If she wanted to borrow any sum of money in expecting the arrangements of Congress, it would not become a stranger, unknown to her, to offer himself for that purpose.

If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

If one takes the middle number, 5, and multiplies it by 3 ( the base number of the magic square of three ), the result is 15, which is also the constant sum of all the rows, columns, and two main diagonals.

* 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 more than one wavelength is present ( as is usually the case ), one must sum or integrate over the spectrum of wavelengths present to get the total luminous intensity.

If the shooter " hits " the point value again ( any value of the dice that sum to the point will do ; the shooter doesn't have to exactly repeat the value combination of the come-out roll ) before rolling a seven, the Pass line wins and a new round starts, usually with the dice staying with the current shooter.

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

Augustine of Hippo in De Civitate Dei writes Si fallor, sum (" If I am mistaken, I am ") ( book XI, 26 ), and also anticipates modern refutations of the concept.

The parallel postulate: If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

If we allow to incorporate the algebraic sign, this becomes a sum and implies that there is a function of state which is conserved over a complete cycle.

If the two input signals are both sinusoids of specified frequencies f < sub > 1 </ sub > and f < sub > 2 </ sub >, then the output of the mixer will contain two new sinsoids that have the sum f < sub > 1 </ sub > + f < sub > 2 </ sub > frequency and the difference frequency absolute value | f < sub > 1 </ sub >-f < sub > 2 </ sub >|.

If the password is correct, then M releases the transferred sum to B ( 3b ), usually minus a small commission.

If R is the direct sum of the rings R < sub > 1 </ sub >,..., R < sub > n </ sub >, then the identity elements of the rings R < sub > i </ sub > are central idempotents in R, pairwise orthogonal, and their sum is 1.

If not, it is a sum of central orthogonal idempotents, which in turn are primitive or sums of more central idempotents, and so on.

If a decomposition exists with each c < sub > i </ sub > a centrally primitive idempotent, then R is a direct sum of the corner rings c < sub > i </ sub > Rc < sub > i </ sub >, each of which is ring irreducible.

If a crest of a wave meets a crest of another wave of the same frequency at the same point, then the magnitude of the displacement is the sum of the individual magnitudes – this is constructive interference.

If a merchant received a loan to fund his shipment, he would pay the lender an additional sum in exchange for the lender's guarantee to cancel the loan should the shipment be stolen or lost at sea.

# If A is a disjoint union of countably many disjoint Lebesgue measurable sets, then A is itself Lebesgue measurable and λ ( A ) is equal to the sum ( or infinite series ) of the measures of the involved measurable sets.

If the field had characteristic p > 0, then − 1 would be the sum of p − 1 ones, but − 1 is not positive ).

If then we obtain a contradiction, because the sum does not exceed which is finite.

On 11 June 1304, with both of them having witnessed the heroic efforts of their countrymen during King Edward's siege of Stirling Castle, Bruce and William Lamberton made a pact that bound them, each to the other, in “ friendship and alliance against all men .” If one should break the secret pact, he would forfeit to the other the sum of ten thousand pounds.

If Wallace was indeed an archer he must have been a professional, worth paying a reasonable sum of money for military services.

If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero.

If any two functions are solutions to Laplace's equation ( or any linear homogeneous differential equation ), their sum ( or any linear combination ) is also a solution.

If one also removes the second postulate (" a line can be extended indefinitely ") then elliptic geometry arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees.

If the orbiting periods were in this relation, the mean motions ( inverse of periods, often expressed in degrees per day ) would satisfy the following

If the phase difference is 180 degrees ( π radians ), then the two oscillators are said to be in antiphase.

If the rotor disk is to be tilted forward ( to gain forward velocity ), its rotation requires that the downward net force on the blade be applied roughly 90 degrees ( depending on blade configuration ) before, or when the blade is to one side of the pilot and rotating forward.

If the inclination is zero or 180 degrees ( π radians ), the azimuth is arbitrary.

If the woman stands upright, the vaginal tube points in an upward-backward direction and forms an angle of slightly more than 45 degrees with the uterus.

If the handle is turned ninety degrees between operating positions, the valve is called a quarter-turn valve.

If directional shear between the surface and the height in the atmosphere at which the barometric pressure measures is greater than 60 degrees, nothing more than flurries can be expected.

If the directional shear between the body of water and the vertical height at which the pressure measures is between 30 and 60 degrees, weak lake-effect bands are possible.

The later invention of non-Euclidean geometry does not resolve this question ; for one might as well ask, " If given the axioms of Riemannian geometry, can an omnipotent being create a triangle whose angles do not add up to more than 180 degrees?

If we rotate the rod again in the same directions so that it will have completed a 720 degrees rotation in total the strings can be untangled without rotating any part ( e. g. by " sliding " the rods and / or stretching the strings ).

If he occupies himself with dthikr, He ( God ) will unveil to the secrets of the vegetable world, then the secrets of the animal world, then the infusion of the world of life-force into lives, then the " surface sign " ( the light of the Divine Names, according to Abdul-Karim al-Jeeli, the book's translator ), then the degrees of speculative sciences, then the world of formation and adornment and beauty, then the degrees of the qutb ( the soul or pivot of the universe-see # 16 ) ( 59 ) Then he will be given the divine wisdom and the power of symbols and authority over the veil and the unveiling.

If the angle of the fault plane is low ( generally less than 20 degrees from the horizontal ) and the displacement of the overlying block is large ( often in the kilometer range ) the fault is called an overthrust.

If the wind in blows parallel to the coast in the southern hemisphere ( such as along the coast of Peru, where the wind blows North ), then Ekman transport produces a net movement of surface water 90 degrees to the left.

If two sides of the triangle are equal to R and the length of the third side, the chord, is given as 100 feet and the angle C opposite the chord is given in degrees, then

If precipitation is falling down in the area, the temperature decreases by several degrees, like most of the Southwestern US.

If a navigator measures the angle to Polaris and finds it to be 10 degrees from the horizon, then he is about 10 degrees north of the equator.

If the curved track turns the cars 60 degrees, and three rail lines meet as three equally spaced spokes, 120 degrees apart, then the curved track sitting on a turn table can be rotated to connect any two of the three rail lines at this junction, creating a triangle junction.

If A is an Toeplitz matrix, then the system has only 2n − 1 degrees of freedom, rather than n < sup > 2 </ sup >.

If the phase of βA < sub > OL </ sub >( f </ sub >< sub > 0dB </ sub >) > − 180 °, the instability condition cannot be met at any frequency ( because its magnitude is going to be < 1 when f = f < sub > 180 </ sub >), and the distance of the phase at f < sub > 0dB </ sub > in degrees above − 180 ° is called the phase margin.

** If there is one unoccupied square before or behind opposing pieces then jumps multiple times over opposing pieces in a single turn forward or backward can and must be made, making angles of 90 degrees.

If the magnetic heading changed downwards by 5 degrees to 228, the Runway would still be Runway 23.