Computational Prospects of Infinity

(1998). A Stroll Through the Gaussian Primes. The American Mathematical Monthly: Vol. 105, No. 4, pp. 327-337.

This thesis offers a philosophical study of computation, which is understood here as a method of abstraction that systematises reality through logico-quantitative means. The thesis challenges the view that computation’s abstractive processes are simple and static ‘formulae’ that capture the world’s dynamism. By engaging with the formal and axiomatic character of computing, it argues that computation is itself dynamic, because it has a potential to actualise itself. This potentiality is theorised in aesthetic terms. Drawing from Deleuze, aesthetics is viewed as an investigation into the conditions of real experience. For Deleuze, these conditions pertain to virtuality, i.e. to the indeterminacy of sensation, and of thought’s immanence to affect. However, through a novel reading of the ontological significance of Gödel’s incompleteness theorems and of Turing’s notion of incomputability, the thesis demonstrates that indeterminacy does not pertain uniquely to virtual life, but rather lies at the axiomatic heart of computational logic. Computation is thus shown to be contingent, because it is always indeterminate. This contingency is formal, not empirical; it is the status of self-sufficient processes of algorithmic determination, which always confront quantitative infinity by means of formal abstraction. Whitehead’s philosophy is used to extend aesthetics from the sensible to the intelligible. Experience is thereby understood as self-actualisation. Its conditions are tied to physical and conceptual operations of determination. Computational processes are addressed as Whitehead’s ‘actual occasions’: as events that constitute themselves through the dynamic processing of eternal and actual data. This dynamism is not pre-determined a priori, and therefore breaks with the computationalist and cognitivist paradigms, which reduce actualisation to universalising prescriptions. This dynamism cannot be flattened onto sense-empirical factuality either. Instead, an aesthetics of contingent computation conceptualises ‘computational actual occasions’ as discrete processes of determination that conclude with the production of a structure or a ‘form’ of actualisation.

Many problems in computational science arise in unbounded domains and thus require an artificial boundary B, which truncates the unbounded exterior domain and restricts the region of interest to a finite computational domain, . It then becomes necessary to impose a boundary condition at B, which ensures that the solution in coincides with the restriction to of the solution in the unbounded region. If we exhibit a boundary condition, such that the fictitious boundary appears perfectly transparent, we shall call it exact. Otherwise it will correspond to an approximate boundary condition and generate some spurious reflection, which travels back and spoils the solution everywhere in the computational domain. In addition to the transparency property, we require the computational effort involved with such a boundary condition to be comparable to that of the numerical method used in the interior. Otherwise the boundary condition will quickly be dismissed as prohibitively expensive and impractical. The constant demand for increasingly accurate, efficient, and robust numerical methods, which can handle a wide variety of physical phenomena, spurs the search for improvements in artificial boundary conditions. In the last decade, the perfectly matched layer (PML) approach [16] has proved a flexible and accurate method for the simulation of waves in unbounded media. Standard PML formulations, however, usually require wave equations stated in their standard second-order form to be reformulated as first-order systems, thereby introducing many additional unknowns. To circumvent this cumbersome and somewhat expensive step we propose instead a simple PML formulation directly for the wave equation in its second-order form. Our formulation requires fewer auxiliary unknowns than previous formulations [23, 94]. Starting from a high-order local nonreflecting boundary condition (NRBC) for single scattering [55], we derive a local NRBC for time-dependent multiple scattering problems, which is completely local both in space and time. To do so, we first develop a high order exterior evaluation formula for a purely outgoing wave field, given its values and those of certain auxiliary functions needed for the local NRBC on the artificial boundary. By combining that evaluation formula with the decomposition of the total scattered field into purely outgoing contributions, we obtain the first exact, completely local, NRBC for time-dependent multiple scattering. Remarkably, the information transfer (of time retarded values) between sub-domains will only occur across those parts of the artificial boundary, where outgoing rays intersect neighboring sub-domains, i.e. typically only across a fraction of the artificial boundary. The accuracy, stability and efficiency of this new local NRBC is evaluated by coupling it to standard finite element or finite difference methods.

