@inproceedings{2157, abstract = {We show that the following algorithmic problem is decidable: given a 2-dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in ℝ3? By a known reduction, it suffices to decide the embeddability of a given triangulated 3-manifold X into the 3-sphere S3. The main step, which allows us to simplify X and recurse, is in proving that if X can be embedded in S3, then there is also an embedding in which X has a short meridian, i.e., an essential curve in the boundary of X bounding a disk in S3 nX with length bounded by a computable function of the number of tetrahedra of X.}, author = {Matoušek, Jiří and Sedgwick, Eric and Tancer, Martin and Wagner, Uli}, booktitle = {Proceedings of the Annual Symposium on Computational Geometry}, location = {Kyoto, Japan}, pages = {78 -- 84}, publisher = {ACM}, title = {{Embeddability in the 3 sphere is decidable}}, doi = {10.1145/2582112.2582137}, year = {2014}, } @article{2158, abstract = {Directional guidance of migrating cells is relatively well explored in the reductionist setting of cell culture experiments. Here spatial gradients of chemical cues as well as gradients of mechanical substrate characteristics prove sufficient to attract single cells as well as their collectives. How such gradients present and act in the context of an organism is far less clear. Here we review recent advances in understanding how guidance cues emerge and operate in the physiological context.}, author = {Majumdar, Ritankar and Sixt, Michael K and Parent, Carole}, journal = {Current Opinion in Cell Biology}, number = {1}, pages = {33 -- 40}, publisher = {Elsevier}, title = {{New paradigms in the establishment and maintenance of gradients during directed cell migration}}, doi = {10.1016/j.ceb.2014.05.010}, volume = {30}, year = {2014}, } @inproceedings{2159, abstract = {Motivated by topological Tverberg-type problems, we consider multiple (double, triple, and higher multiplicity) selfintersection points of maps from finite simplicial complexes (compact polyhedra) into ℝd and study conditions under which such multiple points can be eliminated. The most classical case is that of embeddings (i.e., maps without double points) of a κ-dimensional complex K into ℝ2κ. For this problem, the work of van Kampen, Shapiro, and Wu provides an efficiently testable necessary condition for embeddability (namely, vanishing of the van Kampen ob-struction). For κ ≥ 3, the condition is also sufficient, and yields a polynomial-time algorithm for deciding embeddability: One starts with an arbitrary map f : K→ℝ2κ, which generically has finitely many double points; if k ≥ 3 and if the obstruction vanishes then one can successively remove these double points by local modifications of the map f. One of the main tools is the famous Whitney trick that permits eliminating pairs of double points of opposite intersection sign. We are interested in generalizing this approach to intersection points of higher multiplicity. We call a point y 2 ℝd an r-fold Tverberg point of a map f : Kκ →ℝd if y lies in the intersection f(σ1)∩. ∩f(σr) of the images of r pairwise disjoint simplices of K. The analogue of (non-)embeddability that we study is the problem Tverbergκ r→d: Given a κ-dimensional complex K, does it satisfy a Tverberg-type theorem with parameters r and d, i.e., does every map f : K κ → ℝd have an r-fold Tverberg point? Here, we show that for fixed r, κ and d of the form d = rm and k = (r-1)m, m ≥ 3, there is a polynomial-time algorithm for deciding this (based on the vanishing of a cohomological obstruction, as in the case of embeddings). Our main tool is an r-fold analogue of the Whitney trick: Given r pairwise disjoint simplices of K such that the intersection of their images contains two r-fold Tverberg points y+ and y- of opposite intersection sign, we can eliminate y+ and y- by a local isotopy of f. In a subsequent paper, we plan to develop this further and present a generalization of the classical Haeiger-Weber Theorem (which yields a necessary and sufficient condition for embeddability of κ-complexes into ℝd for a wider range of dimensions) to intersection points of higher multiplicity.}, author = {Mabillard, Isaac and Wagner, Uli}, booktitle = {Proceedings of the Annual Symposium on Computational Geometry}, location = {Kyoto, Japan}, pages = {171 -- 180}, publisher = {ACM}, title = {{Eliminating Tverberg points, I. An analogue of the Whitney trick}}, doi = {10.1145/2582112.2582134}, year = {2014}, } @inproceedings{2160, abstract = {Transfer learning has received a lot of attention in the machine learning community over the last years, and several effective algorithms have been developed. However, relatively little is known about their theoretical properties, especially in the setting of lifelong learning, where the goal is to transfer information to tasks for which no data have been observed so far. In this work we study lifelong learning from a theoretical perspective. Our main result is a PAC-Bayesian generalization bound that offers a unified view on existing paradigms for transfer learning, such as the transfer of parameters or the transfer of low-dimensional representations. We also use the bound to derive two principled lifelong learning algorithms, and we show that these yield results comparable with existing methods.