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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 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]

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]

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 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]

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 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 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]

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]

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]

Our aim in this paper is to study generalizations of the Caginalp phase-field system based on a thermomechanical theory involving two temperatures and a nonlinear coupling. In particular, we prove well-posedness results. More precisely, the existence of a pullback attractor for a nonautonomous parabolic of type Cahn-Hilliard phase-field system. The pullback attractor is a compact set, invariant with respect to the cocycle and which attracts the solutions in the neighborhood of minus infinity, consequently the attractor pullback (or attractor retrograde) exhibits a infinite fractal dimension. [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]

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]

The paper presents some characterizations for the study of dichotomy of dynamical systems. Integral characterizations of Datko type are considered for the notion of uniform dichotomy with growth rates, also called uniform h-dichotomy, for skew-evolution cocycles in Banach spaces.

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 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]

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]

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]

In this paper, we study the cycle structure of a permutation polynomial of the form f (x) = x (q + 1) s 1 (x + x q) s 2 + x s 3 over F q 2 , where (s 1 , s 2 , s 3) ∈ { (q 2 - 2 , 3 , q) , (1 , q 2 - 2 , 1) , (1 , q 2 - 1 , 2) , (1 , 2 , 4) } and q is even. By calculating the sum of all elements in each cycle of a fraction polynomial x x 3 + x 2 + 1 or a linearized polynomial x 2 e + x 2 + x with e ≥ 0 , the cycle structure of f(x) over F q 2 in the first three cases, that is, (s 1 , s 2 , s 3) = { (q 2 - 2 , 3 , q) , (1 , q 2 - 2 , 1) or (1 , q 2 - 1 , 2) } , is characterized. For the case (s 1 , s 2 , s 3) = (1 , 2 , 4) , we give the cycle structure of f(x) over F q 2 for q = 2 2 k with a positive integer k. For q = 2 2 p k with an odd prime p, it needs more techniques to determine the cycle structure of f(x). We only give its cycle length. [ABSTRACT FROM AUTHOR]

We show that for certain non-CM elliptic curves E_{/\mathbb {Q}} such that 3 is an Eisenstein prime of good reduction, a positive proportion of the quadratic twists E_{\psi } of E have Mordell–Weil rank one and the 3-adic height pairing on E_{\psi }(\mathbb {Q}) is non-degenerate. We also show similar but weaker results for other Eisenstein primes. The method of proof also yields examples of middle codimensional algebraic cycles over number fields of arbitrarily large dimension (generalized Heegner cycles) that have non-zero p-adic height. It is not known – though expected – that the archimedian height of these higher-codimensional cycles is non-zero. [ABSTRACT FROM AUTHOR]

We construct a non-trivial |$U(1)/\mathbb {Z}_q$| principal bundle on  T 4 from the compact U (1) lattice gauge field by generalizing Lüscher's constriction so that the cocycle condition contains |$\mathbb {Z}_q$| elements (the 't Hooft flux). The construction requires an admissibility condition on lattice gauge field configurations. From the transition function so constructed, we have the fractional topological charge that is |$\mathbb {Z}_q$| one-form gauge invariant and odd under the lattice time reversal transformation. Assuming a rescaling of the vacuum angle θ → q θ suggested from the Witten effect, our construction provides a lattice implementation of the mixed 't Hooft anomaly between the |$\mathbb {Z}_q$| one-form symmetry and the time reversal symmetry in the U (1) gauge theory with matter fields of charge  |$q\in 2\mathbb {Z}$| when θ = π, which was studied by Honda and Tanizaki [J. High Energy Phys. 12 , 154 (2020)] in the continuum framework. [ABSTRACT FROM AUTHOR]

In this paper we study Butson Hadamard matrices, and codes over finite rings coming from these matrices in logarithmic form, called BH-codes. We introduce a new morphism of Butson Hadamard matrices through a generalized Gray map on the matrices in logarithmic form, which is comparable to the morphism given in a recent note of Ó Catháin and Swartz. That is, we show how, if given a Butson Hadamard matrix over the k th roots of unity, we can construct a larger Butson matrix over the ℓ th roots of unity for any ℓ dividing k, provided that any prime p dividing k also divides ℓ . We prove that a Z p s -additive code with p a prime number is isomorphic as a group to a BH-code over Z p s and the image of this BH-code under the Gray map is a BH-code over Z p (binary Hadamard code for p = 2 ). Further, we investigate the inherent propelinear structure of these codes (and their images) when the Butson matrix is cocyclic. Some structural properties of these codes are studied and examples are provided. [ABSTRACT FROM AUTHOR]

