USING MAPLE FOR VISUALIZATION OF TOPOLOGICAL SUBGROUPOIDS OF X Z X

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1 USING MAPLE FOR VISUALIZATION OF TOPOLOGICAL SUBGROUPOIDS OF X Z X Mădălina Roxana BUNECI, Uniersity Constantin Brâncşi of Târg-Ji Abstract. The prpose of this paper is to present arios ways to isalize topological sbgropoids of the triial gropoid X Z X, where Z is the grop of integer nmbers endowed with the discrete topology and X a topological space. Keywords: topological gropoid; Khalimsky topology; eqialence relation; isalization;. 1. INTRODUCTION We shall se the notation and terminology from [1] - [4]. In [3] we proed that a sbgropoid G of XZX, where Z is the grop of integer nmbers, is characterized by the graph R of an eqialence relation on X and a family of integer nmbers {k, } (,)R satisfying the following conditions: 1. k,0 for all X. 2. If k, 0, then k, + k,w = k,w (mod k, ), else k, + k,w = k,w. We also proed that if k, 0 and, then we may consider k, {0,1,, k, -1} for all sch that (,)R. The gropoid G is represented as G= G ={(,k, +tk,,): tz}, (,)R G where, R Alternatiely, we proe in [4] that G can be completely characterized by X and two fnctions f:xx and k :XZ satisfying the properties 4. f(f())=f() for all X. 5. k(f()) 0 for all X. 6. If k(f()) 0, then k(){0,1,, k(f())-1}. If the fnction f and k are gien, then the relation R is defined by (,)R if and only if f()=f(), k, := k(f()) for all X and k, := ( k()+k(f()-k()) mod k(f()), if k(f()) 0 k() k(), if k(f()) = 0 for all (,)X X with the property that f()=f() and. The prpose of this paper is to proide arios ways to isalize the topological sbgropoids G of XZX assming that X is a topological space and Z has the discrete topology. We se Maple enironment to implement the isalizations. 220 Fiabilitate si Drabilitate - Fiability & Drability No 1/ 2014 Editra Academica Brâncşi, Târg Ji, ISSN X

2 2. VISUALIZATION OF SUBGROUPOIDS OF XZX DATA Let G ={(,k, +tk,,): f()=f(), tz} be a sbgropoid of XZX, where X is a finite set {x 1, x 2,, x n } and Z is the grop of integers. As in [4] we shall se the characterization of G in terms of f:xx and k :XZ in order to implement in Maple the gropoid G, where k()=k,f() for all X. More precisely, we se a list gd of three arrays: gd[1] contains the elements of X (gd[1][i]=x i, i=1..n), gd[2][i] = the index in gd[1] of f(gd[1][i]), i=1..n, gd[3][i] = k(gd[1][i]), i=1..n. The procedre isalization(gd) represent each (r,d)-fibre x j xi G as the rectangle with top left corner (i-1,j) and bottom right corner (i,j-1) filled with a color niqely determined by k and k. x i,x j x i,xi isalization:=proc(gd) local n,no_iso0, maxk, maxk0,i,j,elem,m,c,p,q,z,mk; n:=op(2,op(2,gd[1])); no_iso0:=0; maxk:=0; maxk0:=0; if gd[3][gd[2][i]]>maxk then maxk:=gd[3][gd[2][i]] else if gd[3][gd[2][i]]=0 then no_iso0:=no_iso0+1; for j from 1 to i-1 do if (gd[2][j]=gd[2][i] and abs(gd[3][i]-gd[3][j])>maxk0) then maxk0:= abs(gd[3][i]-gd[3][j]); end do elem:=array(1..n*n);c:=array(0..n+1,0..n+1); for i from 0 to n+1 do for j from 0 to n+1 do c[i,j]:=-2 end do z:= no_iso0/n; mk:= 2*(1-z)/(maxk*(maxk+1)); if gd[3][gd[2][i]]=0 then c[i,j]:=frac(1+(gd[3][i]-gd[3][j])/(2*maxk0+1)*z); else c[i,j]:=z/2+(gd[3][gd[2][i]]*(gd[3][gd[2][i]]-1)/ Fiabilitate si Drabilitate - Fiability & Drability No 1/ 2014 Editra Academica Brâncşi, Târg Ji, ISSN X

