query: the term that describes a function that is symmetric about the y-axis is

pos: Even and odd functions all formula_6 and formula_7 in the domain of formula_5: Geometrically speaking, the graph face of an even function is symmetric with respect to the "y"-axis, meaning that its graph remains unchanged after reflection about the "y"-axis. Examples of even functions are: Again, let formula_4 be a real-valued function of a real variable. Then formula_5 is odd if the following equation holds for all formula_6 and formula_7 in the domain of formula_5: or equivalently if the following equation holds for all formula_6 and formula_7 in the domain of formula_5: Geometrically, the graph of an odd function has rotational symmetry with respect

pos: Symmetry in mathematics "x" and "-x" in the domain of "f": Geometrically speaking, the graph face of an even function is symmetric with respect to the "y"-axis, meaning that its graph remains unchanged after reflection about the "y"-axis. Examples of even functions are x, "x", "x", cos("x"), and cosh("x"). Again, let "f"("x") be a real-valued function of a real variable. Then "f" is odd if the following equation holds for all "x" and "-x" in the domain of "f": or Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation

pos: Constant function polynomial of degree 0 and its general form is formula_11 . This function has no intersection point with the "x"-axis, that is, it has no root (zero). On the other hand, the polynomial formula_12 is the identically zero function. It is the (trivial) constant function and every "x" is a root. Its graph is the "x"-axis in the plane. A constant function is an even function, i.e. the graph of a constant function is symmetric with respect to the "y"-axis. In the context where it is defined, the derivative of a function is a measure of the rate of change

pos: Abel transform actually at "x" = ∞ so that the limits of integration are ±∞ and all lines of sight are parallel to the "x"-axis. Realizing that the radius "r" is related to "x" and "y" via "r" = "x" + "y", it follows that for "x>0". Since "f"("r") is an even function in "x", we may write: which yields the Abel transform of "f(r)". The Abel transform may be extended to higher dimensions. Of particular interest is the extension to three dimensions. If we have an axially symmetric function "f"(ρ",z") where ρ = "x" + "y" is the cylindrical radius, then

pos: Support function instance, a set "A" is point symmetric with respect to the origin if and only "h" is an even function. In general, the support function is not differentiable. However, directional derivatives exist and yield support functions of support sets. If "A" is "compact" and convex, and "h"'("u";"x") denotes the directional derivative of "h" at "u" ≠ "0" in direction "x", we have Here "H"("u") is the supporting hyperplane of "A" with exterior normal vector "u", defined above. If "A" ∩ "H"("u") is a singleton {"y"}, say, it follows that the support function is differentiable at "u" and its gradient coincides

neg: Molecular term symbol situations as, e.g., Π, the ± is omitted. For the Σ states, however, this two-fold degeneracy disappears, and all Σ states are either symmetric under any plane containing the internuclear axis, or antisymmetric. These two situations are labeled as Σ or Σ. Taking the molecular center of mass as origin of coordinates, consider the change of all electrons' position from ("x", "y", "z") to (−"x", −"y", −"z"). If the resulting wave function is unchanged, it is said to be "gerade" (German for even); if the wave function changes sign then it is said to be "ungerade" (odd). For a molecule

neg: Abel transform we may want to know the projection of that function onto a plane parallel to the "z" axis. Without loss of generality, we can take that plane to be the "yz"-plane so that which is just the Abel transform of "f"(ρ",z") in ρ and "y". A particular type of axial symmetry is spherical symmetry. In this case, we have a function "f"("r") where "r" = "x" + "y" + "z". The projection onto, say, the "yz"-plane will then be circularly symmetric and expressible as "F"("s") where "s" = "y" + "z". Carrying out the integration, we have: which is again,

neg: Folium of Descartes to write y = px and solve for x and y in terms of p. This yields the rational parametric equations: formula_6. We can see that the parameter is related to the position on the curve as follows: Another way of plotting the function can be derived from symmetry over y = x. The symmetry can be seen directly from its equation (x and y can be interchanged). By applying rotation of 45° CW for example, one can plot the function symmetric over rotated x axis. This operation is equivalent to a substitution: and yields Plotting in the cartesian system

neg: Statistics doing regression. Least squares applied to linear regression is called ordinary least squares method and least squares applied to nonlinear regression is called non-linear least squares. Also in a linear regression model the non deterministic part of the model is called error term, disturbance or more simply noise. Both linear regression and non-linear regression are addressed in polynomial least squares, which also describes the variance in a prediction of the dependent variable (y axis) as a function of the independent variable (x axis) and the deviations (errors, noise, disturbances) from the estimated (fitted) curve. Measurement processes that generate statistical data

neg: Commutative property the resulting function is symmetric across the line "y = x". As an example, if we let a function "f" represent addition (a commutative operation) so that "f"("x","y") = "x" + "y" then "f" is a symmetric function, which can be seen in the adjacent image. For relations, a symmetric relation is analogous to a commutative operation, in that if a relation "R" is symmetric, then formula_16. In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as "x" (meaning multiply by "x"), and formula_17. These two operators do not commute as may be seen

neg: Multiple integral respect to the -axis, and so the only contribution to the final result is that of the third integral. Therefore the original integral is equal to the area of the disk times 5, or 5. Example 2. Consider the function and as integration region the sphere with radius 2 centered at the origin, The "ball" is symmetric about all three axes, but it is sufficient to integrate with respect to -axis to show that the integral is 0, because the function is an odd function of that variable. This method is applicable to any domain for which: Such a domain

neg: X–Y plotter appear continuous to the naked eye, it in fact is a discrete set of points. "See Plotter for a more detailed explanation of the function of a plotter." X–Y plotter An X–Y plotter is a plotter that operates in two axes of motion ("X" and "Y") in order to draw continuous vector graphics. The term was used to differentiate it from standard plotters which had control only of the "y" axis, the "x" axis being continuously fed to provide a plot of some variable with time. Plotters differ from Inkjet and Laser printers in that a plotter draws a continuous

neg: Riemann zeta function that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line . For sums involving the zeta-function at integer and half-integer values, see rational zeta series. The reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius function : for every complex number with real part greater than 1. There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series. The

neg: V-statistic "x" is a sample. In typical applications the statistical function has a representation as the V-statistic where "h" is a symmetric kernel function. Serfling discusses how to find the kernel in practice. "V" is called a V-statistic of degree "m". A symmetric kernel of degree 2 is a function "h"("x", "y"), such that "h"("x", "y") = "h"("y", "x") for all "x" and "y" in the domain of h. For samples "x", ..., "x", the corresponding V-statistic is defined In examples 1–3, the asymptotic distribution of the statistic is different: in (1) it is normal, in (2) it is chi-squared, and

