3,836 research outputs found
A System of Interaction and Structure II: The Need for Deep Inference
This paper studies properties of the logic BV, which is an extension of
multiplicative linear logic (MLL) with a self-dual non-commutative operator. BV
is presented in the calculus of structures, a proof theoretic formalism that
supports deep inference, in which inference rules can be applied anywhere
inside logical expressions. The use of deep inference results in a simple
logical system for MLL extended with the self-dual non-commutative operator,
which has been to date not known to be expressible in sequent calculus. In this
paper, deep inference is shown to be crucial for the logic BV, that is, any
restriction on the ``depth'' of the inference rules of BV would result in a
strictly less expressive logical system
The geometry of variations in Batalin-Vilkovisky formalism
This is a paper about geometry of (iterated) variations. We explain why no
sources of divergence are built into the Batalin-Vilkovisky (BV) Laplacian,
whence there is no need to postulate any ad hoc conventions such as
"" and "" within BV-approach to quantisation of
gauge systems. Remarkably, the geometry of iterated variations does not refer
at all to the construction of Dirac's -function as a limit of smooth
kernels. We illustrate the reasoning by re-deriving - but not just "formally
postulating" - the standard properties of BV-Laplacian and Schouten bracket and
by verifying their basic inter-relations (e.g., cohomology preservation by
gauge symmetries of the quantum master-equation).Comment: XXI International Conference on Integrable Systems and Quantum
Symmetries (ISQS21) 11-16 June 2013 at CVUT Prague, Czech Republic; 51 pages
(9 figures). - Main Example 2.4 on pp.34-36 retained from arXiv:1302.4388v1,
standard proofs in Appendix A amended and quoted from arXiv:1302.4388v1
(joint with S.Ringers). - Solution to Exercise 11.6 from IHES/M/12/13 by the
same autho
Governing equations of tissue modelling and remodelling: A unified generalised description of surface and bulk balance
Several biological tissues undergo changes in their geometry and in their
bulk material properties by modelling and remodelling processes. Modelling
synthesises tissue in some regions and removes tissue in others. Remodelling
overwrites old tissue material properties with newly formed, immature tissue
properties. As a result, tissues are made up of different "patches", i.e.,
adjacent tissue regions of different ages and different material properties,
within evolving boundaries. In this paper, generalised equations governing the
spatio-temporal evolution of such tissues are developed within the continuum
model. These equations take into account nonconservative, discontinuous surface
mass balance due to creation and destruction of material at moving interfaces,
and bulk balance due to tissue maturation. These equations make it possible to
model patchy tissue states and their evolution without explicitly maintaining a
record of when/where resorption and formation processes occurred. The time
evolution of spatially averaged tissue properties is derived systematically by
integration. These spatially-averaged equations cannot be written in closed
form as they retain traces that tissue destruction is localised at tissue
boundaries.
The formalism developed in this paper is applied to bone tissues, which
exhibit strong material heterogeneities due to their slow mineralisation and
remodelling processes. Evolution equations are proposed in particular for
osteocyte density and bone mineral density. Effective average equations for
bone mineral density (BMD) and tissue mineral density (TMD) are derived using a
mean-field approximation. The error made by this approximation when remodelling
patchy tissue is investigated. The specific time signatures of BMD or TMD
during remodelling events may provide a way to detect these events occurring at
lower, unseen spatial resolutions from microCT scans.Comment: 14 pages, 8 figures. V2: minor stylistic changes, more detailed
derivation of Eqs (30)-(31), additional comments on implication of BMD and
TMD signatures for microCT scan
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