In physics, we use fundamental theories to describe and explain phenomena occurring in nature. Two of the most prominent theories are classical mechanics, pioneered in the 17th century by Sir Isaac Newton, and quantum mechanics which arose in the beginning of the 20th century. While classical and quantum mechanics dissent in various aspects, the most pronounced difference is called entanglement. Entanglement provides us with powerful tools that allow us to perform tasks which are not possible by classical means. Quantum computing studies the possible use of quantum principles for computational tasks. For example the factorisation of large numbers is infeasible with classical computers but can be done efficiently with Shor’s algorithm. The power of quantum computers can be intuitively understood by realizing that its basic unit can encode infinitely many states via the superposition principle. Another significant application of quantum mechanics is quantum key distribution (QKD). The goal is to create two identical strings of random bits, called a key, at two spatially-separated locations. This key is then used to encrypt a message allowing for secret communication. Secret communication is essential in our modern society, in which we use the internet to manage our bank account, buy products in online stores and send personal messages to friends. We communicate over great distances and desire that this communication be secure. The current classical cryptographic protocols use complex mathematical problems such as factorization to create secure keys. The downside of this procedure is that the security is based on the complexity of mathematical problems and relies on assumptions about the computational power of the person who wants to break the cryptographic system. Hence there is every chance that a potential eavesdropper hacks the key - especially if he has access to quantum computers. On the other hand, a key can be obtained by performing appropriate measurements on an entangled state. This provides the means to actually create secret keys with provable security. In order to achieve long-distance QKD, we envision quantum networks whose purpose is to transmit entanglement between two arbitrary parties on earth. A network consists of various quantum mechanical devices, including sources for creating quantum information, memories which allow for the storage of it, as well as quantum gates and projective measurements for processing the information. Quantum networks and quantum computers sound very appealing. However, with the benefits of quantum mechanics there also come great challenges. A central challenge we want to tackle in this thesis is how to certify that one indeed works with quantum mechanical devices. The subtlety here lies in the fact that we humans are classical and thus cannot directly observe quantum features such as entanglement. As a consequence, we desire to employ certification schemes which do not overburden our classical competences. The need for such certifications becomes apparent when considering the following scenario: Basic quantum machines are already available commercially, for example true random number generators. If we purchase such a device that promises to prepare entangled states or act as a quantum gate, we aim at verifying that the promise actually holds. We want to do this without breaking or opening the device since we would lose the warranty or anyway be overstrained by the complexity of the physics involved. At best, the certification should be such that even an unqualified user can conduct it. In this thesis we will discuss how this can be achieved.

Rendering two-dimensional data in the case of rough, complex surfaces is a challenge in computer graphics. Typically, splines with knots and control points are used, and while they yield useful surfaces they can be of poor quality, or can be difficult to apply. Fundamentally splines are local. An alternative mathematical method is to construct global methods which can be tuned to have polynomial behavior, or behave in ways that are not as restrictive, and which can be local, or not, depending on user input. This study examines a hybrid method of stochastic interpolation built around Bernstein functions. This approach is non-polynomial and global, but readily computable and can successfully fit complex two-dimensional surface data to obtain high quality at low computational cost. The representation of parametric surfaces in 3 dimensions can be achieved using approximation, or interpolation using this method. The generation of computational surfaces rendered using OpenGL, shows that this hybrid method of Bernstein function interpolation is a sound approach to surface rendering, and computational issues in achieving speed with accuracy are discussed. The hybrid method is shown to be robust, and can be selectively adjusted to yield controlled smoothing of the surface data. The method enables use of computational stencils of arbitrary size, and permits the construction of infinitely differentiable surfaces if needed.

