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ON STATISTICAL APPROXIMATION PROPERTIES OF MODIFIED q-BERNSTEIN-SCHURER OPERATORS

Title & Authors
ON STATISTICAL APPROXIMATION PROPERTIES OF MODIFIED q-BERNSTEIN-SCHURER OPERATORS
Ren, Mei-Ying; Zeng, Xiao-Ming;

Abstract
In this paper, a kind of modified $\small{q}$-Bernstein-Schurer operators is introduced. The Korovkin type statistical approximation property of these operators is investigated. Then the rates of statistical convergence of these operators are also studied by means of modulus of continuity and the help of functions of the Lipschitz class. Furthermore, a Voronovskaja type result for these operators is given.
Keywords
modified q-Bernstein-Schurer operators;statistical approximation property;modulus of continuity;rate of statistical convergence;Voronovskaja type result;
Language
English
Cited by
1.
Modified (p,q)-Bernstein-Schurer operators and their approximation properties, Cogent Mathematics, 2016, 3, 1
2.
Approximation Properties For Modifiedq-Bernstein-Kantorovich Operators, Numerical Functional Analysis and Optimization, 2015, 36, 9, 1178
3.
On generalized integral Bernstein operators based on q-integers, Applied Mathematics and Computation, 2014, 242, 931
4.
On (p, q)-analogue of modified Bernstein–Schurer operators for functions of one and two variables, Journal of Applied Mathematics and Computing, 2017, 54, 1-2, 1
5.
Some approximation properties of a Durrmeyer variant ofq-Bernstein-Schurer operators, Mathematical Methods in the Applied Sciences, 2016, 39, 18, 5636
6.
Stancu–Schurer–Kantorovich operators based on q-integers, Applied Mathematics and Computation, 2015, 259, 896
7.
Kantorovich variant of a new kind of q -Bernstein-Schurer operators, Mathematical Methods in the Applied Sciences, 2017
8.
qBernstein–Schurer–Durrmeyer type operators for functions of one and two variables, Applied Mathematics and Computation, 2016, 275, 372
9.
Approximation properties of q-Kantorovich-Stancu operator, Journal of Inequalities and Applications, 2015, 2015, 1
10.
A new kind of Bernstein-Schurer-Stancu-Kantorovich-type operators based on q-integers, Journal of Inequalities and Applications, 2017, 2017, 1
11.
q-Bernstein-Schurer-Kantorovich type operators, Bollettino dell'Unione Matematica Italiana, 2015, 8, 3, 169
12.
Approximation properties for King's type modifiedq-Bernstein-Kantorovich operators, Mathematical Methods in the Applied Sciences, 2015, 38, 18, 5242
13.
Weighted statistical convergence based on generalized difference operator involving (p,q)-gamma function and its applications to approximation theorems, Journal of Mathematical Analysis and Applications, 2017, 448, 2, 1633
14.
Approximation of Schurer type q-Bernstein-Kantorovich operators, Journal of Inequalities and Applications, 2015, 2015, 1
References
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O. Dalmanoglu and O. Dogru, On statistical approximation properties of Kantorovich type q-Bernstein operators, Math. Comput. Modelling 52 (2010), no. 5-6, 760-771.

2.
O. Dogru and K. Kanat, On statistical approximation properties of the Kantorovich type Lupas operators, Math. Comput. Modelling 55 (2012), no. 3-4, 1610-1621.

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O. Dogru, On statistical approximation properties of Stancu type bivariate generalization of q-Balas-Szabados operators in: Numerical analysis and approximation theory, 179-194, Casa Cartii de stiinta, Cluj-Napoca, 2006.

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S. Ersan and O. Dogru, Statistical approximation properties of q-Bleimann, Butzer and Hahn operators, Math. Comput. Modelling 49 (2009), no. 7-8, 1595-1606.

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H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244.

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V. Gupta and C. Radu, Statistical approximation properties of q-Baskokov-Kantorovich operators, Cent. Eur. J. Math. 7 (2009), no. 4, 809-818.

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G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its applications, Cambridge University press, Cambridge, UK, Vol. 35, 1990.

8.
A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32 (2002), no. 1, 129-138.

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V. G. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002.

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C. V. Muraru, Note on q-Bernstein-Schurer operators, Stud. Univ. Babes-Bolyai Math. 56 (2011), no. 2, 480-495.

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I. Niven, H. S. Zuckerman, and H. Montgomery, An Introduction to the Theory Numbers, 5th ed. Wiley, New York, 1991.

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M. Orkcu and O. Dogru, Weighted statistical approximation by Kantorovich type q-Szasz-Mirakjan operators, Appl. Math. Comput. 217 (2011), no. 20, 7913-7919.

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G. M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997), no. 1-4, 511-518.

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F. Schurer, Linear positive operators in approximation theory, Math. Inst. Techn. Univ. Delft: Report, 1962.

