We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over $\mathbb {C} ^d$ is defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov–Clark measure theory for noncommutative and contractive rational multipliers.

Noncommutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz algebra, the unital $C^*$-algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz–Toeplitz and Cuntz algebras, and the emerging field of noncommutative rational functions.

In this paper, we compute the Fourier expansion of the Shintani lift of nearly holomorphic modular forms. As an application, we deduce modularity properties of generating series of cycle integrals of nearly holomorphic modular forms.

Hardin and Taylor proved that any function on the reals—even a nowhere continuous one—can be correctly predicted, based solely on its past behavior, at almost every point in time. They showed that one could even arrange for the predictors to be robust with respect to simple time shifts, and asked whether they could be robust with respect to other, more complicated time distortions. This question was partially answered by Bajpai and Velleman, who provided upper and lower frontiers (in the subgroup lattice of $\mathrm{Homeo}^+(\mathbb {R})$) on how robust a predictor can possibly be. We improve both frontiers, some of which reduce ultimately to consequences of Hölder’s Theorem (that every Archimedean group is abelian).

In this paper, we considered the global strong solution to the 3D incompressible micropolar equations with fractional partial dissipation. Whether or not the classical solution to the 3D Navier–Stokes equations can develop finite-time singularity remains an outstanding open problem, so does the same issue on the 3D incompressible micropolar equations. We establish the global-in-time existence and uniqueness strong solutions to the 3D incompressible micropolar equations with fractional partial velocity dissipation and microrotation diffusion with the initial data $(\mathbf {u}_0,\ \mathbf {w}_0)\in H^1(\mathbb {R}^3)$.

In this paper, we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices $UT_n$. For positive integers $q\leq n$, we classify these images on $UT_{n}$ endowed with a particular elementary ${\mathbb {Z}}_{q}$-grading. As a consequence, we obtain the images of multilinear graded polynomials on $UT_{n}$ with the natural ${\mathbb {Z}}_{n}$-grading. We apply this classification in order to give a new condition for a multilinear polynomial in terms of graded identities so that to obtain the traceless matrices in its image on the full matrix algebra. We also describe the images of multilinear polynomials on the graded algebras $UT_{2}$ and $UT_{3}$, for arbitrary gradings. We finish the paper by proving a similar result for the graded Jordan algebra $UJ_{2}$, and also for $UJ_{3}$ endowed with the natural elementary ${\mathbb {Z}}_{3}$-grading.

In the past $20$ years, the enumeration of plane lattice walks confined to a convex cone—normalized into the first quadrant—has received a lot of attention, stimulated the development of several original approaches, and led to a rich collection of results. Most of these results deal with the nature of the associated generating function: for which models is it algebraic, D-finite, D-algebraic? By model, what we mean is a finite collection of allowed steps.

More recently, similar questions have been raised for nonconvex cones, typically the three-quadrant cone $\mathcal {C} = \{ (i,j) : i \geq 0 \text { or } j \geq 0 \}$. They turn out to be more difficult than their quadrant counterparts. In this paper, we investigate a collection of eight models in $\mathcal {C}$, which can be seen as the first level of difficulty beyond quadrant problems. This collection consists of diagonally symmetric models in $\{-1, 0,1\}^2\setminus \{(-1,1), (1,-1)\}$. Three of them are known not to be D-algebraic. We show that the remaining five can be solved in a uniform fashion using Tutte’s notion of invariants, which has already proved useful for some quadrant models. Three models are found to be algebraic, one is (only) D-finite, and the last one is (only) D-algebraic. We also solve in the same fashion the diagonal model $\{ \nearrow , \nwarrow , \swarrow , \searrow \}$, which is D-finite. The three algebraic models are those of the Kreweras trilogy, $\mathcal S=\{\nearrow , \leftarrow , \downarrow \}$, $\mathcal S^*=\{\rightarrow , \uparrow , \swarrow \}$, and $\mathcal S\cup \mathcal S^*$.

Our solutions take similar forms for all six models. Roughly speaking, the square of the generating function of three-quadrant walks with steps in $\mathcal S$ is an explicit rational function in the quadrant generating function with steps in $\mathscr S:= \{(j-i,j): (i,j) \in \mathcal S\}$. We derive various exact or asymptotic corollaries, including an explicit algebraic description of a positive harmonic function in $\mathcal C$ for the (reverses of the) five models that are at least D-finite.

We use tools from free probability to study the spectra of Hermitian operators on infinite graphs. Special attention is devoted to universal covering trees of finite graphs. For operators on these graphs, we derive a new variational formula for the spectral radius and provide new proofs of results due to Sunada and Aomoto using free probability.

With the goal of extending the applicability of free probability techniques beyond universal covering trees, we introduce a new combinatorial product operation on graphs and show that, in the noncommutative probability context, it corresponds to the notion of freeness with amalgamation. We show that Cayley graphs of amalgamated free products of groups, as well as universal covering trees, can be constructed using our graph product.

First, we build a computational procedure to reconstruct the convex body from a pre-given surface area measure. Nontrivially, we prove the convergence of this procedure. Then, the sufficient and necessary conditions of a Sobolev binary function to be a lightness function of a convex body are investigated. Finally, we present a computational procedure to compute the curvature function from a pre-given lightness function, and then we reconstruct the convex body from this curvature function by using the first procedure. The convergence is also proved. The main computations in both procedures are simply about the spherical harmonics.

An explicit formula forthe mean value of $\vert L(1,\chi )\vert ^2$ is known, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p. Bounds on the relative class number of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Lately, the authors obtained that the mean value of $\vert L(1,\chi )\vert ^2$ is asymptotic to $\pi ^2/6$, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors $p\equiv 1\ \ \pmod {2d}$ which are trivial on a subgroup H of odd order d of the multiplicative group $({\mathbb Z}/p{\mathbb Z})^*$, provided that $d\ll \frac {\log p}{\log \log p}$. Bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Here, for a given integer $d_0>1$, we consider the same questions for the nonprimitive odd Dirichlet characters $\chi '$ modulo $d_0p$ induced by the odd primitive characters $\chi $ modulo p. We obtain new estimates for Dedekind sums and deduce that the mean value of $\vert L(1,\chi ')\vert ^2$ is asymptotic to $\frac {\pi ^2}{6}\prod _{q\mid d_0}\left (1-\frac {1}{q^2}\right )$, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p which are trivial on a subgroup H of odd order $d\ll \frac {\log p}{\log \log p}$. As a consequence, we improve the previous bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$. Moreover, we give a method to obtain explicit formulas and use Mersenne primes to show that our restriction on d is essentially sharp.