}, author = {Pentina, Anastasia and Lampert, Christoph}, location = {Beijing, China}, pages = {991 -- 999}, publisher = {ML Research Press}, title = {{A PAC-Bayesian bound for Lifelong Learning}}, volume = {32}, year = {2014}, } @article{2161, abstract = {Repeated pathogen exposure is a common threat in colonies of social insects, posing selection pressures on colony members to respond with improved disease-defense performance. We here tested whether experience gained by repeated tending of low-level fungus-exposed (Metarhizium robertsii) larvae may alter the performance of sanitary brood care in the clonal ant, Platythyrea punctata. We trained ants individually over nine consecutive trials to either sham-treated or fungus-exposed larvae. We then compared the larval grooming behavior of naive and trained ants and measured how effectively they removed infectious fungal conidiospores from the fungus-exposed larvae. We found that the ants changed the duration of larval grooming in response to both, larval treatment and their level of experience: (1) sham-treated larvae received longer grooming than the fungus-exposed larvae and (2) trained ants performed less self-grooming but longer larval grooming than naive ants, which was true for both, ants trained to fungus-exposed and also to sham-treated larvae. Ants that groomed the fungus-exposed larvae for longer periods removed a higher number of fungal conidiospores from the surface of the fungus-exposed larvae. As experienced ants performed longer larval grooming, they were more effective in fungal removal, thus making them better caretakers under pathogen attack of the colony. By studying this clonal ant, we can thus conclude that even in the absence of genetic variation between colony members, differences in experience levels of brood care may affect performance of sanitary brood care in social insects.}, author = {Westhus, Claudia and Ugelvig, Line V and Tourdot, Edouard and Heinze, Jürgen and Doums, Claudie and Cremer, Sylvia}, issn = {0340-5443}, journal = {Behavioral Ecology and Sociobiology}, number = {10}, pages = {1701 -- 1710}, publisher = {Springer}, title = {{Increased grooming after repeated brood care provides sanitary benefits in a clonal ant}}, doi = {10.1007/s00265-014-1778-8}, volume = {68}, year = {2014}, } @inproceedings{2162, abstract = {We study two-player (zero-sum) concurrent mean-payoff games played on a finite-state graph. We focus on the important sub-class of ergodic games where all states are visited infinitely often with probability 1. The algorithmic study of ergodic games was initiated in a seminal work of Hoffman and Karp in 1966, but all basic complexity questions have remained unresolved. Our main results for ergodic games are as follows: We establish (1) an optimal exponential bound on the patience of stationary strategies (where patience of a distribution is the inverse of the smallest positive probability and represents a complexity measure of a stationary strategy); (2) the approximation problem lies in FNP; (3) the approximation problem is at least as hard as the decision problem for simple stochastic games (for which NP ∩ coNP is the long-standing best known bound). We present a variant of the strategy-iteration algorithm by Hoffman and Karp; show that both our algorithm and the classical value-iteration algorithm can approximate the value in exponential time; and identify a subclass where the value-iteration algorithm is a FPTAS. We also show that the exact value can be expressed in the existential theory of the reals, and establish square-root sum hardness for a related class of games.}, author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus}, location = {Copenhagen, Denmark}, number = {Part 2}, pages = {122 -- 133}, publisher = {Springer}, title = {{The complexity of ergodic mean payoff games}}, doi = {10.1007/978-3-662-43951-7_11}, volume = {8573}, year = {2014}, } @inproceedings{2163, abstract = {We consider multi-player graph games with partial-observation and parity objective. While the decision problem for three-player games with a coalition of the first and second players against the third player is undecidable in general, we present a decidability result for partial-observation games where the first and third player are in a coalition against the second player, thus where the second player is adversarial but weaker due to partial-observation. We establish tight complexity bounds in the case where player 1 is less informed than player 2, namely 2-EXPTIME-completeness for parity objectives. The symmetric case of player 1 more informed than player 2 is much more complicated, and we show that already in the case where player 1 has perfect observation, memory of size non-elementary is necessary in general for reachability objectives, and the problem is decidable for safety and reachability objectives. From our results we derive new complexity results for partial-observation stochastic games.