3 irem(gd[3][gd[2][i]]+gd[3][i]-gd[3][j],gd[3][gd[2][i]]))*mk; end do m:=0; m:=m+1; elem[m]:=rectangle([i-1,j],[i,j-1],color=color(hsv,c[i,j],1.,1.)); ; RETURN(display(seq(elem[i],i=1..m),axes=none,style=patchnogrid)) end proc; For each k{1,2,,kmax}, where kmax=max{k, : X} the procedre isalization allocates a he (in HSV model). Ths we can isalize the distribtion of {k, : X} among {1,2,,kmax}. Howeer if kmax is big and the length of the minimal interal containing the set {k, :X} is small, then less details are isible. Therefore in this it wold be better to se a ersion of the isalization procedre that allocates hes (in HSV model) only for each k,. The left image below is obtain sing the first ersion, while the right correspond to the second ersion. Also if n is big less details are isible. In this case it is better to represent only the (r,d)-fibres x j G with i=a 1..b 1 and j=a 2...b 2 where a 1, b 1, a 2 and b 2 are chosen in concordance to the region xi of interest. The pictres below illstrated this sitation: 222 Fiabilitate si Drabilitate - Fiability & Drability No 1/ 2014 Editra Academica Brâncşi, Târg Ji, ISSN X

4 3. VISUALIZATION OF TOPOLOGY OF SUBGROUPOIDS G XZX Let s consider that X is endowed with a topology X and Z is endowed with the discrete topology. Then nder the operations (x, n, y)(y, m, z) = (x, n+m, y) (x, n, y) -1 = (y, -n, x) XZX become a topological gropoid. In the following the nit space of the gropoid XZX, {(x,0,x), xx}, is identified with X. We endow any sbgropoid G XZX with the sbspace topology G coming from XZX. In [1] we started with a topological gropoid (G, τ G ), we introdced a topology τ R (τ G ) on the principal gropoid R associated with G (called transported topology from G) and a new topology τ GR on G (called the modified topology on G with respect to R). Let s recall that a basis for the topology τ R (τ G ) is gien by the family of sets {U(F)} F, where each F is a finite collection of open sbsets of G (i.e. F G ) and U(F) = r,du UF Moreoer a basis for the topology τ GR is gien by 1 V r,d r,d U UF where V rns oer all open sets of G and F rns oer all finite collections of open sbsets of G. 223 Fiabilitate si Drabilitate - Fiability & Drability No 1/ 2014 Editra Academica Brâncşi, Târg Ji, ISSN X

5 Let s se the notation (x,y)(,) if and only the seqence ((x i,y i )) i conerges to (,), where x i = and y i = for all i. According to [2] this is eqialent to (x i ) ii conerges to with respect to X, (y i ) ii conerges to with respect to X and 1. for k, 0: k x,x 0, k x k (k x diides k ) and k x,x k, k x,y 2. for k, =0: (k x,x =0 and k, =k x,y ) or (k x,x 0 and k x,x k, k x,y ) We want to know if (+p,x+q)(,) for p,q{-1,0,1}. Let s assme that +p (respectiely, +q)u for all neighborhood U of (respectiely, ) with respect to X. In this case if p=0 or q=0, then (+p,x+q)(,) if and only the procedre isalization allocates the same he (in HSV model) to G and p G q. If (+1,+1)(,) then we draw a rectangle r1 on the right top corner of the rectangle r associated to G by the procedre isalization. The dimensions p of r1 are ¼ of the dimensions of r and color is that of G q. The below procedre isalization_top(gd) allows s to isalize the topology of G in this way for the case when X is endowed with the indiscrete topology X ={, X}. We present here only the part of the procedre that differs from isalization(gd), meaning the constrction of the array elem containing the rectangles associated to each isalization_top:=proc(gd). m:=0; m:=m+1; elem[m]:=[rectangle([i-1,j],[i,j-1],color=color(hsv,c[i,j],1.,1.))]; for p from -1 to 1 by 2 do for q from -1 to 1 by 2 do if ((c[i+p,j+q]<>-2) and (c[i,j]<>c[i+p,j+q]) and (gd[3][gd[2][i+p]]<>0) and (irem(gd[3][gd[2][i]],gd[3][gd[2][i+p]])=0) and (irem(gd[3][i+p]- gd[3][j+q]-gd[3][i]+gd[3][j],gd[3][gd[2][i+p]])=0)) then elem[m]:=[rectangle([i-1/2+p/4,j-1/2+q/4], [i-1/2+p/2,j-1/2+q/2], color=color(hsv,c[i+p,j+q],1.,1.)),op(elem[m])] ; ;. G. 224 Fiabilitate si Drabilitate - Fiability & Drability No 1/ 2014 Editra Academica Brâncşi, Târg Ji, ISSN X