neg: Symmetric probability distribution the index value formula_9 only in the context of a term formula_10 where formula_11 is some positive integer (usually 1). This quadratic or other even-powered term takes on the same value for formula_12 as for formula_13, giving symmetry about formula_1. Sometimes the density function contains the term formula_15, which also shows symmetry about formula_16 The following distributions are symmetric for all parametrizations. (Many other distributions are symmetric for a particular parametrization.) Symmetric probability distribution In statistics, a symmetric probability distribution is a probability distribution—an assignment of probabilities to possible occurrences—which is unchanged when its probability density function or probability mass

neg: Plane partition of formula_60 A plane partition formula_61 is called symmetric if "formula_61" = "formula_61" for all "i,j" . In other words, a plane partition is symmetric if ("i,j,k")"formula_64" if and only if ("j,i,k")"formula_65". Plane partitions of this type are symmetric with respect to the plane "x = y". Below an example of a symmetric plane partition is given. Attached the matrix is visualised. formula_66 In 1898, Percy A. MacMahon formulated his conjecture about the generating function for symmetric plane partitions which are subsets of "formula_67". This conjecture is called The MacMahon conjecture. The generating function is given by formula_68 Ian G.

neg: Cayley's sextic The equation of the curve in polar coordinates is In Cartesian coordinates the equation is Cayley's sextic may be parametrised (as a periodic function, period π, ℝ→ℝ) by the equations The node is at "t" = ±"π"/3. Cayley's sextic In geometry, Cayley's sextic (sextic of Cayley, Cayley's sextet) is a plane curve, a member of the sinusoidal spiral family, first discussed by Colin Maclaurin in 1718. Arthur Cayley was the first to study the curve in detail and it was named after him in 1900 by Archibald. The curve is symmetric about the "x"-axis ("y" = 0) and self-intersects at

neg: Linearity the point of intersection between the graph of the function and the "y"-axis. Note that this usage of the term "linear" is not the same as in the section above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so if and only if . Hence, if , the function is often called an affine function (see in greater generality affine transformation). In Boolean algebra, a linear function is a function formula_2 for which there exist formula_3 such that Note that if formula_6, the above function is considered affine

neg: Y-intercept Y-intercept In analytic geometry, using the common convention that the horizontal axis represents a variable "x" and the vertical axis represents a variable "y", a "y"-intercept or vertical intercept is a point where the graph of a function or relation intersects the "y"-axis of the coordinate system. As such, these points satisfy "x" = 0. If the curve in question is given as formula_1 the "y"-coordinate of the "y"-intercept is found by calculating formula_2 Functions which are undefined at "x" = 0 have no "y"-intercept. If the function is linear and is expressed in slope-intercept form as formula_3 the constant

neg: Rotation operator (quantum mechanics) construct the following differential equation using the condition formula_41: Similar to the translation operator, if we are given a Hamiltonian formula_19 which rotationally symmetric about the z axis, formula_48 implies formula_49. This result means that angular momentum is conserved. For the spin angular momentum about the y-axis we just replace formula_50 with formula_51 and we get the spin rotation operator formula_52. Operators can be represented by matrices. From linear algebra one knows that a certain matrix formula_53 can be represented in another basis through the transformation where formula_55 is the basis transformation matrix. If the vectors formula_56 respectively formula_57 are

neg: Critical point (mathematics) point of a function must not be confused with the notion of critical point, in some direction, of a curve (see below for a detailed definition). If "g"("x","y") is a differentiable function of two variables, then "g"("x","y") = 0 is the implicit equation of a curve. A critical point of such a curve, for the projection parallel to the "y"-axis (the map ("x", "y") → "x"), is a point of the curve where formula_2 This means that the tangent of the curve is parallel to the "y"-axis, and that, at this point, "g" does not define an implicit function from

neg: Cayley's sextic Cayley's sextic In geometry, Cayley's sextic (sextic of Cayley, Cayley's sextet) is a plane curve, a member of the sinusoidal spiral family, first discussed by Colin Maclaurin in 1718. Arthur Cayley was the first to study the curve in detail and it was named after him in 1900 by Archibald. The curve is symmetric about the "x"-axis ("y" = 0) and self-intersects at "y" = 0, "x" = −"a"/8. Other intercepts are at the origin, at ("a", 0) and with the "y"-axis at ±"a" The curve is the pedal curve (or "roulette") of a cardioid with respect to its cusp.

neg: Optical transfer function confocal microscope in all three-dimensions, confirming the previously noted lower resolution of the wide-field microscope. Note that along the z-axis, for x=y=0, the transfer function is zero everywhere except at the origin. This "missing cone" is a well-known problem that prevents optical sectioning using a wide-field microscope. The two-dimensional optical transfer function at the focal plane can be calculated by integration of the 3D optical transfer function along the z-axis. Although the 3D transfer function of the wide-field microscope (b) is zero on the z-axis for z≠0; its integral, the 2D optical transfer, reaching a maximum at x=y=0. This is

neg: Fundamental theorem of Riemannian geometry showing that a torsion-free metric connection on a Riemannian manifold is necessarily given by the Koszul formula: This proves the uniqueness of the Levi-Civita connection. Existence is proven by showing that this expression is tensorial in "X" and "Z", satisfies the Leibniz rule in "Y", and that hence defines a connection. This is a metric connection, because the symmetric part of the formula in "Y" and "Z" is the first term on the first line; it is torsion-free because the anti-symmetric part of the formula in "X" and "Y" is the first term on the second line. Fundamental theorem of

neg: Fraunhofer diffraction equation and the limits of the integration become 0 and , we get Putting = sin , we get We can write the aperture function as a step function The Fourier–Bessel transform for this function is given by the relationship where is the transform frequency which is equal to and = . Thus, we get The intensity is given by: This known as the Airy diffraction pattern The diffracted pattern is symmetric about the normal axis. An aperture with a Gaussian profile, for example, a photographic slide whose transmission has a Gaussian variation, so that the amplitude at a point in

