Mathematical Sciences Faculty PublicationsCopyright (c) 2021 Susquehanna University All rights reserved.
https://scholarlycommons.susqu.edu/math_fac_pubs
Recent documents in Mathematical Sciences Faculty Publicationsen-usSat, 30 Oct 2021 02:54:05 PDT3600A Sequential Importance Sampling Algorithm for Counting Linear Extensions
https://scholarlycommons.susqu.edu/math_fac_pubs/17
https://scholarlycommons.susqu.edu/math_fac_pubs/17Thu, 28 Oct 2021 15:35:12 PDTAlathea Jensen et al.Local tomography and the Jordan structure of quantum theory
https://scholarlycommons.susqu.edu/math_fac_pubs/16
https://scholarlycommons.susqu.edu/math_fac_pubs/16Wed, 03 Aug 2016 06:00:06 PDT
Using a result of H. Hanche-Olsen, we show that (subject to fairly natural constraints on what constitutes a system, and on what constitutes a composite system), orthodox finite-dimensional complex quantum mechanics with superselection rules is the only non-signaling probabilistic theory in which (i) individual systems are Jordan algebras (equivalently, their cones of unnormalized states are homogeneous and self-dual), (ii) composites are locally tomographic (meaning that states are determined by the joint probabilities they assign to measurement outcomes on the component systems) and (iii) at least one system has the structure of a qubit. Using this result, we also characterize finite dimensional quantum theory among probabilistic theories having the structure of a dagger-monoidal category.
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Alexander Wilce et al.The Surface Evolver
https://scholarlycommons.susqu.edu/math_fac_pubs/15
https://scholarlycommons.susqu.edu/math_fac_pubs/15Mon, 25 Jul 2016 07:24:24 PDT
The Surface Evolver is a computer program that minimizes the energy of a surface subject to constraints. The surface is represented as a simplicial complex. The energy can include surface tension, gravity, and other forms. Constraints can be geometrical constraints on vertex positions or constraints on integrated quantities such as body volumes. The minimization is done by evolving the surface down the energy gradient. This paper describes the mathematical model used and the operations available to interactively modify the surface.
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K. A. BrakkeThe Opaque Cube Problem
https://scholarlycommons.susqu.edu/math_fac_pubs/14
https://scholarlycommons.susqu.edu/math_fac_pubs/14Mon, 25 Jul 2016 07:24:20 PDT
It is a classic puzzle to find the shortest set of curves that intersect all straight lines through a square, and the conjectured solution is still unproven. This paper asks the analogous question for a cube, and comes up with the best known solution.
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K. A. BrakkeMinimal cones on hypercubes
https://scholarlycommons.susqu.edu/math_fac_pubs/13
https://scholarlycommons.susqu.edu/math_fac_pubs/13Mon, 25 Jul 2016 07:24:16 PDT
It is shown that in dimension greater than 4, the minimal area hypersurface separating the faces of a hypercube is the cone over the edges of the hypercube. This constrasts with the cases of two and three dimensions, where the cone is not minimal. For example, a soap film on a cubical frame has a small rounded square in the center. In dimensions over 6, the cone is minimal even if the area separating opposite faces is given zero weight. The proof uses the maximal flow problem that is dual to the minimal surface problem.
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K. A. BrakkeInstability of the wet X soap film
https://scholarlycommons.susqu.edu/math_fac_pubs/12
https://scholarlycommons.susqu.edu/math_fac_pubs/12Mon, 25 Jul 2016 07:09:39 PDT
For idealized, infinitely thin ("dry") soap films, an X is unstable, while for very thick ("wet") soap films it is minimizing. We show that for soap films of relatively small but positive wetness, the X is unstable. Full stability diagrams for the constant liquid fraction case and the constant pressure case are generated. Analogous questions about other singularities remain controversial.
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K. A. Brakke et al.Computation of equilibrium foam structure using the Surface Evolver
https://scholarlycommons.susqu.edu/math_fac_pubs/11
https://scholarlycommons.susqu.edu/math_fac_pubs/11Mon, 25 Jul 2016 07:09:33 PDT
The Surface Evolver has been used to minimise the surface area of various ordered structures for monodisperse foam. Additional features have enabled its application to foams of arbitrary liquid fraction. Early results for the case of dry foam (negligible liquid fraction) produced a structure haveing lower surface area, or energy, than Kelvin's 1887 minimal tetrakaidecahedron. The calculations reported here show that this remains the case when the liquid fraction is finite, up to about 11%, at which point an f.c.c arrangement of the cells becomes preferable.
