f ( t ) = L -1 {F ( s )} 1. 3. 5. 7. 9. 11. 1 t n , n = 1, 2,3,K t sin ( at ) t sin ( at ) sin ( at ) - at cos ( at ) cos ( at ) - at sin ( at ) sin ( at + b ) sinh ( at ) e at sin ( bt ) e at sinh ( bt ) t ne at , n = 1, 2,3,K uc ( t ) = u ( t - c ) Heaviside Function

F ( s ) = L { f ( t )} 1 s n! s n +1

Table of Laplace Transforms

f ( t ) = L -1 {F ( s )}

F ( s ) = L { f ( t )} 1 s-a G ( p + 1) s p +1 1 × 3 × 5L ( 2n - 1) p 2n s 2 s 2 s + a2 s2 - a2 2 n+ 1

2. 4. 6. 8.
2

e at t p , p > -1 t
n- 1 2

p
2s a 2 s + a2 2as
2
3 2

, n = 1, 2,3,K

cos ( at ) t cos ( at ) sin ( at ) + at cos ( at ) cos ( at ) + at sin ( at ) cos ( at + b ) cosh ( at ) e at cos ( bt ) e at cosh ( bt ) f ( ct )

s n F ( s ) - s n-1 f ( 0 ) - s n- 2 f ¢ ( 0 )L - sf ( n- 2) ( 0 ) - f ( n-1) ( 0 )

Table Notes
1. This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas. 2. Recall the definition of hyperbolic functions. et + e - t et - e - t cosh ( t ) = sinh ( t ) = 2 2 3. Be careful when using “normal” trig function vs. hyperbolic functions. The only difference in the formulas is the “+ a2” for the “normal” trig functions becomes a “- a2” for the hyperbolic functions! 4. Formula #4 uses the Gamma function which is defined as G ( t ) = ò e - x x t -1 dx 0 ¥

If n is a positive integer then,

G ( n + 1) = n !

The Gamma function is an extension of the normal factorial function. Here are a couple of quick facts for the Gamma function G ( p + 1) = pG ( p ) p ( p + 1)( p + 2 )L ( p + n - 1) = æ1ö Gç ÷ = p è2ø G ( p + n) G ( p)

...contains at least k records. Recently, several authors have recognized that k-anonymity cannot prevent attribute disclosure. The notion of ‘-diversity has been proposed to address this; ‘-diversity requires that each equivalence class has at least ‘well-represented (in Section 2) values for each sensitive attribute. In this paper, we show that ‘-diversity has a number of limitations. In particular, it is neither necessary nor sufficient to prevent attribute disclosure. Motivated by these limitations, we propose a new notion of privacy called “closeness.” We first present the base model closeness, which requires that the distribution of a sensitive attribute in any equivalence class is close to the distribution of the attribute in the overall table (i.e., the distance between the two distributions should be no more than a threshold t). We then propose a more flexible privacy model called closeness that offers higher utility. We describe our desiderata for designing a distance measure between two probability distributions and present two distance measures. We discuss the rationale for using closeness as a privacy measure and illustrate its advantages through examples and experiments.
Existing System
One problem with l-diversity is that it is limited in its assumption of adversarial knowledge. As we shall explain below, it is possible for an adversary to gain information about a sensitive attribute as long as she...

