The Vortex-like Behavior of the Riemann Zeta Function to the Right of the Critical Strip

Based on an equivalence relation that was established recently on exponential sums, in this paper we study the class of functions that are equivalent to the Riemann zeta function in the half-plane $$\{s\in {\mathbb {C}}:\mathrm{Re}\, s>1\}$$ { s ∈ C : Re s > 1 } . In connection with this class of functions, we first determine the value of the maximum abscissa from which the images of any function in it cannot take a prefixed argument. The main result shows that each of these functions experiments a vortex-like behavior in the sense that the main argument of its images varies indefinitely near the vertical line $$\mathrm{Re}\, s=1$$ Re s = 1 . In particular, regarding the Riemann zeta function $$\zeta (s)$$ ζ ( s ) , for every $$\sigma _0>1$$ σ 0 > 1 we can assure the existence of a relatively dense set of real numbers $$\{t_m\}_{m\ge 1}$$ { t m } m ≥ 1 such that the parametrized curve traced by the points $$(\mathrm{Re} (\zeta (\sigma +it_m)),\mathrm{Im}(\zeta (\sigma +it_m)))$$ ( Re ( ζ ( σ + i t m ) ) , Im ( ζ ( σ + i t m ) ) ) , with $$\sigma \in (1,\sigma _0)$$ σ ∈ ( 1 , σ 0 ) , makes a prefixed finite number of turns around the origin.

Cluster Analysis of Financial Strategies of Companies

Measuring the value of companies and assessing their risk often relies on econometric methods that consider companies as a set of objects under study, homogeneous in the sense of their use of financial strategies. This paper shows that cluster analysis methods can divide companies into classes according to financial strategies that they employ. This indicates that homogeneity can be considered within these classes, while between-class companies should rather be perceived as heterogeneous. The clustering of companies has to be performed on quite a dense set of strategies, which requires a combination of formal and heuristic methods. To divide companies into classes, we used financial coefficients characterizing strategies for the 2030 largest non-financial companies within the time period from 2006 to 2018. As a result, a stable division into seven clusters/strategies was obtained. We revealed that some strategies were more characteristic for the companies of high-tech economy, while others were typical for the companies in basic industries. The dynamics of clusters is characterized by an increase in the share of risky strategies. A good meaningful interpretation of the resulting clustering confirms its consistency. The identified clusters can be used as dummy variables in econometric studies of companies to improve the quality of the results.

College Attendance among Low-Income Youth: Explaining Differences across Wisconsin High Schools

Bolstering low-income students’ postsecondary participation is important to remediate these students’ disadvantages and to improve society’s overall level of education. Recent research has demonstrated that secondary schools vary considerably in their tendencies to send students to postsecondary education, but existing research has not systematically identified the school characteristics that explain this variation. Identifying these characteristics can help improve low-income students’ postsecondary outcomes. We identify relevant characteristics using population-level data from Wisconsin, a mid-size state in the United States. We first show that Wisconsin’s income-based disparities in postsecondary participation are wide, even net of academic achievement. Next, we show that several geographic characteristics of schools help explain between-secondary school variation in low-income students’ postsecondary outcomes. Finally, we test whether a dense set of school organisational features explain any remaining variation. We find that these features explain virtually no variation in secondary schools’ tendencies to send low-income students to postsecondary education.

On the spectrum of a multidimensional periodic magnetic Shrödinger operator with a singular electric potential

We prove absolute continuity of the spectrum of a periodic $n$-dimensional Schrödinger operator for $n\geqslant 4$. Certain conditions on the magnetic potential $A$ and the electric potential $V+\sum f_j\delta_{S_j}$ are supposed to be fulfilled. In particular, we can assume that the following conditions are satisfied. (1) The magnetic potential $A\colon{\mathbb{R}}^n\to{\mathbb{R}}^n$ either has an absolutely convergent Fourier series or belongs to the space $H^q_{\mathrm{loc}}({\mathbb{R}}^n;{\mathbb{R}}^n)$, $2q>n-1$, or to the space $C({\mathbb{R}}^n;{\mathbb{R}}^n)\cap H^q_{\mathrm{loc}}({\mathbb{R}}^n;{\mathbb{R}}^n)$, $2q>n-2$. (2) The function $V\colon{\mathbb{R}}^n\to\mathbb{R}$ belongs to Morrey space ${\mathfrak{L}}^{2,p}$, $p\in \big(\frac{n-1}{2},\frac{n}{2}\big]$, of periodic functions (with a given period lattice), and $$\lim\limits_{\tau\to+0}\sup\limits_{0<r\leqslant\tau}\sup\limits_{x\in{\mathbb{R}}^n}r^2\bigg(\big(v(B^n_r)\big)^{-1}\int_{B^n_r(x)}|{\mathcal{V}}(y)|^pdy\bigg)^{1/p}\leqslant C,$$ where $B^n_r(x)$ is a closed ball of radius $r>0$ centered at a point $x\in{\mathbb{R}}^n$, $B^n_r=B^n_r(0)$, $v(B^n_r)$ is volume of the ball $B^n_r$, $C=C(n,p;A)>0$. (3) $\delta_{S_j}$ are $\delta$-functions concentrated on (piecewise) $C^1$-smooth periodic hypersurfaces $S_j$, $f_j\in L^p_{\mathrm{loc}}(S_j)$, $j=1,\ldots,m$. Some additional geometric conditions are imposed on the hypersurfaces $S_j$, and these conditions determine the choice of numbers $p\geqslant n-1$. In particular, let hypersurfaces $S_j$ be $C^2$-smooth, the unit vector $e$ be arbitrarily taken from some dense set of the unit sphere $S^{n-1}$ dependent on the magnetic potential $A$, and the normal curvature of the hypersurfaces $S_j$ in the direction of the unit vector $e$ be nonzero at all points of tangency of the hypersurfaces $S_j$ and the lines $\{x_0+te\colon t\in\mathbb{R}\}$, $x_0\in{\mathbb{R}}^n$. Then we can choose the number $p>\frac{3n}{2}-3$, $n\geqslant 4$.

