Any polynomial may be decomposed into the product of a constant by a product of irreducible polynomials.

Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.

Any polynomial can be easily expressed as a power series around any center c, albeit one with most coefficients equal to zero.

Any polynomial written in standard form has a unique constant term, which can be considered a coefficient of x < sup > 0 </ sup >.

Any quadratic polynomial over the complex numbers ( polynomials of the form where,, and ∈ ) can be factored into an expression with the form using the quadratic formula.

Any rational function of a real variable can be written as the sum of a polynomial function and a finite number of algebraic fractions.

Any polynomial in D with function coefficients is also a differential operator.

Any single-variable quadratic polynomial may be written as

Any quadratic polynomial with two variables may be written as

Any monic polynomial is the characteristic polynomial of some matrix.

Any general polynomial of degree n

Any holomorphic modular form for the modular group can be written as a polynomial in and.

Any polynomial, and more generally any rational function p ( x )/ q ( x ) where p and q are polynomials, transforms the Riemann sphere by mapping it to itself.

# Any real number can be added to both sides.

# Any real number can be subtracted from both sides.

# Any real number can be multiplied to both sides.

# Any non-zero real number can divide both sides.

Any given individual or organization attempting to implement this model in a real situation would be unable to comply with the three requirements.

* Any Lie group G defines an associated real Lie algebra.

* Any closed interval b of real numbers is Lebesgue measurable, and its Lebesgue measure is the length b − a.

Any real mode operating system ( such as Xenix ) could also be made to run using OS / 2's virtual machine capabilities, subject to certain direct hardware access limitations.

Any Dedekind-complete ordered field is isomorphic to the real numbers.

Any poles on the imaginary axis have real strictly positive residues, and similarly at any zeros on the imaginary axis, the function has a real strictly positive derivative.

* Any field, including fields of rational numbers, real numbers, and complex numbers.

Any non-negative real number can be represented as a base-φ numeral using only the digits 0 and 1, and avoiding the digit sequence " 11 "-this is called a standard form.

Any Taylor series for this function converges not only for x close enough to x < sub > 0 </ sub > ( as in the definition ) but for all values of x ( real or complex ).

Any real analytic function on some open set on the real line can be extended to a complex analytic function on some open set of the complex plane.

Any destruction by the Occupying Power of real or personal property belonging individually or collectively to private persons, or to the State, or to other public authorities, or to social or cooperative organizations, is prohibited, except where such destruction is rendered absolutely necessary by military operations.

Furthermore, in the opening credits there is a mock disclaimer which reads ( in translation ): “ Any resemblance to real events, to persons dead or living, is not accidental.

Any real advantages to the system were far outweighed by the additional cost of manufacture associated with the device.

Any hope of real progress by the Poles was then lost.

Any entity from the real or virtual world that lends itself to dramatic interpretation may be taken up as a subject.

* Any open subset of a finite-dimensional real, and therefore complex, vector space is a diffeological space.

Any public employee who fell under his power would be replaced by someone in whom he saw real ability instead of a mere useful recipient of a favor.

Any complex number z = x + iy can be written as

Any non-trivial true statement about reality is necessarily an abstraction composed of a complex of objects and properties or relations.

* Any narrow complex tachycardia combined with a problem with the conduction system of the heart, often termed " supraventricular tachycardia with aberrancy.

To show that U ( n ) is connected, recall that any unitary matrix A can be diagonalized by another unitary matrix S. Any diagonal unitary matrix must have complex numbers of absolute value 1 on the main diagonal.

Any number system that forms a commutative ring — for instance, the integers, the real numbers, and the complex numbers — can be extended to a wheel in which division by zero is always possible ; however, in such a case, " division " has a slightly different meaning.

Any complex number may be written a + bi, where a and b are real numbers and i is the imaginary unit.

Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions.

* Any new system of romanization, regardless of its merits, makes romanization choices more complex rather than more simple.

Any product where one material slides or rubs over another is affected by complex tribological interactions, whether lubricated like hip implants and other artificial prostheses, or unlubricated as in high temperature sliding wear in which conventional lubricants cannot be used but in which the formation of compacted oxide layer glazes have been observed to protect against wear.

Any of these conditions implies the existence of a unique compatible complex structure.

Through an evaluation and bidding process, the Apple II was chosen by MECC for state schools over other candidates, such as the Radio Shack TRS-80 ; the win was an important early deal in the history of Apple Inc. Any school in the state could buy Apple computers through MECC for a substantial discount without having to go through complex evaluation and purchasing procedures.

Any “ nice ” space such as a manifold or CW complex is semi-locally simply connected.

Any such complex torus has the Gaussian integers as endomorphism ring.

Any conception of a traditional market for making beneficial exchanges, such as an agricultural market or trading pit, or any system where individuals respond to posted prices on a computer screen is woefully inadequate for the extremely large number of often complex trades that will be required.

Any tool that tries to simplify a complex picture will introduce distortions, whether it is a geographic map or a set of scenarios.

Any investment in a war would be expected to bring a return of plunder and prisoners, which became part of the endowment of the palace complex.

Any basis for V over the real numbers serves as a basis for V < sup > C </ sup > over the complex numbers.

Any continuous periodic function can be represented as a complex harmonic motion using its fourier series.