...======== 101 Ideas for Great Table Topics ========
Compiled by Mark LaVergne, DTM, Past International Director of Toastmasters International
Phone: 718-457-8390; Email: MarkLaVergne@aol.com Address: 38-11 Ditmars Boulevard, Astoria, NY 11105 Updated: May 2006 – Background and Acknowledgements on Page 14
----------------------------------------------------------------------------------------------------------------------------------==> TABLE TOPIC IDEA #1 -------------------------------------------------------------------------------Holiday Themes: We are blessed with a wonderful set of holidays in this society. When looking for Table Topic theme ideas, begin by looking at these holidays as sources for theme-based questions. Please find below a partial list of possible themes, sorted roughly by calendar order (January-to-December): --- New Year’s Day --- Dr. Martin Luther King, Jr. Birthday --- Chinese New Year (date varies by year, depending on Chinese Lunar calendar) --- Ramadan (date varies by year, depending on religious calendar) --- African American History Month --- Valentine’s Day --- U.S. President’s Day --- St. Patrick’s Day (March 17) --- Women’s History Month --- Passover (date varies by year, depending on religious calendar) --- Easter (date varies by year, depending on religious calendar) --- Spring is Here (March 21) --- April Fools --- Tax Burden Day – April 15th --- Earth Day --- Cinco de Mayo --- Mother’s Day...

...
For my week one assignment I choose to write about “The Welcome Table” (Walker, A) 1970. What first captured my interest in this short story was a poem listed before the story, the poem was called “For my sister Clara Ward” by (Walker, A) 1970 “I’m going to sit at the Welcome table, Shout my troubles over, Walk and talk with Jesus, Tell God how you treat me, One of these days!” This short story and poem reminds me of going to church with my great grandmother and grandmother.
At that time I didn’t realize how precious it was to have them around. I took for granted having dinner every Sunday after church with those two wonderful ladies, sometime we would have conversations about Jesus and life for hours and hours as we sat around the table. Today I wish they were still here to help guide me through hard times. I find myself walking with my eyes closed listening for their voices for some kind of spiritual direction. Reading “The Welcome Table” allowed my mind to connect with my imagination to what Alice Walker was expressing to the world, and to the readers.
The story focuses on an elderly woman’s life, and after she had worked for many years in many different households, she knew her life was soon coming to an end. Though she felt her life would soon be over her inspiration and focus was looking forward to having a talk with Jesus at the welcome table. The welcome...

...Chapter 7
Laplace Transform
The Laplace transform can be used to solve diﬀerential equations. Besides being a diﬀerent and eﬃcient alternative to variation of parameters and undetermined coeﬃcients, the Laplace method is particularly advantageous for input terms that are piecewise-deﬁned, periodic or impulsive. The direct Laplace transform or the Laplace integral of a function f (t) deﬁned for 0 ≤ t < ∞ is the ordinary calculus integration problem
∞ 0
f (t)e−st dt,
succinctly denoted L(f (t)) in science and engineering literature. The L–notation recognizes that integration always proceeds over t = 0 to t = ∞ and that the integral involves an integrator e−st dt instead of the usual dt. These minor diﬀerences distinguish Laplace integrals from the ordinary integrals found on the inside covers of calculus texts.
7.1 Introduction to the Laplace Method
The foundation of Laplace theory is Lerch’s cancellation law
∞ −st dt 0 y(t)e
=
∞ −st dt 0 f (t)e
(1) L(y(t) = L(f (t))
implies or implies
y(t) = f (t), y(t) = f (t).
In diﬀerential equation applications, y(t) is the sought-after unknown while f (t) is an explicit expression taken from integral tables. Below, we illustrate Laplace’s method by solving the initial value problem y = −1, y(0) = 0. The method obtains a relation L(y(t)) = L(−t), whence Lerch’s cancellation law...

...How to Calculate the Laplace Transform of a Function
TerminologySolving the transformDiscontinuous FunctionsUsing Properties of Laplace Transforms
Edited by Caidoz, Flickety, Zareen, Garshepp and 4 others
The Laplace transform is an integral transform which allows a differential equation to be converted into a (hopefully) simpler algebraic equation, making it easier to solve.
While you can use tables of Laplace Transforms, it is never a bad idea to know how to do the transform yourself.
EditSteps
1Know whether you are trying to find the unilateral (one-sided) Laplace transform or the bilateral (two-sided) Laplace transform of the function. If the type of Laplace transform is not specified, it can be assumed that you should calculate the unilateral version.
A unilateral Laplace transform is defined as:
A bilateral Laplace transform is defined as:
2Put your function, f(t), into the definition of the Laplace transform.
EditMethod 1 of 4: Terminology
1Consider "Laplace Transforms" -- in part it is a system to convert time dependent domain relationships to a set of equations expressed in terms of the Laplace operator 's'. Then, the solution of the original problem is effected by "complex-algebra manipulations" in the 's' or Laplace domain rather than the time...

