Conference rusure::math

>>    Hi, just out of curiosity, I want to ask this question--can a function
>>    be devised, without using the Signum function, such that its results
>>    are:
>>                /
>>                \ 0 if x =/= 0
>>         Z(x) = <
>>                / 1 if x  =  0
>>                \

	You just did it, and without mentioning Signum.

	An alternative is Z(x) = if (x == 0) then 1 else 0.

	Obviously, you have some kind of rules in mind as to what
	"devises" a function, but we have no idea of knowing what
	those are unless you tell us.

	How about:
			 2
		      -nx
		lim  e
	       n->oo

	Dan

    Oh, yes you are right.  What I meant was to devise a function with
    those given results.  I like your answer using a limit; that indeed is
    a good example of what I am looking for.  I never thought of that one
    before at all!
    
    Now is there one without using limits or the e constant?  :-)

I think you are defining the Kronecker (sp? of course) Delta Function.

You might want to attach his name to it.

I once knew a reason why some people call this a distribution and not a 
function, but I have forgotten it.

I think that .1 contains two answers to the question in .2.  You still need
to clarify what you mean by a function.  

> I think you are defining the Kronecker (sp? of course) Delta Function.

	The Kronecker Delta Function has value infinity, not 1, at zero.

> I once knew a reason why some people call this a distribution and not a 
> function, but I have forgotten it.

	The definite integral is 1, and the function is always positive.
There's probably some measure theoretic stuff that I should quote here, but
I don't "do" measure yet.   I've just bought a book, however...

	The indefinite integral is the Heaviside function, H:

		H(x) =  0, x < 0
       			1, x >= 0

	So another function with the desired property is:

		H(x) + H(-x) - 1

Regards,
Andrew.

    RE: .3
    
    > I think that .1 contains two answers to the question in .2.
    
    Let me rephrase my question then:
    
         Are there _any_more_ possible answers (and without limits and e)?
    
    >                                                      You still need
    > to clarify what you mean by a function.
    
    How about if I said "equation" instead?  That is, how simply can you
    write an equation of the form:
    
         Z(x) = (insert math terms here wrt x)
    
    And for further clarification, how about if I say try to do it with
    algebraic, trigonometric, or Calculus-based terms?

    RE: .4
    
    Call me curious as a cat, but how complex is the Heaviside function?

    A function which takes on a value of 1 at zero and zero elsewhere
    (on the real line) can be defined generally as an equivalence class
    of limits of continuous functions, much the same way any real number
    can be defined as an equivalence class of convergent Cauchy sequences.
    [Another definition of the reals is by Dedekind cuts.]

    A continuous function is called a category zero function by Baire.
    A limit of the sum of countably many category zero functions which is
    not also category zero is called a category 1 function and so on.
    Category 1 functions include square waves and the like; category 2
    functions include the function which is 1 for irrational numbers and zero
    for rationals.  Arbitrarily high category functions can be defined.

    The ideas behind this are actually simple, but can be sufficiently
    obfucscated by heavy formalism in "real analysis" texts to seem much
    deeper than they are.  This was also a case where physicists and engineers
    (Dirac, Heaviside...) were ahead of mathemeticians in that they cheerfully
    used these functions formally without rigorous foundations - later the
    rigor was obtained.  It's not *always* the case that mathemeticians
    invent abstractions to be used later.

    A really nice elementary book on distributions - Lighthill, "Generalized
    Functions", Cambridge University Press.

    - Jim

        re .3, .4
        
        The Kronecker delta is a symbol often used in linear
        algebra and matrices, with
        
                          { 1 if i = j
        	delta   = {
        	     ij   { 0 if i =/= j
        
        It probably has generalizations to more than two
        subscripts.  It is not a function of the real line.
        
        What .4 describes is the Dirac delta function (the
        same Dirac mentioned in .-1).
        
        Dan
        

	(1) Dan's correction is correct.   
	(2) delta(Dirac,Kronecker) = 0.   
	(3) My memory is not what I remember it to be.

Andrew

    This is not really the kind of discussion I was hoping to start, but it
    is nonetheless very interesting.  I had hoped that my question would be
    considered as an entertaining math puzzle to see what kinds of answers
    people could come up with.  I was pleased with the answer by Dan in .1
    using a limit and e because it had not occurred to me before.  The use
    of the Heaviside function in .4 is rather intriquing to me, as I can't
    imagine what H(x) looks like.
    
    I don't know if you like the idea of making up specific expressions
    that are equal to Z(x).  That does require some thinking to accomplish.
    
    Well, I hope I have not spoiled the mood and if you still want to
    discuss it in terms of math history, feel free!!  I find it
    educational, although I'm not enough of a math historian to contribute
    any tidbits that you don't already know about.  :-)
    
    Enjoy...                                           
    
    Andy

        Known tricks when playing with functions include
        
                              2n
        f(x) =	lim   lim  cos  (m! pi x)
               m->oo n->oo
        
        If m! x is an integer then the cosine is +/= 1 which when
        raised to the 2n power is 1.  If m! x is not an integer
        then the cosine is between -1 and 1 and so the 2n power
        converges to 0 as n->oo.  Now if x is rational, m! x
        eventually becomes and remains an integer as m-> oo.  For
        x irrational, m! x is never an integer and so the inner
        limit is always 0.  So f(x) = 1 if x is rational and 0 if x
        is irrational.
        
        Another limit that yields Z(x) is:
        
        		1
        Z(x) =  lim  -------
               n->oo       2
                     1 + nx
        
        Dan
        

    Re .10:
    
    A "function" is just a mapping from inputs to outputs -- it does not
    saying anything about _how_ the outputs are computed.
    
    I think what you meant to ask was perhaps something like:  Can the
    function described be written as the composition of elementary
    functions?
    
    That is, can your function be written as the result of combining
    functions such as ordinary arithmetic and other "known" functions.
    
    If that is what you meant to ask, you'll have to tell us what functions
    we're allowed to use.
    
    
    				-- edp

    Re: .12:
    
    Up to you, but I suppose the use of Algebra, Calculus, or Trigonometry
    would be the most interesting methods.  I came up with one using just
    the absolute value and the greatest integer functions about 3 years
    ago.  The use of the absolute value could be eliminated by squaring.
    
    Is this question more trivial than I thought?  Seems like it.

	Since the function Z(x) has a jump discontinuity at zero, any 
	algebraic (+-*/) combination of elementary functions (basically those
	you can find on a good scientific calculator) will either be continuous
	or have a singularity ("infinite" jump).  This assumes functions like
	floor(), ceil(), truncation, int, etc. are not allowed.

	Thus the only way to make it have a finite jump is with a limit such
	as those proposed earlier, or an infinite series, which is itself a
	limit of partial sums.

	Avery

    RE: 14
    
    >                                           This assumes functions like
    > floor(), ceil(), truncation, int, etc. are not allowed.
    
    What assumes that they ARE allowed?  In other words, how do I convey in
    a sweeping sort of way that all these functions are permissible?

Consider the following FORTRAN program. It depends for its utility on the 
combination of logical and integer operations on integers, and that the 
value of .TRUE. is represented as the integer (-1). So dirac(n) is 1 
if n=0, 0 otherwise. - Lynn Yarbrough 

	integer dirac, n
	dirac(n) = (n .ne. 0) + 1
	write (*,*) dirac(0), dirac(1), dirac(-1)
	end