Funcoids and reloids

PDF slides with content similar to this seminar

Seminar
 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 Oct 10 14:49:09 * CESSMASTER (~CESSMASTE@unaffiliated/joelywoely) has joined #mathematics Oct 10 14:50:19 * porton (~porton@87.70.212.21) has joined #mathematics Oct 10 14:51:30 * bavarious (~bavarious@pdpc/supporter/student/bavarious) has joined #mathematics Oct 10 14:53:17 * Rogozi (~Rogozi@h-32-202.A212.priv.bahnhof.se) has joined #mathematics Oct 10 14:53:25 * PhileX (~phx@unaffiliated/philex) has joined #mathematics Oct 10 14:56:37 * afk|eoc (~eoc@pD95610FB.dip0.t-ipconnect.de) has joined #mathematics Oct 10 15:00:02 The topic of this seminar are funcoids and reloids. Oct 10 15:00:08 Funcoids are a generalization of proximity spaces. Also funcoids are a generalization of binary relations, pretopological spaces, preclosure operators. Oct 10 15:00:12 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 Read details in "Funcoids and Reloids" article at http://www.mathematics21.org/binaries/funcoids-reloids.pdf Oct 10 15:00:38 I will reprise the most interesting properties here in the chat. Oct 10 15:00:57 As you can anyway read that article I will skip all proofs. Oct 10 15:01:14 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 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 To continue? Oct 10 15:01:44 <@thermoplyae> you should at least define what a funcoid is Oct 10 15:01:50 See below Oct 10 15:01:58 Most naturally funcoids arrive as a generalization of proximities. Oct 10 15:02:02 First, I will introduce some strange terminology: Oct 10 15:02:09 I will consider "the set of filters" on some set UNIV "the set of all sets". Oct 10 15:02:14 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 I call such reverse ordered filters equated with sets, when appropriate, "filter objects". Oct 10 15:02:25 Note that "the set of all sets" SET is a subset of the set of filter objects. Oct 10 15:02:37 continue? Oct 10 15:02:58 <@thermoplyae> yes, go ahead, someone will interrupt if they have questions Oct 10 15:02:58 I will denote the filter object corresponding to the filter X as "up X". Oct 10 15:03:01 I will denote the set of filter objects as FILT. Oct 10 15:03:10 FILT is an atomistic complete co-brouwerian lattice. (Atoms of FILT correspond to maximal filters.) Oct 10 15:03:26 Now we can introduce funcoids. Oct 10 15:03:30 Let d is a proximity. We can extend it to filter objects by the formula: Oct 10 15:03:34 x d y <=> forall X in x, Y in y: X d Y. Oct 10 15:04:01 Sorry, corrected formula: x d y <=> forall X in up x, Y in up y: X d Y. Oct 10 15:04:22 For every proximity there exists two functions alpha, beta: FILT->FILT such that Oct 10 15:04:25 x d y <=> y /\ alpha(x) != 0 <=> x /\ beta(y) != 0. Oct 10 15:04:29 I call "funcoids" pairs (alpha;beta) of functions FILT->FILT such that Oct 10 15:04:34 y /\ alpha(x) != 0 <=> x /\ beta(y) != 0. Oct 10 15:04:53 I will denote FCD the set of funcoids. Oct 10 15:04:56 I've said above that for every proximity exists a corresponding funcoid. Oct 10 15:05:04 In fact a corresponding (see the formula above) funcoid exists for every binary relation d on SETS such that Oct 10 15:05:07 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 I denote  the first component (alpha) of a funcoid f. Oct 10 15:05:38 (X\/Y) = X \/ Y for every filter objects X and Y. Oct 10 15:05:47 The function  for a funcoid f can be constructed as a (unique) continuation of the function alpha: SET->FILT Oct 10 15:05:55 such that alpha(0)=0 and alpha(X\/Y)=X\/Y. Oct 10 15:06:03 Conversely, any  restricted to SET for a funcoid f conforms to the above formulas for alpha. Oct 10 15:06:19 Any (multivalued) function (binary relation) can be considered as a funcoid where Oct 10 15:06:26 (X) = /\ {f[x] | x in up X}. Oct 10 15:06:38 I will call funcoids corresponding to binary relations discrete. Oct 10 15:06:53 Like binary relations funcoids can be reversed and composed: Oct 10 15:06:59 (alpha;beta)^-1 = (beta;alpha); Oct 