亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

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 consider functions $f: \mathbb{Z} \to \mathbb{R}$ and kernels $u: \{-n, \cdots, n\} \to \mathbb{R}$ normalized by $\sum_{\ell = -n}^{n} u(\ell) = 1$, making the convolution $u \ast f$ a "smoother" local average of $f$. We identify which choice of $u$ most effectively smooths the second derivative in the following sense. For each $u$, basic Fourier analysis implies there is a constant $C(u)$ so $\|\Delta(u \ast f)\|_{\ell^2(\mathbb{Z})} \leq C(u)\|f\|_{\ell^2(\mathbb{Z})}$ for all $f: \mathbb{Z} \to \mathbb{R}$. By compactness, there is some $u$ that minimizes $C(u)$ and in this paper, we find explicit expressions for both this minimal $C(u)$ and the minimizing kernel $u$ for every $n$. The minimizing kernel is remarkably close to the Epanechnikov kernel in Statistics. This solves a problem of Kravitz-Steinerberger and an extremal problem for polynomials is solved as a byproduct.

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.

We continue the study of selection and sorting of $n$ numbers under the adversarial comparator model, where comparisons can be adversarially tampered with if the arguments are sufficiently close. We derive a randomized sorting algorithm that does $O(n \log^2 n)$ comparisons and gives a correct answer with high probability, addressing an open problem of Ajtai, Feldman, Hassadim, and Nelson [AFHN15]. Our algorithm also implies a selection algorithm that does $O(n \log n)$ comparisons and gives a correct answer with high probability. Both of these results are a $\log$ factor away from the naive lower bound. [AFHN15] shows an $\Omega(n^{1+\varepsilon})$ lower bound for both sorting and selection in the deterministic case, so our results also prove a discrepancy between what is possible with deterministic and randomized algorithms in this setting. We also consider both sorting and selection in rounds, exploring the tradeoff between accuracy, number of comparisons, and number of rounds. Using results from sorting networks, we give general algorithms for sorting in $d$ rounds where the number of comparisons increases with $d$ and the accuracy decreases with $d$. Using these algorithms, we derive selection algorithms in $d+O(\log d)$ rounds that use the same number of comparisons as the corresponding sorting algorithm, but have a constant accuracy. Notably, this gives selection algorithms in $d$ rounds that use $n^{1 + o(1)}$ comparisons and have constant accuracy for all $d = \omega(1)$, which still beats the deterministic lower bound of $\Omega(n^{1+\varepsilon})$.

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.

Sparse suffix sorting is the problem of sorting $b=o(n)$ suffixes of a string of length $n$. Efficient sparse suffix sorting algorithms have existed for more than a decade. Despite the multitude of works and their justified claims for applications in text indexing, the existing algorithms have not been employed by practitioners. Arguably this is because there are no simple, direct, and efficient algorithms for sparse suffix array construction. We provide two new algorithms for constructing the sparse suffix and LCP arrays that are simultaneously simple, direct, small, and fast. In particular, our algorithms are: simple in the sense that they can be implemented using only basic data structures; direct in the sense that the output arrays are not a byproduct of constructing the sparse suffix tree or an LCE data structure; fast in the sense that they run in $\mathcal{O}(n\log b)$ time, in the worst case, or in $\mathcal{O}(n)$ time, when the total number of suffixes with an LCP value greater than $2^{\lfloor \log \frac{n}{b} \rfloor + 1}-1$ is in $\mathcal{O}(b/\log b)$, matching the time of the optimal yet much more complicated algorithms [Gawrychowski and Kociumaka, SODA 2017; Birenzwige et al., SODA 2020]; and small in the sense that they can be implemented using only $8b+o(b)$ machine words. Our algorithms are simplified, yet non-trivial, space-efficient adaptations of the Monte Carlo algorithm by I et al. for constructing the sparse suffix tree in $\mathcal{O}(n\log b)$ time [STACS 2014]. We also provide proof-of-concept experiments to justify our claims on simplicity and efficiency.

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.

One of the prominent problems with processing and operating on text data is the non uniformity of it. Due to the change in the dialects and languages, the caliber of translation is low. This creates a unique problem while using NLP in text data; which is the spell variation arising from the inconsistent translations and transliterations. This problem can also be further aggravated by the human error arising from the various ways to write a Proper Noun from an Indian language into its English equivalent. Translating proper nouns originating from Indian languages can be complicated as some proper nouns are also used as common nouns which might be taken literally. Applications of NLP that require addresses, names and other proper nouns face this problem frequently. We propose a method to cluster these spell variations for proper nouns using ML techniques and mathematical similarity equations. We aimed to use Affinity Propagation to determine relative similarity between the tokens. The results are augmented by filtering the token-variation pair by a similarity threshold. We were able to reduce the spell variations by a considerable amount. This application can significantly reduce the amount of human annotation efforts needed for data cleansing and formatting.

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.

While existing work in robust deep learning has focused on small pixel-level $\ell_p$ norm-based perturbations, this may not account for perturbations encountered in several real world settings. In many such cases although test data might not be available, broad specifications about the types of perturbations (such as an unknown degree of rotation) may be known. We consider a setup where robustness is expected over an unseen test domain that is not i.i.d. but deviates from the training domain. While this deviation may not be exactly known, its broad characterization is specified a priori, in terms of attributes. We propose an adversarial training approach which learns to generate new samples so as to maximize exposure of the classifier to the attributes-space, without having access to the data from the test domain. Our adversarial training solves a min-max optimization problem, with the inner maximization generating adversarial perturbations, and the outer minimization finding model parameters by optimizing the loss on adversarial perturbations generated from the inner maximization. We demonstrate the applicability of our approach on three types of naturally occurring perturbations -- object-related shifts, geometric transformations, and common image corruptions. Our approach enables deep neural networks to be robust against a wide range of naturally occurring perturbations. We demonstrate the usefulness of the proposed approach by showing the robustness gains of deep neural networks trained using our adversarial training on MNIST, CIFAR-10, and a new variant of the CLEVR dataset.