text:
# American Institue of Mathematical Sciences

2017, 13(4): 1793-1813. doi: 10.3934/jimo.2017019

## A numerical scheme for pricing American options with transaction costs under a jump diffusion process

 1 Department of Mathematics, Bogor Agricultural University, Kampus IPB Darmaga, Bogor, Jawa Barat 16680, Indonesia 2 Department of of Mathematics & Statistics, Curtin University, GPO Box U1987, WA 6845, Australia

Received  April 2016 Revised  July 2016 Published  December 2016

In this paper we develop a numerical method for a nonlinear partial integro-differential complementarity problem arising from pricing American options with transaction costs when the underlying assets follow a jump diffusion process. We first approximate the complementarity problem by a nonlinear partial integro-differential equation (PIDE) using a penalty approach. The PIDE is then discretized by a combination of a spatial upwind finite differencing and a fully implicit time stepping scheme. We prove that the coefficient matrix of the system from this scheme is an M-matrix and that the approximate solution converges to the viscosity solution to the PIDE by showing that the scheme is consistent, monotone, and unconditionally stable. We also propose a Newton's iterative method coupled with a Fast Fourier Transform for the computation of the discretized integral term for solving the fully discretized system. Numerical results will be presented to demonstrate the convergence rates and usefulness of this method.

Citation: Donny Citra Lesmana, Song Wang. A numerical scheme for pricing American options with transaction costs under a jump diffusion process. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1793-1813. doi: 10.3934/jimo.2017019
##### References:
 [1] A. Almendral, C. W. Oosterlee, Numerical valuation of options with jumps in the underlying, Appl. Math. Comput., 53 (2005), 1-18. doi: 10.1016/j.apnum.2004.08.037. [2] A. Anderson, J. Andresen, Jump diffusion process: volatility smile fitting and numerical methods for option pricing, Rev. Derivat. Res., 4 (2000), 231-262. [3] J. Ankudinova, M. Ehrhardt, On the numerical solution of nonlinear Black-Scholes equations, Computers and Mathematics with Applications, 56 (2008), 799-812. doi: 10.1016/j.camwa.2008.02.005. [4] C. G. Averbuj, Nonlinear differential evolution equation arising in option pricing when including transaction costs: a viscosity solution approach, R. Bras. Eco. de Emp., 12 (2012), 81-90. [5] G. Barles, Convergence of numerical schemes for degenerate parabolic equations arising in finance theory, in: L. C. G. Rogers, D. Talay (Eds), Numerical Methods in Finance, Cambridge University Press, Cambridge, (1997), 1-21. [6] F. Black, M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062. [7] W. Chen, S. Wang, A penalty method for a fractional order parabolic variational inequality governing American put option valuation, Computers & Mathematics with Applications, 67 (2014), 77-90. doi: 10.1016/j.camwa.2013.10.007. [8] W. Chen, S. Wang, A finite difference method for pricing European and American options under a geometric Levy process, Journal of Industrial and Management Optimization, 11 (2015), 241-264. doi: 10.3934/jimo.2015.11.241. [9] R. Company, L. Jodar, J. R. Pintos, A numerical method for European option pricing with transaction costs nonlinear equation, Mathematics and Computer Modelling, 50 (2009), 910-920. doi: 10.1016/j.mcm.2009.05.019. [10] R. Cont, P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC, Boca Raton, FL, 2004. [11] R. Cont, E. Voltchkova, A finite difference scheme for option pricing in jump diffusion and exponential lěvy models, SIAM J. Numer. Anal., 43 (2005), 1596-1626. doi: 10.1137/S0036142903436186. [12] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [13] B. D. üring, M. Fournier, A. J. üngel, High order compact finite difference schemes for a nonlinear Black-Scholes equation, International Journal of Theoretical and Applied Finance, 6 (2003), 767-789. [14] Y. d'Halluin, P. A. Forsyth, K. R. Vetzal, Robust numerical methods for contingent claims under jump diffusion processes, IMA J. Numer. Anal., 25 (2005), 87-112. doi: 10.1093/imanum/drh011. [15] P. Heider, Numerical methods for nonlinear Black-Scholes equations, Applied Mathematical Finance, 17 (2010), 59-81. doi: 10.1080/13504860903075670. [16] S. L. Heston, A closed-form solution for options with stochastic volatility with application to bond and currency options, Rev. Financial Stud., 6 (1993), 327-343. doi: 10.1093/rfs/6.2.327. [17] C. Huang, S. Wang, A power penalty approach to a nonlinear complementary problem, Operations Research Letters, 38 (2010), 72-76. doi: 10.1016/j.orl.2009.09.009. [18] C. Huang, S. Wang, A penalty method for a mixed nonlinear complementarity problem, Nonlinear Analysis TMA, 75 (2012), 588-597. doi: 10.1016/j.na.2011.08.061. [19] C. S. Huang, C. H. Hung, S. Wang, A fitted finite volume method for the valuation of options on assets with stochastic volatilities, Computing, 77 (2006), 297-320. doi: 10.1007/s00607-006-0164-4. [20] J. Hull, Options, Futures, and Other Derivatives, Prentice-Hall, Englewood Cliffs, 2005. [21] J. Hull, A. White, The pricing of options on assets with stochastic volatilities, J. Finance, 42 (1987), 281-300. doi: 10.1111/j.1540-6261.1987.tb02568.x. [22] H. E. Leland, Option pricing and replication with transaction costs, Journal of Finance, 40 (1985), 1283-1301. [23] D.C. Lesmana, S. Wang, An upwind finite difference method for a nonlinear Black-Scholes equation governing European option valuation, Applied Mathematics and Computation, 219 (2013), 8818-8828. doi: 10.1016/j.amc.2012.12.077. [24] D. C. Lesmana, S. Wang, Penalty approach to a nonlinear obstacle problem governing American put option valuation under transaction costs, Applied Mathematics and Computation, 251 (2015), 318-330. doi: 10.1016/j.amc.2014.11.060. [25] W. Li, S. Wang, Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme, Journal of Industrial and Management Optimization, 9 (2013), 365-389. doi: 10.3934/jimo.2013.9.365. [26] C. Van Loan, Computational Frameworks for the Fast Fourier Transform, Frontier in applied mathematics, Vol. 10. SIAM, Philadelphia, PA, 1992. [27] R. C. Merton, Option pricing when underlying stock returns are discontinuous, J. Financial Econ., 3 (1976), 125-144. doi: 10.1016/0304-405X(76)90022-2. [28] A. Mocioalca, Jump diffusion options with transaction costs, Rev. Roumaine Math. Pures Appl., 52 (2007), 349-366. [29] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Engelwood Cliffs, NJ, 1962. [30] S. Wang, A penalty method for a finite-dimensional obstacle problem with derivative constraints, Optimization Letters, 8 (2014), 1799-1811. doi: 10.1007/s11590-013-0651-4. [31] S. Wang, A penalty approach to a discretized double obstacle problem with derivative constraints, Journal of Global Optimization, 62 (2015), 775-790. doi: 10.1007/s10898-014-0262-3. [32] S. Wang, X. Q. Yang, A power penalty method for linear complementarity problems, Operations Research Letters, 36 (2008), 211-214. doi: 10.1016/j.orl.2007.06.006. [33] S. Wang, X. Q. Yang, A power penalty method for a bounded nonlinear complementarity problem, Optimization, 64 (2015), 2377-2394. doi: 10.1080/02331934.2014.967236. [34] S. Wang, X. Q. Yang, K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, Journal of Optimization Theory & Applications, 129 (2006), 227-254. doi: 10.1007/s10957-006-9062-3. [35] S. Wang, S. Zhang, Z. Fang, A superconvergent fitted finite volume method for Black-Scholes equations governing European and American option valuation, Numerical Methods for Partial Differential Equations, 31 (2015), 1190-1208. doi: 10.1002/num.21941. [36] Y.-P. Wang, S.-L. Tao, Application of regularization technique to variational adjoint method: A case for nonlinear convection-diffusion problem, Applied Mathematics and Computation, 218 (2011), 4475-4482. doi: 10.1016/j.amc.2011.10.028. [37] P. Wilmott, J. Dewynne, S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, 1993. [38] D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, 1971. [39] K. Zhang, S. Wang, Pricing options under jump diffusion processes with fitted finite volume method, Applied Mathematics & Computation, 201 (2008), 398-413. doi: 10.1016/j.amc.2007.12.043. [40] K. Zhang, S. Wang, A computational scheme for options under jump diffusion processes, International Journals of Numerical Analysis and Modeling, 6 (2009), 110-123. [41] K. Zhang, S. Wang, Pricing American bond options using a penalty method, Automatica, 48 (2012), 472-479. doi: 10.1016/j.automatica.2012.01.009. [42] X. L. Zhang, Numerical analysis of American option pricing in a jump diffusion model, Math. Oper. Res., 22 (1997), 668-690. doi: 10.1287/moor.22.3.668. [43] Y. Y. Zhou, S. Wang, X. Q. Yang, A penalty approximation method for a semilinear parabolic double obstacle problem, Journal of Global Optimization, 60 (2014), 531-550. doi: 10.1007/s10898-013-0122-6.

