$ and $t=4.0s$, therefore, the change in flux is 0 and the induced emf is zero.

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


text:
# explicit formula for Riemann zeros counting function

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros.

Because there are various explicit formulae whereby prime counting functions can be expressed as infinite sums of sinusoidal functions over the set of zeta zeros, I had always assumed that the same thing would work in the opposite direction. That is, I imagined there was a well-known formula whereby the nontrivial-zeta-zeros-counting function could be expressed as an infinite sum of sinusoidal functions over the set of primes.

However, I've been unable to find such a formula. The only explicit formula I've found for this counting function looks like this:

$f(E) = \frac{1}{\pi} \Im(\ln(\Gamma(1/4 + iE/2)) - \frac{E}{2\pi}\ln(\pi) + \frac{1}{\pi} \Im(\ln(\zeta(1/2 + iE)) + 1$

where there's no explicit involvement of the primes.

Is anyone aware of a formula of the type I'm seeking? I'm quite sure such a thing must exist, since a programmer friend recently looked at this and got curious enough to "illegally" substitute the (truncated) Euler product expression for zeta in the above (it doesn't converge in the critical strip, hence such a substitution is not mathematically valid), just to see what the resulting function would look like. His plots of

$N_m(E) = \frac{1}{\pi} \Im(\ln(\Gamma(1/4 + iE/2)) - \frac{E}{2\pi}\ln(\pi) - \frac{1}{\pi} \sum_{p < m} \Im(\ln(1-p^{-1/2-iE})) + 1$

in three colours [corresponding to m= 100, 1000 and 10000 and where E varies over (4,60)] can be seen here:

http://img593.imageshack.us/img593/9048/nmfunction.gif

The positions of the zeta zeros are clearly visible, so even though this isn't a valid formula, I imagine that there must be something like this which is valid.

-
You might want to change $p$ in the second formula to the more customary $\rho$. At first glance, I assumed you were summing over primes less than $m$. –  Barry Cipra Dec 4 '11 at 17:59
@Barry, I think OP is summing over primes. I think the whole point of the question is whether there is a formula for the number of zeros as a sum over primes. –  Gerry Myerson Dec 4 '11 at 22:12
Gerry, you're quite right. –  Barry Cipra Dec 4 '11 at 23:54

It looks like the exact formula being sought here can be found in A.P. Guinand, "A summation formula in the theory of prime numbers", Proceedings of the London Mathematical Society (2) 50 (1948) 107--119. The first page is visible here without subscription:

You can see the explicit formula in the abstract, although hard to see the fine detail without access to the full PDF. This is a general form involving a function $f$ and an integral transform thereof. The abstract mentions "appropriate conditions" on $f$: I can't see what these are, but with an appropriate choice of this function, the formula displayed would presumably reduce to a fairly straightforward relation between a sum over the primes and a sum over the nontrivial zeros.

-
@Matthew: I took the liberty of adding an image of the formula. –  Joseph O'Rourke Dec 10 '11 at 15:55

Assuming the Riemann Hypothesis, you can use a smooth approximation to the characteristic function of an interval in the Guinand-Weil explicit formula to approximately count the number of zeros of the zeta-function in an interval on the critical line. This expresses the approximate number of such zeros in terms of an integral of your test function and a sum over primes, as you seek. In fact, this can be set-up in such a way that the sum over primes is finite. (This method can be used to give upper and lower bounds for the number of zeros, but not an exact formula.)

The details are (essentially) contained in a paper by Goldston & Gonek "A note on S(t) and the zeros of the Riemann zeta function" available on Dan Goldston's webpage: math.sjsu.edu/~goldston/publications.htm

