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In this paper we construct new families of extremal copositive matrices in arbitrary dimension by an algorithmic procedure. Extremal copositive matrices are organized in relatively open subsets of real-algebraic varieties, and knowing a particular such matrix A allows in principle to obtain the variety in which A is embedded by solving the corresponding system of algebraic equations. We show that if A is a matrix associated to a so-called COP-irreducible graph with stability number equal 3, then by a trigonometric transformation these algebraic equations become linear and can be solved by linear algebra methods. We develop an algorithm to construct and solve the corresponding linear systems and give examples where the variety contains singularities at the initial matrix A. For the cycle graph C 7 we completely characterize the part of the variety which consists of copositive matrices. [ABSTRACT FROM AUTHOR]

Given an extension 0 → V → G → Q → 1 of locally compact groups, with V abelian, and a compatible essentially bijective 1 -cocycle η : Q → V ̂ , we define a dual unitary 2 -cocycle on G and show that the associated deformation of Ĝ is a cocycle bicrossed product defined by a matched pair of subgroups of Q ⋉ V ̂. We also discuss an interpretation of our construction from the point of view of Kac cohomology for matched pairs. Our setup generalizes that of Etingof and Gelaki for finite groups and its extension due to Ben David and Ginosar, as well as our earlier work on locally compact groups satisfying the dual orbit condition. In particular, we get a locally compact quantum group from every involutive nondegenerate set-theoretical solution of the Yang–Baxter equation, or more generally, from every brace structure. On the technical side, the key new points are constructions of an irreducible projective representation of G on L 2 (Q) and a unitary quantization map L 2 (G) → H S (L 2 (Q)) of Kohn–Nirenberg type. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the Avalanche Principle in [Ann. of. Math. (2), 2001, 154(1): 155–203]. [ABSTRACT FROM AUTHOR]

We consider the 4-dimensional N = 1 Lie superconformal algebra and search for completely "symmetric" (in the graded sense) 3-index invariant tensors. The solution we find is unique and we show that the corresponding invariant polynomial cubic in the generalized curvatures of superconformal gravity vanishes. Consequently, the associated Chern-Simons polynomial is a non-trivial anomaly cocycle. We explicitly compute this cocycle to all orders in the independent fields of superconformal gravity and establish that it is BRST equivalent to the so-called superconformal a-anomaly. We briefly discuss the possibility that the superconformal c-anomaly also admits a similar Chern-Simons formulation and the potential holographic, 5-dimensional, interpretation of our results. [ABSTRACT FROM AUTHOR]

Let G${\bf G}$ be a linear algebraic group, not necessarily connected or reductive, over the field of real numbers R${\mathbb {R}}$. We describe a method, implemented on computer, to find the first Galois cohomology set H1(R,G)${\rm H}^1({\mathbb {R}},{\bf G})$. The output is a list of 1‐cocycles in G${\bf G}$. Moreover, we describe an implemented algorithm that, given a 1‐cocycle z∈Z1(R,G)$z\in {\rm Z}^1({\mathbb {R}}, {\bf G})$, finds the cocycle in the computed list to which z$z$ is equivalent, together with an element of G(C)${\bf G}({\mathbb {C}})$ realizing the equivalence. [ABSTRACT FROM AUTHOR]

Following a theorem of Hayes, we give a geometric interpretation of the special value at s = 0 of certain 1 -cocycle on PGL 2 (ℚ) previously introduced by the author. This work yields three main results: an explicit formula for our cocycle at s = 0 , a generalization and a new proof of Hayes' theorem, and an elegant summation formula for the zeroth coefficient of the Ehrhart quasi-polynomial of certain triangles in ℝ 2 . [ABSTRACT FROM AUTHOR]