Graph theory is an important topic in discrete mathematics. It is particularly interesting because it has a wide range of applications. Among the main problems in graph theory, we shall mention the following ones: graph coloring and the Hamiltonian circuit problem. Chapter 1 presents basic definitions of graph theory, such as graph coloring, graph coloring with color-classes of bounded size b, and Hamiltonian circuits and paths. We also present online algorithms and online coloring. Chapter 2 starts with some general remarks about online graph covering with sets of bounded sizes (such as online bounded coloring): we give a simple method for transforming an online covering algorithm into an online bounded covering algorithm, and to derive the performance ratio of the bounded algorithm from the performance ratio of the unbounded algorithm. As will be shown in later chapters, this method often leads to optimal results. Furthermore, some basic preliminary results on online graph covering with sets of bounded size are given: for every graph, the performance ratio is bounded above by 1/2 + b/2 and for b = 2, this bound is optimal. In the second part, online coloring of co-interval graphs is studied. Based on two industrial applications, two different versions of this problem are discussed. In the case where the intervals are presented in increasing order of their left ends, we show that the performance ratio is 1 in the unbounded case and 2 - 1/b in the bounded case. In the case where the intervals may be presented in any order, we show that the performance ratio is at most 3 in the bounded case. Chapter 3 deals with online coloring of permutation and comparability graphs. First, we give a tight analysis of the First-Fit algorithm on bipartite permutation graphs and we show that its performance ratio is O(√n), even for some simple presentation orders. For both classes of graphs, we show that the performance ratio is bounded above by (χ+1)/2 in the unbounded case and that the performance ratio of First-Fit is equal to 1/2 + b/2 in the bounded case. In the second part of this chapter, we study cocoloring of permutation graphs. We show that the performance ratio is n/4 + 1/2 and we give better bounds in some more restricted cocoloring problems. Chapter 4 deals with an application of online coloring: the online Track Assignment Problem. Depending on the assumptions that are made, the Track Assignment Problem can be reduced to coloring permutation or overlap graphs online. We show that when a permutation graph is presented on a latticial plane, from west to east, then the performance ratio is exactly 2 - (min{b,k})-1, where k is the best known upper bound on the bounded chromatic number. We also show that, when a permutation graph is presented on a latticial plan, starting from the origin and growing, simultaneously or not, towards west and east, then the performance ratio is exactly 2 - 1/χ. We also show that online coloring overlap graphs does not have a performance ratio bounded by a constant, even if the overlap graph is bipartite and presented in increasing order of the intervals left ends. In this special case, we show that First-Fit has a tight performance ratio of O(√n). We consider coloring overlap graphs online where the intervals have a bounded size between 1 and a given number M. In this case, we show that the performance ratio can be bounded above by 2√M if M ≤ M0, and by log M (⎡log M / log log M⎤ + 1) if M > M0, M0 being defined by the equation 2√M0 = 3 log(M0). For large values of M, the ratio is O(log2 M / log log M). Chapter 5 is about online coloring of trees, forests and split-graphs. For trees, we show that the performance ratio of online coloring is exactly ½log2(2n) in the unbouded case and at most 1 + ⎣log2(b)⎦/χb in the bounded case. For split-graphs, we show that the performance ratio of online coloring is exactly 1 + 1/χ in the unbounded case and is at most 2 + 1/χb + 3/b in the bounded case. In Chapter 6, we present a class of digraphs: the quasi-adjoint graphs. These are a super class of both the graphs used for a DNA sequencing algorithm in (Blazewicz, Kasprzak, "Computational complexity of isothermic DNA sequencing by hybridization", 2006) and the adjoints. A polynomial recognition algorithm in O(n3), as well as a polynomial algorithm in O(n2 + m2) for finding a Hamiltonian circuit in quasi-adjoint graphs are given. Furthermore, some results about related problems such as finding a Eulerian circuit while respecting some forbidden transitions (a sequence of two consecutive arcs) are discussed.

