Atlas of forms.

The probability of configuration s is the exponential of the sum of local h times s terms and J times paired-state interactions, divided by normalisation constant Z; each state is minus one or plus one.

Graph G contains vertices V and edges E; an adjacency-matrix entry is one when an edge connects vertex i to vertex j, and zero otherwise.

Probability is one divided by one plus the exponential of the negative of one point two x minus zero point five y plus zero point four five x y; the displayed boundary corresponds to probability one half.

The signal s is the sum of three sinusoids. Vector v combines three values separated by delay tau, equal to six intervals on the original grid. Sample entropy is the negative natural logarithm of the ratio A to B, defined for dimension m and tolerance r.

The state is cosine of half theta times state zero plus the exponential of i phi times sine of half theta times state one; the vector coordinates are sine theta cosine phi, sine theta sine phi, and cosine theta.

The surface sums cosine of three x and cosine of three y, multiplied by zero point three four; the squared magnitude of two summed amplitudes includes their individual squared magnitudes and twice the real part of the first conjugated amplitude times the second.

Apply Hadamard to the first qubit and identity to the second in state zero zero, followed by CNOT; the result is the sum of states zero zero and one one, divided by the square root of two.

The lattice is the set of all combinations of the three basis vectors with integer coefficients.

The change in S contains drift mu S times dt and a random term sigma S times dW; the solution multiplies the initial value by the exponential of mu minus half sigma squared, times t, plus sigma W.

The Gaussian density exponent is minus one half of the quadratic form of the deviation from the mean with the inverse covariance; normalisation is two pi times the square root of the determinant of sigma.

The surface contains points whose quadratic form of deviation from the mean, using inverse covariance, equals one; covariance is matrix T times its transpose.

The sum of sine x times cosine y, sine y times cosine z, and sine z times cosine x equals zero.

The sum of the cosines of coordinates x, y and z equals zero.

The product of the cosines of x, y and z minus the product of their sines equals zero.

The sum of the cosines of two pi times each coordinate is zero. Gaussian curvature K is the product of the principal curvatures k one and k two. Their opposite signs make the product negative in the neck.

Displacement along z equals one point one times the sine of zero point seven x times the cosine of zero point four two y.

The first two coordinates describe a circle with radius zero point six four times the hyperbolic cosine of v, while the third coordinate is v; necks are oriented along three directions.

Sixty vertices and ninety edges enclose thirty-two faces: twelve pentagons and twenty hexagons. Vertices minus edges plus faces equals two.

The planar lattice has two atoms per primitive cell and unit bond lengths in this model. The cut direction determines an armchair or zigzag termination.

Each transverse coordinate rotates through an angle that increases linearly along the ribbon axis. The bond list remains identical to the planar ribbon.

The second fragment is translated horizontally and separated vertically from the first. All drawn bonds belong to individual layers.

At each iteration, square complex number z and add c, equal to minus zero point seven four five four three plus zero point one one three zero one times the imaginary unit.

The set is the union of four copies obtained by halving the sum of a point and each vertex; the self-similar dimension is logarithm of four divided by logarithm of two, equal to two.

The set is the union of twenty copies scaled to one third; its self-similar dimension is logarithm of twenty divided by logarithm of three.

Parameter u makes a full turn while v ranges from minus one to one; the width, scaled by zero point three five, makes a half turn around the central circle.

The first two coordinates are u cosine v and u sine v, while height is zero point two three v, coupling rotation with vertical translation.

At height zero point eight v, the circular section has radius zero point eight times the hyperbolic cosine of v; angle u runs around the circle.

A circle of radius zero point three eight revolves around the central axis at distance one; u locates a point around the ring and v around the tube section.