However, if an inaccessible cardinal κ is assumed, then the sets of smaller rank form a model of ZF ( a Grothendieck universe ), and its subsets can be thought of as " classes ".

More generally, if κ is any infinite cardinal, then a product of at most 2 < sup > κ </ sup > spaces with dense subsets of size at most κ has itself a dense subset of size at most κ ( Hewitt – Marczewski – Pondiczery theorem ).

Instead, the consonant before the contracted vowel changes to the aspirated equivalent ( i. e., π → φ, τ → θ, κ → χ ), if possible, and the contracted vowel takes the apostrophe or coronis ( identical to the smooth breathing ).

A cardinal κ is α-inaccessible, for α any ordinal, if and only if κ is inaccessible and for every ordinal β < α, the set of β-inaccessibles less than κ is unbounded in κ ( and thus of cardinality κ, since κ is regular ).

For any ordinal α, a cardinal κ is α-hyper-inaccessible if and only if κ is hyper-inaccessible and for every ordinal β < α, the set of β-hyper-inaccessibles less than κ is unbounded in κ.

Antimony is stable in air at room temperature, but reacts with oxygen if heated to form antimony trioxide, Sb < sub > 2 </ sub > O < sub > 3 </ sub >.

Also, no finite field F is algebraically closed, because if a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > n </ sub > are the elements of F, then the polynomial ( x − a < sub > 1 </ sub >)( x − a < sub > 2 </ sub >) ··· ( x − a < sub > n </ sub >) + 1

The automorphism group of an object X in a category C is denoted Aut < sub > C </ sub >( X ), or simply Aut ( X ) if the category is clear from context.

The color of amethyst has been demonstrated to result from substitution by irradiation of trivalent iron ( Fe < sup > 3 +</ sup >) for silicon in the structure, in the presence of trace elements of large ionic radius, and, to a certain extent, the amethyst color can naturally result from displacement of transition elements even if the iron concentration is low.

Thus, if P is a program in L < sub > 2 </ sub > which is a minimal description of s, then InterpretLanguage ( P ) returns the string s. The length of this description of s is the sum of

After the indecisive < ref name =" British historian Townsend Miller "> British historian Townsend Miller: “ But, if the outcome of < nowiki > battle of </ nowiki > Toro, militarily, is debatable, there is no doubt whatsoever as to its enormous psychological and political effects ” in The battle of Toro, 1476, in History Today, volume 14, 1964, p. 270 </ ref > Battle of Toro in 1476 against King Ferdinand II of Aragon, the husband of Isabella I of Castile, he went to France to obtain the assistance of Louis XI, but finding himself deceived by the French monarch, he returned to Portugal in 1477 in very low spirits.

Thus, if a two-dimensional array has rows and columns indexed from 1 to 10 and 1 to 20, respectively, then replacing B by B + c < sub > 1 </ sub >-− 3 c < sub > 1 </ sub > will cause them to be renumbered from 0 through 9 and 4 through 23, respectively.

For example, if the sorted list to which we apply binary search has n elements, and we can guarantee that each lookup of an element in the list can be done in unit time, then at most log < sub > 2 </ sub > n + 1 time units are needed to return an answer.

His comment on Numbers 23: 19 has a still more polemical tone: “ God is not a man that he should lie ; neither the son of man, that he should repent ; < font face =" times new roman " size = 3 > if a man says: ‘ I am a god ’ he is a liar ; if he says: ‘ I am a son of man ’ he will have cause to regret it ; and if he says, ‘ I will go up to heaven ’ he has said but will not keep his word ” last phrase is borrowed from B ' midbar 23: 19 ( Yer.

In the statement < code >< u > if </ u > token = if < u > then </ u > < u > result </ u >

A convex function | function is convex if and only if its Epigraph ( mathematics ) | epigraph, the region ( in green ) above its graph of a function | graph ( in blue ), is a convex set.

It can then be shown that a real number r is constructible if and only if, given a line segment of unit length, a line segment of length | r | can be constructed with compass and straightedge.

Image: pons_asinorum. png | The bridge of asses theorem states that if A = B then C = D.

Image: Thales ' Theorem Simple. svg | Thales ' theorem: if AC is a diameter, then the angle at B is a right angle.

if y < ( x | 5 ) then write (" y =", y )</ code >

More formally, if a, b, c, and d are all integers, then the fraction < sup > a </ sup >⁄< sub > b </ sub > is irreducible if and only if there is no other equal fraction < sup > c </ sup >⁄< sub > d </ sub > such that | c | < | a | or | d | < | b |, where | a | means the absolute value of a.

In French, the period only follows an abbreviation if the last letter in the abbreviation is not the last letter of its antecedent: " M ." is the abbreviation for " monsieur " while " Mme " is that for " madame ".

Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p ( x ) is true and the set

It is the same angular momentum one would obtain if there were just one particle of mass M moving at velocity V located at the center of mass.

because if M is any algebraically closed field containing K, then the elements of M which are algebraic over K form an algebraic closure of K.

