@unpublished{14601, abstract = {In this work, we address the problem of learning provably stable neural network policies for stochastic control systems. While recent work has demonstrated the feasibility of certifying given policies using martingale theory, the problem of how to learn such policies is little explored. Here, we study the effectiveness of jointly learning a policy together with a martingale certificate that proves its stability using a single learning algorithm. We observe that the joint optimization problem becomes easily stuck in local minima when starting from a randomly initialized policy. Our results suggest that some form of pre-training of the policy is required for the joint optimization to repair and verify the policy successfully.}, author = {Zikelic, Dorde and Lechner, Mathias and Chatterjee, Krishnendu and Henzinger, Thomas A}, booktitle = {arXiv}, title = {{Learning stabilizing policies in stochastic control systems}}, doi = {10.48550/arXiv.2205.11991}, year = {2022}, } @article{7577, abstract = {Weak convergence of inertial iterative method for solving variational inequalities is the focus of this paper. The cost function is assumed to be non-Lipschitz and monotone. We propose a projection-type method with inertial terms and give weak convergence analysis under appropriate conditions. Some test results are performed and compared with relevant methods in the literature to show the efficiency and advantages given by our proposed methods.}, author = {Shehu, Yekini and Iyiola, Olaniyi S.}, issn = {1563-504X}, journal = {Applicable Analysis}, number = {1}, pages = {192--216}, publisher = {Taylor & Francis}, title = {{Weak convergence for variational inequalities with inertial-type method}}, doi = {10.1080/00036811.2020.1736287}, volume = {101}, year = {2022}, } @article{7791, abstract = {Extending a result of Milena Radnovic and Serge Tabachnikov, we establish conditionsfor two different non-symmetric norms to define the same billiard reflection law.}, author = {Akopyan, Arseniy and Karasev, Roman}, issn = {2199-6768}, journal = {European Journal of Mathematics}, number = {4}, pages = {1309 -- 1312}, publisher = {Springer Nature}, title = {{When different norms lead to same billiard trajectories?}}, doi = {10.1007/s40879-020-00405-0}, volume = {8}, year = {2022}, } @unpublished{8125, abstract = {Context, such as behavioral state, is known to modulate memory formation and retrieval, but is usually ignored in associative memory models. Here, we propose several types of contextual modulation for associative memory networks that greatly increase their performance. In these networks, context inactivates specific neurons and connections, which modulates the effective connectivity of the network. Memories are stored only by the active components, thereby reducing interference from memories acquired in other contexts. Such networks exhibit several beneficial characteristics, including enhanced memory capacity, high robustness to noise, increased robustness to memory overloading, and better memory retention during continual learning. Furthermore, memories can be biased to have different relative strengths, or even gated on or off, according to contextual cues, providing a candidate model for cognitive control of memory and efficient memory search. An external context-encoding network can dynamically switch the memory network to a desired state, which we liken to experimentally observed contextual signals in prefrontal cortex and hippocampus. Overall, our work illustrates the benefits of organizing memory around context, and provides an important link between behavioral studies of memory and mechanistic details of neural circuits.SIGNIFICANCEMemory is context dependent — both encoding and recall vary in effectiveness and speed depending on factors like location and brain state during a task. We apply this idea to a simple computational model of associative memory through contextual gating of neurons and synaptic connections. Intriguingly, this results in several advantages, including vastly enhanced memory capacity, better robustness, and flexible memory gating. Our model helps to explain (i) how gating and inhibition contribute to memory processes, (ii) how memory access dynamically changes over time, and (iii) how context representations, such as those observed in hippocampus and prefrontal cortex, may interact with and control memory processes.