If my poor talents can be useful in any other land, they must be of some utility to Italy ; and ought not her claim to be preferred to all others?

" If I have learned a formula in gibberish which in any way jogs my memory so as to enable me in each single case to act as though I had a general idea, what possible utility is there in distinguishing between such a gibberish ... and an idea?

If you move " off " an indifference curve traveling in a northeast direction ( assuming positive marginal utility for the goods ) you are essentially climbing a mound of utility.

If the utility function is of the form then the marginal utility of is and the marginal utility of is.

If the utility function is of the form the marginal utility of is and the marginal utility of is. Where < math >

If a risk averse consumer has a utility function over consumption x, and if is differentiable, then the consumer is not prudent unless the third derivative of utility is positive, that is,.

If they choose they may remain with the utility at the Regulated Rate Option.

If the sender's and recipient's versions of the file have many sections in common, the utility needs to transfer relatively little data to synchronize the files.

If the entertainment value ( or other non-monetary value ) obtained by playing is high enough for a given individual, then the purchase of a lottery ticket could represent a gain in overall utility.

If the aim of law is to make citizens virtuous ( as opposed to maximizing utility or realizing a set of moral rights ), what are the implications for the content of the laws?

If the filesystem becomes fragmented, the only way to defragment it is to use a utility such as Coriolis System's iDefrag, or to wipe the hard drive completely and install the system from scratch.

If the project can't be self sustaining, such as an electric utility in a small village or town, it will not be approved by higher headquarters.

If this equality did not hold, the consumer could increase his / her utility by cutting spending on the good with lower marginal utility per unit of money and increase spending on the other good.

If transferring FileVault data from a previous Mac that uses 10. 4 using the built-in utility to move data to a new machine, the data continues to be stored in the old sparse image format, and the user must turn FileVault off and then on again to re-encrypt in the new sparse bundle format.

If it is assumed that, p < sub > 1 </ sub > < p < sub > 2 </ sub > then the consumer would consume only X < sup > 1 </ sup > as this would maximize his utility.

If two separate pieces of material were used, this would introduce a grain boundary between the semiconductors that severely inhibits its utility by scattering the electrons and holes.

If there is a unique utility maximizing package for each

If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

If the United States were to punish a visiting diplomat without sufficient grounds, U. S. representatives in other countries could receive harsher treatment.

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if is continuous and linear, then there exists an extension of φ which is also continuous and linear and which has the same norm as φ ( see Banach space for a discussion of the norm of a linear map ).

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if z is an element of V not in the closure of U, then there exists a continuous linear map with ψ ( x ) = 0 for all x in U, ψ ( z ) = 1, and || ψ || = 1 / dist ( z, U ).

If ƒ is complex differentiable at every point z < sub > 0 </ sub > in an open set U, we say that ƒ is holomorphic on U. We say that ƒ is holomorphic at the point z < sub > 0 </ sub > if it is holomorphic on some neighborhood of z < sub > 0 </ sub >.

Chinese immigrants were blamed for importing the opium-smoking habit to the U. S. The 1903 blue-ribbon citizens ' panel, the Committee on the Acquirement of the Drug Habit concluded, " If the Chinaman cannot get along without his dope we can get along without him.

“ The U. S. Fish and Wildlife Service-funded Ivory-billed Woodpecker searches will continue through the 2008-09 search season ,” says Laurie Fenwood, Ivory-billed Woodpecker Recovery Team Coordinator for the U. S, Fish and Wildlife Service .… If no birds are confirmed, the Cornell Lab of Ornithology will not send an organized team into the field next year.

If I manage to leave for the U. S., then I will be over there.

If a judge determines that there is sufficient evidence to believe that the defendant committed the crime, it is said that the defendant is " held to answer " or " bound over " ( in U. S. jurisdictions ).

If rotations are also considered distinct, then the pentominoes from the first category count eightfold, the ones from the next three categories ( T, U, V, W, Z ) count fourfold, I counts twice, and X counts only once.

If one considers a version of QCD with N < sub > f </ sub > flavors of massless quarks, then there is a global ( chiral ) flavor symmetry group SU < sub > L </ sub >( N < sub > f </ sub >) × SU < sub > R </ sub >( N < sub > f </ sub >) × U < sub > B </ sub >( 1 ) × U < sub > A </ sub >( 1 ).

If an electron of energy E is incident upon an energy barrier of height U ( z ), the electron wave function is a traveling wave solution,

If the bias is small, we can let U − E ≈ φM in the expression for κ, where φM, the work function, gives the minimum energy needed to bring an electron from an occupied level, the highest of which is at the Fermi level ( for metals at T = 0 kelvins ), to vacuum level.

If U. S. officials can " help clean up his image " and lift the ban on arms sales to the Panamanian Defense Force, Noriega will "' take care of ' the Sandinista leadership for us.

But they soon had another US No. 2 hit with a cover of Simon & Garfunkel's " A Hazy Shade of Winter " from the soundtrack of the film Less Than Zero, and the melancholic " If She Knew What She Wants ", written by Jules Shear, reached the U. S. Top 30 and the German Top 20.

The U. S. Supreme Court explained this, in U. S. Public Workers v. Mitchell: " If granted power is found, necessarily the objection of invasion of those rights, reserved by the Ninth and Tenth Amendments, must fail.

* If S and T are in M with S ⊆ T then T − S is in M and a ( T − S ) =

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

If the object point O is infinitely distant, u1 and u2 are to be replaced by h1 and h2, the perpendicular heights of incidence ; the sine condition then becomes sin u ' 1 / h1 = sin u ' 2 / h2.

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 F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

* If the operation is associative, ( ab ) c = a ( bc ), then the value depends only on the tuple ( a, b, c ).

* If the operation is commutative, ab = ba, then the value depends only on

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

If the sets A and B are equal, this is denoted symbolically as A = B ( as usual ).

If a problem can be shown to be in both NP and co-NP, that is generally accepted as strong evidence that the problem is probably not NP-complete ( since otherwise NP = co-NP ).

If the user pressed keys 1 + 2 = 3 simultaneously the letter " c " appeared.

If the ideals A and B of R are coprime, then AB = A ∩ B ; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals.

If κ is an infinite cardinal number, then cf ( κ ) is the least cardinal such that there is an unbounded function from it to κ ; and cf ( κ ) = the cardinality of the smallest collection of sets of strictly smaller cardinals such that their sum is κ ; more precisely

If the disk was not otherwise prepared with a custom format, ( e. g. for data disks ), 664 blocks would be free after formatting, giving 664 × 254 = 168, 656 bytes ( or almost 165 kB ) for user data.

This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit x, then necessarily x < sup > 2 </ sup > = 2, yet no rational number has this property.

If y = f ( x ) is differentiable at a, then f must also be continuous at a.

If a vector field F with zero divergence is defined on a ball in R < sup > 3 </ sup >, then there exists some vector field G on the ball with F = curl ( G ).

If in the third identity we take H = G, we get that the set of commutators is stable under any endomorphism of G. This is in fact a generalization of the second identity, since we can take f to be the conjugation automorphism.

Linear Diophantine equations take the form ax + by = c. If c is the greatest common divisor of a and b then this is Bézout's identity, and the equation has an infinite number of solutions.

It follows that there are also infinitely many solutions if c is a multiple of the greatest common divisor of a and b. If c is not a multiple of the greatest common divisor of a and b, then the Diophantine equation ax + by = c has no solutions.