Oct 10 14:42:23 * Now talking on #mathematics
Oct 10 14:42:24 * Topic for #mathematics is: Next seminar is on Sun 10 Oct, 12pm PDT: Funcoids and Reloids, by porton | Seminar logs for Oct 3: http://chromotopy.org/geometry-seminar.txt
Oct 10 14:42:24 * Topic for #mathematics set by thermoplyae!~thermo@li154-228.members.linode.com at Sun Oct  3 23:01:22 2010
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Oct 10 14:50:19 * porton (~porton@87.70.212.21) has joined #mathematics
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Oct 10 15:00:02 <porton> The topic of this seminar are funcoids and reloids.
Oct 10 15:00:08 <porton> Funcoids are a generalization of proximity spaces. Also funcoids are a generalization of binary relations, pretopological spaces, preclosure operators.
Oct 10 15:00:12 <porton> Reloids are a generalization of uniform spaces and binary relations.
Oct 10 15:00:16 <@thermoplyae> slow down, bro
Oct 10 15:00:22 * qr (~qr@kokanee.cs.ubc.ca) has joined #mathematics
Oct 10 15:00:30 <porton> Read details in "Funcoids and Reloids" article at http://www.mathematics21.org/binaries/funcoids-reloids.pdf
Oct 10 15:00:38 <porton> I will reprise the most interesting properties here in the chat.
Oct 10 15:00:57 <porton> As you can anyway read that article I will skip all proofs.
Oct 10 15:01:14 <porton> That funcoids are a common generalization of spaces and (multivalued) functions makes them smart for analyzing properties of functions in regard of spaces.
Oct 10 15:01:22 <porton> For example, the statement "f is a continuous function from a space a to a space b" can be written as "f o a <= b o f" in terms of funcoids.
Oct 10 15:01:26 <porton> To continue?
Oct 10 15:01:44 <@thermoplyae> you should at least define what a funcoid is
Oct 10 15:01:50 <porton> See below
Oct 10 15:01:58 <porton> Most naturally funcoids arrive as a generalization of proximities.
Oct 10 15:02:02 <porton> First, I will introduce some strange terminology:
Oct 10 15:02:09 <porton> I will consider "the set of filters" on some set UNIV "the set of all sets".
Oct 10 15:02:14 <porton> I will order the set of filters reversely to the set theoretic inclusion of filters and equate principal filters with corresponding sets.
Oct 10 15:02:19 <porton> I call such reverse ordered filters equated with sets, when appropriate, "filter objects".
Oct 10 15:02:25 <porton> Note that "the set of all sets" SET is a subset of the set of filter objects.
Oct 10 15:02:37 <porton> continue?
Oct 10 15:02:58 <@thermoplyae> yes, go ahead, someone will interrupt if they have questions
Oct 10 15:02:58 <porton> I will denote the filter object corresponding to the filter X as "up X".
Oct 10 15:03:01 <porton> I will denote the set of filter objects as FILT.
Oct 10 15:03:10 <porton> FILT is an atomistic complete co-brouwerian lattice. (Atoms of FILT correspond to maximal filters.)
Oct 10 15:03:26 <porton> Now we can introduce funcoids.
Oct 10 15:03:30 <porton> Let d is a proximity. We can extend it to filter objects by the formula:
Oct 10 15:03:34 <porton> x d y <=> forall X in x, Y in y: X d Y.
Oct 10 15:04:01 <porton> Sorry, corrected formula: x d y <=> forall X in up x, Y in up y: X d Y.
Oct 10 15:04:22 <porton> For every proximity there exists two functions alpha, beta: FILT->FILT such that
Oct 10 15:04:25 <porton> x d y <=> y /\ alpha(x) != 0 <=> x /\ beta(y) != 0.
Oct 10 15:04:29 <porton> I call "funcoids" pairs (alpha;beta) of functions FILT->FILT such that
Oct 10 15:04:34 <porton> y /\ alpha(x) != 0 <=> x /\ beta(y) != 0.
Oct 10 15:04:53 <porton> I will denote FCD the set of funcoids.
Oct 10 15:04:56 <porton> I've said above that for every proximity exists a corresponding funcoid.
Oct 10 15:05:04 <porton> In fact a corresponding (see the formula above) funcoid exists for every binary relation d on SETS such that
Oct 10 15:05:07 <porton> not(0 d X) and not(X d 0) and (X\/Y) d Z <=> X d Z \/ Y d Z and Z d (X\/Y) <=> Z d X \/ Z d Y.