We discuss the growth of the singular values of symplectic transfer matrices associated with ergodic discrete Schrödinger operators in one dimension, with scalar and matrix-valued potentials. While for an individual value of the spectral parameter the rate of exponential growth is almost surely governed by the Lyapunov exponents, this is not, in general, true simultaneously for all the values of the parameter. The structure of the exceptional sets is interesting in its own right, and is also of importance in the spectral analysis of the operators. We present new results along with amplifications and generalisations of several older ones, and also list a few open questions. Here are two sample results. On the negative side, for any square-summable sequence |$p_{n}$| there is a residual set of energies in the spectrum on which the middle singular value (the |$W$| -th out of |$2W$|⁠) grows no faster than |$p_{n}^{-1}$|⁠. On the positive side, for a large class of cocycles including the i.i.d. ones, the set of energies at which the growth of the singular values is not as given by the Lyapunov exponents has zero Hausdorff measure with respect to any gauge function |$\rho (t)$| such that |$\rho (t)/t$| is integrable at zero. The employed arguments from the theory of subharmonic functions also yield a generalisation of the Thouless formula, possibly of independent interest: for each |$k$|⁠ , the average of the first |$k$| Lyapunov exponents is the logarithmic potential of a probability measure. [ABSTRACT FROM AUTHOR]

In this paper we begin the study of set-theoretic type solution of the braid equation. Our theory includes set-theoretical solutions as basic examples. More precisely, the linear solution associated to a set-theoretic solution on a set X can be regarded as coming from the coalgebra kX , where k is a field and the elements of X are grouplike. We introduce and study a broader class of linear solutions associated in a similar way to more general coalgebras. We show that the relationships between set-theoretical solutions, q -cycle sets, q -braces, skew-braces, matched pairs of groups and invertible 1-cocycles remain valid in our setting. [ABSTRACT FROM AUTHOR]

Davydov–Yetter (DY) cohomology classifies infinitesimal deformations of the monoidal structure of tensor functors and tensor categories. In this paper we provide new tools for the computation of the DY cohomology for finite tensor categories and exact functors between them. The key point is to realize DY cohomology as relative Ext groups. In particular, we prove that the infinitesimal deformations of a tensor category C are classified by the 3-rd self-extension group of the tensor unit of the Drinfeld center Z (C) relative to C . From classical results on relative homological algebra we get a long exact sequence for DY cohomology and a Yoneda product for which we provide an explicit formula. Using the long exact sequence and duality, we obtain a dimension formula for the cohomology groups based solely on relatively projective covers which reduces a problem in homological algebra to a problem in representation theory, e.g. calculating the space of invariants in a certain object of Z (C) . Thanks to the Yoneda product, we also develop a method for computing DY cocycles explicitly which are needed for applications in the deformation theory. We apply these tools to the category of finite-dimensional modules over a finite-dimensional Hopf algebra. We study in detail the examples of the bosonization of exterior algebras Λ C k ⋊ C [ Z 2 ] , the Taft algebras and the small quantum group of sl 2 at a root of unity. [ABSTRACT FROM AUTHOR]

This paper proposes a geometrisation of N -manifolds of degree n as n -fold vector bundles equipped with a (signed) S n -symmetry. More precisely, it proves an equivalence between the categories of [ n ] -manifolds and the category of (signed) symmetric n -fold vector bundles, by finding that symmetric n -fold vector bundle cocycles and [ n ] -manifold cocycles are identical. This extends the already known equivalences of [1]-manifolds with vector bundles, and of [2]-manifolds with involutive double vector bundles, where the involution is understood as an S 2 -action. [ABSTRACT FROM AUTHOR]

For a smooth projective variety X over an algebraic number field k a conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map of X is a torsion group. In this article we consider a product X = C 1 × ⋯ × C d of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true for X. For a product X = C 1 × C 2 of two curves over Q with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map J 1 (Q) ⊗ J 2 (Q) → ε CH 0 (C 1 × C 2) is finite, where J i is the Jacobian variety of C i . Our constructions include many new examples of non-isogenous pairs of elliptic curves E 1 , E 2 with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products X = C 1 × ⋯ × C d for which the analogous map ε has finite image. [ABSTRACT FROM AUTHOR]

We consider linear cocycles taking values in \mathrm {SL}_d(\mathbb {R}) driven by homeomorphic transformations of a smooth manifold, in discrete and continuous time. We show that any discrete-time cocycle can be extended to a continuous-time cocycle, while preserving its characteristic properties. We provide a necessary and sufficient condition under which this extension is canonical in the sense that the base is extended to an associated suspension flow and that the discrete-time cocycle is recovered as the time-1 map of the continuous-time cocycle. Further, we refine our general result for the case of (quasi-)periodic driving. We use our findings to construct a non-uniformly hyperbolic continuous-time cocycle in \mathrm {SL}_{2}(\mathbb {R}) over a uniquely ergodic driving. [ABSTRACT FROM AUTHOR]