show all references

##### References:
 [1] A. Almendral, C. W. Oosterlee, Numerical valuation of options with jumps in the underlying, Appl. Math. Comput., 53 (2005), 1-18. doi: 10.1016/j.apnum.2004.08.037. [2] A. Anderson, J. Andresen, Jump diffusion process: volatility smile fitting and numerical methods for option pricing, Rev. Derivat. Res., 4 (2000), 231-262. [3] J. Ankudinova, M. Ehrhardt, On the numerical solution of nonlinear Black-Scholes equations, Computers and Mathematics with Applications, 56 (2008), 799-812. doi: 10.1016/j.camwa.2008.02.005. [4] C. G. Averbuj, Nonlinear differential evolution equation arising in option pricing when including transaction costs: a viscosity solution approach, R. Bras. Eco. de Emp., 12 (2012), 81-90. [5] G. Barles, Convergence of numerical schemes for degenerate parabolic equations arising in finance theory, in: L. C. G. Rogers, D. Talay (Eds), Numerical Methods in Finance, Cambridge University Press, Cambridge, (1997), 1-21. [6] F. Black, M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062. [7] W. Chen, S. Wang, A penalty method for a fractional order parabolic variational inequality governing American put option valuation, Computers & Mathematics with Applications, 67 (2014), 77-90. doi: 10.1016/j.camwa.2013.10.007. [8] W. Chen, S. Wang, A finite difference method for pricing European and American options under a geometric Levy process, Journal of Industrial and Management Optimization, 11 (2015), 241-264. doi: 10.3934/jimo.2015.11.241. [9] R. Company, L. Jodar, J. R. Pintos, A numerical method for European option pricing with transaction costs nonlinear equation, Mathematics and Computer Modelling, 50 (2009), 910-920. doi: 10.1016/j.mcm.2009.05.019. [10] R. Cont, P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC, Boca Raton, FL, 2004. [11] R. Cont, E. Voltchkova, A finite difference scheme for option pricing in jump diffusion and exponential lěvy models, SIAM J. Numer. Anal., 43 (2005), 1596-1626. doi: 10.1137/S0036142903436186. [12] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [13] B. D. üring, M. Fournier, A. J. üngel, High order compact finite difference schemes for a nonlinear Black-Scholes equation, International Journal of Theoretical and Applied Finance, 6 (2003), 767-789. [14] Y. d'Halluin, P. A. Forsyth, K. R. Vetzal, Robust numerical methods for contingent claims under jump diffusion processes, IMA J. Numer. Anal., 25 (2005), 87-112. doi: 10.1093/imanum/drh011. [15] P. Heider, Numerical methods for nonlinear Black-Scholes equations, Applied Mathematical Finance, 17 (2010), 59-81. doi: 10.1080/13504860903075670. [16] S. L. Heston, A closed-form solution for options with stochastic volatility with application to bond and currency options, Rev. Financial Stud., 6 (1993), 327-343. doi: 10.1093/rfs/6.2.327. [17] C. Huang, S. Wang, A power penalty approach to a nonlinear complementary problem, Operations Research Letters, 38 (2010), 72-76. doi: 10.1016/j.orl.2009.09.009. [18] C. Huang, S. Wang, A penalty method for a mixed nonlinear complementarity problem, Nonlinear Analysis TMA, 75 (2012), 588-597. doi: 10.1016/j.na.2011.08.061. [19] C. S. Huang, C. H. Hung, S. Wang, A fitted finite volume method for the valuation of options on assets with stochastic volatilities, Computing, 77 (2006), 297-320. doi: 10.1007/s00607-006-0164-4. [20] J. Hull, Options, Futures, and Other Derivatives, Prentice-Hall, Englewood Cliffs, 2005. [21] J. Hull, A. White, The pricing of options on assets with stochastic volatilities, J. Finance, 42 (1987), 281-300. doi: 10.1111/j.1540-6261.1987.tb02568.x. [22] H. E. Leland, Option pricing and replication with transaction costs, Journal of Finance, 40 (1985), 1283-1301. [23] D.C. Lesmana, S. Wang, An upwind finite difference method for a nonlinear Black-Scholes equation governing European option valuation, Applied Mathematics and Computation, 219 (2013), 8818-8828. doi: 10.1016/j.amc.2012.12.077. [24] D. C. Lesmana, S. Wang, Penalty approach to a nonlinear obstacle problem governing American put option valuation under transaction costs, Applied Mathematics and Computation, 251 (2015), 318-330. doi: 10.1016/j.amc.2014.11.060. [25] W. Li, S. Wang, Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme, Journal of Industrial and Management Optimization, 9 (2013), 365-389. doi: 10.3934/jimo.2013.9.365. [26] C. Van Loan, Computational Frameworks for the Fast Fourier Transform, Frontier in applied mathematics, Vol. 10. SIAM, Philadelphia, PA, 1992. [27] R. C. Merton, Option pricing when underlying stock returns are discontinuous, J. Financial Econ., 3 (1976), 125-144. doi: 10.1016/0304-405X(76)90022-2. [28] A. Mocioalca, Jump diffusion options with transaction costs, Rev. Roumaine Math. Pures Appl., 52 (2007), 349-366. [29] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Engelwood Cliffs, NJ, 1962. [30] S. Wang, A penalty method for a finite-dimensional obstacle problem with derivative constraints, Optimization Letters, 8 (2014), 1799-1811. doi: 10.1007/s11590-013-0651-4. [31] S. Wang, A penalty approach to a discretized double obstacle problem with derivative constraints, Journal of Global Optimization, 62 (2015), 775-790. doi: 10.1007/s10898-014-0262-3. [32] S. Wang, X. Q. Yang, A power penalty method for linear complementarity problems, Operations Research Letters, 36 (2008), 211-214. doi: 10.1016/j.orl.2007.06.006. [33] S. Wang, X. Q. Yang, A power penalty method for a bounded nonlinear complementarity problem, Optimization, 64 (2015), 2377-2394. doi: 10.1080/02331934.2014.967236. [34] S. Wang, X. Q. Yang, K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, Journal of Optimization Theory & Applications, 129 (2006), 227-254. doi: 10.1007/s10957-006-9062-3. [35] S. Wang, S. Zhang, Z. Fang, A superconvergent fitted finite volume method for Black-Scholes equations governing European and American option valuation, Numerical Methods for Partial Differential Equations, 31 (2015), 1190-1208. doi: 10.1002/num.21941. [36] Y.-P. Wang, S.-L. Tao, Application of regularization technique to variational adjoint method: A case for nonlinear convection-diffusion problem, Applied Mathematics and Computation, 218 (2011), 4475-4482. doi: 10.1016/j.amc.2011.10.028. [37] P. Wilmott, J. Dewynne, S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, 1993. [38] D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, 1971. [39] K. Zhang, S. Wang, Pricing options under jump diffusion processes with fitted finite volume method, Applied Mathematics & Computation, 201 (2008), 398-413. doi: 10.1016/j.amc.2007.12.043. [40] K. Zhang, S. Wang, A computational scheme for options under jump diffusion processes, International Journals of Numerical Analysis and Modeling, 6 (2009), 110-123. [41] K. Zhang, S. Wang, Pricing American bond options using a penalty method, Automatica, 48 (2012), 472-479. doi: 10.1016/j.automatica.2012.01.009. [42] X. L. Zhang, Numerical analysis of American option pricing in a jump diffusion model, Math. Oper. Res., 22 (1997), 668-690. doi: 10.1287/moor.22.3.668. [43] Y. Y. Zhou, S. Wang, X. Q. Yang, A penalty approximation method for a semilinear parabolic double obstacle problem, Journal of Global Optimization, 60 (2014), 531-550. doi: 10.1007/s10898-013-0122-6.
Prices of the European call and put options with $a=0.01$ and $b=0.07$
Prices of the European call and put option for different values of the transaction cost parameter
Computed American and European put option prices
Computed American put option prices for different values of the transaction cost parameter
Computed rates of convergence for the call option with $a = 0.01$ and $b=0.07$
 $M$ $N$ $\|\cdot\|_{h,2}$ Ratio$(\|\cdot\|_{h,2})$ 21 11 0.215680 41 21 0.116543 1.85 81 41 0.061550 1.89 161 81 0.031986 1.92 321 161 0.016228 1.97 641 321 0.007861 2.06 1281 641 0.003457 2.27 2561 1281 0.001170 2.96
 $M$ $N$ $\|\cdot\|_{h,2}$ Ratio$(\|\cdot\|_{h,2})$ 21 11 0.215680 41 21 0.116543 1.85 81 41 0.061550 1.89 161 81 0.031986 1.92 321 161 0.016228 1.97 641 321 0.007861 2.06 1281 641 0.003457 2.27 2561 1281 0.001170 2.96
Computed rates of convergence for the put option with $a = 0.01$ and $b=0.07$
 $M$ $N$ $\|\cdot\|_{h,2}$ Ratio$(\|\cdot\|_{h,2})$ 21 11 0.454596 41 21 0.438884 1.04 81 41 0.390547 1.12 161 81 0.327934 1.19 321 161 0.259319 1.26 641 321 0.189478 1.37 1281 641 0.121703 1.56 2561 1281 0.058168 2.09
 $M$ $N$ $\|\cdot\|_{h,2}$ Ratio$(\|\cdot\|_{h,2})$ 21 11 0.454596 41 21 0.438884 1.04 81 41 0.390547 1.12 161 81 0.327934 1.19 321 161 0.259319 1.26 641 321 0.189478 1.37 1281 641 0.121703 1.56 2561 1281 0.058168 2.09
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# Displaying index as subscript on output: e.g. C[i] -> C_i with Notation[...] or Interpretation[..]?

You have all convinced me not to type in formulas with subscripts. However, in order to be able to match the subscripts in my whiteboard math, I figure a 2nd best solution is to want to display instances of C[i] with $C_i$ , only in the output, for a chosen set of $C$ and for any $i$

DisplaySubscripted[val_] :=
Format[val[args__]] := Interpretation[Subscript[val, args], val[args]]
SubscriptVariables[valueList_] := Scan[DisplaySubscripted, valueList];

(* Usage*)
SubscriptVariables[{a, b}]; (* Label the variables I want to display subscripted*)
Solve[a[1] + a[2] - 2 b[2] == c[1], b[2]]
(* This subscripts the a[1], a[2], b[2] and not the c[1], as I hoped.
It also doesn't seem to mess up the symbolics*)

Am I missing something, or is this a side-effect free approach to this problem? (it also works well with Subsuperscript and a[1][2] displaying as $a_1^2$ and it doesn't seem to mess up the superscript with powers)

ALTERNATIVELY: I also thought that Notation may be a way to do this. Easy enough to do this for an arbitrary variable. e.g.

Needs["Notation"];
DisplaySubscripted[var_] :=
Notation[ParsedBoxWrapper[
SubscriptBox[var, "1"]]\[DoubleLongLeftArrow]ParsedBoxWrapper[
RowBox[{var, "[", "1", "]"}]]];
DisplaySubscripted["f"];

f[1] (* Displays in output properly as f_1*)

However, I can't figure out how to match this for arbitrary subscript patterns. The obvious one does not work: Notation[ParsedBoxWrapper[ SubscriptBox[var, "_"]]\[DoubleLongLeftArrow]ParsedBoxWrapper[ RowBox[{var, "[", "_", "]"}]]]

I am also not entirely sure that passing var_ as strings is the best way to do it, but if I use DisplaySubscripted[f] instead, it displays the namespace (i.e. Global'f_1). If the Format approach I gave first works, it seems much simpler.

I think your approach is fine; in fact I just recommended something similar in another answer.

However, for the best handing of using your formatted output as input may want to use MakeBoxes for the reason described by Michael Pilat.

You also might consider putting a list your subscripted symbols in a global variable for easy changes, or Blocking.