}, author = {Podlaski, William F. and Agnes, Everton J. and Vogels, Tim P}, booktitle = {bioRxiv}, publisher = {Cold Spring Harbor Laboratory}, title = {{High capacity and dynamic accessibility in associative memory networks with context-dependent neuronal and synaptic gating}}, doi = {10.1101/2020.01.08.898528}, year = {2022}, } @article{9199, abstract = {We associate a certain tensor product lattice to any primitive integer lattice and ask about its typical shape. These lattices are related to the tangent bundle of Grassmannians and their study is motivated by Peyre's programme on "freeness" for rational points of bounded height on Fano varieties.}, author = {Browning, Timothy D and Horesh, Tal and Wilsch, Florian Alexander}, issn = {1944-7833}, journal = {Algebra & Number Theory}, number = {10}, pages = {2385--2407}, publisher = {Mathematical Sciences Publishers}, title = {{Equidistribution and freeness on Grassmannians}}, doi = {10.2140/ant.2022.16.2385}, volume = {16}, year = {2022}, } @article{9311, abstract = {Partially observable Markov decision processes (POMDPs) are standard models for dynamic systems with probabilistic and nondeterministic behaviour in uncertain environments. We prove that in POMDPs with long-run average objective, the decision maker has approximately optimal strategies with finite memory. This implies notably that approximating the long-run value is recursively enumerable, as well as a weak continuity property of the value with respect to the transition function. }, author = {Chatterjee, Krishnendu and Saona Urmeneta, Raimundo J and Ziliotto, Bruno}, issn = {1526-5471}, journal = {Mathematics of Operations Research}, keywords = {Management Science and Operations Research, General Mathematics, Computer Science Applications}, number = {1}, pages = {100--119}, publisher = {Institute for Operations Research and the Management Sciences}, title = {{Finite-memory strategies in POMDPs with long-run average objectives}}, doi = {10.1287/moor.2020.1116}, volume = {47}, year = {2022}, } @article{9364, abstract = {Let t : Fp → C be a complex valued function on Fp. A classical problem in analytic number theory is bounding the maximum M(t) := max 0≤H

0 there exists a large subset of a ∈ F×p such that for kl a,1,p : x → e((ax+x) / p) we have M(kla,1,p) ≥ (1−ε/√2π + o(1)) log log p, as p→∞. Finally, we prove a result on the growth of the moments of {M (kla,1,p)}a∈F×p. 2020 Mathematics Subject Classification: 11L03, 11T23 (Primary); 14F20, 60F10 (Secondary).}, author = {Bonolis, Dante}, issn = {1469-8064}, journal = {Mathematical Proceedings of the Cambridge Philosophical Society}, number = {3}, pages = {563 -- 590}, publisher = {Cambridge University Press}, title = {{On the size of the maximum of incomplete Kloosterman sums}}, doi = {10.1017/S030500412100030X}, volume = {172}, year = {2022}, } @article{9649, abstract = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f : Rd → Rd−n. A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation T of the ambient space Rd. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine triangulation T . This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary.}, author = {Boissonnat, Jean-Daniel and Wintraecken, Mathijs}, issn = {1615-3383}, journal = {Foundations of Computational Mathematics }, pages = {967--1012}, publisher = {Springer Nature}, title = {{The topological correctness of PL approximations of isomanifolds}}, doi = {10.1007/s10208-021-09520-0}, volume = {22}, year = {2022}, } @article{9794, abstract = {Lymph nodes (LNs) comprise two main structural elements: fibroblastic reticular cells that form dedicated niches for immune cell interaction and capsular fibroblasts that build a shell around the organ. Immunological challenge causes LNs to increase more than tenfold in size within a few days. Here, we characterized the biomechanics of LN swelling on the cellular and organ scale. We identified lymphocyte trapping by influx and proliferation as drivers of an outward pressure force, causing fibroblastic reticular cells of the T-zone (TRCs) and their associated conduits to stretch. After an initial phase of relaxation, TRCs sensed the resulting strain through cell matrix adhesions, which coordinated local growth and remodeling of the stromal network. While the expanded TRC network readopted its typical configuration, a massive fibrotic reaction of the organ capsule set in and countered further organ expansion. Thus, different fibroblast populations mechanically control LN swelling in a multitier fashion.