We introduce and study the weak Glimm property for $\mathrm{C}^{*}$-algebras, and also a property we shall call (HS$_0$). We show that the properties of being nowhere scattered and residual (HS$_0$) are equivalent for any $\mathrm{C}^{*}$-algebra. Also, for a $\mathrm{C}^{*}$-algebra with the weak Glimm property, the properties of being purely infinite and weakly purely infinite are equivalent. It follows that for a $\mathrm{C}^{*}$-algebra with the weak Glimm property such that the absolute value of every nonzero, square-zero, element is properly infinite, the properties of being (weakly, locally) purely infinite, nowhere scattered, residual (HS$_0$), residual (HS$_{\text {t}}$), and residual (HI) are all equivalent, and are equivalent to the global Glimm property. This gives a partial affirmative answer to the global Glimm problem, as well as certain open questions raised by Kirchberg and Rørdam.

In this article, we construct some examples of noncommutative projective Calabi–Yau schemes by using noncommutative Segre products and quantum weighted hypersurfaces. We also compare our constructions with commutative Calabi–Yau varieties and examples constructed in Kanazawa (2015, Journal of Pure and Applied Algebra 219, 2771–2780). In particular, we show that some of our constructions are essentially new examples of noncommutative projective Calabi–Yau schemes.

In this paper, we solved an open problem raised by Cecil and Ryan (2015, Geometry of Hypersurfaces, Springer Monographs in Mathematics, Springer, New York, p. 531) by proving the nonexistence of non-Hopf Ricci-semisymmetric real hypersurfaces in $\mathbb {C}P^{2}$ and $\mathbb {C}H^{2}$.

Determining the range of complex maps plays a fundamental role in the study of several complex variables and operator theory. In particular, one is often interested in determining when a given holomorphic function is a self-map of the unit ball. In this paper, we discuss a class of maps in $\mathbb {C}^N$ that generalize linear fractional maps. We then proceed to determine precisely when such a map is a self-map of the unit ball. In particular, we take a novel approach, obtaining numerous new results about this class of maps along the way.

We give explicit presentations of the integral equivariant cohomology of the affine Grassmannians and flag varieties in type A, arising from their natural embeddings in the corresponding infinite (Sato) Grassmannian and flag variety. These presentations are compared with results obtained by Lam and Shimozono, for rational equivariant cohomology of the affine Grassmannian, and by Larson, for the integral cohomology of the moduli stack of vector bundles on .

Although Naimark dilation theorem was originally stated in 1940, it still finds many important applications in various areas. The objective of this paper is to introduce a method for explicitly constructing the vectors of complementary frames in the Naimark dilation theorem, specifically in cases where the initial Parseval frame contains a Riesz basis as a subset. These findings serve as a foundation for the construction of dual frames.

Williamson’s theorem states that for any $2n \times 2n$ real positive definite matrix A, there exists a $2n \times 2n$ real symplectic matrix S such that $S^TAS=D \oplus D$, where D is an $n\times n$ diagonal matrix with positive diagonal entries known as the symplectic eigenvalues of A. Let H be any $2n \times 2n$ real symmetric matrix such that the perturbed matrix $A+H$ is also positive definite. In this paper, we show that any symplectic matrix $\tilde {S}$ diagonalizing $A+H$ in Williamson’s theorem is of the form $\tilde {S}=S Q+\mathcal {O}(\|H\|)$, where Q is a $2n \times 2n$ real symplectic as well as orthogonal matrix. Moreover, Q is in symplectic block diagonal form with the block sizes given by twice the multiplicities of the symplectic eigenvalues of A. Consequently, we show that $\tilde {S}$ and S can be chosen so that $\|\tilde {S}-S\|=\mathcal {O}(\|H\|)$. Our results hold even if A has repeated symplectic eigenvalues. This generalizes the stability result of symplectic matrices for non-repeated symplectic eigenvalues given by Idel, Gaona, and Wolf [Linear Algebra Appl., 525:45–58, 2017].

Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weight $k_{1}$ and $k_{2}$ for the full modular group $SL(2,\mathbb {Z})$, respectively. Suppose that $\lambda _{f\times f\times f}(n)$ and $\lambda _{g\times g\times g}(n)$ are the n-th Dirichlet coefficient of the triple product L-functions $L(s,f\times f\times f)$ and $L(s,g\times g\times g)$. In this paper, we consider the sign changes of the sequence $\{\lambda _{f\times f\times f}(n)\}_{n\geq 1}$ and $\{\lambda _{f\times f\times f}(n)\lambda _{g\times g\times g}(n)\}_{n\geq 1}$ in short intervals and establish quantitative results for the number of sign changes for $n \leq x$, which improve the previous results.

