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Miller indices of a plane (hkl) and a direction [hkl].The intercepts on the axes are at a/ h, b/ k and c/ l. The International Union of Crystallography (IUCr) gives the following definition: "The law of rational indices states that the intercepts, OP, OQ, OR, of the natural faces of a crystal form with the unit-cell axes a, b, c are inversely proportional to prime integers, h, k, l.
The free indices in a tensor expression always appear in the same (upper or lower) position throughout every term, and in a tensor equation the free indices are the same on each side. Dummy indices (which implies a summation over that index) need not be the same, for example:
The contact goniometer was the first instrument used to measure the interfacial angles of crystals. The International Union of Crystallography (IUCr) gives the following definition: "The law of the constancy of interfacial angles (or 'first law of crystallography') states that the angles between the crystal faces of a given species are constant, whatever the lateral extension of these faces ...
Sylvester's law of inertia (quadratic forms) Sylvester–Gallai theorem (plane geometry) Symmetric hypergraph theorem (graph theory) Symphonic theorem (triangle geometry) Synge's theorem (Riemannian geometry) Sz.-Nagy's dilation theorem (operator theory) Szegő limit theorems (mathematical analysis) Szemerédi's theorem (combinatorics)
In Cartesian coordinates (x, y, z), let the region y < 0 have refractive index n 1, intrinsic admittance Y 1, etc., and let the region y > 0 have refractive index n 2, intrinsic admittance Y 2, etc. Then the xz plane is the interface, and the y axis is normal to the interface (see diagram).
In mathematics, a law is a formula that is always true within a given context. [1] Laws describe a relationship , between two or more expressions or terms (which may contain variables ), usually using equality or inequality , [ 2 ] or between formulas themselves, for instance, in mathematical logic .
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These are the three main logarithm laws/rules/principles, [3] from which the other properties listed above can be proven. Each of these logarithm properties correspond to their respective exponent law, and their derivations/proofs will hinge on those facts. There are multiple ways to derive/prove each logarithm law – this is just one possible ...