neg: Quartic function the polynomial Substituting the by their values in term of the , this polynomial may be expanded in a polynomial in whose coefficients are symmetric polynomials in the . By the fundamental theorem of symmetric polynomials, these coefficients may be expressed as polynomials in the coefficients of the monic quartic. If, for simplification, we suppose that the quartic is depressed, that is , this results in the polynomial This polynomial is of degree six, but only of degree three in , and so the corresponding equation is solvable by the method described in the article about cubic function. By substituting

neg: Hankel transform of order zero is essentially the -dimensional Fourier transform of a circularly symmetric function. Consider a -dimensional function of the radius vector . Its Fourier transform is: With no loss of generality, we can pick a polar coordinate system such that the vector lies on the axis(in K-space). The Fourier transform is now written in these polar coordinates as: where is the angle between the and vectors. If the function happens to be circularly symmetric, it will have no dependence on the angular variable and may be written . The integration over may be carried out, and the Fourier transform

neg: Lorenz curve function "L"("F"), where "F", the cumulative portion of the population, is represented by the horizontal axis, and "L", the cumulative portion of the total wealth or income, is represented by the vertical axis. For a population of size "n", with a sequence of values "y", "i" = 1 to "n", that are indexed in non-decreasing order ( "y" ≤ "y"), the Lorenz curve is the continuous piecewise linear function connecting the points ( "F", "L" ), "i" = 0 to "n", where "F" = 0, "L" = 0, and for "i" = 1 to "n": For a discrete probability function

neg: Best response response correspondences for the stag hunt game. The dotted line in Figure 1 shows the optimal probability that player Y plays 'Stag' (in the y-axis), as a function of the probability that player X plays Stag (shown in the x-axis). In Figure 2 the dotted line shows the optimal probability that player X plays 'Stag' (shown in the x-axis), as a function of the probability that player Y plays Stag (shown in the y-axis). Note that Figure 2 plots the independent and response variables in the opposite axes to those normally used, so that it may be superimposed onto the

neg: Binomial coefficient formula to convert trigonometric functions to complex exponentials, expanding using the binomial theorem, and integrating term by term. For a fixed , the ordinary generating function of the sequence formula_101 is For a fixed , the ordinary generating function of the sequence formula_103 is The bivariate generating function of the binomial coefficients is Another, symmetric, bivariate generating function of the binomial coefficients is A symmetric exponential bivariate generating function of the binomial coefficients is: In 1852, Kummer proved that if "m" and "n" are nonnegative integers and "p" is a prime number, then the largest power of "p" dividing formula_108

neg: Abel transform in photofragment-ion imaging and photoelectron imaging. Among recent most notable extensions of inverse Abel transformation are the Onion Peeling and Basis Set Expansion (BASEX) methods of photoelectron and photoion image analysis. In two dimensions, the Abel transform "F"("y") can be interpreted as the projection of a circularly symmetric function "f"("r") along a set of parallel lines of sight which are a distance "y" from the origin. Referring to the figure on the right, the observer (I) will see where "f"("r") is the circularly symmetric function represented by the gray color in the figure. It is assumed that the observer is

neg: Lieb–Thirring inequality the wave function is symmetric, instead of anti-symmetric, such that for all formula_56, the constant formula_59 has to be replaced by formula_68. Inequality () describes the minimum kinetic energy necessary to achieve a given density formula_69 with formula_1 particles in formula_7 dimensions. If formula_72 was proven to hold, the right-hand side of () for formula_73 would be precisely the kinetic energy term in Thomas–Fermi theory. The inequality can be compared to the Sobolev inequality. M. Rumin derived the kinetic energy inequality () (with a smaller constant) directly without the use of the Lieb–Thirring inequality. The kinetic energy inequality plays an

neg: Y-intercept term formula_4 is the "y"-coordinate of the "y"-intercept. Some 2-dimensional mathematical relationships such as circles, ellipses, and hyperbolas can have more than one "y"-intercept. Because functions associate "x" values to no more than one "y" value as part of their definition, they can have at most one "y"-intercept. Analogously, an "x"-intercept is a point where the graph of a function or relation intersects with the "x"-axis. As such, these points satisfy "y"=0. The zeros, or roots, of such a function or relation are the "x"-coordinates of these "x"-intercepts. Unlike "y"-intercepts, functions of the form "y" = "f"("x") may contain multiple

neg: Optical transfer function Period patterns can be shifted by any amount, depending on the aberration in the system. This is generally the case with non-rotational-symmetric aberrations. The hue of the colors of the surface plots in the above figure indicate phase. It can be seen that, while for the rotational symmetric aberrations the phase is either 0 or π and thus the transfer function is real valued, for the non-rotational symmetric aberration the transfer function has an imaginary component and the phase varies continuously. While optical resolution, as commonly used with reference to camera systems, describes only the number of pixels in an

neg: Sokhotski–Plemelj theorem proof is as follows. For the first term, we note that is a nascent delta function, and therefore approaches a Dirac delta function in the limit. Therefore, the first term equals ∓"i" "f"(0). For the second term, we note that the factor approaches 1 for |"x"| ≫ "ε", approaches 0 for |"x"| ≪ ε, and is exactly symmetric about 0. Therefore, in the limit, it turns the integral into a Cauchy principal value integral. In quantum mechanics and quantum field theory, one often has to evaluate integrals of the form where "E" is some energy and "t" is time. This

neg: Supertoroid radii in the "X" and "Y" directions. With "s"="t"=1 and "a"="b"="R" one obtains the ordinary torus with major radius "R" and minor radius 1, with the center at the origin and rotational symmetry about the "Z" axis. In general, the supertorus defined as above spans the intervals formula_4 in "X", formula_5 in "Y", and formula_6 in "Z". The whole shape is symmetric about the planes "X"=0, "Y"=0, and "Z"=0. The hole runs in the "Z" direction and spans the intervals formula_7 in "X" and formula_8 in "Y". A curve of constant "u" on this surface is an horizontal Lamé curve

neg: Stellar aberration (derivation from Lorentz transformation) light ray (=path of the light signal which is a straight line) and the y-axis whereby θ is positive if the y-axis would have to rotate counter-clockwise to coincide with the light ray. Then the derivation of the formula of angle θ' for the motion along the y-axis is the same as the derivation of the formula of angle δ' for the motion along the x-axis. Hence: formula_54 The symmetric form is: formula_55 and the approximate formula is: formula_56 As formula_57 and as formula_58 and formula_59 one gets: formula_60 and therefore: formula_61 and hence formula_62 Since formula_63 one also gets:

neg: Sum of normally distributed random variables the CDF for "Z" will be This integral is over the half-plane which lies under the line "x"+"y" = "z". The key observation is that the function is radially symmetric. So we rotate the coordinate plane about the origin, choosing new coordinates formula_24 such that the line "x"+"y" = "z" is described by the equation formula_25 where formula_26 is determined geometrically. Because of the radial symmetry, we have formula_27, and the CDF for "Z" is This is easy to integrate; we find that the CDF for "Z" is To determine the value formula_30, note that we rotated the plane so

neg: Quadratrix of Hippias describe the quadratrix as simple function rather than planar curve, it is advantageous to switch the "y"-axis and the "x"-axis, that is to place the side on "y"-axis rather than on the "x"-axis. Then the quadratrix is given by the following function "f"(x): The trisection of an arbitrary angle using only ruler and compasses is impossible. However, if the quadratrix is allowed as an additional tool, it is possible to divide an arbitrary angle into "n" equal segments and hence a trisection ("n" = 3) becomes possible. In practical terms the quadratrix can be drawn with the help of a

neg: Champernowne distribution Champernowne distribution In statistics, the Champernowne distribution is a symmetric, continuous probability distribution, describing random variables that take both positive and negative values. It is a generalization of the logistic distribution that was introduced by D. G. Champernowne. Champernowne developed the distribution to describe the logarithm of income. The Champernowne distribution has a probability density function given by where formula_2 are positive parameters, and "n" is the normalizing constant, which depends on the parameters. The density may be rewritten as using the fact that formula_4 The density "f"("y") defines a symmetric distribution with median "y", which has tails somewhat heavier

neg: Reflection symmetry a given property of the object form a group. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa). The symmetric function of a two-dimensional figure is a line such that, for each perpendicular constructed, if the perpendicular intersects the figure at a distance 'd' from the axis along the perpendicular, then there exists another intersection of the shape and the perpendicular, at the same distance 'd' from the axis, in the opposite direction along the perpendicular. Another way to

neg: Parent function graph of "y" = "x" by translating +2 units along the X axis and +3 units along Y axis. This is because the equation can also be written as "y" − 3 = ("x" − 2). For many trigonometric functions, the parent function is usually a basic sin("x"), cos("x"), or tan("x"). For example, the graph of "y" = "A" sin("x") + "B" cos("x") can be obtained from the graph of "y" = sin("x") by translating it through an angle α along the positive X axis (where tan(α) = ), then stretching it parallel to the Y axis using a stretch

neg: Symmetry of second derivatives the second partial derivatives are not continuous at (0,0), and the symmetry fails. In fact, along the "x"-axis the "y"-derivative is formula_29, and so: In contrast, along the "y"-axis the "x"-derivative formula_31, and so formula_32. That is, formula_33 at (0, 0), although the mixed partial derivatives do exist, and at every other point the symmetry does hold. The above function, written in a cylindrical coordinate system, can be expressed as showing that the function oscillates four times when traveling once around an arbitrarily small loop containing the origin. Intuitively, therefore, the local behavior of the function at formula_35 cannot be

neg: Volatility smile the put options on a particular underlying stock price. The Z-axis represents implied volatility in percent, and X and Y axes represent the option delta, and the days to maturity. Note that to maintain put-call parity, a 20 delta put must have the same implied volatility as an 80 delta call. For this surface, we can see that the underlying symbol has both volatility skew (a tilt along the delta axis), as well as a volatility term structure indicating an anticipated event in the near future. An implied volatility surface is "static": it describes the implied volatilities at a given

neg: Multiple integral integral is equal to twice the integral over one half of the domain, as the integrals over the two halves of the domain are equal. Example 1. Consider the function integrated over the domain a disc with radius 1 centered at the origin with the boundary included. Using the linearity property, the integral can be decomposed into three pieces: The function is an odd function in the variable and the disc is symmetric with respect to the -axis, so the value of the first integral is 0. Similarly, the function is an odd function of , and is symmetric with

neg: BKL singularity gives rise to the following transformation "T" of the unit square: formula_94 (with formula_95, formula_96, cf. ). The density "P"("x", "y") defines the invariant measure for this transformation. It is natural to suppose that "P"("x", "y") is a symmetric function of "x" and "y". This means that the measure is invariant with respect to the transformation "S"("x", "y") = ("y", "x") and hence with respect to the product "ST" with "ST"("x", "y") = ("x″", "y″") and formula_97 Evidently "ST" has a first integral "H" = 1/"x" + "y". On the line "H" = const ≡ c the transformation has the

neg: Social constructionism facts by the constitutive rule: X counts as Y in C. "The Y terms has to assign a new "status" that the object does not already have just in virtue of satisfying the Y term; and there has to be collective agreement, or at least acceptance, both in the imposition of that status on the stuff referred to by the X term and about the function that goes with that status. Furthermore, because the physical features brute facts specified by the X term are insufficient by themselves to guarantee the fulfillment of the assigned function specified by the Y term,

neg: Rotational–vibrational spectroscopy function of the two hydrogens has three symmetric ortho states and one antisymmetric para states. For the three ortho states, the rotational wave function must be antisymmetric corresponding to odd J, and for the one para state it is symmetric corresponding to even J. The population of the odd J levels are therefore three times higher than the even J levels, and alternate line intensities are in the ratio 3:1. These molecules have equal moments of inertia about any axis, and belong to the point groups T (tetrahedral AX) and O (octahedral AX). Molecules with these symmetries have a dipole

neg: Bowling ball to be adopted. For various formulaic purposes, physicists divide rotation into three components, assigning portions to x, y and z axis that are mutually perpendicular. For bowling, the x-axis can be assigned to a line that is parallel to the foul line, the y-axis to the line parallel to the boards, and the z-axis to the vertical. Forward-roll is rotation about the x-axis, side-roll is rotation about the y-axis and mid-roll (or spin) is rotation about the z-axis. The pure full-roller delivery is a combination of forward- and side-roll only. Semi-rollers include spin. Spinners may have very little side roll.