}, author = {Assen, Frank P and Abe, Jun and Hons, Miroslav and Hauschild, Robert and Shamipour, Shayan and Kaufmann, Walter and Costanzo, Tommaso and Krens, Gabriel and Brown, Markus and Ludewig, Burkhard and Hippenmeyer, Simon and Heisenberg, Carl-Philipp J and Weninger, Wolfgang and Hannezo, Edouard B and Luther, Sanjiv A. and Stein, Jens V. and Sixt, Michael K}, issn = {1529-2916}, journal = {Nature Immunology}, pages = {1246--1255}, publisher = {Springer Nature}, title = {{Multitier mechanics control stromal adaptations in swelling lymph nodes}}, doi = {10.1038/s41590-022-01257-4}, volume = {23}, year = {2022}, } @article{9955, abstract = {Neurons can change their classical neurotransmitters during ontogeny, sometimes going through stages of dual release. Here, we explored the development of the neurotransmitter identity of neurons of the avian nucleus isthmi parvocellularis (Ipc), whose axon terminals are retinotopically arranged in the optic tectum (TeO) and exert a focal gating effect upon the ascending transmission of retinal inputs. Although cholinergic and glutamatergic markers are both found in Ipc neurons and terminals of adult pigeons and chicks, the mRNA expression of the vesicular acetylcholine transporter, VAChT, is weak or absent. To explore how the Ipc neurotransmitter identity is established during ontogeny, we analyzed the expression of mRNAs coding for cholinergic (ChAT, VAChT, and CHT) and glutamatergic (VGluT2 and VGluT3) markers in chick embryos at different developmental stages. We found that between E12 and E18, Ipc neurons expressed all cholinergic mRNAs and also VGluT2 mRNA; however, from E16 through posthatch stages, VAChT mRNA expression was specifically diminished. Our ex vivo deposits of tracer crystals and intracellular filling experiments revealed that Ipc axons exhibit a mature paintbrush morphology late in development, experiencing marked morphological transformations during the period of presumptive dual vesicular transmitter release. Additionally, although ChAT protein immunoassays increasingly label the growing Ipc axon, this labeling was consistently restricted to sparse portions of the terminal branches. Combined, these results suggest that the synthesis of glutamate and acetylcholine, and their vesicular release, is complexly linked to the developmental processes of branching, growing and remodeling of these unique axons.}, author = {Reyes‐Pinto, Rosana and Ferrán, José L. and Vega Zuniga, Tomas A and González‐Cabrera, Cristian and Luksch, Harald and Mpodozis, Jorge and Puelles, Luis and Marín, Gonzalo J.}, issn = {1096-9861}, journal = {Journal of Comparative Neurology}, number = {2}, pages = {553--573}, publisher = {Wiley}, title = {{Change in the neurochemical signature and morphological development of the parvocellular isthmic projection to the avian tectum}}, doi = {10.1002/cne.25229}, volume = {530}, year = {2022}, } @article{9977, abstract = {For a Seifert fibered homology sphere X we show that the q-series invariant Zˆ0(X; q) introduced by Gukov-Pei-Putrov-Vafa, is a resummation of the Ohtsuki series Z0(X). We show that for every even k ∈ N there exists a full asymptotic expansion of Zˆ0(X; q) for q tending to e 2πi/k, and in particular that the limit Zˆ0(X; e 2πi/k) exists and is equal to the WRT quantum invariant τk(X). We show that the poles of the Borel transform of Z0(X) coincide with the classical complex Chern-Simons values, which we further show classifies the corresponding components of the moduli space of flat SL(2, C)-connections.}, author = {Mistegaard, William and Andersen, Jørgen Ellegaard}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {2}, pages = {709--764}, publisher = {Wiley}, title = {{Resurgence analysis of quantum invariants of Seifert fibered homology spheres}}, doi = {10.1112/jlms.12506}, volume = {105}, year = {2022}, } @article{10000, abstract = {Inhibition or targeted deletion of histone deacetylase 3 (HDAC3) is neuroprotective in a variety neurodegenerative conditions, including retinal ganglion cells (RGCs) after acute optic nerve damage. Consistent with this, induced HDAC3 expression in cultured cells shows selective toxicity to neurons. Despite an established role for HDAC3 in neuronal pathology, little is known regarding the mechanism of this pathology.}, author = {Schmitt, Heather M. and Fehrman, Rachel L. and Maes, Margaret E and Yang, Huan and Guo, Lian Wang and Schlamp, Cassandra L. and Pelzel, Heather R. and Nickells, Robert W.