The diagonalizable operator is the special case of this in which Af for each i.

For each i, let Af.

When Af for each i, we shall have Af, because the operator Af will then be 0 on the range of Af.

The family moved north across the Ohio River to free ( i. e., non-slave ) territory and made a new start in what was then Perry County but is now Spencer County, Indiana.

This religion preaches the view of Ahimsa – to live and let live, thereby not harming sentient beings, i. e. uncompromising reverence for all life.

The importance lies in sacrifice for the sake of the greater good ; Islam considers those practicing i ' thar as abiding by the highest degree of nobility.

This is similar to the notion of chivalry, but unlike that European concept, in i ' thar attention is focused on everything in existence.

A constant concern for Allah ( i. e. God ) results in a careful attitude towards people, animals, and other things in this world.

This concept was emphasized by Sufi mystics like Rabia al-Adawiyya who paid attention to the difference between dedication to Allah ( i. e. God ) and dedication to people.

For many Muslims, i ' thar must be practiced as a religious obligation during specific Islamic holidays.

However, i ' thar is also still an Islamic ideal to which all Muslims should strive to adhere at all times.

These two defunct phyla were the Myxomycota ( i. e., plasmodial slime molds, now classified in the taxon Myxogastria ), and Acrasiomycota ( i. e. cellular slime molds, now divided into the taxa Acrasida and Dictyosteliida ).

Compared to earlier telegraph codes, the proposed Bell code and ASCII were both ordered for more convenient sorting ( i. e., alphabetization ) of lists and added features for devices other than teleprinters.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

If the appellate court does find a legal defect in the decision " below " ( i. e., in the lower court ), it may " modify " the ruling to correct the defect, or it may nullify (" reverse " or " vacate ") the whole decision or any part of it.

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

If the resulting four kilobyte table size is too large for a given target platform, the table lookup operation can be performed with a single 256-entry 32-bit ( i. e. 1 kilobyte ) table by the use of circular rotates.

If a description of s, d ( s ), is of minimal length ( i. e. it uses the fewest number of characters ), it is called a minimal description of s. Thus, the length of d ( s ) ( i. e. the number of characters in the description ) is the Kolmogorov complexity of s, written K ( s ).

If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their perpendicular height of incidence, i. e. their distance from the axis.

If the ratio a '/ a be sufficiently constant, as is often the case, the above relation reduces to the condition of Airy, i. e. tan w '/ tan w = a constant.

If we define r < sub > i </ sub > as the displacement of particle i from the center of mass, and v < sub > i </ sub > as the velocity of particle i with respect to the center of mass, then we have

If ( m, n ) is regular and M and N have i and j prime factors respectively, then ( m, n ) is said to be of type ( i, j ).

If the auction is passed out, i. e. no bids are made and only four passes are called, the hands are abandoned and the turn to deal passes in rotation.

If a Banach algebra has unit 1, then 1 cannot be a commutator ; i. e., for any x, y ∈ A.

If your side has two aces and a void, then you are not at risk of losing the first two tricks, so long as ( a ) your void is useful ( i. e., does not duplicate the function of an ace that your side holds ) and ( b ) you are not vulnerable to the loss of the first two tricks in the fourth suit ( because, for instance, one of the partnership hands holds a singleton in that suit or the protected king, giving your side second round control ).

If one eigenvalue is negative ( i. e., an imaginary frequency ), then the stationary point is a transition structure.

If both high priority substituents are on the same side of the double bond, i. e. in the cis configuration, then the stereoisomer is assigned a Z or Zusammen configuration.

If a 4, 5, 6, 8, 9, or 10 is thrown on the come-out roll ( i. e., if a point is set ), most casinos allow pass line bettors to take odds by placing from one to five times ( and at some casinos, up to 100 times ) the pass line bet behind the pass line.

If two cofinal subsets of B have minimal cardinality ( i. e. their cardinality is the cofinality of B ), then they are order isomorphic to each other.

* If is the norm ( usually noted as ) defined in the sequence space ℓ < sup >∞</ sup > of all bounded sequences ( which also matches the non-linear distance measured as the maximum of distances measured on projections into the base subspaces, without requiring the space to be isotropic or even just linear, but only continuous, such norm being definable on all Banach spaces ), and is lower triangular non-singular ( i. e., ) then

If the glass is cooled quickly in the absence of a strong light source by putting ice on the glass or placing it in the freezer with the door almost closed, it turns backwards ( i. e. the silver sides trail ).

However, it spins in the opposite direction ( i. e. from the leg side to the off side ), easily confusing batsmen, making it a very effective weapon.

Most batsmen manage to score at an average of four runs an over ( i. e. four runs in the six ball over ).

