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In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages

Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives. First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, … Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincare compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincare disc for the separable quadratic polynomial differential systems.

The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GU m ( q ) × GU n ( q ) on M m × n ( q 2 ) (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form A A ∗ and A ∗ A over a finite field and A A ⊤ and A ⊤ A over arbitrary fields.

This paper is a follow-up to the recent paper of Pillichshammer and Sonnleitner (2020) [12] . We show that the isotropic discrepancy of a lattice point set is at most d 2 2 ( d + 1 ) times its spectral test, thereby correcting the dependence on the dimension d and an inaccuracy in the proof of the upper bound in Theorem 2 of the mentioned paper. The major task is to bound the volume of the neighbourhood of the boundary of a convex set contained in the unit cube. Further, we characterize averages of the distance to a lattice point set in terms of the spectral test. As an application, we infer that the spectral test – and with it the isotropic discrepancy – is crucial for the suitability of the lattice point set for the approximation of Sobolev functions.

A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new definitions, results, and examples. We introduce the notion of topological emergence of a dynamical system, which essentially evaluates how big the set of all its ergodic probability measures is. On the other hand, the metric emergence of a particular reference measure (usually Lebesgue) quantifies how non-ergodic this measure is. We prove fundamental properties of these two emergences, relating them with classical concepts such as Kolmogorov's $\epsilon$-entropy of metric spaces and quantization of measures. We also relate the two types of emergences by means of a variational principle. Furthermore, we provide several examples of dynamics with high emergence. First, we show that the topological emergence of some standard classes of hyperbolic dynamical systems is essentially the maximal one allowed by the ambient. Secondly, we construct examples of smooth area-preserving diffeomorphisms that are extremely non-ergodic in the sense that the metric emergence of the Lebesgue measure is essentially maximal. These examples confirm that super-polynomial emergence indeed exists, as conjectured in the paper [Berger]. Finally, we prove that such examples are locally generic among smooth diffeomorphisms., Comment: v3: Final version; to appear in Advances in Mathematics

Let f be a polynomial in n variables over some number field and Z a subscheme of affine n-space. The notion of motivic oscillation index of f at Z was initiated by Cluckers in [7] and Cluckers-Mustaţǎ-Nguyen in [12] . In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta function of f has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of l-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if f is a polynomial ‘of Thom-Sebastiani type’ with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of A − D − E type. In an appendix, we answer affirmatively a recent question of Cluckers-Mustaţǎ-Nguyen in [12] on poles of maximal order of twisted Igusa's local zeta functions.

The goal of this paper is to provide a categorical framework that leads to the definition of shtukas a la Drinfeld and of excursion operators a la V. Lafforgue. We take as the point of departure the Hecke action of Rep ( G ˇ ) on the category Shv ( Bun G ) of sheaves on Bun G , and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace.

In this paper, we generalize the arithmetic functions τ and τ e in algebraic number fields and we find some properties of these functions.

A partial flag variety P K of a Kac–Moody group G has a natural stratification into projected Richardson varieties. When G is a connected reductive group, a Bruhat atlas for P K was constructed in He et al. (2013): P K is locally modelled with Schubert varieties in some Kac–Moody flag variety as stratified spaces. The existence of Bruhat atlases implies some nice combinatorial and geometric properties on the partial flag varieties and the decomposition into projected Richardson varieties. A Bruhat atlas does not exist for partial flag varieties of an arbitrary Kac–Moody group due to combinatorial and geometric reasons. To overcome obstructions, we introduce the notion of Birkhoff–Bruhat atlas. Instead of the Schubert varieties used in a Bruhat atlas, we use the J -Schubert varieties for a Birkhoff–Bruhat atlas. The notion of the J -Schubert varieties interpolates Birkhoff decomposition and Bruhat decomposition of the full flag variety (of a larger Kac–Moody group). The main result of this paper is the construction of a Birkhoff–Bruhat atlas for any partial flag variety P K of a Kac–Moody group. We also construct a combinatorial atlas for the index set Q K of the projected Richardson varieties in P K . As a consequence, we show that Q K has some nice combinatorial properties. This gives a new proof and generalizes the work of Williams (2007) in the case where the group G is a connected reductive group.