}, issn = {1552-5783}, journal = {Investigative Ophthalmology and Visual Science}, number = {10}, publisher = {Association for Research in Vision and Ophthalmology}, title = {{Increased susceptibility and intrinsic apoptotic signaling in neurons by induced HDAC3 expression}}, doi = {10.1167/IOVS.62.10.14}, volume = {62}, year = {2021}, } @inproceedings{10002, abstract = {We present a faster symbolic algorithm for the following central problem in probabilistic verification: Compute the maximal end-component (MEC) decomposition of Markov decision processes (MDPs). This problem generalizes the SCC decomposition problem of graphs and closed recurrent sets of Markov chains. The model of symbolic algorithms is widely used in formal verification and model-checking, where access to the input model is restricted to only symbolic operations (e.g., basic set operations and computation of one-step neighborhood). For an input MDP with n vertices and m edges, the classical symbolic algorithm from the 1990s for the MEC decomposition requires O(n2) symbolic operations and O(1) symbolic space. The only other symbolic algorithm for the MEC decomposition requires O(nm−−√) symbolic operations and O(m−−√) symbolic space. A main open question is whether the worst-case O(n2) bound for symbolic operations can be beaten. We present a symbolic algorithm that requires O˜(n1.5) symbolic operations and O˜(n−−√) symbolic space. Moreover, the parametrization of our algorithm provides a trade-off between symbolic operations and symbolic space: for all 0<ϵ≤1/2 the symbolic algorithm requires O˜(n2−ϵ) symbolic operations and O˜(nϵ) symbolic space ( O˜ hides poly-logarithmic factors). Using our techniques we present faster algorithms for computing the almost-sure winning regions of ω -regular objectives for MDPs. We consider the canonical parity objectives for ω -regular objectives, and for parity objectives with d -priorities we present an algorithm that computes the almost-sure winning region with O˜(n2−ϵ) symbolic operations and O˜(nϵ) symbolic space, for all 0<ϵ≤1/2 .}, author = {Chatterjee, Krishnendu and Dvorak, Wolfgang and Henzinger, Monika H and Svozil, Alexander}, booktitle = {Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science}, isbn = {978-1-6654-4896-3}, issn = {1043-6871}, keywords = {Computer science, Computational modeling, Markov processes, Probabilistic logic, Formal verification, Game Theory}, location = {Rome, Italy}, pages = {1--13}, publisher = {Institute of Electrical and Electronics Engineers}, title = {{Symbolic time and space tradeoffs for probabilistic verification}}, doi = {10.1109/LICS52264.2021.9470739}, year = {2021}, } @inproceedings{10004, abstract = {Markov chains are the de facto finite-state model for stochastic dynamical systems, and Markov decision processes (MDPs) extend Markov chains by incorporating non-deterministic behaviors. Given an MDP and rewards on states, a classical optimization criterion is the maximal expected total reward where the MDP stops after T steps, which can be computed by a simple dynamic programming algorithm. We consider a natural generalization of the problem where the stopping times can be chosen according to a probability distribution, such that the expected stopping time is T, to optimize the expected total reward. Quite surprisingly we establish inter-reducibility of the expected stopping-time problem for Markov chains with the Positivity problem (which is related to the well-known Skolem problem), for which establishing either decidability or undecidability would be a major breakthrough. Given the hardness of the exact problem, we consider the approximate version of the problem: we show that it can be solved in exponential time for Markov chains and in exponential space for MDPs.