10 15:07:03 (alpha2;beta2) o (alpha1;beta1) = (alpha2 o alpha1; beta1 o beta2). Oct 10 15:07:14 Reverse of a funcoid corresponding to a proximity is equal to this funcoid (because proximities are symmetric). Oct 10 15:07:28 We have =  o ; Oct 10 15:07:31 (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 I will denote x[f]y <=> y /\ (x) != 0 <=> x /\ (y) != 0 for every funcoid f and a filter objects x and y. Oct 10 15:07:53 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 [f^-1] = [f]^-1. Oct 10 15:08:07 For every value of  exists no more than one funcoid f. Oct 10 15:08:09 For every value of [f] exists no more than one funcoid f. Oct 10 15:08:15 Moreover a funcoid f is uniquely determined by values of  on sets. Oct 10 15:08:31 Oh, that funcoids are a generalization of pretopological spaces is provided by the formula: Oct 10 15:08:35 X = \/ { alpha(x) | x in X } for a set X where alpha is a pretopology (a function UNIV->FILT). Oct 10 15:08:54 Funcoids form a complete brouwerian atomistic lattice by the order f <= g <=> [f] <= [g]. Oct 10 15:09:02 I'll skip the formulas for calculating  and [g] for join of funcoids. Oct 10 15:09:05 More properties of funcoids: Oct 10 15:09:16 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 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 The identity funcoid I_a = (a/\; a/\) is defined for a filter object a. Oct 10 15:09:41 Reverse of identity funcoid is identity funcoid. Oct 10 15:09:46 x[I_a]y <=> x/\y/\a != 0 for every filter objects x and y. Oct 10 15:09:57 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 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 Image and domain of a funcoid are filter object. Oct 10 15:10:17 (I recall that that's a generalization of image and domain of a binary relation being sets.) Oct 10 15:10:25 "dom f" is the join of all atoms a of FILT such that <f>a != 0. Oct 10 15:10:39 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 I will denote "atoms X" atoms under a lattice element X. Oct 10 15:11:07 <f>x = \/ { <f>X | X in atoms x } for every filter object x. Oct 10 15:11:11 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 Funcoids can be considered as a continuation of certain functions and relations defined on atomic filter objects. Oct 10 15:11:45 I skip that here for simplicity. Refer to the above mentioned article. Oct 10 15:11:53 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 Direct product of filter objects a and b is such funcoid aXb that Oct 10 15:12:06 x[aXb]y <=> x/\a != 0 /\ y/\b != 0 for every filter objects x and y. Oct 10 15:12:11 We have <aXb>x = b if x/\b != 0 and <aXb> = 0 if x/\b = 0. Oct 10 15:12:33 f /\ (aXb) = I_a o f o I_b. Oct 10 15:12:36 f /\ (aXb) != 0 <=> a[f]b. Oct 10 15:12:43 (a0Xb0)/\(a1Xb1) = (a0/\a1)X(b0/\b1). Oct 10 15:12:52 not too fast? Oct 10 15:13:18 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 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 atoms(f\/g) = atoms f \/ atoms g for every funcoids f and g. Oct 10 15:13:35 For every funcoids f,g,h and a set R of funcoids we have Oct 10 15:13:40 f/\(g\/h) = (f/\g) \/ (f/\h); Oct 10 15:13:46 f \/ /\R = /\ { f\/g | g in R }. Oct 10 15:13:51 Thus the lattice of funcoids is co-brouwerian. Oct 10 15:13:53 * kilimanjaro (~kilimanja@unaffiliated/kilimanjaro) has joined #mathematics Oct 10 15:14:10 I will call co-complete such funcoid f that <f>|_SET : SET->SET. Oct 10 15:14:10 * Pthing (~pthing@cpc2-pres4-0-0-cust1130.pres.cable.virginmedia.com) has joined #mathematics Oct 10 15:14:13 I will call complete a funcoid reverse of a co-complete funcoid. Oct 10 15:14:17 * Pthing (~pthing@cpc2-pres4-0-0-cust1130.pres.cable.virginmedia.com) has left #mathematics ("surge surge vigila, semper esto paratus") Oct 10 15:14:20 A funcoid f is complete if <f>A = \/{ <f>(a) | a in A } for every set A. Oct 10 15:14:20 * xerox (~xerox@unaffiliated/xerox) has joined #mathematics Oct 10 15:14:31 A funcoid f is complete if <f>A = \/{ <f>(a) | a in A } for every set A. Oct 10 15:14:34 There