Matematisk institutt
http://hdl.handle.net/10852/6
Sat, 24 Oct 2020 10:36:03 GMT2020-10-24T10:36:03ZThe Fourier-Stieltjes algebra of a C*-dynamical system II
http://hdl.handle.net/10852/80643
The Fourier-Stieltjes algebra of a C*-dynamical system II
Bedos, Erik Christopher; Conti, Roberto
We continue our study of the Fourier–Stieltjes algebra associated to a twisted (unital, discrete) C*-dynamical system and discuss how the various notions of equivalence of such systems are reflected at the algebra level. As an application, we show that the amenability of a system, as defined in our previous work, is preserved under Morita equivalence.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/806432020-01-01T00:00:00ZRobust preconditioning for coupled Stokes–Darcy problems with the Darcy problem in primal form
http://hdl.handle.net/10852/80608
Robust preconditioning for coupled Stokes–Darcy problems with the Darcy problem in primal form
Holter, Karl Erik; Kuchta, Miroslav; Mardal, Kent-Andre
The coupled Darcy–Stokes problem is widely used for modeling fluid transport in physical systems consisting of a porous part and a free part. In this work we consider preconditioners for monolithic solution algorithms of the coupled Darcy–Stokes problem, where the Darcy problem is in primal form. We employ the operator preconditioning framework and utilize a fractional solver at the interface between the problems to obtain order optimal schemes that are robust with respect to the material parameters, i.e. the permeability, viscosity and Beavers–Joseph–Saffman condition. Our approach is similar to that of Holter et al. (2020), but since the Darcy problem is in primal form, expressing mass conservation at the interface involves the normal derivative, which introduces some mathematical challenges. These challenges will be specifically addressed in this paper, in particular we will employ fractional Laplacians at the interface. Numerical experiments illustrating the performance are provided. The preconditioner is posed in non-standard Sobolev spaces which may be perceived as an obstacle for its use in applications. However, we detail the implementational aspects and show that the preconditioner is quite feasible to realize in practice.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/806082020-01-01T00:00:00ZDiscontinuous Galerkin methods for multiphase flow
http://hdl.handle.net/10852/80602
Discontinuous Galerkin methods for multiphase flow
Landet, Tormod Ravnanger
This thesis describes a finite element method for simulation of free-surface flows, such as ocean waves, using the discontinuous Galerkin method. Free-surface flows where there is a large difference in density between two immiscible fluids will have a large jump in the magnitude of the momentum field at the interface between the two fluids. Such discontinuities create problems for higher-order discretisations, which are more computationally efficient where the solution is smooth, but need careful handling near discontinuities in order to avoid Gibbs oscillations. This thesis shows how slope limiting can be used to stabilise the momentum equation in an exactly mass conserving discontinuous Galerkin finite element discretisation of the Navier–Stokes equation. The stabilised method is tested on a range of well known 2D and 3D free-surface-flow test cases. The results are in good agreement with published experimental results, and also give the expected higher-order convergence rates for smooth solutions.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/806022020-01-01T00:00:00ZProjecting onto Helson matrices in Schatten classes
http://hdl.handle.net/10852/80560
Projecting onto Helson matrices in Schatten classes
Brevig, Ole Fredrik; Miheisi, Nazar
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10852/805602021-01-01T00:00:00ZComputations in A1-homotopy theory. Contractibility and enumerative geometry
http://hdl.handle.net/10852/80256
Computations in A1-homotopy theory. Contractibility and enumerative geometry
Pauli, Sabrina
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/802562020-01-01T00:00:00ZSome new theoretical and computational results around the Jacobian conjecture
http://hdl.handle.net/10852/80184
Some new theoretical and computational results around the Jacobian conjecture
Truong, Tuyen Trung
In this paper, we study a so-called Condition C1 on square matrices with complex coefficients and a weaker Condition C2. For Druzkowski maps Condition C2 is equivalent to the Jacobian conjecture. We show that these conditions satisfy many good properties and in particular are satisfied by a dense subset of the set of square matrices of a given rank [Formula: see text]. Based on this, we propose a heuristic argument for the truth of the Jacobian conjecture. We propose some new equivalent formulations and some generalizations of the Jacobian conjecture, and some approaches (including computer algebra and numerical methods) toward resolving it. We show that some of these equivalent formulations are automatically satisfied by generic Druzkowski matrices. Applications and experimental results are included.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/801842020-01-01T00:00:00ZVariance and interest rate risk in unit-linked insurance policies
http://hdl.handle.net/10852/80155
Variance and interest rate risk in unit-linked insurance policies
