Trivial Coefficients Research Articles

Cluster automorphisms and quasi-automorphisms

We study the relation between the cluster automorphisms and the quasi-automorphisms of a cluster algebra A. We prove that under some mild condition, satisfied for example by every skew-symmetric cluster algebra, the quasi-automorphism group of A is isomorphic to a subgroup of the cluster automorphism group of Atriv, and the two groups are isomorphic if A has principal or universal coefficients; here Atriv is the cluster algebra with trivial coefficients obtained from A by setting all frozen variables equal to the integer 1.We also compute the quasi-automorphism group of all finite type and all skew-symmetric affine type cluster algebras, and show in which types it is isomorphic to the cluster automorphism group of Atriv.

Active Coefficient Detection Maximum Correntropy Criterion Algorithm for Sparse Channel Estimation Under Non-Gaussian Environments

In this paper, a kind of active coefficient detection (ACD)-based maximum correntropy criterion (MCC) algorithm is proposed to estimate a sparse multi-path channel under the non-Gaussian environments. The proposed ACD-based MCC algorithms are realized by developing an active coefficient detection mechanism, which can distinguish the active taps within the sparse channels and find out the position and the number of active taps. Therefore, only the active taps coefficient is updated while the trivial channel coefficients are set to be zeros. Various computer simulation experiments are carried out to investigate the performance of the proposed ACD-based MCC algorithms under different impulsive noises. The achieved simulation results prove that the proposed ACD-based MCC algorithms are effective and outperform the previous adaptive filtering algorithms for the sparse channel estimation with regard to both the convergence and the estimation error.

The (not so) trivial lifting in two dimensions

When generating cutting-planes for mixed-integer programs from multiple rows of the simplex tableau, the usual approach has been to relax the integrality of the non-basic variables, compute an intersection cut, then strengthen the cut coefficients corresponding to integral non-basic variables using the so-called trivial lifting procedure. Although of polynomial-time complexity in theory, this lifting procedure can be computationally costly in practice. For the case of two-row relaxations, we present a practical algorithm that computes trivial lifting coefficients in constant time, for arbitrary maximal lattice-free sets. Computational experiments confirm that the algorithm works well in practice.

Lévy–Khintchine decompositions for generating functionals on algebras associated to universal compact quantum groups

We study the first and second cohomology groups of the [Formula: see text]-algebras of the universal unitary and orthogonal quantum groups [Formula: see text] and [Formula: see text]. This provides valuable information for constructing and classifying Lévy processes on these quantum groups, as pointed out by Schürmann. In the case when all eigenvalues of [Formula: see text] are distinct, we show that these [Formula: see text]-algebras have the properties (GC), (NC) and (LK) introduced by Schürmann and studied recently by Franz, Gerhold and Thom. In the degenerate case [Formula: see text], we show that they do not have any of these properties. We also compute the second cohomology group of [Formula: see text] with trivial coefficients — [Formula: see text] — and construct an explicit basis for the corresponding second cohomology group for [Formula: see text] (whose dimension was known earlier, thanks to the work of Collins, Härtel and Thom).

F-polynomial formula from continued fractions

For cluster algebras from surfaces, there is a known formula for cluster variables and F-polynomials in terms of the perfect matchings of snake graphs. If the cluster algebra has trivial coefficients, there is also a known formula for cluster variables in terms of continued fractions. In this paper, we extend this result to cluster algebras with principal coefficients by producing a formula for the F-polynomials in terms of continued fractions.

Infinitely many knots with the trivial (2,1)-cable Γ-polynomial

For coprime integers [Formula: see text] and [Formula: see text], the [Formula: see text]-cable [Formula: see text]-polynomial of a knot is the [Formula: see text]-polynomial of the [Formula: see text]-cable knot of the knot, where the [Formula: see text]-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. In this paper, we show that there exist infinitely many knots with the trivial [Formula: see text]-cable [Formula: see text]-polynomial, that is, the [Formula: see text]-cable [Formula: see text]-polynomial of the trivial knot. Moreover, we see that the knots have the trivial [Formula: see text]-polynomial, the trivial first coefficient HOMFLYPT and Kauffman polynomials.

The Minimal Exact Crossed Product

Given a locally compact group G , we study the smallest exact crossed-product functor (A,G,\alpha)\mapsto A\rtimes_{\mathcal E} G on the category of G - C^* -dynamical systems. As an outcome, we show that the smallest exact crossed-product functor is automatically Morita compatible, and hence coincides with the functor \rtimes_{\mathcal E} as introduced by P. Baum et al. in their reformulation of the Baum–Connes conjecture [Ann. K -Theory 1, No. 2, 155–208 (2016; Zbl 1331.46064)]. We show that the corresponding group algebra C_{\mathcal E}^*(G) always coincides with the reduced group algebra, thus showing that the new formulation of the Baum–Connes conjecture coincides with the classical one in the case of trivial coefficients.