are several ways to characterize complete funcoids which I skip here. Oct 10 15:14:41 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 A funcoid is discrete iff it is both complete and co-complete. Oct 10 15:15:00 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 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 _llll_: A generalization of general topology Oct 10 15:15:55 to continue? Oct 10 15:16:05 <_llll_> no, tell me more about what you are generalising, and why Oct 10 15:16:51 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 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 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 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 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? Oct 10 15:19:51 * burned (~burned@174.110.137.18) has joined #mathematics Oct 10 15:20:46 _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 Can I continue the lesson? Oct 10 15:21:08 yes, please Oct 10 15:21:16 <_llll_> porton: this seems like a big waste of time to me Oct 10 15:21:24 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 they said that to einstein Oct 10 15:21:35 A funcoid is monovalued iff it maps atoms of FILT to atoms or empty set. Oct 10 15:21:40 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 Thus a discrete funcoid is monovalued iff the corresponding binary relation is monovalued (a function). Oct 10 15:21:46 when he used the sheer power of intellect to create general relativity Oct 10 15:21:52 they tried to silence him Oct 10 15:21:55 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 (I skip definitions of T_n here.) Oct 10 15:22:11 "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 In the case if f is monovalued and entirely defined this formula can be rewritten in two equivalent ways: Oct 10 15:22:21 "a <= f^-1 o b o f" or "f o a o f^-1 <= b". Oct 10 15:22:38 Now to reloids: Oct 10 15:22:45 Reloids are simply filter objects on the lattice of of binary relations (on the set UNIV). Oct 10 15:22:50 Thus reloids are a generalization of uniform spaces and of binary relations. Oct 10 15:23:08 The reverse reloid is defined in in the natural way. Oct 10 15:23:11 The composition of two reloids is defined by the formula: Oct 10 15:23:13 g o f = /\ { G o F | F in f, G in g }. Oct 10 15:23:28 Sorry, corrected: g o f = /\ { G o F | F in up f, G in up g }. Oct 10 15:23:37 (h o g) o f = h o (g o f) for every reloids f,g,h. Oct 10 15:23:40 Also f^-1^-1 = f and (g o f)^-1 = f^-1 o g^-1. Oct 10 15:23:57 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 Direct product of filter objects is also defined in the theory of reloids: Oct 10 15:24:04 aXb = /\ { AXB | A in up a, B in up B }. Oct 10 15:24:06 AXB is just the cartesian product for sets A and B. Oct 10 15:24:18 aXb = \/ { AXB | A in atoms a, B in atoms b }. Oct 10 15:24:24 (a0Xb0)/\(a1Xb1) = (a0/\a1)X(b0/\b1). Oct 10 15:24:38 I will call a reloid "convex" if it is a union of direct products of filter objects. Oct 10 15:24:44 Non-convex reloids exist. Oct 10 15:24:53 I will call restricting of a reloid f to a filter object a the reloid f|_a = f/\(aX1). Oct 10 15:24:56 Domain and range of a reloid are defined as follows: Oct 10 15:24:59 dom f = /\{dom F | F in up f}; im f = /\{im F | F in up f}. Oct 10 15:25:02 f <= aXb <=> dom f <= a /\ im f <= b. Oct 10 15:25:13 I call identity reloid for the filter object a the reloid I_a = (=)|_a. Oct 10 15:25:16 The reverse of identity reloid is identity reloid. Oct 10 15:25:24 f|_a = f o I_a for every reloid f and filter object a. Oct 10 15:25:27 (g o f)|_a = g o f|_a. Oct 10 15:25:29 f /\ (aXb) = I_a o f o I_b. Oct 10 15:25:45 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 I will call monovalued such a reloid f that f o f^-1 <= I_{im f}. Oct 10 15:25:57 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 _llll_: Which facts? Oct 10 15:26:18 <_llll_> so why did you bother typing all this in? Oct 10 15:26:37 porton, what are you working towards exactly? Oct 10 15:26:49 <@thermoplyae> limits i thought he said Oct 10 15:26:51 _llll_: I will continue. You may exit from the chat, if you want Oct 10 15:27:36 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 then by all means continue Oct 10 15:28:41 ... 