Special partial matchings (SPMs) are a generalisation of Brentis special matchings. Let a pircon be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Mariettis zircons. We prove that every open interval in a pircon is a PL ball or a PL sphere. It is then demonstrated that Bruhat orders on certain twisted identities and quasiparabolic W-sets constitute pircons. Together, these results extend a result of Can, Cherniaysky, and Twelbeck, prove a conjecture of Hultman, and confirm a claim of Rains and Vazirani.

When acts on the flag variety of , the orbits are in bijection with fixed point free involutions in the symmetric group . In this case, the associated Kazhdan-Lusztig-Vogan polynomials can be indexed by pairs of fixed point free involutions , where denotes the Bruhat order on . We prove that these polynomials are combinatorial invariants in the sense that if is a poset isomorphism of upper intervals in the Bruhat order on fixed point free involutions, then for all v amp;gt;= u.

The schedule for the jobs in a real-time system can have a huge impact on how the system behave. Since real-time systems are common in safety applications it is important that the scheduling is done in a valid way. Furthermore, one can enhance the performance of the applications by minimizing data latency and jitter. A challenge is that jobs in real-time systems usually have complex constraints making it too time consuming to minimize data latency and jitter to optimality. The purpose of this report is to investigate the possibility of creating high quality schedules using heuristics, with the goal to keep the computational time under one minute. This will be done by comparing three different algorithms that will be used on real scheduling instances provided by the company Arcticus. The first algorithm is a greedy heuristic, the second one a local search and the third one is a metaheuristic, simulated annealing. The results indicate that the data latency can be reduced whilst keeping the computational time below one minute.

In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.

In the hyperbolic research community, there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability. In the first part of the series [6], the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly. By applying this technique, the authors demonstrate that a pure continuous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way. In this work, we extend this investigation to the nonlinear case and focus on entropy conservation. By switching to entropy variables, we provide an estimation of the boundary operators also for nonlinear problems, that guarantee conservation. In numerical simulations, we verify our theoretical analysis.

Authentication is an indispensable part of Quantum Cryptography, which is an unconditionally secure key distribution technique based on the laws of nature. Without proper authentication, Quantum Cryptography is vulnerable to “man-in-the-middle” attacks. Therefore, to guarantee unconditional security of any Quantum Cryptographic protocols, the authentication used must also be unconditionally secure. The standard in Quantum Cryptography is to use theWegman-Carter authentication, which is unconditionally secure and is based on the idea of universal hashing.

In this thesis, we first investigate properties of a Strongly Universal hash function family to facilitate understanding the properties of (classical) authentication used in Quantum Cryptography. Then, we study vulnerabilities of a recently proposed authentication protocol intended to rule out a "man-in-the-middle" attack on Quantum Cryptography. Here, we point out that the proposed authentication primitive is not secure when used in a generic Quantum Cryptographic protocol. Lastly, we estimate the lifetime of authentication using encrypted tags when the encryption key is partially known. Under simplifying assumptions, we derive that the lifetime is linearly dependent on the length of the authentication key. Experimental results that support the theoretical results are also presented.

Secure message authentication is an important part of Quantum Key Distribution. In this paper we analyze special properties of a Strongly Universal2 hash function family, an understanding of which is important in the security analysis of the authentication used in Quantum Cryptography. We answer the following question: How much of Alices message does Eve need to influence so that the message along with its tag will give her enough information to create the correct tag for her message?

In this paper, we review and comment on "A novel protocol-authentication algorithm ruling out a man-in-the-middle attack in quantum cryptography" [M. Peev et al., Int. J. Quant. Inf. 3 (2005) 225]. In particular, we point out that the proposed primitive is not secure when used in a generic protocol, and needs additional authenticating properties of the surrounding quantum-cryptographic protocol.

In this thesis the relationship between Gröbner bases and algebraic coding theory is investigated, and especially applications towards linear codes, with Reed-Müller codes as an illustrative example. We prove that each linear code can be described as a binomial ideal of a polynomial ring, and that a systematic encoding algorithm for such codes is given by the remainder of the information word computed with respect to the reduced Gröbner basis. Finally we show how to apply the representation of a code by its corresponding polynomial ring ideal to construct a class of codes containing the so called primitive Reed-Müller codes, with a few examples of this result.