In this paper, we consider random invariant densities and the mean ergodic theorem for Markov operator cocycles which are applicable to quenched type random dynamical systems. We give necessary and sufficient conditions for the existence of random invariant densities for Markov operator cocycles and establish the mean ergodic theorem for generalized linear operator cocycles over a weakly sequentially complete Banach space. The advantage of the result is that we show the implication of weak precompactness for almost every environment to strong convergence in the global sense. [ABSTRACT FROM AUTHOR]

In this paper, we introduce in a braided setting the notion of left module for an invertible 1-cocycle and we prove some categorical equivalences between categories of modules associated to an invertible 1-cocycle and categories of modules associated to Hopf braces. [ABSTRACT FROM AUTHOR]

It is known that the Lyapunov exponent for multifrequency analytic cocycles is weak-Hölder continuous in cocycle for certain Diophantine frequencies, and that this implies certain regularity of the integrated density of states in energy for Jacobi operators. In this paper, we establish the pointwise modulus of continuity in both cocycle and frequency and obtain analogous regularity of the integrated density of states in energy, potential, and frequency. [ABSTRACT FROM AUTHOR]

Let Γ be a subgroup of finite index of SL 2 ⁡ (퐙) . We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian J Γ of the corresponding modular curve X Γ . Our main tool is the explicit description, in terms of modular symbols, of what we call Eisenstein cycles. The latter are representations of relative homology classes over which integration of any holomorphic differential forms vanishes. Our approach relies in an essential way on the specific case Γ ⊂ Γ ⁢ (2) , where we can consider convenient generalized Jacobians instead of J Γ . We relate the Eisenstein classes to the scattering constants attached to Eisenstein series. Finally, we illustrate our approach by considering Fermat curves. [ABSTRACT FROM AUTHOR]

Given a locally compact group G = Q ⋉ V such that V is Abelian and such that the action of Q on the Pontryagin dual V ^ has a free orbit of full measure, we construct a family of unitary dual 2-cocycles Ω ω (aka non-formal Drinfel’d twists) whose equivalence classes [ Ω ω ] ∈ H 2 (G ^ , T) are parametrized by cohomology classes [ ω ] ∈ H 2 (Q , T) . We prove that the associated locally compact quantum groups are isomorphic to cocycle bicrossed product quantum groups associated with a pair of subgroups of the dual semidirect product Q ⋉ V ^ , both isomorphic to Q, and to a pentagonal cocycle Θ ω explicitly given in terms of the group cocycle ω . [ABSTRACT FROM AUTHOR]

Let $\alpha $ be a complex-valued $2$ -cocycle of a finite group $G.$ A new concept of strict $\alpha $ -regularity is introduced and its basic properties are investigated. To illustrate the potential use of this concept, a new proof is offered to show that the number of orbits of G under its action on the set of complex-valued irreducible $\alpha _N$ -characters of N equals the number of $\alpha $ -regular conjugacy classes of G contained in $N,$ where N is a normal subgroup of $G.$ [ABSTRACT FROM AUTHOR]

Let H be a subgroup of a finite group G and let $\alpha $ be a complex-valued $2$ -cocycle of $G.$ Conditions are found to ensure there exists a nontrivial element of H that is $\alpha $ -regular in $G.$ However, a new result is established allowing a prime by prime analysis of the Sylow subgroups of $C_G(x)$ to determine the $\alpha $ -regularity of a given $x\in G.$ In particular, this result implies that every $\alpha _H$ -regular element of a normal Hall subgroup H is $\alpha $ -regular in $G.$ [ABSTRACT FROM AUTHOR]

Avila's Almost Reducibility Conjecture (ARC) is a powerful statement linking purely analytic and dynamical properties of analytic one-frequency cocycles. It is also a fundamental tool in the study of spectral theory of analytic one-frequency Schrödinger operators, with many striking consequences, allowing to give a detailed characterization of the subcritical region. Here we give a proof, completely different from Avila's, for the important case of Schrödinger cocycles with trigonometric polynomial potentials and non-exponentially approximated frequencies, allowing, in particular, to obtain all the desired spectral consequences in this case. [ABSTRACT FROM AUTHOR]

We prove the Hölder continuity of the Lyapunov exponent for quasi-periodic Schrödinger cocycles under the assumptions that the potential is of large C 2 cosine type and the frequency is not super-Liouvillean. [ABSTRACT FROM AUTHOR]

