1,431 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Towards a General Complex Systems Model of Economic Sanctions with Some Results Outlining Consequences of Sanctions on the Russian Economy and the World

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    The main purpose of this paper is to present a complex nonlinear modelling approach to analyzing mixed capitalist economic systems. An application of a more elaborate version of this model is to explore the consequences of sanctions on the Russian economy and evaluate the model’s predictive successes or failures. Furthermore, the formal expanded nonlinear model presented in the appendix may be seen as an initial step to put the analysis of economic sanctions within a formal complex socio-economic systems framework. The results obtained from this structural complex multisectoral model so far seem fairly accurate in terms of agreement with measured values of observable economic variables. The political consequences are uncertain and are to be explored separately in a companion paper and ultimately in a book length treatment. Methodologically, the paper also presents the case for using Social Accounting Matrix (SAM)-based models for understanding problems of analyzing sanctions in an economywide context. Linear as well as Nonlinear models are presented in the appendix. The nonlinear modelling approach might prove to be especially relevant for studying the properties of multiple equilibria and complex dynamics

    Characteristic polynomials and eigenvalues of tensors

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    We lay the geometric foundations for the study of the characteristic polynomial of tensors. For symmetric tensors of order d≥3d \geq 3 and dimension 22 and symmetric tensors of order 33 and dimension 33, we prove that only finitely many tensors share any given characteristic polynomial, unlike the case of symmetric matrices and the case of non-symmetric tensors. We propose precise conjectures for the dimension of the variety of tensors sharing the same characteristic polynomial, in the symmetric and in the non-symmetric setting.Comment: 25 pages, comments are welcom

    Stabilisation for varieties in polynomial functors

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    Implicit Loss of Surjectivity and Facial Reduction: Theory and Applications

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    Facial reduction, pioneered by Borwein and Wolkowicz, is a preprocessing method that is commonly used to obtain strict feasibility in the reformulated, reduced constraint system. The importance of strict feasibility is often addressed in the context of the convergence results for interior point methods. Beyond the theoretical properties that the facial reduction conveys, we show that facial reduction, not only limited to interior point methods, leads to strong numerical performances in different classes of algorithms. In this thesis we study various consequences and the broad applicability of facial reduction. The thesis is organized in two parts. In the first part, we show the instabilities accompanied by the absence of strict feasibility through the lens of facially reduced systems. In particular, we exploit the implicit redundancies, revealed by each nontrivial facial reduction step, resulting in the implicit loss of surjectivity. This leads to the two-step facial reduction and two novel related notions of singularity. For the area of semidefinite programming, we use these singularities to strengthen a known bound on the solution rank, the Barvinok-Pataki bound. For the area of linear programming, we reveal degeneracies caused by the implicit redundancies. Furthermore, we propose a preprocessing tool that uses the simplex method. In the second part of this thesis, we continue with the semidefinite programs that do not have strictly feasible points. We focus on the doubly-nonnegative relaxation of the binary quadratic program and a semidefinite program with a nonlinear objective function. We closely work with two classes of algorithms, the splitting method and the Gauss-Newton interior point method. We elaborate on the advantages in building models from facial reduction. Moreover, we develop algorithms for real-world problems including the quadratic assignment problem, the protein side-chain positioning problem, and the key rate computation for quantum key distribution. Facial reduction continues to play an important role for providing robust reformulated models in both the theoretical and the practical aspects, resulting in successful numerical performances

    Geometric Data Analysis: Advancements of the Statistical Methodology and Applications