In this thesis we study the Cauchy problem for elliptic equations. It arises in many areas of application in science and engineering as a problem of reconstruction of solutions to elliptic equations in a domain from boundary measurements taken on a part of the boundary of this domain. The Cauchy problem for elliptic equations is known to be ill-posed.

We use an iterative regularization method based on alternatively solving a sequence of well-posed mixed boundary value problems for the same elliptic equation. This method, based on iterations between Dirichlet-Neumann and Neumann-Dirichlet mixed boundary value problems was first proposed by Kozlov and Maz’ya [13] for Laplace equation and Lame’ system but not Helmholtz-type equations. As a result different modifications of this original regularization method have been proposed in literature. We consider the Robin-Dirichlet iterative method proposed by Mpinganzima et.al [3] for the Cauchy problem for the Helmholtz equation in bounded domains.

We demonstrate that the Robin-Dirichlet iterative procedure is convergent for second order elliptic equations with variable coefficients provided the parameter in the Robin condition is appropriately chosen. We further investigate the convergence of the Robin-Dirichlet iterative procedure for the Cauchy problem for the Helmholtz equation in a an unbounded domain. We derive and analyse the necessary conditions needed for the convergence of the procedure.

In the numerical experiments, the precise behaviour of the procedure for different values of k² in the Helmholtz equation is investigated and the results show that the speed of convergence depends on the choice of the Robin parameters, μ₀ and μ₁. In the unbounded domain case, the numerical experiments demonstrate that the procedure is convergent provided that the domain is truncated appropriately and the Robin parameters, μ₀ and μ₁ are also chosen appropriately.

Cauchy problems for elliptic equations arise in applications in science and engineering. These problems often involve finding important information about an elliptical system from indirect or incomplete measurements. Cauchy problems for elliptic equations are known to be disadvantaged in the sense that a small pertubation in the input can result in a large error in the output. Regularization methods are usually required in order to be able to find stable solutions. In this thesis we study the Cauchy problem for elliptic equations in both bounded and unbounded domains using iterative regularization methods. In Paper I and II, we focus on an iterative regularization technique which involves solving a sequence of mixed boundary value well-posed problems for the same elliptic equation. The original version of the alternating iterative technique is based on iterations alternating between Dirichlet-Neumann and Neumann-Dirichlet boundary value problems. This iterative method is known to possibly work for Helmholtz equation. Instead we study a modified version based on alternating between Dirichlet-Robin and Robin-Dirichlet boundary value problems. First, we study the Cauchy problem for general elliptic equations of second order with variable coefficients in a limited domain. Then we extend to the case of unbounded domains for the Cauchy problem for Helmholtz equation. For the Cauchy problem, in the case of general elliptic equations, we show that the iterative method, based on Dirichlet-Robin, is convergent provided that parameters in the Robin condition are chosen appropriately. In the case of an unbounded domain, we derive necessary, and sufficient, conditions for convergence of the Robin-Dirichlet iterations based on an analysis of the spectrum of the Laplacian operator, with boundary conditions of Dirichlet and Robin types.

In the numerical tests, we investigate the precise behaviour of the Dirichlet-Robin iterations, for different values of the wave number in the Helmholtz equation, and the results show that the convergence rate depends on the choice of the Robin parameter in the Robin condition. In the case of unbounded domain, the numerical experiments show that an appropriate truncation of the domain and an appropriate choice of Robin parameter in the Robin condition lead to convergence of the Robin-Dirichlet iterations.

In the presence of noise, additional regularization techniques have to implemented for the alternating iterative procedure to converge. Therefore, in Paper III and IV we focus on iterative regularization methods for solving the Cauchy problem for the Helmholtz equation in a semi-infinite strip, assuming that the data contains measurement noise. In addition, we also reconstruct a radiation condition at infinity from the given Cauchy data. For the reconstruction of the radiation condition, we solve a well-posed problem for the Helmholtz equation in a semi-infinite strip. The remaining solution is obtained by solving an ill-posed problem. In Paper III, we consider the ordinary Helmholtz equation and use seperation of variables to analyze the problem. We show that the radiation condition is described by a non-linear well-posed problem that provides a stable oscillatory solution to the Cauchy problem. Furthermore, we show that the ill–posed problem can be regularized using the Landweber’s iterative method and the discrepancy principle. Numerical tests shows that the approach works well.

