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Zero, Infinity, and the Empty Set


    We discuss how we treat the above three concepts and how they are related in various theories.
  1. (Set theory: The infimum of the empty set) When we say a set is empty, we must know what the universal set is. Although we may use logic to prove inf = +, it is simpler to interpret the formula as inf {+} = +.
        If f a a.e., then a is called an essential upper bound of f.
    Let S = {aR| m(f -1(a,+)) = 0}, where m denotes the Lebesque measure on R.
    If S , let M = inf S.
    Then M is called the least essential upper bound of f. We write M = ess. sup f.
    I do not like the definition given in [Zyg, vol.1, p.18, l.10-l.13] because it is confusing.
    If S = , we may prove inf S = {+} by the following three methods:
    (1). From the definition M = inf S, we may want to let M = + because every aR is an upper bound of S.
    (2). S = the set of the essential upper bounds of f, so M = the greatest lower bound of the set of essential upper bounds of f. Since the set of the essential upper bounds of f is empty, every aR is a lower bound of the set of the essential upper bounds of f.
    (3). If S = , m(f -1(a,+)) > 0 for every aR. Namely, every aR is not an essential upper bound of f. By common sense, we must set ess. sup f = +. This is because aR cannot be an upper bound no matter how large a is.
        The first two approaches solve the problem by logic, while the third approach solves the problem by common sense. Since we fail to define the universal set, the considerations of all the three approaches are not comprehensive. It is better and simpler to define ess. sup f = inf (S{+}) when we consider essential upper bounds of f.


  2. (Algebraic number theory)
    1. 0 = ; 0 can only divide 0 [Edw2, p.139, l.-10-l.-7].
    2. The empty divisor will be denoted by I [Edw2, p.139, l.17-l.18]. If A is a divisor, then A0 is defined to be I.
    3. The degree of the polynomial 0 is - [Edw2, p.40, l.11].

  3. (Continued fractions)
        0/0 can be meaningful [Perr, p.194, l.8-l.19]


  4. (Complex analysis)
        For the spherical representation, is the north pole and 0 is the south pole [Con, p.8, Figure]. 0 and are interchangeable through the inversion w = 1/z.


  5. In complex variables and continued fractions [Perr, p.232, l.10-l.18], we prefer to interpret "" as "convergent in the broad sense" rather than "divergent". By doing so, we can make series and infinite continued fractions behave the same way [Perr, p.232, l.-19-l.-7]. Thus, the concept of convergence in the broad sense enables us to achieve some kind of unification.
    The rule of thumb: If is considered a quantity as in physics, "" means "divergent"; if is considered a number, point, or position, then "" means "convergent in the broad sense".


  6. (Confluent hypergeometric functions)
       Since earlier factors in the numerators can be cancelled with the later factors in the denominators [Wat, p.103, l.6-l.12], 0/0 can be meaningful.
    Remark. Let N be a natural number. The confluent hypergeometric series 1F1(-N;r; z) is a terminating series if r is neither 0 nor a negative integer, and is an infinite series if r = -N, -N-1, -N-2, [Wat, p.103, (3)].


  7. (Improper tools) A hyperbolic paraboloid has center at (,,) [Fin, p.285, l.-7] because Cartesian coordinates lack ability to distinguish infinities of all directions; if we use spherical coordinates instead, there will be no common midpoint for the chords through the origin.


  8. At the first glance it seems strange that when q passes from -a2+d to -a2-d, the shape of surfaces makes an abrupt change: from the region of the yz-plane inside the disk to that outside [Smy, p.122, l.12-l.13]. However, if we treat w=1/z as the identity transformation, |z| < 1 would be the same as |z| > 1.


  9. Differentiation of an integral at a singularity: [Grif, p.157, Problem 3.42 (b); Jack, p.35, l.2-l.-4]