Baños, David; Lagunas, Marc; Ortiz-Latorre, Salvador
One of the risks derived from selling long-term policies that any insurance company has arises from interest rates. In this paper, we consider a general class of stochastic volatility models written in forward variance form. We also deal with stochastic interest rates to obtain the risk-free price for unit-linked life insurance contracts, as well as providing a perfect hedging strategy by completing the market. We conclude with a simulation experiment, where we price unit-linked policies using Norwegian mortality rates. In addition, we compare prices for the classical Black-Scholes model against the Heston stochastic volatility model with a Vasicek interest rate model.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/801552020-01-01T00:00:00ZFully-dynamic risk-indifference prices and no-good-deal bounds
http://hdl.handle.net/10852/79946
Fully-dynamic risk-indifference prices and no-good-deal bounds
Bion-Nadal, Jocelyne; Di Nunno, Giulia
The seller's risk-indifference price evaluation is studied. We propose a dynamic risk-indifference pricing criterion derived from fully-dynamic risk measures on the $L_p$-spaces for $p\in [1,\infty]$. The concept of fully-dynamic risk measures extends the one of dynamic risk measures by adding the actual possibility of changing the risk perspectives over time. This family is then characterized by a double time index. Our framework fits well the study of both short- and long-term investments. In this paper we analyze whether the risk-indifference pricing criterion actually provides a proper convex price system. It turns out that, depending on $p$, this is not always the case. Then an extension of the framework beyond $L_p$ becomes necessary. Furthermore, we consider the relationship of the fully-dynamic risk-indifference price with no-good-deal bounds. We shall provide necessary and sufficient conditions on the fully-dynamic risk measures so that the corresponding risk-indifference prices satisfy the no-good-deal bounds. Remarkably, the use of no-good-deal bounds also provides a method to select the risk measures and thus construct a proper fully-dynamic risk-indifference price system within the $L_2$-spaces.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/799462020-01-01T00:00:00ZTwenty-seven questions about the cubic surface
http://hdl.handle.net/10852/79537
Twenty-seven questions about the cubic surface
Ranestad, Kristian; Sturmfels, Bernd
We present a collection of research questions on cubic surfaces in 3-space. These questions inspired the present collection of papers. This article serves as the introduction to the issue. The number of questions is meant to match the number of lines on a cubic surface. We end with a list of problems that are open.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/795372020-01-01T00:00:00ZModelling the joint behaviour of electricity prices in interconnected markets
http://hdl.handle.net/10852/79483
Modelling the joint behaviour of electricity prices in interconnected markets
Christensen, Troels Sønderby; Benth, Fred Espen
The liberalization of energy markets worldwide during recent decades has introduced severe implications for the price formation in these markets. Especially within the European day-ahead electricity markets, increased physical connections between different market areas and a joint effort on optimizing the aggregate social welfare have led to highly connected markets. Consequently, observing the exact same hourly day-ahead prices for two or more interconnected electricity markets in Europe happens frequently. This affects the modelling of such prices and in turn the valuation of derivatives written on prices from these market areas. In this paper, we propose a joint model for day-ahead electricity prices in interconnected markets composed of a combination of transformed Ornstein–Uhlenbeck processes. We discuss the properties of the model and propose an estimation procedure based on filtering techniques. Furthermore, the properties of the model reveal that analytical prices are attainable for, e.g., forwards and spread options.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/794832020-01-01T00:00:00ZOn complex dynamics in a Purkinje and a ventricular cardiac cell model
http://hdl.handle.net/10852/79454
On complex dynamics in a Purkinje and a ventricular cardiac cell model
Erhardt, André Henrik; Solem, Susanne
Cardiac muscle cells can exhibit complex patterns including irregular behaviour such as chaos or (chaotic) early afterdepolarisations (EADs), which can lead to sudden cardiac death. Suitable mathematical models and their analysis help to predict the occurrence of such phenomena and to decode their mechanisms. The focus of this paper is the investigation of dynamics of cardiac muscle cells described by systems of ordinary differential equations. This is generically performed by studying a Purkinje cell model and a modified ventricular cell model. We find chaotic dynamics with respect to the leak current in the Purkinje cell model, and EADs and chaos with respect to a reduced fast potassium current and an enhanced calcium current in the ventricular cell model — features that have been experimentally observed and are known to exist in some models, but are new to the models under present consideration. We also investigate the related monodomain models of both systems to study synchronisation and the behaviour of the cells on macro-scale in connection with the discovered features. The models show qualitatively the same behaviour to what has been experimentally observed. However, for certain parameter settings the dynamics occur within a non-physiological range.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10852/794542021-01-01T00:00:00ZStrengthened convexity of positive operator monotone decreasing functions
