Search

We prove rotations-reducibility for close to constant quasi-periodic S L (2 , R) cocycles in one frequency in the finite regularity and smooth cases, and derive some applications to quasi-periodic Schrödinger operators. [ABSTRACT FROM AUTHOR]

The approach for Poisson bialgebras characterized by Manin triples with respect to the invariant bilinear forms on both the commutative associative algebras and the Lie algebras is not available for giving a bialgebra theory for transposed Poisson algebras. Alternatively, we consider Manin triples with respect to the commutative 2-cocycles on the Lie algebras instead. Explicitly, we first introduce the notion of anti-pre-Lie bialgebras as the equivalent structure of Manin triples of Lie algebras with respect to the commutative 2-cocycles. Then we introduce the notion of anti-pre-Lie Poisson bialgebras, characterized by Manin triples of transposed Poisson algebras with respect to the bilinear forms which are invariant on the commutative associative algebras and commutative 2-cocycles on the Lie algebras, giving a bialgebra theory for transposed Poisson algebras. Finally the coboundary cases and the related structures such as analogues of the classical Yang–Baxter equation and -operators are studied. [ABSTRACT FROM AUTHOR]

We prove the large deviation theorem (LDT) for quasi-periodic dynamically defined Gevrey Schrödinger cocycles with weak Liouville frequency. We show that the associated Lyapunov exponent is log-Hölder continuous, while the frequency satisfies β(ω) = 0. [ABSTRACT FROM AUTHOR]

The values of the Dedekind-Rademacher cocycle at certain real quadratic arguments are shown to be global p-units in the narrow Hilbert class field of the associated real quadratic field, as predicted by the conjectures of Darmon-Dasgupta (2006) and Darmon-Vonk (2021). The strategy for proving this result combines the approach of prior work of the authors (2021) with one crucial extra ingredient: the study of infinitesimal deformations of irregular Hilbert Eisenstein series of weight 1 in the anti-parallel direction. [ABSTRACT FROM AUTHOR]

We introduce the notion of pullback along a measurable cocycle and we use it to extend the Borel invariant studied by Bucher, Burger and Iozzi to the world of measurable cocycles. The Borel invariant is constant along cohomology classes and has bounded absolute value. This allows to define maximal cocycles. We conclude by proving that maximal cocycles are actually trivializable to the restriction of the irreducible representation. [ABSTRACT FROM AUTHOR]

This work investigates the stability properties of Lyapunov exponents of transfer operator cocycles from a measure-theoretic perspective. Our results focus on so-called Blaschke product cocycles, a class of random dynamical systems amenable to rigorous analysis. We show that prevalence of stability is related to the dimension of the base system's domain, $ \Omega $. When $ \Omega = S^1 $, we show that stability is prevalent among smooth monic quadratic Blaschke product cocycles fixing the origin by constructing a so-called probe. For higher dimensional $ \Omega $, we show that a probe does not exist, thus providing strong evidence that stability is not prevalent in this setting. Finally, through a perturbative method we show that almost every smooth Blaschke product cocycle fixing the origin is stable. [ABSTRACT FROM AUTHOR]

This paper is a follow-up to Positselski and Št'ovíček (Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity. Electronic preprint arXiv:2212.09639 [math.AG]). We consider two algebraic settings of comodules over a coring and contramodules over a topological ring with a countable base of two-sided ideals. These correspond to two (noncommutative) algebraic geometry settings of certain kind of stacks and ind-affine ind-schemes. In the context of a coring C over a noncommutative ring A, we show that all A-flat C -comodules are ℵ 1 -directed colimits of A-countably presentable A-flat C -comodules. In the context of a complete, separated topological ring R with a countable base of neighborhoods of zero consisting of two-sided ideals, we prove that all flat R -contramodules are ℵ 1 -directed colimits of countably presentable flat R -contramodules. We also describe arbitrary complexes, short exact sequences, and pure acyclic complexes of A-flat C -comodules and flat R -contramodules as ℵ 1 -directed colimits of similar complexes of countably presentable objects. The arguments are based on a very general category-theoretic technique going back to an unpublished 1977 preprint of Ulmer and rediscovered in Positselski (Notes on limits of accessible categories. Electronic preprint arXiv:2310.16773 [math.CT]). Applications to cotorsion periodicity and coderived categories of flat objects in the respective settings are discussed. In particular, in any acyclic complex of cotorsion R -contramodules, all the contramodules of cocycles are cotorsion. [ABSTRACT FROM AUTHOR]

