420 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    On Chevalley group schemes over function fields: quotients of the Bruhat-Tits building by {}\{\wp\}-arithmetic subgroups

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    Let G\mathbf{G} be a reductive Chevalley group scheme (defined over Z\mathbb{Z}). Let C\mathcal{C} be a smooth, projective, geometrically integral curve over a field F\mathbb{F}. Let \wp be a closed point on C\mathcal{C}. Let AA be the ring of functions that are regular outside {}\lbrace \wp \rbrace. The fraction field kk of AA has a discrete valuation ν=ν:k×Z\nu=\nu_{\wp}: k^{\times} \rightarrow \mathbb{Z} associated to \wp. In this work, we study the action of the group G(A) \textbf{G}(A) of AA-points of G\mathbf{G} on the Bruhat-Tits building X=X(G,k,ν)\mathcal{X}=\mathcal{X}(\textbf{G},k,\nu_\wp) in order to describe the structure of the orbit space G(A)\X \textbf{G}(A)\backslash \mathcal{X}. We obtain that this orbit space is the ``gluing'' of a closed connected CW-complex with some sector chambers. The latter are parametrized by a set depending on the Picard group of C{}\mathcal{C} \smallsetminus \{\wp\} and on the rank of G\mathbf{G}. Moreover, we observe that any rational sector face whose tip is a special vertex contains a subsector face that embeds into this orbit space. We deduce, from this description, a writing of G(A)\mathbf{G}(A) as a free product with amalgamation. We also obtain a counting of the Γ\Gamma-conjugacy classes of maximal unipotent subgroups contained in a finite index subgroup ΓG(A)\Gamma \subseteq \mathbf{G}(A), together with a description of these maximal unipotent subgroups.Comment: Comments are welcom

    Maintaining Expander Decompositions via Sparse Cuts

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    In this article, we show that the algorithm of maintaining expander decompositions in graphs undergoing edge deletions directly by removing sparse cuts repeatedly can be made efficient. Formally, for an mm-edge undirected graph GG, we say a cut (S,S)(S, \overline{S}) is ϕ\phi-sparse if EG(S,S)<ϕmin{volG(S),volG(S)}|E_G(S, \overline{S})| < \phi \cdot \min\{vol_G(S), vol_G(\overline{S})\}. A ϕ\phi-expander decomposition of GG is a partition of VV into sets X1,X2,,XkX_1, X_2, \ldots, X_k such that each cluster G[Xi]G[X_i] contains no ϕ\phi-sparse cut (meaning it is a ϕ\phi-expander) with O~(ϕm)\tilde{O}(\phi m) edges crossing between clusters. A natural way to compute a ϕ\phi-expander decomposition is to decompose clusters by ϕ\phi-sparse cuts until no such cut is contained in any cluster. We show that even in graphs undergoing edge deletions, a slight relaxation of this meta-algorithm can be implemented efficiently with amortized update time mo(1)/ϕ2m^{o(1)}/\phi^2. Our approach naturally extends to maintaining directed ϕ\phi-expander decompositions and ϕ\phi-expander hierarchies and thus gives a unifying framework while having simpler proofs than previous state-of-the-art work. In all settings, our algorithm matches the run-times of previous algorithms up to subpolynomial factors. Moreover, our algorithm provides stronger guarantees for ϕ\phi-expander decompositions. For example, for graphs undergoing edge deletions, our approach is the first to maintain a dynamic expander decomposition where each updated decomposition is a refinement of the previous decomposition, and our approach is the first to guarantee a sublinear ϕm1+o(1)\phi m^{1+o(1)} bound on the total number of edges that cross between clusters across the entire sequence of dynamic updates