Oct 10 15:05:25 <porton> I denote <f> the first component (alpha) of a funcoid f.
Oct 10 15:05:38 <porton> <f>(X\/Y) = <f>X \/ <f>Y for every filter objects X and Y.
Oct 10 15:05:47 <porton> The function <f> for a funcoid f can be constructed as a (unique) continuation of the function alpha: SET->FILT
Oct 10 15:05:55 <porton> such that alpha(0)=0 and alpha(X\/Y)=X\/Y.
Oct 10 15:06:03 <porton> Conversely, any <f> restricted to SET for a funcoid f conforms to the above formulas for alpha.
Oct 10 15:06:19 <porton> Any (multivalued) function (binary relation) can be considered as a funcoid where
Oct 10 15:06:26 <porton> <f>(X) = /\ {f[x] | x in up X}.
Oct 10 15:06:38 <porton> I will call funcoids corresponding to binary relations discrete.
Oct 10 15:06:53 <porton> Like binary relations funcoids can be reversed and composed:
Oct 10 15:06:59 <porton> (alpha;beta)^-1 = (beta;alpha);
Oct 10 15:07:03 <porton> (alpha2;beta2) o (alpha1;beta1) = (alpha2 o alpha1; beta1 o beta2).
Oct 10 15:07:14 <porton> Reverse of a funcoid corresponding to a proximity is equal to this funcoid (because proximities are symmetric).
Oct 10 15:07:28 <porton> We have <g o f> = <g> o <f>;
Oct 10 15:07:31 <porton> (h o g) o f = h o (g o f) and f^-1^-1=f and  (g o f)^-1 = f^-1 o g^-1.
Oct 10 15:07:47 <porton> I will denote x[f]y <=> y /\ <f>(x) != 0 <=> x /\ <f^-1>(y) != 0 for every funcoid f and a filter objects x and y.
Oct 10 15:07:53 <porton> x\/y [f] z <=> x[f]z \/ y[f]z and z [f] x\/y <=> z[f]z \/ z[f]y for every filter objects x,y,z
Oct 10 15:08:02 <porton> [f^-1] = [f]^-1.
Oct 10 15:08:07 <porton> For every value of <f> exists no more than one funcoid f.
Oct 10 15:08:09 <porton> For every value of [f] exists no more than one funcoid f.
Oct 10 15:08:15 <porton> Moreover a funcoid f is uniquely determined by values of <f> on sets.
Oct 10 15:08:31 <porton> Oh, that funcoids are a generalization of pretopological spaces is provided by the formula:
Oct 10 15:08:35 <porton> <f>X = \/ { alpha(x) | x in X } for a set X where alpha is a pretopology (a function UNIV->FILT).
Oct 10 15:08:54 <porton> Funcoids form a complete brouwerian atomistic lattice by the order f <= g <=> [f] <= [g].
Oct 10 15:09:02 <porton> I'll skip the formulas for calculating <f> and [g] for join of funcoids.
Oct 10 15:09:05 <porton> More properties of funcoids:
Oct 10 15:09:16 <porton> x[g o f]z iff there exist atomic filter object y such that x[f]y and y[g]z.
Oct 10 15:09:22 <porton> If f,g,h are funcoids, then f o (g\/h) = f o g \/ f o h and (g\/h) o f = g o f \/ h o f.
Oct 10 15:09:34 <porton> The identity funcoid I_a = (a/\; a/\) is defined for a filter object a.
Oct 10 15:09:41 <porton> Reverse of identity funcoid is identity funcoid.
Oct 10 15:09:46 <porton> x[I_a]y <=> x/\y/\a != 0 for every filter objects x and y.
Oct 10 15:09:57 <porton> I'll define restricting of a funcoid f to a filter object a by the formula: f|_a = f o I_a.
Oct 10 15:10:03 <porton> Image of a funcoid is defined by the formula: im f = <f>1. Domain by the formula: dom f = im f^-1.
Oct 10 15:10:06 <porton> Image and domain of a funcoid are filter object.
Oct 10 15:10:17 <porton> (I recall that that's a generalization of image and domain of a binary relation being sets.)
Oct 10 15:10:25 <porton> "dom f" is the join of all atoms a of FILT such that <f>a != 0.
