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We define a new topological polynomial extending the Bollobas-Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behavior under partial duality. This allows to write down a completely explicit combinatorial evaluation of the polynomials, occurring in the parametric representation of the non-commutative Grosse-Wulkenhaar quantum field theory. An explicit solution of the parametric representation for commutative field theories based on the Mehler kernel is also provided.

ABSTRACT                                                                                       This paper is concerned with introducing and studying the M -space by using the mixed degree systems which are the core concept in this paper. The necessary and sufficient condition for the equivalence of two reflexive M -spaces is super imposed. In addition, the m -derived graphs, m -open graphs, m -closed graphs, m -interior operators, m -closure operators and M -subspace are introduced. From an M -space, a unique supratopological space is introduced. Furthermore, the m -continuous ( m -open and m -closed) functions are defined and the fundamental theorem of the m -continuity is provided. Finally, the m -homeomorphism is defined and some of its properties are investigated. 

Geometric, topological and graph theory modeling and analysis of biomolecules are of essential importance in the conceptualization of molecular structure, function, dynamics, and transport. On the one hand, geometric modeling provides molecular surface and structural representation, and offers the basis for molecular visualization, which is crucial for the understanding of molecular structure and interactions. On the other hand, it bridges the gap between molecular structural data and theoretical/mathematical models. Topological analysis and modeling give rise to atomic critical points and connectivity, and shed light on the intrinsic topological invariants such as independent components (atoms), rings (pockets) and cavities. Graph theory analyzes biomolecular interactions and reveals biomolecular structure-function relationship. In this paper, we review certain geometric, topological and graph theory apparatuses for biomolecular data modeling and analysis. These apparatuses are categorized into discrete and continuous ones. For discrete approaches, graph theory, Gaussian network model, anisotropic network model, normal mode analysis, quasi-harmonic analysis, flexibility and rigidity index, molecular nonlinear dynamics, spectral graph theory, and persistent homology are discussed. For continuous mathematical tools, we present discrete to continuum mapping, high dimensional persistent homology, biomolecular geometric modeling, differential geometry theory of surfaces, curvature evaluation, variational derivation of minimal molecular surfaces, atoms in molecule theory and quantum chemical topology. Four new approaches, including analytical minimal molecular surface, Hessian matrix eigenvalue map, curvature map and virtual particle model, are introduced for the first time to bridge the gaps in biomolecular modeling and analysis.

Finite simple graphs are a playground for classical areas of mathematics. We illustrate this by looking at some theorems. These are slightly enhanced preparation notes for a talk given at the joint AMS meeting of January 16, 2014 in Baltimore.

We investigate the relationship between the universal topological polynomials for graphs in mathematics and the parametric representation of Feynman amplitudes in quantum field theory. In this first paper we consider translation invariant theories with the usual heat-kernel-based propagator. We show how the Symanzik polynomials of quantum field theory are particular multivariate versions of the Tutte polynomial, and how the new polynomials of noncommutative quantum field theory are particular versions of the Bollob\'as-Riordan polynomials.

We investigate the relationship between the universal topological polynomials for graphs in mathematics and the parametric representation of Feynman amplitudes in quantum field theory. In this first paper we consider translation invariant theories with the usual heat-kernel-based propagator. We show how the Symanzik polynomials of quantum field theory are particular multivariate versions of the Tutte polynomial, and how the new polynomials of noncommutative quantum field theory are particular versions of the Bollob\'as-Riordan polynomials.

( Wiley Series In Discrete Mathematics And Optimization) Jonathan L. Gross, Thomas W. Tucker, Topological Graph Theory, Wiley Interscience ( 1987)

( Wiley Series In Discrete Mathematics And Optimization) Jonathan L. Gross, Thomas W. Tucker, Topological Graph Theory, Wiley Interscience ( 1987)

( Wiley Series In Discrete Mathematics And Optimization) Jonathan L. Gross, Thomas W. Tucker, Topological Graph Theory, Wiley Interscience ( 1987)

Reputedly inspired by the Sherlock Holmes stories, Austrian criminal jurist and examining magistrate Hans Gross wrote the first handbook on criminal investigation. This treatise covers everything from the qualities of a good investigating officer and how to utilize various experts, to tactics employed by criminals, how to analyze footprints and blood stains, and ways that criminals perpetrate crimes. Some of the remarks relate directly to India, such as disguising one's caste.<br><br>Volume 1 (of 3) consists of Part 1 of the 4 parts in the work. - Summary by TriciaG

Reputedly inspired by the Sherlock Holmes stories, Austrian criminal jurist and examining magistrate Hans Gross wrote the first handbook on criminal investigation. This treatise covers everything from the qualities of a good investigating officer and how to utilize various experts, to tactics employed by criminals, how to analyze footprints and blood stains, and ways that criminals perpetrate crimes. Some of the remarks relate directly to India, such as disguising one's caste.<br><br>Volume 2 (of 3) consists of Parts 2 and 3 of the 4 parts in the work. - Summary by TriciaG

Reputedly inspired by the Sherlock Holmes stories, Austrian criminal jurist and examining magistrate Hans Gross wrote the first handbook on criminal investigation. This treatise covers everything from the qualities of a good investigating officer and how to utilize various experts, to tactics employed by criminals, how to analyze footprints and blood stains, and ways that criminals perpetrate crimes. Some of the remarks relate directly to India, such as disguising one's caste.<br><br>Volume 3 (of 3) consists of Part 4 of the 4 parts in the work. - Summary by TriciaG