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• 1.
Matematiska Institutionen, Umeå Universitet, Umeå, Sweden.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Orthogonal Latin Rectangles2008In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 17, no 4, p. 519-536Article in journal (Refereed)

We use a greedy probabilistic method to prove that, for every ε > 0, every m × n Latin rectangle on n symbols has an orthogonal mate, where m = (1 − ε)n. That is, we show the existence of a second Latin rectangle such that no pair of the mn cells receives the same pair of symbols in the two rectangles.

• 2.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Institutionen för matematik och matematisk statistik, Umeå univeristet. Institutionen för matematik och matematisk statistik, Umeå univeristet.
Factors of r-partite graphs and bounds for the strong chromatic number2010In: Ars combinatoria, ISSN 0381-7032, Vol. 95, p. 277-287Article in journal (Refereed)

We give an optimal degree condition for a tripartite graph to have a spanning subgraph consisting of complete graphs of order 3. This result is used to give an upper bound of 2 Delta for the strong chromatic number of n vertex graphs with Delta >= n/6.

• 3.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Department of Mathematics, University of Bristol, Bristol, United Kingdom. Department of Mathematics, Uppsala Universitet, Uppsala, Sweden. Mathematics Institute, University of Warwick, Coventry, United Kingdom.
Multifractal analysis of non-uniformly hyperbolic systems2010In: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 177, no 1, p. 125-144Article in journal (Refereed)

We prove a multifractal formalism for Birkhoff averages of continuous functions in the case of some non-uniformly hyperbolic maps, which includes interval examples such as the Manneville-Pomeau map.

• 4.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Department of Mathematics, Rutgers University, Piscataway, NJ, United States. Department of Mathematics, Rutgers University, Piscataway, NJ, United States.
Factors in random graphs2008In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 33, no 1, p. 1-28Article in journal (Refereed)

Let H be a fixed graph on v vertices. For an n-vertex graph G with n divisible by v, an H-factor of G is a collection of n/v copies of H whose vertex sets partition V (G).

In this work, we consider the threshold thH(n) of the property that an Erds-Rényi random graph (on n points) contains an H-factor. Our results determine thH(n) for all strictly balanced H.

The method here extends with no difficulty to hypergraphs. As a corollary, we obtain the threshold for a perfect matching in random k-uniform hypergraph, solving the well-known Shamir's problem.

• 5.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics. Mathematics Department, Uppsala University, Uppsala, Sweden.
Institut für Informatik, Universität Leipzig, Leipzig, Germany . Institut für Informatik, Universität Leipzig, Leipzig, Germany . Mathematics Department, Uppsala University, Uppsala, Sweden .
Tuning positive feedback for signal detection in noisy dynamic environments2012In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 309, p. 88-95Article in journal (Refereed)

Learning from previous actions is a key feature of decision-making. Diverse biological systems, from neuronal assemblies to insect societies, use a combination of positive feedback and forgetting of stored memories to process and respond to input signals. Here we look how these systems deal with a dynamic two-armed bandit problem of detecting a very weak signal in the presence of a high degree of noise. We show that by tuning the form of positive feedback and the decay rate to appropriate values, a single tracking variable can effectively detect dynamic inputs even in the presence of a large degree of noise. In particular, we show that when tuned appropriately a simple positive feedback algorithm is Fisher efficient, in that it can track changes in a signal on a time of order L(h)= (vertical bar h vertical bar/sigma)(-2), where vertical bar h vertical bar is the magnitude of the signal and sigma the magnitude of the noise.

• 6.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
School of Engineering and Applied Sciences, Harvard University.
A Slime Mold Solver for Linear Programming Problems2012In: How the World Computes: Turing Centenary Conference and 8th Conference on Computability in Europe, CiE 2012, Cambridge, UK, June 18-23, 2012. Proceedings / [ed] S. Barry Cooper , Anuj Dawar and Benedikt Löwe, Springer, 2012, p. 344-354Conference paper (Refereed)

Physarum polycephalum (true slime mold) has recently emerged as a fascinating example of biological computation through morphogenesis. Despite being a single cell organism, experiments have observed that through its growth process, the Physarum is able to solve various minimum cost flow problems. This paper analyzes a mathematical model of the Physarum growth dynamics. We show how to encode general linear programming (LP) problems as instances of the Physarum. We prove that under the growth dynamics, the Physarum is guaranteed to converge to the optimal solution of the LP. We further derive an efficient discrete algorithm based on the Physarum model, and experimentally verify its performance on assignment problems.

• 7.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Department of Mathematics, Uppsala University, Uppsala, Sweden.
Square summability of variations and convergence of the transfer operator2008In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 28, no 4, p. 1145-1151Article in journal (Refereed)

In this paper we study the one-sided shift operator on a state space defined by a finite alphabet. Using a scheme developed by Walters [P. Walters. Trans. Amer. Math. Soc. 353(1) (2001), 327-347], we prove that the sequence of iterates of the transfer operator converges under square summability of variations of the g-function, a condition which gave uniqueness of a g-measure in our earlier work [A. Johansson and A. Öberg. Math. Res. Lett. 10(5-6) (2003), 587-601]. We also prove uniqueness of the so-called G-measures, introduced by Brown and Dooley [G. Brown and A. H. Dooley. Ergod. Th. & Dynam. Sys. 11 (1991), 279-307], under square summability of variations.

• 8.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Square summability of variations of g-functions and uniqueness of g-measures2003In: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 10, no 5, p. 587-601Article in journal (Refereed)

We prove uniqueness of $g$-measures for $g$-functions satisfying quadratic summability of variations. Our result is in contrast to the situation of, \eg, the one-dimensional Ising model with long-range interactions, since $\ell_1$-summability of variations is required for general potentials. We illustrate this difference with some examples. To prove our main result we use a product martingale argument. We also give conditions for uniqueness of general $G$-measures, \ie, the case for general potentials, based on our investigation of the probabilistic case involving $g$-functions.