-

The formula is given by Guinand as said in his answer by Matthew Watkins but it can be found in page 111 of the paper and reads: assuming Riemann Hypothesis and $T>0$ $$\frac12(N(T+0)+N(T-0))=\frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}-$$ $$- \frac{1}{\pi}\lim_{N\to\infty}\Bigl[\sum_{n=1}^N\Lambda(n)\frac{\sin(T\log n)}{n^{1/2} \log n}-\int_1^N \frac{\sin(T\log t)}{t^{1/2}\log t}dt-$$ $$-\frac{\sin(T\log N)}{\log N}\Bigl(\sum_{n=1}^N\Lambda(n)n^{-1/2}-2N^{1/2} \Bigr)\Bigr]+$$ $$+\frac{1}{2\pi}\left(\operatorname{\rm am}\Gamma\left(\frac12+iT\right)-T\log T+T\right)+$$ $$+\frac{1}{\pi}\arctan(2T)-\frac{1}{4\pi}\arctan(\sinh(\pi T)),$$ where $\operatorname{\rm am} \Gamma(\frac12+it)$ is defined by making $\operatorname{\rm am} \Gamma(\frac12)=0$ and continuing analytically along any path not meeting the real axis.

Of course the sums where $\Lambda(n)$ appears can be considered sum about primes.

-
What is am? argument? –  Marc Palm Feb 9 '14 at 11:55
@MarcPalm Yes, but I maintained the notation in Guinand –  juan Feb 9 '14 at 19:54

Guinand's paper is accessible at archive.org (p.111)

-
I hope you get enough "reputation" to be able to post comments. –  Chandan Singh Dalawat Dec 26 '11 at 13:45

By accumulating/integrating the result from a variant of the Fourier transform of the von Mangoldt function I believe it should be possible:

Clear[f]
scale = 1000000;
f = Range[scale];

f[[1]] = N@MangoldtLambda[1];
Monitor[Do[
f[[i]] = N@MangoldtLambda[i] + f[[i - 1]], {i, 2, scale}], i]

xres = .002;
xlist = Exp[Range[0, Log[scale], xres]];
tmax = 60;
tres = .015;
s = 1/2;
Monitor[errList1 =
Table[((xlist^(-s + I t).(f[[Floor[xlist]]] - xlist)))*(s +
I t), {t, Range[0, 60, tres]}];, t]

Print["Variant of the Fourier transform of the von Mangoldt function"]
g1 = ListLinePlot[Accumulate[Re[errList1]/Length[xlist]],
DataRange -> {0, 60}, PlotRange -> {-.3, 300}, Axes -> True,
Filling -> Axis];
Show[g1, ImageSize -> Large]

I have put a picture of the result here:

http://mobiusfunction.files.wordpress.com/2012/07/zeta-zero-counting-function.png

The scale is wrong though and I don't know how to do it by integration, but the steps at x-values of imaginary parts of Riemann zeta zeros should all be roughly the same.

Edit 12.8.2012: The integration of the formula in the program above is not too hard to do:

Clear[xlist, s, t, g];
Integrate[((xlist^(-s + I t)*(g - xlist)))*(-s + I t), t]
Print["g=f[[Floor[xlist]]]"]
Clear[f]
scale = 1000000;
f = Range[scale];

f[[1]] = N@MangoldtLambda[1];
Monitor[Do[
f[[i]] = N@MangoldtLambda[i] + f[[i - 1]], {i, 2, scale}], i]

xres = .002;
xlist = Exp[Range[0, Log[scale], xres]];
xlist[[1]] = 1.001;
tmax = 60;
tres = .015;
s = 1/2;
Monitor[errList1 =
Table[Total[((f[[Floor[xlist]]] - xlist) xlist^(-s +
I t) (I + (I s + t) Log[xlist]))/Log[xlist]^2], {t,
Range[0, 60, tres]}];, t]

Print["Variant of the Fourier transform of the von Mangoldt function"]
g1 = ListLinePlot[Re[errList1]/Length[xlist] - 1,
DataRange -> {0, 60}, PlotRange -> {-0, 3.7}, Axes -> True];
Show[g1, ImageSize -> Full]

I have put picture of the result here:

http://mobiusfunction.files.wordpress.com/2012/08/zeta-zero-counting-august-2012.png

I don't know if it is entirely correct and the scaling is wrong. The minus one seems to make the counting start at the first zeta zero.

Edit 5.2.2014:

Using this as a starting point:

$$\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\mu(d)}{d^{(s-1)}}.$$

Where $\Lambda(n)$ is the von Mangoldt function.