Gerbes and higher gerbes are geometric cocycles representing higher degree cohomology classes, and are attracting considerable interest in differential geometry and mathematical physics. We prove that a 2-gerbe has a torsion Dixmier–Douady class if and only if the gerbe has locally constant cocycle data. As an application, we give an alternative description of flat twisted vector bundles in terms of locally constant transition maps. These results generalize to n-gerbes for n = 1 and n ≥ 3 , providing insights into the structure of higher gerbes and their applications to the geometry of twisted vector bundles. [ABSTRACT FROM AUTHOR]

In this paper, we mainly research Rota-Baxter co-operators on commutative Hopf algebras. We get a new Hopf algebra and some Hopf co-braces via Rota-Baxter co-operators on commutative Hopf algebras. Finally, we generalize Rota-Baxter co-operators to relative Rota-Baxter co-operators of Hopf algebras, construct a Hopf cobrace by a relative Rota-Baxter co-operator of commutative Hopf algebra, and establish the relation of relative Rota-Baxter co-operators and bijective 1-cocycles. [ABSTRACT FROM AUTHOR]

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]

This article studies past attractors and future repellers of topological cocycles conducted by a group G satisfying Ore's condition. The limit behaviour depends on the Green $ \mathcal {L} $ L-pre-order on a right reversible semigroup S of G. An existence and uniqueness theorem for past attractors is proved. A topological method of extending functions is used in order to describe the past attractors by means of skew-product prolongations in the extended space. Illustrative examples of past attractor and future repeller are provided. [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]

A specialization of a K3 surface with Picard rank one to a K3 with rank two defines a vanishing class of order two in the Brauer group of the general K3 surface. We give the B-field invariants of this class. We apply this to the K3 double plane defined by a cubic fourfold with a plane. The specialization of such a cubic fourfold whose group of codimension two cycles has rank two to one which has rank three induces such a specialization of the double planes. We determine the Picard lattice of the specialized double plane as well as the vanishing Brauer class and its relation to the natural 'Clifford' Brauer class. This provides more insight in the specializations. It allows us to explicitly determine the K3 surfaces associated with infinitely many of the conjecturally rational cubic fourfolds obtained as such specializations. [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 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]

We consider Hölder continuous $ GL(d,{\mathbb R}) $-valued cocycles, and more generally linear cocycles, over an accessible volume-preserving center-bunched partially hyperbolic diffeomorphism. We study the regularity of a conjugacy between two cocycles. We establish continuity of a measurable conjugacy between any constant $ GL(d,{\mathbb R}) $-valued cocycle and its perturbation. We deduce this from our main technical result on continuity of a measurable conjugacy between a fiber bunched linear cocycle and a cocycle with a certain block-triangular structure. The latter class covers constant cocycles with one Lyapunov exponent. We also establish a result of independent interest on continuity of measurable solutions for twisted vector-valued cohomological equations over partially hyperbolic systems. In addition, we give more general versions of earlier results on regularity of invariant subbundles, Riemannian metrics, and conformal structures. [ABSTRACT FROM AUTHOR]

This manuscript extends the nonautonomous notions of attractor-repeller pair and Morse decomposition from the setting of nonautonomous dynamical systems to the setting of topological cocycles. It takes into consideration a general notion of cocycle governed by a group action $ G\times P\rightarrow P $. The limit behavior is determined by the directional attractors depending on a filter basis on the subsets of a semigroup $ S\subset G $. The Conley methodology of constructing Morse decompositions is extended by means of directional attractor-repeller pairs. The main result shows the existence of attractor-repeller pairs for topological cocycles on compact spaces. The properties of attractor-repeller pairs and Morse decompositions are presented. [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]

In this article, under some conditions, we prove that every generalized Jordan triple derivation associated with Hochschild 2-cocycle in a 2-torsion free semiprime ring is a generalized derivation associated with Hochschild 2-cocycle. [ABSTRACT FROM AUTHOR]