neg: Rotational partition function to temperature formula_27. In the high temperature limit approximation, the mean thermal rotational energy of a linear rigid rotor is formula_28. For a diatomic molecule with a center of symmetry, such as formula_29 or formula_30 (i.e. formula_31 point group), rotation of a molecule by formula_32 radian about an axis perpendicular to the molecule axis and going through the center of mass will interchange pairs of equivalent atoms. The spin–statistics theorem of quantum mechanics requires that the total molecular wavefunction be either symmetric or antisymmetric with respect to this rotation depending upon whether an even or odd number of pairs of

neg: Conjectural variation reaction function and most importantly later its slope. To solve for a symmetric equilibrium, where both firms have the same CV, we simply note that the reaction function will pass through the "x=y" line so that: formula_9 so that in symmetric equilibrium formula_10 and the equilibrium price is formula_11. If we have the Cournot-Nash conjecture, formula_8, then we have the standard Cournot equilibrium with formula_13. However, if we have the Bertrand conjecture formula_14, then we obtain the perfectly competitive outcome with price equal to marginal cost (which is zero here). If we assume the joint-profit maximizing conjecture formula_15 then both

neg: Survival function The graphs below show examples of hypothetical survival functions. The x-axis is time. The y-axis is the proportion of subjects surviving. The graphs show the probability that a subject will survive beyond time t. For example, for survival function 1, the probability of surviving longer than t = 2 months is 0.37. That is, 37% of subjects survive more than 2 months. For survival function 2, the probability of surviving longer than t = 2 months is 0.97. That is, 97% of subjects survive more than 2 months. Median survival may be determined from the survival function. For example, for

neg: Carlson symmetric form symmetric polynomials in formula_39, formula_40, formula_41, formula_76 and (again) formula_76 are in full However, it is possible to simplify the formulae for formula_88, formula_89 and formula_90 using the fact that formula_91. Expressing the integrand in terms of these polynomials, performing a multidimensional Taylor expansion and integrating term-by-term as before... As with formula_36, by expanding about the mean value of the arguments, more than half the terms (those involving formula_62) are eliminated. In general, the arguments x, y, z of Carlson's integrals may not be real and negative, as this would place a branch point on the path of integration, making

neg: Clifford torus rotation operator to the second circle, since that circle will only have one independent axis "z" available to it after the first circle consumes "x" and "y". Stated another way, a torus embedded in R is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R. The relationship is similar to that of projecting the edges of a cube onto a sheet of paper. Such a projection creates a lower-dimensional image that accurately captures the connectivity of the cube edges, but also requires the arbitrary selection and removal of one of the three fully symmetric and interchangeable

neg: Rotational spectroscopy quantum number "J" is associated with the total angular momentum of the molecule. For a given value of J, there is a 2"J"+1- fold degeneracy with the quantum number, "M" taking the values +"J" ...0 ... -"J". The third quantum number, "K" is associated with rotation about the principal rotation axis of the molecule. In the absence of an external electrical field, the rotational energy of a symmetric top is a function of only J and K and, in the rigid rotor approximation, the energy of each rotational state is given by where formula_58 and formula_59 for a "prolate" symmetric

neg: Parallel axis theorem obtain The first term is the inertia matrix ["I"] relative to the center of mass. The second and third terms are zero by definition of the center of mass R, And the last term is the total mass of the system multiplied by the square of the skew-symmetric matrix ["d"] constructed from d. The result is the parallel axis theorem, where d is the vector from the reference point S to the center of mass R. In order to compare formulations of the parallel axis theorem using skew-symmetric matrices and the tensor formulation, the following identities are useful. Let ["R"]

neg: Astigmatism (optical systems) that include the optical axis are "meridional" planes. It is common to simplify problems in radially-symmetric optical systems by choosing object points in the vertical ("y") plane only. This plane is then sometimes referred to as "the" meridional plane. The second special plane is the "sagittal plane". This is defined as the plane, orthogonal to the tangential plane, which contains the object point being considered and intersects the optical axis at the entrance pupil of the optical system. This plane contains the chief ray, but does not contain the optic axis. It is therefore a "skew" plane, in other words

neg: Symmetric matrix "V" such that "V""BV" is diagonal with non-negative real entries. Thus is complex symmetric with "C""C" real. Writing with "X" and "Y" real symmetric matrices, . Thus . Since "X" and "Y" commute, there is a real orthogonal matrix "W" such that both "WXW" and "WYW" are diagonal. Setting , the matrix "UAU" is complex diagonal. Post-multiplying "U" by another diagonal matrix the diagonal entries can be made to be real and non-negative. Since their squares are the eigenvalues of "A""A", they coincide with the singular values of "A". (Note, about the eigen-decomposition of a complex symmetric matrix "A", the

neg: Receiver operating characteristic alarms) on non-linearly transformed x- and y-axes. The transformation function is the quantile function of the normal distribution, i.e., the inverse of the cumulative normal distribution. It is, in fact, the same transformation as zROC, below, except that the complement of the hit rate, the miss rate or false negative rate, is used. This alternative spends more graph area on the region of interest. Most of the ROC area is of little interest; one primarily cares about the region tight against the y-axis and the top left corner – which, because of using miss rate instead of its complement, the

neg: Logarithmic growth bacterial growth phase, the number of new cells appearing are proportional to the population. This terminological confusion between logarithmic growth and exponential growth may be explained by the fact that exponential growth curves may be straightened by plotting them using a logarithmic scale for the growth axis. Logarithmic growth In mathematics, logarithmic growth describes a phenomenon whose size or cost can be described as a logarithm function of some input. e.g. "y" = "C" log ("x"). Note that any logarithm base can be used, since one can be converted to another by multiplying by a fixed constant. Logarithmic growth is

neg: Derivative chooses a function, denoted "f", which is a function of one real number. That is, Once a value of "x" is chosen, say "a", then determines a function "f" that sends "y" to : In this expression, "a" is a "constant", not a "variable", so "f" is a function of only one real variable. Consequently, the definition of the derivative for a function of one variable applies: The above procedure can be performed for any choice of "a". Assembling the derivatives together into a function gives a function that describes the variation of "f" in the "y" direction: This is

neg: Parallel axis theorem relationship is called the parallel axis theorem. Consider the inertia matrix [I] obtained for a rigid system of particles measured relative to a reference point S, given by where r defines the position of particle "P", "i" = 1, ..., "n". Recall that ["r" − "S"] is the skew-symmetric matrix that performs the cross product, for an arbitrary vector y. Let R be the center of mass of the rigid system, then where d is the vector from the reference point S to the center of mass R. Use this equation to compute the inertia matrix, Expand this equation to

neg: Complex normal distribution that the matrix is also non-negative definite. Matrices formula_8 and formula_9 can be related to the covariance matrices of formula_15 and formula_16 via expressions and conversely The probability density function for complex normal distribution can be computed as where formula_20 and "P" = − "RC". The characteristic function of complex normal distribution is given by where the argument formula_22 is a "k"-dimensional complex vector. The "'circularly-symmetric normal distribution" corresponds to the case of zero mean and zero relation matrix, "μ=0", "C=0". If is circularly-symmetric complex normal, then the vector vec["X Y"] is multivariate normal with covariance structure where and .