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    Data analysis has become fundamental to our society and comes in multiple facets and approaches. Nevertheless, in research and applications, the focus was primarily on data from Euclidean vector spaces. Consequently, the majority of methods that are applied today are not suited for more general data types. Driven by needs from fields like image processing, (medical) shape analysis, and network analysis, more and more attention has recently been given to data from non-Euclidean spaces–particularly (curved) manifolds. It has led to the field of geometric data analysis whose methods explicitly take the structure (for example, the topology and geometry) of the underlying space into account. This thesis contributes to the methodology of geometric data analysis by generalizing several fundamental notions from multivariate statistics to manifolds. We thereby focus on two different viewpoints. First, we use Riemannian structures to derive a novel regression scheme for general manifolds that relies on splines of generalized Bézier curves. It can accurately model non-geodesic relationships, for example, time-dependent trends with saturation effects or cyclic trends. Since Bézier curves can be evaluated with the constructive de Casteljau algorithm, working with data from manifolds of high dimensions (for example, a hundred thousand or more) is feasible. Relying on the regression, we further develop a hierarchical statistical model for an adequate analysis of longitudinal data in manifolds, and a method to control for confounding variables. We secondly focus on data that is not only manifold- but even Lie group-valued, which is frequently the case in applications. We can only achieve this by endowing the group with an affine connection structure that is generally not Riemannian. Utilizing it, we derive generalizations of several well-known dissimilarity measures between data distributions that can be used for various tasks, including hypothesis testing. Invariance under data translations is proven, and a connection to continuous distributions is given for one measure. A further central contribution of this thesis is that it shows use cases for all notions in real-world applications, particularly in problems from shape analysis in medical imaging and archaeology. We can replicate or further quantify several known findings for shape changes of the femur and the right hippocampus under osteoarthritis and Alzheimer's, respectively. Furthermore, in an archaeological application, we obtain new insights into the construction principles of ancient sundials. Last but not least, we use the geometric structure underlying human brain connectomes to predict cognitive scores. Utilizing a sample selection procedure, we obtain state-of-the-art results

    Chern correspondence for higher principal bundles

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    The classical Chern correspondence states that a choice of Hermitian metric on a holomorphic vector bundle determines uniquely a unitary 'Chern connection'. This basic principle in Hermitian geometry, later generalized to the theory of holomorphic principal bundles, provides one of the most fundamental ingredients in modern gauge theory, via its applications to the Donaldson-Uhlenbeck-Yau Theorem. In this work we study a generalization of the Chern correspondence in the context of higher gauge theory, where the structure group of the bundle is categorified. For this, we define connective structures on a multiplicative gerbe and propose a natural notion of complexification for an important class of 2-groups. Using this, we put forward a new notion of higher connection which is well-suited for describing holomorphic principal 2-bundles for these 2-groups, and establish a Chern correspondence in this way. As an upshot of our construction, we unify two previous notions of higher connections in the literature, namely those of adjusted connections and of trivializations of Chern-Simons 2-gerbes with connection.Comment: 48 page

    A relative Yau-Tian-Donaldson conjecture and stability thresholds

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    Generalizing Fujita-Odaka invariant, we define a function δ~\tilde{\delta} on a set of generalized bb-divisors over a smooth Fano variety. This allows us to provide a new characterization of uniform KK-stability. A key role is played by a new Riemann-Zariski formalism for KK-stability. For any generalized bb-divisor D\mathbf{D}, we introduce a (uniform) D\mathbf{D}-log KK-stability notion. We prove that the existence of a unique K\"ahler-Einstein metric with prescribed singularities implies this new KK-stability notion when the prescribed singularities are given by the generalized bb-divisor D\mathbf{D}. We connect the existence of a unique K\"ahler-Einstein metric with prescribed singularities to a uniform D\mathbf{D}-log Ding-stability notion which we introduce. We show that these conditions are satisfied exactly when δ~(D)>1\tilde{\delta}(\mathbf{D})>1, extending to the D\mathbf{D}-log setting the δ\delta-valuative criterion of Fujita-Odaka and Blum-Jonsson. Finally we prove the strong openness of the uniform D\mathbf{D}-log Ding stability as a consequence of the strong continuity of δ~\tilde{\delta}.Comment: Comments are welcome

    Pointwise convergence for the Schr\"odinger equation [After Xiumin Du and Ruixiang Zhang]

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    This expository essay accompanied the author's presentation at the S\'eminaire Bourbaki on 01 April 2023. It describes the breakthrough work of Du--Zhang on the Carleson problem for the Schr\"odinger equation, together with background material in multilinear harmonic analysis.Comment: 72 pages, 6 figures, comments welcome

    Sum-Of-Squares Lower Bounds for the Minimum Circuit Size Problem

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    We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean function f: {0,1}? ? {0,1}, SoS requires degree ?(s^{1-?}) to prove that f does not have circuits of size s (for any s > poly(n)). As a corollary we obtain that there are no low degree SoS proofs of the statement NP ? P/poly. We also show that for any 0 < ? < 1 there are Boolean functions with circuit complexity larger than 2^{n^?} but SoS requires size 2^{2^?(n^?)} to prove this. In addition we prove analogous results on the minimum monotone circuit size for monotone Boolean slice functions. Our approach is quite general. Namely, we show that if a proof system Q has strong enough constraint satisfaction problem lower bounds that only depend on good expansion of the constraint-variable incidence graph and, furthermore, Q is expressive enough that variables can be substituted by local Boolean functions, then the MCSP problem is hard for Q
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