We further develop the abstract algebraic logic approach to input/output logic initiated in \cite{wollic22}, where the family of selfextensional logics was proposed as a general background environment for input/output logics. In this paper, we introduce and discuss the generalizations of several types of permission (negative, dual negative, static, dynamic), as well as their interactions with normative systems, to various families of selfextensional logics, thereby proposing a systematic approach to the definition of normative and permission systems on nonclassical propositional bases.
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We consider the cubic nonlinear Schr\"odinger equation with a spatially rough potential, a key equation in the mathematical setup for nonlinear Anderson localization. Our study comprises two main parts: new optimal results on the well-posedness analysis on the PDE level, and subsequently a new efficient numerical method, its convergence analysis and simulations that illustrate our analytical results. In the analysis part, our results focus on understanding how the regularity of the solution is influenced by the regularity of the potential, where we provide quantitative and explicit characterizations. Ill-posedness results are also established to demonstrate the sharpness of the obtained regularity characterizations and to indicate the minimum regularity required from the potential for the NLS to be solvable. Building upon the obtained regularity results, we design an appropriate numerical discretization for the model and establish its convergence with an optimal error bound. The numerical experiments in the end not only verify the theoretical regularity results, but also confirm the established convergence rate of the proposed scheme. Additionally, a comparison with other existing schemes is conducted to demonstrate the better accuracy of our new scheme in the case of a rough potential.
Quantifying the heterogeneity is an important issue in meta-analysis, and among the existing measures, the $I^2$ statistic is most commonly used. In this paper, we first illustrate with a simple example that the $I^2$ statistic is heavily dependent on the study sample sizes, mainly because it is used to quantify the heterogeneity between the observed effect sizes. To reduce the influence of sample sizes, we introduce an alternative measure that aims to directly measure the heterogeneity between the study populations involved in the meta-analysis. We further propose a new estimator, namely the $I_A^2$ statistic, to estimate the newly defined measure of heterogeneity. For practical implementation, the exact formulas of the $I_A^2$ statistic are also derived under two common scenarios with the effect size as the mean difference (MD) or the standardized mean difference (SMD). Simulations and real data analysis demonstrate that the $I_A^2$ statistic provides an asymptotically unbiased estimator for the absolute heterogeneity between the study populations, and it is also independent of the study sample sizes as expected. To conclude, our newly defined $I_A^2$ statistic can be used as a supplemental measure of heterogeneity to monitor the situations where the study effect sizes are indeed similar with little biological difference. In such scenario, the fixed-effect model can be appropriate; nevertheless, when the sample sizes are sufficiently large, the $I^2$ statistic may still increase to 1 and subsequently suggest the random-effects model for meta-analysis.
We introduce data structures and algorithms to count numerical inaccuracies arising from usage of floating numbers described in IEEE 754. Here we describe how to estimate precision for some collection of functions most commonly used for array manipulations and training of neural networks. For highly optimized functions like matrix multiplication, we provide a fast estimation of precision and some hint how the estimation can be strengthened.
We address the problem of the best uniform approximation of a continuous function on a convex domain. The approximation is by linear combinations of a finite system of functions (not necessarily Chebyshev) under arbitrary linear constraints. By modifying the concept of alternance and of the Remez iterative procedure we present a method, which demonstrates its efficiency in numerical problems. The linear rate of convergence is proved under some favourable assumptions. A special attention is paid to systems of complex exponents, Gaussian functions, lacunar algebraic and trigonometric polynomials. Applications to signal processing, linear ODE, switching dynamical systems, and to Markov-Bernstein type inequalities are considered.
We consider a class of symmetry hypothesis testing problems including testing isotropy on $\mathbb{R}^d$ and testing rotational symmetry on the hypersphere $\mathcal{S}^{d-1}$. For this class, we study the null and non-null behaviors of Sobolev tests, with emphasis on their consistency rates. Our main results show that: (i) Sobolev tests exhibit a detection threshold (see Bhattacharya, 2019, 2020) that does not only depend on the coefficients defining these tests; and (ii) tests with non-zero coefficients at odd (respectively, even) ranks only are blind to alternatives with angular functions whose $k$th-order derivatives at zero vanish for any $k$ odd (even). Our non-standard asymptotic results are illustrated with Monte Carlo exercises. A case study in astronomy applies the testing toolbox to evaluate the symmetry of orbits of long- and short-period comets.
The numerical approximation of dynamic poroelasticity, modeling flow in deformable porous media, by a family of continuous space-time finite element methods is investigated. Equal order approximation in space without any further stabilization is used for the displacement and pore pressure variable. Optimal order $L^\infty(L^2)$ error estimates are proved and numerically confirmed.
Within the nonparametric diffusion model, we develop a multiple test to infer about similarity of an unknown drift $b$ to some reference drift $b_0$: At prescribed significance, we simultaneously identify those regions where violation from similiarity occurs, without a priori knowledge of their number, size and location. This test is shown to be minimax-optimal and adaptive. At the same time, the procedure is robust under small deviation from Brownian motion as the driving noise process. A detailed investigation for fractional driving noise, which is neither a semimartingale nor a Markov process, is provided for Hurst indices close to the Brownian motion case.
For the Euler scheme of the stochastic linear evolution equations, discrete stochastic maximal $ L^p $-regularity estimate is established, and a sharp error estimate in the norm $ \|\cdot\|_{L^p((0,T)\times\Omega;L^q(\mathcal O))} $, $ p,q \in [2,\infty) $, is derived via a duality argument.
We consider the problem of estimating log-determinants of large, sparse, positive definite matrices. A key focus of our algorithm is to reduce computational cost, and it is based on sparse approximate inverses. The algorithm can be implemented to be adaptive, and it uses graph spline approximation to improve accuracy. We illustrate our approach on classes of large sparse matrices.
Non-classical generalizations of classical modal logic have been developed in the contexts of constructive mathematics and natural language semantics. In this paper, we discuss a general approach to the semantics of non-classical modal logics via algebraic representation theorems. We begin with complete lattices $L$ equipped with an antitone operation $\neg$ sending $1$ to $0$, a completely multiplicative operation $\Box$, and a completely additive operation $\Diamond$. Such lattice expansions can be represented by means of a set $X$ together with binary relations $\vartriangleleft$, $R$, and $Q$, satisfying some first-order conditions, used to represent $(L,\neg)$, $\Box$, and $\Diamond$, respectively. Indeed, any lattice $L$ equipped with such a $\neg$, a multiplicative $\Box$, and an additive $\Diamond$ embeds into the lattice of propositions of a frame $(X,\vartriangleleft,R,Q)$. Building on our recent study of "fundamental logic", we focus on the case where $\neg$ is dually self-adjoint ($a\leq \neg b$ implies $b\leq\neg a$) and $\Diamond \neg a\leq\neg\Box a$. In this case, the representations can be constrained so that $R=Q$, i.e., we need only add a single relation to $(X,\vartriangleleft)$ to represent both $\Box$ and $\Diamond$. Using these results, we prove that a system of fundamental modal logic is sound and complete with respect to an elementary class of bi-relational structures $(X,\vartriangleleft, R)$.
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