neg: Cardiac function curve Cardiac function curve A cardiac function curve is a graph showing the relationship between right atrial pressure (x-axis) and cardiac output (y-axis). Superimposition of the cardiac function curve and venous return curve is used in one hemodynamic model. It shows a steep relationship at relatively low filling pressures and a plateau, where further stretch is not possible and so increases in pressure have little effect on output. The pressures where there is a steep relationship lie within the normal range of right atrial pressure (RAP) found in the healthy human during life. This range is about -1 to +2 mmHg.

neg: Centered trochoid ways of generating an ellipse, both of which involve a circle rolling inside a circle with twice the diameter. If additionally, next to formula_48, formula_60, then formula_61 in both cases and the two ways of generating the curve are the same. In this case the curve is simply formula_62 or a segment of the x-axis. Likewise, if formula_63, then formula_64 or formula_65. The circle is symmetric about the origin, so both of these give the same pair of circles. In this case the curve is simply formula_66: a segment of the y-axis. So the case formula_67 is an exception (in

neg: Rodrigues' rotation formula has unit 2-norm. The previous rotation formula in matrix language is therefore Note the coefficient of the leading term is "now" 1, in this notation. Factorizing the allows the compact expression where is the rotation matrix through an angle counterclockwise about the axis , and the identity matrix. This matrix is an element of the rotation group of , and is an element of the Lie algebra generating that Lie group (note that is skew-symmetric, which characterizes ). In terms of the matrix exponential, To see that the last identity holds, one notes that characteristic of a one-parameter subgroup, i.e.

neg: Busemann function 3"h"-neighbourhood of the origin: in fact if "z" lies in both neighbourhoods, let "x" and "y" be the orthogonal projections of "z" onto the "x"- and "y"-axes; then so taking projections onto the "y"-axis, ; hence . Hence , so that "f" is quasi-equivalent to the identity, as claimed. Given two distinct points "z", "w" on the unit circle or real axis there is a unique hyperbolic geodesic ["z","w"] joining them. It is given by the circle (or straight line) which cuts the unit circl unit circle or real axis orthogonally at those two points. Given four distinct points "a",

neg: Meijer G-function of transform kernels if, for any suitable function "f"("z") or any suitable function "g"("z"), the following two relationships hold simultaneously: The pair of kernels is said to be symmetric if "k"("z","y") = "h"("z","y"). showed that the functions: are an asymmetric pair of transform kernels, where "γ" > 0, "n" − "p" = "m" − "q" > 0, and: along with further convergence conditions. In particular, if "p" = "q", "m" = "n", "a" + "b" = 0 for "j" = 1, 2, ..., "p" and "c" + "d" = 0 for "j" = 1, 2, ..., "m", then the pair

neg: Frobenius manifold everywhere. A family of "commutative products *" on "TM" is equivalent to a section "A" of "S"(T"M") ⊗ "TM" via We require in addition the property Therefore, the composition "g"∘"A" is a symmetric 3-tensor. This implies in particular that a linear Frobenius manifold ("M", "g", *) with constant product is a Frobenius algebra "M". Given ("g", "T", "A"), a "local potential Φ" is a local smooth function such that for all flat vector fields "X", "Y", and "Z". A "Frobenius manifold" ("M", "g", *) is now a flat Riemannian manifold ("M", "g") with symmetric 3-tensor "A" that admits everywhere a

neg: Relativistic angular momentum the off-diagonal shear terms of the spacelike part of the stress–energy tensor. If the spacetime supports a Killing vector field tangent to a circle, then the angular momentum about the axis is conserved. One also wishes to study the effect of a compact, rotating mass on its surrounding spacetime. The prototype solution is of the Kerr metric, which describes the spacetime around an axially symmetric black hole. It is obviously impossible to draw a point on the event horizon of a Kerr black hole and watch it circle around. However, the solution does support a constant of the system that

neg: Parallel axis theorem be the skew symmetric matrix associated with the position vector R = ("x", "y", "z"), then the product in the inertia matrix becomes This product can be computed using the matrix formed by the outer product [R R] using the identify where ["E"] is the 3 × 3 identity matrix. Also notice, that where tr denotes the sum of the diagonal elements of the outer product matrix, known as its trace. Parallel axis theorem The parallel axis theorem, also known as Huygens–Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine

neg: Biological motion perception space-time plots used for describing object motion. Space-time plots with time at the y axis and the spatial dimension at the x axis, define velocity of an object by the slope of the line. Information about an object's motion can be detected by spatio-temporal filters. In this biological motion model, motion is detected similarly but replaces the spatial dimension for posture space along the x axis, and body motion is detected by using posturo-temporal filters rather than spatio-temporal filters. "Posturo-Temporal Filters" Neural responses are first normalized as described by where formula_23 describes the neural response; formula_24 describes the preferred posture

neg: Kampyle of Eudoxus Kampyle of Eudoxus The Kampyle of Eudoxus (Greek: καμπύλη [γραμμή], meaning simply "curved [line], curve") is a curve with a Cartesian equation of from which the solution "x" = "y" = 0 is excluded. In polar coordinates, the Kampyle has the equation Equivalently, it has a parametric representation as This quartic curve was studied by the Greek astronomer and mathematician Eudoxus of Cnidus (c. 408 BC – c.347 BC) in relation to the classical problem of doubling the cube. The Kampyle is symmetric about both the "x"- and "y"-axes. It crosses the "x"-axis at (±"a",0). It has inflection points at

neg: Pieri's formula Pieri's formula In mathematics, Pieri's formula, named after Mario Pieri, describes the product of a Schubert cycle by a special Schubert cycle in the Schubert calculus, or the product of a Schur polynomial by a complete symmetric function. In terms of Schur functions "s" indexed by partitions λ, it states that where "h" is a complete homogeneous symmetric polynomial and the sum is over all partitions λ obtained from μ by adding "r" elements, no two in the same column. By applying the ω involution on the ring of symmetric functions, one obtains the dual Pieri rule for multiplying an

