Finite Kernel Research Articles

Convex co‐compact representations of 3‐manifold groups

AbstractA representation of a finitely generated group into the projective general linear group is called convex co‐compact if it has finite kernel and its image acts convex co‐compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3‐manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic or Euclidean Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples.

Embedded Processing for Extended Depth of Field Imaging Systems: From Infinite Impulse Response Wiener Filter to Learned Deconvolution.

Many works in the state of the art are interested in the increase of the camera depth of field (DoF) via the joint optimization of an optical component (typically a phase mask) and a digital processing step with an infinite deconvolution support or a neural network. This can be used either to see sharp objects from a greater distance or to reduce manufacturing costs due to tolerance regarding the sensor position. Here, we study the case of an embedded processing with only one convolution with a finite kernel size. The finite impulse response (FIR) filter coefficients are learned or computed based on a Wiener filter paradigm. It involves an optical model typical of codesigned systems for DoF extension and a scene power spectral density, which is either learned or modeled. We compare different FIR filters and present a method for dimensioning their sizes prior to a joint optimization. We also show that, among the filters compared, the learning approach enables an easy adaptation to a database, but the other approaches are equally robust.

Conditions on the monodromy for a surface group extension to be CAT(0)

In order to determine when surface-by-surface bundles are non-positively curved, Llosa Isenrich and Py [Math. Ann. 380 (2021), pp. 449–485] give a necessary condition: given a surface-by-surface group G G with infinite monodromy, if G G is CAT(0) then the monodromy representation is injective. We extend this to a more general result: Let G G be a group with a normal surface subgroup R R . Assume G / R G/R satisfies the property that for every infinite normal subgroup Λ \Lambda of G / R G/R , there is an infinite finitely generated subgroup Λ 0 > Λ \Lambda _0>\Lambda so that the centralizer C G / R ( Λ 0 ) C_{G/R}(\Lambda _0) is finite. We then prove that if G G is CAT(0) with infinite monodromy, then the monodromy representation has a finite kernel. This applies in particular if G / R G/R is acylindrically hyperbolic.

Some relational structures with polynomial growth and their associated algebras II. Finite generation.

The profile of a relational structure $R$ is the function $\varphi_R$ which counts for every nonnegative integer $n$ the number, possibly infinite, $\varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $\varphi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $\mathbb{K}.\mathcal A$ introduced by P. J. Cameron. In a previous paper, we studied the relationship between the properties of a relational structure $R$ and those of its age algebra, particularly when $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups or the rings of quasisymmetric polynomials. The main theorem of this paper characterizes combinatorially when the age algebra is finitely generated in this setting. For tournaments, this boils down to the profile being bounded. We further investigate how far the well known algebraic properties of invariant rings and quasisymmetric polynomials extend to age algebras; notably, we explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra. For a homogeneous structure with a profile bounded by a polynomial, Cameron conjectured in the early eighties that the profile is asymptotically polynomial; Macpherson further conjectured that the age algebra is finitely generated. This was proven recently by Falque and the second author. The combined results support the conjecture that---assuming finite kernel---profiles bounded by a polynomial are asymptotically polynomial, and give hope for a complete characterization of when the age algebra is finitely generated.

On endomorphisms of automatic groups

We extend the definition of the bounded reduction property to endomorphisms of automatic group and find conditions for it to hold. We study endomorphisms with L-quasiconvex image and prove that those with finite kernel satisfy a synchronous version of the bounded reduction property. Finally, we use these techniques to prove L-quasiconvexity of the equalizer of two endomorphisms under certain (strict) conditions.

The Role of the Instrumental Response in 21 cm Epoch of Reionization Power Spectrum Gridding Analyses

Reconstruction of the sky brightness measured by radio interferometers is typically achieved through gridding techniques, or histograms in spatial Fourier space. For Epoch of Reionization (EoR) 21 cm power spectrum measurements, extreme levels of gridding resolution are required to reduce spectral contamination, as explored in other works. However, the role of the shape of the Fourier space spreading function, or kernel, also has consequences in reconstructed power spectra. We decompose the instrumental Murchison Widefield Array (MWA) beam into a series of Gaussians and simulate the effects of finite kernel extents and differing shapes in gridding/degridding for optimal map making analyses. For the MWA, we find that the kernel must extend out to 0.001–0.0001% of the maximum value in order to measure the EoR using foreground avoidance. This requirement changes depending on beam shape, with compact kernels requiring far smaller extents for similar contamination levels at the cost of less-optimal errors. However, simple calibration using pixelated degridding results, regardless of shape of the kernel, cannot recover the EoR due to catastrophic errors caused by the pixel resolution. Including an opaque horizon with widefield beams also causes significant spectral contamination via a beam–horizon interaction that creates an infinitely extended kernel in Fourier space, which cannot be represented well. Thus, our results indicate that simple calibration via degridded models and optimal map making for extreme widefield instrumentation are not feasible.