Applications of the Laurent-Stieltjes constants for Dirichlet $L$-series

The Laurent-Stieltjes constants $\gamma_{n}(\chi)$ are, up to a trivial coefficient, the coefficients of the Laurent expansion of the usual Dirichlet $L$-series: when $\chi$ is non-principal, $(-1)^{n}\gamma_{n}(\chi)$ is simply the value of the $n$-th derivative of $L(s,\chi)$ at $s=1$. In this paper, we give an approximation of the Dirichlet $L$-functions in the neighborhood of $s=1$ by a short Taylor polynomial. We also prove that the Riemann zeta function $\zeta(s)$ has no zeros in the region $|s-1|\leq 2.2093$, with $0\leq \Re{(s)}\leq 1$. This work is a continuation of [24].

Universal α-central extensions of Hom-Leibniz n-algebras

ABSTRACTWe construct homology with trivial coefficients of Hom-Leibniz n-algebras. We introduce and characterize universal ()-central extensions of Hom-Leibniz n-algebras. In particular, we show their interplay with the zero-th and first homology with trivial coefficients. When we recover the corresponding results on universal central extensions of Hom-Leibniz algebras. The notion of non-abelian tensor product of Hom-Leibniz n-algebras is introduced and we establish its relationship with universal central extensions. A generalization of the concept and properties of unicentral Leibniz algebras to the setting of Hom-Leibniz n-algebras is developed.

K-THEORY FOR THE C*-ALGEBRAS OF THE SOLVABLE BAUMSLAG–SOLITAR GROUPS

AbstractWe provide a new computation of the K-theory of the group C*-algebra of the solvable Baumslag–Solitar group BS(1, n) (n ≠ 1); our computation is based on the Pimsner–Voiculescu 6-terms exact sequence, by viewing BS(1, n) as a semi-direct product ℤ[1/n] ⋊ ℤ. We deduce from it a new proof of the Baum–Connes conjecture with trivial coefficients for BS(1, n).

A note on bounded-cohomological dimension of discrete groups

Bounded-cohomological dimension of groups is a relative of classical cohomological dimension, defined in terms of bounded cohomology with trivial coefficients instead of ordinary group cohomology. We will discuss constructions that lead to groups with infinite bounded-cohomological dimension, and we will provide new examples of groups with bounded-cohomological dimension equal to 0. In particular, we will prove that every group functorially embeds into an acyclic group with trivial bounded cohomology.

Visual Tracking via Sparse Representation with Reliable Structure Constraint

In this letter, we present a novel visual tracking algorithm based on sparse representation. In contrast to just use the target templates and the trivial templates to sparsely represent the target, we propose to further constrain the model with a set of discriminative weight maps. These weight maps contain the reliable structures of the target object. They help the model penalize the trivial template coefficients depending on the reliable structures of the target object. Then, the target object can be well represented by a sparse set of target templates together with a sparse set of target weight maps. We propose a unified objective function to integrate these two sparse representation problems together. This optimization problem can be well solved by the proposed iteration manner and a customized accelerated proximal gradient method. Furthermore, a novel weight map constructing method is proposed based on consistent motion property and forward–backward errors. Plenty of qualitative and quantitative evaluations demonstrate that our method performs favorably against the state-of-the-art methods in a wide range of tracking scenarios.

Modular Curvature and Morita Equivalence

The curvature of the noncommutative torus $T^2_\theta$ ($\theta$ irrational) endowed with a noncommutative conformal metric has been the focus of attention of several recent works. Continuing the approach taken in the paper [A. Connes and H. Moscovici, http://arxiv.org/abs/1110.3500] we extend the study of the curvature to twisted Dirac spectral triples constructed out of Heisenberg bimodules that implement the Morita equivalence of the $C^*$-algebra $A_\theta = C(T^2_\theta)$ with other toric algebras $A_{\theta'}$. In the enlarged context the conformal metric on $T^2_\theta$ is exchanged with an arbitrary Hermitian metric on the Heisenberg $(A_\theta, A_{\theta'})$-bimodule $E'$ for which ${\rm End}_{A_{\theta'}}(E') = A_\theta $. We prove that the Ray-Singer log-determinant of the corresponding Laplacian, viewed as a functional on the space of all Hermitian metrics on $E'$, attains its extremum at the unique Hermitian metric whose corresponding connection has constant curvature. The gradient of the log-determinant functional gives rise to a noncommutative analogue of the Gaussian curvature. The genuinely new outcome of this paper is that the latter is shown to be independent of any Heisenberg bimodule $E'$ such that $A_\theta = {\rm End}_{A_{\theta'}}(E')$, and in this sense it is Morita invariant. To prove the above results we extend Connes' pseudodifferential calculus to Heisenberg modules. The twisted version, which offers more flexibility even in the case of trivial coefficients, could potentially be applied to other problems in the elliptic theory on noncommutative tori. A noteworthy technical feature is that we systematize the computation of the resolvent expansion for elliptic differential operators on noncommutative tori to an extent which makes the (previously employed) computer assistance unnecessary.