    On linear, fractional, and submodular optimization

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    In this thesis, we study four fundamental problems in the theory of optimization. 1. In fractional optimization, we are interested in minimizing a ratio of two functions over some domain. A well-known technique for solving this problem is the Newton– Dinkelbach method. We propose an accelerated version of this classical method and give a new analysis using the Bregman divergence. We show how it leads to improved or simplified results in three application areas. 2. The diameter of a polyhedron is the maximum length of a shortest path between any two vertices. The circuit diameter is a relaxation of this notion, whereby shortest paths are not restricted to edges of the polyhedron. For a polyhedron in standard equality form with constraint matrix A, we prove an upper bound on the circuit diameter that is quadratic in the rank of A and logarithmic in the circuit imbalance measure of A. We also give circuit augmentation algorithms for linear programming with similar iteration complexity. 3. The correlation gap of a set function is the ratio between its multilinear and concave extensions. We present improved lower bounds on the correlation gap of a matroid rank function, parametrized by the rank and girth of the matroid. We also prove that for a weighted matroid rank function, the worst correlation gap is achieved with uniform weights. Such improved lower bounds have direct applications in submodular maximization and mechanism design. 4. The last part of this thesis concerns parity games, a problem intimately related to linear programming. A parity game is an infinite-duration game between two players on a graph. The problem of deciding the winner lies in NP and co-NP, with no known polynomial algorithm to date. Many of the fastest (quasi-polynomial) algorithms have been unified via the concept of a universal tree. We propose a strategy iteration framework which can be applied on any universal tree

    Optimal Sufficient Requirements on the Embedded Ising Problem in Polynomial Time

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    One of the central applications for quantum annealers is to find the solutions of Ising problems. Suitable Ising problems, however, need to be formulated such that they, on the one hand, respect the specific restrictions of the hardware and, on the other hand, represent the original problems which shall actually be solved. We evaluate sufficient requirements on such an embedded Ising problem analytically and transform them into a linear optimization problem. With an objective function aiming to minimize the maximal absolute problem parameter, the precision issues of the annealers are addressed. Due to the redundancy of several constraints, we can show that the formally exponentially large optimization problem can be reduced and finally solved in polynomial time for the standard embedding setting where the embedded vertices induce trees. This allows to formulate provably equivalent embedded Ising problems in a practical setup.Comment: 34 pages, 3 figure

    Efficient Distributed Decomposition and Routing Algorithms in Minor-Free Networks and Their Applications

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    In the LOCAL model, low-diameter decomposition is a useful tool in designing algorithms, as it allows us to shift from the general graph setting to the low-diameter graph setting, where brute-force information gathering can be done efficiently. Recently, Chang and Su [PODC 2022] showed that any high-conductance network excluding a fixed minor contains a high-degree vertex, so the entire graph topology can be gathered to one vertex efficiently in the CONGEST model using expander routing. Therefore, in networks excluding a fixed minor, many problems that can be solved efficiently in LOCAL via low-diameter decomposition can also be solved efficiently in CONGEST via expander decomposition. In this work, we show improved decomposition and routing algorithms for networks excluding a fixed minor in the CONGEST model. Our algorithms cost poly(logn,1/ϵ)\text{poly}(\log n, 1/\epsilon) rounds deterministically. For bounded-degree graphs, our algorithms finish in O(ϵ1logn)+ϵO(1)O(\epsilon^{-1}\log n) + \epsilon^{-O(1)} rounds. Our algorithms have a wide range of applications, including the following results in CONGEST. 1. A (1ϵ)(1-\epsilon)-approximate maximum independent set in a network excluding a fixed minor can be computed deterministically in O(ϵ1logn)+ϵO(1)O(\epsilon^{-1}\log^\ast n) + \epsilon^{-O(1)} rounds, nearly matching the Ω(ϵ1logn)\Omega(\epsilon^{-1}\log^\ast n) lower bound of Lenzen and Wattenhofer [DISC 2008]. 2. Property testing of any additive minor-closed property can be done deterministically in O(logn)O(\log n) rounds if ϵ\epsilon is a constant or O(ϵ1logn)+ϵO(1)O(\epsilon^{-1}\log n) + \epsilon^{-O(1)} rounds if the maximum degree Δ\Delta is a constant, nearly matching the Ω(ϵ1logn)\Omega(\epsilon^{-1}\log n) lower bound of Levi, Medina, and Ron [PODC 2018].Comment: To appear in PODC 202

