The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler – Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

Many early PLCs did not have accompanying programming terminals that were capable of graphical representation of the logic, and so the logic was instead represented as a series of logic expressions in some version of Boolean format, similar to Boolean algebra.

The series is a challenging one for the viewer, because the concentration required to follow the facial expressions over time must last for minutes or more.

They are differentiated from most other TI graphing calculators by their computer algebra system, which allows symbolic manipulation of algebraic expressions —— equations can be solved in terms of variables, whereas the TI-83 / 84 series can only give a numeric result.

); arbitrary functions of expressions ; optimization ; derivatives, integrals, simplifications, sums, and products of expressions ; truncated series with expressions as coefficients, matrices of expressions, and so on.

In 1987 Hewlett-Packard introduced the first hand held calculator CAS with the HP-28 series, and it was possible, for the first time in a calculator, to arrange algebraic expressions, differentiation, limited symbolic integration, Taylor series construction and a solver for algebraic equations.

The expressions for the present value of such payments are summations of geometric series.

Then, each situation that could cause that effect is added to the tree as a series of logic expressions.

More general and powerful list-building facilities are provided by " list comprehensions " ( previously known as " ZF expressions "), which come in two main forms: an expression applied to a series of terms, e. g.:

Featured a series of interconnected exhibitions celebrating diverse expressions of creativity and the independent spirit of rock ' n ' roll.

Seeing the dismayed expressions of her " parents ", combined with a series of sudden realizations, including the fact that the family photo album contains no pictures of her as a child, she arrives at the shocking reality that she, too, is a robot, albeit much more emotionally sophisticated than the ones that were dismantled.

Also, not all expressions that are meaningful as functions of x are meaningful as expressions designating formal power series ; negative and fractional powers of x are examples of this.

As had been the case for earlier series, a number of alternative heads displaying a range of expressions were created for main character puppets, including " smilers ", " frowners " and " blinkers ".

They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in various expressions for various special functions.

* Lie to Me, a TV series based on behavior analysts who read lies through facial expressions and body language.

In 1868, following the Battle of Ueno, Yoshitoshi made the series Kaidai hyaku sensō in which he portrays contemporary soldiers as historical figures in a semi-western style, using close-up and unusual angles, often shown in the heat of battle with desperate expressions.

Fox – Wright functions are a generalization of generalized hypergeometric functions where the Pochhammer symbols in the series expression are generalised to gamma functions of linear expressions in the index n.

The quality of the animation is surprisingly impressive ; some static backgrounds are the primary concession to a small-screen budget and the fluid character movements and expressions are vastly superior to those of, say, the Timon and Pumbaa TV cartoon series ," and gave the film 3½ stars out of 4.

He notes that the character designs fit well with series ' visual style and the characters ' expressions were convincing.

But I have compared its text with already published commentaries on the 1960 series of Godkin lectures at Harvard, from which the book was derived, and I can with confidence challenge the gist of C. P. Snow's incautious tale ''.

About that same time John Crosby's TV series on the popular arts proved again that giving jazz ample breathing space is one of the most sensible things a producer can do.

Joan Hickson played the part of the home help in this film and can claim to have appeared in two Miss Marple series.

* podcasts. ie In this series of podcasts on Mayo's Heritage you can hear an excerpt from John Mc Hugh's Heritage Tour of Achill ( ref Programme 6 & Programme 8 )

They can be obtained analytically, meaning that the resulting orbitals are products of a polynomial series, and exponential and trigonometric functions.

This structure can be also seen as a series of concentric circles, where the gospel begins in the center, Jerusalem, and is expanding ever outward to Judea & Samaria, Syria, Asia Minor, Europe, and eventually to Rome.

This unstable intermediate can lead to electron " leakage ", when electrons jump directly to oxygen and form the superoxide anion, instead of moving through the normal series of well-controlled reactions of the electron transport chain.

A Most Valuable Player ( MVP ) award is given to the outstanding player in each series, though voters can consider performances made during the divisional series.

Watterson opposed the structure publishers imposed on Sunday newspaper cartoons: the standard cartoon starts with a large, wide rectangle featuring the cartoon's logo or a throwaway panel tangential to the main area so that newspapers pressed for space can remove the top third of the cartoon if they wish ; the rest of the strip is presented in a series of rectangles of different widths.

( The Taylor series indicates that − J < sub > 1 </ sub >( x ) is the derivative of J < sub > 0 </ sub >( x ), much like − sin x is the derivative of cos x ; more generally, the derivative of J < sub > n </ sub >( x ) can be expressed in terms of J < sub > n ± 1 </ sub >( x ) by the identities below.

The Bessel functions can be expressed in terms of the generalized hypergeometric series as

This orthogonality relation can then be used to extract the coefficients in the Fourier – Bessel series, where a function is expanded in the basis of the functions J < sub > α </ sub >( x u < sub > α, m </ sub >) for fixed α and varying m.

This can be proved ( for integer t ) by using the formula for geometric series: ( using y = 1-x )

* Borzoi can also been seen in cameo roles in the films Love at First Bite, Legends of the Fall (" Notchee Boy "), Excalibur, All Dogs Go to Heaven ( 1989 ), Bride of Frankenstein, Easter Parade, Wolfen, Ziegfeld Follies, Onegin ( 1999 ), Gangs of New York ( 2002 ), Chaplin, The Avengers ( TV series ), JAG, Maverick ( 1994 ), Sleepy Hollow, Last Action Hero, and A Knights Tale ( on the DVD deleted scenes ).

This is DNA which contains units named genes that can produce proteins through a code ( genetic code ) in which a series of triplets ( codons ) of four possible nucleotides are translated into one of twenty possible amino acids.

In cryptography, a cipher ( or cypher ) is an algorithm for performing encryption or decryption — a series of well-defined steps that can be followed as a procedure.

Second, the common law evolves through a series of gradual steps, that gradually works out all the details, so that over a decade or more, the law can change substantially but without a sharp break, thereby reducing disruptive effects.

Dot matrix printers can either be character-based or line-based ( that is, a single horizontal series of pixels across the page ), referring to the configuration of the print head.

With river currents slowed by the dams, and the Columbia converted from wild river to a series of slackwater pools, the journey can take several months, which increases the mortality rate.

This is a protective mechanism for the heart, to compensate for an SA node that is no longer handling the pacemaking activity, and is one of a series of backup sites that can take over pacemaker function when the SA node fails to do so.

In the latter, only taxa associated with a rank can be named, yet there are not enough ranks to name a long series of nested clades ; ranks determine the form of names, so names must in many cases change when a name is inserted into such a series ; and taxon names cannot be defined in a way that guarantees them to refer to clades.

In the manga and anime series Ghost in the Shell, the protagonist Motoko Kusanagi is the fully prosthetic leader of an anti-terrorist force, who lives in a future Japan where the majority of adults are cyborgs and can connect wirelessly to the Internet for real-time communication and data research.

A series of 1998 research indicates that each item added to an information display can significantly affect the overall experience of the user.