Lecture Notes for the whole course
... Assume we want to arrange n objects in a line, the n objects are of
k diﬀerent types, and objects of the same type are
indistinguishable. Let ai be the number of objects of type i . Then
the number of diﬀerent arrangements is:
a1 !a2 ! . . . ak !
NORMALITY OF NUMBERS GENERATED BY THE VALUES OF
... is normal in base 10. These examples correspond to the choice f (x) = x in (1.1). Davenport
and Erdös  considered the case where f (x) is a polynomial whose values at x = 1, 2, . . . are
always integers and showed that in this case the numbers θq (f ) and τq (f ) are normal. For f (x) a
Graphical Representation of Canonical Proof: Two case studies
... of freely permutable inferences. A proof system that is free of bureaucracy is called
canonical for a logic. In this dissertation two canonical proof systems are presented,
for two logics: a notion of proof nets for additive linear logic with units, and ‘classical
proof forests’, a graphical formali ...
Announcement as effort on topological spaces
... 3Kϕ again means that the agent comes to know ϕ, but in the interpretation
that there is a formula ψ such that after announcing it the agent knows ϕ. What
becomes true or known by an agent after an announcement can be expressed in
this language without explicit reference to the announced formula.
Principia Logico-Metaphysica (Draft/Excerpt)
... A constant is any expression that is either an individual constant or an n-place
relation constant (n 0). A variable is any expression that is either an individual variable or an n-place relation variable (n 0). The expressions listed in
the column labeled ‘Less Formal’ are often used as replacement ...
Introduction to Computational Logic
... This time the claim involves a boolean variable x and the proof proceeds by case
analysis on x. Since reflexivity performs simplification automatically, we have
omitted the tactic simpl.
It is important that with Coq you step back and forth in the proof script and
observe what happens. This way you ...
Title: Asymptotic distribution of integers with certain prime
... 2.2. Ordinal counting functions. It might seem surprising at first sight that the
counting function M2,2 is related to studying asymptotic properties of transfinite ordinals. Since transfinite ordinals rarely show up in a number-theoretic context we will
explain some features of this connection in i ...
In mathematics, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic.Calculations with infinitesimals were widely used before Karl Weierstrass sought to replace them with the (ε, δ)-definition of limit starting in the 1870s. (See history of calculus.) For almost one hundred years thereafter, mathematicians like Richard Courant viewed infinitesimals as being naive and vague or meaningless.Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt and Jerzy Łoś. According to Jerome Keisler, ""Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century.""