We consider a probability space M on which an ergodic flow φ t : M → M is defined. We study a family of continuous-time linear cocycles, referred to as kinetic , that are associated with solutions of the second-order linear homogeneous differential equation x ¨ + α (φ t (ω)) x ˙ + β (φ t (ω)) x = 0. Here, the parameters α and β evolve along the φ t -orbit of ω ∈ M. Our main result states that for a generic subset of kinetic continuous-time linear cocycles, where generic means a Baire second category with respect to an L p -like topology on the infinitesimal generator, the Lyapunov spectrum is trivial. [ABSTRACT FROM AUTHOR]

We provide a complete classification for regular subalgebras B ⊂ M of injective factors satisfying a natural relative commutant condition. We show that such subalgebras are classified by their associated amenable discrete measured groupoid = B ⊂ M and the action modnew(α) of on the flow of weights induced by the cocycle action (α , u) of on B. We obtain a similar result for triple inclusions A ⊂ B ⊂ M , where M is an injective factor, A is a Cartan subalgebra of M and B ⊂ M is regular, showing that such inclusions are also classified by their associated groupoid = B ⊂ M and the induced action on the flow of weights. Given such a discrete measured amenable groupoid , we also construct a model action of on a field of Cartan inclusions with prescribed action on the associated field of flows. [ABSTRACT FROM AUTHOR]

An exceptional cycle in a triangulated category with Serre functor is a generalization of a spherical object. Suppose that A and B are Gorenstein algebras, given a perfect exceptional n -cycle E ⁎ in K b (A - proj) and a perfect exceptional m -cycle F ⁎ in K b (B - proj) , we construct an A - B -bimodule N , and prove the product E ⁎ ⊠ F ⁎ is an exceptional (n + m − 1) -cycle in K b (Λ - proj) , where Λ = ( A N 0 B ). Using this construction, one gets many new exceptional cycles which is unknown before for certain class of algebras. [ABSTRACT FROM AUTHOR]

A theorem of Guglielmi and Zennaro implies that if the uniform norm growth of a locally constant G L 2 (R) -cocycle on the full shift is not exponential, then it must be either bounded or linear, with no other possibilities occurring. We give an alternative proof of this result and demonstrate that its conclusions do not hold for Lipschitz continuous cocycles over the full shift on two symbols. [ABSTRACT FROM AUTHOR]

A well-known result of Verstraëte [23] shows that for each integer k ≥ 2 every graph G with average degree at least 8 k contains cycles of k consecutive even lengths, the shortest of which is of length at most twice the radius of G. We establish two extensions of Verstraëte's result for linear cycles in linear r -uniform hypergraphs. We show that for any fixed integers r ≥ 3 and k ≥ 2 , there exist constants c 1 = c 1 (r) and c 2 = c 2 (r) , such that every n -vertex linear r -uniform hypergraph G with average degree d (G) ≥ c 1 k contains linear cycles of k consecutive even lengths, the shortest of which is of length at most 2 ⌈ log ⁡ n log ⁡ (d (G) / k) − c 2 ⌉. In particular, as an immediate corollary, we retrieve the current best known upper bound on the linear Turán number of C 2 k r with improved coefficients. Furthermore, we show that for any fixed integers r ≥ 3 and k ≥ 2 , there exist constants c 3 = c 3 (r) and c 4 = c 4 (r) such that every n -vertex linear r -uniform hypergraph with average degree d (G) ≥ c 3 k , contains linear cycles of k consecutive lengths, the shortest of which has length at most 6 ⌈ log ⁡ n log ⁡ (d (G) / k) − c 4 ⌉ + 6. In both cases for given average degree d , the length of the shortest cycles cannot be improved up to the constant factors c 2 , c 4. [ABSTRACT FROM AUTHOR]

In their 2019 paper, Lee and Park presented a formula for the arithmetic Chern–Simons invariant. This formula gives a relation between this invariant and the local Hasse invariants at certain finite primes. Given a number field having real embeddings, we present alternative formulas to give relations between the arithmetic Chern–Simons invariant and the local Hasse invariants at certain primes including all real ones. As an application, we propose a new problem which concerns the existence of a certain 3 -cocycle. If the answer to this problem is positive, the obtained statement is an analogue of the Albert–Brauer–Hasse–Noether theorem. [ABSTRACT FROM AUTHOR]