Something like this:

MakeBoxes[a : h_[args__], fmt_] /;
MemberQ[$subs, Unevaluated @ h] := ToBoxes[Interpretation[Subscript[h, args], a], fmt]$subs = Hold[a, b];

expr = Solve[a[1] + a[2] - 2 b[2] == c[1], b[2]]

Block[{$subs = {c}}, ToString[expr, StandardForm]]  Notes: • I used ToString to cause MakeBoxes to trigger inside the Block; this is similar to what I did for Returning an unevaluated expression with values substituted in. • I put the Symbols inside Hold so that global definitions would not conflict; the MakeBoxes definition is written to handle this. A list can also be used as shown in the Block example. ## Extension Matching symbols by pattern was requested. I believe it is best to use string patterns, and require all symbols (in the list) to be in String form: MakeBoxes[a : h_[args__], fmt_] := ToBoxes[Interpretation[Subscript[h, args], a], fmt] /; MemberQ[Names @$subs, ToString @ Unevaluated @ h]

$subs = {"a", "b*"}; expr = Solve[a[1] + a[2] - 2 b[2] + bx[1]^2 == c[1], b[2]]  The patterns in $subs can be anything that is accepted by Names.

If you want to handle both literal Symbols and string pattern you will need to convert the Symbols to strings. The easiest way would be Names[ToString /@ $subs] but this would preclude using StringExpression patterns and would fail to hold Symbols unevaluated; I recommend just using strings as above. In version 7 if I Copy As > LaTeX the expression above I get LaTeX with subscripts:$b_2\to \frac{1}{2} \left(a_1+a_2+\left(\text{bx}_1\right){}^2-c[1]\right)$In version 9 this does not work right; you could leave out Interpretion to fix the problem at the expense of being able to copy&paste the expression within Mathematica: MakeBoxes[a : h_[args__], fmt_] := ToBoxes[Subscript[h, args], fmt] /; MemberQ[Names@$subs, ToString@Unevaluated@h]

## Refactored full definitions

Here is refactored code for the generation of your full set of definitions.

• The code is far more concise and easily modified; one can make a change in one place rather than a dozen.

• I noted that repeated calls to Names was the cause of a considerable slow-down so I changed that to StringMatchQ.

• I Protected the symbols {bar, hat, tilde, vec, underbar, plus, minus, star} because I don't think you'll want those accidentally evaluating to something else.

• I rewrote definitions that call Subsuperscript to use argsub_ rather than argsub__ as the latter could result in invalid input to Subsuperscript.

• If you use the following code outside of package, without Begin and End, you should add SetAttributes[makeDef, HoldAll] to prevent things like h from possibly evaluating incorrectly.

The Symbols used before Begin are created in the present (typically Global) context; the usage messages are therefore not superfluous, but may be replaced with merely $scriptedconstants;$scriptedfunctions; if desired.

With[{symbols := Sequence[bar, hat, tilde, vec, underbar, plus, minus, star]},
Unprotect @ symbols; ClearAll @ symbols; Protect @ symbols;
]

$scriptedconstants::usage = "a list of string patterns for symbols: constants";$scriptedfunctions::usage = "a list of string patterns for symbols: functions";

Begin["formattingRules"];

makeDef[pat_, body_] := (
MakeBoxes[a : pat, fmt_] := ToBoxes[Interpretation[body, a], fmt] /;
StringMatchQ[ToString @ Unevaluated @ h, $scriptedconstants]; MakeBoxes[a : pat[sub_], fmt_] := ToBoxes[Interpretation[body[sub], a], fmt] /; StringMatchQ[ToString @ Unevaluated @ h,$scriptedfunctions]
)

set1 = {
bar      -> OverBar,
hat      -> OverHat,
tilde    -> OverTilde,
vec      -> OverVector,
underbar -> UnderBar,
plus     -> SuperPlus,
minus    -> SuperMinus,
star     -> SuperStar
};

set2 = {
plus  -> "+",
minus -> "-",
star  -> "*"
};

makeDef[h_[#], #2[h]] & @@@ set1;
makeDef[h_[argssub__][#], Subscript[#2[h], argssub]] & @@@ Take[set1, 5];
makeDef[h_[argsub_][#], Subsuperscript[h, argsub, #2]] & @@@ set2;
makeDef[h_[argssub__], Subscript[h, argssub]];

End[];

(You may notice a funny offset in lines containing $ above; this is a known bug in the SE rendering.) • The only additional feature I would be interested in is whether we could have it match patterns. e.g. $displayscripted = {a_} where ab[1] becomes $ab_1$ Aug 22 '13 at 16:13
• @jlperla I recommend a string pattern directly, rather than a Pattern object: $displayscripted = {"a*"} -- if this is acceptable I will adapt my answer to include it. Aug 22 '13 at 16:16 • Perfectly acceptable. Aug 22 '13 at 17:38 • Sorry, just realized there is 1 more potential issue. The goal would also be to copy/paste as latex. I try to latex copy the displayed output from things like a[1], displaying as$a_1$, it copies as a[1]. If I do things like: ToString[a[1],StandardForm and then latex copy/paste the output it is closer, but still wacky. Finally, if I put in my other forms such as a[1][2] to$a_1^2$or a[star] to$a^{*}$I get an error like Recursion depth of 4096 exceeded or it simply doesn't copy properly. Aug 22 '13 at 17:59 • My implementation of$a^{*}$is MakeBoxes[a : h_[star], fmt_] /; MemberQ[$variablelist, Unevaluated@h] := ToBoxes[Interpretation[SuperStar[h], a], fmt]; for example. Aug 22 '13 at 18:00

Summarizing all of the above, following code implements the list of formatting rules. The features are:

• Displays in output based on the set of formatting tags. e.g. a[1] displays as $a_1$
• Matches string patterns such as "a*" to ab[1]
• Copy/paste in the notebook of the output a_1 actually copies a[1] as it should.
• ToLatex takes an expression and generates the latex with the formatting. i.e. a[1] becomes a_1 in latex.
• Special cases for hats, bars, etc. e.g. a[star] displays as $a^{*}$
• Rules include variations for both variables and functions. i.e. $\hat{f}_l(z)$ and $a_l$
• Can handle derivatives $f[l]'[z]$ to $f_l'(z)$
• Can handle functions with/without subscripts. e.g. $f[z]$ and $f_l[z]$ simultaneously if $z$ is flagged as a parameter.

Usage:

$scriptedconstants = {a, ab};$scriptedfunctions = {f};
$scriptedfunctionsvars = {x}; val = {a[1], ab[1], a[1, 2], a[bar], a[hat], a[tilde], a[vec], a[underbar], a[plus], a[minus], a[star], a[l][bar], a[l][hat], a[l][tilde], a[l][vec], a[l][underbar], a[l][plus], a[l][minus], a[l][star]} val2 = {f[x], D[f[x], x], f[l][x, z], f[l], f[l]'[z], D[f[l][z], z, z], D[f[l][x, z], z], D[f[hat][x, z], z], D[f[l][hat][x, z], z], f[1][z], f[1][z_], f[bar][z], f[bar][x, z], f[hat][z], f[tilde][z], f[vec][z], f[underbar][z], f[plus][z], f[minus][z], f[star][z], f[l][bar][z], f[l][hat][z], f[l][tilde][z], f[l][vec][z], f[l][underbar][z], f[l][plus][z], f[l][minus][z], f[l][star][z]} val // ToLatex val2 // ToLatex  This displays as: $$\left\{a_1,\text{ab}_1,a_{1,2},\bar{a},\hat{a},\tilde{a},\overset{\rightharpoonup }{a},\underline{a},a^+,a^-,a^*,\bar{a}_l,\hat{a}_l,\tilde{a}_l,\overset{\rightharpoonup }{a}_l,\underline{a}_l,\text{Subsuperscript}[a,l,+],\text{Subsuperscript}[a,l,-],\text{Subsuperscript}[a,l,*]\right\}$$ (the super/subscripts here actually show correctly in the FrontEnd, but are the only thing which don't translate to Latex well. An independent problem from this code) $$\left\{f(x),f'(x),f_l(x,z),f_l,f_l'(z),f_l''(z),f_l{}^{(0,1)}(x,z),\hat{f}^{(0,1)}(x,z),\hat{f}_l{}^{(0,1)}(x,z),f_1(z),f_1(\text{z\_}),\bar{f}(z),\bar{f}(x,z),\hat{f}(z),\tilde{f}(z),\overset{\rightharpoonup }{f}(z),\underline{f}(z),f^+(z),f^-(z),f^*(z),\bar{f}_l(z),\hat{f}_l(z),\tilde{f}_l(z),\overset{\rightharpoonup }{f}_l(z),\underline{f}_l(z),\text{Subsuperscript}[f,l,+](z),\text{Subsuperscript}[f,l,-](z),\text{Subsuperscript}[f,l,*](z)\right\}$$ And the ToLatex @val2 generates as: \left{f(x),f'(x),f_l(x,z),f_l,f_l'(z),f_l''(z),f_l{}^{(0,1)}(x,z),\hat{f}^{(0,1)}(x,z),\hat{f}_l{}^{(0,1)}(x,z),f_1(z),f_1(\text{z$\_$}),\bar{f}(z),\bar{f}(x,z),\hat{f}(z),\tilde{f}(z),\overset{\rightharpoonup }{f}(z),\underline{f}(z),f^+(z),f^-(z),f^*(z),\bar{f}_l(z),\hat{f}_l(z),\tilde{f}_l(z),\overset{\rightharpoonup }{f}_l(z),\underline{f}_l(z),\text{Subsuperscript}f,l,+,\text{Subsuperscript}f,l,-,\text{Subsuperscript}f,l,*\right} The full listing of code for this, slightly modified version from that of mr-wizard is: NotScriptedVarQ[z_] := ! MemberQ[$scriptedfunctionsvars, z, Infinity];