In matches where the number of overs is not limited, therefore, a team unable to win may be able to ' save the draw ' by managing to not complete its final innings ( i. e. by having batsmen left at the end of play who are ' not out ').

In statistical terms, Sutcliffe was one of the most successful Test batsmen ever ; his completed career batting average was 60. 73 which is the highest by any English batsman and the sixth-highest worldwide ( i. e., of Test batsmen with 10 completed innings ).

In 1930, Sutcliffe was the leading Englishman in the first-class batting averages behind Don Bradman ( i. e., of batsmen with 10 completed innings ).

Bowling strike rate is a measure of how frequently a bowler achieves the primary goal of bowling, namely taking wickets ( i. e. getting batsmen out ).

Small's 1772 aggregate of 213 runs from six innings would give him an average of 35. 50 if all his innings were completed ( i. e., the scorecards in 1772 do not confirm the not out batsmen ).

Such processes are usually followed or preceded by events that do involve heat transfer ( i. e. are non-adiabatic ).

Non-probate property -- i. e., life insurance policies -- do not abate.

In the classical limit, i. e. at large values of or at small density of states — when wave functions of particles practically do not overlap — both the Bose – Einstein or Fermi – Dirac distribution become the Boltzmann distribution.

Hypomanic episodes do not go to the full extremes of mania ( i. e., do not usually cause severe social or occupational impairment, and are without psychosis ), and this can make bipolar II more difficult to diagnose, since the hypomanic episodes may simply appear as a period of successful high productivity and is reported less frequently than a distressing, crippling depression.

Given any expression involving complex numbers, bras, kets, inner products, outer products, and / or linear operators ( but not addition ), written in bra-ket notation, the parenthetical groupings do not matter ( i. e., the associative property holds ).

Moreover ( and more embarrassingly, although this is essentially trivial ), mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they don't use the *- symbol, but an overline ( which the physicists reserve to averages ) to denote conjugate-complex numbers, i. e. for scalar products mathematicians usually write

These categories surely have some objects that are " special " in a certain way, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered to be atomic, i. e., we do not know whether an object A is a set, a topology, or any other abstract concept – hence, the challenge is to define special objects without referring to the internal structure of those objects.

One could then unplug the C64 itself from the drives ( i. e. from the first drive in the daisy chain ) and do something else with the computer as the drives proceeded to copy the entire disk.

Bolts do not have fletching, i. e. feathered ends like those commonly seen on arrows.

Odd-powered raw moments, however, do not exist at all ( i. e. are undefined ), which is distinctly different from existing with the value of infinity.

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant ( i. e., elements of their transfer function C ( s ), P ( s ), and F ( s ) do not depend on time ), the systems above can be analysed using the Laplace transform on the variables.

As part of this notice procedure, there may have to be several notices, first a notice giving class members the opportunity to opt out of the class, i. e. if individuals wish to proceed with their own litigation they are entitled to do so, only to the extent that they give timely notice to the class counsel or the court that they are opting out.

Fourth, in " limited fund " cases, a class action ensures that all plaintiffs receive relief and that early-filing plaintiffs do not raid the fund ( i. e., the defendant ) of all its assets before other plaintiffs may be compensated.

A copy of the order, with a " penal notice "— i. e. notice informing the recipient that if they do not comply they are subject to imprisonment — is served on the person concerned.

Games do not have to stop at the avatar-player level, but current implementations aiming for more immersive playing space ( i. e. Laser tag ) take the form of augmented reality rather than cyberspace, fully immersive virtual realities remaining impractical.

Some linguists do not differentiate between languages and dialects, i. e. languages are dialects and vice versa.

To achieve this a backup operation is done occasionally or continuously, where each desired database state ( i. e., the values of its data and their embedding in database's data structures ) is kept within dedicated backup files ( many techniques exist to do this effectively ).

Others, however, feel they have been given a " label " that invites social stigma and discrimination ( i. e. mentalism ), or one that they simply do not feel is accurate.

for i ← from 0 to LENGTH ( words ) do

for each ( A → α • Bβ, i ) in chart do

Tackling in Australian Rules is unique in that if tackled player does not dispose of the ball having had opportunity to do so ( called the holding the ball rule ), or concedes a penalty in the tackle ( i. e. tries to throw the ball when he was only allowed to " handball " or kick it ) the tackler is rewarded with a free kick.

First, if planar regions separated by the graph are not triangulated, i. e. do not have exactly three edges in their boundaries, we can add edges without introducing new vertices in order to make every region triangular, including the unbounded outer region.

In the Rex Stout novel " Counterfeit for Murder " ( found in the < i > Homicide Trinity </ i >), character Hattie Annis states that Nero Wolfe " would do fine for Falstaff.