Structures of W(2.2) Lie conformal algebra

Abstract The purpose of this paper is to study W(2, 2) Lie conformal algebra, which has a free ℂ[∂]-basis {L, M} such that [ L λ L ] = ( ∂ + 2 λ ) L , [ L λ M ] = ( ∂ + 2 λ ) M , [ M λ M ] = 0 $\begin{equation}[{L_\lambda }L] = (\partial + 2\lambda )L,[{L_\lambda }M] = (\partial + 2\lambda )M,[{M_\lambda }M] = 0]\end{equation}$ . In this paper, we study conformal derivations, central extensions and conformal modules for this Lie conformal algebra. Also, we compute the cohomology of this Lie conformal algebra with coefficients in its modules. In particular, we determine its cohomology with trivial coefficients both for the basic and reduced complexes.

Higher central extensions and cohomology

We establish a Galois-theoretic interpretation of cohomology in semi-abelian categories: cohomology with trivial coefficients classifies central extensions, also in arbitrarily high degrees. This allows us to obtain a duality, in a certain sense, between “internal” homology and “external” cohomology in semi-abelian categories. These results depend on a geometric viewpoint of the concept of a higher central extension, as well as the algebraic one in terms of commutators.

The Higson–Mackey analogy for finite extensions of complex semisimple groups

In the 1970s, George Mackey pointed out an analogy that exists between tempered representations of semisimple Lie groups and unitary representations of associated semidirect product Lie groups. More recently, Nigel Higson refined Mackey’s analogy into a one-to-one correspondence for connected complex semisimple groups, and in doing so obtained a novel verification of the Baum–Connes conjecture with trivial coefficients for such groups. Here we extend Higson’s results to any Lie group with finitely many connected components whose connected component of the identity is complex semisimple. Our methods include Mackey’s description of unitary representations of group extensions involving projective unitary representations, as well as the notion of twisted crossed product C^* -algebra introduced independently by Green and Dang Ngoc.

Iterated bar complexes and [formula omitted]-homology with coefficients

The first author proved in a previous paper that the n-fold bar construction for commutative algebras can be generalized to En-algebras, and that one can calculate En-homology with trivial coefficients via this iterated bar construction. We extend this result to En-homology and En-cohomology of a commutative algebra A with coefficients in a symmetric A-bimodule.

A class of Schrödinger–Virasoro type Lie conformal algebras

In this paper, a class of Lie conformal algebras associated to a Schrödinger–Virasoro type Lie algebra is constructed, which is nonsimple and can be regarded as an extension of the Virasoro conformal algebra. Then conformal derivations, second cohomology group with trivial coefficients and conformal modules of rank 1 of this Lie conformal algebra are investigated.

$$p$$ p -Adic automorphic $$L$$ L -functions on $$\text {GL}(3)$$ GL ( 3 )

We construct \(p\)-adic \(L\)-functions attached to cohomological cuspidal automorphic representations \(\pi \) of \(\text {GL}_3\) over \({\mathbb Q}\). These functions will be defined as \(p\)-adic Mellin transforms of \(h\)-admissible measures. Our strategy is to generalize the construction of \(p\)-adic \(L\)-functions given in Mahnkopf (Compos. Math. 124(3):253–304, 2000) for cuspidal representations of \(\text {GL}_3\) which are cohomological with respect to the trivial coefficient system. This mainly relies on Harder’s method of computing bounds for the denominators of certain Eisenstein cohomology classes, cf. Harder (Kohomologie Arithmetischer Gruppen, 1987). Finally, we establish a functional equation for the \(p\)-adic \(L\)-functions. This, in particular, requires the construction of \(p\)-adic \(L\)-functions attached to the critical integers on the right-hand side of the functional equation.

Topics in cocyclic development of pairwise combinatorial designs

This is an abstract of the PhD thesis Topics in Cocyclic Development of Pairwise Combinatorial Designs written by Ronan Egan under the supervision of Dane Flannery at the School of Mathematics, Statistics, and Applied Mathematics, National University of Ireland, Galway and submitted in July 2015. This thesis is a compilation of results dealing with cocyclic development of pairwise combinatorial designs. Motivated by a classification of the indexing and extension groups of the Paley Hadamard matrices due to de Launey and Stafford, we investigate cocyclic development of the so-called generalized Sylvester (or Drake) Hadamard matrices. We describe the automorphism groups and derive strict conditions on possible indexing groups, addressing research problems of de Launey and Flannery in doing so. The shift action, discovered by Horadam, is a certain action of any finite group on the set of its 2-cocycles with trivial coefficients, which preserves both cohomological equivalence and orthogonality. We answer questions posed by Horadam about the shift action, in particular regarding its fixed points. One of our main innovations is the concept of linear shift representation. We give an algorithm for calculating the matrix group representation of a shift action, which enables us to compute with the action in a natural setting. We prove detailed results on reducibility, and discuss the outcomes of some computational experiments, including searches for orthogonal cocycles. Using the algorithms developed for shift representations, and other methods, we classify up to equivalence all cocyclic BH(n, p)s where p is an odd prime (necessarily dividing n) and np ≤ 100. This was