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In this paper we study the orbit closure problem for a reductive group $G\subseteq GL(X)$ acting on a finite dimensional vector space $V$ over $\C$. We assume that the center of $GL(X)$ lies within $G$ and acts on $V$ through a fixed non-trivial character. We study points $y,z\in V$ where (i) $z$ is obtained as the leading term of the action of a 1-parameter subgroup $\lambda (t)\subseteq G$ on $y$, and (ii) $y$ and $z$ have large distinctive stabilizers $K,H \subseteq G$. Let $O(z)$ (resp. $O(y)$) denote the $G$-orbits of $z$ (resp. $y$), and $\overline{O(z)}$ (resp. $\overline{O(y)}$) their closures, then (i) implies that $z\in \overline{O(y)}$. We address the question: under what conditions can (i) and (ii) be simultaneously satisfied, i.e, there exists a 1-PS $\lambda \subseteq G$ for which $z$ is observed as a limit of $y$. Using $\lambda$, we develop a leading term analysis which applies to $V$ as well as to ${\cal G}= Lie(G)$ the Lie algebra of $G$ and its subalgebras ${\cal K}$ and ${\cal H}$, the Lie algebras of $K$ and $H$ respectively. Through this we construct the Lie algebra $\hat{\cal K} \subseteq {\cal H}$ which connects $y$ and $z$ through their Lie algebras. We develop the properties of $\hat{\cal K}$ and relate it to the action of ${\cal H}$ on $\overline{N}=V/T_z O(z)$, the normal slice to the orbit $O(z)$. We examine the case of {\em alignment} when a semisimple element belongs to both ${\cal H}$ and ${\cal K}$, and the conditions for the same. We illustrate some consequences of alignment. Next, we examine the possibility of {\em intermediate $G$-varieties} $W$ which lie between the orbit closures of $z$ and $y$, i.e. $\overline{O(z)} \subsetneq W \subsetneq O(y)$. These have a direct bearing on representation theoretic as well as geometric properties which connect $z$ and $y$.

In 1986, Flagg and Friedman \cite{ff} gave an elegant alternative proof of the faithfulness of G\"{o}del translation $(\cdot)^\Box$ of Heyting arithmetic $\bf HA$ to Shapiro's epistemic arithmetic $\bf EA$. In \S 2, we shall prove the faithfulness of $(\cdot)^\Box$ without using stability, by introducing another translation from an epistemic system to corresponding intuitionistic system which we shall call \it the modified Rasiowa-Sikorski translation\rm . That is, this introduction of the new translation simplifies the original Flagg and Friedman's proof. In \S 3, we shall give some applications of the modified one for the disjunction property ($\mathsf{DP}$) and the numerical existence property ($\mathsf{NEP}$) of Heyting arithmetic. In \S 4, we shall show that epistemic Markov's rule $\mathsf{EMR}$ in $\bf EA$ is proved via $\bf HA$. So $\bf EA$ $\vdash \mathsf{EMR}$ and $\bf HA$ $\vdash \mathsf{MR}$ are equivalent. In \S 5, we shall give some relations among the translations treated in the previous sections. In \S 6, we shall give an alternative proof of Glivenko's theorem. In \S 7, we shall propose several (modal-)epistemic versions of Markov's rule for Horsten's modal-epistemic arithmetic $\bf MEA$. And, as in \S 4, we shall study some meta-implications among those versions of Markov's rules in $\bf MEA$ and one in $\bf HA$. Friedman and Sheard gave a modal analogue $\mathsf{FS}$ (i.e. Theorem in \cite{fs}) of Friedman's theorem $\mathsf{F}$ (i.e. Theorem 1 in \cite {friedman}): \it Any recursively enumerable extension of $\bf HA$ which has $\mathsf{DP}$ also has $\mathsf{NPE}$\rm . In \S 8, we shall propose a modified version of \it Fundamental Conjecture \rm $\mathsf{FC}$ ($\mathsf{FS} \Longrightarrow \mathsf{F}$) proposed by the author as $\Delta_0$-Fundamental Conjecture. In \S 9, I shall give some discussions and my philosophy.

We provide explicit convergence rates for Chernoff-type approximations of convex monotone semigroups which have the form $S(t)f=\lim_{n\to\infty}I(\frac{t}{n})^n f$ for bounded continuous functions $f$. Under suitable conditions on the one-step operators $I(t)$ regarding the time regularity and consistency of the approximation scheme, we obtain $\|S(t)f-I(\frac{t}{n})^n f\|_\infty\leq cn^{-\gamma}$ for bounded Lipschitz continuous functions $f$, where $c\geq 0$ and $\gamma>0$ are determined explicitly. Moreover, the mapping $t\mapsto S(t)f$ is H\"older continuous. These results are closely related to monotone approximation schemes for viscosity solutions but are obtained independently by following a recently developed semigroup approach to Hamilton-Jacobi-Bellman equations which uniquely characterizes semigroups via their $\Gamma$-generators. The different approach allows to consider convex rather than sublinear equations and the results can be extended to unbounded functions by modifying the norm with a suitable weight function. Furthermore, up to possibly different consistency errors for the operators $I(t)$, the upper and lower bound for the error between the semigroup and the iterated operators are symmetric. The abstract results are applied to Nisio semigroups and limit theorems for convex expectations.