Edit 4.2.2014:

The Dirichlet series:

$$f(t) = (1-\sum\limits_{n=1}^{n=k} \frac{1}{1.2\log(k)} \frac{1}{n} \zeta(1/2+i \cdot t)\sum\limits_{d|n} \frac{\mu(d)}{d^{(1/2+i \cdot t-1)}})^{12}$$

leads to a sequence of numbers that when accumulated gives the plot:

Mathematica code for the plot above:

scale = 400;
Print["Counting to 60"]
Monitor[g1 =
ListLinePlot[
Accumulate[(1 -
Table[Re[
Zeta[1/2 - I*k]*
Total[Table[
Total[MoebiusMu[Divisors[n]]/Divisors[n]^(1/2 - I*k - 1)]/
n/Log[scale]/1.2, {n, 1, scale}]]], {k, 0 + 1/1000, 60,
N[1/6]}])^12], DataRange -> {0, 60},
PlotRange -> {-0.15, 15}], Floor[k]]

Edit 13.2.2014:

(*program start*)
scale = 300;
Print["Counting to 60"]
Monitor[g1 =
ListLinePlot[
0.69*Accumulate[
Table[Exp[-Re[
Zeta[1/2 - I*k]*
Total[Table[
Total[MoebiusMu[Divisors[n]]/
Divisors[n]^(1/2 - I*k - 1)]/n, {n, 1, scale}]]]], {k,
0 + 1/1000, 60, N[1/6]}]], DataRange -> {0, 60},
PlotRange -> {-0.15, 15}];, Floor[k]]
Show[g1, ListPlot[Table[{N[Im[ZetaZero[n]]], n}, {n, 1, 13}],
PlotStyle -> Black, Filling -> Axis]]
(*program end*)

The plot below from the program above is something like:

$$f(k)=\exp \left(-\Re\left(\zeta \left(\frac{1}{2}-i k\right) \sum _{n=1}^{\text{scale}} \frac{\sum\limits_{d|n}\frac{\mu (d(n))}{d(n)^{-i k+\frac{1}{2}-1}}}{n}\right)\right)$$

-

This may not help you, but: With Excel I calculated the Discrete Fourier Transform of a function = unit spikes at locations of the logarithms of a few hundred primes. The zeta zeros (imaginary part) show up as spikes. But it only works for small numbers of primes and zeros. For a better approximation, I reversed the explicit formula in Riemann's, making the spike heights = ln(p)/sqrt(p), and included powers of primes at 1/exponent as much, like the Von Mangoldt function. That gave much cleaner results, valid for thousands of primes and zeros. However it does diverge (increasing oscillations) for really big values, because logs of primes get close (more than the zeta zeros in Reimann's formula). I'm leaving out some details. If you or anyone is interested, email me at robertjwalcott@yahoo.com for the spreadsheets or graphs.

-


text:
# recounting rationals

## 587 days ago by calkin

N=10^6 a=[1] for i in srange(1,N): if is_even(i): a.append(a[i/2]+a[i/2-1]) else: a.append(a[(i-1)/2])
first_occur={1:1}
first_occur.has_key(2)
 False False
for i in srange(N): if first_occur.has_key(a[i])==False: first_occur.update({a[i]:i})
list_plot_loglog(first_occur)
list_plot(first_occur)

The sequence 1,1,2,1,3,2,3,1,... gives rise to the sequence of fractions

1/1

1/2 , 2/1

1/3 , 3/2 , 2/3 , 3/1

1/4 , 4/3 , 3/5 , 5/2 , 2/5 , 5/3 , 3/4 , 4/1

How can we look at just the left half of the tree?

This is the half that hangs from the node labeled 1/2.