We introduce a new second-kind integral equation method to solve direct rough surface scattering problems in two dimensions. This approach is based, in part, upon the bounded obstacle scattering method that was originally presented in Bruno et al. [2004] and is discussed in an appendix of this thesis. We restrict our attention to problems in which time-harmonic acoustic or electromagnetic plane waves scatter from rough surfaces that are perfectly reflecting, periodic and at least twice continuously differentiable; both sound-soft and sound-hard type acoustic scattering cases---correspondingly, transverse-electric and transverse-magnetic electromagnetic scattering cases---are treated. Key elements of our algorithm include the use of infinitely continuously differentiable windowing functions that comprise partitions of unity, analytical representations of the integral equation’s solution (taking into account either the absence or presence of multiple scattering) and spectral quadrature formulas. Together, they provide an efficient alternative to the use of the periodic Green’s function found in the kernel of most solvers’ integral operators, and they strongly mitigate the rapidly increasing computational complexity that is typically borne as the frequency of the incident field increases. After providing a complete description of our solver and illustrating its usefulness through some preliminary examples, we rigorously prove its convergence. In particular, the super-algebraic convergence of the method is established for problems with infinitely continuously differentiable scattering surfaces. We additionally show that accuracies within prescribed tolerances are achieved with fixed computational cost as the frequency increases without bound for cases in which no multiple reflections occur. We present extensive numerical data demonstrating the convergence, accuracy and efficiency of our computational approach for a wide range of scattering configurations (sinusoidal, multi-scale and simulated ocean surfaces are considered). These results include favorable comparisons with other leading integral equation methods as well as the non-convergent Kirchhoff approximation. They also contain analyses of sets of cases in which the major physical parameters associated with these problems (i.e., surface height, wavenumber and incidence angle) are systematically varied. As a result of these tests, we conclude that the proposed algorithm is highly competitive and robust: it significantly outperforms other leading numerical methods in many cases of scientific and practical relevance, and it facilitates rapid analyses of a wide variety of scattering configurations.

Computer simulations of molecules and atoms are useful tools in soft matter research. Physical chemistry has profited significantly from the insight provided by the use of classical molecular dynamics simulations. In these simulations, atoms are modeled as point masses and their propagation in time and space is described by iterative solving of Newton’s equations of motion. A major challenge lies in the fact that the computational resources required to simulate systems at this resolution render atomistical simulations prohibitively expensive already at comparatively small time and length scales. One approach to make simulations of given systems more efficient in computational terms is to coarse-grain the model, i.e., to reduce the spatial resolution by merging atoms into larger interaction sites. The procedure of coarse-graining consists of two steps: the definition of a mapping between the scales of resolution and the determination of suitable potentials for the interactions between the sites of the coarse-grained model. Especially the second step is challenging because coarse-grained model needs to reproduce the physical behavior of the underlying (fine-grained) reference model as close as possible to retain its predictive quality. Several methods which share an approach described as systematic, bottom-up coarse-graining, have been published in the literature to determine interactions in the coarse-grained model from interactions in fine-grained (mostly atomistic) reference models of the systems of interest. The transferability of a coarse-grained model, i.e, the capability of accurately reproducing predictions of the reference model at varying state points is especially dependent on the method chosen for the parameterization of the coarse-grained model. Among the existing systematic coarse-graining methods, the conditional reversible work (CRW) method achieves a high degree of transferability, while being conceptually simple, straightforward to implement and computationally efficient. In this thesis, studies are presented which aim at an extension of the applications and systems of CRW-parameterized models. In a comparative study, the CRW method is used, among others, for the study of vapor-liquid equilibria and the thermodynamics of mixing with coarse-grained models of hexane and perfluorohexane. Results confirm the strong dependence of model transferability on the coarse-graining method chosen for its parameterization and show that the CRW models are transferable with respect to temperature, transfer from the interface to the bulk, transfer from the vapor to the liquid phase, and composition of a binary mixture. In the existing literature, the CRW method has only been applied to systems of apolar hydrocarbons. This thesis presents studies in which CRW models are parameterized for systems of weakly polar organic molecules and ionic liquids. The resulting CRW models are transferable to the same degree as those of apolar systems. Another major challenge in the simulation with coarse-grained models is the reproduction of dynamic properties in accordance with fine-grained reference models. In general, the time scales of relaxation are smaller in coarser models and this leads to an effective ‘speed-up’ of these simulations, a behavior which is related to a loss of dissipative degrees of freedom in the coarser model. The effective impact of these degrees of freedom on the dynamics of the system can be simulated through the insertion of dissipative pair interactions into the coarse-grained model. These interactions can be parameterized, like the interaction potential energy, in a bottom-up manner from simulations with the fine-grained model. This approach, which is based on the Mori-Zwanzig projection operator formalism, has been successfully utilized in several recent publications to parameterize coarse-grained models that consistently model the dynamics of model systems at low density. However, the approach relies on assumptions on the nature of the system which are not fully satisfied at the higher density typical for soft matter systems. Most importantly, it is assumed that the degrees of freedom removed from the model upon coarse-graining relax infinitely fast in comparison with those retained in the coarse-grained model (complete time scale separation). In this thesis, an application of such a procedure for a dynamically consistent coarse-grained model is presented for realistic model systems of soft matter. The aim of this study is to evaluate whether Mori-Zwanzig-based coarse-grained models can be used for the simulation of realistic soft matter systems, in which time scale separation is not complete. To this end, coarse-grained models are parameterized for model systems of different chain length and the predictions of the dynamics produced by the coarse-grained system are compared to those of an atomistic reference model. The self-diffusion coefficients of these systems can be reproduced to a good degree, whereas dynamics properties with a smaller characteristic time scale are less well reproduced with the Mori-Zwanzig coarse-grained model, a finding that can be related to the increasing deviation of the system’s state from the assumed complete time scale separation.

