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A Model of Parametric Dependent Type Theory in Bridge/Path Cubical Sets [article]

Andreas Nuyts
2017 arXiv   pre-print
In chapter 3, we introduce the category BPCube of bridge/path cubes, and its presheaf category Psh(BPCube) of bridge/path cubical sets.  ...  In chapter 4, we define discrete types and show that they form a model of dependent type theory. We prove some infrastructural results.  ...  CHAPTER 3 Bridge/path cubical sets In this chapter, we move from the general presheaf model to the category Cube of cubical sets and the category BPCube of bridge/path cubical sets. roughout, we assume  ... 
arXiv:1706.04383v2 fatcat:o4a6p44ajzhjpgbnde3rwmlyt4

Parametric quantifiers for dependent type theory

Andreas Nuyts, Andrea Vezzosi, Dominique Devriese
2017 Proceedings of the ACM on Programming Languages  
We prove soundness of our type system by enhancing existing iterated reflexive graph (cubical set) models of dependently typed parametricity by distinguishing between edges that express relatedness of  ...  In order to expose the problem that occurs in dependent type theory, we will elaborate an example that shows the power of parametricity in System F, but which breaks down in dependent type theory.  ...  Andreas Nuyts and Dominique Devriese hold a Ph.D. Fellowship and a Postdoctoral Mandate (resp.) from the Research Foundation -Flanders (FWO).  ... 
doi:10.1145/3110276 dblp:journals/pacmpl/NuytsVD17 fatcat:tmebic2krvc4jg5qc4nwhh2qfy

Parametric Cubical Type Theory [article]

Evan Cavallo, Robert Harper
2019 arXiv   pre-print
We exhibit a computational type theory which combines the higher-dimensional structure of cartesian cubical type theory with the internal parametricity primitives of parametric type theory, drawing out  ...  interpretation of inductive types.  ...  Conversations with the directed type theory group at the 2017 Mathematics Research Communities workshop and with Emily Riehl and Christian Sattler afterwards were also instrumental to the first author's  ... 
arXiv:1901.00489v2 fatcat:rqbzj52wu5dujp4mchcxdxmm7y

Internal Parametricity for Cubical Type Theory [article]

Evan Cavallo, Robert Harper
2021 arXiv   pre-print
We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives.  ...  We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, observe how cubical equality regularizes parametric type theory, and examine the similarities and  ...  This puts parametric type theory in especially close correspondence with the Bezem-Coquand-Huber (BCH) cubical set model [BCH13] , the first constructive model of univalent type theory.  ... 
arXiv:2005.11290v5 fatcat:rnbnor535jcbbeb74726r7mltm

Presheaf Models of Relational Modalities in Dependent Type Theory [article]

Andreas Nuyts
2018 arXiv   pre-print
This report is an extension of 'A Model of Parametric Dependent Type Theory in Bridge/Path Cubical Sets' (Nuyts, arXiv:1706.04383).  ...  The purpose of this text is to prove all technical aspects of our model for dependent type theory with parametric quantifiers (Nuyts, Vezzosi and Devriese, 2017) and with degrees of relatedness (Nuyts  ...  Bridge/path cubical sets In this chapter, we move from the general presheaf model to the category Cube of cubical sets and the category BPCube of bridge/path cubical sets. roughout, we assume the existence  ... 
arXiv:1805.08684v1 fatcat:weesuc7g5ffo5clxx3yf43s76u

Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty

Toni Lassila, Andrea Manzoni, Alfio Quarteroni, Gianluigi Rozza
2013 Mathematical Modelling and Numerical Analysis  
The results obtained in 2D are compared with simulations in a 3D geometry but without model reduction or the robust framework. Mathematics Subject Classification. 35Q93, 49Q10, 76D05.  ...  We also consider a reduced order modelling framework based on reduced basis methods in order to make the robust design problem computationally feasible.  ...  Bypass grafts can provide blood flow through an alternative bridging path in order to overcome critically occluded arteries.  ... 
doi:10.1051/m2an/2012059 fatcat:y7esmp2ixzbitnoczccxjua6si

A three-dimensional dynamic analysis scheme for the interaction between trains and curved railway bridges

Elias G. Dimitrakopoulos, Qing Zeng
2015 Computers & structures  
Proposed approach With the aim to avoid an ad hoc treatment of the VBI problem, the proposed approach deploys a matrix formulation, and results in a set of equations of motion (Eq. (23)) which are condensed  ...  For a different bridge and/or different vehicle-models or numbers of vehicles, one has only to implement the same matrix equations, presented later in Eq. (23), for the pertinent matrices.  ...  In particular, the paper shows that the centrifugal and Coriolis forces, arising due to the curvature of the bridge/path, govern the dynamics (e.g. the dynamic amplification factor), along the radial direction  ... 
doi:10.1016/j.compstruc.2014.12.002 fatcat:t2dwpqdf2nanrdl6hjgwzoxj34