N=10^2 a=[1,2] for i in srange(2,N): if is_even(i): a.append(a[i/2]+a[i/2-1]) else: a.append(a[(i-1)/2])
print a
 [1, 2, 3, 2, 5, 3, 5, 2, 7, 5, 8, 3, 8, 5, 7, 2, 9, 7, 12, 5, 13, 8, 11, 3, 11, 8, 13, 5, 12, 7, 9, 2, 11, 9, 16, 7, 19, 12, 17, 5, 18, 13, 21, 8, 19, 11, 14, 3, 14, 11, 19, 8, 21, 13, 18, 5, 17, 12, 19, 7, 16, 9, 11, 2, 13, 11, 20, 9, 25, 16, 23, 7, 26, 19, 31, 12, 29, 17, 22, 5, 23, 18, 31, 13, 34, 21, 29, 8, 27, 19, 30, 11, 25, 14, 17, 3, 17, 14, 25, 11] [1, 2, 3, 2, 5, 3, 5, 2, 7, 5, 8, 3, 8, 5, 7, 2, 9, 7, 12, 5, 13, 8, 11, 3, 11, 8, 13, 5, 12, 7, 9, 2, 11, 9, 16, 7, 19, 12, 17, 5, 18, 13, 21, 8, 19, 11, 14, 3, 14, 11, 19, 8, 21, 13, 18, 5, 17, 12, 19, 7, 16, 9, 11, 2, 13, 11, 20, 9, 25, 16, 23, 7, 26, 19, 31, 12, 29, 17, 22, 5, 23, 18, 31, 13, 34, 21, 29, 8, 27, 19, 30, 11, 25, 14, 17, 3, 17, 14, 25, 11]

Assignment: figure out how to list all the fractions that appear on the tree hanging from 1/2 instead of 1/1.

Let us try to compute the matrix representing the fraction in the kth position of the tree hanging off a/b: if the fraction is

$\frac{ea+fb}{ga +hb}$

then the matrix is

$M_k = \begin{pmatrix} e & f\\ g& h \end{pmatrix}$

The matrices $M_k$ satisfy the recurrences

$M_{2k+1} = \begin{pmatrix} 1 & 0 \\ 1 & 1\end{pmatrix} M_k$

$M_{2k+2} = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} M_k$

N=2^18 M=[matrix(2,2 ,[1,0,0,1])] a=[2] for i in srange(1,N): if is_even(i): A=M[(i-2)/2] A = matrix(2,2,[1,1,0,1])*A M.append(A) else: A=M[(i-1)/2] A = matrix(2,2,[1,0,1,1])*A M.append(A) val=matrix(1,2,[0,1])*A*matrix(2,1,[1,2]) a.append(val[0,0])
#list_plot(a)
first_occur={2:1}
for i in srange(N): if first_occur.has_key(a[i])==False: first_occur.update({a[i]:i})
list_plot_loglog(first_occur)

What does this picture show us?  There appear to still be some gaps in the positions in which first occurences happen.  Remember, the vertical axis is showing where the first occurence of n takes place.

We see it even more clearly in the list_plot with non-log scales: the gaps are more noticeable.

list_plot(first_occur)
b=first_occur.values()
list_plot(b)
b.sort()
b[:10]
 [1, 1, 3, 5, 9, 11, 15, 17, 19, 21] [1, 1, 3, 5, 9, 11, 15, 17, 19, 21]
list_plot(b)
print len(b) print b[-1],N
 4596 240929 262144 4596 240929 262144
len(first_occur)
 4596 4596

Questions:

1. Where are the gaps in the sorted list of positions where first occurrences happen?  Are powers of 2 involved?  How long are the biggest gaps? Where on the tree do these positions correspond to?

2. In the list of numbers which first occur in the first N values of the sequence, which numbers are "missing"?  Which numbers are "latecomers" in the sequence a[n].

3. How fast does the list of numbers which first occur in the first N values of the sequence grow as a function of N?


text:
# The differential and smooth sections

I am working on a question regarding a smooth section, which seems quite intuitive in a Euclidian space.

Let $M$ be a smooth manifold an $c:(0,1) \to M$ a smooth curve. Show, that $\dot c:(0,1) \to TM, \, t \mapsto dc_t(1)$ is a smooth section along (?) c, which means, that $\pi \circ \dot c = c$.