Paper IV is an extension of the theory from Paper III to the case of variable coefficients. Theoretical analysis of this Cauchy problem shows that, with suitable bounds on the coefficients, can iterative regularization methods be used to stabilize the ill-posed Cauchy problem.

The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers k², in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of k² in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

We consider the Helmholtz equation with a variable coefficient in a semi-infinite strip. Homogeneous Neumann conditions are prescribed on a part of the boundary of the strip. Our aim is to find the unknown function in the Dirichlet boundary condition on the remaining part of the boundary from measurements taken on a segment inside the semi-infinite strip. We assume that the radiation condition at infinity is unknown and must be found also. The main difficulty here is the variable coefficient in the Helmholtz equation which does not allow to apply the method of separation of variable as was done in [2]. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements in non-uniform mediums. We split the problem in two parts. One consists of finding the radiation condition and this problem is well-posed. We derive the equation for finding the parameter of the radiation condition, that holds at infinity. The second one is to find the Dirichlet data and this problem is ill-posed. We present an operator equation with compact operator for finding the Dirichlet data. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).

We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well.

We consider the Cauchy problem for the Helmholtz equation with a domain in with N cylindrical outlets to infinity with bounded inclusions in . Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Mazya proposed an alternating iterative method for solving Cauchy problems associated with elliptic, selfadjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Berntsson, Kozlov, Mpinganzima and Turesson (2018) for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in R-2 that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters mu(0) and mu(1), the Robin-Dirichlet alternating iterative procedure is convergent.

We study various boundary and inner regularity questions for p(.)-(super)harmonic functions in Euclidean domains. In particular, we prove the Kellogg property and introduce a classification of boundary points for p(.)-harmonic functions into three disjoint classes: regular, semiregular and strongly irregular points. Regular and especially semiregular points are characterized in many ways. The discussion is illustrated by examples. Along the way, we present a removability result for bounded p(.)-harmonic functions and give some new characterizations of W-0(1,p(.)) spaces. We also show that p(.)-superharmonic functions are lower semicontinuously regularized, and characterize them in terms of lower semicontinuously regularized supersolutions.

In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Caratheodory and Nakki. Modulus ends and prime ends, defined by means of the p-modulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.

This thesis addresses topics in sparse least squares computation. A stable method for solving the least squares problem, min ||Ax-b||₂ is based on the QR factorization.Here we have addressed the difficulty for storing the orthogonal matrix Q. Using traditional methods, the number of nonzero elements in Q makes it in many cases not feasible to store. Using the multifrontal technique when computing the QR factorization,Q may be stored and used more efficiently. A new user friendly Matlab implementation is developed.

When a row in A is dense the factor R from the QR factorization may be completely dense. Therefore problems with dense rows must be treated by special techniques. The usual way to handle dense rows is to partition the problem into one sparse and one dense subproblem. The drawback with this approach is that the sparse subproblem may be more ill-conditioned than the original problem or even not have a unique solution. Another method, useful for problems with few dense rows, is based on matrix stretching, where the dense rows are split into several less dense rows linked then together with new artificial variables. We give and analyze the conditioning of the matrix obtained by this method and show that no ill-conditioned subproblem arise.

In many least squares problems upper and lower bounds on the variables have to be satisfied at the solution. This type of problem arises, for example, in reconstruction problems in geodesy and tomography. Here methods based on direct factorization methods for sparse matrix computation are explored. Two completely different approaches for solving the problem are discussed and compared, i.e. active set methods and primal-dual interior-point methods based on Mehrotra's predictor-corrector path following method. An active set block method suitable for sparse problems is developed and a convergence proof is presented. The choice of barrier parameter, multiple corrections and finite termination for the interior-point method are discussed. Numerical comparison is given of the active set method, the interior-point method, together with an trust region method based on the interior-reflective Newton implemented in the optimization toolbox for MATLAB. The numerical tests show that the block active set method is faster and gives better accuracy for both nondegenerate and degenerate problems.

