Nice Functions Research Articles

Hochschild homology of reductive $p$-adic groups

Consider a reductive p -adic group G , its (complex-valued) Hecke algebra \mathcal H (G) , and the Harish-Chandra–Schwartz algebra \mathcal S (G) . We compute the Hochschild homology groups of \mathcal H (G) and of \mathcal S (G) , and we describe the outcomes in several ways. Our main tools are algebraic families of smooth G -representations. With those we construct maps from HH_{n} (\mathcal H (G)) and HH_{n} (\mathcal S(G)) to modules of differential n -forms on affine varieties. For n = 0 , this provides a description of the cocentres of these algebras in terms of nice linear functions on the Grothendieck group of finite length (tempered) G -representations. It is known from [J. Algebra 606 (2022), 371–470] that every Bernstein ideal \mathcal H (G)^{\mathfrak s} of \mathcal H (G) is closely related to a crossed product algebra of the form \mathcal O (T)\rtimes W . Here \mathcal O (T) denotes the regular functions on the variety T of unramified characters of a Levi subgroup L of G , and W is a finite group acting on T . We make this relation even stronger by establishing an isomorphism between HH_{*} (\mathcal H (G)^{\mathfrak s}) and HH_{*} (\mathcal O (T)\rtimes W) , although we have to say that in some cases it is necessary to twist \mathbb{C} [W] by a 2-cocycle. Similarly, we prove that the Hochschild homology of the two-sided ideal \mathcal S (G)^{\mathfrak s} of \mathcal S (G) is isomorphic to HH_{*} (C^{\infty} (T_{u})\rtimes W) , where T_{u} denotes the Lie group of unitary unramified characters of L . In these pictures of HH_{*} (\mathcal H (G)) and HH_{*} (\mathcal S (G)) , we also show how the Bernstein centre of \mathcal H (G) acts. Finally, we derive similar expressions for the (periodic) cyclic homology groups of \mathcal H (G) and of \mathcal S (G) and we relate that to topological K-theory.

Zappa–Szép products for partial actions of groupoids on left cancellative small categories

We study groupoid actions on left cancellative small categories and their associated Zappa–Szép products. We show that certain left cancellative small categories with nice length functions can be seen as Zappa–Szép products. We compute the associated tight groupoids, characterizing important properties of them, like being Hausdorff, effective, and minimal. Finally, we determine amenability of the tight groupoid under mild, reasonable hypotheses.

Constructive Decomposition of Any L^{1}left( a,bright) Function as Sum of a Strongly Convergent Series of Integrable Functions Each One Positive or Negative Exactly in Open Sets

Researchers dealing with real functions fleft( cdot right) in L^{1}left( a,bright) are often challenged with technical difficulties on trying to prove statements involving the positive f^{,+}left( cdot right) and negative f^{,-}left( cdot right) parts of these functions. Indeed, the set of points where fleft( cdot right) is positive (resp. negative) is just Lebesgue measurable, and in general these two sets may both have positive measure inside each nonempty open subinterval of left( a,bright) . To remedy this situation, we regularize these sets through open sets. More precisely, for each zero-average fleft( cdot right) in L^{,1}left( a,bright) , we construct, explicitly, a series of functions overset{frown }{f}_{i}left( cdot right) having sum fleft( cdot right) — a.e. and in L^{1}left( a,bright) — in such a way that, for each iin left{ ,0,1,2,ldots , right} , there exist two disjoint open sets where overset{frown }{f}_{i}left( cdot right) ge 0 a.e. and overset{frown }{f}_{i}left( cdot right) le 0 a.e., respectively, while overset{frown }{f}_{i}left( cdot right) =0 a.e. elsewhere. Moreover, its primitive int ^{t}fleft( cdot right) becomes the sum of a strongly convergent series of nice AC functions. Applications to calculus of variations & optimal control appear in our next papers.