makeDef[pat_, body_] := (
MakeBoxes[a : pat, fmt_] := ToBoxes[Interpretation[body, a], fmt] /;
MemberQ[Union[$scriptedconstants,$scriptedfunctions], Unevaluated @ h];
MakeBoxes[a : pat[sub_], fmt_] := ToBoxes[Interpretation[body[sub], a], fmt] /;
MemberQ[$scriptedfunctions, Unevaluated @ h] ) set1 = { bar -> OverBar, hat -> OverHat, tilde -> OverTilde, vec -> OverVector, underbar -> UnderBar,(* After here subsuper is ncessary to have both annotations*) plus -> SuperPlus, minus -> SuperMinus, star -> SuperStar }; set2 = { plus -> "+", minus -> "-", star -> "*" }; makeDef[h_Symbol[#], #2[h]] & @@@ set1; makeDef[h_Symbol[argssub__][#], Subscript[#2[h], argssub]] & @@@ Take[set1, 5]; makeDef[h_Symbol[argsub_][#], Subsuperscript[h, argsub, #2]] & @@@ set2; makeDef[h_Symbol[argssub__?NotScriptedVarQ], Subscript[h, argssub]];  • I am glad you have a solution. Would you like me to add a condensed version of your rule creation code to my answer? Aug 23 '13 at 4:49 • By all means, please edit or move things into your answer if you prefer. I think this is a very useful utility for people. So far, the only missing pattern I have found is that while F[h][z] matched my function pattern I also needed F[h]'[z] to match to$F_h'(z)\$ so I could write an ODE in terms of F[h] Aug 23 '13 at 5:08
• Okay; tomorrow. Aug 23 '13 at 6:08
• Thanks. If you have any ideas of the best/idiomatic way to create the as a package it would also be appreciated. I had trouble with contexts, etc when I tried it Aug 23 '13 at 16:30
• When you say "ask a package" do you just mean a convenient way to load these definitions? If that is so you shouldn't need things like BeginPackage`; if on the other hand you only want the definitions to be active only in certain contexts or the like you'll have to clearly specify what you want. Aug 23 '13 at 17:23


text:
# American Institute of Mathematical Sciences

March  2010, 13(2): 249-267. doi: 10.3934/dcdsb.2010.13.249

## Bifurcation curves in discontinuous maps

 1 University of Urbino, Department of Economics and Quantitative Methods, Via Saffi 42, 61029 Urbino, Italy, Italy 2 Marche Polytechnic University, Department of Economics, Piazzale Martelli 8, 60121 Ancona, Italy

Received  October 2008 Revised  February 2009 Published  December 2009

Several discrete-time dynamic models are ultimately expressed in the form of iterated piecewise linear functions, in one- or two- dimensional spaces. In this paper we study a one-dimensional map made up of three linear pieces which are separated by two discontinuity points, motivated by a dynamic model arising in social sciences. Starting from the bifurcation structure associated with one-dimensional maps with only one discontinuity point, we show how this is modified by the introduction of a second discontinuity point, and we give the analytic expressions of the bifurcation curves of the principal tongues (or tongues of first degree) for the family of maps considered, which depends on five parameters.
Citation: Gian-Italo Bischi, Laura Gardini, Fabio Tramontana. Bifurcation curves in discontinuous maps. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 249-267. doi: 10.3934/dcdsb.2010.13.249
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2020 Impact Factor: 1.327


text:
2015 CMS Winter Meeting

McGill University, December 4 - 7, 2015

Fibrations, Mirror Symmetry and Calabi-Yau Geometry
Org: Charles Doran (Alberta), Andreas Malmendier (Utah State) and Alan Thompson (Waterloo)
[PDF]

LARA ANDERSON, Virginia Tech
Heterotic and F-theory Compactifications and Geometry  [PDF]

We systematically analyze a broad class of dual heterotic and F-theory models that give rise to six- and four-dimensional supergravity theories, and compare geometric constraints on the two sides of the duality. In this talk we will demonstrate that both theories together give new insight into the space of possible effective theories arising from string theory. We will describe explicit new results regarding elliptic and K3-fibration structures and string dualities.

MBOYO ESOLE, Harvard University
Geometric Engineering of SU(3) SU(2) through Elliptic Fibrations: Resolutions and Flops.  [PDF]

I will review the different ways to engineer a SU(2)SU(3) gauge theory using elliptic fibrations and discuss the resulting geometry. In particular, I will present many new non-Kodaira fibers that naturally occur in this setting. I will also discuss the network of flops between different nonsingular elliptic fibrations sharing the same Jacobian (Weierstrass model).

ANTONELLA GRASSI, University of Pennsylavania
Birational Geometry of Higher dimensional elliptic fibrations  [PDF]

I wil present comparison and differences between Calabi Yau threefolds and fourfolds which are elliptically fibered.

ANDREW HARDER, University of Alberta
Tyurin degenerations, K3 fibrations and the Batyrev-Borisov construction  [PDF]

If a Calabi-Yau threefold degenerates to the normal crossings union of a pair of quasi-Fano threefolds, then mirror symmetry predicts that there is a corresponding K3 fibration on the mirror Calabi-Yau threefold. I will show that this prediction is compatible with the Batyrev-Borisov mirror construction, and I will discuss how the resulting K3 fibration reflects the mirror-dual degeneration.

ATSUSHI KANAZAWA, Harvard CMSA
Holomorphic symplectic geometry of the space of Bridgeland stability conditions  [PDF]

For a Calabi-Yau triangulated category D of odd dimension, the space Stab(D) of Bridgeland stability conditions on D is naturally a holomorphic symplectic manifold. I will discuss geometry of Lagrangian submanifolds of Stab(D) in comparison with the Donagi-Markman integrable system. I will also explain our attempt to identify the gauged Kahler moduli space inside Stab(D) when D comes from Calabi-Yau geometry. This is a joint work with Y.-W. Fan and S.-T. Yau.

ALEX MOLNAR, Queen's University
On Calabi-Yau threefolds of CM-type  [PDF]

The first known examples of elliptic curves satisfying the Birch and Swinnerton-Dyer conjecture were elliptic curves with complex multiplication. With this in mind, one may try to study the Beilinson-Bloch conjecture for (rigid) Calabi-Yau threefolds by first examining (rigid) Calabi-Yau threefolds of CM-type.

We will discuss an elementary construction of such threefolds (over the complex numbers) towards a classification up to birational equivalence, as well as the limits of this approach and an expected classification. We will also briefly mention how, specifically, the CM-type allows us to achieve arithmetic results.

SIMON ROSE, Copenhagen University
Quasi-modularity of generalized sum-of-divisors functions  [PDF]

In this talk I will present a generalization of the sum-of-divisors introduce by P. A. MacMahon, and show how they are quasi-modular forms. I will finish with some speculation as to the geometric significance of these functions.

EMANUEL SCHEIDEGGER, Freiburg

EGOR SHELUKHIN, Institute for Advanced Study
Non-trivial Hamiltonian fibrations via K-theory quantization  [PDF]

We show how quantization of families with values in K-theory can detect non-trivial Hamiltonian fibrations, yielding examples that are not detected by previous methods (the characteristic classes of Reznikov for example). We also upgrade a theorem of Spacil on the cohomology-surjectivity of a natural map of classifying spaces by providing it with an "almost" weak retraction. Joint work with Yasha Savelyev.

WASHINGTON TAYLOR, MIT
Classification and enumeration of elliptic Calabi-Yau threefolds and fourfolds  [PDF]

Recent work motivated by physics has led to progress in understanding elliptic Calabi-Yau threefolds and fourfolds, using new mathematical and computational tools for analyzing the geometry of the bases that support such fibrations. This talk will give an introduction to some aspects of this research program, including: the identification (work with D. Morrison) of irreducible geometric structures in the base geometry that facilitate the classification of allowed bases, connections between codimension two singularities and representation theory, Mordell-Weil groups, a systematic approach to enumerating elliptic Calabi-Yau threefolds with large $h_{2,1}$, and a Monte Carlo study of $\sim 10^{50}$ distinct toric threefold bases that support elliptic Calabi-Yau fourfolds. A brief description will also be given of applications to physics including hints at how the observed standard model of particle physics may emerge from "typical" features of Calabi-Yau fourfolds.

URSULA WHITCHER, University of Wisconsin-Eau Claire
Arithmetic Mirror Symmetry and Isogenies  [PDF]

Arithmetic mirror symmetry is a relationship between the number of points on appropriately chosen mirror pairs of Calabi-Yau varieties over finite fields. We investigate whether arithmetic mirror relationships observed for diagonal pencils in weighted projective spaces can be extended to mirror families obtained via the Batyrev-Borisov construction. Our results show that arithmetic mirror symmetry is controlled by an isogeny structure. This talk describes joint work with Christopher Magyar.


text:
Not sure if i've got the trig functions correct

I'm doing A2 Edexcel maths and i keep on forgetting the trig functions so can someone take a look and tell me if i've got it right.

So far:

Sin^2(x)=cos^2(x)=1

so:

sin^2(x)=1-cos^2(x)

cos^2(x)=1-sin^2(x)

this is where i get abit stuck

(sin^2(x)+cos^2(x)=1)/cos^2(x) = tan^2(x)+1=sec^2(x)

(sin^2(x)+cos^2(x)=1)/sin^2(x) = tan^2(x)+1=sec^2(x)

i'm not sure if i've got them wrong or if i'm meant to divide by sin(x) or sin^2(x)

just looked and i dont think i should have divided by sin^2(x) because the powers on sin/cos would be wrong.... i think

just looked and i dont think i should have divided by sin^2(x) because the powers on sin/cos would be wrong.... i think
$$\frac{a^2}{b^2} = (\frac{a}{b})^2, b \neq 0$$

Redbelly98
Staff Emeritus
Homework Helper
(sin^2(x)+cos^2(x)=1)/cos^2(x) = tan^2(x)+1=sec^2(x)

(sin^2(x)+cos^2(x)=1)/sin^2(x) = tan^2(x)+1=sec^2(x)
tan = sin/cos.

1st equation here looks good.
2nd equation: you get a cos/sin term, that is not = tan.

tiny-tim
Homework Helper
I'm doing A2 Edexcel maths and i keep on forgetting the trig functions …
Hi groom03!

hmm … how to remember trigonometric identities … ?

I always find that the safest plan is to write down what I think the formula is, and then multiply by cos2 or sin2 to check it.

For example, if I get confused and think cosec2 + cot2 = 1, then I multiply by sin2 and get 1 + cos2 = sin2 … which it isn't!

tan = sin/cos.

1st equation here looks good.
2nd equation: you get a cos/sin term, that is not = tan.

i keep on getting sin/cos and cos/sin wrong, and i just realised the second equation should equal cosec^2(x).

Integral
Staff Emeritus
Gold Member
I'm doing A2 Edexcel maths and i keep on forgetting the trig functions so can someone take a look and tell me if i've got it right.