neg: Quadratic function then the hyperbola's major axis (through its vertices) is horizontal, while if the ordinate is positive then the hyperbola's major axis is vertical. If formula_34 then the equation formula_32 describes either a circle or other ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola formula_33 is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points. To iterate a function formula_37, one applies the function repeatedly, using the output from one iteration as the input to the next. One cannot always

neg: Critical point (mathematics) "x" to "y" (see implicit function theorem). If ("x", "y") is such a critical point, then "x" is the corresponding critical value. Such a critical point is also called a bifurcation point, as, generally, when "x" varies, there are two branches of the curve on a side of "x" and zero on the other side. It follows from these definitions that the function "f"("x") has a critical point "x" with critical value "y", if and only if ("x", "y") is a critical point of its graph for the projection parallel to the "x"-axis, with the same critical value "y". For

neg: Anderson's theorem Pr), suppose that "X" : Ω → R is an R-valued random variable with probability density function "f" : R → [0, +∞) and that "Y" : Ω → R is an independent random variable. The probability density functions of many well-known probability distributions are "p"-concave for some "p", and hence unimodal. If they are also symmetric (e.g. the Laplace and normal distributions), then Anderson's theorem applies, in which case for any origin-symmetric convex body "K" ⊆ R. Anderson's theorem In mathematics, Anderson's theorem is a result in real analysis and geometry which says that the integral of an integrable,

neg: Geopotential model above (that are divided by "r" in ()) take very large values when for example kilometers and seconds are used as units. In the literature it is common to introduce some arbitrary "reference radius" "R" close to Earth's radius and to work with the dimensionless coefficients and to write the potential as \ P_n(\sin\theta)</math> which is rotational symmetric around the z-axis is an harmonic function If formula_11 is a solution to the differential equation with "m" ≥ 1 one has the potential where "a" and "b" are arbitrary constants is a harmonic function that depends on φ and therefore is

neg: Chain rule for Kolmogorov complexity logarithmic term: (An exact version, "KP"("x", "y") = "KP"("x") + "KP"("y"|"x"*) + O(1), holds for the prefix complexity "KP", where "x*" is a shortest program for "x".) It states that the shortest program printing "X" and "Y" is obtained by concatenating a shortest program printing "X" with a program printing "Y" given "X", plus at most a logarithmic factor. The results implies that algorithmic mutual information, an analogue of mutual information for Kolmogorov complexity is symmetric: "I(x:y) = I(y:x) + O(log K(x,y))" for all "x,y". The ≤ direction is obvious: we can write a program to produce "x" and "y"

neg: Covariance function Covariance function In probability theory and statistics, covariance is a measure of how much two variables change together, and the covariance function, or kernel, describes the spatial or temporal covariance of a random variable process or field. For a random field or stochastic process "Z"("x") on a domain "D", a covariance function "C"("x", "y") gives the covariance of the values of the random field at the two locations "x" and "y": The same "C"("x", "y") is called the autocovariance function in two instances: in time series (to denote exactly the same concept except that "x" and "y" refer to locations

neg: Lifting-line theory For a symmetric wing, the even terms of the series coefficients are identically equal to 0, and so can be dropped. When the aircraft is rolling, an additional term can be added that adds the wing station distance multiplied by the rate of roll to give additional angle of attack change. Equation 3 then becomes: formula_66 where Note that y can be negative, which introduces non-zero even coefficients in the equation that must be accounted for. The effect of ailerons can be accounted by simply changing formula_68 term in Equation 3. For non-symmetric controls such as ailerons the formula_68 term

neg: Inverse demand function solving for Q gives Q = 20. So 20 is the profit maximizing quantity: to find the profit-maximizing price simply plug the value of Q into the inverse demand equation and solve for P. The inverse demand function is the form of the demand function that appears in the famous Marshallian Scissors diagram. The function appears in this form because economists place the independent variable on the y-axis and the dependent variable on the x-axis. The slope of the inverse function is ∆P/∆Q. This fact should be kept in mind when calculating elasticity. The formula for elasticity is (∆Q/∆P) ×

neg: Error function function are defined by The general recurrence formula is They have the power series from which follow the symmetry properties and Error function In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as: In statistics, for nonnegative values of "x", the error function has the following interpretation: for a random variable "Y" that is normally distributed with mean 0 and variance 1/2, erf("x") describes the probability of "Y" falling in the range [−"x", "x"]. The name

neg: Molecular orbital of either two atomic s-orbitals or two atomic p-orbitals. An MO will have σ-symmetry if the orbital is symmetric with respect to the axis joining the two nuclear centers, the internuclear axis. This means that rotation of the MO about the internuclear axis does not result in a phase change. A σ* orbital, sigma antibonding orbital, also maintains the same phase when rotated about the internuclear axis. The σ* orbital has a nodal plane that is between the nuclei and perpendicular to the internuclear axis. A MO with π symmetry results from the interaction of either two atomic p orbitals

neg: Spherical cap which encompasses all of the Tropics. The spheroidal dome is obtained by sectioning off a portion of a spheroid so that the resulting dome is circularly symmetric (having an axis of rotation), and likewise the ellipsoidal dome is derived from the ellipsoid. Generally, the formula_43-dimensional volume of a hyperspherical cap of height formula_3 and radius formula_1 in formula_43-dimensional Euclidean space is given by where formula_48 (the gamma function) is given by formula_49. The formula for formula_50 can be expressed in terms of the volume of the unit n-ball formula_51 and the hypergeometric function formula_52 or the regularized incomplete beta function

neg: Flight dynamics (fixed-wing aircraft) angle "β" is the angle between the velocity vector and the projection of the aircraft longitudinal axis onto the "x","y"-plane, which describes whether there is a lateral component to the aircraft velocity, also known as sideslip. The angle of attack "α" is the angle between the "x","y"-plane and the aircraft longitudinal axis and, among other things, is an important variable in determining the magnitude of the force of lift. In analyzing the stability of an aircraft, it is usual to consider perturbations about a nominal steady flight state. So the analysis would be applied, for example, assuming: The speed, height

neg: Mercer's theorem by the formula "T" is a compact operator (actually it is even a Hilbert–Schmidt operator). If the kernel "K" is symmetric, by the spectral theorem, "T" has an orthonormal basis of eigenvectors. Those eigenvectors that correspond to non-zero eigenvalues can be arranged in a sequence {"e"} (regardless of separability). Theorem. If "K" is a symmetric positive-definite kernel on("X", "M", μ), then where the convergence in the "L" norm. Note that when continuity of the kernel is not assumed, the expansion no longer converges uniformly. In mathematics, a real-valued function "K(x,y)" is said to fulfill Mercer's condition if for all square-integrable