In this thesis, a collection of novel numerical techniques culminating in a fast, parallel method for the direct numerical simulation of incompressible viscous flows around surfaces immersed in unbounded fluid domains is presented. At the core of all these techniques is the use of the fundamental solutions, or lattice Green’s functions, of discrete operators to solve inhomogeneous elliptic difference equations arising in the discretization of the three-dimensional incompressible Navier-Stokes equations on unbounded regular grids. In addition to automatically enforcing the natural free-space boundary conditions, these new lattice Green’s function techniques facilitate the implementation of robust staggered-Cartesian-grid flow solvers with efficient nodal distributions and fast multipole methods. The provable conservation and stability properties of the appropriately combined discretization and solution techniques ensure robust numerical solutions. Numerical experiments on thin vortex rings, low-aspect-ratio flat plates, and spheres are used verify the accuracy, physical fidelity, and computational efficiency of the present formulations.

Machine learning, as a subarea of artificial intelligence, is widely believed to reshape the human world in the coming decades. This thesis is focused on both the unsupervised and supervised self-organising transparent machine learning techniques. One particularly interesting aspect is the transparent self-organising deep learning systems. Traditional data analysis approaches and most of the machine learning algorithms are built upon the basis of probability theory and statistics. The solid mathematical foundation of the probability theory and statistics guarantees the good properties of these learning algorithms when the amount of data tends to infinity and all the data comes from the same distribution. However, the prior assumptions of the random nature and same distribution imposed on the data generation model are often too strong and impractical in real applications. Moreover, traditional machine learning algorithms also require a number of free parameters to be predefined. However, without any prior knowledge of the problem, which is often the case in real situations, the performance of the algorithms can be largely influenced by the improper choice. Deep learning-based approaches are currently the state-of-the-art techniques in the fields of machine learning and computer vision. However, they are also suffering from a number of deficiencies including the computational burden of training using huge amount of data, lack of transparency and interpretation, ad hoc decisions about the internal structure, no proven convergence for the adaptive versions that rely on reinforcement learning, limited parallelisation and offline training, etc. These shortcomings largely all hinder the wider applications of the deep learning in real situations. The novel approaches presented in this thesis are developed within the Empirical Data Analytics framework, which is an alternative, but more advanced computational methodology to the traditional approaches based on the ensemble properties and mutual distribution of the empirical discrete observations. The novel self-organising transparent machine learning algorithms presented in this work for clustering, regression, classification and anomaly detection are autonomous, self-organising, data-driven and free from user- and problem- specific parameters. They do not impose any data generation models on the data a priori, but are driven by the empirically observed data and are able to produce the objective results without prior knowledge of the problems. In addition, they are highly efficient and suitable for large-scale static/streaming data processing. The newly proposed self-organising transparent deep learning systems are able to achieve human-level performance comparable to or even