Oct 10 15:10:39 <porton> Funcoids equipped with two filter objects, a superobject of the domain and a superobject of the image form a category, with identity funcoids being identity morphisms.
Oct 10 15:11:02 <porton> I will denote "atoms X" atoms under a lattice element X.
Oct 10 15:11:07 <porton> <f>x = \/ { <f>X | X in atoms x } for every filter object x.
Oct 10 15:11:11 <porton> x[f]y <=> exists X in atoms x, Y in atoms y: x[f]y.
Oct 10 15:11:32 * Rogozi (~Rogozi@h-32-202.A212.priv.bahnhof.se) has left #mathematics
Oct 10 15:11:42 <porton> Funcoids can be considered as a continuation of certain functions and relations defined on atomic filter objects.
Oct 10 15:11:45 <porton> I skip that here for simplicity. Refer to the above mentioned article.
Oct 10 15:11:53 <porton> A generalization of direct (Cartesian) product of two sets is direct product of two filter objects as defined in the theory of funcoids:
Oct 10 15:12:02 <porton> Direct product of filter objects a and b is such funcoid aXb that
Oct 10 15:12:06 <porton> x[aXb]y <=> x/\a != 0 /\ y/\b != 0 for every filter objects x and y.
Oct 10 15:12:11 <porton> We have <aXb>x = b if x/\b != 0 and <aXb> = 0 if x/\b = 0.
Oct 10 15:12:33 <porton> f /\ (aXb) = I_a o f o I_b.
Oct 10 15:12:36 <porton> f /\ (aXb) != 0 <=> a[f]b.
Oct 10 15:12:43 <porton> (a0Xb0)/\(a1Xb1) = (a0/\a1)X(b0/\b1).
Oct 10 15:12:52 <porton> not too fast?
Oct 10 15:13:18 <porton> If a is an atomic filter object then f|_a = a X <f>a.
Oct 10 15:13:21 <@thermoplyae> nope, this is fine
Oct 10 15:13:23 <porton> A funcoid is an atom of the lattice of funcoids iff it is a direct product of two atomic filter objects.
Oct 10 15:13:30 <porton> atoms(f\/g) = atoms f \/ atoms g for every funcoids f and g.
Oct 10 15:13:35 <porton> For every funcoids f,g,h and a set R of funcoids we have
Oct 10 15:13:40 <porton> f/\(g\/h) = (f/\g) \/ (f/\h);
Oct 10 15:13:46 <porton> f \/ /\R = /\ { f\/g | g in R }.
Oct 10 15:13:51 <porton> Thus the lattice of funcoids is co-brouwerian.
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Oct 10 15:14:10 <porton> I will call co-complete such funcoid f that <f>|_SET : SET->SET.
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Oct 10 15:14:13 <porton> I will call complete a funcoid reverse of a co-complete funcoid.
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Oct 10 15:14:20 <porton> A funcoid f is complete if <f>A = \/{ <f>(a) | a in A } for every set A.
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Oct 10 15:14:31 <porton> A funcoid f is complete if <f>A = \/{ <f>(a) | a in A } for every set A.
Oct 10 15:14:34 <porton> There are several ways to characterize complete funcoids which I skip here.
Oct 10 15:14:41 <porton> To specify a complete funcoid f it is enough to specify <f> on one-element sets, values of <f> on one element sets can be specified arbitrarily.
Oct 10 15:14:54 <porton> A funcoid is discrete iff it is both complete and co-complete.
Oct 10 15:15:00 <porton> I will skip discussion of "completion" and "co-completion" of funcoids turning arbitrary funcoids into complete and co-complete funcoids.
Oct 10 15:15:19 <porton> I will call monovalued such a funcoid f that f o f^-1 <= I_{im f}.
Oct 10 15:15:21 <_llll_> there seems to be a lack of motivation for all this, what's the point?
Oct 10 15:15:37 <porton> _llll_: A generalization of general topology
Oct 10 15:15:55 <porton> to continue?
Oct 10 15:16:05 <_llll_> no, tell me more about what you are generalising, and why
Oct 10 15:16:51 <porton> I generalize proximity spaces, pretopology spaces, preclosures, and uniformities (as well as binary relations)
Oct 10 15:17:05 <_llll_> i dont care what you are generalisaing, *why* do those things need generalising?
Oct 10 15:17:19 <porton> The purpose of this is the ability to express topological properties with algebraic formulas
Oct 10 15:17:40 <_llll_> could you give a simple example of that?