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K. A. Brakke et al.Numerical Solution of Soap Film Dual Problems
https://scholarlycommons.susqu.edu/math_fac_pubs/10
https://scholarlycommons.susqu.edu/math_fac_pubs/10Mon, 25 Jul 2016 07:09:27 PDT
The soap film problem is to minimize area, and its dual is to maximize the flux of a divergenceless bounded vectorfield. This paper discretizes the continuous problem and solves it numerically. This gives upper and lower bounds on the area of the globally minimizing film. In favorable cases, the method can be used to discover previously unknown films. No initial assumptions about the topology of the film are needed. The paired calibration or covering space model of soap films is used to enable representation of films with singularities.
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K. A. BrakkeSoap films and covering spaces
https://scholarlycommons.susqu.edu/math_fac_pubs/9
https://scholarlycommons.susqu.edu/math_fac_pubs/9Mon, 25 Jul 2016 07:09:20 PDT
A new mathematical model of soap films is proposed, called the "covering space model." The two sides of a film are modelled as currents on different sheets of a covering space branching along the film boundary. Hence a film may be seen as the minimal cut separating one sheet of the covering space from the others. The film is thus the oriented boundary of one sheet, which represents the exterior of the film. As oriented boundaries, films may be calibrated with differential forms on the covering space, a version of the min-cut, max-flow duality of network theory. This model applies to unoriented films, films with singularities, films touching only part of a knotted curve, films that deformation retract to their boundaries, and other examples that have proved troublesome for previous soap film models.
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K. A. BrakkeMinimal Surfaces, Corners, and Wires
https://scholarlycommons.susqu.edu/math_fac_pubs/8
https://scholarlycommons.susqu.edu/math_fac_pubs/8Mon, 25 Jul 2016 07:09:14 PDT
Weierstrass representations are given for minimal surfaces that have free boundaries on two planes that meet at an arbitrary dihedral angle. The contact angles of a surface on the planes may be different. These surfaces illustrate the behavior of soapfilms in convex and nonconvex corners. They can also be used to show how a boundary wire can penetrate a soapfilm with a free end, as in the overhand knot surface. They should also cast light on the behavior of capillary surfaces.
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K. A. BrakkeInstability of the wet cube cone soap film
https://scholarlycommons.susqu.edu/math_fac_pubs/7
https://scholarlycommons.susqu.edu/math_fac_pubs/7Fri, 22 Jul 2016 09:33:50 PDT
A "dry" conical soap film on a cubical frame is well known not to be stable. Recent experimental evidence seems to indicate that adding liquid to form "Plateau borders" stabilizes the conical film, perhaps to arbitrarily low liquid volumes. This paper presents numerical simulation evidence that the wet cone is unstable for low enough liquid volume, with the critical volume fraction being about 0.000274.
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K. A. BrakkeCalculations of and evidence for chain packing stress in inverse lyotropic bicontinuous cubic phases
https://scholarlycommons.susqu.edu/math_fac_pubs/6
https://scholarlycommons.susqu.edu/math_fac_pubs/6Fri, 22 Jul 2016 09:08:12 PDT
Inverse bicontinuous cubic lyotropic phases are a complex solution to the dilemma faced by all self-assembled water-amphiphile systems: how to satisfy the incompatible requirements for uniform interfacial curvature and uniform molecular packing. The solution reached in this case is for the water-amphiphile interfaces to deform hyperbolically onto triply periodic minimal surfaces. We have previously suggested that although the molecular packing in these structures is rather uniform the relative phase behavior of the gyroid, double diamond, and primitive inverse bicontinuous cubic phases can be understood in terms of subtle differences in packing frustration. In this work, we have calculated the packing frustration for these cubics under the constraint that their interfaces have constant mean curvature. We find that the relative packing stress does indeed differ between phases. The gyroid cubic has the least packing stress, and at low water volume fraction, the primitive cubic has the greatest packing stress. However, at very high water volume fraction, the double diamond cubic becomes the structure with the greatest packing stress. We have tested the model in two ways. For a system with a double diamond cubic phase in excess water, the addition of a hydrophobe may release packing frustration and preferentially stabilize the primitive cubic, since this has previously been shown to have lower curvature elastic energy. We have confirmed this prediction by adding the long chain alkane tricosane to 1-monoolein in excess water. The model also predicts that if one were able to hydrate the double diamond cubic to high water volume fractions, one should destabilize the phase with respect to the primitive cubic. We have found that such highly swollen metastable bicontinuous cubic phases can be formed within onion vesicles. Data from monoelaidin in excess water display a well-defined transition, with the primitive cubic appearing above a water volume fraction of 0.75. Both of these results lend support to the proposition that differences in the packing frustration between inverse bicontinuous cubic phases play a pivotal role in their relative phase stability.