Dynamics of the meromorphic families $f_\lambda=\lambda \tan^pz^q$

This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite. Here we look at a generalization of the family of polynomials $P_a(z)=z^{d-1}(z- \frac{da}{(d-1)})$, the family $f_{\lambda}=\lambda \tan^p z^q$. These functions have a super-attractive fixed point, and, depending on $p$, one or two asymptotic values. Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are required here. Adding transcendental methods to standard ones, we give a description of the dynamical properties; in particular we prove the Julia set of a hyperbolic map is either connected and locally connected or a Cantor set. We also give a description of the parameter plane of the family $f_{\lambda}$. Again there are similarities to and differences from the parameter plane of the family $P_a$ and again there are new techniques. In particular, we prove there is dense set of points on the boundaries of the hyperbolic components that are accessible along curves and we characterize these points.

On the Recurrent C 0 -Semigroups, Their Existence, and Some Criteria

In this paper, recurrent C 0 -semigroups are introduced and investigated. It is proved that, despite hypercyclic C 0 -semigroups, recurrent C 0 -semigroups can be found on finite-dimensional Banach spaces. Some criteria are stated for recurrence, which is based on open sets, neighborhoods of zero, and special eigenvectors. It is established that having a dense set of recurrent vectors is a sufficient and necessary condition for a C 0 -semigroup to be recurrent. Moreover, the direct sum of recurrent C 0 -semigroups is investigated.

Quantitative Global-Local Mixing for Accessible Skew Products

We study global-local mixing for a family of accessible skew products with an exponentially mixing base and non-compact fibers, preserving an infinite measure. For a dense set of almost periodic global observables, we prove rapid mixing, and for a dense set of global observables vanishing at infinity, we prove polynomial mixing. More generally, we relate the speed of mixing to the “low frequency behavior” of the spectral measure associated to our global observables. Our strategy relies on a careful choice of the spaces of observables and on the study of a family of twisted transfer operators.

Shapes of hyperbolic triangles and once-punctured torus groups

Let $$\Delta $$ Δ be a hyperbolic triangle with a fixed area $$\varphi $$ φ . We prove that for all but countably many $$\varphi $$ φ , generic choices of $$\Delta $$ Δ have the property that the group generated by the $$\pi $$ π -rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $$\varphi \in (0,\pi ){\setminus }\mathbb {Q}\pi $$ φ ∈ ( 0 , π ) \ Q π , a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $$\mathfrak {C}_\theta $$ C θ of singular hyperbolic metrics on a torus with a single cone point of angle $$\theta =2(\pi -\varphi )$$ θ = 2 ( π - φ ) , and answer an analogous question for the holonomy map $$\rho _\xi $$ ρ ξ of such a hyperbolic structure $$\xi $$ ξ . In an appendix by Gao, concrete examples of $$\theta $$ θ and $$\xi \in \mathfrak {C}_\theta $$ ξ ∈ C θ are given where the image of each $$\rho _\xi $$ ρ ξ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds.

Dense and σ-Porous Subsets in Some Families of Darboux Functions

G. Ivanova and E. Wagner-Bojakowska shown that the set of Darboux quasi-continuous functions with nowhere dense set of discontinuity points is dense in the metric space of Darboux quasi-continuous functions with the supremum metric. We prove that this set also is σ-strongly porous in such space. We obtain the symmetrical result for the family of strong Świątkowski functions, i.e., that the family of strong Świątkowski functions with nowhere dense set of discontinuity points is dense (thus, “large”) and σ-strongly porous (thus, asymmetrically, “small”) in the family of strong Świątkowski functions.