better than the deep convolutional neural networks on image classification problems with the merits of being fully transparent, self-evolving, highly efficient, parallelisable and human-interpretable. More importantly, the proposed deep learning systems have the ability of starting classification from the very first image of each class in the same way as humans do. Numerical examples based on numerous challenging benchmark problems and comparisons conducted with the state-of-the-art approaches presented in this thesis demonstrated the validity and effectiveness of the proposed new machine learning algorithms and deep learning systems and show their potential for real applications.

Pipelines are very vulnerable infrastructure components to geohazard-induced ground deformation and failure. How soil transmits loads on pipelines and vice versa, known as soil-pipe interaction (SPI), thus is very important for the assessment and design of resilient pipeline systems. In the first part, this work proposes a simplified macroelement designed to capture SPI in cohesionless soils subjected to arbitrary loading normal to the pipeline axis. We present the development of a uniaxial hysteresis model that can capture the smooth nonlinear reaction force-relative displacement curves (FDCs) of SPI problems. Using the unscented Kalman filter, we derived the model parameter κ that controls the smoothness of the transition zone from linear to plastic using published experimental data. We extended this uniaxial model to biaxial loading effects and showed that the macroelement can capture effects such as pinching and shear-dilation coupling. The model input parameters were calibrated using finite element (FE) analyses validated by experiments. The FDCs of the biaxial model were verified by comparison with FE and smoothed-particle hydrodynamic (SPH) simulations for different loading patterns: cyclic uniaxial, 0-shaped, 8-shaped, and transient loading. Accounting for smooth nonlinearity, hysteresis, pinching, and coupling effects, the proposed biaxial macroelement shows good agreement with FE and SPH analyses, while maintaining the computational efficiency and simplicity of beam-on-nonlinear-Winkler foundation models, as well as a small number of input parameters. Next, this work presents analytical solutions for computing frequency-domain axial and in-plane soil impedance functions (SIFs) for an infinitely long rigid circular structure buried horizontally in homogeneous elastic half-space. Using Hankel— and Bessel—Fourier series expansion, we solved a mixed-boundary-value problem considering a harmonic displacement at the structure boundary and traction-free boundary condition at the half-space free surface. We then verified our analytical solutions using results obtained from FE simulations. The SIFs of a buried structure in a homogeneous elastic half-space calculated by these two approaches are in perfect agreement with each other. In addition, we used analytical solutions and FE simulations to comprehensively investigate factors that affect the SIFs in homogeneous and two-layered half-spaces, respectively. The parametric study shows that SIFs of buried structures in elastic half-space primarily depend on frequency of excitation, shear modulus and Poisson's ratio of the half-space, burial depth and radius of the structure. In a two-layered soil domain, SIFs depend also on material contrast and the distance from the structure location to the interface between soil layers. Lastly, it demonstrates how the SIFs obtained previously can be incorporated into a reduced-order model to analyze SPI problems, specifically a straight pipe subjected to Rayleigh surface wave propagating through homogeneous and heterogeneous elastic half-spaces. Calculated displacement time histories at the control points are shown to agree well with those computed by direct two-dimensional FE analyses.