We present a decomposition of rational twisted G-equivariant K-theory, G a finite group, into cyclic group equivariant K-theory groups of fixed point spaces. This generalises the untwisted decomposition by Atiyah and Segal [J. Geom. Phys. 6 (1989), pp. 671–677] as well as the decomposition by Adem and Ruan [Comm. Math. Phys. 237 (2003), pp. 533–556] for twists coming from group cocycles. [ABSTRACT FROM AUTHOR]

In this paper, we show that one‐dimensional discrete multifrequency quasiperiodic Schrödinger operators with smooth potentials demonstrate ballistic motion on the set of energies on which the corresponding Schrödinger cocycles are smoothly reducible to constant rotations. The proof is performed by establishing a local version of strong ballistic transport on an exhausting sequence of subsets on which reducibility can be achieved by a conjugation uniformly bounded in the Cℓ‐norm. We also establish global strong ballistic transport under an additional integral condition on the norms of conjugation matrices. The latter condition is quite mild and is satisfied in many known examples. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

Let G be a split, simple, simply connected, algebraic group over Q . The degree 4, weight 2 motivic cohomology group of the classifying space BG of G is identified with Z . We construct cocycles representing the generator, known as the second universal motivic Chern class. If G = SL (m) , there is a canonical cocycle, defined by Goncharov (Explicit construction of characteristic classes. Advances in Soviet mathematics, 16, vol 1. Special volume dedicated to I.M.Gelfand's 80th birthday, pp 169–210, 1993). For any group G, we define a collection of cocycles parametrised by cluster coordinate systems on the space of G -orbits on the cube of the principal affine space G / U . Cocycles for different clusters are related by explicit coboundaries, constructed using cluster transformations relating the clusters. The cocycle has three components. The construction of the last one is canonical and elementary; it does not use clusters, and provides the motivic generator of H 3 (G (C) , Z (2)) . However to lift it to the whole cocycle we need cluster coordinates: construction of the first two components uses crucially the cluster structure of the moduli spaces A (G , S) related to the moduli space of G -local systems on S . In retrospect, it partially explains why cluster coordinates on the space A (G , S) should exist. The construction has numerous applications, including explicit constructions of the universal extension of the group G by K 2 , the line bundle on Bun (G) generating its Picard group, Kac–Moody groups, etc. Another application is an explicit combinatorial construction of the second motivic Chern class of a G -bundle. It is a motivic analog of the work of Gabrielov et al. (1974), for any G . We show that the cluster construction of the measurable group 3-cocycle for the group G (C) , provided by our motivic cocycle, gives rise to the quantum deformation of its exponent. [ABSTRACT FROM AUTHOR]

We prove quenched versions of a central limit theorem, a large deviations principle as well as a local central limit theorem for expanding on average cocycles. This is achieved by building an appropriate modification of the spectral method for nonautonomous dynamics developed by [D. Dragičević, G. Froyland, C. Gonzàlez-Tokman and S. Vaienti, A spectral approach for quenched limit theorems for random expanding dynamical systems, Commun. Math. Phys.360 (2018) 1121–1187], to deal with the case of random dynamics that exhibits nonuniform decay of correlations, which are ubiquitous in the context of the multiplicative ergodic theory. [ABSTRACT FROM AUTHOR]

Our goal is to describe superconformal structures on super Riemann surfaces (SRSs) based on data assigned to a fatgraph. We start from the complex structures on punctured (1 | 1) -supermanifolds, characterizing the corresponding moduli and the deformations using Strebel differentials and certain Čech cocycles for a specific covering, which we reproduce from fatgraph data, consisting of U (1) -graph connection and odd parameters at the vertices. Then, we consider dual (1 | 1) -supermanifolds and related superconformal structures for N = 2 super Riemann surfaces. The superconformal structures, N = 1 SRS, are computed as the fixed points of involution on the supermoduli space of N = 2 SRS. [ABSTRACT FROM AUTHOR]

It is proved that the generic extensions of a dynamical system inherit the triviality of pairwise independent self-joinings. This property is related to well-known problems of joining theory and to Rokhlin's famous multiple mixing problem. [ABSTRACT FROM AUTHOR]