• 9.
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.
Department of Mathematics, Uppsala University, Uppsala, Sweden. Mathematics Institute, University of Warwick, Coventry, United Kingdom.
Countable state shifts and uniqueness of g-measures2007In: American Journal of Mathematics, ISSN 0002-9327, E-ISSN 1080-6377, Vol. 129, no 6, p. 1501-1511Article in journal (Refereed)

In this paper we present a new approach to studying $g$-measures which is based upon local absolute continuity. We extend an earlier result that square summability of variations of $g$ ensures uniqueness of $g$-measures. The first extension is to the case of countably many symbols. The second extension is to some cases where $g \geq 0$, relaxing the earlier requirement that $\inf g>0$.

• 10.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.
Department of Mathematics, Uppsala University, Uppsala, Sweden. Mathematics Institute, University of Warwick, Coventry, UK.
Ergodic Theory of Kusuoka Measures2017In: Journal of Fractal Geometry, ISSN 2308-1309, Vol. 4, no 2, p. 185-214Article in journal (Refereed)

In the analysis on self-similar fractal sets, the Kusuoka measure plays an important role. Here we investigate the Kusuoka measure from an ergodic theoretic viewpoint, seen as an invariant measure on a symbolic space. Our investigation shows that the Kusuoka measure generalizes Bernoulli measures and their properties to higher dimensions of an underlying finite dimensional vector space. Our main result is that the transfer operator on functions has a spectral gap when restricted to a certain Banach space that contains the Hölder continuous functions, as well as the highly discontinuous g" role="presentation">g-function associated to the Kusuoka measure. As a consequence, we obtain exponential decay of correlations. In addition, we provide some explicit rates of convergence for a family of generalized Sierpinski gaskets.

• 11.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electrical Engineering, Mathematics and Science, Mathematics.
Uppsala University. University of Warwick, Coventry, UK.
Phase transitions in long-range Ising models and an optimal condition for factors of g-measures2019In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 39, no 5, p. 1317-1330Article in journal (Refereed)

We weaken the assumption of summable variations in a paper by Verbitskiy [On factors of g-measures. Indag. Math. (N.S.) 22 (2011), 315-329] to a weaker condition, Berbee's condition, in order for a one-block factor (a single-site renormalization) of the full shift space on finitely many symbols to have a g-measure with a continuous g-function. But we also prove by means of a counterexample that this condition is (within constants) optimal. The counterexample is based on the second of our main results, where we prove that there is a critical inverse temperature in a one-sided long-range Ising model which is at most eight times the critical inverse temperature for the (two-sided) Ising model with long-range interactions.

• 12.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics. Uppsala University, Department of Mathematics.
Uppsala University, Department of Mathematics. University of Warwick, Mathematics Institute.
Unique Bernoulli g-measures2012In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 14, no 5, p. 1599-1615Article in journal (Refereed)

We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a g-measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique g-measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the g-measure.

• 13.
Department of Mathematics, Uppsala University.
Department of Mathematics, Uppsala University. Department of Mathematics, Uppsala University.
A first principles derivation of animal group size distributions2011In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 283, no 1, p. 35-43Article in journal (Refereed)

Several empirical studies have shown that the animal group size distribution of many species can be well fit by power laws with exponential truncation. A striking empirical result due to Niwa is that the exponent in these power laws is one and the truncation is determined by the average group size experienced by an individual. This distribution is known as the logarithmic distribution. In this paper we provide first principles derivation of the logarithmic distribution and other truncated power laws using a site-based merge and split framework. In particular, we investigate two such models. Firstly, we look at a model in which groups merge whenever they meet but split with a constant probability per time step. This generates a distribution similar, but not identical to the logarithmic distribution. Secondly, we propose a model, based on preferential attachment, that produces the logarithmic distribution exactly. Our derivation helps explain why logarithmic distributions are so widely observed in nature. The derivation also allows us to link splitting and joining behavior to the exponent and truncation parameters in power laws.

• 14.
Mathematics Department, Uppsala University, Uppsala, Sweden.
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics. Mathematics Department, Uppsala University, Uppsala, Sweden. PRESTO, JST, Kawaguchi, Saitama, Japan. Future University Hakodate, Hakodate, Japan. Mathematics Department, Uppsala University, Uppsala, Sweden.
Current-reinforced random walks for constructing transport networks2013In: Journal of the Royal Society Interface, ISSN 1742-5689, E-ISSN 1742-5662, Vol. 10, no 80, p. 20120864-Article in journal (Refereed)

Biological systems that build transport networks, such as trail-laying ants and the slime mould Physarum, can be described in terms of reinforced random walks. In a reinforced random walk, the route taken by 'walking' particles depends on the previous routes of other particles. Here, we present a novel form of random walk in which the flow of particles provides this reinforcement. Starting from an analogy between electrical networks and random walks, we show how to include current reinforcement. We demonstrate that current-reinforcement results in particles converging on the optimal solution of shortest path transport problems, and avoids the self-reinforcing loops seen in standard density-based reinforcement models. We further develop a variant of the model that is biologically realistic, in the sense that the particles can be identified as ants and their measured density corresponds to those observed in maze-solving experiments on Argentine ants. For network formation, we identify the importance of nonlinear current reinforcement in producing networks that optimize both network maintenance and travel times. Other than ant trail formation, these random walks are also closely related to other biological systems, such as blood vessels and neuronal networks, which involve the transport of materials or information. We argue that current reinforcement is likely to be a common mechanism in a range of systems where network construction is observed.

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