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polyvalm

Matrix polynomial evaluation

Syntax

``Y = polyvalm(p,X)``

Description

example

````Y = polyvalm(p,X)` returns the evaluation of polynomial `p` in a matrix sense. This evaluation is the same as substituting matrix `X` in the polynomial, `p`.```

Examples

collapse all

Find the characteristic polynomial of a Pascal Matrix of order 4.

`X = pascal(4)`
```X = 1 1 1 1 1 2 3 4 1 3 6 10 1 4 10 20 ```
`p = poly(X)`
```p = 1.0000 -29.0000 72.0000 -29.0000 1.0000 ```

The characteristic polynomial is

Pascal matrices have the property that the vector of coefficients of the characteristic polynomial is the same forward and backward (palindromic).

Substitute the matrix, `X`, into the characteristic equation, `p`. The result is very close to being a zero matrix. This example is an instance of the Cayley-Hamilton theorem, where a matrix satisfies its own characteristic equation.

`Y = polyvalm(p,X)`
```Y = 1.0e-10 * -0.0013 -0.0063 -0.0104 -0.0241 -0.0048 -0.0217 -0.0358 -0.0795 -0.0114 -0.0510 -0.0818 -0.1805 -0.0228 -0.0970 -0.1553 -0.3396 ```

Input Arguments

collapse all

Polynomial coefficients, specified as a vector. For example, the vector `[1 0 1]` represents the polynomial ${x}^{2}+1$, and the vector `[3.13 -2.21 5.99]` represents the polynomial $3.13{x}^{2}-2.21x+5.99$.

Data Types: `single` | `double`
Complex Number Support: Yes

Input matrix, specified as a square matrix.

Data Types: `single` | `double`
Complex Number Support: Yes

Output Arguments

collapse all

Output polynomial coefficients, returned as a row vector.


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The Dirichlet Convolution μ * 1 = ꙇ

# The Dirichlet Convolution μ * 1 = ꙇ

Recall from The Möbius Function page that the Möbius function is an arithmetic function defined by:

(1)
\begin{align} \quad \mu(n) = \left\{\begin{matrix}0 & \mathrm{if} \: n \: \mathrm{is \: NOT \: square free.}\\ (-1)^{\omega (n)} & \mathrm{if} \: n \: \mathrm{is \: square \: free.} \end{matrix}\right. \end{align}

We proved that $\mu$ is a multiplicative function. We will now prove a very important Dirichlet convolution, which says the Dirichlet convolution with $\mu$ and $1$ is the $\iota$ function.

 Proposition 1: $\mu * 1 = \iota$.
• Proof: Since $\mu$ and $1$ are both multiplicative functions, their Dirichlet convolution $\mu * 1$ is also multiplicative. So we only need to show that $\mu * 1$ and $\iota$ agree on prime powers. Let $p$ be a prime and let $k \in \mathbb{N}$. Then:
(2)
\begin{align} \quad \mu(1) = 1 \end{align}
• And:
(3)
\begin{align} \quad (\mu * 1)(p^k) &= \sum_{d|p^k} \mu (d) \\ &= \mu(1) + \mu(p) + \mu(p^2) + ... + \mu(p^k) \\ &= 1 + (-1) + 0 + ... + 0 \\ &= 0 \end{align}
• Therefore we see that $(\mu * 1)(n) = 1$ if $n = 1$ and $(\mu *1)(n) = 0$ if $n \geq 1$. Hence:
(4)


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Article Contents
Article Contents

# Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model

• In this paper, we consider a Kermack-McKendrick epidemic model with nonlocal dispersal. We find that the existence and nonexistence of traveling wave solutions are determined by the reproduction number. To prove the existence of nontrivial traveling wave solutions, we construct an invariant cone in a bounded domain with initial functions being defined on, and apply Schauder's fixed point theorem as well as limiting argument. Here, the compactness of the support set of dispersal kernel is needed when passing to an unbounded domain in the proof. Moreover, the nonexistence of traveling wave solutions is obtained by Laplace transform if the speed is less than the critical velocity.
Mathematics Subject Classification: 35K57, 37C65, 92D30.

 Citation:

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Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.doi: 10.1016/j.jde.2007.11.002. [7] J. Coville, J. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.doi: 10.1137/060676854. [8] J. Coville and L. Dupaigne, On a nonlocal reaction diffusion equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755.doi: 10.1017/S0308210504000721. [9] P. De Mottoni, E. Orlandi and A. Tesei, Asymptotic behavior for a system describing epidemics with migration and spatial spread of infection, Nonlinear Anal., 3 (1979), 663-675.doi: 10.1016/0362-546X(79)90095-6. [10] A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459-482.doi: 10.1017/S0308210507000455. [11] A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structure epidemic models, Arch. Ration. Mech. Anal., 195 (2010), 311-331.doi: 10.1007/s00205-008-0203-8. [12] P. Fife, An integrodifferential analog of semilinear parabolic PDEs, in "Partial Differential Equations and Applications" in "Lect. Notes" Pure Appl. Math., 177, Dekker, New York, (1996), 137-145. [13] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in "Trends in Nonlinear Analysis," Springer, Berlin, (2003), 153-191. [14] R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369.doi: 10.1111/j.1469-1809.1937.tb02153.x. [15] J. García-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion, Commun. Pure Appl. Anal., 8 (2009), 2037-2053.doi: 10.3934/cpaa.2009.8.2037. [16] V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. J. Appl. Math., 17 (2006), 221-232.doi: 10.1017/S0956792506006462. [17] V. Hutson, S. Martinez, K. Mischailow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.doi: 10.1007/s00285-003-0210-1. [18] V. Hutson, W. Shen and G. T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain J. Math., 38 (2008), 1147-1175.doi: 10.1216/RMJ-2008-38-4-1147. [19] Y. Hosono and B. Ilyas, Travelling waves for a simple diffusive epidemic model, Math. Model Meth. Appl. Sci., 5 (1995), 935-966.doi: 10.1142/S0218202595000504. [20] C. Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596.doi: 10.3934/dcds.2010.26.551. [21] C.-Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047-2072.doi: 10.3934/dcdsb.2012.17.2047. [22] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London Ser. B, 115 (1927), 700-721.doi: 10.1098/rspa.1927.0118. [23] W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinerity, 19 (2006), 1253-1273.doi: 10.1088/0951-7715/19/6/003. [24] W.-T. Li, Y.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real Word Appl., 11 (2010), 2302-2313.doi: 10.1016/j.nonrwa.2009.07.005. [25] G. Lin, W.-T. Li and M.-J. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414.doi: 10.3934/dcdsb.2010.13.393. [26] J. D. Murray, "Mathematical Biology, II, Spatial Models and Biomedical Applications," Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. [27] S. Ma, Travelling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314.doi: 10.1006/jdeq.2000.3846. [28] J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.doi: 10.1016/S0025-5564(03)00041-5. [29] S. Pan, Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity, J. Math. Anal. Appl., 346 (2008), 415-424.doi: 10.1016/j.jmaa.2008.05.057. [30] S. Pan, W.-T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.doi: 10.1007/s00033-007-7005-y. [31] S. Pan, W.-T. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Anal., 72 (2010), 3150-3158.doi: 10.1016/j.na.2009.12.008. [32] K. Schumacher, Travelling-front solutions for integro-differential equations.I., J. Reine Angew. Math., 316 (1980), 54-70.doi: 10.1515/crll.1980.316.54. [33] W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 15 (2010), 747-795.doi: 10.1016/j.jde.2010.04.012. [34] Y.-J. Sun, W.-T. Li and Z.-C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011), 551-581.doi: 10.1016/j.jde.2011.04.020. [35] Y.-J. Sun, W.-T. Li and Z.-C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity, Nonlinear Anal., 74 (2011), 814-826.doi: 10.1016/j.na.2010.09.032. [36] X. Wang, Metastability and stability of patterns in a convolution model for phase transitions, J. Differential Equations, 183 (2002), 434-461.doi: 10.1006/jdeq.2001.4129. [37] Z.-C. Wang, W.-T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Diff. Eqns., 20 (2008), 573-607.doi: 10.1007/s10884-008-9103-8. [38] Z.-C. Wang, W.-T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc., 361 (2009), 2047-2084.doi: 10.1090/S0002-9947-08-04694-1. [39] Z.-C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A, 466 (2010), 237-261.doi: 10.1098/rspa.2009.0377. [40] Z.-C. Wang and J. Wu, Traveling waves in a bio-reactor model with stage-structure, J. Math. Anal. Appl., 385 (2012), 683-692.doi: 10.1016/j.jmaa.2011.06.084. [41] D. V. Widder, "The Laplace Transform," Princeton University Press, Princeton, NJ, 1941. [42] H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925-953.doi: 10.2977/prims/1260476648. [43] H. Yagisita, Existence of traveling waves for a nonlocal monostable equation: an abstract approach, Publ. Res. Inst. Math. Sci., 45 (2009), 955-979.doi: 10.2977/prims/1260476649. [44] L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neural networks, J. Differential Equations, 197 (2004), 162-196.doi: 10.1016/S0022-0396(03)00170-0. [45] G.-B. Zhang, W.-T. Li and Z.-C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations, 252 (2012), 5096-5124.doi: 10.1016/j.jde.2012.01.014.


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 Transseries, nest-series, and other exotic series representations for tetration mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 11/28/2009, 06:50 AM (This post was last modified: 11/28/2009, 07:42 AM by mike3.) Hmm. My hypothesis that the double-sum, or at least the exp-series, can only provide continuum sums of functions for "double-exponential type" or less seems to be wrong. Indeed, it seems that if an exp-series exists and converges for a function, then its continuum sum does too. Consider the triple-exponential function $f(x) = \exp^3(x) = e^{e^{e^x}}$. This can be expressed as an exp-series $e^{e^{e^x}} = \sum_{n=0}^{\infty} a_n e^{nx}$ where $a_n$ are the coefficients of the Taylor series of $e^{e^x}$ at $x = 0$ (MacLaurin series). Taking the continuum sum gives, \begin{align}\sum_{n=0}^{x-1} e^{e^{e^n}} &= a_0 x + \sum_{n=1}^{\infty} \frac{a_n}{e^{n} - 1} \left(e^{nx} - 1\right) \\ &= \left(a_0 - \sum_{n=1}^{\infty} \frac{a_n}{e^{n} - 1}\right) x + \sum_{n=1}^{\infty} \frac{a_n}{e^{n} - 1} e^{nx}\end{align} As the coefficients of both sums are smaller than those of the original sum (because $e^{n} - 1 > 0$ for all $n > 0$), if the original series converges at 0 and at the point $x$, so does this (see series comparison test). Since we have an expression for $e^{e^{e^x}}$, we have achieved its continuum sum and the proof (or disproof, insofar as my original hypothesis that such a series could not yield a continuum sum of something faster than a double exponential, but as a proof this proves even more, namely that any convergent exp-series' continuum sum also converges, unlike the case with Taylor series summed via direct application of Faulhaber's formula) is complete. As exp-series look to be a special case of nested series, this suggests even 2 layers of nesting may be able to represent tetration and continuum-sum it, though there's still no proof for that. The special case of exp-series themselves do not appear useful for doing tetration with Ansus' formula, however, for two reasons: any function constructed with them is $2\pi i$-periodic, yet tetration seems not to be given pretty much every "good" extension there is -- though they may be able to express tetration for the base $b = e^{e^{1-e}}$ whose regular tetration is periodic with the required period (and so the exp-series can be recovered via the Fourier series), but a single base isn't very useful. And, we can't even represent $f(x) = x$ as an exp-series, thus we can't even continuum-sum one exp-series to another exp-series so this is not very useful insofar as trying to iteratively apply Ansus' formula to generate tetrationals goes! « Next Oldest | Next Newest »