Many planning problems involve choosing a set of optimal decisions for a system in the face of uncertainty of elements that may play a central role in the way the system is analyzed and operated. During the past decade, there has been a renewed interest in the modelling, analysis, and solution of such problems due to a remarkable development of both new theoretical results and novel computational techniques in stochastic optimization. A prominent approach is to develop upper and lower bounding approximations to the problem along with procedures to sharpen bounds until an acceptable tolerance is satisfied. The contributions of this dissertation are concerned with the latter approach. The thesis first studies the stochastic linear programming problem with randomness in both the objective coefficients and the constraints. A convex concave saddle property of the value function is utilized to derive new bounding techniques which generalize previously known results. These approximations require discretizing bounded domains of the random variables in such a way that tight upper and lower bounds result. Such techniques will prove attractive with the recent advances in large scale linear programming. The above results are also extended to obtain new upper and lower bounds when the domains of random variables are unbounded. While these bounds are tight, the approximating models are large-scale deterministic linear programs. In particular, with a proposed order-cone decomposition for the domains, these linear programs are well-structured, thus enabling one to use efficient techniques for solution, such as parallel computation. The thesis next considers convex stochastic programs. Using aggregation concepts from the deterministic literature, new bounds are developed for the problem which are computable using standard convex programming algorithms. Finally, the discussion is focused on a stochastic convex program arising in a certain resource allocation problem. Exploiting the problem structure, bounds are developed via the Karush-Kuhn-Tucker conditions. Rather than discretizing domains, these approximations advocate replacing difficult multidimensional integrals by a series of simple univariate integrals. Such practice allows one to preserve differentiability properties so that smooth convex programming methods can be applied for solution.

The partial string avoidability problem, also known as partial word avoidability, is stated as follows: given a finite set of strings with possible ``holes'' (undefined symbols), determine whether there exists any two-sided infinite string containing no substrings from this set, assuming that a hole matches every symbol. The problem is known to be NP-hard and in PSPACE, and this paper establishes its PSPACE-completeness. Next, string avoidability over the binary alphabet is interpreted as a version of conjunctive normal form (CNF) satisfiability problem (SAT), with each clause having infinitely many shifted variants. Non-satisfiability of these formulas can be proved using variants of classical propositional proof systems, augmented with derivation rules for shifting constraints (such as clauses, inequalities, polynomials, etc). Two results on their proof complexity are established. First, there is a particular formula that has a short refutation in Resolution with shift, but requires classical proofs of exponential size (Resolution, Cutting Plane, Polynomial Calculus, etc.). At the same time, exponential lower bounds for shifted versions of classical proof systems are established.

Computer-aided design of electromagnetic devices often involves solving an unbounded elliptic boundary-value problem. Ballooning and infinitesimal scaling, two methods for dealing with the exterior of elliptic boundary-value problems, are shown to be sufficiently closely related for either method to be considered a generalisation of the other. They are shown to differ in their computational complexity, however, so there may be clear practical advantages in favour of either method even though the results obtained may not differ at all.

This study proposes a three-dimensional grid-free method to simulate particle-laden gas flows. It is based on a vortex method. The flow region is not resolved into computational grids, but the gas vorticity field is discretized by vortex elements. The behavior of the vortex element and the particle motion are simultaneously calculated by using the Lagrangian approach. Eight cubic cells are locally allocated around each particle to compute the effect of the particle motion on the gas flow. In each cell, the change in the vorticity due to the particle is calculated, and it is considered by generating a vortex element or changing the strength of the existing vortex elements. This study also applies the grid-free method to simulate a free fall of small spherical solid particles. The particles, initially arranged within a spherical region in an unbounded quiescent air, are made to fall, and their fall induces the air flow around them. The particles are accelerated by the induced downward air flow just after the commencement of their fall. After the acceleration, they are whirled up by a vortex ring produced around the downward air flow. These results are in good agreement with the existing ones, demonstrating the validity of the proposed grid-free method.