So far:

Sin^2(x)=cos^2(x)=1
I am guessing that this is a typo. You surely mean:

$$\sin^2(x) + \cos^2(x) =1$$


text:
# Debugging

• -pg

Generate extra code to write profile information suitable for the analysis program gprof. You must use this option when compiling the source files you want data about, and you must also use it when linking.

• -p

Generate extra code to write profile information suitable for the analysis program prof. You must use this option when compiling the source files you want data about, and you must also use it when linking.


text:
# zbMATH — the first resource for mathematics

## Le Bruyn, Lieven

Compute Distance To:
 Author ID: le-bruyn.lieven Published as: Le Bruyn, L.; Le Bruyn, Lieven
 Documents Indexed: 94 Publications since 1981, including 6 Books
all top 5

#### Co-Authors

 58 single-authored 7 Van den Bergh, Michel 7 van Oystaeyen, Freddy 3 Bocklandt, Raf 2 De Laet, Kevin 2 Jespers, Eric 2 Procesi, Claudio 2 Van de Weyer, Geert 1 Adriaenssens, Jan 1 Alev, Jacques 1 Bauwens, Kristien 1 Belmans, Pieter 1 Bessenrodt, Christine 1 Caenepeel, Stefaan 1 Cvetko-Vah, Karin 1 Dăscălescu, Sorin 1 Hemelaer, Jens 1 Molenberghs, Geert 1 Ooms, Alfons I. 1 Reichstein, Zinovy B. 1 Schofield, Aidan 1 Seelinger, George F. 1 Symens, Stijn 1 Teranishi, Yasuo 1 Verschoren, Alain H. M. J.
all top 5

#### Serials

 17 Journal of Algebra 16 Communications in Algebra 5 Journal of Pure and Applied Algebra 5 Bulletin de la Société Matheḿatique de Belgique. Série B 4 Proceedings of the American Mathematical Society 4 Journal of Algebra and its Applications 3 Transactions of the American Mathematical Society 2 Israel Journal of Mathematics 2 Glasgow Mathematical Journal 2 Mathematische Zeitschrift 2 Proceedings of the London Mathematical Society. Third Series 2 Linear Algebra and its Applications 1 Advances in Mathematics 1 Bulletin des Sciences Mathématiques. Deuxième Série 1 Inventiones Mathematicae 1 Manuscripta Mathematica 1 Memoirs of the American Mathematical Society 1 $$K$$-Theory 1 Publicacions Matemàtiques 1 Nieuw Archief voor Wiskunde. Vierde Serie 1 Algebras and Representation Theory 1 AMA. Algebra Montpellier Announcements 1 International Journal of Mathematics and Computer Science 1 NATO ASI Series. Series C. Mathematical and Physical Sciences 1 Pure and Applied Mathematics (Boca Raton) 1 Journal of Noncommutative Geometry
all top 5

#### Fields

 82 Associative rings and algebras (16-XX) 52 Algebraic geometry (14-XX) 18 Nonassociative rings and algebras (17-XX) 14 Linear and multilinear algebra; matrix theory (15-XX) 12 Commutative algebra (13-XX) 9 Group theory and generalizations (20-XX) 3 Number theory (11-XX) 3 Category theory; homological algebra (18-XX) 2 Field theory and polynomials (12-XX) 2 Differential geometry (53-XX) 1 General and overarching topics; collections (00-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 $$K$$-theory (19-XX) 1 Partial differential equations (35-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Computer science (68-XX) 1 Quantum theory (81-XX)


text:
# Dr. Eric Werner

### HomePublicationsSeminars/TalksBio

Title: Modal Logic of Games

Author: Eric Werner, Ph.D.

Contact: eric.werner@cellnomica.com

Abstract

It is shown that modal reasoning in games has its own logic. The logic of games is a temporal modal logic that also depends on information. For, the axioms that are valid in the temporal modal logic depend on the information conditions that hold in the game. It is shown that von Neumann games with perfect information have a particular modal logic. Other logics of games hold given imperfect information. The limitations of the classical mathematical definition of games are described, and the way to possible extensions of game theory for games with non-von Neumann modal logics is indicated. The results obtained apply to temporal modal reasoning within any open, indeterministic domain where the information about the domain may be imperfect. \end{abstract}

Introduction

Modal reasoning in games has its own logic. The logic of games is information and time dependent. The axioms that are valid in the temporal modal logic depend on the information conditions that obtain in the game. It is shown that von Neumann games with perfect information have a particular modal logic. The logic of games with imperfect information depends on the information conditions that obtain in the game. The limits of the classical mathematical definition of a game are described, and the way to possible extensions of game theory for games with non-von Neumann modal logics is indicated. The results obtained are quite general, and apply to the temporal modal reasoning of any agent with possibly imperfect information about an open, indeterministic domain.

The semantics developed has some novel features not found in traditional semantics of modal logic (e.g., [Kripke 63], [Montague 76]). The semantics is an information based semantics that integrates the concept of an angent's information state. This step brings the sematnics of modal logic closer to the work in communication theory [Werner 88a] and economic theory [von Neumann \& Morgenstern 47], [Myerson 88]. Furthermore, the logics and semantics investigated have a close relationship to the logics used to investigate distributed systems [Halpern 87].

Also of interest is that the accessiblity relation is relativized not just to information but to time. The general properties of temporal accessibility relations are investigated. This makes the semantics adaptible. With minor modifications it can serve as a semantics for tensed modal logics, i.e., logics that contain both tense and modal operators. This allows one to investigate the meaning and logic of expressions like $P \Box \alpha$ = " It was necessary that $\alpha$" or $\diamond F\alpha$ = "It is possible that it will be the case that $\alpha$" [Werner 89a].

We start with a description of the temporal modal language. Then we define information states and the information based semantics. We then look at temporal accessibility relations in general. Such relations are then used to generalize the information-based semantics to what we call the abstract semantics. Next the axioms of information-based temporal modal logic are presented. Completeness and consistency theorems are proven. Finally, the results are applied to game theory.


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# Study of $B^0(\bar{} \to \pi^0pi^0$, $B^\pm \pi^\pm\pi^0$ and $B^\pm \to K^\pm\pi^0$ decays

Abstract : We present a preliminary measurement of the time-dependent CP-violating asymmetry in decays of neutral B mesons to the final states D*-+ pi+- using approximately 178 million BB events recorded by the BaBar experiment at the PEP-II e+e- storage ring. Events containing these decays are selected with a partial reconstruction technique, in which only the high-momentum pi+- from the B decay and the low-momentum pi-+ from the D*-+ decay are used. % We measure the amplitude of the asymmetry to be $-0.041 \pm 0.016 (stat.) \pm 0.010 (syst.)$ and determine bounds on $|\sin(2 \beta + \gamma)|$.
Document type :
Conference papers

http://hal.in2p3.fr/in2p3-00087617
Contributor : Dominique Girod Connect in order to contact the contributor
Submitted on : Wednesday, July 26, 2006 - 8:30:06 AM
Last modification on : Sunday, June 26, 2022 - 11:46:07 AM

### Citation

B. Aubert, T. Abe, G. S. Abrams, T. Adye, Mohamed-Salem Ahmed, et al.. Study of $B^0(\bar{} \to \pi^0pi^0$, $B^\pm \pi^\pm\pi^0$ and $B^\pm \to K^\pm\pi^0$ decays. 32nd International Conference on High Energy Physics (ICHEP 2004), Aug 2004, Beijing, China. ⟨in2p3-00087617⟩

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text:
Last edited by Mezigore
Friday, July 31, 2020 | History

4 edition of On the principle of limiting absorptions found in the catalog.

# On the principle of limiting absorptions

## by D. M. Eidus

Published by Courant Institute of Mathematical Sciences, New York University in New York .
Written in English

Edition Notes

The Physical Object ID Numbers Statement tr. From the Russian by Clyde D. Hill. Contributions Russian, tr. From the, Hill, Clyde D. Pagination 67 p. Number of Pages 67 Open Library OL17971620M

15 of the third edition of Principles of Federal Appropriations Law, in conjunction with GAO, Principles of Federal Appropriations Law: Annual Update to the Third Edition, GAOSP (Washington, D.C.: Mar. ), remain the most currently available material on the topics discussed therein. Both Principles .   Think & Grow Rich chapter summary – Napoleon Hill (Principles Of Think & Grow Rich) The well-known & long time best selling book by Napoleon Hill is a classic that contains lessons for anyone who wants to find their purpose in life. If you want to feel like a success or just feel there has to be something better to life than this, you must read this book.

Hutchinson, Michael. The Case of Windy Lake. Second Story Press, Micheal Hutchinson is a citizen of the Misipawistik Cree Nation in the Treaty 5 territory and is no stranger to the Canadian media. He’s worked as a print reporter for The Calgary Straight and Aboriginal Times and became the host of APTN national news. Somehow in his busy life, he has found time to write two amazing. SUPPORT LARB. The Los Angeles Review of Books is a (c)(3) nonprofit. Help us create the kind of literary community you’ve always dreamed of.

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### On the principle of limiting absorptions by D. M. Eidus Download PDF EPUB FB2

Search the world's most comprehensive index of full-text books. My library. A LIMITING ABSORPTION PRINCIPLE F OR SEP ARA TED DIRA C OPERA TORS WITH WIGNER V ON NEUMANN TYPE POTENTIALS HORST BEHNCKE y AND PETER REJTO 1. In tro duction. Sev eral authors form ulated a limiting absorption principle for sev eral classes of op erators.

F or the ph ysical signi cance of this principle w e refer to the pap er of Eidus [Ei] and. In mathematics, the limiting absorption principle (LAP) is a concept from operator theory and scattering theory that consists of choosing the "correct" resolvent of a linear operator at the essential spectrum based on the behavior of the resolvent near the essential spectrum.

The term is often used to indicate that the resolvent, when considered not in the original space (which is usually the. The limiting absorption principle asserts that if H is a suitable Schrödinger operator, and f lives in a suitable weighted L 2 space (namely On the principle of limiting absorptions book, 1/2 + \sigma}}$$for some$${\sigma > 0}$$), then the functions$${R(\lambda + i \varepsilon) f:= (H - \lambda - i \varepsilon)^{-1} f}$$converge in another weighted L 2 space$${H^{0, -1/2 - \sigma}} to the unique solution u of the Cited by:   Mourre’s result was later improved and generalized in various direc- tions (we mention among many others the papers [10,11,14,19,20]), the most general framework being probably the one exposed in the book [1].

The proofs of the abstract limiting absorption principle all rely on a clever differential inequal- ity in ε on a family of operators Cited by: 44 • Flame Emission -> it measures the radiation emitted by the excited atoms that is related to concentration.