In this paper we recast the Serret theorem about a characterization of palindromic continued fractions in the context of polynomial continued fractions. Then, using the relation between symmetric tridiagonal matrices and polynomial continued fractions we give a quick exposition of the mathematical aspect of the perfect quantum state transfer problem., 19 pages

Assume E is an imaginary quadratic field and O is its ring of integers. For each positive integer m, let I m be the free Hermitian lattice of rank m over O having an orthonormal basis. For each positive integer n, let S O ( n ) be the set of all Hermitian lattices of rank n over O that can be represented by some I m . Denote by g O ( n ) the smallest positive integer g such that each Hermitian lattice in S O ( n ) can be represented by I g . In this paper, we shall provide an explicit upper bound for g O ( n ) for all imaginary quadratic fields E and all positive integers n.

We show in this paper that the VC-dimension of the family of d-dimensional axis-parallel boxes and cubes on the d-dimensional torus are both asymptotically d log 2 ⁡ d . This is especially surprising as in most other examples the VC-dimension usually grows linearly with d in similar settings.

This paper presents various functional-geometric aspects of the logarithmic Sobolev capacity Cap log ⁡ , γ , p - a capacity generated by the logarithmic Sobolev space W p log ⁡ , γ , where ( γ , p ) ∈ ( 0 , ∞ ) × [ 1 , ∞ ) . This new nonlinear-inhomogeneous-nontrivial capacity is closely related to the logarithmic Hausdorff capacity. Not only continuously sharp embedding property of W p log ⁡ , γ into certain logarithmic weighted Lebesgue space and the tracing inequalities are established, but also the logarithmic perimeter and the Lebesgue measure are utilized to: reformulate Cap log ⁡ , γ , p ; find the first variation of the logarithmic perimeter via the corresponding logarithmic mean curvature; characterize the hypersurface of a constant logarithmic mean curvature.

For any $k\geq 1$, this paper studies the number of polynomials having $k$ irreducible factors (counted with or without multiplicities) in $\mathbf{F}_q[t]$ among different arithmetic progressions. We obtain asymptotic formulas for the difference of counting functions uniformly for $k$ in a certain range. In the generic case, the bias dissipates as the degree of the modulus or $k$ gets large, but there are cases when the bias is extreme. In contrast to the case of products of $k$ prime numbers, we show the existence of complete biases in the function field setting, that is the difference function may have constant sign. Several examples illustrate this new phenomenon., The exposition has been improved, we now present the case of the number of irreducible factors both counting and not counting multiplicities. We also add some results on the possible values of the bias

In previous work, we used an $\infty$-categorical version of ultraproducts to show that, for a fixed height $n$, the symmetric monoidal $\infty$-categories of $E_{n,p}$-local spectra are asymptotically algebraic in the prime $p$. In this paper, we prove the analogous result for the symmetric monoidal $\infty$-categories of $K_{p}(n)$-local spectra, where $K_{p}(n)$ is Morava $K$-theory at height $n$ and the prime $p$. This requires $\infty$-categorical tools suitable for working with compactly generated symmetric monoidal $\infty$-categories with non-compact unit. The equivalences that we produce here are compatible with the equivalences for the $E_{n,p}$-local $\infty$-categories., Comment: 33 pages

In this paper we study exponential tractability of multivariate approximation problem for weighted Korobov spaces in the worst case setting. The considered algorithms are constructive and use the class Λ std of only function evaluations. We give matching necessary and sufficient conditions for notions of EC-quasi-polynomial tractability and EC-uniform weak tractability in terms of two weight parameters of the problem.

Pairwise learning naturally arises from machine learning tasks such as AUC maximization, ranking, and metric learning. In this paper we propose a new pairwise learning algorithm based on the additive noise regression model, which adopts the pairwise Huber loss and applies effectively even to the situation where the noise only satisfies a weak moment condition. Owing to the robustness of Huber loss function, this new method is resistant to heavy-tailed errors or outliers in the response variable. We establish a comparison theorem to characterize the gap between the excess generalization error and the prediction error. We derive the error bounds and convergence rates under appropriate conditions. It is worth mentioning that all the results are established under the ( 1 + ϵ ) -th moment condition of the noise variable. It is rather weak particularly in the case of ϵ 1 , which means the noise variable does not even admit a finite variance.

