Blackboard bold

Blackboard bold is a style of writing bold symbols on a blackboard by doubling certain strokes, commonly used in mathematical lectures, and the derived style of typeface used in printed mathematical texts. The style is most commonly used to represent the number sets (natural numbers), (integers), (rational numbers), (real numbers), and (complex numbers). To imitate a bold typeface on a typewriter, a character can be typed over itself (called double-striking); symbols thus produced are called double-struck, and this name is sometimes adopted for blackboard bold symbols, for instance in Unicode glyph names. In typography, a typeface with characters that are not solid is called inline, handtooled, or open face. Traditionally, various symbols were indicated by boldface in print but on blackboards and in manuscripts "by wavy underscoring, or enclosure in a circle, or even by wavy overscoring". Most typewriters have no dedicated bold characters at all. To produce a bold effect on a typewriter, a character can be double-struck with or without a small offset. By the mid 1960s typewriter accessories such as the "Doublebold" could automatically double-strike every character while engaged. While this method makes a character bolder, and can effectively emphasize words or passages, in isolation a double-struck character is not always clearly different from its single-struck counterpart. Blackboard bold originated from the attempt to write bold symbols on typewriters and blackboards that were legible but distinct, perhaps starting in the late 1950s in France, and then taking hold at the Princeton University mathematics department in the early 1960s. Mathematical authors began typing faux-bold letters by double-striking them with a significant offset or over-striking them with the letter I, creating new symbols such as IR, IN, CC, or ZZ; at the blackboard, lecturers began writing bold symbols with certain doubled strokes. The notation caught on: blackboard bold spread from classroom to classroom and is now used around the world.

Moments of the number of points in a bounded set for number field lattices

Maryna Viazovska, Nihar Prakash Gargava, Vlad Serban

We examine the moments of the number of lattice points in a fixed ball of volume

V

for lattices in Euclidean space which are modules over the ring of integers of a number field

K

. In particular, denoting by

ω_K

the number of roots of unity in

K

, we ...

arXiv2023

On distributional autoregression and iterated transportation

Moments of the number of points in a bounded set for number field lattices

A decomposition of multicorrelation sequences for commuting transformations along primes

Moments of the number of points in a bounded set for number field lattices

A decomposition of multicorrelation sequences for commuting transformations along primes

On distributional autoregression and iterated transportation