}, author = {Chatterjee, Krishnendu and Doyen, Laurent}, booktitle = {Lecture Notes in Computer Science}, location = {Copenhagen, Denmark}, number = {Part 2}, pages = {110 -- 121}, publisher = {Springer}, title = {{Games with a weak adversary}}, doi = {10.1007/978-3-662-43951-7_10}, volume = {8573}, year = {2014}, } @article{2164, abstract = {Neuronal ectopia, such as granule cell dispersion (GCD) in temporal lobe epilepsy (TLE), has been assumed to result from a migration defect during development. Indeed, recent studies reported that aberrant migration of neonatal-generated dentate granule cells (GCs) increased the risk to develop epilepsy later in life. On the contrary, in the present study, we show that fully differentiated GCs become motile following the induction of epileptiform activity, resulting in GCD. Hippocampal slice cultures from transgenic mice expressing green fluorescent protein in differentiated, but not in newly generated GCs, were incubated with the glutamate receptor agonist kainate (KA), which induced GC burst activity and GCD. Using real-time microscopy, we observed that KA-exposed, differentiated GCs translocated their cell bodies and changed their dendritic organization. As found in human TLE, KA application was associated with decreased expression of the extracellular matrix protein Reelin, particularly in hilar interneurons. Together these findings suggest that KA-induced motility of differentiated GCs contributes to the development of GCD and establish slice cultures as a model to study neuronal changes induced by epileptiform activity. }, author = {Chai, Xuejun and Münzner, Gert and Zhao, Shanting and Tinnes, Stefanie and Kowalski, Janina and Häussler, Ute and Young, Christina and Haas, Carola and Frotscher, Michael}, journal = {Cerebral Cortex}, number = {8}, pages = {2130 -- 2140}, publisher = {Oxford University Press}, title = {{Epilepsy-induced motility of differentiated neurons}}, doi = {10.1093/cercor/bht067}, volume = {24}, year = {2014}, } @inproceedings{2167, abstract = {Model-based testing is a promising technology for black-box software and hardware testing, in which test cases are generated automatically from high-level specifications. Nowadays, systems typically consist of multiple interacting components and, due to their complexity, testing presents a considerable portion of the effort and cost in the design process. Exploiting the compositional structure of system specifications can considerably reduce the effort in model-based testing. Moreover, inferring properties about the system from testing its individual components allows the designer to reduce the amount of integration testing. In this paper, we study compositional properties of the ioco-testing theory. We propose a new approach to composition and hiding operations, inspired by contract-based design and interface theories. These operations preserve behaviors that are compatible under composition and hiding, and prune away incompatible ones. The resulting specification characterizes the input sequences for which the unit testing of components is sufficient to infer the correctness of component integration without the need for further tests. We provide a methodology that uses these results to minimize integration testing effort, but also to detect potential weaknesses in specifications. While we focus on asynchronous models and the ioco conformance relation, the resulting methodology can be applied to a broader class of systems.}, author = {Daca, Przemyslaw and Henzinger, Thomas A and Krenn, Willibald and Nickovic, Dejan}, booktitle = {IEEE 7th International Conference on Software Testing, Verification and Validation}, isbn = {978-1-4799-2255-0}, issn = {2159-4848}, location = {Cleveland, USA}, publisher = {IEEE}, title = {{Compositional specifications for IOCO testing}}, doi = {10.1109/ICST.2014.50}, year = {2014}, } @article{2168, abstract = {Many species have an essentially continuous distribution in space, in which there are no natural divisions between randomly mating subpopulations. Yet, the standard approach to modelling these populations is to impose an arbitrary grid of demes, adjusting deme sizes and migration rates in an attempt to capture the important features of the population. Such indirect methods are required because of the failure of the classical models of isolation by distance, which have been shown to have major technical flaws. A recently introduced model of extinction and recolonisation in two dimensions solves these technical problems, and provides a rigorous technical foundation for the study of populations evolving in a spatial continuum. The coalescent process for this model is simply stated, but direct simulation is very inefficient for large neighbourhood sizes. We present efficient and exact algorithms to simulate this coalescent process for arbitrary sample sizes and numbers of loci, and analyse these algorithms in detail.