The categorical Gini correlation, $\rho_g$, was proposed by Dang et al. to measure the dependence between a categorical variable, $Y$ , and a numerical variable, $X$. It has been shown that $\rho_g$ has more appealing properties than current existing dependence measurements. In this paper, we develop the jackknife empirical likelihood (JEL) method for $\rho_g$. Confidence intervals for the Gini correlation are constructed without estimating the asymptotic variance. Adjusted and weighted JEL are explored to improve the performance of the standard JEL. Simulation studies show that our methods are competitive to existing methods in terms of coverage accuracy and shortness of confidence intervals. The proposed methods are illustrated in an application on two real datasets.

This work is concerned with cone-beam computed tomography with circular source trajectory, where the reconstruction inverse problem requires an accurate knowledge of source, detector and rotational axis relative positions and orientations. We address this problem as a preceding step of the reconstruction process directly from the acquired projections. The method estimates both the detector shift (orthogonal to focal and rotational axes) and the in-plane detector rotation, relative to source and rotational axis. The obtained algorithm is based on a fan-beam symmetry condition and the variable projection optimization approach with a low computational cost. Therefore, the alignment problem for fan-beam tomography is addressed as well. The methods are validated with simulated and real industrial tomographic data with code examples available for both fan- and cone-beam geometries.

We introduce the extremal range, a local statistic for studying the spatial extent of extreme events in random fields on $\mathbb{R}^2$. Conditioned on exceedance of a high threshold at a location $s$, the extremal range at $s$ is the random variable defined as the smallest distance from $s$ to a location where there is a non-exceedance. We leverage tools from excursion-set theory to study distributional properties of the extremal range, propose parametric models and predict the median extremal range at extreme threshold levels. The extremal range captures the rate at which the spatial extent of conditional extreme events scales for increasingly high thresholds, and we relate its distributional properties with the bivariate tail dependence coefficient and the extremal index of time series in classical Extreme-Value Theory. Consistent estimation of the distribution function of the extremal range for stationary random fields is proven. For non-stationary random fields, we implement generalized additive median regression to predict extremal-range maps at very high threshold levels. An application to two large daily temperature datasets, namely reanalyses and climate-model simulations for France, highlights decreasing extremal dependence for increasing threshold levels and reveals strong differences in joint tail decay rates between reanalyses and simulations.

The stability of an approximating sequence $(A_n)$ for an operator $A$ usually requires, besides invertibility of $A$, the invertibility of further operators, say $B, C, \dots$, that are well-associated to the sequence $(A_n)$. We study this set, $\{A,B,C,\dots\}$, of so-called stability indicators of $(A_n)$ and connect it to the asymptotics of $\|A_n\|$, $\|A_n^{-1}\|$ and $\kappa(A_n)=\|A_n\|\|A_n^{-1}\|$ as well as to spectral pollution by showing that $\limsup {\rm Spec}_\varepsilon A_n= {\rm Spec}_\varepsilon A\cup{\rm Spec}_\varepsilon B\cup{\rm Spec}_\varepsilon C\cup\dots$. We further specify, for each of $\|A_n\|$, $\|A_n^{-1}\|$, $\kappa(A_n)$ and ${\rm Spec}_\varepsilon A_n$, under which conditions even convergence applies.

This study proposes an interpretable neural network-based non-proportional odds model (N$^3$POM) for ordinal regression. N$^3$POM is different from conventional approaches to ordinal regression with non-proportional models in several ways: (1) N$^3$POM is designed to directly handle continuous responses, whereas standard methods typically treat de facto ordered continuous variables as discrete, (2) instead of estimating response-dependent finite coefficients of linear models from discrete responses as is done in conventional approaches, we train a non-linear neural network to serve as a coefficient function. Thanks to the neural network, N$^3$POM offers flexibility while preserving the interpretability of conventional ordinal regression. We establish a sufficient condition under which the predicted conditional cumulative probability locally satisfies the monotonicity constraint over a user-specified region in the covariate space. Additionally, we provide a monotonicity-preserving stochastic (MPS) algorithm for effectively training the neural network. We apply N$^3$POM to several real-world datasets.

Synthesis consists in deciding whether a given labeled transition system (TS) $A$ can be implemented by a net $N$ of type $\tau$. In case of a negative decision, it may be possible to convert $A$ into an implementable TS $B$ by applying various modification techniques, like relabeling edges that previously had the same label, suppressing edges/states/events, etc. It may however be useful to limit the number of such modifications to stay close to the original problem, or optimize the technique. In this paper, we show that most of the corresponding problems are NP-complete if $\tau$ corresponds to the type of flip-flop nets or some flip-flop net derivatives.

We improve the previously best known upper bounds on the sizes of $\theta$-spherical codes for every $\theta<\theta^*\approx 62.997^{\circ}$ at least by a factor of $0.4325$, in sufficiently high dimensions. Furthermore, for sphere packing densities in dimensions $n\geq 2000$ we have an improvement at least by a factor of $0.4325+\frac{51}{n}$. Our method also breaks many non-numerical sphere packing density bounds in smaller dimensions. This is the first such improvement for each dimension since the work of Kabatyanskii and Levenshtein~\cite{KL} and its later improvement by Levenshtein~\cite{Leven79}. Novelties of this paper include the analysis of triple correlations, usage of the concentration of mass in high dimensions, and the study of the spacings between the roots of Jacobi polynomials.