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Gemma C. Shearman et al.Clefting in a Pumpkin Balloon
https://scholarlycommons.susqu.edu/math_fac_pubs/5
https://scholarlycommons.susqu.edu/math_fac_pubs/5Fri, 22 Jul 2016 09:00:01 PDT
NASA's development of a large payload, high altitude, long duration balloon, the Ultra Long Duration Balloon, centers on a pumpkin shape super-pressure design. Under certain circumstances, it has been observed that a pumpkin balloon may be unable to pressurize into the desired cyclically symmetric equilibrium configuration, settling into a distorted, undesired state instead. In this paper, we will use th concept of stability to classify equilibrium shapes of fully pressurized/fully deployed strained ball oons.
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K. A. Brakke et al.Silicon Die Self-alignment on a Wafer: Stable and Unstable Modes
https://scholarlycommons.susqu.edu/math_fac_pubs/4
https://scholarlycommons.susqu.edu/math_fac_pubs/4Fri, 22 Jul 2016 08:43:54 PDT
3D integration is the key to advanced microelectronic systems. Die-to-wafer assembly is a necessary step to reach full integration. Self-assembly methods are promising due to their parallel aspect which overcomes the main difficulties of the current techniques. The aim of this work is the understanding of the mechanisms of self-alignment with an evaporating droplet technique and the investigation the stable and unstable modes. Using the Surface Evolver software, we analyze the causes for misalignments of the system and their evolution.
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J. Berthier et al.Self-alignment of silicon chips on wafers: a capillary approach
https://scholarlycommons.susqu.edu/math_fac_pubs/3
https://scholarlycommons.susqu.edu/math_fac_pubs/3Fri, 22 Jul 2016 08:34:19 PDT
As the limits of Moores law are approached, three-dimensional integration appears as the key to advanced microelectronic systems. Die-to-wafer assembly appears to be an unavoidable step to reach full integration. While robotic methods experience difficulties to accommodate fabrication speed and alignment accuracy, self-assembly methods are promising due to their parallel aspect, which overcomes the main difficulties of current techniques. The aim of this work is the understanding of the mechanisms of self-alignment with an evaporating droplet technique. Stable and unstable modes are examined. Causes for misalignments of chips on wafers and their evolution are investigated with the help of the SURFACE EVOLVER numerical software. Precautions for suitable alignment are proposed.
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J. Berthier et al.Stabilization of the tilt motion during capillary self-alignment of rectangular chips
https://scholarlycommons.susqu.edu/math_fac_pubs/2
https://scholarlycommons.susqu.edu/math_fac_pubs/2Fri, 22 Jul 2016 08:26:24 PDT
Capillary self-alignment (CSA) has emerged as a convenient technique to assemble solid objects. In thistechnique a liquid droplet forces a mobile solid plate or chip to align with its counterpart on a solid substrate. It has been widely investigated for applications such as 3D microelectronics and assembly of optical components. It is now thought that it could be a solution for surface mounting and packaging technologies. For 3D microelectronics, where square or rectangular chips are used, it has been found that amongst the four displacement modes, i.e. shift, twist, lift and tilt, only the tilt mode was unstable (not restoring). In particular, tilting of a floating square or rectangular chip may trigger a direct contactbetween the plate and the pad that impedes alignment. In this text, an analysis of the tilt mode is firstpresented. Second, it is demonstrated that tilt can be stabilized by incorporating specific geometrical features such as lyophilic bands patterned on the substrate.
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J. Berthier et al.Tensor virial equation of evolving surfaces in sintering of aggregates of particles by diffusion
https://scholarlycommons.susqu.edu/math_fac_pubs/1
https://scholarlycommons.susqu.edu/math_fac_pubs/1Fri, 22 Jul 2016 08:26:18 PDT
The moment of inertia tensor is a quantity that characterizes the morphology of aggregates of particles. The deviatoric components indicate the anisotropy of the aggregate, and its compactness is described by the isotropic component, i.e. the second moment of inertia, which is related to the radius of gyration. The equation of motion of the moment of inertia tensor is proposed for the sintering and coalescence of crystalline particles by bulk diffusion and surface diffusion. Simulations of the evolution of aggregates of particles (linear chains, rings and branched chains) show that the aggregates become more compact and more isotropic structures, driven by the surface energy tensor or the surface force density. The tensor virial equation for diffusion is applicable also to evolution of pores, precipitates and inclusions embedded in a surrounding matrix.
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K. A. Brakke et al.