We replace a ring with a small ℂ-linear category 풞, seen as a ring with several objects in the sense of Mitchell. We introduce Fredholm modules over this category and construct a Chern character taking values in the cyclic cohomology of 풞. We show that this categorified Chern character is homotopy invariant and is well behaved with respect to the periodicity operator in cyclic cohomology. For this,we also obtain a description of cocycles and coboundaries in the cyclic cohomology of 풞 (and more generally, in the Hopf cyclic cohomology of a Hopf-module category) by means of DG-semicategories equipped with a trace on endomorphism spaces. [ABSTRACT FROM AUTHOR]

We generalize Bloch's map on torsion cycles from algebraically closed fields to arbitrary fields. While Bloch's map over algebraically closed fields is injective for zero-cycles and for cycles of codimension at most two, we show that the generalization to arbitrary fields is only injective for cycles of codimension at most two but, in general, not for zero-cycles. Our result implies that Jannsen's cycle class map in integral ℓ-adic continuous étale cohomology is, in general, not injective on torsion zero-cycles over finitely generated fields. This answers a question of Scavia and Suzuki. [ABSTRACT FROM AUTHOR]

Let ℓ be a prime and G a pro-ℓ group with torsion-free abelianization. We produce group-theoretic analogues of the Johnson/Morita cocycle for G-in the case of surface groups, these cocycles appear to refine existing constructions when ℓ = 2. We apply this construction to the pro-ℓ étale fundamental groups of smooth curves to obtain Galois-cohomological analogues of these cocycles, which represent what we call the "modified diagonal" and "Johnson" classes, and discuss their relationship to work of Hain and Matsumoto in the case where the curve is proper. We analyze many of the fundamental properties of these classes and use them to give two examples of non-hyperelliptic curves whose Ceresa class has torsion image under the ℓ-adic Abel-Jacobi map. [ABSTRACT FROM AUTHOR]

Let Γ be an irreducible lattice in a product of two locally compact groups and assume that Γ is densely embedded in a profinite group K. We give necessary conditions which imply that the left translation action Γ Õ K is "virtually" cocycle superrigid: any cocycle wW Γ x K > Δ with values in a countable group Δ is cohomologous to a cocycle which factors through the map Γ x>Γ x K0 for some finite quotient groupK0 ofK. As a corollary, we deduce that any ergodic profinite action of Γ D SL2(ZOES-1]) is virtually cocycle superrigid and virtuallyW-superrigid for any finite nonempty set of primes S. [ABSTRACT FROM AUTHOR]

We prove that if - is a group of polynomial growth, then each delocalized cyclic cocycle on the group algebra has a representative of polynomial growth. For each delocalized cocycle, we thus define a higher analogue of Lott's delocalized eta invariant and prove its convergence for invertible differential operators.We also use a determinant map construction of Xie and Yu to prove that if ' is of polynomial growth, then there is a well-defined pairing between delocalized cyclic cocycles and K-theory classes of C*-algebraic secondary higher invariants. When this K-theory class is that of a higher rho invariant of an invertible differential operator, we show this pairing is precisely the aforementioned higher analogue of Lott's delocalized eta invariant. As an application of this equivalence, we provide a delocalized higher Atiyah--Patodi--Singer index theorem, given that M is a compact spin manifold with boundary, equipped with a positive scalar metric g and having fundamental group which is finitely generated and of polynomial growth. [ABSTRACT FROM AUTHOR]

Cycle sets are known to give non-degenerate unitary solutions of the Yang-Baxter equation and linear cycle sets are enriched versions of these algebraic systems. The paper explores the recently developed cohomology and extension theory for linear cycle sets. We derive a four term exact sequence relating 1-cocycles, second cohomology and certain groups of automorphisms arising from central extensions of linear cycle sets. This is an analogue of a similar exact sequence for group extensions known due to Wells. We also relate the exact sequence for linear cycle sets with that for their underlying abelian groups via the forgetful functor and also discuss generalities on dynamical 2-cocycles. [ABSTRACT FROM AUTHOR]

We show that the spectral action, when perturbed by a gauge potential, can be written as a series of Chern--Simons actions and Yang--Mills actions of all orders. In the odd orders, generalized Chern--Simons forms are integrated against an odd (b,B)-cocycle, whereas, in the even orders, powers of the curvature are integrated against (b,B)-cocycles that are Hochschild cocycles as well. In both cases, the Hochschild cochains are derived from the Taylor series expansion of the spectral action Tr(f(D+V)) in powers of V=πD(A), but unlike the Taylor expansion we expand in increasing order of the forms in A. This extends [Connes--Chamseddine 2006], which computes only the scale-invariant part of the spectral action, works in dimension at most 4, and assumes the vanishing tadpole hypothesis. In our situation, we obtain a truly infinite odd (b,B)-cocycle. The analysis involved draws from recent results in multiple operator integration, which also allows us to give conditions under which this cocycle is entire, and under which our expansion is absolutely convergent. As a consequence of our expansion and of the gauge invariance of the spectral action, we show that the odd (b,B)-cocycle pairs trivially with K1. [ABSTRACT FROM AUTHOR]