 Messages In This Thread Transseries, nest-series, and other exotic series representations for tetration - by mike3 - 11/26/2009, 09:46 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by Daniel - 11/26/2009, 03:57 PM RE: Transseries, nest-series, and other exotic series representations for tetration - by bo198214 - 11/26/2009, 04:42 PM RE: Transseries, nest-series, and other exotic series representations for tetration - by Daniel - 11/29/2009, 09:09 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by bo198214 - 11/29/2009, 09:38 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by Daniel - 12/01/2009, 02:56 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by bo198214 - 12/01/2009, 09:08 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by tommy1729 - 12/01/2009, 10:22 PM RE: Transseries, nest-series, and other exotic series representations for tetration - by mike3 - 11/27/2009, 01:29 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by andydude - 11/28/2009, 04:56 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by mike3 - 11/28/2009, 06:36 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by mike3 - 11/28/2009, 06:50 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by kobi_78 - 12/14/2009, 07:17 PM

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# Simple question.

1. Jan 19, 2005

### Erin_Sharpe

I have no idea where to begin for this one:

How do you dtermine if sulfate ions are present in tap water?

Thanks!

2. Jan 19, 2005

### chem_tr

Barium sulfate has a very low solubility in water, so concentrated sample of tap water may be tested with an excess of barium ions, say, barium nitrate. An additional test which can be done after precipitation occurs is that barium sulfate can be dissolved in hot concentrated sulfuric acid, but this is very dangerous. Precipitation might be enough for simple cases.

Sulfate ion is IR-active, so an infrared of the sample without considering water peaks might also be helpful; a peak system belonging to S=O vibration occurs at around 1000 cm-1.

3. Jan 20, 2005

### poolwin2001

$(CH_{3}COO)_{2}SO_{4}$is also insoluble.So if on addn of a little lead acetate,a white ppt is formed,it indicates presence of sulfate(sulphate in my place) ions .

Last edited: Jan 20, 2005
4. Jan 20, 2005

### poolwin2001

A cheecky answer will be by spectroscopy.

5. Jan 20, 2005

### chem_tr

Poolwin2001, are you sure the structure you wrote your first post is not PbSO4? Barium sulfate is less soluble than lead sulfate, but the latter is also okay by the way.

6. Jan 20, 2005

### poolwin2001

Oops.I guess you are right .It should be PbSO4.
Sorry for the error erin.
Thanks for correcting it.

7. Jan 21, 2005

### archie

SO4 ion detection

The simplest and probably cheapest way to determine the presence of sulphate ion in the water is by using Barium Chloride soln. ( BaCl2 is preferrable than Barium Nitrate because it's more water soluble). Dissolve about 5 to 10 gram of BaCl2.2H2O in 100 ml distilled water. To test the water, just add several drops of BaCl2 soln. If there's white precipitate and cloudy appearance then big chance that your water contains sulphate ion.
Other alternative is by using Lead (II) Nitrate or Silver Nitrate, but these two as not as reliable as BaCl2 because the solubility of the sulphate of lead and silver is greater than BaSO4.


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DIODES

# Diodes

• DIACs
• LEDs
• PIN Diodes
• TVS Diodes
• ## Overview

Diodes are passive semiconductor components consisting of a single P-N junction. Their main property is that they only allow conduction of current in one direction, which makes them a useful component in many electronic circuits.

## Parameters

These are some of the important parameters shared by all types of diode (in a rough order of importance):

ParameterSymbolDescription
Maximum Continuous Forward Current$$I_{f(cont)}$$The maximum continuous current the diode can withstain, usually limited by overheating. Typically 20-500V.
Forward Voltage$$V_f$$The forward voltage drop, usually rated at maximum continuous current ($$I_f$$ ). An ideal diode would have no forward voltage drop. Schottky diodes have the lowest forward voltage drop of any diode. Generally, the smaller the forward voltage drop, the larger the reverse-leakage. The higher the temperature, the smaller the forward voltage drop. Typically 0.3-1.2V.
Reverse-leakage Current$$I_R$$The leakage current when the diode is reverse-biased at the stand-off voltage. An ideal diode would have no reverse-leakage current. Generally, the smaller the reverse-leakage current, the larger the forward voltage drop. The higher the temperature, the higher the reverse-leakage current. Typically 10nA-1mA.
Peak-surge forward current$$I_{FSM}$$The forward current the diode can handle for a small amount of time. The exact time depends on the standard used to calculate this value (usually JEDEC). The is normally so you can determine the diode can handle inrush current/inductive energy pulses of a particular circuit.