http://hdl.handle.net/10852/79331
Strengthened convexity of positive operator monotone decreasing functions
Kirihata, Megumi; Yamashita, Makoto
We prove a strengthened form of convexity for operator monotone decreasing positive functions defined on the positive real numbers. This extends Ando and Hiai's work to allow arbitrary positive maps instead of states (or the identity map), and functional calculus by operator monotone functions defined on the positive real numbers instead of the logarithmic function.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/793312020-01-01T00:00:00ZBacktracking Gradient Descent Method and Some Applications in Large Scale Optimisation. Part 2: Algorithms and Experiments
http://hdl.handle.net/10852/79322
Backtracking Gradient Descent Method and Some Applications in Large Scale Optimisation. Part 2: Algorithms and Experiments
Truong, Tuyen Trung; Nguyen, Hang-Tuan
In this paper, we provide new results and algorithms (including backtracking versions of Nesterov accelerated gradient and Momentum) which are more applicable to large scale optimisation as in Deep Neural Networks. We also demonstrate that Backtracking Gradient Descent (Backtracking GD) can obtain good upper bound estimates for local Lipschitz constants for the gradient, and that the convergence rate of Backtracking GD is similar to that in classical work of Armijo. Experiments with datasets CIFAR10 and CIFAR100 on various popular architectures verify a heuristic argument that Backtracking GD stabilises to a finite union of sequences constructed from Standard GD for the mini-batch practice, and show that our new algorithms (while automatically fine tuning learning rates) perform better than current state-of-the-art methods such as Adam, Adagrad, Adadelta, RMSProp, Momentum and Nesterov accelerated gradient. To help readers avoiding the confusion between heuristics and more rigorously justified algorithms, we also provide a review of the current state of convergence results for gradient descent methods. Accompanying source codes are available on GitHub.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/793222020-01-01T00:00:00ZNumerical simulations of a dust grain in a flowing magnetized plasma
http://hdl.handle.net/10852/78645
Numerical simulations of a dust grain in a flowing magnetized plasma
Darian, Diako; Miloch, Wojciech Jacek; Mortensen, Mikael; Miyake, Yohei; Usui, Hideyuki
The effect of an external magnetic field on the formation of the wake in the potential distribution behind a dust grain is studied with self-consistent Particle-In-Cell numerical simulations. The collisionless plasma flow is aligned with the magnetic field. It is demonstrated that the topology of the wakefield is significantly affected by the magnetization degree of plasma and by the ion flow speed. The external magnetic field acts to reduce the potential enhancements in the wake and leads to splitting of the wake pattern across the symmetry axis. For high ion flow speeds, a strong magnetization of plasma suppresses the potential enhancements and results in a narrow negative potential line along the symmetry axis, parallel to the ion flow direction, in the wake.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10852/786452019-01-01T00:00:00ZGas-liquid slug flow in a horizontal concentric annulus, a comparison of numerical simulations and experimental data
http://hdl.handle.net/10852/78641
Gas-liquid slug flow in a horizontal concentric annulus, a comparison of numerical simulations and experimental data
Friedemann, Christopher; Mortensen, Mikael; Nossen, Jan
Multiphase flow simulations were run in OpenFOAM utilizing InterFoam, a volume of fluid type solver. A concentric annulus geometry was constructed and periodic boundary conditions were applied to alleviate the requirement for a longer domain. The simulations were run in 3, 5 and 7m long domains with the annulus dimensioned such that the outer and inner cylinder diameter were 0.1 and 0.05m respectively. The 4 individual mesh designs were constructed such that the coarsest mesh consists of 102k cells/m while the finest mesh was generated with 400k cells/m. Each mesh was significantly refined within 0.005m of both the inner and outer cylinder wall. The simulation data was compared with experimental pressure and holdup data collected at Institute for Energy Technology in Norway. The 3 and 7m domains reproduced slug frequencies to within 9% of the experiment results of 1.43 Hz for all mesh densities. Comparatively, the 5m domain has larger errors with respect to slugging frequency (22–27%). The 5m case performs poorly, probably due to an artificial restraint introduced by the limitation of available liquid which is set as oil = 0.53 for all cases. The αoil restriction combined with the domain length determines the amount of liquid in the system. This interaction of factors means that the domain length is an important parameter when preparing the simulation. The pressure data display a stronger dependence on the mesh quality in comparison to the slug frequency analysis. The 3m domain with a 400k cells/m mesh resulted in a maximum and minimum pressure gradient of 1783.5 and 803.9 Pa/m, compared to the experiment values of 1785 and 822 Pa/m, which are within 3% of the expected results.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10852/786412019-01-01T00:00:00ZExperiments on air entrainment produced by a circular free falling jet