}, author = {Kelleher, Jerome and Etheridge, Alison and Barton, Nicholas H}, journal = {Theoretical Population Biology}, pages = {13 -- 23}, publisher = {Academic Press}, title = {{Coalescent simulation in continuous space: Algorithms for large neighbourhood size}}, doi = {10.1016/j.tpb.2014.05.001}, volume = {95}, year = {2014}, } @article{2169, author = {Barton, Nicholas H and Novak, Sebastian and Paixao, Tiago}, journal = {PNAS}, number = {29}, pages = {10398 -- 10399}, publisher = {National Academy of Sciences}, title = {{Diverse forms of selection in evolution and computer science}}, doi = {10.1073/pnas.1410107111}, volume = {111}, year = {2014}, } @article{2170, abstract = { Short-read sequencing technologies have in principle made it feasible to draw detailed inferences about the recent history of any organism. In practice, however, this remains challenging due to the difficulty of genome assembly in most organisms and the lack of statistical methods powerful enough to discriminate between recent, nonequilibrium histories. We address both the assembly and inference challenges. We develop a bioinformatic pipeline for generating outgroup-rooted alignments of orthologous sequence blocks from de novo low-coverage short-read data for a small number of genomes, and show how such sequence blocks can be used to fit explicit models of population divergence and admixture in a likelihood framework. To illustrate our approach, we reconstruct the Pleistocene history of an oak-feeding insect (the oak gallwasp Biorhiza pallida), which, in common with many other taxa, was restricted during Pleistocene ice ages to a longitudinal series of southern refugia spanning the Western Palaearctic. Our analysis of sequence blocks sampled from a single genome from each of three major glacial refugia reveals support for an unexpected history dominated by recent admixture. Despite the fact that 80% of the genome is affected by admixture during the last glacial cycle, we are able to infer the deeper divergence history of these populations. These inferences are robust to variation in block length, mutation model and the sampling location of individual genomes within refugia. This combination of de novo assembly and numerical likelihood calculation provides a powerful framework for estimating recent population history that can be applied to any organism without the need for prior genetic resources.}, author = {Hearn, Jack and Stone, Graham and Bunnefeld, Lynsey and Nicholls, James and Barton, Nicholas H and Lohse, Konrad}, journal = {Molecular Ecology}, number = {1}, pages = {198 -- 211}, publisher = {Wiley-Blackwell}, title = {{Likelihood-based inference of population history from low-coverage de novo genome assemblies}}, doi = {10.1111/mec.12578}, volume = {23}, year = {2014}, } @inproceedings{2171, abstract = {We present LS-CRF, a new method for training cyclic Conditional Random Fields (CRFs) from large datasets that is inspired by classical closed-form expressions for the maximum likelihood parameters of a generative graphical model with tree topology. Training a CRF with LS-CRF requires only solving a set of independent regression problems, each of which can be solved efficiently in closed form or by an iterative solver. This makes LS-CRF orders of magnitude faster than classical CRF training based on probabilistic inference, and at the same time more flexible and easier to implement than other approximate techniques, such as pseudolikelihood or piecewise training. We apply LS-CRF to the task of semantic image segmentation, showing that it achieves on par accuracy to other training techniques at higher speed, thereby allowing efficient CRF training from very large training sets. For example, training a linearly parameterized pairwise CRF on 150,000 images requires less than one hour on a modern workstation.}, author = {Kolesnikov, Alexander and Guillaumin, Matthieu and Ferrari, Vittorio and Lampert, Christoph}, booktitle = {Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)}, editor = {Fleet, David and Pajdla, Tomas and Schiele, Bernt and Tuytelaars, Tinne}, location = {Zurich, Switzerland}, number = {PART 3}, pages = {550 -- 565}, publisher = {Springer}, title = {{Closed-form approximate CRF training for scalable image segmentation}}, doi = {10.1007/978-3-319-10578-9_36}, volume = {8691}, year = {2014}, } @inproceedings{2172, abstract = {Fisher Kernels and Deep Learning were two developments with significant impact on large-scale object categorization in the last years. Both approaches were shown to achieve state-of-the-art results on large-scale object categorization