A Reply to the Comment by Carlos Luiz Ryff.

The two-photon interferometric experiment proposed by J. D. Franson [Phys. Rev. Lett. 62, 2205 (1989)] is often treated as a “Bell test of local realism.” However, it has been suggested that this is incorrect due to the 50% postselection performed even in the ideal gedanken version of the experiment. Here we present a simple local hidden variable model of the experiment that successfully explains the results obtained in usual realizations of the experiment, even with perfect detectors. Furthermore, we also show that there is no such model if the switching of the local phase settings is done at a rate determined by the internal geometry of the interferometers.

Let R be a noetherian ring, alpha an ideal of R, and M an R-module. We prove that for a finite module M, if H-alpha(i)(M) is minimax for all i andgt;= r andgt;= 1, then H-alpha(i)(M) is artinian for i andgt;= r. A local-global principle for minimax local cohomology modules is shown. If H-alpha(i)(M) is coatomic for i andlt;= r (M finite) then H-alpha(i)(M) is finite for i andlt;= r. We give conditions for a module which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing theorems are proved for local cohomology modules.

Let R be a Noetherian ring, a an ideal of R, M an R-module and n a non-negative integer. In this paper we first study the finiteness properties of the kernel and the cokernel of the natural map f: Ext(R)(n) (R/alpha, M) -andgt; Hom(R)(R/alpha, H-alpha(n) (M)), under some conditions on the previous local cohomology modules. Then we get some corollaries about the associated primes and Artinianness of local cohomology modules. Finally we will study the asymptotic behavior of the kernel and the cokernel of the natural map in the graded case.

Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about the artinianness of some special local cohomology modules in the graded case.

Let R be a noetherian ring, alpha an ideal of R such that dim R/alpha = 1 and M a finite R-module. We will study cofiniteness and some other properties of the local cohomology modules H-alpha(i)(M). For an arbitrary ideal alpha and an R-module M (not necessarily finite), we will characterize alpha-cofinite artinian local cohomology modules. Certain sets of coassociated primes of top local cohomology modules over local rings are characterized.

The membership of the local cohomology modules H-a(n) (M) of a module M in certain Serre subcategories of the category of modules is studied from below (i < n) and from above (i > n). Generalizations of depth and regular sequences are defined. The relation of these notions to local cohomology are found. It is shown that the membership of the local cohomology modules of a finite module in a Serre subcategory in the upper range just depends on the support of the module. (C) 2008 Elsevier Inc. All rights reserved.

Summary. Questions of existence and uniqueness for discrete frictional quasi-static incremental problems, rate problems and wedging problems are discussed. Various methods to compute critical bounds for the coefficient of friction which guarantee existence and uniqueness are described, as well as the sharpness of the bounds and their interdependence.

Knot theory is the mathematical study of knots. In this thesis we study knots and one of its applications in DNA. Knot theory sits in the mathematical field of topology and naturally this is where the work begins. Topological concepts such as topological spaces, homeomorphisms, and homology are considered. Thereafter knot theory, and in particular, knot theoretical invariants are examined, aiming to provide insights into why it is difficult to answer the question "How can we tell knots appart?". In knot theory invariants such as the bracket polynomial, the Jones polynomial and tricolorability are considered as well as other helpful results like Seifert surfaces. Lastly knot theory is applied to DNA, where it will shed light on how certain enzymes interact with the genome.