However he has expressed hope for its resurrection, saying " It ’ s only going to be dead if it doesn ’ t talk about the messages of life as much as the messages of death and non-movement ", citing artists such as NYOil, M. I. A.

By contrast, if M ( t ) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M ( t ) is discontinuous.

Thus, if ( M, d ) and ( N, d ) are two metric spaces, and, then there is a constant k such that

Such a definition can be formulated in terms of equivalence classes of smooth functions on M. Informally, we will say that two smooth functions f and g are equivalent at a point x if they have the same first-order behavior near x.

The algebra M < sub > n </ sub >( C ) of n-by-n matrices over C becomes a C *- algebra if we consider matrices as operators on the Euclidean space, C < sup > n </ sup >, and use the operator norm ||.|| on matrices.

There is also a traditional convention in English that surnames beginning Mc and M are listed as if those prefixes were written Mac.

Other MFM formats were possible if their characteristics were added to the CP / M C128-specific source code ( available from Commodore ) and the CP / M operating system were re-assembled.

In mathematical analysis, a metric space M is called complete ( or a Cauchy space ) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

But what you could not know, of course, was how smoothly the Victorian Fitzgerald was to lead into an American Fitzgerald of my own vintage under whose banner we adolescents were to come, if not of age, then into a bright, taut semblance of it.

William Wimsatt and Cleanth Brooks, it seems to me, have a penetrating insight into the way in which this control is effected: `` For if we say poetry is to talk of beauty and love ( and yet not aim at exciting erotic emotion or even an emotion of Platonic esteem ) and if it is to talk of anger and murder ( and yet not aim at arousing anger and indignation ) -- then it may be that the poetic way of dealing with these emotions will not be any kind of intensification, compounding, or magnification, or any direct assault upon the affections at all.

What I want is to have this evidence come before Congress and if the Attorney General does not report it, as I am very sure he won't, as he has refused to do anything of the kind, I then wish that a committee of seven Representatives be appointed with power to take the evidence.

The Acropolis was unique in the world and if that imcomparable work flooded by moonlight wasn't enough for both natives and tourists, then they were quite simply barbarians and the hell with them.

Sturley quoted Quiney as having written on November 1 that if he had `` more monei presente much might be done to obtaine our Charter enlargd, ij faires more, with tole of corne, bestes, and sheepe, and a matter of more valewe then all that ''.

If man is actually the product of his environment and if science can discover the laws of human nature and the ways in which environment determines what people do, then someone -- a someone probably standing outside traditional systems of values -- can turn around and develop completely efficient means for controlling people.

And if you bore each other then, heaven help you.

He invited Mr. Case to stop by to say hello if he ever visited the academy and then added that he was on the managerial staff of the freshman football team

The prevailing official attitude then seemed to be that fallout, if not exactly good for you, might not be much worse than a bad cold.

And if the foreigners fighting in the Katanga Army are mercenaries then Lafayette and Von Steuben were mercenaries too, as were also the members of the Lafayette Escadrille in the early part of World War 1, and of Chennault's Flying Tigers in the early days of World War 2.

Rachel came close to the bed, bent as if she would kiss him, then moved away.

and then I was adding my own voice to the crescendo of sound, hurling more vile language than I ever thought I knew, sobbing and shouting, and aware that if I had passed water before, it was not enough, for my pants were soaking wet.

`` I'll have a drink, then, if you'll have one with me ''.

And, if we follow the Rayburn pattern, as consciously or by an instinctual political sense I like to think I have followed it, then the very nature of our loyalty to our own immediate areas must necessarily be reflected in the devotion of our services to our country.

During nighttime hours, because of the intense skywave propagation then prevailing, no large number of stations can be permitted to operate on one of these channels, if the wide area service for which these frequencies are assigned is to be rendered satisfactorily by the dominant stations which must be relied upon to render it.

My husband enters the motel office, signs up for a room, and then solemnly asks the proprieter if he accepts pets.

`` We, the Subscribers, do agree, that as soon as a convenient Number of Persons have subscribed to this, or a similar Writing, We will present a petition to the Hon'ble General Court of the Commonwealth of Massachusetts, praying for an Act incorporating into a Body politic the subscribers to such Writing with Liberty to build such a Bridge, and a Right to demand a Toll equal to that received at Malden Bridge, and on like Terms, and if such an Act shall be obtained, then we severally agree each with the others, that we will hold in the said Bridge the several shares set against our respective Names, the whole into two hundred shares being divided, and that we will pay such sums of Money at such Times and in such Manners, as by the said proposed Corporation, shall be directed and required ''.

You can save all that shimming time if you set your windows in one, two, three order -- first the stud on one side, then the window, then the stud on the other side.

( C ) if Af is the operator induced on Af by T, then the minimal polynomial for Af is Af.

Note also that if Af, then Af is divisible by the polynomial p, because Af contains each Af as a factor.

It is clear that each vector in the range of Af is in Af for if **ya is in the range of Af, then Af and so Af because Af is divisible by the minimal polynomial P.