Oct 10 15:17:56 <_llll_> does it solve some problem in topology?
Oct 10 15:18:17 <porton> For example "f o a <= b o f" is a formula which expresses the statement "f is a continuous function from a space a to a space b"
Oct 10 15:18:40 <_llll_> apart from having fewer letters, what's the advantage of that?
Oct 10 15:19:02 <porton> Below I will refer to a definition of limit of arbitrary (discontinuous) function. I think this all is wort that.
Oct 10 15:19:49 <porton> Also in my works there is defined one formula for all kinds of continuity: continuity, proximal continuity, uniform continuity. I think this unification is worth
Oct 10 15:19:49 <_llll_> can you give an example of a discontinuous function that has a limit according to the new definition?
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Oct 10 15:20:46 <porton> _llll_: I hope we will be able to use my theory in the future to solve present problems. But I haven't dived into the topic of open problems in general topology. So I don't know
Oct 10 15:20:58 <porton> Can I continue the lesson?
Oct 10 15:21:08 <kilimanjaro> yes, please
Oct 10 15:21:16 <_llll_> porton: this seems like a big waste of time to me
Oct 10 15:21:24 <porton> Again: I will call monovalued such a funcoid f that f o f^-1 <= I_{im f}.
Oct 10 15:21:29 <_llll_> you can generalise anything in a million ways
Oct 10 15:21:29 <kilimanjaro> they said that to einstein
Oct 10 15:21:35 <porton> A funcoid is monovalued iff it maps atoms of FILT to atoms or empty set.
Oct 10 15:21:40 <porton> A funcoid f is monovalued iff <f^-1> is distributive over \/.
Oct 10 15:21:43 <_llll_> and einstein solved some interesting things
Oct 10 15:21:46 <porton> Thus a discrete funcoid is monovalued iff the corresponding binary relation is monovalued (a function).
Oct 10 15:21:46 <kilimanjaro> when he used the sheer power of intellect to create general relativity
Oct 10 15:21:52 <kilimanjaro> they tried to silence him
Oct 10 15:21:55 <porton> Funcoids can be T_0, T_1, and T_2 separable. T_0 and T_2 separability is defined through T_1 separability.
Oct 10 15:22:06 <porton> (I skip definitions of T_n here.)
Oct 10 15:22:11 <porton> "A function f is continuous from space a to space b" is formalized by the formula: f o a <= b o f.
Oct 10 15:22:16 <porton> In the case if f is monovalued and entirely defined this formula can be rewritten in two equivalent ways:
Oct 10 15:22:21 <porton> "a <= f^-1 o b o f" or "f o a o f^-1 <= b".
Oct 10 15:22:38 <porton> Now to reloids:
Oct 10 15:22:45 <porton> Reloids are simply filter objects on the lattice of of binary relations (on the set UNIV).
Oct 10 15:22:50 <porton> Thus reloids are a generalization of uniform spaces and of binary relations.
Oct 10 15:23:08 <porton> The reverse reloid is defined in in the natural way.
Oct 10 15:23:11 <porton> The composition of two reloids is defined by the formula:
Oct 10 15:23:13 <porton> g o f = /\ { G o F | F in f, G in g }.
Oct 10 15:23:28 <porton> Sorry, corrected: g o f = /\ { G o F | F in up f, G in up g }.
Oct 10 15:23:37 <porton> (h o g) o f = h o (g o f) for every reloids f,g,h.
Oct 10 15:23:40 <porton> Also f^-1^-1 = f and (g o f)^-1 = f^-1 o g^-1.
Oct 10 15:23:57 <porton> Conjecture: If f,g,h are reloids, then f o (g\/h) = f o g \/ f o h and (g\/h) o f = g o f \/ h o f.
Oct 10 15:24:01 <porton> Direct product of filter objects is also defined in the theory of reloids:
Oct 10 15:24:04 <porton> aXb = /\ { AXB | A in up a, B in up B }.
Oct 10 15:24:06 <porton> AXB is just the cartesian product for sets A and B.
Oct 10 15:24:18 <porton> aXb = \/ { AXB | A in atoms a, B in atoms b }.
Oct 10 15:24:24 <porton> (a0Xb0)/\(a1Xb1) = (a0/\a1)X(b0/\b1).