Syftet med detta arbete är att undersöka hur komvuxelever beskriver orsaker till sina upplevda matematiksvårigheter och hur de värderar sina lärares strategier för att hjälpa dem. Studien använde sig av semistrukturerade intervjuer för att undersöka matematiksvårigheter hos fem komvuxelever som har haft svårt med ämnet matematik under sin grundskole- eller gymnasietid. En gemensam nämnare visade sig vara att de under sin grundskoletid inte hade motivation för ämnet matematik och det ansåg de flesta av eleverna har haft en negativ inverkan på deras matematiklärande. Bristen på motivation ledde till att de missade väsentliga delar i den grundläggande matematiken trots att de fick hjälp och stöd av sina matematiklärare på grundskolan och gymnasiet. Samtliga elever berättade att deras otillräckliga grundkunskaper i matematik hade betydelse för deras svårigheter idag. Elevers brister från tidigare årskurser gör det svårt för eleverna att tillgodose sig utökade kunskaper. Alla ansåg också att det var svårt med matematikspråket. Med matematikspråk avsåg de begrepp, ord och symboler som har betydelse för att förstå matematik. Informanterna berättade att variation i undervisningen kunde ha påverkat deras motivation och inverkat positivt på deras intresse för matematik.Alla intervjuade elever sade sig idag ha varit missnöjda med den hjälp och det stöd de fick av sina Komvuxlärare, vilket betyder till att de nu hade mer svårigheter i matematik än tidigare.

Ahmad, Ohlson, and von Rosen (2011a) present asymptotic distribution of a one-sample test statistic under non-normality, when the data are high dimensional, i.e., when the dimension of the vector, p, may exceed the sample size, n. This paper extends the case to a two-sample statistic to test the difference of mean vectors of two independent multivariate distributions, again under high-dimensional set up. Using the asymptotic theory of U-statistics, and under mild assumptions on the traces of the unknown covariance matrices, the statistic is shown to follow an approximate normal distribution when n and p are large. However, no relationship between n and p is assumed. An extension to the paired case is given, which, being essentially a one-sample statistic, supplements the asymptotic results obtained in Ahmad, Ohlson, and von Rosen (2011a).

A test statistic is considered for testing a hypothesis for the mean vector for multivariate data, when the dimension of the vector, p, may exceed the number of vectors, n, and the underlying distribution need not necessarily be normal. With n, p large, and under mild assumptions, the statistic is shown to asymptotically follow a normal distribution. A by product of the paper is the approximate distribution of a quadratic form, based on the reformulation of well-known Box's approximation, under high-dimensional set up.

Test statistics for sphericity and identity of the covariance matrix are presented, when the data are multivariate normal and the dimension, p, can exceed the sample size, n. Using the asymptotic theory of U-statistics, the test statistics are shown to follow an approximate normal distribution for large p, also when p >> n. The statistics are derived under very general conditions, particularly avoiding any strict assumptions on the traces of the unknown covariance matrix. Neither any relationship between n and p is assumed. The accuracy of the statistics is shown through simulation results, particularly emphasizing the case when p can be much larger than n. The validity of the commonly used assumptions for high-dimensional set up is also briefly discussed.

Test statistics for sphericity and identity of the covariance matrix are presented, when the data are multivariate normal and the dimension, p, can exceed the sample size, n. Under certain mild conditions mainly on the traces of the unknown covariance matrix, and using the asymptotic theory of U-statistics, the test statistics are shown to follow an approximate normal distribution for large p, also when p and#8811;n. The accuracy of the statistics is shown through simulation results, particularly emphasizing the case when p can be much larger than n. A real data set is used to illustrate the application of the proposed test statistics.

Ahmad et al. (in press) presented test statistics for sphericity and identity of the covariance matrix of a multivariate normal distribution when the dimension, p, exceeds the sample size, n. In this note, we show that their statistics are robust to normality assumption, when normality is replaced with certain mild assumptions on the traces of the covariance matrix. Under such assumptions, the test statistics are shown to follow the same asymptotic normal distribution as under normality for large p, also whenp greater thangreater than n. The asymptotic normality is proved using the theory of U-statistics, and is based on very general conditions, particularly avoiding any relationship between n and p.

A test statistic is considered for testing a hypothesis for the mean vector for multivariate data, when the dimension of the vector, p, may exceed the number of vectors, n, and the underlying distribution need not necessarily be normal. With n,p→∞, and under mild assumptions, but without assuming any relationship between n and p, the statistic is shown to asymptotically follow a chi-square distribution. A by product of the paper is the approximate distribution of a quadratic form, based on the reformulation of the well-known Box's approximation, under high-dimensional set up. Using a classical limit theorem, the approximation is further extended to an asymptotic normal limit under the same high dimensional set up. The simulation results, generated under different parameter settings, are used to show the accuracy of the approximation for moderate n and large p.