http://hdl.handle.net/10852/78501
Experiments on air entrainment produced by a circular free falling jet
Ramirez De La Torre, Reyna Guadalupe; Jensen, Atle; Kuchta, Miroslav
The process of a circular free falling jet entering an idle pool was studied with the objective of determining a relation between the naturally occurring disturbances on the jet surface and air entrainment. To this end the instabilities of the free falling jet were characterized and compared with the bubble count distribution and an estimated amount of entrained air in the plume. Different jet lengths were considered. The aeration process was captured through image sequences. Individ- ual analysis of each disturbance in the jet surface was made. Our results show that both the bubble count and entrained air have a linear relation with the steepness of the jet disturbances. Moreover, the wave-jet ratio is introduced, which defines an entrainment condition for all the studied cases.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/785012020-01-01T00:00:00ZFailure of the integral Hodge conjecture for threefolds of Kodaira dimension zero
http://hdl.handle.net/10852/78411
Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero
Ottem, John Christian; Benoist, Olivier
We prove that the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth projective complex threefolds.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/784112020-01-01T00:00:00ZA pencil of Enriques surfaces with non-algebraic integral Hodge classes
http://hdl.handle.net/10852/78403
A pencil of Enriques surfaces with non-algebraic integral Hodge classes
Ottem, John Christian; Suzuki, Fumiaki
We prove that there exists a pencil of Enriques surfaces defined over Q with non-algebraic integral Hodge classes of non-torsion type. This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question of Murre on the universality of the Abel-Jacobi maps in codimension three.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/784032020-01-01T00:00:00ZApparent diffusion coefficient estimates based on 24 hours tracer movement support glymphatic transport in human cerebral cortex
http://hdl.handle.net/10852/78400
Apparent diffusion coefficient estimates based on 24 hours tracer movement support glymphatic transport in human cerebral cortex
Valnes, Lars Magnus; Mitusch, Sebastian; Ringstad, Geir; Eide, Per Kristian; Funke, Simon Wolfgang; Mardal, Kent-Andre
The recently proposed glymphatic system suggests that bulk flow is important for clearing waste from the brain, and as such may underlie the development of e.g. Alzheimer’s disease. The glymphatic hypothesis is still controversial and several biomechanical modeling studies at the micro-level have questioned the system and its assumptions. In contrast, at the macro-level, there are many experimental findings in support of bulk flow. Here, we will investigate to what extent the CSF tracer distributions seen in novel magnetic resonance imaging (MRI) investigations over hours and days are suggestive of bulk flow as an additional component to diffusion. In order to include the complex geometry of the brain, the heterogeneous CSF flow around the brain, and the transport over the time-scale of days, we employed the methods of partial differential constrained optimization to identify the apparent diffusion coefficient (ADC) that would correspond best to the MRI findings. We found that the computed ADC in the cortical grey matter was 5–26% larger than the ADC estimated with DTI, which suggests that diffusion may not be the only mechanism governing transport.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/784002020-01-01T00:00:00ZMeasure-theoretic Uniformity and the Suslin Functional
http://hdl.handle.net/10852/78391
Measure-theoretic Uniformity and the Suslin Functional
Normann, Dag
Given a set A in the unit interval and the associated Lebesgue measure λ, it is a natural question whether we may (in some sense) compute the measure λ(A) in terms of the set A. Under the moniker measure theoretic uniformity, Tanaka and Sacks have (independently) provided a positive answer for the well-known class of hyperarithmetical sets of reals, and provided a basis theorem for such sets of positive measure. The hyperarithmetical sets are exactly the sets computable in terms of the functional 2E, in the sense of Kleene’s S1–S9. In turn, Kleene’s 2E essentially corresponds to arithmetical comprehension as in ACA0. In this paper, we generalise the aforementioned results to the ‘next level’, namely Π11-CA0, in the form of the Suslin functional, or the equivalent hyperjump. We also generalise the Tanaka-Sacks basis theorem to sets of positive measure that are semi-computability relative to the Suslin functional. Finally, we discuss similar generalisations for infinite time Turing machines.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/783912020-01-01T00:00:00Z