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A root of a polynomial P(z) is a number z_i such that P(z_i)=0. The fundamental theorem of algebra states that a polynomial P(z) of degree n has n roots, some of which may be ...

The ith Pontryagin class of a vector bundle is (-1)^i times the ith Chern class of the complexification of the vector bundle. It is also in the 4ith cohomology group of the ...

Let A be a C^*-algebra, then a linear functional f on A is said to be positive if it is a positive map, that is f(a)>=0 for all a in A_+. Every positive linear functional is ...

You buy 100 pounds of potatoes and are told that they are 99% water. After leaving them outside, you discover that they are now 98% water. The weight of the dehydrated ...

The algebra A is called a pre-C^*-algebra if it satisfies all conditions to be a C^*-algebra except that its norm need not be complete.

The problem of forecasting future values X_(t+tau) (tau>0) of a weakly stationary process {X_t} from the known values X_s (s<=t).

The present value v_n of a single payment made at n periods in the future is v_n=p/((1+r)^n), (1) where n is the number of periods until payment, p is the payment amount, and ...

A primary ideal is an ideal I such that if ab in I, then either a in I or b^m in I for some m>0. Prime ideals are always primary. A primary decomposition expresses any ideal ...

Find two numbers such that x^2=y^2 (mod n). If you know the greatest common divisor of n and x-y, there exists a high probability of determining a prime factor. Taking small ...

The prime subfield of a field F is the subfield of F generated by the multiplicative identity 1_F of F. It is isomorphic to either Q (if the field characteristic is 0), or ...