datasets, such as ImageNet. Conceptually, however, they are perceived as very different and it is not uncommon for heated debates to spring up when advocates of both paradigms meet at conferences or workshops. In this work, we emphasize the similarities between both architectures rather than their differences and we argue that such a unified view allows us to transfer ideas from one domain to the other. As a concrete example we introduce a method for learning a support vector machine classifier with Fisher kernel at the same time as a task-specific data representation. We reinterpret the setting as a multi-layer feed forward network. Its final layer is the classifier, parameterized by a weight vector, and the two previous layers compute Fisher vectors, parameterized by the coefficients of a Gaussian mixture model. We introduce a gradient descent based learning algorithm that, in contrast to other feature learning techniques, is not just derived from intuition or biological analogy, but has a theoretical justification in the framework of statistical learning theory. Our experiments show that the new training procedure leads to significant improvements in classification accuracy while preserving the modularity and geometric interpretability of a support vector machine setup.}, author = {Sydorov, Vladyslav and Sakurada, Mayu and Lampert, Christoph}, booktitle = {Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition}, location = {Columbus, USA}, pages = {1402 -- 1409}, publisher = {IEEE}, title = {{Deep Fisher Kernels – End to end learning of the Fisher Kernel GMM parameters}}, doi = {10.1109/CVPR.2014.182}, year = {2014}, } @inproceedings{2173, abstract = {In this work we introduce a new approach to co-classification, i.e. the task of jointly classifying multiple, otherwise independent, data samples. The method we present, named CoConut, is based on the idea of adding a regularizer in the label space to encode certain priors on the resulting labelings. A regularizer that encourages labelings that are smooth across the test set, for instance, can be seen as a test-time variant of the cluster assumption, which has been proven useful at training time in semi-supervised learning. A regularizer that introduces a preference for certain class proportions can be regarded as a prior distribution on the class labels. CoConut can build on existing classifiers without making any assumptions on how they were obtained and without the need to re-train them. The use of a regularizer adds a new level of flexibility. It allows the integration of potentially new information at test time, even in other modalities than what the classifiers were trained on. We evaluate our framework on six datasets, reporting a clear performance gain in classification accuracy compared to the standard classification setup that predicts labels for each test sample separately. }, author = {Khamis, Sameh and Lampert, Christoph}, booktitle = {Proceedings of the British Machine Vision Conference 2014}, location = {Nottingham, UK}, publisher = {BMVA Press}, title = {{CoConut: Co-classification with output space regularization}}, year = {2014}, } @article{2174, abstract = {When polygenic traits are under stabilizing selection, many different combinations of alleles allow close adaptation to the optimum. If alleles have equal effects, all combinations that result in the same deviation from the optimum are equivalent. Furthermore, the genetic variance that is maintained by mutation-selection balance is 2μ/S per locus, where μ is the mutation rate and S the strength of stabilizing selection. In reality, alleles vary in their effects, making the fitness landscape asymmetric and complicating analysis of the equilibria. We show that that the resulting genetic variance depends on the fraction of alleles near fixation, which contribute by 2μ/S, and on the total mutational effects of alleles that are at intermediate frequency. The inpplayfi between stabilizing selection and mutation leads to a sharp transition: alleles with effects smaller than a threshold value of 2 remain polymorphic, whereas those with larger effects are fixed. The genetic load in equilibrium is less than for traits of equal effects, and the fitness equilibria are more similar. We find p the optimum is displaced, alleles with effects close to the threshold value sweep first, and their rate of increase is bounded by Long-term response leads in general to well-adapted traits, unlike the case of equal effects that often end up at a suboptimal fitness peak. However, the particular peaks to which the populations converge are extremely sensitive to the initial states and to the speed of the shift of the optimum trait value.