}, author = {Chatterjee, Krishnendu and Doyen, Laurent}, booktitle = {Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science}, isbn = {978-1-6654-4896-3}, issn = {1043-6871}, keywords = {Computer science, Heuristic algorithms, Memory management, Automata, Markov processes, Probability distribution, Complexity theory}, location = {Rome, Italy}, pages = {1--13}, publisher = {Institute of Electrical and Electronics Engineers}, title = {{Stochastic processes with expected stopping time}}, doi = {10.1109/LICS52264.2021.9470595}, year = {2021}, } @article{10005, abstract = {We study systems of nonlinear partial differential equations of parabolic type, in which the elliptic operator is replaced by the first-order divergence operator acting on a flux function, which is related to the spatial gradient of the unknown through an additional implicit equation. This setting, broad enough in terms of applications, significantly expands the paradigm of nonlinear parabolic problems. Formulating four conditions concerning the form of the implicit equation, we first show that these conditions describe a maximal monotone p-coercive graph. We then establish the global-in-time and large-data existence of a (weak) solution and its uniqueness. To this end, we adopt and significantly generalize Minty’s method of monotone mappings. A unified theory, containing several novel tools, is developed in a way to be tractable from the point of view of numerical approximations.}, author = {Bulíček, Miroslav and Maringová, Erika and Málek, Josef}, issn = {1793-6314}, journal = {Mathematical Models and Methods in Applied Sciences}, keywords = {Nonlinear parabolic systems, implicit constitutive theory, weak solutions, existence, uniqueness}, number = {09}, publisher = {World Scientific}, title = {{On nonlinear problems of parabolic type with implicit constitutive equations involving flux}}, doi = {10.1142/S0218202521500457}, volume = {31}, year = {2021}, } @phdthesis{10007, abstract = {The present thesis is concerned with the derivation of weak-strong uniqueness principles for curvature driven interface evolution problems not satisfying a comparison principle. The specific examples being treated are two-phase Navier-Stokes flow with surface tension, modeling the evolution of two incompressible, viscous and immiscible fluids separated by a sharp interface, and multiphase mean curvature flow, which serves as an idealized model for the motion of grain boundaries in an annealing polycrystalline material. Our main results - obtained in joint works with Julian Fischer, Tim Laux and Theresa M. Simon - state that prior to the formation of geometric singularities due to topology changes, the weak solution concept of Abels (Interfaces Free Bound. 9, 2007) to two-phase Navier-Stokes flow with surface tension and the weak solution concept of Laux and Otto (Calc. Var. Partial Differential Equations 55, 2016) to multiphase mean curvature flow (for networks in R^2 or double bubbles in R^3) represents the unique solution to these interface evolution problems within the class of classical solutions, respectively. To the best of the author's knowledge, for interface evolution problems not admitting a geometric comparison principle the derivation of a weak-strong uniqueness principle represented an open problem, so that the works contained in the present thesis constitute the first positive results in this direction. The key ingredient of our approach consists of the introduction of a novel concept of relative entropies for a class of curvature driven interface evolution problems, for which the associated energy contains an interfacial contribution being proportional to the surface area of the evolving (network of) interface(s). The interfacial part of the relative entropy gives sufficient control on the interface error between a weak and a classical solution, and its time evolution can be computed, at least in principle, for any energy dissipating weak solution concept. A resulting stability estimate for the relative entropy essentially entails the above mentioned weak-strong uniqueness principles. The present thesis contains a detailed introduction to our relative entropy approach, which in particular highlights potential applications to other problems in curvature driven interface evolution not treated in this thesis.}, author = {Hensel, Sebastian}, issn = {2663-337X}, pages = {300}, publisher = {Institute of Science and Technology Austria}, title = {{Curvature driven interface evolution: Uniqueness properties of weak solution concepts}}, doi = {10.15479/at:ista:10007}, year = {2021}, } @unpublished{10011, abstract = {We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] - and hence any classical solution to mean curvature flow - is unique in the class of our new varifold solutions. This is in sharp contrast to the case of Brakke flows, which a priori may disappear at any given time and are therefore fatally non-unique. Finally, we propose an extension of the solution concept to the multi-phase case which is at least guaranteed to satisfy a weak-strong uniqueness principle.