A Kreĭn string is (essentially) a pair \(\mathsf{S}[L,m]\) where \(0<L\le \infty \) and \(m:[0,L)\rightarrow [0,\infty )\) is nondecreasing. Each string gives rise to an operator model, the Kreĭn-Feller differential operator \(-D_mD_x\) acting in the space \(L^2(dm)\). This operator has a selfadjoint realization which is nonnegative. Provided that \(L+\lim _{x\rightarrow L}m(x)<\infty \), this realization has discrete spectrum and, when \((\lambda _n)\) denotes the sequence of positive eigenvalues arranged increasingly, then $$\begin{aligned} \lim \frac{n}{\sqrt{\lambda _n}}=\frac{1}{\pi }\int \limits _0^L\sqrt{m'(x)}\,dx \,. \end{aligned}$$ We show that for a class of strings defined by a weaker growth restriction the spectrum is discrete, the integral on the right side is still finite, and the asymptotic behaviour of the eigenvalues is determined by the above formula.

We prove that the Eliahou-Kervaire resolution of a Cohen-Macaulay stable monomial is supported by a regular CW complex whose underlying space is a closed ball. We also show that the modified Eliahou-Kervaire resolutions of variants of a Borel fixed ideal (e.g., a squarefree strongly stable ideal) are supported by regular CW complexes, and their underlying spaces are closed balls in the Cohen-Macaulay case.

We study the approximate controllability of partial fractional neutral stochastic functional integro-differential inclusions with state-dependent delay under the assumptions that the corresponding linear system is approximately controllable. Using fractional calculus, stochastic analysis theory, and the fixed-point technique with the properties of analytic \(\alpha \)-resolvent operators, a new set of sufficient conditions for approximate controllability of fractional stochastic functional integro-differential inclusions are formulated and proved. The results in this paper are generalization and continuation of the recent results on this issue. An example is provided to show the application of our result.

This paper is concerned with the existence of \(\alpha \)-mild solutions for a class of fractional stochastic integro-differential evolution equations with nonlocal initial conditions in a real separable Hilbert space. We assume that the linear part generates a compact, analytic and uniformly bounded semigroup, the nonlinear part satisfies some local growth conditions in Hilbert space \(\mathbb {H}\) and the nonlocal term satisfies some local growth conditions in fractional power space \(\mathbb {H}_\alpha \). The result obtained in this paper improves and extends some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract result.

A generalized Cauchy problem for nonlinear hyperbolic functional differential systems is considered. A theorem on the existence of weak solutions is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions of this system is proved by using a method of successive approximations. It is shown a result on the differentiability of solutions with respect to initial functions. This is the main result of the paper.

Let \(X\) be a Banach space. Nakamura and Hashimoto in (Proc Japan Acad Ser A 87:77–82, 2011), we showed that for every \(f\in L_{1}(\mathbb {R})\), $$\begin{aligned} \lim _{h\rightarrow 0}\int \limits _{\mathbb {R}}\left| \frac{f(t+h)-f(t)}{h}\right| \, dt=\hbox {ess} V_{1}(f). \end{aligned}$$ In this paper, we are concerned with the limit $$\begin{aligned} \lim _{h\rightarrow 0}\int \limits _{\mathbb {R}}\left\| \frac{f(t+h)-f(t)}{h}\right\| ^p\, dt \end{aligned}$$ (*) for \(f\in L_1^{\hbox {loc}}(\mathbb {R},X)\). We show that the limit \((*)\) coincides with the essential \(p\)-variation of \(f\) in the sense of F. Riesz, and we give characterizations of functions with bounded essential \(p\)-variation, i.e, \(\hbox {ess} V_p(f,X)<\infty \).

In this note we extend connectedness results to formal properties of inverse images under proper maps of Schubert varieties and of the diagonal in products of projective rational homogeneous spaces.

We characterize the boundedness, compactness and weak compactness of Volterra operators $$V_{g}(f)(z) := \int _{0}^{z}f({\zeta })g^{\prime }({\zeta })\,d{\zeta }$$ acting between different weighted spaces of type $$H^{\infty }_{v}$$ in terms of the symbol function $$g$$ , for the case when $$v$$ is a quasi-normal weight, a notion weaker than normality. Then we apply the characterization of compactness to analyze the behavior of semigroups of composition operators on $$H^{\infty }_{v}$$ .

For the Hardy space $$H^p(\mathbb{K }),\;$$ $$ 0<p\le 1$$ and $$\mathbb K $$ being the Laguerre hypergroup, we shall establish a Hardy’s type inequality associated with Laguerre Fourier transform for the strip $$\frac{Q}{2}(2-p)<\sigma < \frac{Q}{2}+\frac{p}{2}(N+1),$$ where $$ N=\left[Q\left(\frac{1}{p}-1\right)\right]$$ is the greatest integer not exceeding $$Q\left(\frac{1}{p}-1\right)$$ and $$Q $$ is the homogenous dimension of $$\mathbb{K }.$$

In this paper, we investigate the NP-hard absolute value equations (AVE) Ax − |x| = b and establish two computational enclosures for the solution of AVE. The condition for enclosing the solution and guaranteeing the existence of solutions of AVE is presented in a given domain. Based on this condition, new algorithms are designed to compute an interval to enclose the unknown solution of AVE. Numerical results are reported to support the theoretical analysis in the paper.