Some other important properties of diodes are their ability to prevent conduction until a certain, configurable breakdown voltage, and a conduction current that is related to the square of the voltage across it. On schematics, they have the designator “D”, and the following symbol (for a general diode).

Diodes come in many component packages, one of the most common being the through-hole DO-41 package. They also come in standard SMD packages. It is a good idea to add polarity marks to the silkscreen layer on diode footprints. The picture below shows polarity marks being added to a diode with a 0603 footprint.

## Polarity

Most diodes have their polarity marked with a single line near the cathode (the “more negative” end when conducting current). They will let current flow from anode to cathode but not in the other direction.

## Schottky Diodes

Schottky diodes are special diodes that have a lower voltage drop than standard diodes (typically 0.3V instead of 0.7V). They are used in applications where the input voltage is small, and in high power applications in where the power consumption of the diode needs to be kept to a minimum. The standard schematic symbol for a schottky diode is shown below (note the curls on the bar, which differs it from the standard diode symbol.

## Photo Diodes

Note these are not be confused with photo-transistors, which are similar, but technically not photo diodes. Photo diodes have a faster response time than photo-transistors.

## Avalanche Photo-diodes (APDs)

Avalanche photo-diodes (APDs) are constructed in a similar manner to PIN diodes. The major difference is that they are operated with a much larger reverse voltage (100-200V for silicon based ones). This causes the avalanche effect (impact ionization) whenever photons strike the sensor, giving a current-gain of around 100. The current gain is roughly proportional to the applied reverse voltage, and for this reason some special avalanche diodes have been made which have a reverse breakdown voltage of over 1500V, allowing much higher gains (e.g. 1000).

Sometimes they can be operated above their maximum reverse voltage for short periods of time, giving even larger gains! When operated in this fashion, it is called Geiger mode.

APDs are used in range-finders and optical communications.

## Bridge Rectifiers

Bridge rectifiers are 4 diodes connected in such a way that they rectify an AC voltage waveform into a DC one. The image below shows a bridge rectifier being used after a transformer to convert $$12VAC$$ into $$12VDC$$. Note that the frequency of the ripple will be twice the AC input frequency (2x $$50Hz = 100Hz$$).

Bridge rectifiers can have snubber elements attached to each diode. This helps reduce the high-frequency noise which can be induced when the diodes themselves switch on/off, due the leakage inductance and parasitic capacitance of the transformer (which cause oscillations when the diodes essentially change the output impedance). Typical values for the snubber circuit are a 47pF capacitor in series with a 2kR resistor.

## Temperature Sensors

Diodes can be used as temperature sensors, as their forward voltage changes depending on the temperature. Most 3-pin active linear temperature sensors use a diode for the temperature measurement, along with additional circuitry to linearise and scale the reading. See the Temperature Sensors page for more information.

## Steering Diodes

Steering diodes is a name given to a configuration of two or more diodes that changes the direction of current depending on the polarity of the waveform.

They can be used to provide transient ESD protection.

They can be used alongside a potentiometer and 555 timer to create a PWM circuit.

## Can Diodes Share Current?

Diodes have a negative resistive thermal co-efficient, that is, as they warm up, their resistance decreases. This means that if you connect two or more diodes in parallel to share the current, one will heat up a bit faster than the other, start to conduct more, heat up even further, start to conduct even more, e.tc e.t.c, until one is conduction almost all the current. This even occurs when the diodes are “identical”, due to the fact their will be small differences in any two identical diodes.One way to prevent one diode from gobbling all the current is to add current-sharing resistors to each diode leg. They should be identical in resistance and have to drop at least $$0.3-0.4V$$ (when the diode has a nominal voltage drop of around $$0.7V$$) to be effective.

## Zener Diodes

### Overview

Zener diodes are diodes which have a specified reverse blocking voltage at which they breakdown and begin to conduct. They are similar to TVS diodes, but generally have a more defined and precise breakdown voltage, but a lower power rating. The allows Zeners to be used a shunt-style voltage regulators to power small circuits and as such, are sometimes called voltage regulator diodes. Shunt voltage references are similar in concept to zener diodes, except that they are more precise but can’t dissipate as much power.

Uses for zener diodes include:

• Low power/simple voltage reference
• Over-voltage protection for low power applications (use TVS diodes to dissipate high energy voltage spikes)
• To turn on a sub-circuit once a certain voltage level is reached (e.g. an LED in a simple battery charging circuit)

You can purchase Zeners with a reverse voltage drop as low as $$1.8V$$ all the way to above $$100V$$. For voltage drops less than 1.8V, you can stack (i.e. place in series) multiple normal or schottky diodes in forward bias.

### How To Read A Zener Diode Datasheet

A zener voltage $$V_Z$$ is given at a Zener test current $$I_{ZT}$$. $$V_Z$$ is the voltage the Zener regulates to. The test current typically a current large enough to overcome the “knee” in the voltage vs. current curve, and put the Zener into it’s “voltage regulation” state (where the voltage stays relatively stable with large changes in current).

### Regulation Performance And Dynamic Resistance

Low voltage (1-4V) Zener diodes are notoriously bad at voltage regulation due to their high dynamic resistance compared to their high-voltage siblings.