C<sub>2</sub> Invariants of Hourglass Chains via Quadratic Denominator Reduction

We introduce families of four-regular graphs consisting of chains of hourglasses which are attached to a finite kernel. We prove a formula for the $c_2$ invariant of these hourglass chains which only depends on the kernel. For different kernels these hourglass chains typically give rise to different $c_2$ invariants. An exhaustive search for the $c_2$ invariants of hourglass chains with kernels that have a maximum of ten vertices provides Calabi-Yau manifolds with point-counts which match the Fourier coefficients of modular forms whose weights and levels are [4,8], [4,16], [6,4], and [9,4]. Assuming the completion conjecture, we show that no modular form of weight 2 and level $\leq1000$ corresponds to the $c_2$ of such hourglass chains. This provides further evidence in favour of the conjecture that curves are absent in $c_2$ invariants of $\phi^4$ quantum field theory.

Regret Bounds for Online Kernel Selection in Continuous Kernel Space

Regret bounds of online kernel selection in a finite kernel set have been well studied, having at least an order O( √ NT) of magnitude after T rounds, where N is the number of candidate kernels. But it is still an unsolved problem to achieve sublinear regret bounds of online kernel selection in a continuous kernel space under different learning frameworks. In this paper, to represent different learning frameworks of online kernel selection, we divide online kernel selection approaches in a continuous kernel space into two categories according to the order of selection and training at each round. Then we construct a surrogate hypothesis space that contains all the candidate kernels with bounded norms and inner products, representing the continuously varying hypothesis space. Finally, we decompose the regrets of the proposed online kernel selection categories into different types of instantaneous regrets in the surrogate hypothesis space, and derive optimal regret bounds of order O( √ T) of magnitude under mild assumptions, independent of the cardinality of the continuous kernel space. Empirical studies verified the correctness of the theoretical regret analyses.

Schwartz homologies of representations of almost linear Nash groups

Let G be an almost linear Nash group, namely, a Nash group that admits a Nash homomorphism with finite kernel to some GLk(R). A homology theory (the Schwartz homology) is established for the category of smooth Fréchet representations of G of moderate growth. Frobenius reciprocity and Shapiro's lemma are proved in this category. As an application, we give a criterion for automatic extensions of Schwartz homologies of Schwartz sections of a tempered G-vector bundle.

On approximation properties of semidirect products of groups

Let R be a class of groups closed under taking semidirect products with finite kernel and fully residually R-groups. We prove that R contains all R-by-{finitely generated residually finite} groups. It follows that a semidirect product of a finitely generated residually finite group with a surjunctive group is surjunctive. This remained unknown even for direct products of a surjunctive group with the integers Z.

Hysteretic behavior of spatially coupled phase-oscillators

Motivated by phenomena related to biological systems such as the synchronously flashing swarms of fireflies, we investigate a network of phase oscillators evolving under the generalized Kuramoto model with inertia. A distance-dependent, spatial coupling between the oscillators is considered. Zeroth and first order kernel functions with finite kernel radii were chosen to investigate the effect of local interactions. The hysteretic dynamics of the synchronization depending on the coupling parameter was analyzed for different kernel radii. Numerical investigations demonstrate that (1) locally locked clusters develop for small coupling strength values, (2) the hysteretic behavior vanishes for small kernel radii, (3) the ratio of the kernel radius and the maximal distance between the oscillators characterizes the behavior of the network.

Fullness of crossed products of factors by discrete groups

AbstractLet M be an arbitrary factor and $\sigma : \Gamma \curvearrowright M$ an action of a discrete group. In this paper, we study the fullness of the crossed product $M \rtimes _\sigma \Gamma $. When Γ is amenable, we obtain a complete characterization: the crossed product factor $M \rtimes _\sigma \Gamma $ is full if and only if M is full and the quotient map $\overline {\sigma } : \Gamma \rightarrow {\rm out}(M)$ has finite kernel and discrete image. This answers the question of Jones from [11]. When M is full and Γ is arbitrary, we give a sufficient condition for $M \rtimes _\sigma \Gamma $ to be full which generalizes both Jones' criterion and Choda's criterion. In particular, we show that if M is any full factor (possibly of type III) and Γ is a non-inner amenable group, then the crossed product $M \rtimes _\sigma \Gamma $ is full.

An efficient primal-dual interior point method for linear programming problems based on a new kernel function with a finite exponential-trigonometric barrier term

In this paper, we first propose a new finite exponential-trigonometric kernel function that has finite value at the boundary of the feasible region. Then by using some simple analysis tools, we show that the new kernel function has exponential convexity property. We prove that the large-update primal-dual interior-point method based on this kernel function for solving linear optimization problems has $$O\left( \sqrt{n}\log n\log \frac{n}{\epsilon }\right)$$ iteration bound in the worst case when the barrier parameter is taken large enough. Moreover, the numerical results reveal that the new finite exponential-trigonometric kernel function has better results than the other kernel functions.