Oct 10 15:24:38 <porton> I will call a reloid "convex" if it is a union of direct products of filter objects.
Oct 10 15:24:44 <porton> Non-convex reloids exist.
Oct 10 15:24:53 <porton> I will call restricting of a reloid f to a filter object a the reloid f|_a = f/\(aX1).
Oct 10 15:24:56 <porton> Domain and range of a reloid are defined as follows:
Oct 10 15:24:59 <porton> dom f = /\{dom F | F in up f}; im f = /\{im F | F in up f}.
Oct 10 15:25:02 <porton> f <= aXb <=> dom f <= a /\ im f <= b.
Oct 10 15:25:13 <porton> I call identity reloid for the filter object a the reloid I_a = (=)|_a.
Oct 10 15:25:16 <porton> The reverse of identity reloid is identity reloid.
Oct 10 15:25:24 <porton> f|_a = f o I_a for every reloid f and filter object a.
Oct 10 15:25:27 <porton> (g o f)|_a = g o f|_a.
Oct 10 15:25:29 <porton> f /\ (aXb) = I_a o f o I_b.
Oct 10 15:25:45 <porton> Reloids equipped with two filter objects, a superobject of the domain and a superobject of the image form a category, with identity reloids being identity morphisms.
Oct 10 15:25:54 <porton> I will call monovalued such a reloid f that f o f^-1 <= I_{im f}.
Oct 10 15:25:57 <porton> There are some conjectures about monovalued reloids.
Oct 10 15:26:03 <_llll_> but no facts?
Oct 10 15:26:07 <_llll_> and no itnerest?
Oct 10 15:26:11 <_llll_> *interest?
Oct 10 15:26:16 <porton> _llll_: Which facts?
Oct 10 15:26:18 <_llll_> so why did you bother typing all this in?
Oct 10 15:26:37 <burned> porton, what are you working towards exactly?
Oct 10 15:26:49 <@thermoplyae> limits i thought he said
Oct 10 15:26:51 <porton> _llll_: I will continue. You may exit from the chat, if you want
Oct 10 15:27:36 <porton> burned: _A generalization of general topology_ is my main topic. By the way I define and research limits of discontinuous functions.
Oct 10 15:28:19 <burned> then by all means continue
Oct 10 15:28:41 <porton> ... When I was studying in a university, I have solved some infinite sum using the theory of discontinuous limits. (I don't remember the details, but this shows that my theory can be of practical use.)
Oct 10 15:28:58 <porton> I skip discussion of complete reloids and completion of reloids.
Oct 10 15:29:14 <porton> Now to relationships of funcoids and reloids:
Oct 10 15:29:19 <porton> Every reloid f induces a funcoid (FCD)f by the following formulas:
Oct 10 15:29:22 <porton> x[(FCD)f]y <=> forall F in up f: x[F]y.
Oct 10 15:29:24 <porton> <(FCD)f>x = /\{<F>x | F in up f}.
Oct 10 15:29:38 <porton> (FCD)f = f for a binary relation f.
Oct 10 15:29:42 <porton> x[(FCD)f]y <=> (xXy)/\f != 0;
Oct 10 15:30:00 <porton> (FCD)(g o f) = (FCD)g o (FCD)f.
Oct 10 15:30:04 <porton> (FCD)(aXb) = aXb.
Oct 10 15:30:21 <porton> Every funcoid f induces a reloid in two ways, intersection of _outward_ relations and union of _inward_ direct products of filter objects:
Oct 10 15:30:24 <porton> (RLD)_out f = /\ up f; (RLD)_in f = \/ {aXb | a,b in FILT, aXb <= f}.
Oct 10 15:30:33 <porton> (RLD)_in f = \/ {aXb | a,b are atomic filter objects, aXb <= f}.
Oct 10 15:31:16 <porton> (RLD)_out f = f for every binary relation f.
Oct 10 15:31:21 <porton> A funcoid is greater inward than outward: (RLD)_out f <= (RLD)_in f for every reloid f.
Oct 10 15:31:28 <porton> (FCD)(RLD)_in f = f for every funcoid f.
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Oct 10 15:31:34 <porton> (RLD)_in (aXb) = aXb for filter objects a,b.
Oct 10 15:31:39 <porton> It is a question whether (RLD)_out (aXb) = aXb.
Oct 10 15:32:02 <porton> (RLD)_out I_a = I_a. Generally (RLD)_in I_a != I_a.