### Simple Voltage-Limiting Circuit With A Zener Diode

You can build a simple voltage limiting circuit from a Zener diode, a NPN BJT transistor, and a couple of resistors. The schematic below shows an example of this, used to limit the maximum voltage to the $$V_{in}\,$$ pin of the ADP8140 LED driver IC.

The voltage at $$V_{in}\,$$ is regulated to approximately $$V_Z - 0.7V$$. The current through $$R_Z$$ is:

$$I_{RZ} = \frac{V_{CC} - V_Z}{R_Z}$$

## Manufacturer Part Numbers

• 1N58xx: Common Schottky diode family.
• BZX384: Series of Zeners in a SOD-323 package from Nexperia.
• BZX384-B: ±2% tolerance range.
• BZX384-C: ±5% tolerance range.
• MM3Z: Family of Zener diodes from Fairchild Semiconductor (now On Semiconductor).
• MMSZ52: Family of Zener diodes from Diodes Incorporated.


text:
# RapidIdentity Product Guide: New UI

##### Record Mappings

The Record Mappings define how to construct records in the Metaverse namespace from Consumer or Provider records. This menu does not vary by Connection Type. Existing record mappings can be deleted, and new ones can be added. Clicking the Details button in the rightmost column provides access to record mapping settings.

###### General Settings

This section defines whether the record mapping is enabled, its description, and priority.

### Note

Priority can be set as any number between -231 and 231-1. Lower numbers are considered higher priority.

###### Record Types

This section shows the Source Type Name (from the Data Provider) or the Target Type Name (from the Data Consumer), and provides Studio Administrators the option to specify a field name in the source type to be used to construct a partition.

### Note

For every value of the field specified for partitioning in each source record, one target record will be constructed. This means that one source record can be used to construct potentially many target records.

###### Primary Source Filter

This is where you would restrict the source records used by the record mappings.

###### Field Mappings

The Field Mappings section defines how to construct the fields in the target record. Clicking the Details button in the rightmost column provides access to more granular field mapping settings.

##### General Settings

Enable or disable this specific field mapping and define the target fields.

### Critical Fields

The recordStatus field name is used to tell RapidIdentity how to process the records, and can be any value (normative is active or inactive). This will automatically be set up properly for new apps. If any record has a recordStatus of delete at the end of a template job that governs it, it will be removed from the database.

The recordKey field name is a unique identifier used for matching records. This field will need to be manually mapped to the correct source ID field value.

Table 349. Field Mapping Inputs

Field

Description

Target Field Name

Select the relevant field's name from the drop-down list. Each of the fields from the Field Definitions for the target record will be available in this list.

Value Type

The type of value that will appear in this field. Supported options are FIELD, STATIC, and EXPRESSION.

Value

This varies by Value Type. For each:

FIELD: The name that the target field expects from the source field

### Note

If there is a reference chain attached to the FIELD value, append a dot to the source field and further field names to use those values from the source records.

STATIC: Fills a static string for each record

EXPRESSION: Applies to expression values

##### Events

Events define when the field mapping will be applied.

Table 350. Template Events Options

Field

Description

On Create

Used when a source record is found with no matching target

On Update

Used when a source record is found that has a matching target

On Delete

Used when a source record cannot be found for a target

Once all of the settings have been configured, run the associated Provider Jobs to pull the data from the provider and populate the Metaverse namespace or associated Consumer Jobs to push the updated data to a consumer.


text:
# Solving $\cos(2x)=-\sin(x)$

Given $\cos(2x)=-\sin(x)$:

A. Solve the equation algebraically for the exact value of the solution(s) on the interval $[0,2\pi]$

B. Verify the answer(s) in part A using ZERO or INTERSECT features of your graphing calculator.

Any help is appreciated

• Hint: $\cos (2x)=1-2\sin^2 x$. Use this and you have a quadratic equation in the variable $\sin x$. – David Mitra May 12 '14 at 22:17
• Since you can't do much with the linear term $\ -\sin x \$ , write $\ \cos 2x \$ in the form $\ 1 - 2 \sin^2 x \$ . You will now have a quadratic equation in $\ \sin x \$ . – colormegone May 12 '14 at 22:17
• And for the graphing calculator part, there are any number of videos like this one (youtube.com/watch?v=tIAtBsfgXxc) to help you, I'm sure. – grantfgates May 12 '14 at 22:20

Recall the trigonometric identity $\cos(2x)=1-2\sin^2x$. Now we can rewrite our equation in terms of sine. $$1-2\sin^2x=-\sin x$$ $$2\sin^2x-\sin x-1=0$$ Notice something? This is a quadratic equation in sine, which we can solve for. Let $\sin x=y$. $$2y^2-y-1=0$$ $$(y-1)(2y+1)=0$$ $$y=1, \ -\frac 12$$ Reverse the substitution. $$\sin x=1, \ -\frac 12$$ Break this up into two cases. $$\sin x=1$$ $$\sin x=-\frac 12$$ Let's solve $\sin x=1$ first. I know that one solution is $\frac{\pi}2$. This is also the only solution in the interval $[0, \ 2\pi]$ $$x=\frac{\pi}2$$ Now for $\sin x=-\frac 12$. Using the CAST rule, I know that one or more solutions will be between $\pi$ and $2\pi$. These solutions are $\frac{7\pi}{6}$ and $\frac{11\pi}{6}$.
Therefore the solutions are: $$\color{green}{x=\frac{\pi}{2}, \ \frac{7\pi}{6}, \ \frac{11\pi}{6}}$$
To find the solutions graphically, first move $\cos(2x)$ to the right hand side. $$0=-\cos(2x)-\sin(x)$$ Replace $0$ with $y$. $$y=-\cos(2x)-\sin(x)$$ This is the equation of the graph, with the zeroes as the solutions. Here is what the graph looks like:
The zeroes within the interval $[0, \ 2\pi]$ are $\frac{\pi}{2},$ $\frac{7\pi}{6}$, and $\frac{11\pi}{6}$, which is the same as our solutions.