• $\pi \circ \dot c = c$ ist obviously true, as $\dot c$ maps $t$ to a vector $(c(t), dc_t(1))$ in $TM$
• Now the smoothness, which is the difficult part for me. In general: Is there a suitable way for daily life to calculate the differential? I have come across different definitions (with equivalence classes of curves, the algebraic definition, as the "Jacobian matrix" of the charts), but nevertheless I still don't really grasp the meaning and the connection to the Euclidian case (despite sending tangential vectors of one tangent space to another), particularily when it comes to questions like the one above. So what I have got so far is, that $\dot c$ is smooth, when $\tilde \varphi \circ \dot c$ is smooth as a function from $(0,1) \to \mathbb{R}^{2m}$ where $m$ is the dimension of $M$ and $\tilde \varphi$ the to $\varphi$ corresponding chart of $TM$. $$(\tilde \varphi \circ \dot c)(t) = (\varphi(c(t)), \xi)$$ with $\xi$ given by the representation $dc_t(1) = \xi^i \frac{\partial}{\partial x^I}\big|_t$ What is $\xi$ exactly? $\phi(c(t))$ is smooth, as $c(t)$ is smooth and I guess, that $\xi$ is smooth, because it is given by the partial derivatives of $c(t)$, but the partial derivatives of $c$. Is that correct? I there a general approach to questions like this with the differential

I know this feeling pretty well, it's really confusing, until it starts to make sense! I am not an expert, but maybe my student-like point of view can help you a bit.

Notation: I use $T$ to denote the tangential (or differential) of a map, i.e. for $f: M \to N$, we get the tangential as $Tf: TM \to TN$. The restriction to a point $x$ is then given by $T_x f: T_xM \to T_{f(x)}N$. For tangent spaces, I use the directional derivatives as the definition. The Jacobian of a real valued function at point $x$ is denoted by $Df(x)$.

Let me define $U = (0,1)$ to avoid some confusion. My approach would be to write $\dot{c}$ as a composition of smooth maps $U \rightarrow TU \rightarrow TM$, where the first map is given by $t \mapsto (t,1)$ and the second map is given by the tangential of $c$.

Smoothness of the first map: The first map is smooth. To check this, we compute the chart representation. If we take $\mathrm{id}_U: U \to (0,1)$ as a chart, then the chart for $TU$ is given by $T \mathrm{id}_U: TU \to (0,1) \times \mathbb{R}$. With resprect to these charts the map is then given by $$(0,1) \to (0,1) \times \mathbb{R} : t \mapsto (t,1),$$ which is a smooth map.

Smoothness of the second map: If $c : U \rightarrow M$ is smooth, then also it tangential $Tc : TU \rightarrow TM$ is a smooth map between the two manifolds $TU$ and $TM$. (Is this already known at the point of this excercise?)

What is $\xi$ exactly?

Let us fix $t$, then $\dot{c}(t) =: (c(t),v(t)) \in T_{c(t)}M$ is the tangent vector we are interested in.

To get some coordinates of $v(t)$, we first need a basis for the vector space $T_{c(t)}M$. Like in your question, we take a chart around $c(t)$, say $$\varphi: U \to \mathbb{R}^n: p \mapsto (x^1(p), \dots, x^n(p)),$$ where $U$ is an open set containing $c(t)$.

Now each individual component $x^i$ of the chart is related to a tangent vector, denoted by $\frac{\partial}{\partial x^i}\big|_p$. Its defining property is being the corresponding directional derivative after applying the chart:

$$\frac{\partial}{\partial x^i}\big|_p(f) = (\partial_i (f \circ \varphi^{-1}) ) ( \varphi(p) ),\quad \text{for all } f\in C^{\infty}(M).$$

In this way the chart $\tilde{\varphi}$ is constructed. An explicit formula for the coefficients is then given by

$$\xi^i(t) = D( x^i \circ c ) ( t ) = \frac{\partial (x^i \circ c)}{\partial t}\big|_t.$$

Hence you are right with the intuition, that the coefficients are smooth in $t$, as being partial derivatives $$\dot{c}(t) = \sum_i \frac{\partial (x^i \circ c)}{\partial t}\big|_t \frac{\partial}{\partial x^i} \big|_{c(t)}$$