Partial Linear Eigenvalue Statistics for Non-Hermitian Random Matrices

For an $n \times n$ independent-entry random matrix $X_n$ with eigenvalues $\lambda_1, \dots, \lambda_n$, the seminal work of Rider and Silverstein [Ann. Probab., 34 (2006), pp. 2118--2143] asserts that the fluctuations of the linear eigenvalue statistics $\sum_{i=1}^n f(\lambda_i)$ converge to a Gaussian distribution for sufficiently nice test functions $f$. We study the fluctuations of $\sum_{i=1}^{n-K} f(\lambda_i)$, where $K$ randomly chosen eigenvalues have been removed from the sum. In this case, we identify the limiting distribution and show that it need not be Gaussian. Our results hold for the case when $K$ is fixed as well as for the case when $K$ tends to infinity with $n$. The proof utilizes the predicted locations of the eigenvalues introduced by E. Meckes and M. Meckes, [Ann. Fac. Sci. Toulouse Math. (6), 24 (2015), pp. 93--117]. As a consequence of our methods, we obtain a rate of convergence for the empirical spectral distribution of $X_n$ to the circular law in Wasserstein distance, which may be of independent interest.

Primal Lower Nice Functions in Reflexive Smooth Banach Spaces

In the present work, we extend, to the setting of reflexive smooth Banach spaces, the class of primal lower nice functions, which was proposed, for the first time, in finite dimensional spaces in [Nonlinear Anal. 1991, 17, 385–398] and enlarged to Hilbert spaces in [Trans. Am. Math. Soc. 1995, 347, 1269–1294]. Our principal target is to extend some existing characterisations of this class to our Banach space setting and to study the relationship between this concept and the generalised V-prox-regularity of the epigraphs in the sense proposed recently by the authors in [J. Math. Anal. Appl. 2019, 475, 699–29].

A support theorem for SLE curves

For all $\kappa > 0$, we show that the support of SLE$_\kappa$ curves is the closure in the sup-norm of the set of Loewner curves driven by nice (e.g. smooth) functions. It follows that the support is the closure of the set of simple curves starting at $0$.

Maximal commutators and commutators of potential operators in new vanishing Morrey spaces

We study mapping properties of commutators in certain vanishing subspaces of Morrey spaces, which were recently used to solve the delicate problem of describing the closure of nice functions in Morrey norm. We show that the vanishing properties defining those subspaces are preserved under the action of maximal commutators and commutators of fractional integral operators.

Signal reconstruction by conjugate gradient algorithm based on smoothing $$l_1$$-norm

The $$l_1$$-norm regularized minimization problem is a non-differentiable problem and has a wide range of applications in the field of compressive sensing. Many approaches have been proposed in the literature. Among them, smoothing $$l_1$$-norm is one of the effective approaches. This paper follows this path, in which we adopt six smoothing functions to approximate the $$l_1$$-norm. Then, we recast the signal recovery problem as a smoothing penalized least squares optimization problem, and apply the nonlinear conjugate gradient method to solve the smoothing model. The algorithm is shown globally convergent. In addition, the simulation results not only suggest some nice smoothing functions, but also show that the proposed algorithm is competitive in view of relative error.

Approximation in generalized Morrey spaces

Abstract In this paper we study the approximation of functions from generalized Morrey spaces by nice functions. We introduce a new subspace whose elements can be approximated by infinitely differentiable compactly supported functions. This provides, in particular, an explicit description of the closure of the set of such functions in generalized Morrey spaces.

Characterizations of A2 matrix power weights

In the scalar setting, the power functions |x|γ, for −1<γ<1, are the canonical examples of A2 weights. In this paper, we study two types of power functions in the matrix setting, with the goal of obtaining canonical examples of A2 matrix weights. We first study Type 1 matrix power functions, which are n×n matrix functions whose entries are of the form a|x|γ. Our main result characterizes when these power functions are A2 matrix weights and has two extensions to Type 1 power functions of several variables. We also study Type 2 matrix power functions, which are n×n matrix functions whose eigenvalues are of the form a|x|γ. We find necessary conditions for these to be A2 matrix weights and give an example showing that even nice functions of this form can fail to be A2 matrix weights.