The existence of the Ginzburg-Landau vortex number is established for any configuration with finite action. As a consequence, Bogomol'nyi's formula for the critical action is valid for any finite action configuration.

We suggest a method of constructing decompositions of a topological space X having an open subset homeomorphic to the space (R-n , tau), where n is an integer greater than= 1 and tau is any admissible extension of the Euclidean topology of R-n (in particular, X can be a finite-dimensional separable metrizable manifold), into a countable family F of sets (dense in X and zero-dimensional in the case of manifolds) such that the union of each non-empty proper subfamily of F does not have the Baire property in X.

We study the algebra of semigroups of sets (i.e. families of sets closed under finite unions) and its applications. For each n greater than 1 we produce two finite nested families of pairwise different semigroups of sets consisting of subsets of R" without the Baire property.

The variational capacity cap(p) in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every E subset of R-n, infx is an element of R(n)cap(p)(E boolean AND B(x, r), B(x, 2r))/cap(p)(B(x, r), B(x, 2r)) is either zero or tends to 1 as r -amp;gt; infinity. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincare inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in R-n. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.

Achieving food security will require closing yield gaps in many regions, including Pakistan. Although fertilizer subsidies have facilitated increased nitrogen (N) application rates, many staple crop yields have yet to reach their maximum potential. Considering that current animal manure and human excreta (bio-supply) recycling rates are low, there is substantial potential to increase the reuse of nutrients in bio-supply. We quantified 2010 crop N, phosphorus (P), and potassium (K) needs along with bio-supply nutrient availability for Pakistani districts, and compared these values to synthetic fertilizer use and costs. We found that synthetic fertilizer use combined with low bio-supply recycling resulted in a substantial gap between nutrient supply and P and K crop needs, which would cost 3 billion USD to fill with synthetic fertilizers. If all bio-supply was recycled, it could eliminate K synthetic fertilizer needs and decrease N synthetic fertilizer needs to 43% of what was purchased in 2010. Under a full recycling scenario, farmers would still require an additional 0.28 million tons of synthetic P fertilizers, costing 2.77 billion USD. However, it may not be prohibitively expensive to correct P deficiencies. Pakistan already spends this amount of money on fertilizers. If funds used for synthetic N were reallocated to synthetic P purchases in a full bio-supply recycling scenario, crop needs could be met. Most recycling could happen within districts, with only 6% of bio-supply requiring between-district transport when optimized to meet national N crop needs. Increased recycling in Pakistan could be a viable way to decrease yield gaps.

Increased recycling of nutrient-rich organic waste to meet crop nutrient needs is an essential component of a more sustainable food system. However, agricultural specialization continues to pose a significant challenge to balancing crop nutrient needs and the nutrient supply from animal manure and human excreta locally. For Sweden, this study found that recycling all excreta (in 2007) could meet up to 75% of crop nitrogen and 81% of phosphorus needs, but that this would exceed crop potassium needs by 51%. Recycling excreta within municipalities could meet 63% of crop P nutrient needs, but large regional differences and imbalances need to be corrected to avoid over or under fertilizing. Over 50% of the total nitrogen and phosphorus in excreta is contained in just 40% of municipalities, and those have a surplus of excreta nutrients compared to crop needs. Reallocation of surpluses (nationally optimized for phosphorus) towards deficit municipalities, would cost 192 million USD (for 24 079 km of truck travel). This is 3.7 times more than the total NPK fertilizer value being transported. These results indicate that Sweden could reduce its dependence on synthetic fertilizers through investments in excreta recycling, but this would likely require valuing also other recycling benefits.