text:
A presentation of a group of order 12

Show that the presentation $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ defines a group of order $12$.

I tried to let $d=ab\Rightarrow G=\langle d,c\mid d^2 =c^3 = 1, c^2d=dcdc\rangle$. But I don't know how to find the order of the new presentation. I mean I am not sure how the elements of new $G$ look like. (For sure not on the form $c^id^j$ and $d^kc^l$ otherwise $|G|\leq 5$).

Is it good step to reduce the number of generators or not necessary?

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Direct proof.

$N:=\langle a,b \rangle$ is clearly a normal subgroup of $G$ with $G/N = \langle c : c^3 = 1 \rangle = C_3$, and $N$ is a quotient of the Klein four-group $V_4 = \langle a,b : a^2=b^2=1, ab=ba \rangle$. Therefore $|G|$ divides $12$. Now one checks that the permutations $a=(12)(34)$, $b = (14)(23)$, $c = (123) \in A_4$ satisfy the relations which define $G$, and that they generate $A_4$. Hence we get a surjective homomorphism $G \to A_4$. Since $|G|$ divides $12$, this shows that $G \cong A_4$ is an isomorphism and $|G|=12$.

A more conceptual proof using semidirect products.

The automorphism group of the Klein four-group $V_4$ is $S_3$ (since you can permute $a,b,ab$ freely, you can also see this from linear algebra applied to $V_4 \cong \mathbb{F}_2^2$). In particular there is an automorphism $c$ of order $3$, namely the one mapping $a \mapsto b, b \mapsto ab, ab \mapsto a$. This means that the cyclic group $C_3 = \langle c : c^3 = 1 \rangle$ of order $3$ acts on this group. The corresponding semidirect product $V_4 \rtimes C_3$ has the desired presentation $\langle a,b,c : a^2=b^2=c^3=1, ab=ba, c a c^{-1} = b, c b c^{-1} = ab \rangle$. See MO/96078 for the universal property and the resulting group presentation of a semidirect product. But the usual construction of the semidirect product $N \rtimes H$ shows that $|N \rtimes H|= |N| |H|$. In particular, $G \cong V_4 \rtimes C_3$ has order $12$.

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Before editing was easier to follow –  Ronald Aug 17 '13 at 19:02
this is very clear and useful .. thanks a lot –  Ronald Aug 17 '13 at 19:34
But since $N$ is normal why can't we directly use $|G|=|N||\frac{G}{N}|$ hence $|G|=12$? –  Ronald Aug 17 '13 at 19:55
A priori $N$ is only a quotient of $V_4$ (since $a,b \in N$ satisfy $a^2=b^2=1$ and $ab=ba$). One has to prove that $N=V_4$ (equivalently that $|N|=4$). For example, one has to check that $a \neq 1$ and that $b \neq 1$. This is only possible by giving a specific group where the relations for $a,b,c$ hold, but $a=b^2$ does not hold. In fact, there is no general algorithm for groups which checks whether a relation follows from other relations (as compared to the case of monoids). –  Martin Brandenburg Aug 17 '13 at 20:55

Although my approach is not the way you expected, it is a nice way for your group. This way is called Coset enumeration or Todd-Coxeter Algorithm. You have see $12$ rows completed as follows. In fact, I found $[G:\langle e\rangle]=12$:

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It looks simple,, I will learn this way .. thank you –  Ronald Aug 17 '13 at 19:35
It's quicker to do the coset enumeration using the subgroup $H:=\langle c \rangle$ and prove that $|G:H|=4$. –  Derek Holt Aug 17 '13 at 21:47
Nice work, Babak! +1 –  amWhy Aug 18 '13 at 0:18
@Babak Great! +1 –  Mahdi Khosravi Aug 18 '13 at 5:41
@MahdiKhosravi: G.M. Mahdiand Thanks . I think this the first time someone did this way for solving this special question. It took time to be completed but it was enjoyable reaching the target. Moreover, what Derek commented means a lot for me and show me he had a look at mine here. I am glad of that. :-) –  Babak S. Aug 18 '13 at 5:45

Consider $D := \langle a, b\rangle$ and $C = \langle c \rangle$. Note that

$$ab = ba \implies D = \langle a \rangle \langle b \rangle$$

so $|D| \leq 4$. Further, the last two relations tell you that $c \in N_{G}(D)$ (the normalizer in $G$ of $D$), so $C \leq N_{G}(D)$. It follows that $DC \leq G$, but $DC$ contains every generator of $C$; therefore, $DC = G$. We then have

$$|G| = |DC| = \frac{|D||C|}{|D \cap C|} \leq \frac{|D||C|}{1} \leq \frac{4 \cdot 3}{1} = 12$$

It just remains to find a group of order $12$ satisfying the given relations - so how many groups of order $12$ do you know?

