Mathematically, two Luenberger observers can be used, if is properly selected, using, for example, positive systems properties: one for the upper bound ( that ensures that converges to zero from above when, in the absence of noise and uncertainty ), and a lower bound ( that ensures that converges to zero from below ).

Mathematically, the inverse relation may be stated with causal relation as:

Mathematically, Bayes ' theorem gives the relationship between the probabilities of and, and, and the conditional probabilities of given and given, and.

Mathematically, the Levenshtein distance between two strings is given by where

Mathematically, linear least squares is the problem of approximately solving an overdetermined system of linear equations, where the best approximation is defined as that which minimizes the sum of squared differences between the data values and their corresponding modeled values.

Mathematically, the difference between artificial perspective projection ( perspective projection onto a flat surface ) and natural perspective projection ( perspective projection onto a spherical surface ) is a distortion resulting from the small-angle approximation which results from projecting onto a flat surface.

Mathematically, if each question is structured to eliminate half the objects, 20 questions will allow the questioner to distinguish between 2 < sup > 20 </ sup > or 1, 048, 576 objects.

Mathematically, a quantum operation is a linear map Φ between spaces of trace class operators on Hilbert spaces H and G such that

Mathematically, the cyclomatic complexity of a structured program is defined with reference to the control flow graph of the program, a directed graph containing the basic blocks of the program, with an edge between two basic blocks if control may pass from the first to the second.

Mathematically, this is represented as a ' hopping integral ' or ' transfer integral ' between neighboring atoms, which can be viewed as the physical principle that creates electron bands in crystalline materials, due to overlapping between atomic orbitals.

Mathematically, the number of points in the inner circle is equal to the number of points in the outer circle, i. e. there is a bijection between them.

Mathematically, the relationship between a cell and its neighbors, located within an area of influence, can be defined by a coupling law, and this is what primarily determines the behavior of the processor.

Mathematically, the return from a short position is equivalent to that of owning ( being " long ") a negative amount of the instrument.

Mathematically, we can describe the reciprocal lattice as the set of all vectors K that satisfy the above identity for all lattice point position vectors R. This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice, which reveals the Pontryagin duality of their respective vector spaces.

Mathematically, helicity is the sign of the projection of the spin vector onto the momentum vector: left is negative, right is positive.

Mathematically, this is because in reversible processes, the change in entropy of the cold reservoir is the negative of that of the hot reservoir ( i. e., ), keeping the overall change of entropy zero.

Mathematically, the extremely wideband excitation provided by the spark gap was bandpass filtered by the self-oscillating antenna side circuit, which because of its simple construction, also had a relatively broad and badly controlled filter characteristic.

Mathematically the switching mixer is not much different from a multiplying mixer, just because instead of the LO sine wave term we would use the signum function.

Mathematically ideal drums with membranes of these two different shapes ( but otherwise identical ) would sound the same, because the eigenfrequency | eigenfrequencies are all equal, so the Timbre # Spectra | timbral spectra would contain the same overtones.

Mathematically, nominal numbering is a one-to-one function from a set of objects being named to the set of numerals: it is a function because each object is assigned a numeral, and it is one-to-one because different objects are assigned different numerals.

Mathematically, this says that the state vector is confined to one of the two eigenspaces of P, and is not allowed to range over the entire Hilbert space.

Mathematically this invariance can be ensured in one of two ways: by treating the four-vectors as Euclidean vectors and multiplying time by the square root of ; or by keeping time a real quantity and embedding the vectors in a Minkowski space.

Mathematically, the factor of two relating physical angles to their counterparts in Stokes space derives from the use of second-order moments and correlations, and incorporates the loss of information due to absolute phase invariance.

Let this property be represented by just one scalar variable, q, and let the volume density of this property ( the amount of q per unit volume V ) be ρ, and the all surfaces be denoted by S. Mathematically, ρ is a ratio of two infinitesimal quantities:

Mathematically, the operator uses two 3 × 3 kernels which are convolved with the original image to calculate approximations of the derivatives-one for horizontal changes, and one for vertical.

Mathematically, the modulus of resilience can be expressed by the product of the square of the yield stress divided by two times the Young's modulus.

Mathematically speaking, two quantum states and are separated by a selection rule if for any given Hamiltonian, while they are separated by a superselection rule if for all physical observables.

Mathematically, the number of strands is the greatest common divisor of the number of leads and the number of bights ; the knot may be tied with a single strand if and only if the two numbers are coprime.

Mathematically stated, a collision attack finds two different messages m1 and m2, such that hash ( m1 ) = hash ( m2 ).

Mathematically stated, given two different prefixes p1, p2, the attack finds two appendages m1 and m2 such that hash ( p1 ∥ m1 ) = hash ( p2 ∥ m2 ) ( where ∥ is the concatenation operation ).

Mathematically, the process of constructing a flownet consists of contouring the two harmonic or analytic functions of potential and stream function.

Mathematically, the carbonate anion concentration is counted twice due to its ability to neutralize two protons, while bicarbonate is counted once as it can neutralize one proton.

Mathematically, the operator uses two 3 × 3 kernels which are convolved with the original image to calculate approximations of the derivatives-one for horizontal changes, and one for vertical.

Mathematically, the URR is closely related to Kt / V, and the two quantities can be derived from another with more or less precision, depending on the amount of additional information available about a given dialysis session.

Mathematically, the MCDM problem corresponding to the above arguments can be represented as

Mathematically, this means that for a causal linear system to be stable all of the poles of its transfer function must have negative-real values, i. e. the real part of all the poles are less than zero.

Mathematically speaking, magnetic fields and electric fields are convertible with relative motion as a 2nd-order tensor or bivector.

Mathematically, the formula above entails that all incomes are at least the lower bound x < sub > m </ sub >, which is positive.

Mathematically it is a manifold consisting of " events " which are described by some type of coordinate system.

David Sklansky, in his book The Theory of Poker, states " Mathematically, the optimal bluffing strategy is to bluff in such a way that the chances against your bluffing are identical to the pot odds your opponent is getting.

Mathematically they are understood through parallel transport.

Mathematically, classical physics equations are ones in which Planck's constant does not appear.

Mathematically, corrosion rates of glasses are characterized by normalized corrosion rates of elements NR < sub > i </ sub > ( g / cm < sup > 2 </ sup >· d ) which are determined as the ratio of total amount of released species into the water M < sub > i </ sub > ( g ) to the water-contacting surface area S ( cm < sup > 2 </ sup >), time of contact t ( days ) and weight fraction content of the element in the glass f < sub > i </ sub >:

Mathematically speaking, there are seventeen basic patterns, described as wallpaper groups, that can be used to tile an infinite plane.

Mathematically speaking, a sphere and a plane are not isometric, even locally.

Mathematically, removing the rotational symmetry by placing the " 20 " at the top, there are 19 !, or 121, 645, 100, 408, 832, 000 possible dartboards.

Mathematically, majorans may be modeled by allowing them to propagate through a material while all other Standard Model forces are fixed to an orbifold point.

Mathematically, dislocations are a type of topological defect, sometimes called a soliton.

Mathematically, spin networks are related to group representation theory and can be used to construct knot invariants such as the Jones polynomial.

Mathematically, there are exactly q < sup > m </ sup > possible messages of length m, and each such message can be regarded as a vector of length m. The encoding scheme converts an m-dimensional vector into an n-dimensional vector.

Mathematically, barrel and pincushion distortion are quadratic, meaning they increase as the square of distance from the center.