6 The following two pictres are the representations of a gropoid G XZX, where X={1,2,,n} retrned by the procedre isalization (left) and isalization_top (right). We see, for instance that (4,4)(3,3), (4,4)(5,5), (4,6)(5,5), (4,6)(3,5), etc. The below procedre isalization_topk(gd) allows s to isalize the topology of G the case when X={x 1, x 2,, x n } is endowed with the topology X with basis {{x 2k+1 },k=0..[n/2]} {{x 2k-1, x 2k, x 2k+1 }X, k=1..[n/2]} (Khalimsky topology see [5], [6] and [7]). For an (r,d)- fibre 225 G we modify the color of the rectangle associated to G by the procedre isalization by decreasing the satration (in the HSV model) by 1/8 for each odd index of, in X. We present here only the part of the procedre that differs from isalization(gd), meaning the constrction of the array elem containing the rectangles associated to each G. isalization_topk:=proc(gd)... m:=0; m:=m+1; elem[m]:=[rectangle([i-1,j],[i,j-1],color=color(hsv,c[i,j],1- irem(i,2)/8-irem(j,2)/8,1.))]; if irem(i,2)=0 and irem(j,2)=0 then for p from -1 to 1 by 2 do for q from -1 to 1 by 2 do if ((c[i+p,j+q]<>-2) and (c[i,j]<>c[i+p,j+q]) and (gd[3][gd[2][i+p]]<>0) and (irem(gd[3][gd[2][i]],gd[3][gd[2][i+p]])=0) and (irem(gd[3][i+p]- gd[3][j+q]-gd[3][i]+gd[3][j],gd[3][gd[2][i+p]])=0)) then elem[m]:=[rectangle([i-1/2+p/4,j-1/2+q/4], [i-1/2+p/2,j-1/2+q/2], color=color(hsv,c[i+p,j+q], 1-irem(i+p,2)/8-irem(j+q,2)/8,1.)),op(elem[m])] ; Fiabilitate si Drabilitate - Fiability & Drability No 1/ 2014 Editra Academica Brâncşi, Târg Ji, ISSN X

7 ; ; The following two pictres are the representations of a gropoid G XZX, where X={1,2,,n} retrned by the procedre isalization_top (left) and isalization_topk (right). We see, for instance that (2,2)(1,1) if X is endowed with the indiscrete topology bt not with the Khalimsky topology. BIBLIOGRAPHY [1] M. Bneci, Topological gropoids with locally compact fibres, Topology Proceedings 37 (2011), [2] M. Bneci, Gropoids and irreersible discrete dynamical systems II, Fiabilitate şi drabilitate (Fiability & drability), No. 1/2012, [3] M. Bneci, Gropoid redctions associated to discrete dynamical systems, Annals of the Constantin Brâncşi Uniersity of Târg-Ji. Engineering Series. No. 3(2012), [4] M. Bneci, Using Maple to represent the sbgropoids of triial gropoid X Z X, Fiabilitate şi drabilitate (Fiability & drability), Spplement No 1 (2013), [5] E.D. Khalimsky, Pattern analysis of n-dimensional digital images, in: Proc. of the IEEE Internat. Conf. on Systems, Man and Cybernetics, 1986, [6] K. Y. Kong, Concepts of digital topology, Topology and its applications, 46 (1992), [7] V. Koalesky, Finite Topology as Applied to Image Analysis, Compter Vision, Graphics and Image Processing, 46 (2) (1989), Fiabilitate si Drabilitate - Fiability & Drability No 1/ 2014 Editra Academica Brâncşi, Târg Ji, ISSN X