In this paper, we show that the expansions of functions from $L^p$-Paley–Wiener type spaces in terms of the prolate spheroidal wave functions converge almost everywhere for $1<p<\infty$, even in the cases when they might not converge in $L^p$-norm. We thereby consider the classical Paley–Wiener spaces $P{W}_c^p\subset L^p\left(\mathbb{R}\right)$ of functions whose Fourier transform is supported in $[-c,c]$ and Paley–Wiener-like spaces $B_{\alpha ,c}^p\subset L^p(0,\infty )$ of functions whose Hankel transform ${\mathcal{H}}^{\alpha }$ is supported in $[0,c]$. As a side product, we show the continuity of the projection operator $P_c^{\alpha }f:={\mathcal{H}}^{\alpha }({\chi }_{[0,c]}\cdot {\mathcal{H}}^{\alpha }f)$ from $L^p(0,\infty )$ to $L^q(0,\infty )$, $1<p\le q<\infty$.

We determine the condition on a given lens space having a realization as a closure of homology cobordism over a planar surface with a given number of boundary components. As a corollary, we see that every lens space is represented as a closure of homology cobordism over a planar surface with three boundary components. In the proof of this corollary, we use Chebotarev’s density theorem.

We define ${\mathrm{SL}}_r$-opers in the setup of vector bundles on curves with a parabolic structure over a divisor. Basic properties of these objects are investigated.

In this paper, we study deformation in free probability theory. Our setting is systems of specific Banach-space operators acting on a $C^{\ast }$-algebra ${\mathcal{X}}_{\phi }$ generated by mutually free, infinitely-many semicircular elements. In particular, we give a constructive classification for the cases where those operators are generated by certain $\ast$-homomorphisms on ${\mathcal{X}}_{\phi }$. Our main results not only classify and characterize specific types of the operators, but they also show how such operators deform the free probability on ${\mathcal{X}}_{\phi }$. In particular, we outline how properties of those operators affect the semicircular law.

Peres and Virág in 2005 studied the zeros of the random power series ${\sum }_{n=1}^{\infty }{a}_n{z}^{n-1}$ in the unit disc whose coefficients $\left(a_n\right)$ are independent and identically distributed symmetrical Gaussian complex random variables. They discovered that its zeros constitute a determinant point process and derived many other interesting related properties. In this paper, we study the general Gaussian power series where the independence requirement is relaxed by taking $f\left(z\right)={\sum }_{n=1}^{\infty }{\Delta }_n{z}^n$, where $\left({\Delta }_n\right)$ is the classical fractional Gaussian noise of index $0\le H<1$. The intensity of the point process of the zeros of $f$ is used to derive important properties. As for the case of independent variables (corresponding to $H=1/2$), the zeros cluster around the unit circle, but in any domain inside the open unit disc, the dependent case (that is, $H\ne 1/2$) yields fewer zeros than the classical case of independent random variables. This implies that the clustering around the unit circle is faster than in the independent case. Moreover, in contrast to independent random variables, the zeros of $f$ are generally asymmetrically distributed in the unit circle. However, asymptotically, the two point processes are equivalent except near the sole point $z=1$. An interesting open problem is whether this point process is also a determinantal process.

Let $\left\{a\right(x){\}}_{x=1}^{\infty }$ be a positive, real-valued, lacunary sequence. This note shows that the pair correlation function of the fractional parts of the dilations $\alpha a\left(x\right)$ is Poissonian for Lebesgue almost every $\alpha \in \mathbb{R}$. By using harmonic analysis, our result—irrespective of the choice of the real-valued sequence $\left\{a\right(x){\}}_{x=1}^{\infty }$—can essentially be reduced to showing that the number of solutions to the Diophantine inequality $$\left|n_1\right(a\left(x_1\right)-a\left(y_1\right))-n_2(a\left(x_2\right)-a\left(y_2\right)\left)\right|<1$$ in integer six-tuples $(n_1,n_2,x_1,x_2,y_1,y_2)$ located in the box $[-N,N{]}^6$ with the “excluded diagonals”; that is, $$x_1\ne y_1,x_2\ne y_2,(n_1,n_2)\ne (0,0),$$ is at most $N^{4-\delta }$ for some fixed $\delta >0$, for all sufficiently large $N$.

Let $R=K[X_1,\dots ,X_n]$, where $K$ is a field of characteristic zero, and let $A_n\left(K\right)$ be the $n$th Weyl algebra over $K$. We give standard grading on $R$ and $A_n\left(K\right)$. Let $I$, $J$ be homogeneous ideals of $R$. Let $M=H_I^i\left(R\right)$ and $N=H_J^j\left(R\right)$ for some $i,j$. We show that ${Ext}_{A_n\left(K\right)}^{\nu }(M,N)$ is concentrated in degree zero for all $\nu \ge 0$ (i.e., ${Ext}_{A_n\left(K\right)}^{\nu }(M,N{)}_l=0$ for $l\ne 0$). This proves a conjecture stated in Part I of this paper (T. J. Puthenpurakal and J. Singh, On derived functors of graded local cohomology modules, Math. Proc. Cambridge Philos. Soc. 167 (2018), no. 3, 549–565).

The definition of the usual $p$th Weyl semi-norm for sequences is extended to the case of $(C,\alpha )$ averages for $0<\alpha \le 1$ and the $(p,\alpha )$-Besicovitch sequences are defined similarly to the classical case $\alpha =1$. We study the effects of $(p,\alpha )$-Besicovitch sequences with non-integral orders as good modulators. The major finding is the almost everywhere convergence of $(C,\alpha )$ ergodic averages with discrete and continuous $(p,\alpha )$-Besicovitch modulators for Dunford–Schwartz operators. The results have the additional advantage that they are sufficiently general to give as corollaries a (new) weighted Abelian ergodic theorem and the a.e. convergence of random $(C,\alpha )$-ergodic averages for Dunford–Schwartz operators.