Given the germ of an analytic function on affine space with a smooth critical locus, we prove that the constancy of the reduced cohomology of the Milnor fiber in lowest possible non-trivial degree off a codimension two subset of the critical locus implies that the vanishing cycles are concentrated in lowest degree and are constant. [ABSTRACT FROM AUTHOR]

Recently, Gekeler proved that the group of invertible analytic functions modulo constant functions on Drinfeld's upper half space is isomorphic to the dual of an integral generalized Steinberg representation. In this note, we show that the group of invertible functions is the dual of a universal extension of that Steinberg representation. As an application, we show that lifting obstructions of rigid analytic theta cocycles of Hilbert modular forms in the sense of Darmon–Vonk can be computed in terms of L -invariants of the associated Galois representation. The same argument applies to theta cocycles for definite unitary groups. [ABSTRACT FROM AUTHOR]

We prove that the real cohomology of semi-simple Lie groups admits boundary values, which are measurable cocycles on the Furstenberg boundary. This generalises known invariants such as the Maslov index on Shilov boundaries, the Euler class on projective space, or the hyperbolic ideal volume on spheres. In rank one, this leads to an isomorphism between the cohomology of the group and of this boundary model. In higher rank, additional classes appear, which we determine completely. [ABSTRACT FROM AUTHOR]

A cohomology theory associated to an n-Lie algebra and a representation space of it is introduced. It is shown that this cohomology theory classifies generalized derivations of n-Lie algebras as 1-cocycles, and inner generalized derivations as 1-coboundaries. [ABSTRACT FROM AUTHOR]

For a non-autonomous dynamics with discrete time defined by a tempered sequence of upper-triangular matrices, we obtain lower and upper bounds for the Grobman regularity coefficients. We also give two applications of these results: we obtain an upper bound for the Grobman coefficients of the exterior powers of a tempered sequence, and we give a simple proof of Oseledets' multiplicative ergodic theorem for cocycles over a measure-preserving transformation without using Kingman's subadditive ergodic theorem. [ABSTRACT FROM AUTHOR]

In this paper we study different types of compact global attractors for non-autonomous (cocycle) dynamical systems with noncompact phase space of driving system. We establish the relations between them. [ABSTRACT FROM AUTHOR]

We show that the class B, of discrete groups which satisfy the conclusion of Popa's cocycle superrigidity theorem for Bernoulli actions, is invariant under measure equivalence. We generalize this to the setting of discrete probability measure preserving (p.m.p.) groupoids, and as a consequence we deduce that any nonamenable lattice in a product of two noncompact, locally compact second countable groups must belong to B. We also introduce a measure-conjugacy invariant called weak Pinsker entropy and show that, if G is a group in the class B, then weak Pinsker entropy is an orbit-equivalence invariant of every essentially free p.m.p. action of G. [ABSTRACT FROM AUTHOR]

Abstract This paper concerns the long term behavior of the stochastic two-dimensional g-Navier–Stokes equations with additive noise defined on a sequence of expanding domains, where the ultimate domain is unbounded and of Poincaré type. We prove that the weak continuity is uniform with respect to all expanding cocycles, which yields the equi-asymptotic compactness by using an energy equation method. Finally, we show the existence of a random attractor for the equation on each domain and the upper semi-continuity of random attractors as the bounded domain is expanded to the unbounded ultimate domain.

When analytic quasi-periodic cocycles are defined on a high dimension torus, their Lyapunov exponents have perturbative positivity and continuity. In this article, we study a class of analytic quasi-periodic Jacobi cocycles defined on a two dimension torus. We show that in the non-perturbative large coupling regimes, the Lyapunov exponent is positive for any frequency and weak Holder continuous for the full-measured frequency.

Rigid meromorphic cocycles were introduced by Darmon and Vonk as a conjectural p-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of SL 2 (Z [ 1 / p ]) which can be evaluated at real quadratic irrationalities, and the values thus obtained are conjectured to lie in algebraic extensions of the base field. In this article, we present a construction of cohomology classes inspired by that of Darmon–Vonk, in which SL 2 (Z [ 1 / p ]) is replaced by an order in an indefinite quaternion algebra over a totally real number field F. These quaternionic cohomology classes can be evaluated at elements in almost totally complex extensions K of F, and we conjecture that the corresponding values lie in algebraic extensions of K. We also report on extensive numerical evidence for this algebraicity conjecture. [ABSTRACT FROM AUTHOR]