Symbolic optimal expected time reachability computation and controller synthesis for probabilistic timed automata

In this paper we consider the problem of computing the optimal (minimum or maximum) expected time to reach a target and the synthesis of an optimal controller for a probabilistic timed automaton (PTA). Although this problem admits solutions that employ the digital clocks abstraction or statistical model checking, symbolic methods based on zones and priced zones fail due to the difficulty of incorporating probabilistic branching in the context of dense time. We work in a generalisation of the setting introduced by Asarin and Maler for the corresponding problem for timed automata, where simple and nice functions are introduced to ensure finiteness of the dense-time representation. We find restrictions sufficient for value iteration to converge to the optimal expected time on the uncountable Markov decision process representing the semantics of a PTA. We formulate Bellman operators on the backwards zone graph of a PTA and prove that value iteration using these operators equals that computed over the PTA's semantics. This enables us to extract an ε-optimal controller from value iteration in the standard way.

Fluctuations of linear statistics of half-heavy-tailed random matrices

In this paper, we consider a Wigner matrix A with entries whose cumulative distribution decays as x−α with 2<α<4 for large x. We are interested in the fluctuations of the linear statistics N−1Trφ(A), for some nice test functions φ. The behavior of such fluctuations has been understood for both heavy-tailed matrices (i.e. α<2) in Benaych-Georges (2014) and light-tailed matrices (i.e. α>4) in Bai and Silverstein (2009). This paper fills in the gap of understanding it for 2<α<4. We find that while linear spectral statistics for heavy-tailed matrices have fluctuations of order N−1/2 and those for light-tailed matrices have fluctuations of order N−1, the linear spectral statistics for half-heavy-tailed matrices exhibit an intermediate α-dependent order of N−α/4.

Atomic decomposition for &amp;lt;italic&amp;gt;&amp;mu;&amp;lt;/italic&amp;gt;-Bloch space in C&amp;lt;sup&amp;gt;&amp;lt;italic&amp;gt;n&amp;lt;/italic&amp;gt;&amp;lt;/sup&amp;gt;

Let μ be a normal function on [0; 1). In this paper, we discuss several problems on the normal weight Bloch type space β μ ( B ) in the unit ball B. First, we give an integral representation for functions in β μ ( B ). Second, we prove that β μ ( B ) can be identi ed with the dual space of the normal weight Bergman type space A ν 1 ( B ). Finally, we give and prove the main result, i.e., every function in the μ -Bloch space β μ ( B ) can be decomposed into a series of very nice functions (called atoms).

Partial Linear Eigenvalue Statistics for Wigner and Sample Covariance Random Matrices

Let \(M_n\) be an \(n \times n\) Wigner or sample covariance random matrix, and let \(\mu _1(M_n), \mu _2(M_n), \ldots , \mu _n(M_n)\) denote the randomly ordered eigenvalues of \(M_n\). We study the fluctuations of the partial linear eigenvalue statistics $$\begin{aligned} \sum _{i=1}^{n-k} f(\mu _i(M_n)) \end{aligned}$$ as \(n \rightarrow \infty \) for sufficiently nice test functions \(f\). We consider both the cases when \(k\) is fixed and when \(\min \{k,n-k\}\) tends to infinity with \(n\).

Symmetries groups of nice Morse functions on surfaces

Symmetries groups of nice Morse functions on surfaces

Prox-regular sets and epigraphs in uniformly convex Banach spaces: Various regularities and other properties

We continue the study of prox-regular sets that we began in a previous work in the setting of uniformly convex Banach spaces endowed with a norm both uniformly smooth and uniformly convex (e.g., L p , W m,p spaces). We prove normal and tangential regularity properties for these sets, and in particular the equality between Mordukhovich and proximal normal cones. We also compare in this setting the proximal normal cone with different Holderian normal cones depending on the power types s, q of moduli of smoothness and convexity of the norm. In the case of sets that are epigraphs of functions, we show that J-primal lower regular functions have prox-regular epigraphs and we compare these functions with Poliquin's primal lower nice functions depending on the power types s, q of the moduli. The preservation of prox-regularity of the intersection of finitely many sets and of the inverse image is obtained under a calmness assumption. A conical derivative formula for the metric projection mapping of prox-regular sets is also established. Among other results of the paper it is proved that the Attouch-Wets convergence preserves the uniform r-prox-regularity property and that the metric projection mapping is in some sense continuous with respect to this convergence for such sets.