For $x = h(p)$ and $v = \sum_i \eta^i(v) \frac{\partial}{\partial x^i} \big|_p \in T_p M$ we get the formula for the coefficients of

$$T f( v ) = \sum_j \xi^j( Tf(v) ) \frac{\partial}{\partial y^j} \big|_{f(x)},$$ given by

$$\xi( Tf(v) ) = D(k \circ f \circ h^{-1})(x)[\eta(v)] = \left( \sum_{j=1}^m \frac{\partial (k^i \circ f \circ h^{-1})}{\partial x^j}\big|_{x} \cdot \eta^j(v) \right)_{i=1,\dots,n} \in \mathbb{R}^n$$

Like you see, this is not really an equation you want to use often, but on the other hand it is just the normal derivative and some chain rules. In my experience, you can usually take advantage of the special kind of manifold you are working on. Since $T_x f$ is a linear map, you 'just' need to find out how it acts on a basis of $T_x M$. For example if you know how it acts on the vectors of a special charts, say polar coordinates, then you can find the coefficients of $T_x f$ directly, without the need to compute the derivative.

Anyway, quite often the questions are of general nature and the numeric values are of less interest. And if you are lucky, you maybe can use some theorems and the calculus related to differential forms, Lie derivatives, Riemannian metrics, Stokes etc... I guess that is the point where all these abstract notations really show off their power.

Is there a general approach?

This question is beyond my knowledge. Quite often you can use diagrams to proof certain statements in a nice way. For example the tangential $T$ acts like a functor on the category of smooth manifolds and there exists many 'canonical' chart independent constructions and objects, which can be used.

Proving smoothness is often no really needed nor interesting, since almost all objects are smooth by construction and you just combine them. But in the beginning you spend a lot of time proving all these basics, which will later be used without any notice.

• Thanks a lot for the effort. It really helps. Do you have an example for polar coordinates as you mentioned above, because I haven't seen any example at all yet. – Jonas W. May 27 '18 at 17:49
• A constructed example: $c:U=(0,1)\to \mathbb{R}^2: \lambda \mapsto (\sin(\lambda),\cos(\lambda))^T$, then $T_{\lambda} c\big( v \frac{\partial}{\partial t}\big|_\lambda \big) = v \cdot \cos(\lambda) \frac{\partial}{\partial x^1}\big|_{c(\lambda)} - v \cdot \sin(\lambda) \frac{\partial}{\partial x^2}\big|_{c(\lambda)}.$ But in polar coordinates $\varphi \circ c~(\lambda) = (0,\lambda)$ (with $\varphi$ denoting the chart), hence $T_{\lambda} c\big( v \frac{\partial}{\partial t}\big|_\lambda \big) = v \frac{\partial}{\partial \phi}\big|_{c(\lambda)}$. – Steffen Plunder May 28 '18 at 2:21
• Note that, in contrast to the Jacobian matrices, both expressions describe the same point in $T\mathbb{R}^2$. The chain rule, which is usually needed to translate tangent vector between different coordinate systems, is already included in the definition of the tangent vectors. In this way using the notions of differential geometry may help to clarify in settings of many different coordinates. This is for example useful in rigid body motion or elasticity. – Steffen Plunder May 28 '18 at 2:25
• Alright, thanks a lot again! – Jonas W. May 28 '18 at 17:05


text:
# 10 Mod 3

10 Mod 3. First need to divide the dividend by the divisor: 3 10 = 0. 30. Next we take the whole part of the quotient (0) and multiply that by the divisor (10): 0 x 10 = 0.

As you can see, the answer to 3 mod 10 is 3. Compute answers using wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history. 10 mod 3 mods if you; Giant mutant geckos, plasma guns that dissolve your enemies, thunderstorms and bullet time are just a few of the many additions made by this collection of top updates. Used in cómbination, they cán bring new iife to the capitai wasteland ánd turn your cópy of fallout 3 into a desperate fight for survival. Given two positive numbers, a and n, a modulo n (a % n, abbreviated as a mod n) is the remainder of the euclidean division of a by n, where a is the dividend and n is the divisor. Basically, python modulo operation is used to get the remainder of a division. The modulo operator ( %) is considered an arithmetic operation, along with