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Some Conjectures For Exponential Matching Polynomials

Some Conjectures For Exponential \\[0.3cm] Matching Polynomials\\[0.9cm] \end{;bf}; \end{;Large}; \begin{;large}; DRAGUTIN SVRTAN\\[0.9cm] {;\em Faculty of Natural Sciences and Mathematics};\\ {;\em Department of Mathematics};\\ {;\em Bijenička cesta 30};\\ {;\em 10000 Zagreb, Croatia};\\ {;\em dsvrtan@math.hr};\\ {;\em http://www.math.pmf.unizg.hr};\\ [0.7cm] \end{;large}; \end{;center}; In 2001 Sir M. F. Atiyah formulated a conjecture C1 and later with P. Sutcliffe two stronger conjectures C2 and C3. These conjectures, inspired by physics (spin-statistics theorem of quantum mechanics), are geometrically defined for any configuration of points in the Euclidean three space. The conjecture C1 is proved for n = 3, 4 and for general n only for some special configurations (M.F. Atiyah, M. Eastwood and P. Norbury, D.Djoković). In a lengthy preprint [5] we have verified the conjectures C2 and C3 for parallelograms, cyclic quadrilaterals and some infinite families of tetrahedra. We have also proposed a strengthening of the conjecture C3 for configurations of four points (Four Points Conjectures).(The present author found a new geometric fact for arbitrary tetrahedra and a proof of C2 and C3 for arbitrary four points in the Euclidean three space, but not yet for the hyperbolic case !). A number of conjectures for almost collinear configurations are stated in [5] and proven for n up to nine. In this talk we explain our recent variations of the original conjecture C3 (for almost collinear configurations ) involving naturally exponential matching polynomial inequalities which turn out to be stronger than for the ordinary matching polynomials. \\[0.1cm] {;\bf References and Literature for Further Reading};\\ \newcommand\itm[2]{;\parbox[t]{;1cm};{;#1};\parbox[t] {;14cm};{;#2};\\[1mm] }; \itm{;[1]}; {;Atiyah M, Sutcliffe P, The Geometry of Point Particles. arXiv: hep-th/0105179 (32 pages). Proc.R.Soc.Lond. A (2002) 458, 1089-115};.\\ \itm{;[2]}; {;Atiyah M, Sutcliffe P, Polyhedra in Physics, Chemistry and Geometry, arXiv: math- ph/03030701 (22 pages), "Milan J.Math." 71:33-58 (2003)};\\ \itm{;[3]}; {;Eastwood M., Norbury P. A proof of Atiyah's conjecture on configurations of four points in Euclidean three space, Geometry and Topology 5(2001) 885-893.};\\ \itm{;[4]}; {;Svrtan D, Urbiha I, Atiyah-Sutcliffe Conjectures for almost Collinear Configurations and Some New Conjectures for Symmetric Functions, arXiv: math/0406386 (23 pages).}; \\ \itm{;[5]};. {;Svrtan D, Urbiha I, Verification and Strengthening of the Atiyah-Sutcliffe Conjectures for Several Types of Configurations, arXiv: math/0609174 (49 pages).}; %*************** PLEASE DO NOT REMOVE THIS STAR LINE ************** \end{;document};

exponential matching polynomials

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