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Arakelyan's **theorem** - **Theorem**

Arakelyan's **theorem** states that for every f continuous in E and holomorphic in the interior of E and for every ε > 0 there exists g holomorphic in Ω such that |g − f| < ε on E if and only if Ω * \ E is connected and locally connected.

Abel's **theorem** - **Theorem**

The same **theorem** holds for complex power series

PBR **theorem** - **Theorem**

This theorem, which first appeared as an arXiv preprint and was subsequently published in Nature Physics, concerns the interpretational status of pure quantum states. Under the classification of hidden variable models of Harrigan and Spekkens, the interpretation of the quantum wavefunction can be categorized as either ψ-ontic if "every complete physical state ontic state in the theory is consistent with only one pure quantum state" and ψ-epistemic "if there exist ontic states that are consistent with more than one pure quantum state." The PBR **theorem** proves that either the quantum state is ψ-ontic, or else non-entangled quantum states violate the assumption of preparation independence, which would entail action at a distance.

Kharitonov's **theorem** - **Theorem**

Kharitonov's **theorem** is useful in the field of robust control, which seeks to design systems that will work well despite uncertainties in component behavior due to measurement errors, changes in operating conditions, equipment wear and so on.

Green's **theorem** - **Theorem**

In physics, Green's **theorem** finds many applications. One is solving two-dimensional flow integrals, stating that the sum of fluid outflowing from a volume is equal to the total outflow summed about an enclosing area. In plane geometry, and in particular, area surveying, Green's **theorem** can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.

Ginsberg's **theorem** - **Theorem**

The "theorem" is given as a restatement of the consequences of the zeroth, first, second, and third laws of thermodynamics, with regard to the usable energy of a closed system:

UTM **theorem** - **Theorem**

The **theorem** states that a partial computable function u of two variables exists such that, for every computable function f of one variable, an e exists such that for all x. This means that, for each x, either f(x) and u(e,x) are both defined and are equal, or are both undefined.

Abel's **theorem** - **Theorem**

Note that G(z) is continuous on the real closed interval [0,t] for t < 1, by virtue of the uniform convergence of the series on compact subsets of the disk of convergence. Abel's **theorem** allows us to say more, namely that G(z) is continuous on [0,1].

Varignon's **theorem** - **Theorem**

The Varignon parallelogram exists even for a skew quadrilateral, and is planar whether the quadrilateral is planar or not. The **theorem** can be generalized to the midpoint polygon of an arbitrary polygon.

Foster's **theorem** - **Theorem**

Consider an irreducible discrete-time Markov chain on a countable state space S having a transition probability matrix P with elements p ij for pairs i, j in S. Foster's **theorem** states that the Markov chain is positive recurrent if and only if there exists a Lyapunov function, such that and

Tunnell's **theorem** - **Theorem**

Tunnell's **theorem** states that supposing n is a congruent number, if n is odd then 2A n = B n and if n is even then 2C n = D n. Conversely, if the Birch and Swinnerton-Dyer conjecture holds true for elliptic curves of the form, these equalities are sufficient to conclude that n is a congruent number.

UTM **theorem** - **Theorem**

The **theorem** thus shows that, defining φ e (x) as u(e, x), the sequence φ 1, φ 2 , … is an enumeration of the partial computable functions. The function u in the statement of the **theorem** is called a universal function.

Löwenheim–Skolem **theorem** - **Theorem**

The **theorem** is often divided into two parts corresponding to the two bullets above. The part of the **theorem** asserting that a structure has elementary substructures of all smaller infinite cardinalities is known as the downward Löwenheim–Skolem Theorem. The part of the **theorem** asserting that a structure has elementary extensions of all larger cardinalities is known as the upward Löwenheim–Skolem Theorem.

Gabriel–Popescu **theorem** - **Theorem**

Let A be a Grothendieck category (an AB5 category with a generator), G a generator of A and R be the ring of endomorphisms of G; also, let S be the functor from A to Mod-R (the category of right R-modules) defined by S(X) = Hom(G,X). Then the Gabriel–Popescu **theorem** states that S is full and faithful and has an exact left adjoint.

Angle bisector **theorem** - **Theorem**

The angle bisector **theorem** is commonly used when the angle bisectors and side lengths are known. It can be used in a calculation or in a proof.

Lee–Yang **theorem** - **Theorem**

In the original Ising model case considered by Lee and Yang, the measures all have support on the 2 point set −1, 1, so the partition function can be considered a function of the variable ρ = e πz. With this change of variable the Lee–Yang **theorem** says that all zeros ρ lie on the unit circle.

Pappus's area **theorem** - **Theorem**

The **theorem** generalizes the Pythagorean **theorem** twofold. Firstly it works for arbitrary triangles rather than only for right angled ones and secondly it uses parallelograms rather than squares. For squares on two sides of an arbitrary triangle it yields a parallelogram of equal area over the third side and if the two sides are the legs of a right angle the parallelogram over the third side will be square as well. For a right-angled triangle, two parallelograms attached to the legs of the right angle yield a rectangle of equal area on the third side and again if the two parallelograms are squares then the rectangle on the third side will be a square as well.

Poincaré–Bendixson **theorem** - **Theorem**

A weaker version of the **theorem** was originally conceived by Henri Poincaré, although he lacked a complete proof which was later given by.

Kelvin–Stokes **theorem** - **Theorem**

The main challenge in a precise statement of Stokes' **theorem** is in defining the notion of a boundary. Surfaces such as the Koch snowflake, for example, are well-known not to exhibit a Riemann-integrable boundary, and the notion of surface measure in Lebesgue theory cannot be defined for a non-Lipschitz surface. One (advanced) technique is to pass to a weak formulation and then apply the machinery of geometric measure theory; for that approach see the coarea formula. In this article, we instead use a more elementary definition, based on the fact that a boundary can be discerned for full-dimensional subsets of

Lee–Yang **theorem** - **Theorem**

The Lee–Yang **theorem** states that if the Hamiltonian is ferromagnetic and all the measures dμ j have the Lee-Yang property, and all the numbers z j have positive real part, then the partition function is non-zero. :In particular if all the numbers z j are equal to some number z, then all zeros of the partition function (considered as a function of z) are imaginary.