Recycling essential plant nutrients like nitrogen (N), phosphorus (P), and potassium (K) from organic waste such as human and animal excreta will be an essential part of sustainable food systems and a circular economy. However, transportation is often cited as a major barrier to increased recycling as organic waste is heavy and bulky, and distances between areas of abundant waste may be far from areas with a need for fertilizers. We investigated the effect of increased input data spatial resolution to an optimization model on the weight, distance, and spatial patterns of transport. The model was run in Sweden and in Pakistan to examine cost-effectiveness of transporting excess excreta to areas of crop need after local recycling. Increasing the resolution of input data from political boundaries (municipalities and districts) to 0.083 decimal grids increased the amount of N requiring transport by 12% in Pakistan and increased P requiring transport by 14% in Sweden. The average distance decreased by 67% (to 44 km) in Pakistan but increased by 1 km in Sweden. Further increasing the resolution to 5 km grids in Sweden decreased the average transportation distance by 9 km (down to 123 km). In both countries, increasing resolution also decreased the number of long-distance heavy transports, and as such costs did not increase as much as total distance and weight transported. Ultimately, transportation in Pakistan seemed financially beneficial: the cost of transport only represented 13% of the NPK fertilizer value transported, and total recycling could even cover 78% of additional fertilizer purchases required. In Sweden, the cost of transporting excreta did not seem cost effective without valuing other potential benefits of increased recycling: costs were three times higher than the fertilizer value transported in excreta at the 5 km resolution. In summary, increasing input data resolution created a more realistic picture of recycling needs. This also highlighted more favorable cost to fertilizer value ratios which could make it easier to move forward with industry and government partners to facilitate productive recycling. Our analysis shows that in both countries increased recycling can result in better spatial nutrient balances.

Detta arbete utgår från tanken att utveckla undervisningen genom att integrera matematik med  andra ämnen. Projektet har genomförts genom att designa och utvärdera lektioner tillsammans med slöjd- och teknik- och hemkunskapslärare. Studien har utförts i åk 7–9, på en skola i Västmanland och en skola i Östergötland.

Arbete syftar till att eleverna på högstadiet ska kunna dra nytta av sina matematiska kunskaper i reella situationer. Lektionerna bidrar till att utveckla elevernas tänkande och förmåga att kunna lösa olika problem.

Vi har observerat lektioner, som de deltagande lärarna har genomfört, och då noterat moment som kan användas för vårt projekt. Efter det har vi intervjuat dem och de har fått bedöma sina kunskaper i den pedagogiska modellen, samt vilka möjligheter de har att genomföra integrering med andra ämnen.

Enkätundersökning som vi genomfört, har visat deltagarnas intresse och engagemang för denna metod samt att de har motivation, att fortsätta och vidareutveckla sin undervisning enligt denna modell.

We propose a variance reduced algorithm for solving monotone variational inequalities. Without assuming strong monotonicity, cocoercivity, or boundedness of the domain, we prove almost sure convergence of the iterates generated by the algorithm to a solution. In the monotone case, the ergodic average converges with the optimal O(1/k) rate of convergence. When strong monotonicity is assumed, the algorithm converges linearly, without requiring the knowledge of strong monotonicity constant. We finalize with extensions and applications of our results to monotone inclusions, a class of non-monotone variational inequalities and Bregman projections.

This article presents a characterization of the possible resolutions toa Fermi problem from the Fermi problem Activity Templates (FpAT). The articlepresents the theoretical foundation based on previous research on Fermi prob-lems in Mathematics Education and other fields of research that allows theidentification of the four main activities that can be used when solving them.These activities are reasoned estimation, experimentation, searching for datafrom reliable sources, and statistical data collection and processing. From theseactivities, the FpAT are specified and their possibilities as tools for the manage-ment and design of classroom activities are discussed. Specific examples ofclassroom proposals are included to support both the work of students and theactivity design and management options enabled by the FpAT to promoteproblem solving and mathematical modelling.

In this paper we analyse 11 pairs of pre-service primary teachers’ solutions to twoFermi problems. The solutions were expressed using the descriptive and analytic FPATframework (Fermi problem Activity Template), and our analysis focus on characterizingthe FPAT representations produced by the pre-service teachers. The results show thatalmost half of the produced FPATs in principle solved the problems but that only threesolutions provided enough detail to be practically implementable. Multiple keyconstructs were used to structure the solutions in the first problem, but not so in thesecond problem. The variation of different activities and ways of working suggested insolving the sub-problems in the two problems also differ in a significant way.