• Atomic Absorption -> it measures the radiation absorbed by the unexcited atoms that are determined. •Atomic absorption depends only upon the number of unexcited atoms, the absorption intensity is not directly affected by the.

Absorption, in wave motion, the transfer of the energy of a wave to matter as the wave passes through it. The energy of an acoustic, electromagnetic, or other wave is proportional to the square of its amplitude—i.e., the maximum displacement or movement of a point on the wave; and, as the wave.

UV spectroscopy is an important tool in analytical chemistry. The other name of UV (Ultra-Violet) spectroscopy is Electronic spectroscopy as it involves the promotion of the electrons from the ground state to the higher energy or excited state.

In this article I will explain the basic principle, working and all the applications of UV spectroscopy. absorptions may be decreased by molecular symmetry, spectrometer limitations, and spectroscopic selection rules.

One selection rule that influences the intensity of infrared absorptions, is th at a change in dipole moment should occur for a vibration to absorb infrared energy. Absorption bands associated with C=O bond stretching areusually very.

1 J. Seader and E. Henley, Separation Process Principles, Wiley,pg. leaving a dryer where solid cellulose acetate fibers, wet with water and acetone, are dried. Acetone is removed by a tray absorber using water as the absorbent. The percentage of acetone removed from the air stream is.

The book An Essay on the Principle of Population was first published anonymously inbut the author was soon identified as Thomas Robert book warned of future difficulties, on an interpretation of the population increasing in geometric progression (so as to double every 25 years) while food production increased in an arithmetic progression, which would leave a difference.

An Essay on the Principle of Population An Essay on the Principle of Population, as it Affects the Future Improvement of Society with Remarks on the Speculations of Mr. Godwin, M. Condorcet, and Other Writers. Thomas Malthus London Printed for J. Johnson, in St. Paul’s Church-Yard Upon first inspection, a typical infrared spectrum can be visually divided into two regions.

The left half, above cm-1, usually contains relatively few peaks, but some very diagnostic information can be foundalkane C-H stretching absorptions just below cm-1 demonstrate the presence of saturated carbons, and signals just above cm-1 demonstrate unsaturation.

• introduce you to the principles underlying accounting • enable you to apply, interpret and explain key accounting techniques • provide a broad understanding of the theory and practice of ﬁnancial and management accounting. The unit is intended both for non-specialist students, and as a foundation for further study in the area.

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I would suggest you. Consequently, NIR absorptions need higher energy than a fundamental, and few overtones can be observed in this region. On the contrary to overtones arising from a group of fundamental absorptions, combinations, appearing because of the allocation of energy between two or more fundamental MIR bands that get excited simultaneously, are largely seen.

Ecological principles are basic assumptions (or beliefs) about ecosystems and how they function that are informed by the ecological concepts.

Ecological principles use ecological concepts (which are understood to be true) to draw key conclusions that can then guide human applications (section 3) aimed at. There are two versions of Thomas Robert Malthus’s Essay on the Principle of Population.

The first, published anonymously inwas so successful that Malthus soon elaborated on it under his real name. * The rewrite, culminating in the sixth edition ofwas a scholarly expansion and generalization of the ing his success with [ ]. Overcoming Limiting Beliefs. Techniques > > Overcoming Limiting Beliefs.

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28 Practical Pharmacology for the Pharmacy Technician Safe and effective drug treatment is not only a function of the physical and chemical prop-erties of drugs, but also a function of how the human body responds to the administration.The book consists of two parts: the theory of absorption and the calculation of absorbers.

Part I covers basic knowledge on diffusion and the theory of mass transfer in binary and multicomponent systems. Significant stress is laid on diffusion theory because this forms the basis for the absorption process. In the next chapters the fundamentals.To mention other works that have helped me get first principles from different fields- NN Taleb’s ‘Incerto’ series consisting of 5 books is an exposition of philosophy, probability-risk.


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# latex math symbols

This default can be changed using the mathtext.default rcParam. Mathematical operators and symbols in Unicode This page was last edited on 8 June 2020, at 16:51 (UTC). . . There are The ctan alreadyprovides a huge list with currently 5913 symbols, which you can download here. . An accent command may precede any symbol to add an accent above it. inside a pair of dollar signs (\$). For example, to write the sum of from to small: The solution is to precede the bracket with \left and \right to inform Simply append the abbreviation of the Trigonometric function after the single backslash (\). This is a small change from regular TeX, where the dollar sign in in the following \imath is used to avoid the extra dot over the i: You can also use a large number of the TeX symbols, as in \infty, You can play around with the commands shown here in the sandbox below and insert your equation into your LaTeX document. . Instead I'm trying to limit this list to the most common math symbols and commands. mathtext also provides a way to use custom fonts for math. In this regard, the package provides \bigpumpkin \greatpumpkin This is Note in the example above Note that Mathtext can use DejaVu Sans (default), DejaVu Serif, the Computer Modern fonts , you could do: Fractions, binomials, and stacked numbers can be created with the See the usetex tutorial for more . stix: STIX (designed to blend well with Times). [3] . patient users only. . You can try testing these Trigonometric functions commands directly on our online LaTeX compilerfor a better understanding. Note that you can set the integral boundaries by using the underscore _ and circumflex ^ symbol as seen below. To make it text, by setting it to regular. \frac{}{}, \binom{}{} and \genfrac{}{}{}{}{}{} commands, a Roman font have shortcuts. . that is not contained in your custom fonts, you can set the rcParam If you want to use a math symbol Of course LaTeX is able to typeset matrices as well. In particular, characters such as: have special meaning outside of math mode in TeX. They can be loosely grouped as follows. Note that you do not need to have TeX installed, since Matplotlib ships I don't want to provide a complete list of LaTeX symbols on this website. fairly tricky to use, and should be considered an experimental feature for Each parameter should be set to a fontconfig font descriptor (as defined in the font command: More conveniently, many commonly used function names that are typeset in obscure symbols in the STIX fonts), Unicode characters can also be used: Here is an example illustrating many of these features in context. follows: Here "s" and "t" are variable in italics font (default), "sin" is in Roman . . sign. . flag. . is a fairly direct adaptation of the layout algorithms in Donald Knuth's information. . Gallery generated by Sphinx-Gallery, $\frac{3}{4} \binom{3}{4} \stackrel{}{}{0}{}{3}{4}$, $\left(\frac{5 - \frac{1}{x}}{4}\right)$, $s(t) = \mathcal{A}\mathrm{sin}(2 \omega t)$. For this purpose LaTeX offers the following environments. in TeX. . the parser that those brackets encompass the entire object. . Columns are separated with ampersand & and rows with a double backslash \\ (the linebreak command). The following table provides a comprehensive list/guide of Trigonometric functions along with their LaTeX command. (from (La)TeX), STIX fonts (with are designed respectively: Note that special care needs to be taken to place parentheses and brackets . You can try testing these Relational symbols/operators commands directly on our online LaTeX compiler for a better understanding.ss.


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Hardcover | $80.00 Short | £55.95 | ISBN: 9780262042215 | 528 pp. | 6 x 9 in | 45 illus.| November 2004 Paperback |$32.00 Short | £22.95 | ISBN: 9780262541770 | 528 pp. | 6 x 9 in | 45 illus.| November 2004

# Transforming Enterprise

The Economic and Social Implications of Information Technology

## Overview

Innovators across all sectors of society are using information and communication technology to reshape economic and social activity. Even after the boom—and despite the bust—the process of structural change continues across organizational boundaries. Transforming Enterprise considers the implications of this change from a balanced, post-bust perspective. Original essays examine first the impact on the economy as a whole, and, in particular, the effect on productivity; next, the role of information technology in creating and using knowledge—especially knowledge that leads to innovation; then, new organizational models, as seen in the interlocking and overlapping networks made possible by the Internet. The authors also analyze structural changes in specific sectors, including the effect of information technology on the automotive industry, demand-driven production and flexible value chains in the personal computer industry, and new models of outsourced manufacturing in the electronics industry. The final essays examine the societal implications of the diverse ways that information technologies are used—across individuals, groups, communities, and nations, considering questions of access and the digital divide.


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# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Hamiltonian square roots of skew-Hamiltonian matrices revisited. (English) Zbl 0978.15010

Denote by ${M}_{n}\left(𝐂\right)$ and ${I}_{n}$ respectively the set of $n×n$ complex matrices and the identity matrix of order $n$. Set $J=\left(\begin{array}{cc}0& {I}_{n}\\ {I}_{n}& 0\end{array}\right)$. A matrix $H\in {M}_{2n}\left(𝐂\right)$ is Hamiltonian (resp. skew-Hamiltonian) if $H=\left(\begin{array}{cc}E& F\\ G& Y\end{array}\right)$ with blocks in ${M}_{n}\left(𝐂\right)$ where ${F}^{T}=F$, ${G}^{T}=G$, $Y=-{E}^{T}$ (resp. ${F}^{T}=-F$, ${G}^{T}=-G$, $Y={E}^{T}$); one has ${\left(JH\right)}^{T}=JH$ (resp. ${\left(JH\right)}^{T}=-JH$). A matrix $S\in {M}_{2n}\left(𝐂\right)$ is symplectic if ${S}^{T}JS=J$.

The author proves that every skew-Hamiltonian matrix can be brought into skew-Hamiltonian Jordan form $\left(\begin{array}{cc}K& 0\\ 0& {K}^{T}\end{array}\right)$ where $K\in {M}_{n}\left(𝐂\right)$ is in complex Jordan form, by a symplectic similarity transformation. This is the complex analog of a similar result of H. Fassbender, D. S. Mackey, N. Mackey, and H. Xu in the case of real matrices [Linear Algebra Appl. 287, No. 1-3, 125-159 (1999; Zbl 0940.15017)]. The author proves also that every skew-Hamiltonian matrix has a Hamiltonian square root, that two similar skew-Hamiltonian matrices $U$, $V$ are symplectic similar, i.e. $U={S}^{-1}VS$, and that every nonsingular matrix $Z\in {M}_{2n}\left(𝐂\right)$ is representable in the forms $WS$ and ${S}^{\text{'}}{W}^{\text{'}}$; here $S$, ${S}^{\text{'}}$ are symplectic and $W$, ${W}^{\text{'}}$ are skew-Hamiltonian.

