The collection

Every essay — page 5

Page 5 of 34, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Analysis

Limits, curves, and what happens when the going does not stop.

The slopes the equation demands, and the curves that obey them. A field of short segments whose slope at each point is 0.9 times the height there, with 3 solution curves integrated through it; each doubles over an interval of 0.770 wherever that interval is taken.

The equation with only one answer

The flow of a linear equation, and the matrix that runs it for one unit of time. Paths of points moving so that their velocity is [0.25, −1.2, 1.2, 0.25] applied to their position, with the position after time 1 marked on each; the matrix taking start to finish is e^A.

The exponential of a square

The constant that counts what does not happen

A set with no interval in it and half its length left, after 6 stages. Stages of removing a shrinking middle from every surviving interval, with the total length left printed at each stage, and the middle-thirds construction of the same depth drawn beneath for comparison.

No interval in it, and length to spare

The rationals covered by intervals of total length 0.1800. Intervals of rapidly shrinking length placed around the rationals of the unit interval in the order they are listed, with the union of them drawn as a single band beneath.

Covering a set from outside

Two indicators, and the upper sum that will not come down. A partition of the unit interval drawn against the middle-thirds set and against a set of positive length, above a chart of each one's upper sum as the partition is refined.

Which functions can be added up

The classes, a selection from them, and the translates that cannot have a length. Points of several classes of the unit interval under translation by rationals, drawn one class per row, above rows showing rational translates of a selection that never overlap.

A set that has no size at all

A staircase with no steps. The Cantor function drawn to several stages: a continuous non-decreasing curve from nought to one which is constant on every interval of the complement of the middle-thirds set, so its whole rise happens on a set of measure zero.

A staircase with no steps

A line under every point of x². The curve x² with 4 tangent lines drawn, each extended across the whole interval and each staying below the curve throughout.

A line under every point

x² and its conjugate. Two panels: the curve x² with tangent lines of several slopes, and the conjugate function plotted against slope, whose value at each slope is the intercept of the corresponding tangent.

The function seen from its tangents

One minimum, or several. Two curves side by side with their local minima marked: a convex one with a single minimum, and a fourth-power well with 2.

Where the guarantee stops

Every point of a hull, as a mixture of three of 11 points. A scatter of points with its convex hull outlined, and several interior points each shown inside a triangle of three of the scattered points, found by trying every triple.

Three points, however many there are

Two convex sets 1.50 apart, and the line that separates them. Two convex polygons with a straight line drawn between them, together with the shortest segment joining the two sets, whose perpendicular bisector the line is.

A wall between two bodies

What the degree-5 sum costs, and what the bound claims. The error of the degree-5 Taylor polynomial of sin x against x, on a logarithmic scale, with Lagrange's bound drawn above it. The bound exceeds the error by a factor of 1.8 at the right-hand end.

An error with an unknown in it

1/(1 + x²), expanded about 1.2. 1/(1 + x²) with Taylor sums of degree 2, 6, 14 about x = 1.2 rather than about zero. The interval they converge on reaches 1.562 either side of the centre.

The centre is a choice

11 points, equally spaced. 1/(1 + x²) and the polynomial of degree 10 through 11 of its points, spaced evenly across the interval. The worst error is 2.48e-1, at x = -2.350.

The points that ruin the fit

ln(1 + x) past its radius, with a denominator allowed. ln(1 + x), its Taylor sum of degree 8, and its Padé approximants of order 2 and 4. At x = 3 the Taylor sum is out by 5.95e+2 and the highest-order approximant by 2.97e-4.

A denominator that reaches past the radius

A series that gets better, then worse. The error of Euler's series against the number of terms kept, at x = 0.05 and 0.1 and 0.2. At 0.05 the error falls to 1.1e-8 at 20 terms and then climbs without limit. At 0.1 the error falls to 1.8e-4 at 10 terms and then climbs without limit. At 0.2 the error falls to 1.8e-2 at 5 terms and then climbs without limit.

A series that converges nowhere

sin(2πkx) for k up to 10, and the largest gap between every pair. Members of the sequence sin 2πkx drawn on one pair of axes, beside a table of the largest vertical gap between every two members, none of which is less than one.

The subsequence that has to exist

Partial sums of x − x²/2 + x³/3 − … on [0, 1]. Partial sums of the power series x − x²/2 + x³/3 − … drawn on the interval from 0 to 1 with the function the series sums to inside the interval, and the values at x = 1 marked.

A sum read from inside

Inscribed polygons in a quarter circle, and the length they climb towards. Four polygons inscribed in a quarter circle with increasing numbers of corners, each drawn over the curve, with its length beneath it — the lengths increase towards the curve's own.

Which curves have a length at all

One curve, 3 parametrisations, 3 lengths. Several maps from an interval with the same image drawn side by side, each with marks at equal parameter steps and its length beneath it — the same point set reported at different lengths.

The length belongs to the journey

n / 2ⁿ held under a geometric series. A bar for each term of the series n / 2ⁿ with a decaying geometric curve above them, the curve lying above every bar from term 2 onwards.

The series everything else is measured against

Runs of doubling length in 1/n^2, and the geometric series that bounds them. A bar for the total of each run of terms of 1/n to the 2, with an outlined bar above it for the bound obtained by replacing every term in the run with its largest, the bounds forming a geometric series.

The repair at the boundary