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.

However, in what newspapers such as the Wall Street Journal later call the " math wars ", organizations such as Mathematically Correct complain that some elementary texts based on the standards, including Mathland, have almost entirely abandoned any instruction of traditional arithmetic in favor of cutting, coloring, pasting, and writing.

Mathematically, Williams or McLaren could have won the championship at the very last race.

Mathematically oriented quantitative analysts tend to have more of a reliance on numerical analysis, and less of a reliance on statistics and econometrics.

Mathematically, quadrant II and quadrant IV have coordinates ( 0, 0 ) in a coordinate plane where ε is the horizontal axis, and µ is the vertical axis.

Mathematically, this requires a stiffness matrix to have full rank.

Mathematically, the mass matrix of the structure must have full rank.

Such omissions were criticized by such groups as Mathematically Correct, and some states and districts have since abandoned this experiment, though it remains widely used.

Mathematically, in order to have a reasonable chance to obtain one molecule of the original extract, the patient would have to consume an amount of the remedy roughly 10 < sup > 321 </ sup > times the number of atoms in the observable universe.

Mathematically it may look as though all of the fields are vector-valued ( in addition to being operator-valued ), since they all have several components, can be multiplied by matrices, etc., but physicists assign a more specific meaning to the word: a vector is something which transforms like a four-vector under Lorentz transformations, and a scalar is something which does not transform under Lorentz transformations.

Mathematically, homogeneity has the connotation of invariance, as all components of the equation have the same degree of value whether or not each of these components are scaled to different values, for example, by multiplication or addition.

Mathematically, a Pareto distribution has often been used to quantify the distribution of wealth, since it models an unequal distribution.

He founded the Johns Hopkins University Center for Talented Youth ( CTY ), as well as a related research project, the Study of Mathematically Precocious Youth ( SMPY ), whose work has, since 1980, been supplemented by the Julian C. Stanley Study of Exceptional Talent ( SET ), which provides academic assistance to gifted children.

Mathematically, the input-output relationship of many devices should be described by a polynomial or Taylor series, as shown below.

The instrument used for assessing environmental thermal comfort in terms of operative temperature is called a eupatheoscope and was invented by A. F. Dufton in 1929 .< ref > Glossary of Meteorology, American Meteorological Society, < http :// amsglossary. allenpress. com / glossary / browse? s = e & p = 42 >, accessed Sept 2010 </ ref > Mathematically, operative temperature can be shown as ;

Mathematically, this can be represented for n species as:

Mathematically, a baseband modulated signal may be approximated by a sinusoidal continuous wave signal with a frequency f < sub > m </ sub >.

Mathematically the 0-1-knapsack problem can be formulated as:

Mathematically the bounded knapsack problem can be formulated as:

Mathematically, the potential of the ion cores can be treated by various models, the simplest being the nearly free electron model.

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, a neuron's network function is defined as a composition of other functions, which can further be defined as a composition of other functions.

Mathematically, the equation may be rearranged to compute torque for a given power output.

Mathematically, the modulated wave can be written in the form:

Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions.

Mathematically ( using the Laplace transform ) the impedance of this network can be written as:

Mathematically, the E field can be thought of as a function that associates a vector with every point in space.

Mathematically, convection can be described by the convection – diffusion equation or the generic scalar transport equation.

Mathematically, the pressure of a mixture of gases can be defined as the summation

Mathematically corrected punch motions can be programmed to simulate any type and model of production tablet press.

Mathematically, it refers to a property of solutions to the Schrödinger equation ; since the Schrödinger equation is linear, any linear combination of solutions to a particular equation will also be a solution of it.

Mathematically, assuming the terminator lies on a great circle, such a crescent will actually be the figure bounded by a half-ellipse and a half-circle, with the major axis of the ellipse coinciding with a diameter of the semicircle.

Mathematically, this can be expressed by means of the signal to noise ratio ( SNR or S / N ).

Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U. The region U may be a set in some Euclidean space, Minkowski space, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order.

Mathematically, it may be represented as follows:

Mathematically, the Lorentz group may be described as the generalized orthogonal group O ( 1, 3 ), the matrix Lie group which preserves the quadratic form

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

Mathematically, we shall not distinguish the experiment of three dice tossed once from that of one die tossed three times.

Mathematically, the total electromagnetic energy density ( radiation energy density ) in thermodynamic equilibrium from Planck's law is

Mathematically, the light received from stars as a function of star distance in a hypothetical fractal cosmos is:

Mathematically, this follows from:

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.

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

Mathematically gifted, he was very young when he joined Enrico Fermi's team in Rome as one of the " Via Panisperna boys ", who took their name from the street address of their laboratory.

Mathematically, the climatological annual cycle is commonly estimated from observational data or model output by taking the average of all Januaries, all Februaries, and so forth.

Mathematically talented, Lamarr also co-invented — with composer George Antheil — an early technique for spread spectrum communications and frequency hopping, necessary for wireless communication from the pre-computer age to the present day.

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, 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, apart from discretization ( whole euros both for income and for tax ), the tax is a continuous, convex, piecewise linear function of income.

Mathematically, the axiom states that the probability of selecting item i from a pool of j items is given by:

Mathematically, texture filtering is a type of anti-aliasing, but it filters out high frequencies from the texture fill whereas other AA techniques generally focus on visual edges.

Mathematically a stock can be seen as an accumulation or integration of flows over time-with outflows subtracting from the stock.

Mathematically, the problem of recovering conductivity from surface measurements of current and potential is a non-linear inverse problem and is severely ill-posed.

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

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 eliminated from the chase.