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It is $A_4$ thanks:-) –  Ronald Aug 17 '13 at 20:10
But I need some explanation where did you use that $C \leq N_{G}(D)$ please –  Ronald Aug 17 '13 at 20:21
@Danial In order to conclude that $CD \leq G$. –  user61527 Aug 17 '13 at 20:28
Isn't $D$ is normal, and so $DH\leq G$ for any $H\leq G$? –  Ronald Aug 17 '13 at 20:31
Also, Is any subgroup of a normalizer is normal in $G$? Or is it a fact that if $A\leq N_G(B)$ then $AB\leq G$? –  Ronald Aug 17 '13 at 20:33

If you note in the original presentation that the exponents of $a,b,c$ are $2,2,3$ whose product is $12$ a natural strategy would be to try to put a general element in the form $a^pb^qc^r$ and then show that these elements are distinct.

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Yes, then is the original presentation easier to find the order? –  Ronald Aug 17 '13 at 18:47
It depends what you mean by "easier" - the first presentation shows that the group generated by $a, b$ is normal and reveals a lot of structure. I almost deleted my answer when I saw Martin Brandenburg's, which uses that structure so well. The second presentation doesn't tell you much about how to multiply elements, while with the first it is easy to see that you can push all the $c$s to the end and find a standard form for any given element. –  Mark Bennet Aug 17 '13 at 18:56
This is very useful answer and comment. I am lucky you didn't delete it. thank you –  Ronald Aug 17 '13 at 18:58

We can construct a directed graph called the Cayley diagram of the group $G := \langle a,b,c | a^2=b^2=c^3=1, ab=ba, ca=bc, cb=abc \rangle$ with respect to the generators $a,b,c$. The vertices of this digraph will be the group elements. The set of arcs will be of of the form $\{(g,gs): g \in G, s \in \{a,b,c\} \}$. After this digraph is constructed completely, each vertex $g \in G$ will have 3 outgoing arcs, labeled with colors $a,b,c$, and 3 incoming arcs with colors $a,b,c$, and the number of vertices will be the order of the group.

To construct this digraph, start with the identity vertex 1. Draw an arc from 1 to $a$, and label the arc as $a$. This is an edge-colored arc, denoted $(1,a,a)$, where the third coordinate refers to the color of the arc $(1,a)$. Add a new arc from vertex $a$ to vertex $a^2$ with color $a$, and notice that the relation $a^2=1$ forces us to identify the vertex $a^2$ and $1$. So we really have a directed cycle of length 2 at this point. Another directed cycle of length 2 arises due to arcs $(1,b,b)$ and $(b,1,b)$. Next we draw a directed cycle of length 3 on vertices $1,c,c^2$.

Next start at vertex $a$ and draw its 3 outgoing arcs, one of which will $(a,b,ab)$. Draw three outgoing arcs from $b$, one of which will be $(b,a,ba)$. Since $ab=ba$, we identify (i.e. merge) these two vertices $ab$ and $ba$. In addition, we start at the existing vertices (say the vertex 1), and enforce each relation at each vertex. Thus, starting at 1, the directed path $ab$ of length 2 and the directed path $ba$ of length 2 must end up at the same vertex. Similarly, $ca$ and $bc$ end up at the same vertex, as do $cb$ and $abc$.

Continue this process until every vertex has three outgoing and three incoming arcs, and until each relation is enforced at each vertex. We identify vertices whenever a relation forces us to do so. The number of vertices will then end up being the order of the group. I constructed this digraph in your example and verified it has order 12.

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Does this process also really prove that the order is exactly $12$ and not just $\leq 12$? –  Martin Brandenburg Aug 18 '13 at 8:04
@MartinBrandenburg It depends on what AG. means by "verified" in the last line. This gives us 12 group elements, and we need to verify that each are non-equal. –  user1729 Aug 18 '13 at 10:06
I believe this process does give the exact value of $|G|$ for any presentation. Let $F$ be the free group generated by the symbols $a,b,c, \ldots$, and let $N$ be the smallest normal subgroup of $F$ generated by the relators $r_1,r_2,\ldots$. (In the above example, $r_1=a^2, \ldots, r_6 = cbc^{-1}b^{-1}a^{-1}$). Then $G$ is defined to be the quotient group $F/N$. It can be shown that two words $w_1$ and $w_2$ are equivalent in $G$ iff $w_2$ can be obtained from $w_1$ by a sequence of insertions and deletions of the form $x x^{-1}, x^{-1} x, r_i, r_i^{-1}$, which the procedure accounts for. –  AG. Aug 18 '13 at 12:04
In other words, it suffices to count the number of vertices of the final digraph to obtain the order of the group. Distinct vertices of the final digraph would automatically correspond to distinct group elements, so no further verification is necessary. –  AG. Aug 18 '13 at 13:13

The presentation can be rewritten as $\langle d,c\mid d^2 =c^3 = (cd)^3=1 \rangle$. This is the standard presentation for the symmetry group (rotations) of the regular tetrahedron where $d$ represents a rotation about an edge, $c$ represents a rotation about a face and $cd$ represents a rotation about a vertex.

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