Functoriality Properties of the Dual Group

Let G be a connected reductive group. Previously, it was shown that for any G -variety X one can define the dual group G^\vee_X which admits a natural homomorphism with finite kernel to the Langlands dual group G^\vee of G . Here, we prove that the dual group is functorial in the following sense: if there is a dominant G -morphism X\to Y or an injective G -morphism Y\to X then there is a unique homomorphism with finite kernel G^\vee_Y\to G^\vee_X which is compatible with the homomorphisms to G^\vee .

Integrable structure of products of finite complex Ginibre random matrices

We consider the squared singular values of the product of M standard complex Gaussian matrices. Since the squared singular values form a determinantal point process with a particular Meijer G-function kernel, the gap probabilities are given by a Fredholm determinant based on this kernel. It was shown by Strahov (2014) that a hard edge scaling limit of the gap probabilities is described by Hamiltonian differential equations which can be formulated as an isomonodromic deformation system similar to the theory of the Kyoto school. We generalize this result to the case of finite matrices by first finding a representation of the finite kernel in integrable form. As a result we obtain the Hamiltonian structure for finite size matrices and formulate it in terms of a (M+1)×(M+1) matrix Schlesinger system. The case M=1 reproduces the Tracy and Widom theory which results in the Painlevé V equation for the (0,s) gap probability. Some integrals of motion for M=2 are identified, and a coupled system of differential equations in two unknowns is presented which uniquely determines the corresponding (0,s) gap probability.

An explicit determination of the Springer morphism

ABSTRACTLet G be a simply connected semisimple algebraic groups over ℂ and let ρ:G→GL(Vλ) be an irreducible representation of G of highest weight λ. Suppose that ρ has finite kernel. Springer defined an adjoint-invariant regular map with Zariski dense image from the group to the Lie algebra, 𝜃λ:G→𝔤, which depends on λ. This map, 𝜃λ, takes the maximal torus T of G to its Lie algebra 𝔱. Thus, for a given simple group G and an irreducible representation Vλ, one may write , where we take the simple coroots as a basis for 𝔱. We give a complete determination for these coefficients ci(t) for any simple group G as a sum over the weights of the torus action on Vλ.

Short-ranged memory model with preferential growth.

In this work we introduce a variant of the Yule-Simon model for preferential growth by incorporating a finite kernel to model the effects of bounded memory. We characterize the properties of the model combining analytical arguments with extensive numerical simulations. In particular, we analyze the lifetime and popularity distributions by mapping the model dynamics to corresponding Markov chains and branching processes, respectively. These distributions follow power laws with well-defined exponents that are within the range of the empirical data reported in ecologies. Interestingly, by varying the innovation rate, this simple out-of-equilibrium model exhibits many of the characteristics of a continuous phase transition and, around the critical point, it generates time series with power-law popularity, lifetime and interevent time distributions, and nontrivial temporal correlations, such as a bursty dynamics in analogy with the activity of solar flares. Our results suggest that an appropriate balance between innovation and oblivion rates could provide an explanatory framework for many of the properties commonly observed in many complex systems.

Convex cocompactness in pseudo-Riemannian hyperbolic spaces

Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups $$\mathrm {PO}(p,q)$$ by considering their action on the associated pseudo-Riemannian hyperbolic space $$\mathbb {H}^{p,q-1}$$ in place of the Riemannian symmetric space. Following work of Barbot and Merigot in anti-de Sitter geometry, we find an intimate connection between Anosov representations and a natural notion of convex cocompactness in this setting.

Abelianization of the F-divided fundamental group scheme

Let (X , x0) be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for (X , x0) produces a homomorphism from the abelianization of the F-divided fundamental group scheme of X to the F-divided fundamental group of the Albanese variety of X. We prove that this homomorphism is surjective with finite kernel. The kernel is also described.

Transitivity degrees of countable groups and acylindrical hyperbolicity

We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to answer a question of Garion and Glassner on the existence of highly transitive faithful actions of mapping class groups. It also implies that in various geometric and algebraic settings, the transitivity degree of an infinite group can only take two values, namely $1$ and $\infty$. Here by transitivity degree of a group we mean the supremum of transitivity degrees of its faithful permutation representations. Further, for any countable group $G$ admitting a highly transitive faithful action, we prove the following dichotomy: Either $G$ contains a normal subgroup isomorphic to the infinite alternating group or $G$ resembles a free product from the model theoretic point of view. We apply this theorem to obtain new results about universal theory and mixed identities of acylindrically hyperbolic groups. Finally, we discuss some open problems.