}, author = {Hensel, Sebastian and Laux, Tim}, booktitle = {arXiv}, keywords = {Mean curvature flow, gradient flows, varifolds, weak solutions, weak-strong uniqueness, calibrated geometry, gradient-flow calibrations}, title = {{A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness}}, doi = {10.48550/arXiv.2109.04233}, year = {2021}, } @unpublished{10013, abstract = {We derive a weak-strong uniqueness principle for BV solutions to multiphase mean curvature flow of triple line clusters in three dimensions. Our proof is based on the explicit construction of a gradient-flow calibration in the sense of the recent work of Fischer et al. [arXiv:2003.05478] for any such cluster. This extends the two-dimensional construction to the three-dimensional case of surfaces meeting along triple junctions.}, author = {Hensel, Sebastian and Laux, Tim}, booktitle = {arXiv}, title = {{Weak-strong uniqueness for the mean curvature flow of double bubbles}}, doi = {10.48550/arXiv.2108.01733}, year = {2021}, } @article{10015, abstract = {Auxin plays a dual role in growth regulation and, depending on the tissue and concentration of the hormone, it can either promote or inhibit division and expansion processes in plants. Recent studies have revealed that, beyond transcriptional reprogramming, alternative auxincontrolled mechanisms regulate root growth. Here, we explored the impact of different concentrations of the synthetic auxin NAA that establish growth-promoting and -repressing conditions on the root tip proteome and phosphoproteome, generating a unique resource. From the phosphoproteome data, we pinpointed (novel) growth regulators, such as the RALF34-THE1 module. Our results, together with previously published studies, suggest that auxin, H+-ATPases, cell wall modifications and cell wall sensing receptor-like kinases are tightly embedded in a pathway regulating cell elongation. Furthermore, our study assigned a novel role to MKK2 as a regulator of primary root growth and a (potential) regulator of auxin biosynthesis and signalling, and suggests the importance of the MKK2 Thr31 phosphorylation site for growth regulation in the Arabidopsis root tip.}, author = {Nikonorova, N and Murphy, E and Fonseca de Lima, CF and Zhu, S and van de Cotte, B and Vu, LD and Balcerowicz, D and Li, Lanxin and Kong, X and De Rop, G and Beeckman, T and Friml, Jiří and Vissenberg, K and Morris, PC and Ding, Z and De Smet, I}, issn = {2073-4409}, journal = {Cells}, keywords = {primary root, (phospho)proteomics, auxin, (receptor) kinase}, publisher = {MDPI}, title = {{The Arabidopsis root tip (phospho)proteomes at growth-promoting versus growth-repressing conditions reveal novel root growth regulators}}, doi = {10.3390/cells10071665}, volume = {10}, year = {2021}, } @article{10023, abstract = {We study the temporal dissipation of variance and relative entropy for ergodic Markov Chains in continuous time, and compute explicitly the corresponding dissipation rates. These are identified, as is well known, in the case of the variance in terms of an appropriate Hilbertian norm; and in the case of the relative entropy, in terms of a Dirichlet form which morphs into a version of the familiar Fisher information under conditions of detailed balance. Here we obtain trajectorial versions of these results, valid along almost every path of the random motion and most transparent in the backwards direction of time. Martingale arguments and time reversal play crucial roles, as in the recent work of Karatzas, Schachermayer and Tschiderer for conservative diffusions. Extensions are developed to general “convex divergences” and to countable state-spaces. The steepest descent and gradient flow properties for the variance, the relative entropy, and appropriate generalizations, are studied along with their respective geometries under conditions of detailed balance, leading to a very direct proof for the HWI inequality of Otto and Villani in the present context.}, author = {Karatzas, Ioannis and Maas, Jan and Schachermayer, Walter}, issn = {1526-7555}, journal = {Communications in Information and Systems}, keywords = {Markov Chain, relative entropy, time reversal, steepest descent, gradient flow}, number = {4}, pages = {481--536}, publisher = {International Press}, title = {{Trajectorial dissipation and gradient flow for the relative entropy in Markov chains}}, doi = {10.4310/CIS.2021.v21.n4.a1}, volume = {21}, year = {2021}, }