Codes from generalized Hadamard matrices have already been introduced. Here we deal with these codes when the generalized Hadamard matrices are cocyclic. As a consequence, a new class of codes that we call generalized Hadamard full propelinear codes turns out. We prove that their existence is equivalent to the existence of central relative (v, w, v, v/w)-difference sets. Moreover, some structural properties of these codes are studied and examples are provided. Some of the propelinear codes constructed for the examples perform better than any comparable linear code. [ABSTRACT FROM AUTHOR]

HIV-1 employs a rich arsenal of viral factors throughout its life cycle and co-opts intracellular trafficking pathways. This exquisitely coordinated process requires precise manipulation of the host microenvironment, most often within defined subcellular compartments. The virus capitalizes on the host by modulating cell-surface proteins and cleverly exploiting nuclear import pathways for post entry events, among other key processes. Successful virus–cell interactions are indeed crucial in determining the extent of infection. By evolving defenses against host restriction factors, while simultaneously exploiting host dependency factors, the life cycle of HIV-1 presents a fascinating montage of an ongoing host–virus arms race. Herein, we provide an overview of how HIV-1 exploits native functions of the host cell and discuss recent findings that fundamentally change our understanding of the post-entry replication events. [ABSTRACT FROM AUTHOR]

We study cocycles (non-autonomous dynamical systems) satisfying a certain squeezing condition with respect to the quadratic form of a bounded self-adjoint operator acting in a Hilbert space. We prove that (under additional assumptions) the orthogonal projector maps the fibres of some invariant set, containing bounded trajectories, in a one-to-one manner onto the negative subspace of the operator. This allows to reduce interesting dynamics onto this invariant set, which in some cases can be considered as a kind of inertial manifold for the cocycle. We consider applications of the reduction principle for periodic cocycles. For such cocycles we give an extension of the Massera second theorem, obtain the conditions for the existence of a Lyapunov stable periodic trajectory and prove convergence-type results, which we apply to study nonlinear periodic in time delayed-feedback equations posed in a proper Hilbert space and parabolic problems with a nonlinear periodic in time boundary control. The required operator is obtained as a solution to certain operator inequalities with the use of Yakubovich-Likhtarnikov frequency theorem for C 0 -semigroups and its properties are established from the Lyapunov inequality and dichotomy of the linear part of the problem. [ABSTRACT FROM AUTHOR]

This paper concerns the long term behavior of the stochastic two-dimensional g-Navier–Stokes equations with additive noise defined on a sequence of expanding domains, where the ultimate domain is unbounded and of Poincaré type. We prove that the weak continuity is uniform with respect to all expanding cocycles, which yields the equi-asymptotic compactness by using an energy equation method. Finally, we show the existence of a random attractor for the equation on each domain and the upper semi-continuity of random attractors as the bounded domain is expanded to the unbounded ultimate domain. [ABSTRACT FROM AUTHOR]

Recently E.R. Alvares, M.M. Alves and M.J. Redondo introduced a cohomology for a group G with values in a module over the partial group algebra K par (G) , which is different from the partial group cohomology defined earlier by the first two named authors of the present paper. Given a unital partial action α of G on a (unital) algebra A we consider A as a K par (G) -module in a natural way and study the globalization problem for the cohomology in the sense of Alvares-Alves-Redondo with values in A. The problem is reduced to an extendibility property of cocycles. Furthermore, assuming that A is a product of blocks, we prove that any cocycle is globalizable, and globalizations of cohomologous cocycles are also cohomologous. As a consequence we obtain that the Alvares-Alves-Redondo cohomology group H p a r n (G , A) is isomorphic to the usual cohomology group H n (G , M (B)) , where M (B) is the multiplier algebra of B and B is the algebra under the enveloping action of α. [ABSTRACT FROM AUTHOR]

We present an Eilenberg-MacLane-type description for the first, second and third spaces of the spectrum defined by a symmetric monoidal category. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the problem of dichotomic behaviors of dynamical systems described by discrete-time skew evolution cocycles in Banach spaces. We study two concepts of uniform dichotomy: uniform exponential dichotomy and uniform polynomial dichotomy. Some characterizations of these notions and connections between these concepts are given.

We generalize Sczech's Eisenstein cocycle for G L (n) over totally real extensions of Q to finite extensions of imaginary quadratic fields. By evaluating the cocycle on certain cycles, we parametrize complex values of Hecke L -functions previously considered by Colmez, giving a cohomological interpretation of his algebraicity result on special values of the L -functions. [ABSTRACT FROM AUTHOR]

We study the Atkin Ut operator for Drinfeld cusp forms. In particular, we define newforms and oldforms of level Γ0(t) and we study basic properties of their slopes. Moreover, we find an explicit formula for the matrix associated to the action of Ut on Γ1(t)-invariant cusp forms using Teitelbaum's interpretation as harmonic cocycles. [ABSTRACT FROM AUTHOR]