}, author = {De Vladar, Harold and Barton, Nicholas H}, journal = {Genetics}, number = {2}, pages = {749 -- 767}, publisher = {Genetics Society of America}, title = {{Stability and response of polygenic traits to stabilizing selection and mutation}}, doi = {10.1534/genetics.113.159111}, volume = {197}, year = {2014}, } @article{2175, abstract = {The cerebral cortex, the seat of our cognitive abilities, is composed of an intricate network of billions of excitatory projection and inhibitory interneurons. Postmitotic cortical neurons are generated by a diverse set of neural stem cell progenitors within dedicated zones and defined periods of neurogenesis during embryonic development. Disruptions in neurogenesis can lead to alterations in the neuronal cytoarchitecture, which is thought to represent a major underlying cause for several neurological disorders, including microcephaly, autism and epilepsy. Although a number of signaling pathways regulating neurogenesis have been described, the precise cellular and molecular mechanisms regulating the functional neural stem cell properties in cortical neurogenesis remain unclear. Here, we discuss the most up-to-date strategies to monitor the fundamental mechanistic parameters of neuronal progenitor proliferation, and recent advances deciphering the logic and dynamics of neurogenesis.}, author = {Postiglione, Maria P and Hippenmeyer, Simon}, issn = {1748-6971}, journal = {Future Neurology}, number = {3}, pages = {323 -- 340}, publisher = {Future Science Group}, title = {{Monitoring neurogenesis in the cerebral cortex: an update}}, doi = {10.2217/fnl.14.18}, volume = {9}, year = {2014}, } @article{2176, abstract = {Electron microscopy (EM) allows for the simultaneous visualization of all tissue components at high resolution. However, the extent to which conventional aldehyde fixation and ethanol dehydration of the tissue alter the fine structure of cells and organelles, thereby preventing detection of subtle structural changes induced by an experiment, has remained an issue. Attempts have been made to rapidly freeze tissue to preserve native ultrastructure. Shock-freezing of living tissue under high pressure (high-pressure freezing, HPF) followed by cryosubstitution of the tissue water avoids aldehyde fixation and dehydration in ethanol; the tissue water is immobilized in â ̂1/450 ms, and a close-to-native fine structure of cells, organelles and molecules is preserved. Here we describe a protocol for HPF that is useful to monitor ultrastructural changes associated with functional changes at synapses in the brain but can be applied to many other tissues as well. The procedure requires a high-pressure freezer and takes a minimum of 7 d but can be paused at several points.}, author = {Studer, Daniel and Zhao, Shanting and Chai, Xuejun and Jonas, Peter M and Graber, Werner and Nestel, Sigrun and Frotscher, Michael}, journal = {Nature Protocols}, number = {6}, pages = {1480 -- 1495}, publisher = {Nature Publishing Group}, title = {{Capture of activity-induced ultrastructural changes at synapses by high-pressure freezing of brain tissue}}, doi = {10.1038/nprot.2014.099}, volume = {9}, year = {2014}, } @inproceedings{2177, abstract = {We give evidence for the difficulty of computing Betti numbers of simplicial complexes over a finite field. We do this by reducing the rank computation for sparse matrices with to non-zero entries to computing Betti numbers of simplicial complexes consisting of at most a constant times to simplices. Together with the known reduction in the other direction, this implies that the two problems have the same computational complexity.}, author = {Edelsbrunner, Herbert and Parsa, Salman}, booktitle = {Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms}, location = {Portland, USA}, pages = {152 -- 160}, publisher = {SIAM}, title = {{On the computational complexity of betti numbers reductions from matrix rank}}, doi = {10.1137/1.9781611973402.11}, year = {2014}, } @article{2178, abstract = {We consider the three-state toric homogeneous Markov chain model (THMC) without loops and initial parameters. At time T, the size of the design matrix is 6 × 3 · 2T-1 and the convex hull of its columns is the model polytope. We study the behavior of this polytope for T ≥ 3 and we show that it is defined by 24 facets for all T ≥ 5. Moreover, we give a complete description of these facets. From this, we deduce that the toric ideal associated with the design matrix is generated by binomials of degree at most 6. Our proof is based on a result due to Sturmfels, who gave a bound on the degree of the generators of a toric ideal, provided the normality of the corresponding toric variety. In our setting, we established the normality of the toric variety associated to the THMC model by studying the geometric properties of the model polytope.}, author = {Haws, David and Martin Del Campo Sanchez, Abraham and Takemura, Akimichi and Yoshida, Ruriko}, journal = {Beitrage zur Algebra und Geometrie}, number = {1}, pages = {161 -- 188}, publisher = {Springer}, title = {{Markov degree of the three-state toric homogeneous Markov chain model}}, doi = {10.1007/s13366-013-0178-y}, volume = {55}, year = {2014}, }