##### MSC:
 15A24 Matrix equations and identities 15A21 Canonical forms, reductions, classification 15A57 Other types of matrices (MSC2000)


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# soffice.bin on Windows command line [closed]

LibreOffice v4.0.4.2

I have a series of questions about converting documents to PDF with LibreOffice on the command line.

1. Can LibreOffice convert a multi page tiff image into a multi page PDF file?
(edit: No it cannot at this time. Use GhostScript instead.)

I am using the following command:
soffice.bin -headless -convert-to pdf:writer_pdf_Export -outdir files/ files/multi_page.tif

The About Import and Export Filters (help.libreoffice.org/Common/About_Import_and_Export_Filters) page seems to suggest that it can, but I am unable to make it work. The page states "Multipage-TIFFs are allowed when graphics are imported or exported in TIFF format". However, when I try to convert the tiff file to pdf only page 1 is saved. All subsequent pages in the tiff are dropped.

2. Can soffice.bin report an error code and error message for a failed command?
(edit: This has been reported as a bug: fdo#67009)

Given the following valid command with an invalid input file:
soffice.bin -headless -convert-to pdf:writer_pdf_Export -outdir files/ files/I_do_not_exist.xls

LibreOffice silently fails with no indication that anything went wrong. No warning/error diagnostic is printed to STDOUT or STDERR, and the Windows %ERRORLEVEL% variable remains unchanged. Is there an option to enable verbose error reporting for soffice? Is there a better way of doing this on Windows? Am I going about this the wrong way?

3. What is the best way to pass a large number of input files without exceeding the command line length?
(edit: Use file globbing and send batches of files to soffice.bin)

I am batch converting many thousands of files and will certainly run afoul of CMD.exe's 8192 character command line length limit if I tried to list all the files out.

From what I can tell I only have the following two options available:
Call soffice.bin with each individual file one-by-one (easy to implement, but very slow since LibreOffice will startup and shut down for each file)
Call soffice.bin with batches of files (faster than calling it for each individual file, but requires more calculations to ensure the command line character limit is not exceeded).

Are there any other ways to pass input files to soffice.bin?
For example, GhostScript allows you to specify a name of a file with '@' appended to the beginning, '@files.txt', that has all the input files listed on ...

edit retag reopen merge delete

### Closed for the following reason the question is answered, right answer was accepted by Alex Kemp close date 2015-11-03 23:43:36.510646

Sort by » oldest newest most voted

A1. The quote from the referenced help page is in a context relating to Draw / Impress rather than the conversion of graphics to PDF. While this may not matter at a low level it is also worth noting that the help pages are not always as accurate as they could be. To complicate your particular case, there are known differences (e.g., fdo#40186, fdo#42871, and fdo#46026 to indicate just a few) between the headless and UI forms of filters.

Example multi-page TIFFs can be found here (10 pages) and here (2 pages). LO Draw cannot open the former, but it will open the first page of the second graphic. This is probably indicative of limitations in the multi-page TIFF handling ability of Draw and the related graphics filters in LO. The first example file fails silently when attempting to convert to PDF in headless mode (refer A2 below). The second example file comes from bug fdo#63722 and I can confirm that using headless mode to try and convert this particular file to PDF results in only the first page being output.

I would look at alternative tools (e.g., ImageMagick, ghostscript, pdftk, etc.) to handle multi-page TIFFs. In a worst-case scenario (given TIFF variability in production) split them into single-page TIFFs:

\$ convert multipage.tif single%d.tif  # this is using the ImageMagick tool "convert"

...then convert them to single-page PDFs, then combine them. Refer A4 below.

A2. The only related bug report I could find about this issue is fdo#59756 although it deals with a specific bind-to-port failure. The overall failure to respond with an error message is similar. I don't think you are doing anything wrong, but it would not hurt to raise a bug specific to this issue. Matters like the parameter checking example you provide (i.e., missing input file) I would think of as being an EasyHack.

A3. The only thing I can recommend here for Windows is file globbing e.g., soffice.bin -headless -convert-to pdf:writer_pdf_Export -outdir files/ *.odt. You could even use *.* but you run the risk of overwriting output where there are multiple identically named files of different types (extensions). You could work through directories in this manner or script a solution. The exact approach will depend upon the nature of your files, their arrangement ...

more

Thank you for your help. I have gone ahead and opened some bug reports and updated the original question. I will mark this as solved (such as it is).

( 2013-07-17 21:07:54 +0200 )edit

bug reports:

soffice.bin usage prints to modal window instead of STDOUT - fdo#67010

more


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# 3 carbon atoms and 8 hydrogen atoms formula in ghana

A sample of propane, C3H8, contains 10.6 moles of …

For example, propane has a formula unit of {eq}\displaystyle C_3H_8 {/eq}, which means that in every 1 molecule of propane, we need 3 atoms of carbon and 8 atoms of hydrogen. Answer and Explanation:

Alkanes: Molecular and Structural Formulas

Each carbon atom is also attached to sufficient hydrogen atoms to produce a total of four single covalent bonds about itself. Thus octane, an eight‐carbon alkane, has a molecular formula of C 8 H 18 and a structural formula of

Chapter 6 Practice Questions | StudyHippo

1) The chemical formula clearly indies the relationship between the mass of each element in the formula. false 2) Mass is used as a method of counting atoms true 3) One hundred nickels must have the same mass as one hundred pennies. false 4) The numerical

4CO 5CO

2 atoms of oxygen 2 atoms of hydrogen 5 total atoms Be(HCO) 3 2 1 atom of beryllium 2 atoms of hydrogen 2 atoms of carbon 6 atoms of oxygen 11 total atoms Pull compound up. subscript Chemical formula chemical equation Hydroxide Write the chemical OH

Stoichiometry: Calculations with Chemical Formulas and …

Formula Weight (FW) •A formula weight is the sum of the atomic weights for the atoms in a chemical formula. •So, the formula weight of calcium chloride, CaCl 2, would be Ca: 1(40.1 amu) + Cl: 2(35.5 amu) 111.1 amu •Formula weights are generally

The ratio of carbon atoms to hydrogen atoms to …

22/5/2017· The ratio of carbon atoms to hydrogen atoms to oxygen atoms in a molecule of dicyclohexyl maleate is 4 to 6 to 1. What is its molecular formula if its molar mass is 280 g? Chemistry The Mole Concept Empirical and Molecular Formulas

CARBON AND ITS COMPOUNDS.ppt - Google Slides

The atomic nuer of carbon is 6, its electronic arrangement is 2,4, it has 4 valence electrons. It can attain stability by gaining 4 electrons, losing 4 electrons or sharing 4 electrons with other atoms. It does not gain 4 electrons because it is difficult for the 6 protons

C70 fullerene - Wikipedia

C70 fullerene is the fullerene molecule consisting of 70 carbon atoms. It is a cage-like fused-ring structure which reseles a rugby ball, made of 25 hexagons and 12 pentagons, with a carbon atom at the vertices of each polygon and a bond along each polygon edge. A related fullerene molecule, named buckminsterfullerene (C60 fullerene

Iupac Naming And Formulae | Organic Molecules | Siyavula

Reduce the structural representation to the molecular formula There are 4 carbon atoms, 8 hydrogen atoms and 1 oxygen atom so the molecular formula is $$\text{C}_{4}\text{H}_{8}\text{O}$$. (Remeer that there is no structural information given by the molecular

CBSE 9, Chemistry, CBSE- Atoms and Molecules, MCQ

6.022 * 10 23 atoms of C = 24g of carbon IV. 6.022 * 10 23 atoms of H = 1g of hydrogen atoms STUDY MATERIAL FOR CBSE CLASS 9 CHEMISTRY Chapter 1 - Atoms and Molecules Chapter 2 - Structure of Atom Chapter 3 - Is Matter Around Us Pure

How many atoms of hydrogen are in 1 mole of water …

25/5/2015· There are 2 hydrogen atoms in the molecule so that will be twice the constant 1.2044283 x 10^24 atoms. 1 1 Joe 5 years ago 1 mole of water is Avogadro''s nuer of molecules (6.022 * 10^23 molecules). In each molecule of water is 2 atoms of hydrogen 0 1 Lv

Carbon–hydrogen bond - Wikipedia

The carbon-hydrogen bond (C–H bond) is a bond between carbon and hydrogen atoms that can be found in many organic compounds. [1] This bond is a covalent bond meaning that carbon shares its outer valence electrons with up to four hydrogens. This completes

2.4 Chemical Formulas – Chemistry

This formula indies that a molecule of acetic acid contains two carbon atoms, four hydrogen atoms, and two oxygen atoms. The ratio of atoms is 2:4:2. Dividing by the lowest common denominator (2) gives the simplest, whole-nuer ratio of atoms, 1:2:1, so the empirical formula is CH 2 O. Note that a molecular formula is always a whole-nuer multiple of an empirical formula.

Chemical formula - Wikipedia

Structural formula for butane.Examples of other chemical formulae for butane are the empirical formula C 2 H 5, the molecular formula C 4 H 10 and the condensed (or semi-structural) formula CH 3 CH 2 CH 2 CH 3. The simplest types of chemical formulae are called empirical formulas, which use letters and nuers indiing the numerical proportions of atoms of each type.

Carbon Molecule - Edinformatics

Ethane contains two carbon atoms and six hydrogen atoms. In chemistry we use a molecular formula to show how many atoms of each element are present in a molecule. A molecular formula however does not show the structure of the molecule.

Atoms, elements and compounds test questions - KS3 …

11/9/2018· Which of these contains two carbon atoms and six hydrogen atoms? CH 4 C 2 H 6 C 2 H 4 8 Which of these contains four hydrogen atoms? CH 3 OH H 2 O H 2 O 2 9 How many atoms …

Formula | Article about formula by The Free Dictionary

The formula of a well-known compound, water, is H 2 O. Water is made up of molecules, and the formula shows that each molecule consists of two atoms of hydrogen, H, bonded to an atom of oxygen, O. The subscript 2 indies that there are two atoms of hydrogen in the molecule; where no subscript appears, as after the O, the subscript 1 is implied.

Lakhmir Singh Chemistry Class 10 Solutions Carbon and …

(c) By how many carbon atoms and hydrogen atoms do any two adjacent homologues differ ? Solution : (a) (i) C 4 H 8 (ii) C 3 H 8 (iii) C 3 H 4 (b) 14 u (c) 1 carbon atom and 2 hydrogen atoms i.e. a CH 2 group. Lakhmir Singh Chemistry Class 10 Solutions

CHAPTER 8 Chemical Equations and Reactions

17/8/2014· carbon or hydrogen atoms. Usually, the elements hydrogen and oxygen are balanced only after balancing all other elements in an equation. (You will read more about the rules of balancing equations later in the chapter.) Thus, we begin by counting carbon atoms.