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 I skip discussion of complete reloids and completion of reloids. Oct 10 15:29:14 Now to relationships of funcoids and reloids: Oct 10 15:29:19 Every reloid f induces a funcoid (FCD)f by the following formulas: Oct 10 15:29:22 x[(FCD)f]y <=> forall F in up f: x[F]y. Oct 10 15:29:24 <(FCD)f>x = /\{<F>x | F in up f}. Oct 10 15:29:38 (FCD)f = f for a binary relation f. Oct 10 15:29:42 x[(FCD)f]y <=> (xXy)/\f != 0; Oct 10 15:30:00 (FCD)(g o f) = (FCD)g o (FCD)f. Oct 10 15:30:04 (FCD)(aXb) = aXb. Oct 10 15:30:21 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 (RLD)_out f = /\ up f; (RLD)_in f = \/ {aXb | a,b in FILT, aXb <= f}. Oct 10 15:30:33 (RLD)_in f = \/ {aXb | a,b are atomic filter objects, aXb <= f}. Oct 10 15:31:16 (RLD)_out f = f for every binary relation f. Oct 10 15:31:21 A funcoid is greater inward than outward: (RLD)_out f <= (RLD)_in f for every reloid f. Oct 10 15:31:28 (FCD)(RLD)_in f = f for every funcoid f. Oct 10 15:31:32 * arborist (~arborist@82.113.106.31) has joined #mathematics Oct 10 15:31:34 (RLD)_in (aXb) = aXb for filter objects a,b. Oct 10 15:31:39 It is a question whether (RLD)_out (aXb) = aXb. Oct 10 15:32:02 (RLD)_out I_a = I_a. Generally (RLD)_in I_a != I_a. Oct 10 15:32:16 (FCD) is a lower adjoint of (RLD)_in. Oct 10 15:32:20 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 can you show us how (RLD)_out I_a= I_a? Oct 10 15:32:59 burned: I don't remember the proof. You may consult my article Oct 10 15:33:09 <_llll_> rofl Oct 10 15:33:29 Continuous reloids are defined analogously to continuous funcoids. Oct 10 15:33:30 * burned blinks Oct 10 15:33:33 (In fact I have a general theory of continuity in http://www.mathematics21.org/binaries/funcoids-reloids.pdf). Oct 10 15:33:44 Now I am busy with creating the theory of pointfree generalization of funcoids. Oct 10 15:33:51 And afterward I am going to research n-ary (multidimensional) funcoids. Oct 10 15:33:57 The theory of funcoids also allows to define limits including... Oct 10 15:34:03 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 _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 _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? Oct 10 15:35:24 * toed (~toad@unaffiliated/toed) has joined #mathematics Oct 10 15:35:31 _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 _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 _llll_: I think they will find uses in engineering, but I'm not sure Oct 10 15:37:03 porton, no they won't Oct 10 15:37:11 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 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 +or just Oct 10 15:38:36 _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 _llll_: Sorry, I cannot explain this quickly Oct 10 15:39:52 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 _llll_: Please exit from the chat and I'll continue Oct 10 15:41:05 The theory of funcoids also allows to define limits including... Oct 10 15:41:09 generalized limits of arbitrary (discontinuous) functions. Oct 10 15:41:14 I have yet not researched how to do (for example) binary operations on limits of discontinuous functions. Oct 10 15:41:18 But I expect that it will be always lim f + lim g = lim(f+g). Oct 10 15:41:23 To the definition of limits of funcoids: Oct 10 15:41:27 A filter object F converges to a filter object A regarding a funcoid f when F <= <f>A. Oct 10 15:41:34 (This generalizes the standard definition of filter convergent to a point or to a set.) Oct 10 15:41:46 A funcoid f converges to a filter object A regarding a funcoid g iff Oct 10 15:41:49 im f <= <g>A that is if im f converges to A regarding g. Oct 10 15:41:59 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 The standard theorem about convergence of composition