Diagram hypothesis as a technique for enhancing science and arithmetic educational program.

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Problem(s). school arithmetic: dull? excessively confused? to technical?various subjects taught in school: to isolated from one another? from the genuine life?possible arrangements?. Fun in school. fun and math/science - a contradiction?can you draw the photo navigating every line just once?
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Chart hypothesis as a strategy for enhancing science and arithmetic educational program Franka M. Brückler , Dep t. of Mathematics, University of Zagreb (Croatia) Vladimir Stilinović , Dep t. of Chemistry, University of Zagreb (Croatia)

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Problem(s) school arithmetic: dull? excessively muddled? to specialized? different subjects instructed in school: to isolated from each other? from the genuine living? conceivable arrangements?

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Fun in school fun and math/science - a disagreement? can you draw the photo navigating every line just once? – Eulerian visits is it conceivable to cross a chessboard with a knight so that every field is gone by once? – Hamiltonian circuits

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Graphs vertices (set V) and edges (set E) – drawn as focuses and lines the arrangement of edges in an (undirected) diagram can be considered as a subset of P (V) comprising of one-and two-part sets history: Euler, Cayley

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Basic ideas nearness – u , v nearby if { u , v } edge vertex degrees – number of neighboring vertices ways – successions u 1 u 2 ... u n to such an extent that each { u i , u i+1 } is and edge + no different edges circuits – shut ways cycles – circuits with all vertices seeming just once basic diagrams – no circles and no numerous edges associated charts – each two vertices associated by a way trees – associated diagram without cycles

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Graphs in science sub-atomic (auxiliary) charts (regularly: hydrogen-supressed) level of a vertex = valence of iota

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response charts – union of the sub-atomic charts of the supstrate and the item 0 : 1 C 2 : 1 2 : 1 C Diels-Alder response C 1 : 2 0 : 1 C 2 : 1

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Mathematical trees develop in science sub-atomic charts of non-cyclic mixes are trees case: alkanes fundamental reality about trees: |V| = |E| + 1 essential actuality about diagrams: 2|E| = whole of all vertex degrees 5–isobutyl–3–isopropyl–2,3,7,7,8-pentamethylnonan

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Alkanes: C n H m no circuits & no various securities  tree number of vertices: v = n + m n vertices with degree 4, m vertices mind degree 1 number of edges: e = (4 n + m )/2 for each tree e = v – 1 4 n + m = 2 n + 2 m – 2  m = 2 n + 2  a recipe C n H m speaks to an alkane just if m = 2 n + 2 methane CH 4 ethane C 2 H 6 propane C 3 H 8

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Topological lists properties of substances depend of their synthetic organization, as well as of the state of their particles descriptors of sub-atomic size, shape and fanning relationships to specific properties of substances (physical properties, compound reactivity, natural movement… )

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Wiener record – 1947. total of separations between all sets of vertices in a H-supressed diagram; just for trees; created to decide parrafine breaking points Randić record – 1975. Great connection capacity for some physical & biochem properties Hosoya record – p ( k ) is the quantity of routes for picking k non-neighboring edges from the diagram; p (0)=1, p (1)=|E|

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topological files and breaking points of a few essential amines

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conceivable activities for students: clearly: to figure a file from an offered chart to locate a normal estimation of the breaking point of an essential amine not recorded in a table , and contrasting it with a test esteem. Such an activity gives the understudy an immaculate perspective of how a property of a su b position may rely on upon its sub-atomic structure

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Examples 2-methylbutane W = 0,5 ((1+2+2+3)+(1+1+1+2)+(1+1+2+2)+(1+2+3+3)+ (1+2+2+3)) = 18: There are four edges, and two methods for picking two non neighboring edges so Z = p (0) + p (1) + p (2) = 1 + 4 + 2 = 7

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For isoprene W isn\'t characterized, since its sub-atomic chart isn\'t a tree Randić list is and Hosoya list is  Z = 1 + 6 + 6 = 13. For cyclohexane W isn\'t characterized, since its atomic chart isn\'t a tree Randić file is and Hosoya file is  Z = 1 + 6 + 18 + 2 = 27.

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Enumeration issues truly the primary utilization of diagram hypothesis to science (A. Cayley, 1870ies) initially: count of isomers i.e. mixes with the same exact recipe, however diverse line and/or stereochemical equation speculation: tallying every conceivable particle for a given arrangement of supstituents and deciding the quantity of isomers for each supstituent mix (Polya count hypothesis) in spite of the fact that there is more combinatorics and gathering hypothesis than chart hypothesis in the arrangement, the beginning stage is the sub-atomic diagram

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Cayley\'s specification of trees 1875. endeavored count of isomeric alkanes C n H 2n+2 and alkyl radicals C n H 2n+1 understood the issues are equal to list of trees/established trees built up a producing capacity for identification of established trees 1881. enhanced the technique for trees

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P ó lya list strategy 1937. – methodical technique for count bunch hypothesis, combinatorics, diagram hypothesis cycle record of a change bunch: aggregate of all cycle sorts of components in the gathering, isolated by the request of the gathering cycle kind of a component is spoken to by a term of the structure x 1 a x 2 b x 3 c ..., where an is the quantity of altered focuses (1-cycles), b is the quantity of transpositions (2-cycles), c is the quantity of 3-cycles and so forth when the symmetry gathering of an atom (considered as a chart) is resolved, utilize the cycle file of the gathering and substitute all x i - s with entireties of An i with An extending through conceivable substituents

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2 3 Example 1 4 6 5 what number of chlorobenzenes are there? what number of isomers of different sorts? consider every single conceivable stage of vertices that can hold a H or a Cl iota that outcome in isomorphic diagrams (for the most part, symmetries of the atomic chart that is inserted regarding geometrical properties) of 6!=720 conceivable stages just 12 don\'t change the adjacencies

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2 3 1 4 5 6 2 1 6 3 5 4 6 1 5 2 4 3 1 symmetry comprising od 6 1-cycles: 1 · x 1 6 2 symmetries (left and right pivot for 60°) comprising od 1 6-cycle: 2 · x 6 1 2 symmetries (left and right turn for 120°) comprising od 2 3-cycles: 2 · x 3 2

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6 3 5 2 1 4 1 3 6 2 5 3 symmetries (diagonals as mirrors) comprising od 2 1-cycles and 2-cycles: 3 · x 1 2 · x 2 4 symmetries (1 revolution for 180° and 3 mirror-operations with mirrors = bisectors of oposite pages) comprising od 3 2-cycles: 4 · x 2 3 summing the terms  cycle record

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substitute x i = H i + Cl i into Z(G)  i.e. there is one and only chlorobenzene with 0, 1, 5 or 6 hydrogen particles and there are 3 isomers with 4 hydrogen iotas, with 3 hydrogen particles and with 2 hydrogen particles

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Planarity and chirality planar charts: conceivable to implant into the plane so edges meet just in vertices an atom is chiral on the off chance that it is not consistent to its mirror picture topological chirality: there is no homeomorphism changing the atom into its mirror picture if the atom is topologically chiral then the comparing diagram is non-planar

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