In this paper, we study the existence of large orbits of odd-order subgroups of a finite solvable linear group G, and present some applications on bounding the odd-order parts of the chief series of an arbitrary linear group G. We also find applications to Gluck's conjecture as well as a related question of Navarro.

We obtain estimates for the $L^{p}$-norm of the short-time Fourier transform (STFT) for functions in modulation spaces, providing information about the concentration on a given subset of $\mathbb{R}^{2}$, leading to deterministic guarantees for perfect reconstruction using convex optimization methods. More precisely, we will obtain large sieve inequalities of the Donoho-Logan type, but instead of localizing the signals in regions $T\times W$ of the time-frequency plane using the Fourier transform to intertwine time and frequency, we will localize the representation of the signals in terms of the short-time Fourier transform in sets $\Delta $ with arbitrary geometry. At the technical level, since there is no proper analogue of Beurling's extremal function in the STFT setting, we introduce a new method, which rests on a combination of an argument similar to Schur's test with an extension of Seip's local reproducing formula to general Hermite windows. When the windows are Hermite functions, we obtain local reproducing formulas for polyanalytic Fock spaces which lead to explicit large sieve constant estimates and, as a byproduct, to a reconstruction formula for $f\in L^{2}(\mathbb{R})$ from its STFT values on arbitrary discs. A discussion on optimality follows, along the lines of Donoho-Stark paper on uncertainty principles and signal recovery. We also consider the case of discrete Gabor systems, vector-valued STFT transforms and rephrase the results in terms of the polyanalytic Bargmann-Fock transforms.

In this paper, moment estimates of Holder norm type are obtained for the solution to stochastic differential equation driven by fractional Brownian motion whose drift is measurable and has linear growth. As application, we prove the existence of a density for this kind of stochastic equation which extends the validity of a method introduced by N. Fournier and J. Printems (2010) and developed by M. Romito (2018).

Given a nonautonomous and nonlinear differential equation (0.1) x ′ = A ( t ) x + f ( t , x ) t ≥ 0 , on an arbitrary Banach space X, we formulate very general conditions for the associated linear equation x ′ = A ( t ) x and for the nonlinear term f : [ 0 , + ∞ ) × X → X under which the above system satisfies an appropriate version of the shadowing property. More precisely, we require that x ′ = A ( t ) x admits a very general type of dichotomy, which includes the classical hyperbolic behavior as a very particular case. In addition, we require that f is Lipschitz in the second variable with a sufficiently small Lipschitz constant. Our general framework enables us to treat various settings in which no shadowing result has been previously obtained. Moreover, we are able to recover and refine several known results. We also show how our main results can be applied to the study of the shadowing property for higher order differential equations. Finally, we conclude the paper by presenting discrete time versions of our results.

In this paper order estimates of the Kolmogorov widths for some weighted Sobolev classes in a weighted Lebesgue space are obtained. The weighted Sobolev classes are defined by a restriction on the weighted L p 1 -norm of the highest order derivative and the weighted L p 0 -norm of the function.

Quite recently, Bor (2021) [22] has proved two main theorems dealing with absolute Riesz summability factors of infinite series and Fourier series by using an almost increasing sequence. In this paper, we have generalized these theorems to the | A , p n | k summability method.

This paper begins a series studying D-modules on the Feigin-Frenkel semi-infinite flag variety from the perspective of the Beilinson-Drinfeld factorization (or chiral) theory. Here we calculate Whittaker-twisted cohomology groups of Zastava spaces, which are certain finite-dimensional subvarieties of the affine Grassmannian. We show that such cohomology groups realize the nilradical of a Borel subalgebra for the Langlands dual group in a precise sense, following earlier work of Feigin-Finkelberg-Kuznetsov-Mirkovic and Braverman-Gaitsgory. Moreover, we compare this geometric realization of the Langlands dual group to the standard one provided by (factorizable) geometric Satake.