Integrated material practice in free-form timber structures

Tom Svilans, Mette Ramsgaard Thomsen, Martin Tamke
2021 Zenodo  
The simulation of material performance and the shift from models as representational tools to functional ones has opened up new methods of working between digital model and physical material.  ...  The ability to process large amounts of data quickly has made computation commonplace in the design and manufacture of buildings, especially in iterative digital design workflows.  ...  While these are indicative of the state-of-the-art in both architectural practice and industrial timber production, the knowledge gathered from secondments and collaboration with these partners is limited  ... 
doi:10.5281/zenodo.4545123 fatcat:z4u5ikmtmnbdnj4zl4c4exteyu

Higher Inductive Types and Internal Parametricity for Cubical Type Theory

Evan Cavallo
thebehavior of internal parametricity, and applying internal parametricity as a tool to solve difficult problems in cubical type theory.  ...  We realize a system of generalized quotients for cubical type theory originally conceived in homotopy type theory, called higher inductive types, that merges the concepts of inductive type and quotient  ...  Their system is interpreted in a presheaf category PSh(BPCube) (bridge-path cubical sets), which is cohesive over the category PSh( c ) of cartesian cubical sets.  ... 
doi:10.1184/r1/14555691.v1 fatcat:couyfgkz7za6xl4zhsvlo4t7eu

Efficient Pricing of CPPI using Markov Operators [article]

Louis Paulot, Xavier Lacroze
2009 arXiv   pre-print
Provided the underlying follows a homogeneous process, the path-dependent CPPI strategy is reformulated into a Markov process in one variable, which allows to use efficient linear algebra techniques.  ...  Some gap risk due to extreme events is often kept by the issuer of the product: a put option on the CPPI strategy is included in the product.  ...  Both of them are also acknowledged for a careful reading of this paper and useful comments. This work was supported by Sophis Technology.  ... 
arXiv:0901.1218v1 fatcat:f5oeww3nlvbmjgwjutgeygk6lq

Optimal exercise of American options under stock pinning [article]

Bernardo D'Auria, Eduardo García-Portugués, Abel Guada-Azze
2020 arXiv   pre-print
These settings came up as a first approach of optimally exercising an option within the so-called "stock pinning" scenario.  ...  In order to do so, we solve the discounted optimal stopping problem endowed with the gain function G(x) = (S - x)^+ and a Brownian bridge whose final value equals S.  ...  Inferring the boundary using one third (n = 66, N = 200) of the Brownian bridge path, for T = 1, S = 10, X 0 = 10, λ = 0, and σ = 1.  ... 
arXiv:1903.11686v3 fatcat:e4n5c46ihzahdoqarvo2qrtany

Multidimensional Markov-Functional and Stochastic Volatility Interest Rate Modelling Multidimensional Markov-Functional and Stochastic Volatility Interest Rate Modelling

Linus Kaisajuntti
and that a careful implementation using Sobol sequences and Brownian bridge path construction methods could achieve far faster convergence, with a lot weaker dependence on dimension.  ...  Extending the model with a CEV type local volatility function is of course in theory an easy task however this implies that the Fourier transform technique can no longer be used and hence efficient pricing  ...  To sum up; if the SABR model is a good representation of the market, then the parameters of the quadratic polynomial in Σ(f, f ) in the r.h.s. of (21) may be estimated using a set of market snapshots.  ... 

Computational Optimal Transport [article]

Gabriel Peyré, Marco Cuturi
2020 arXiv   pre-print
Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on  ...  The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle).  ...  Let P be an extremal point of the polytope U(a, b).  ... 
arXiv:1803.00567v4 fatcat:zgannw6i6beqde5bx7pj62uyry

Theoretical Study of Electron Transport and Trapping in Solvated Titanium Dioxide Nanoparticles

Jing Zhang
Those advances enable Jaguar to be used in a wide range of materials science problems that previously were not accessible to our approach.In second half, by employing a large cluster, a hybrid DFT model  ...  Those results advance our understanding of the effect of the cation and solvent on the density of states (DOS) of electron traps, and establish the plausibility that ambipolar model plays a role in electron  ...  The complexity of the experimental systems, and uncertainties in the calculations of various types, make direct comparison of theory and experiment difficult to achieve in an unambiguous fashion.  ... 
doi:10.7916/d89p37z9 fatcat:r3au7veukvffbhax2bsoujf23y