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We obtain several new comparison results on the distance between zeros and local extrema of solutions for the second order delay differential equation \begin{equation*} x^{\prime \prime }(t)+p(t)x(t-\tau (t))=0,~~t\geq s\text{ }\ \end{equation*} where $\tau :\mathbb{R}\rightarrow \lbrack 0,+\infty )$, $p:\mathbb{R}% \rightarrow \mathbb{R}$ are Lebesgue measurable and uniformly essentially bounded, including the case of a sign-changing coefficient. We are thus able to calculate upper bounds on the semicycle length, which guarantee that an oscillatory solution is bounded or even tends to zero. Using the estimates of the distance between zeros and extrema, we investigate the classification of solutions in the case $p(t)\leq 0,t\in \mathbb{R}.$ Comment: 23 pages

We introduce an algorithmic approach based on generating tree method for enumerating the inversion sequences with various pattern-avoidance restrictions. For a given set of patterns, we propose an algorithm that outputs either an accurate description of the succession rules of the corresponding generating tree or an ansatz. By using this approach, we determine the generating trees for the pattern-classes $I_n(000, 021), I_n(100, 021)$, $I_n(110, 021), I_n(102, 021)$, $I_n(100,012)$, $I_n(011,201)$, $I_n(011,210)$ and $I_n(120,210)$. Then we use the kernel method, obtain generating functions of each class, and find enumerating formulas. Lin and Yan studied the classification of the Wilf-equivalences for inversion sequences avoiding pairs of length-three patterns and showed that there are 48 Wilf classes among 78 pairs. In this paper, we solve six open cases for such pattern classes. Comment: 20 pages, 2 figures. Section 5 added. Typos in Equation 4.16 and Theorem 4.4 corrected. The authors thank Jay Pantone for pointing out them

We prove that the theory of the models constructible using finitely many cofinality quantifiers - $C_{\lambda_{1},...,\lambda_{n}}^{*}$ and $C_{<\lambda_{1},...,<\lambda_{n}}^{*}$ for $\lambda_{1},...,\lambda_{n}$ regular cardinals - is set-forcing absolute under the assumption of class many Woodin cardinals, and is independent of the regular cardinals used. Towards this goal we prove some properties of the generic embedding induced from the stationary tower restricted to $<\mu$-closed sets. Comment: 10 pages

For two independent, almost surely finite random variables, independence of their minimum (time) and the event that one of them is either greater, equal or less than the other (cause) is completely characterized. It is shown that, other than for trivial cases where, almost surely, one random variable is greater than or equal to the other, this happens if and only if both random variables are distributed like the same strictly increasing function of two independent random variables, where either both are exponentially distributed or both are geometrically distributed. This is then easily generalized to the multivariate case.

We study the scenery reconstruction problem on the $d$-dimensional torus, proving that a criterion on Fourier coefficients obtained by Matzinger and Lember (2006) for discrete cycles applies also in continuous spaces. In particular, with the right drift, Brownian motion can be used to reconstruct any scenery. To this end, we prove an injectivity property of an infinite Vandermonde matrix. Comment: 16 pages, 3 figures

We construct a Borel maximal eventually different family.

We prove that a Tychonoff space $X$ is an Ascoli space (resp., a sequentially Ascoli space) if and only if for each Banach space $E$, every $k$-continuous and almost $k$-compact (resp., almost $k$-sequential) map $T$ form $X$ into the Banach dual $E'$ of $E$ is continuous.

We propose an accurate numerical scheme for approximating the solution of the two dimensional acoustic wave problem. We use machine learning to find a stencil suitable even in the presence of high wavenumbers. The proposed scheme incorporates physically informed elements from the field of optimized numerical schemes into a convolutional optimization machine learning algorithm.