Oct 10 15:32:16 <porton> (FCD) is a lower adjoint of (RLD)_in.
Oct 10 15:32:20 <porton> Continuous reloids are defined analogously to continuous funcoids.
Oct 10 15:32:24 <_llll_> why is it a question worth considering? what a waste of time
Oct 10 15:32:28 <burned> can you show us how (RLD)_out I_a= I_a?
Oct 10 15:32:59 <porton> burned: I don't remember the proof. You may consult my article
Oct 10 15:33:09 <_llll_> rofl
Oct 10 15:33:29 <porton> Continuous reloids are defined analogously to continuous funcoids.
Oct 10 15:33:30 * burned blinks
Oct 10 15:33:33 <porton> (In fact I have a general theory of continuity in http://www.mathematics21.org/binaries/funcoids-reloids.pdf).
Oct 10 15:33:44 <porton> Now I am busy with creating the theory of pointfree generalization of funcoids.
Oct 10 15:33:51 <porton> And afterward I am going to research n-ary (multidimensional) funcoids.
Oct 10 15:33:57 <porton> The theory of funcoids also allows to define limits including...
Oct 10 15:34:03 <porton> generalized limits of arbitrary (discontinuous) functions.
Oct 10 15:34:05 <_llll_> whya re you going to reserch n-ary (multidimensional) funcoids?
Oct 10 15:34:15 <_llll_> why not research something else?
Oct 10 15:34:36 <porton> _llll_: These are useful for such things as binary operations on limits of discontinuous functions
Oct 10 15:34:47 <_llll_> what does that mean?
Oct 10 15:35:02 <_llll_> useful in what way? or are you just making this up as you go along
Oct 10 15:35:08 <_llll_> because this seems worthless snake oil so far
Oct 10 15:35:13 <porton> _llll_: I don't want to research something other now because my research of funcoids is quite productive
Oct 10 15:35:21 <_llll_> productive? for what?
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Oct 10 15:35:31 <porton> _llll_: For new results
Oct 10 15:35:38 <_llll_> results about what?
Oct 10 15:35:49 <_llll_> why should *anyone* care?
Oct 10 15:36:24 <porton> _llll_: Because it is just a beautiful theory. And limits of discontinuous functions seems also practical
Oct 10 15:36:26 * antonfire (~me@169.232.171.227) has joined #mathematics
Oct 10 15:36:34 <_llll_> practical for what?
Oct 10 15:36:53 <porton> _llll_: I think they will find uses in engineering, but I'm not sure
Oct 10 15:37:03 <burned> porton, no they won't
Oct 10 15:37:11 <burned> I'm sorry but they won't
Oct 10 15:37:17 <_llll_> can you tell me what the limit, in youre sense, of f:R-->R where f(x)=0 x<0 and f(x)=1, x>=0 is?
Oct 10 15:37:28 * mirror- (~regul@157-114.dsl.iskon.hr) has joined #mathematics
Oct 10 15:38:03 <_llll_> or how about f(x)=0 for x!=0 and f(x)=A for some A.  what is the "limit" of f? is it always 0? or does it depend on A?
Oct 10 15:38:13 <burned> everything you have done is either a relabeling of something old, just flat out crazy nonsense, and not in the "I don't understand this so its crazy" way, in the "please start taking anti-psychotics" way
Oct 10 15:38:24 <burned> +or just
Oct 10 15:38:36 <porton> _llll_: It is a little complex abstract object. So I can't just quickly present it. I want to introduce the definition of my generalized limit below. Using this definition you'd be able to calculate the generalized limits of your functions.
Oct 10 15:38:37 <_llll_> or feel free to pick any other discontinuous fuction with a "natural" limit
Oct 10 15:38:54 <_llll_> but *why* introduce a definition? how do we know it's a sensible definition
Oct 10 15:39:07 <_llll_> anyone can write down a list of symbols and say "i call this the limit"
Oct 10 15:39:12 <_llll_> doesnt mean it's useful
Oct 10 15:39:25 <_llll_> why is your theory useful?
Oct 10 15:39:31 <porton> _llll_: Sorry, I cannot explain this quickly
Oct 10 15:39:52 <porton> Maybe who doesn't want to hear will exit the chat and I'll continue?