holds. Oct 10 15:42:11 The standard theorem about convergence of a continuous function extends to funcoids. Oct 10 15:42:18 Limit of a funcoid is its point (that is a trivial atomic filter object) of convergence. Oct 10 15:42:33 Generalized limit (for discontinuous functions): Oct 10 15:42:36 Here we need some settings: Oct 10 15:42:39 Let m and n are funcoids, G is a group of funcoids. Oct 10 15:42:46 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 We require that a and every r in G commutes that is m o r = r o m. Oct 10 15:43:00 We require that for every y in UNIV we have n >= <n>{y} X <n>{y}. (1) Oct 10 15:43:03 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 We are going to consider (generalized) limits of arbitrary functions acting from m to n. Oct 10 15:43:10 (The functions in consideration are not required to be continuous.) Oct 10 15:43:25 Most typically G is the group of translations of some topological vector space. Oct 10 15:43:31 Generalized limit is defined by the following formula: Oct 10 15:43:34 xlim f = { n o f o r | r in G }. Oct 10 15:43:47 We will assume that the function f is defined on <m>{x}. Oct 10 15:43:49 xlim_x f = xlim f|_{<m>{x}}. Oct 10 15:43:54 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 I finished Oct 10 15:44:03 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 Also subscribe to my blog: http://portonmath.wordpress.com Oct 10 15:44:23 so what's the limit of the function _llll_ mentioned? Oct 10 15:45:29 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 Well, D is the group of shifts of the real line Oct 10 15:46:22 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 what if we were working on the 1-sphere, with the group of translations, would everything still work? Oct 10 15:47:05 It is all about hiding the complexity of underlined object inside simple theorems Oct 10 15:47:22 how about sin(1/x), can you take the limit of this at x->0? Oct 10 15:47:22 ... This is just like object oriented programming Oct 10 15:47:58 Object oriented programming (and abstraction in general) boils down to something concrete. Oct 10 15:48:01 burned: Sorry, what is 1-sphere? Isn't it just two points of real line? Oct 10 15:48:03 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 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 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 please stop trolling antonfire Oct 10 15:49:09 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 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 _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 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 _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 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 I'd rather we continue Oct 10 15:54:39 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 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 thermoplyae: No, 0^0 isn't a joke Oct 10 15:56:41 <@thermoplyae> o Oct 10 15:56:49 Is the next lector here? Oct 10 15:57:09 No, he's unavailable right now Oct 10 15:57:21 he had to reschedule. Oct 10 15:57:36 can you fill in? Oct 10 16:02:20 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 < _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 I was disconnected again Oct 10 16:04:01 i don't know you said it Oct 10 16:08:00 ... and again Oct 10 16:11:53 just because nobody is talking does not mean you were disconnected from the internet Oct 10 16:13:49 â¤ Oct 10 16:19:49 * bavarious (~bavarious@pdpc/supporter/student/bavarious) has left #mathematics ("bye") Oct 10 16:23:57 * porton has quit (Quit: Leaving) Oct 10 16:35:19 * Kasadkad (~Kasadkad@98.117.96.47) has joined #mathematics Oct 10 16:37:23 * thermoplyae has changed the topic to: Next seminar: TBD. Join #math for mathematical discussion. Oct 10 16:37:25 * thermoplyae sets mode +m #mathematics Oct 10 16:37:28 * thermoplyae removes channel operator status from thermoplyae Oct 10 16:39:22 * thermoplyae (~thermo@li154-228.members.linode.com) has left #mathematics