    Disease progression and genetic risk factors in the primary tauopathies

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    The primary tauopathies are a group of progressive neurodegenerative diseases within the frontotemporal lobar degeneration spectrum (FTLD) characterised by the accumulation of misfolded, hyperphosphorylated microtubule-associated tau protein (MAPT) within neurons and glial cells. They can be classified according to the underlying ratio of three-repeat (3R) to four-repeat (4R) tau and include Pick’s disease (PiD), which is the only 3R tauopathy, and the 4R tauopathies the most common of which are progressive supranuclear palsy (PSP) and corticobasal degeneration (CBD). There are no disease modifying therapies currently available, with research complicated by the wide variability in clinical presentations for each underlying pathology, with presentations often overlapping, as well as the frequent occurrence of atypical presentations that may mimic other non-FTLD pathologies. Although progress has been made in understanding the genetic contribution to disease risk in the more common 4R tauopathies (PSP and CBD), very little is known about the genetics of the 3R tauopathy PiD. There are two broad aims to this thesis; firstly, to use data-driven generative models of disease progression to try and more accurately stage and subtype patients presenting with PSP and corticobasal syndrome (CBS, the most common presentation of CBD), and secondly to identify genetic drivers of disease risk and progression in PiD. Given the rarity of these disorders, as part of this PhD I had to assemble two large cohorts through international collaboration, the 4R tau imaging cohort and the Pick’s disease International Consortium (PIC), to build large enough sample sizes to enable the required analyses. In Chapter 3 I use a probabilistic event-based modelling (EBM) approach applied to structural MRI data to determine the sequence of brain atrophy changes in clinically diagnosed PSP - Richardson syndrome (PSP-RS). The sequence of atrophy predicted by the model broadly mirrors the sequential spread of tau pathology in PSP post-mortem staging studies, and has potential utility to stratify PSP patients on entry into clinical trials based on disease stage, as well as track disease progression. To better characterise the spatiotemporal heterogeneity of the 4R tauopathies, I go on to use Subtype and Stage Inference (SuStaIn), an unsupervised machine algorithm, to identify population subgroups with distinct patterns of atrophy in PSP (Chapter 4) and CBS (Chapter 5). The SuStaIn model provides data-driven evidence for the existence of two spatiotemporal subtypes of atrophy in clinically diagnosed PSP, giving insights into the relationship between pathology and clinical syndrome. In CBS I identify two distinct imaging subtypes that are differentially associated with underlying pathology, and potentially a third subtype that if confirmed in a larger dataset may allow the differentiation of CBD from both PSP and AD pathology using a baseline MRI scan. In Chapter 6 I investigate the association between the MAPT H1/H2 haplotype and PiD, showing for the first time that the H2 haplotype, known to be strongly protective against developing PSP or CBD, is associated with an increased risk of PiD. This is an important finding and has implications for the future development of MAPT isoform-specific therapeutic strategies for the primary tauopathies. In Chapter 7 I perform the first genome wide association study (GWAS) in PiD, identifying five genomic loci that are nominally associated with risk of disease. The top two loci implicate perturbed GABAergic signalling (KCTD8) and dysregulation of the ubiquitin proteosome system (TRIM22) in the pathogenesis of PiD. In the final chapter (Chapter 8) I investigate the genetic determinants of survival in PiD, by carrying out a Cox proportional hazards genome wide survival study (GWSS). I identify a genome-wide significant association with survival on chromosome 3, within the NLGN1 gene. which encodes a synaptic scaffolding protein located at the neuronal pre-synaptic membrane. Loss of synaptic integrity with resulting dysregulation of synaptic transmission leading to increased pathological tau accumulation is a plausible mechanism though which NLGN1 dysfunction could impact on survival in PiD

    Online embedding of metrics

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    We study deterministic online embeddings of metrics spaces into normed spaces and into trees against an adaptive adversary. Main results include a polynomial lower bound on the (multiplicative) distortion of embedding into Euclidean spaces, a tight exponential upper bound on embedding into the line, and a (1+ϵ)(1+\epsilon)-distortion embedding in \ell_\infty of a suitably high dimension.Comment: 15 pages, no figure

    From telescopes to frames and simple groups

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    We introduce the notion of a telescope of groups. Very roughly a telescope is a directed system of groups that contains various commuting images of some fixed group BB. Telescopes are inspired from the theory of groups acting on rooted trees. Imitating known constructions of branch groups, we obtain a number of examples of BB-telescopes and discuss several applications. We give examples of 22-generated infinite amenable simple groups. We show that every finitely generated residually finite (amenable) group embeds into a finitely generated (amenable) LEF simple group. We construct 22-generated frames in products of finite simple groups and show that there are Grothendieck pairs consisting of amenable groups and groups with property (τ)(\tau). We give examples of automorphisms of finitely generated, residually finite, amenable groups that are not inner, but become inner in the profinite completion. We describe non-elementary amenable examples of finitely generated, residually finite groups all of whose finitely generated subnormal subgroups are direct factors.Comment: 41 pages, comments welcom

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum
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