We develop the theory of Free Integro-Differential Algebras (FIDA) extending the powerful technique of Free Differential Algebras constructed by D. Sullivan. We extend the analysis beyond the superforms to integral- and pseudo-forms used in supergeometry. It is shown that there are novel structures that might open the road to a deeper understanding of the geometry of supergravity. We apply the technique to some models as an illustration and we provide a complete analysis for D = 11 supergravity. There, it is shown how the Hodge star operator for supermanifolds can be used to analyze the set of cocycles and to build the corresponding FIDA. A new integral form emerges which plays the role of the truly dual to 4-form F (4) and we propose a new variational principle on supermanifolds. [ABSTRACT FROM AUTHOR]

We derive a criterion for the positivity of the maximal Lyapunov exponent of generic mixed random-quasiperiodic linear cocycles, a model introduced in a previous work. This result is applicable to cocycles corresponding to Schrödinger operators with randomly perturbed quasiperiodic potentials. Moreover, we establish an average uniform convergence to the Lyapunov exponent in the Oseledets theorem. [ABSTRACT FROM AUTHOR]

The BRST structure of the extended Bondi-Metzner-Sachs symmetry group of asymptotically flat manifolds is investigated using the recently introduced framework of the Beltrami field parametrization of four-dimensional metrics. The latter identifies geometrically the two physical degrees of freedom of the graviton as fundamental fields. The graded BRST BMS4 nilpotent differential operator relies on four horizontality conditions giving a Lagrangian reformulation of the asymptotic BMS4 symmetry. A series of cocycles is found which indicate the possibility of anomalies for three-dimensional Lagrangian theories to be built in the null boundaries of asymptotically flat spaces from the principle of BRST BMS4 invariance. [ABSTRACT FROM AUTHOR]

In this article, we consider the ergodic optimization of the top Lyapunov exponent. We prove that there is a unique maximising measure of top Lyapunov exponent for typical matrix cocyles. By using the results we obtain, we prove that in any nonuniquely ergodic minimal dynamical system, the Lyapunov-irregular points are typical for typical matrix cocyles. [ABSTRACT FROM AUTHOR]

We show that braided Cherednik algebras introduced by Bazlov and Berenstein are cocycle twists of rational Cherednik algebras of the imprimitive complex reflection groups |$G(m,p,n)$|⁠ , when m is even. This gives a new construction of mystic reflection groups which have Artin–Schelter regular rings of quantum polynomial invariants. As an application of this result, we show that a braided Cherednik algebra has a finite-dimensional representation if and only if its rational counterpart has one. [ABSTRACT FROM AUTHOR]

Answering a question by Chatterji–Druţu–Haglund, we prove that, for every locally compact group $G$ , there exists a critical constant $p_G \in [0,\infty ]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($0) with unbounded orbits if and only if $p \geq p_G$. A similar result holds for the existence of proper continuous affine isometric actions on $L_p$ spaces. Using a representation of cohomology by harmonic cocycles, we also show that such unbounded orbits cannot occur when the linear part comes from a measure-preserving action, or more generally a state-preserving action on a von Neumann algebra and $p>2$. We also prove the stability of this critical constant $p_G$ under $L_p$ measure equivalence, answering a question of Fisher. [ABSTRACT FROM AUTHOR]

Let be a G L d (ℝ) -valued cocycle over a subshift of finite type. Under a certain twisting assumption, we prove that has a uniform Lyapunov exponent if and only if the largest Lyapunov exponent of at all periodic points equals. Under the typicality assumption, we give two checkable criteria for deciding whether has uniform singular value exponents. [ABSTRACT FROM AUTHOR]

We consider pinching cocycles taking values in the space of homeomorphisms of the circle over an hyperbolic base. Using the invariance principle of Malicet, we prove that the cocycles having nonzero exponents of contraction are dense. In this paper, we generalize some common notions an results known of linear cocycles and cocycles of diffeomorphisms, to the nonlinear non-differentiable case. [ABSTRACT FROM AUTHOR]

Lopéz de Medrano-Rincón-Shaw defined Chern-Schwartz-MacPher-son cycles for an arbitrary matroid {M} and proved by an inductive geometric argument that the unsigned degrees of these cycles agree with the coefficients of T({M};x,0), where T({M};x,y) is the Tutte polynomial associated to {M}. Ardila-Denham-Huh recently utilized this interpretation of these coefficients in order to demonstrate their log-concavity. In this note we provide a direct calculation of the degree of a matroid Chern-Schwartz-MacPherson cycle by taking its stable intersection with a generic tropical linear space of the appropriate codimension and showing that the weighted point count agrees with the Gioan-Las Vergnas refined activities expansion of the Tutte polynomial. [ABSTRACT FROM AUTHOR]