Oct 10 15:40:03 <_llll_> given that only you ever heard of funcoids and reloids, and only you think theya re improtant, what is the point of the seminar
Oct 10 15:40:20 <_llll_> you typing meaingless symbols is just wasting eevryone's time
Oct 10 15:40:28 <CESSMASTER> I am interested, and I would appreciate if you would stop being disruptive so that the rest of us could learn
Oct 10 15:40:34 <porton> _llll_: Please exit from the chat and I'll continue
Oct 10 15:41:05 <porton> The theory of funcoids also allows to define limits including...
Oct 10 15:41:09 <porton> generalized limits of arbitrary (discontinuous) functions.
Oct 10 15:41:14 <porton> I have yet not researched how to do (for example) binary operations on limits of discontinuous functions.
Oct 10 15:41:18 <porton> But I expect that it will be always lim f + lim g = lim(f+g).
Oct 10 15:41:23 <porton> To the definition of limits of funcoids:
Oct 10 15:41:27 <porton> A filter object F converges to a filter object A regarding a funcoid f when F <= <f>A.
Oct 10 15:41:34 <porton> (This generalizes the standard definition of filter convergent to a point or to a set.)
Oct 10 15:41:46 <porton> A funcoid f converges to a filter object A regarding a funcoid g iff
Oct 10 15:41:49 <porton> im f <= <g>A that is if im f converges to A regarding g.
Oct 10 15:41:59 <porton> A funcoid converges to a filter object A on a filter object B regarding a funcoid g iff f|_B converges to A regarding g.
Oct 10 15:42:07 <porton> The standard theorem about convergence of composition holds.
Oct 10 15:42:11 <porton> The standard theorem about convergence of a continuous function extends to funcoids.
Oct 10 15:42:18 <porton> Limit of a funcoid is its point (that is a trivial atomic filter object) of convergence.
Oct 10 15:42:33 <porton> Generalized limit (for discontinuous functions):
Oct 10 15:42:36 <porton> Here we need some settings:
Oct 10 15:42:39 <porton> Let m and n are funcoids, G is a group of funcoids.
Oct 10 15:42:46 <porton> Let D is such a set that forall r in G: im r <= D /\ forall x,y in D exists r in G: r(x)=y.
Oct 10 15:42:47 * ChanServ has changed the topic to: meaningless gibberish will continue until further notice
Oct 10 15:42:50 <porton> We require that a and every r in G commutes that is m o r = r o m.
Oct 10 15:43:00 <porton> We require that for every y in UNIV we have n >= <n>{y} X <n>{y}. (1)
Oct 10 15:43:03 <porton> The formula (1) usually works when n is a proximity. It does not work if n is a pretopology or preclosure.
Oct 10 15:43:06 <porton> We are going to consider (generalized) limits of arbitrary functions acting from m to n.
Oct 10 15:43:10 <porton> (The functions in consideration are not required to be continuous.)
Oct 10 15:43:25 <porton> Most typically G is the group of translations of some topological vector space.
Oct 10 15:43:31 <porton> Generalized limit is defined by the following formula:
Oct 10 15:43:34 <porton> xlim f = { n o f o r | r in G }.
Oct 10 15:43:47 <porton> We will assume that the function f is defined on <m>{x}.
Oct 10 15:43:49 <porton> xlim_x f = xlim f|_{<m>{x}}.
Oct 10 15:43:54 <porton> For T_1-separable funcoid n the limits of continuous funcoids bijectively correspond to the points of the space UNIV.
Oct 10 15:44:00 <porton> I finished
Oct 10 15:44:03 <porton> Again read http://www.mathematics21.org/binaries/funcoids-reloids.pdf and more generally http://www.mathematics21.org/algebraic-general-topology.html
Oct 10 15:44:05 <porton> Also subscribe to my blog: http://portonmath.wordpress.com
Oct 10 15:44:23 <antonfire> so what's the limit of the function _llll_ mentioned?
Oct 10 15:45:29 <porton> antonfire: substitute the values to the formula. You have an exercise :-) (I also could consider it as an exercise for me as I can't solve this in one second.)
Oct 10 15:45:48 <porton> Well, D is the group of shifts of the real line
Oct 10 15:46:22 <antonfire> so you have a method for taking limits of noncontinuous functions, but you haven't used it to take a limit of the most obvious noncontinuous function to try?
Oct 10 15:46:36 <burned> what if we were working on the 1-sphere, with the group of translations, would everything still work?