In deformation-rigidity theory, it is often important to know whether certain bimodules are weakly contained in the coarse bimodule. Consider a bimodule H over the group algebra C[Γ] with Γ a discrete group. The starting point of this paper is that if a dense set of the so-called coefficients of H is contained in the Schatten Sp class p ∈ [2,∞), then the n-fold tensor power HΓ⊗n for n ≥ p/2 is quasi-contained in the coarse bimodule. We apply this to gradient bimodules associated with the carré du champ of a symmetric quantum Markov semi-group. For Coxeter groups, we give a number of characterizations of having coefficients in Sp for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient-Sp property introduced by the second named author, (2) smallness at infinity of a natural compactification of the Coxeter group, and for a large class of Coxeter groups, (3) walks in the Coxeter diagram called parity paths. We derive several strong solidity results. In particular, we extend current strong solidity results for right-angled Hecke von Neumann algebras beyond right-angled Coxeter groups that are small at infinity. Our general methods also yield a concise proof of a result by Sinclair for discrete groups admitting a proper cocycle into a p-integrable representation. [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 introduce the concept of braided anti-flexible bialgebra and construct cocycle bicrossproduct anti-flexible bialgebras. As an application, we solve the extending problem for anti-flexible bialgebras by using some non-abelian cohomology theory. [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]

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]

In a recent survey, Schmidt compiled equivalences between generalized bent functions, group invariant Butson Hadamard matrices, and abelian splitting relative difference sets. We establish a broader network of equivalences by considering Butson matrices that are cocyclic rather than strictly group invariant. This result has several applications; for example, to the construction of Boolean functions whose expansions are generalized partially bent functions, including cases where no bent function can exist. [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]

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 extend Katok's result on 'the approximation of hyperbolic measures by horseshoes' to Banach cocycles. More precisely, let f be a $C^r(r>1)$ diffeomorphism of a compact Riemannian manifold M , preserving an ergodic hyperbolic measure $\mu $ with positive entropy, and let $\mathcal {A}$ be a Hölder continuous cocycle of bounded linear operators acting on a Banach space $\mathfrak {X}$. We prove that there is a sequence of horseshoes for f and dominated splittings for $\mathcal {A}$ on the horseshoes, such that not only the measure theoretic entropy of f but also the Lyapunov exponents of $\mathcal {A}$ with respect to $\mu $ can be approximated by the topological entropy of f and the Lyapunov exponents of $\mathcal {A}$ on the horseshoes, respectively. As an application, we show the continuity of sub-additive topological pressure for Banach cocycles. [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]

Unifying various generalizations of Reynolds operators, the relative cocycle weighted Reynolds operators are studied. We give a characterization of relative cocycle weighted Reynolds operators in the context of pre-Lie algebras. We construct an explicit graded Lie algebra whose Maurer-Cartan elements are given by a relative cocycle weighted Reynolds operator, which makes it possible to construct a cohomology for relative cocycle weighted Reynolds operators. This cohomology can be seen as the cohomology of a certain pre-Lie algebra with coefficients in a suitable representation. Finally, we introduce the notion of NS-pre-Lie algebras and show NS-pre-Lie algebras induce pre-Lie algebras and L-dendriform algebras. [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]

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 this paper, we extend the expansivity, pseudo trajectory tracing property and hyperbolicity of linear dynamical systems for the random view point. We show that to a random linear cocycle A , it is expansive if and only if it has the generalized pseudo trajectory tracing property. Moreover, we show that A is topologically stable if and only if it is structurally stable. [ABSTRACT FROM AUTHOR]

In this paper, we construct a class of special Jacobi cocycles, some element of which is a measurable function defined on the high dimensional torus. We prove that no matter the underlying dynamics is the shift or the skew-shift, the non-perturbative positive Lyapunov exponent holds for any irrational frequency, when the coupling number is large. What's more, if the frequency becomes to be the Diophantine number, then the Lyapunov exponent is weak continuity in the energy. [ABSTRACT FROM AUTHOR]

Lyapunov exponent plays an important role in the dynamics of quasi-periodic cocycles, and admits an asymptotic formula for Schrödinger cocycle with large parameter (see [12] for analytic case, and see [15] for finitely smooth case). In this paper, we establish similar asymptotic formulas of Lyapunov exponent for two classes of important quasi-periodic cocycles: Szegő cocycles and Schrödinger Mosaic 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]

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]