neg: Error function Error function In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as: In statistics, for nonnegative values of "x", the error function has the following interpretation: for a random variable "Y" that is normally distributed with mean 0 and variance 1/2, erf("x") describes the probability of "Y" falling in the range [−"x", "x"]. The name "error function" and its abbreviation were proposed by J. W. L. Glaisher in 1871 on account of its connection with "the

neg: Leave-one-out error Leave-one-out error formula_1 formula_6, with formula_4and formula_5 going to zero for formula_9 X and Y ⊂ R being respectively an input and an output space, we consider a training set formula_10 of size m in formula_11 drawn i.i.d. from an unknown distribution D. A learning algorithm is a function formula_12 from formula_13 into formula_14which maps a learning set S onto a function formula_15 from X to Y. To avoid complex notation, we consider only deterministic algorithms. It is also assumed that the algorithm formula_16 is symmetric with respect to S, i.e. it does not depend on the order of the

neg: Delta potential wave function that is anti-symmetric about the midpoint where = – is called "ungerade" (shown in green in the diagram). They represent an approximation of the two lowest discrete energy states of the three-dimensional formula_31 and are useful in its analysis. Analytical solutions for the energy eigenvalues for the case of symmetric charges are given by: where "W" is the standard Lambert W function. Note that the lowest energy corresponds to the symmetric solution formula_33. In the case of "unequal" charges, and for that matter the three-dimensional molecular problem, the solutions are given by a "generalization" of the Lambert W

neg: Photoconductive atomic force microscopy translation and rotation relative to the AFM tip, thus movement of the tip is generated in the x and y directions When voltage is applied of the inside of the tube, movement in the z-direction is implemented. The relationship between the movement of the piezo-tube and the direction of the displacement of the AFM tip assumes that the tube is perfectly symmetric. When no voltage is applied to the tube the z-axis bisects the tube, sample and sample stage symmetrically. When a voltage is applied to the exterior of the tube (x and y motion), the expansion of the tube

neg: Two-dimensional space Wessel (1745–1818). Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. In mathematics, analytic geometry (also called Cartesian geometry) describes every point in two-dimensional space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin. They are usually labeled "x" and "y". Relative to these axes, the position of any point in two-dimensional space is given by an ordered pair of real numbers, each number giving the distance of that point from the origin measured along the given axis, which

neg: Champernowne distribution than a normal distribution. In the special case formula_5 it is the Burr Type XII density. When formula_6, which is the density of the standard logistic distribution. If the distribution of "Y", the logarithm of income, has a Champernowne distribution, then the density function of the income "X" = exp("Y") is where "x" = exp("y") is the median income. If λ = 1, this distribution is often called the Fisk distribution, which has density Champernowne distribution In statistics, the Champernowne distribution is a symmetric, continuous probability distribution, describing random variables that take both positive and negative values. It is a

neg: Symmetric closure Symmetric closure In mathematics, the symmetric closure of a binary relation "R" on a set "X" is the smallest symmetric relation on "X" that contains "R". For example, if "X" is a set of airports and "xRy" means "there is a direct flight from airport "x" to airport "y"", then the symmetric closure of "R" is the relation "there is a direct flight either from "x" to "y" or from "y" to "x"". Or, if "X" is the set of humans and "R" is the relation 'parent of', then the symmetric closure of "R" is the relation ""x" is a

neg: Rotational spectroscopy or solid phases due to the presence of intermolecular forces. Rotation about each unique axis is associated with a set of quantized energy levels dependent on the moment of inertia about that axis and a quantum number. Thus, for linear molecules the energy levels are described by a single moment of inertia and a single quantum number, formula_1, which defines the magnitude of the rotational angular momentum. For nonlinear molecules which are symmetric rotors (or symmetric tops - see next section), there are two moments of inertia and the energy also depends on a second rotational quantum number, formula_2, which

neg: Surjective function o "P"(~). Surjective function In mathematics, a function "f" from a set "X" to a set "Y" is surjective (or onto), or a surjection, if for every element "y" in the codomain "Y" of "f" there is at least one element "x" in the domain "X" of "f" such that "f"("x") = "y". It is not required that "x" be unique; the function "f" may map one or more elements of "X" to the same element of "Y". The term "surjective" and the related terms "injective" and "bijective" were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians

neg: Surjective function Surjective function In mathematics, a function "f" from a set "X" to a set "Y" is surjective (or onto), or a surjection, if for every element "y" in the codomain "Y" of "f" there is at least one element "x" in the domain "X" of "f" such that "f"("x") = "y". It is not required that "x" be unique; the function "f" may map one or more elements of "X" to the same element of "Y". The term "surjective" and the related terms "injective" and "bijective" were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who under

neg: Anderson's theorem of "x". Anderson's theorem also has an interesting application to probability theory. Let "K" be a convex body in "n"-dimensional Euclidean space R that is symmetric with respect to reflection in the origin, i.e. "K" = −"K". Let "f" : R → R be a non-negative, symmetric, globally integrable function; i.e. Suppose also that the super-level sets "L"("f", "t") of "f", defined by are convex subsets of R for every "t" ≥ 0. (This property is sometimes referred to as being unimodal.) Then, for any 0 ≤ "c" ≤ 1 and "y" ∈ R, Given a probability space (Ω, Σ,

neg: Semi-log plot since taking logs of both sides gives This can easily be seen as a line in slope-intercept form with formula_3 as the slope and formula_4 as the vertical intercept. To facilitate use with logarithmic tables, one usually takes logs to base 10 or e, or sometimes base 2: The term log-lin is used to describe a semi-log plot with a logarithmic scale on the "y"-axis, and a linear scale on the "x"-axis. Likewise, a lin-log plot uses a logarithmic scale on the "x"-axis, and a linear scale on the "y"-axis. Note that the naming is "output-input" ("y"-"x"), the opposite order

neg: Conic section are symmetric about both the -axis and -axis (for the circle, ellipse and hyperbola), or about the -axis only (for the parabola). The rectangular hyperbola, however, is instead symmetric about the lines and . These standard forms can be written parametrically as, In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section (though it may be degenerate), and all conic sections arise in this way. The most general equation is of the form with all coefficients real numbers and not all zero. The above equation can be written in matrix notation