...Laplace Transformation Laplace transformation is a Mathematical tool which can be used to solve several problems in science and engineering. The transformed was first introduced by Pierre-Simon Laplace a French Mathematician, in the year 1790 in his work on probability theorem. Application of Laplace Transform The Laplace transform technique is applicable in many fields of science and technology such as: Control Engineering Communication Signal Analysis and Design Image Processing System Analysis Solving Differential Equations (ordinary and partial)
Advantages of Laplace transformation A Laplace transformation technique reduces the solutions of an ordinary differential equation to the solution of an algebraic equation. When the Laplace transform technique is applied to a PDE, it reduces the number of independent variable by one. With application of Laplace transform, particular solution of differential equation is obtained directly without necessity of first determining general solution.
Periodic Function
A real valued function ������(������) is said to be periodic with period ������ > 0 if for all ������, ������ ������ + ������ = ������(������) , and T is the least of such values. For example, sin ������ and cos ������ are periodic functions with period 2π. tan ������ and cot ������ are periodic functions with...

...Q-A. Find the Laplace transform of the following functions 1. f (t) = t − 1, 0 < t < 3; 7, t > 3. 2. f (t) = cos t − 0,
2π 3
, 0 2π . 3
2π ; 3
4, 0 < t < 1; −2, 1 < t < 3; 3. f (t) = 5, t > 3. 5. f (t) = 3t3 + e−2t + t 3 7. f (t) = cos3 2t 9. f (t) = sin (3t + 5) 11. f (t) = e−3t sin2 t 13. f (t) = 7T 15. f (t) = e−3t (cos (4t) + 3 sin (4t)) 17. f (t) = teat 19. f (t) = t sin2 3t 21. f (t) = t2 e−2t cos t 23. f (t) = t cos (7t + 9) 25. f (t) = 27. f (t) = sin2 t t e−t sin t t
1 2 2
4
4. f (t) = 6. f (t) =
sin t, 0 < t < π; 0, t > π. eat − 1 a
8. f (t) = cosh3 3t 10. f (t) = t2 sin (at) 12. f (t) = e−t cos2 t 14. f (t) = cat+b 16. f (t) = 4e−2t cosh (3t) 18. f (t) = t2 cos (3t) 20. f (t) = teat sin (at) 22. f (t) = t2 eat sin (105t) 24. f (t) = t2 e−3t cos (2t) 26. f (t) = sin t t
1 2
28. f (t) = et u t − u t−
1 2
29. f (t) = t −
30. f (t) = cos t u(t − π) 32. f (t) = (1 + 2t + 3t2 − 4t3 ) u(t − 2)
31. f (t) = t2 u(t − 3)
33. f (t) = t − t u(t − π) + sin t u(t − 2π) 34. f (t) = 2t u(t − 2)
Q-B. Find the Laplace inverse transform of the following functions 1. F (s) = 3. F (s) = 5. F (s) = 7. F (s) = s s4 + 4a4 s 2+s+1 s 4s + 12 s2 + 8s + 16 5s − 10 9s2 − 16 s+1 s−1 2. F (s) = 4. F (s) = 6. F (s) = 8. F (s) = s+2 s2 − 4s + 13 (s2 − 1)2 s5 2s − 1 s4 + s2 + 1 1 16 + 4s 1 − s2 1+s s
9. F (s) = log
10. F (s) = log 12. F (s) = 14. F (s) = 16. F (s) =
11. F (s) = cot−1 (s + 1) 13. F (s) = 15. F (s)...