EXAMPLE EXERCISE 4.1 Change of Physical State

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text:
How does LP token accrue yield?

The yield tokenization and yield accrual mechanism for LP tokens are similar to that of Compound's c-tokens.

In a typical X * Y = K AMM like Sushiswap / Uniswap, k represents the size of the pool. Upon liquidity addition, the liquidity provider receives LP tokens to represent his share of the pool. The share of the pool is quantified by sqrt(K), a unit called "Liquidity".

$Liquidity = sqrt(qtyTokenX*qtyTokenY)$

LP tokens represent the amount of Liquidity that an LP is entitled to. Learn more.

Swap fee is incurred on the transaction sender whenever a swap occurs. The swap fee added to the pool increases the size of the pool, increasing the value of Liquidity. Hence, when swaps occur, the Liquidity owned by each LP increases.

In other words, the amount of Liquidity per unit LP token increases as swaps occur. (1 LP tokens will be worth more unit of Liquidity)

LP Token Yield Tokenization

1 YT-LP represents the yield from 1 unit of Liquidity.

Yield for LP position is equivalent to the swap fees generated in a fixed time period. As swap fee incurred is added to the liquidity pool, yield accrued will be in the form of LP tokens.

1 OT represents the ownership of 1 unit of Liquidity (redeemable for 1 unit of Liquidity at expiry, NOT number of LP tokens).

To simply illustrate this, when LP token is deposited into Pendle, the amount of Liquidity it represents at that point in time is represented by OT. As swap happens, YT holders receive the extra Liquidity that the LP token accrues.

Example

The following example is to illustrate the concept explained above.

1st Jan 2022:

Let's say that at the very start, the PENDLE/ETH pool on Sushiswap has 10 ETH and 1000 PENDLE. We'll subsequently refer to this pool as PE.

• Amount of liquidity of the pool = sqrt(10*1000) = 100 liquidity

• Inititially, the total amount of LPs is also exactly 100 LPs

• 1 LP = 1 liquidity = 0.1 ETH + 10 PENDLE

1st July 2022

After some swaps, the PE pool could have 11 ETH and 1100 PENDLE

• Amount of liquidity of the pool now = sqrt(11*1100) = 110 liquidity

• The total amount of LPs is still 100 LPs (we assume no-one has entered or left the pool, for the sake of simplicity)

• 1 LP = 1.1 liquidity = 0.11 ETH + 11 PENDLE

Let's say user A tokenizes 10 PE SLP (=11 liquidity) in Pendle now, A will receive 11 OT-PE-Dec2022 and 11 YT-PE-Dec2022. A transfers the 11 YT to B

• By holding 11 OT-PE-Dec2022, A is entitled to receive 11 liquidity after the expiry (which will be paid in terms of LP tokens)

• Currently, 11 liquidity is equivalent to 1.1ETH + 110 PENDLE (10% of the Sushiswap pool)

• By holding 11 YT-PE-Dec2022, B is entitled to receive swap fees (in terms of LP tokens) from the moment B started holding the YT, until the expiry. If B transfers away the YT at some point, B will only receive the swap fees up to that point

1st Jan 2023:

After more swapping on Sushiswap happened, the PE pool could have 6 ETH and 2400 PENDLE

• Amount of liquidity of the pool = sqrt(6 * 2400) = 120

• The total amount of LPs is still 100 LPs

• 1 LP = 1.2 liquidity = 0.06 ETH + 24 PENDLE

A can now redeem his 11 OT-PE-Dec2022:

• 11 liquidity is quivalent to 11/120 = 9.17% of the Sushiswap pool = 0.55 ETH + 220 PENDLE

• 11 liquidity is equivalent to 9.17 LP now

• As such, A will get back 9.17 LP token

B has held 11 YT-PE-Dec2022 from 1st July 2022 to 1st Jan 2023, and will get the swap fees on 11 liquidity for this whole period:

• 1st July 2022: 11 liquidity was equivalent to 10 LPs

• 1st Jan 2023: 10 LPs now will be worth 10*1.2 = 12 liquidity. Therefore, the swap fees gained was 12-11 = 1 liquidity. Exchanged into LPs, this is equivalent to 1/12 = 0.83 LPs

We can see that 9.17 + 0.83 = 10. Essentially, the ownership of 10 LPs on 1st Jul 2022 was split into the ownership of the underlying liquidity for OT, and the swap fees for YT.


text:
# [Unit conversion] From r/min to period (msec)

1. Nov 24, 2012

### JJ91

1. The problem statement, all variables and given/known data
I would like to know how a 4000 r/min is converted to period of t=2.5mSec

2. Relevant equations
Speed of motor = 4000 r/min
Build-up time of current = 2.5msec
Frequency = 60Hz

3. The attempt at a solution
$\frac{1}{4000}$=0.25msec not 2.5msec

OR
Assuming it means 4000RPM which is equivalent to 419 rad/sec we obtain:
$\frac{1}{419}$=2.39msec not 2.5msec

OR
If frequency is 60 Hz, and 1-phase is conducting for 60° we can take period as:
t=$\frac{1}{f}$=$\frac{1}{60}$=16.67msec
Conducting for $\frac{1}{3}$ of the cycle (60°), t = 5.55msec

Where have I made a mistake ?

2. Nov 24, 2012

### Staff: Mentor

If T = 2.5ms, then the frequency is 400Hz. There is something wrong with your initial statement. Where did it come from?

3. Nov 24, 2012

### JJ91

It comes from "Electric Machinery" by Mr. Fitzgerald

At the very bottom of the screen a period, T, of build-up of current is defined as 2.5msec.

4. Nov 24, 2012

### SteamKing

Staff Emeritus
For N = 4000 rpm, then n = 4000/60 rps = 66.67 rev/s

1 rev = 2 pi radians

omega = 2 pi * 66.67 = 133.33 pi radians/s

T = 1/omega = .0024 s = 2.4 msec

5. Nov 24, 2012

### SteamKing

Staff Emeritus
Disregard previous post:

For N = 4000 rpm, then frequency f = 4000 / 60 = 66.67 cycles / s

T = 1/f = 0.015 s = 1.5 msec

6. Nov 25, 2012

### JJ91

All right, thank you very much. I will use the very first formula, still it isn't exact 2.5msec.

7. Nov 25, 2012

### Staff: Mentor

The solution gives the expression:

$\theta_m = -\frac{\pi}{3} + \omega_m t$

Plug in your value for $\omega_m$ and solve for t that makes $\theta_m = 0$.

8. Nov 25, 2012

### JJ91

This is the final correct solution.

Thanks, thread can be now closed.


text:
Abstract

MEMS are miniaturized sensors or actuators and are essential to enabling “smart systems” to interact with their physical environment. These devices add the “ears,” “eyes,” “noses,” and “touch” to these systems. The system-level modeling of MEMS requires considering not only the multi-physical behavior of these devices but also their electronic readout circuitry and packaging. This chapter describes a methodology for MEMS system-level design and its implementation in commercially available software. We introduce the Coventor MEMS+® environment for creating system-aware MEMS models, an approach based on a library of 3-D, high-order parametric finite elements. Model-order reduction techniques are employed to reduce the complexity of the multi degree-of-freedom models, to speed up simulation time and to provide a path for designing MEMS together with electronic hardware. The influence of the package surrounding the MEMS device can be simulated by combining traditional finite element simulations with the new methodology of MEMS+. Finally, we discuss the virtual co-development of embedded software and MEMS hardware.

Citation

  @Inbook{Schröpfer2016,
author="Schr{\"o}pfer, Gerold
and Lorenz, Gunar
and Krust, Arnaud
and Vernay, Beno{\^i}t
and Breit, Stephen
and Mehdaoui, Alexandre
and Sanginario, Alessandro",
editor="Bombieri, Nicola
and Poncino, Massimo
and Pravadelli, Graziano",
title="MEMS System-Level Modeling and Simulation in Smart Systems",
bookTitle="Smart Systems Integration and Simulation",
year="2016",
publisher="Springer International Publishing",
address="Cham",
pages="145--168",
isbn="978-3-319-27392-1",
doi="10.1007/978-3-319-27392-1_6",
url="http://dx.doi.org/10.1007/978-3-319-27392-1_6"
}

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text:
The mutate_msd function designs the necessary set of primers for the desired mutations. Note that you can also select TGG in saturation mutagenesis to apply the 22c trick.

mutate_msd(
input_sequence,
codon = "NDT",
prefix = "TT",
restriction_enzyme = "GGTCTC",
suffix = "A",
vector = c("AATG", "AAGC"),
replacements,
replacement_range = 3,
binding_min_length = 4,
binding_max_length = 9,
target_temp = 60,
fragment_min_size = 100
)

## Arguments

input_sequence The sequence which should be modified. This is an object of type character containing the sequence. The desired type of MSD mutation [default: NDT] Additional nucleobases in 5' position of the recognition site [default: TT] Recognition site sequence of the respective restriction enzyme [default: GGTCTC] Spacer nucleotides matching the cleavage pattern of the enzyme [default: A] Four basepair overhangs complementary to the created overhangs in the acceptor vector [default: c("AATG", "AAGC")] The desired substitutions. Can be a numeric vector or a list of character vectors with position and codon. See vignette("MSD3", package = "goldenmutagenesis") for examples. Maximum distance in amino acid residues between two randomization sites to be incoporated into a single primer (reverse, end of the fragment) - has a cascading effect for following mutations [default: 3] The minimal threshold value of the length of the template binding sequence in amino acid residues [default: 4] Maximal length of the binding sequence in amino acid residues [default: 9] Melting temperature of the binding sequence in print('\u00B0')C [default: 60] Minimal size of a generated gene fragment in base pairs [default 100]

## Value

An object of class Primerset with the designed Primers.

## Examples

#Load the setup of the MSD vignette and design the primers
data(MSD_BsaI_setup_lv2)
print(mutations)#> [1] 137 143 147 232 234print(recognition_site_bsai)#> [1] "GGTCTC"primers<-mutate_msd(input_sequence, prefix="TT" ,
restriction_enzyme=recognition_site_bsai, suffix="A",
vector=c("AATG", "AAGC"), replacements=mutations, replacement_range=5,
binding_min_length=4 , binding_max_length=9, target_temp=60,
fragment_min_size=60 )