Oct 10 15:47:05 <porton> It is all about hiding the complexity of underlined object inside simple theorems
Oct 10 15:47:22 <burned> how about sin(1/x), can you take the limit of this at x->0?
Oct 10 15:47:22 <porton> ... This is just like object oriented programming
Oct 10 15:47:58 <antonfire> Object oriented programming (and abstraction in general) boils down to something concrete.
Oct 10 15:48:01 <porton> burned: Sorry, what is 1-sphere? Isn't it just two points of real line?
Oct 10 15:48:03 <antonfire> Otherwise it's an abstraction of nothing at all.
Oct 10 15:48:04 <_llll_> no, this is about you not understanding basic mathematics
Oct 10 15:48:44 <antonfire> If your theory doesn't give you answers to specific questions that you claim it gives answers to, then your claim is wrong.
Oct 10 15:49:06 <antonfire> If you haven't even tried to use your theory to answer some such specific questions then why the hell are you giving a seminar on it?
Oct 10 15:49:06 <toed> please stop trolling antonfire
Oct 10 15:49:09 <porton> burned: I can write a description of any limit, but not in a few seconds
Oct 10 15:49:20 <CESSMASTER> you guys are being jerks
Oct 10 15:49:35 <antonfire> It's the only way I know to fulfill myself
Oct 10 15:49:40 <@thermoplyae> loveable jerks
Oct 10 15:49:49 <_llll_> porton: why did you call the mess of symbols a limit?
Oct 10 15:50:00 <_llll_> porton: why not call it the integral of a function?
Oct 10 15:50:21 <porton> _llll_: Because for continuous functions it bijectively corresponds to limits in traditional sense!
Oct 10 15:50:33 <_llll_> can you prove that?
Oct 10 15:50:40 <_llll_> or is this another "i forgot"
Oct 10 15:50:53 <porton> Yes, see not so complex proof in my above mentioned article
Oct 10 15:51:11 <_llll_> no-one is going to read your tedious website
Oct 10 15:51:55 <_llll_> you seem to be doing a great disservice to mathematics by continuing this "research"
Oct 10 15:51:58 <porton> _llll_: This proof requires some lemmas. Do you really want I'd put that proof in the chat?
Oct 10 15:52:32 <_llll_> do what you want.  so far you've failed to convince anyone that there is any point reading what you type
Oct 10 15:53:24 <porton> As such, maybe we'll stop the conversation about my theory and switch to someone who was going to take a lesson about 0^0?
Oct 10 15:54:24 <burned> I'd rather we continue
Oct 10 15:54:39 <burned> I find it intriguing to think in new and exciting ways
Oct 10 15:55:07 <_llll_> he's going to tell us about 0^0 now?
Oct 10 15:55:26 <antonfire> hopefully
Oct 10 15:55:27 <_llll_> dont tell me, 0^0 is a reloid?
Oct 10 15:55:37 <_llll_> and 1/0 is a funcoid
Oct 10 15:55:44 <@thermoplyae> he was referring to someone else in #n-m who had jokes about giving a talk on 0^0
Oct 10 15:55:53 <@thermoplyae> or at least i think it was a joke, idk
Oct 10 15:56:09 <porton> thermoplyae: No, 0^0 isn't a joke
Oct 10 15:56:41 <@thermoplyae> o
Oct 10 15:56:49 <porton> Is the next lector here?
Oct 10 15:57:09 <burned> No, he's unavailable right now
Oct 10 15:57:21 <burned> he had to reschedule.
Oct 10 15:57:36 <antonfire> can you fill in?
Oct 10 16:02:20 <porton> I was disconnected from Internet. The last I heard was "antonfire> can you fill in?" Was something said without me?
Oct 10 16:03:06 <toed> < _llll_> i'm sorry i was so hard on your, please go on
Oct 10 16:03:07 <_llll_> they offered you the abel prize, but unfortunately there was a deadline
Oct 10 16:03:48 <_llll_> (so hard on your what?)
Oct 10 16:03:58 <porton> I was disconnected again
Oct 10 16:04:01 <toed> i don't know you said it
Oct 10 16:08:00 <porton> ... and again
Oct 10 16:11:53 <antonfire> just because nobody is talking does not mean you were disconnected from the internet
Oct 10 16:13:49 <kilimanjaro> ❤
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Oct 10 16:37:23 * thermoplyae has changed the topic to: Next seminar: TBD.  Join #math for mathematical discussion.
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