Given a quasiperiodic cocycle in s l (2 , R) sufficiently close to a constant, we prove that it is almost reducible in ultradifferentiable class under an adapted arithmetic condition on the frequency vector. [ABSTRACT FROM AUTHOR]

We study the geometry and holonomy of semi-Riemannian, time-like metric cones that are indecomposable, i.e., which do not admit a local decomposition into a semi-Riemannian product. This includes irreducible cones, for which the holonomy can be classified, as well as non-irreducible cones. The latter admit a parallel distribution of null k-planes, and we study the cases k = 1 in detail. We give structure theorems about the base manifold and in the case when the base manifold is Lorentzian, we derive a description of the cone holonomy. This result is obtained by a computation of certain cocycles of indecomposable subalgebras in SD (1, n - 1). [ABSTRACT FROM AUTHOR]

We continue the investigation of the connection between the genus expansion of matrix models and the ħ expansion of integrable hierarchies. In this paper, we focus on the BKP hierarchy, which corresponds to the infinite-dimensional Lie algebra of type B. We consider the genus expansion of such important solutions as Brézin–Gross–Witten (BGW) model, Kontsevich model, and generating functions for spin Hurwitz numbers with completed cycles. We show that these partition functions with inserted parameter ħ , which controls the genus expansion, are solutions of the ħ -BKP hierarchy with good quasi-classical behavior. ħ -BKP language implies the algorithmic prescription for ħ -deformation of the mentioned models in terms of hypergeometric BKP τ -functions and gives insight into the similarities and differences between the models. Firstly, the insertion of ħ into the Kontsevich model is similar to the one in the BGW model, though the Kontsevich model seems to be a very specific example of hypergeometric τ -function. Secondly, generating functions for spin Hurwitz numbers appear to possess a different prescription for genus expansion. This property of spin Hurwitz numbers is not the unique feature of BKP: already in the KP hierarchy, one can observe that generating functions for ordinary Hurwitz numbers with completed cycles are deformed differently from the standard matrix model examples. [ABSTRACT FROM AUTHOR]

A close relation between categories of color Lie algebras and sets of pairs of homomorphisms of modules of Lie algebras is discussed, which is applicable to construct color Lie algebras from Lie algebras and their modules. It is also shown that such construction yields equivalent categories of color Lie algebras under an equivalent relation between symmetric bicharacters. As an application we construct and classify all simple color Lie algebras from the three dimensional simple complex Lie algebra and its finite-dimensional simple modules. [ABSTRACT FROM AUTHOR]

A heap is a set with a certain ternary operation that is self-distributive and exemplified by a group with the operation (x , y , z) ↦ x y − 1 z. We introduce and investigate framed link invariants using heaps. In analogy with the knot group, we define the fundamental heap of framed links using group presentations. The fundamental heap is determined for some classes of links such as certain families of torus and pretzel links. We show that for these families of links there exist epimorphisms from fundamental heaps to Vinberg and Coxeter groups, implying that corresponding groups are infinite. A relation to the Wirtinger presentation is also described. The cocycle invariant is defined using ternary self-distributive (TSD) cohomology, by means of a state sum that uses ternary heap 2 -cocycles as weights. This invariant corresponds to a rack cocycle invariant for the rack constructed by doubling of a heap, while colorings can be regarded as heap morphisms from the fundamental heap. For the construction of the invariant, first computational methods for the heap cohomology are developed. It is shown that the cohomology splits into two types, called degenerate and nondegenerate, and that the degenerate part is one-dimensional. Subcomplexes are constructed based on group cosets, that allow computations of the nondegenerate part. Computations of the cocycle invariants are presented using the cocycles constructed, and conversely, it is proved that the invariant values can be used to derive algebraic properties of the cohomology. [ABSTRACT FROM AUTHOR]

This paper presents integral charaterizations for nonuniform dichotomy with growth rates and their correspondents for the particular cases of nonuniform exponential dichotomy and nonuniform polynomial dichotomy of skew-evolution cocycles in Banach spaces. The connections between these three concepts are presented. [ABSTRACT FROM AUTHOR]