Deviation Inequalities for Centered Additive Functionals of Recurrent Harris Processes Having General State Space

Let X be a Harris recurrent strong Markov process in continuous time with general Polish state space E, having invariant measure μ. In this paper we use the regeneration method to derive non asymptotic deviation bounds for $$P_{x}\biggl(\biggl|\int_0^tf(X_s)\,ds\biggr|\geq t^{\frac{1}{2}+\eta}\varepsilon \biggr)$$ in the positive recurrent case, for nice functions f with μ(f)=0 (f must be a charge). We generalize these bounds to the fully null-recurrent case in the moderate deviations regime. We obtain a Gaussian concentration bound for all functions f which are a charge. The rate of convergence is expressed in terms of the deterministic equivalent of the process. The main ingredient of the proof is Nummelin splitting in continuous time, which allows one to introduce regeneration times for the process on an enlarged state space.

3-manifolds with Yamabe invariant greater than that of $\Bbb{RP}\sp 3$

We show that, for all nonnegative integers $k, l, m$ and $ n$, the Yamabe invariant of $$ \#(\mathbb{RP}^3)\# \ell(\mathbb{RP}^2 \#m(S^2 \times S^1)\#n(S^2 \tilde \times S^1)$$ is equal to the Yamabe invariant of $\mathbb{RP}^3$, provided $k + \ell \geq 1$. We then complete the classification (started by Bray and the second author) of all closed 3-manifolds with Yamabe invariant greater than that of $\mathbb{RP}^3$. More precisely, we show that such maniforlds are either $S^3$ or finite connected sums $ \# m (S^2 \times S^1) \# n (S^2 \tilde \times S^1)$, where $S^2 \tilde \times S^1$ is the nonorientable $S^2$-bundle over $S^1$. A key ingredient is Aubin’s Lemma [3], which says that if the Yamabe constant is positive, then it is strictly less than the Yamabe constant of any of its non-trivial finite conformal coverings. This lemma, combined with inverse mean curvature flow and with analysis of the Green’s function for the conformal Laplacians on specific finite and normal infinite Riemannian coverings, will al- low us to construct a family of nice test functions on the finite coverings and thus prove the desired result.

Two Banach spaces of atoms for stable wavelet frame expansions

It is well known that for sufficiently nice wavelet functions (e.g., Schwartz functions with a few vanishing moments) the regularity of the wavelet transform allows to recover any L 2 -function in a stable way from its samples over any sufficiently dense, irregular sampling set. Equivalently, the (irregular) set of affine transforms of the given wavelet function forms a frame for L 2 ( R d ) . In the present paper a systematic treatment of mild sufficient conditions for the validity of such a statement is provided on the basis of two new Banach spaces of functions, to be denoted by F 0 ( R d ) and F 1 ( R d ) . Their norms turn out to be highly suitable for the description of perturbation results. Given an irregular wavelet frame using an atom from one of these spaces implies that a new system, based using sufficiently close irregular set (in the sense of small jitter error), and using sufficiently small modification of the atom (in terms of one of the two norms), is an irregular wavelet frame of similar quality. Whereas, it is shown that the perturbation may occur in the sense that every parameter is allowed to be perturbed by the same amount for atoms in F 0 ( R d ) with arbitrary time-scale sequences, one is allowed to modify wavelet frames for atoms from the strictly larger class F 1 ( R d ) in a similar way if the sampling pattern forms an affine lattice (similar to classical wavelet systems).

Integration of Primal Lower Nice Functions in Hilbert Spaces

In this paper, we obtain some integration results from subdifferential inclusions for primal lower nice functions by using the Moreau envelopes. A general result concerns an enlarged subdifferential inclusion. It says that, for g primal lower nice at x, the inclusion \(\partial f \subset g + \gamma \mathbb{B}\) around x entails that, for any γ′]0; γ[, f − g is γ′∈- Lipschitz continuous on an appropriate neighborhood of x.