Given an integer $k\ge 2$, let $\omega _k(n)$ denote the number of primes that divide n with multiplicity exactly k. We compute the density $e_{k,m}$ of those integers n for which $\omega _k(n)=m$ for every integer $m\ge 0$. We also show that the generating function $\sum _{m=0}^\infty e_{k,m}z^m$ is an entire function that can be written in the form $\prod _{p} \bigl (1+{(p-1)(z-1)}/{p^{k+1}} \bigr )$; from this representation we show how to both numerically calculate the $e_{k,m}$ to high precision and provide an asymptotic upper bound for the $e_{k,m}$. We further show how to generalize these results to all additive functions of the form $\sum _{j=2}^\infty a_j \omega _j(n)$; when $a_j=j-1$ this recovers a classical result of Rényi on the distribution of $\Omega (n)-\omega (n)$.

In 1993, E. Lutwak established a minimax inequality for inscribed cones of origin symmetric convex bodies. In this work, we re-prove Lutwak’s result using a maxmin inequality for circumscribed cylinders. Furthermore, we explore connections between the maximum volume of inscribed double cones of a centered convex body and the minimum volume of circumscribed cylinders of its polar body.

I provide simplified proofs for each of the following fundamental theorems regarding selection principles:

1. (1) The Quasinormal Convergence Theorem, due to the author and Zdomskyy, asserting that a certain, important property of the space of continuous functions on a space is actually preserved by Borel images of that space.

2. (2) The Scheepers Diagram Last Theorem, due to Peng, completing all provable implications in the diagram.

3. (3) The Menger Game Theorem, due to Telgársky, determining when Bob has a winning strategy in the game version of Menger’s covering property.

4. (4) A lower bound on the additivity of Rothberger’s covering property, due to Carlson.

The simplified proofs lead to several new results.

In this work, we study the existence of solutions of nonlinear fractional coupled system of $\varphi $-Hilfer type in the frame of Banach spaces. We improve a property of a measure of noncompactness in a suitably selected Banach space. Darbo’s fixed point theorem is applied to obtain a new existence result. Finally, the validity of our result is illustrated through an example.

We obtain a new upper bound for Neumann eigenvalues of the Laplacian on a bounded convex domain in Euclidean space. As an application of the upper bound, we derive universal inequalities for Neumann eigenvalues of the Laplacian.

We study Toeplitz operators on the space of all real analytic functions on the real line and the space of all holomorphic functions on finitely connected domains in the complex plane. In both cases, we show that the space of all Toeplitz operators is isomorphic, when equipped with the topology of uniform convergence on bounded sets, with the symbol algebra. This is surprising in view of our previous results, since we showed that the symbol map is not continuous in this topology on the algebra generated by all Toeplitz operators. We also show that in the case of the Fréchet space of all holomorphic functions on a finitely connected domain in the complex plane, the commutator ideal is dense in the algebra generated by all Toeplitz operators in the topology of uniform convergence on bounded sets.

We discuss the class of functions, which are well approximated on compacta by the geometric mean of the eigenvalues of a unital (completely) positive map into a matrix algebra or more generally a type $II_1$ factor, using the notion of a Fuglede–Kadison determinant. In two variables, the two classes are the same, but in three or more noncommuting variables, there are generally functions arising from type $II_1$ von Neumann algebras, due to the recently established failure of the Connes embedding conjecture. The question of whether or not approximability holds for scalar inputs is shown to be equivalent to a restricted form of the Connes embedding conjecture, the so-called shuffle-word-embedding conjecture.

We prove that every topologically stable homeomorphism with global attractor of $\mathbb {R}^n$ is topologically stable on its global attractor. The converse is not true. On the other hand, if a homeomorphism with global attractor of a locally compact metric space is expansive and has the shadowing property, then it is topologically stable. This extends the Walters stability theorem (Walters, On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems, 1978, pp. 231–244).

We provide conditions under which a generalized shift-invariant (GSI) system can be approximated by a GSI system for which the generators have compact support in the Fourier domain. The approximation quality will be measured in terms of the Bessel bound (upper frame bound) for the difference between the two GSI systems. In particular, this leads to easily verifiable conditions for a perturbation of a GSI system to preserve the frame property.

We prove a comparison theorem between Greenberg–Benois $\mathcal {L}$-invariants and Fontaine–Mazur $\mathcal {L}$-invariants. Such a comparison theorem supplies an affirmative answer to a speculation of Besser–de Shalit.

In this note, we start on the study of the sufficient conditions for the boundedness of Hausdorff operators

$$ \begin{align*}(\mathcal{H}_{K,\mu}f)(z):=\int_{\mathbb{D}}K(w)f(\sigma_w(z))d\mu(w)\end{align*} $$

$\mu $

$\mu $

$\mathbb {D}$

$\sigma _w(z)$

$\mathbb {D}$

We settle the question of how to compute the entry and leaving arcs for turnpikes in autonomous variational problems, in the one-dimensional case using the phase space of the vector field associated with the Euler equation, and the initial/final and/or the transversality condition. The results hinge on the realization that extremals are the contours of a well-known function and that the transversality condition is (generically) a curve. An approximation algorithm is presented, and an example is included for completeness.