scholarly journals Holomorphic volume forms on representation varieties of surfaces with boundary

2020 ◽  
Vol 3 ◽  
pp. 341-380
Author(s):  
Michael Heusener ◽  
Joan Porti
Keyword(s):  
Representation Varieties ◽  
Volume Forms

scholarly journals Conservative finite-volume forms of the Saint-Venant equations for hydrology and urban drainage

2019 ◽  
Vol 23 (3) ◽  
pp. 1281-1304 ◽  
Author(s):  
Ben R. Hodges
Keyword(s):  
Free Surface ◽  
Finite Volume ◽  
Source Term ◽  
Weighting Function ◽  
Bottom Topography ◽  
Urban Drainage ◽  
Venant Equation ◽  
Saint Venant Equations ◽  
Volume Forms ◽  
Conservative Flux

Abstract. New integral, finite-volume forms of the Saint-Venant equations for one-dimensional (1-D) open-channel flow are derived. The new equations are in the flux-gradient conservation form and transfer portions of both the hydrostatic pressure force and the gravitational force from the source term to the conservative flux term. This approach prevents irregular channel topography from creating an inherently non-smooth source term for momentum. The derivation introduces an analytical approximation of the free surface across a finite-volume element (e.g., linear, parabolic) with a weighting function for quadrature with bottom topography. This new free-surface/topography approach provides a single term that approximates the integrated piezometric pressure over a control volume that can be split between the source and the conservative flux terms without introducing new variables within the discretization. The resulting conservative finite-volume equations are written entirely in terms of flow rates, cross-sectional areas, and water surface elevations – without using the bottom slope (S0). The new Saint-Venant equation form is (1) inherently conservative, as compared to non-conservative finite-difference forms, and (2) inherently well-balanced for irregular topography, as compared to conservative finite-volume forms using the Cunge–Liggett approach that rely on two integrations of topography. It is likely that this new equation form will be more tractable for large-scale simulations of river networks and urban drainage systems with highly variable topography as it ensures the inhomogeneous source term of the momentum conservation equation is Lipschitz smooth as long as the solution variables are smooth.

scholarly journals On representation varieties of 3–manifold groups

2017 ◽  
Vol 21 (4) ◽  
pp. 1931-1968 ◽  
Author(s):  
Michael Kapovich ◽  
John Millson
Keyword(s):  
Representation Varieties

3. Forms of matter

2019 ◽  
pp. 19-38
Author(s):  
Geoff Cottrell
Keyword(s):  
Thermal Motion ◽  
Chemical Bonds ◽  
Intermediate States ◽  
State Of Matter ◽  
Volume Forms ◽  
Repulsion And Attraction

Solids, liquids, and gases are the great states of matter; a solid has a shape and a volume, a liquid has a volume but no shape, and a gas has neither shape nor volume. ‘Forms of matter’ explains how these different states arise from a competition between opposites: thermal motion driving particles apart and the attractive forces between atoms pulling them together, repulsion and attraction. The ‘glue’ that holds electrons to atoms, brings atoms together to form molecules, and draws molecules together to make solids and liquids, is electricity. Chemical bonds, crystals, intermediate states, and plasma—the fourth state of matter—are discussed.

Projective geometry on the bundle of volume forms

1998 ◽  
Vol 62 (1-2) ◽  
pp. 66-83 ◽  
Author(s):  
Paul F. Dhooghe ◽  
Annouk Van Vlierden
Keyword(s):  
Projective Geometry ◽  
Volume Forms

scholarly journals TOPOLOGY OF THE REPRESENTATION VARIETIES WITH BOREL MOLD FOR UNSTABLE CASES

2011 ◽  
Vol 91 (1) ◽  
pp. 55-87 ◽  
Author(s):  
KAZUNORI NAKAMOTO ◽  
TAKESHI TORII
Keyword(s):  
Configuration Space ◽  
Cohomology Group ◽  
Cohomology Ring ◽  
Picard Group ◽  
Homotopy Equivalent ◽  
Representation Variety ◽  
Stable Case ◽  
Representation Varieties

AbstractIn this paper we show that, in the stable case, when m≥2n−1, the cohomology ring H*(Repn(m)B) of the representation variety with Borel mold Repn(m)B and $H^{\ast }(F_{n}(\mathbb {C}^m)) \otimes H^{\ast }(\mathrm {Flag}(\mathbb { C}^n)) \otimes \Lambda (s_{1}, \ldots , s_{n-1})$ are isomorphic as algebras. Here the degree of si is 2m−3 when 1≤i<n. In the unstable cases, when m≤2n−2, we also calculate the cohomology group H*(Repn(m)B) when n=3,4 . In the most exotic case, when m=2 , Rep n (2)B is homotopy equivalent to Fn (ℂ2)×PGL n (ℂ) , where Fn (ℂ2) is the configuration space of n distinct points in ℂ2. We regard Rep n (2)B as a scheme over ℤ, and show that the Picard group Pic (Rep n (2)B) of Rep n (2)B is isomorphic to ℤ/nℤ. We give an explicit generator of the Picard group.

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