The aim of this paper is to combine several Ivev-like modal systems characterized by 4-valued non-deterministic matrices (Nmatrices) with IDM4, a 4-valued expansion of Belnap-Dunn's logic FDE with an implication introduced by Pynko in 1999. In order to to this, we introduce a new methodology for combining logics which are characterized by means of swap structures, based on what we call superposition of snapshots. In particular, the combination of IDM4 with Tm, the 4-valued Ivlev's version of KT, will be analyzed with more details. From the semantical perspective, the idea is to combine the 4-valued swap structures (Nmatrices) for Tm (and several of its extensions) with the 4-valued twist structure (logical matrix) for IDM4. This superposition produces a universe of 6 snapshots, with 3 of them being designated. The multioperators over the new universe are defined by combining the specifications of the given swap and twist structures. This gives origin to 6 different paradefinite Ivlev-like modal logics, each one of them characterized by a 6-valued Nmatrix, and conservatively extending the original modal logic and IDM4. This important feature allows us to consider the proposed construction as a genuine technique for combining logics. In addition, it is possible to define in the combined logics a classicality operator in the sense of logics of evidence and truth (LETs). A sound and complete Hilbert-style axiomatization is also presented for the 6 combined systems, as well as a very simple Prolog program which implements the swap structures semantics for the 6 systems, which gives a decision procedure for satisfiability, refutability and validity of formulas in these logics.
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In this paper, I present three closed-form approximations of the two-sample Pearson Bayes factor. The techniques rely on some classical asymptotic results about gamma functions. These approximations permit simple closed-form calculation of the Pearson Bayes factor in cases where only the summary statistics are available (i.e., the t-score and degrees of freedom).
We investigate the randomized decision tree complexity of a specific class of read-once threshold functions. A read-once threshold formula can be defined by a rooted tree, every internal node of which is labeled by a threshold function $T_k^n$ (with output 1 only when at least $k$ out of $n$ input bits are 1) and each leaf by a distinct variable. Such a tree defines a Boolean function in a natural way. We focus on the randomized decision tree complexity of such functions, when the underlying tree is a uniform tree with all its internal nodes labeled by the same threshold function. We prove lower bounds of the form $c(k,n)^d$, where $d$ is the depth of the tree. We also treat trees with alternating levels of AND and OR gates separately and show asymptotically optimal bounds, extending the known bounds for the binary case.
The Graded of Membership (GoM) model is a powerful tool for inferring latent classes in categorical data, which enables subjects to belong to multiple latent classes. However, its application is limited to categorical data with nonnegative integer responses, making it inappropriate for datasets with continuous or negative responses. To address this limitation, this paper proposes a novel model named the Weighted Grade of Membership (WGoM) model. Compared with GoM, our WGoM relaxes GoM's distribution constraint on the generation of a response matrix and it is more general than GoM. We then propose an algorithm to estimate the latent mixed memberships and the other WGoM parameters. We derive the error bounds of the estimated parameters and show that the algorithm is statistically consistent. The algorithmic performance is validated in both synthetic and real-world datasets. The results demonstrate that our algorithm is accurate and efficient, indicating its high potential for practical applications. This paper makes a valuable contribution to the literature by introducing a novel model that extends the applicability of the GoM model and provides a more flexible framework for analyzing categorical data with weighted responses.
In this paper we show that using implicative algebras one can produce models of set theory generalizing Heyting/Boolean-valued models and realizability models of (I)ZF, both in intuitionistic and classical logic. This has as consequence that any topos which is obtained from a Set-based tripos as the result of the tripos-to-topos construction hosts a model of intuitionistic or classical set theory, provided a large enough strongly inaccessible cardinal exists.
B-series and generalizations are a powerful tool for the analysis of numerical integrators. An extension named exotic aromatic B-series was introduced to study the order conditions for sampling the invariant measure of ergodic SDEs. Introducing a new symmetry normalization coefficient, we analyze the algebraic structures related to exotic B-series and S-series. Precisely, we prove the relationship between the Grossman-Larson algebras over exotic and grafted forests and the corresponding duals to the Connes-Kreimer coalgebras and use it to study the natural composition laws on exotic S-series. Applying this algebraic framework to the derivation of order conditions for a class of stochastic Runge-Kutta methods, we present a multiplicative property that ensures some order conditions to be satisfied automatically.
Achieving accurate approximations to solutions of large linear systems is crucial, especially when those systems utilize real-world data. A consequence of using real-world data is that there will inevitably be missingness. Current approaches for dealing with missing data, such as deletion and imputation, can introduce bias. Recent studies proposed an adaptation of stochastic gradient descent (SGD) in specific missing-data models. In this work, we propose a new algorithm, $\ell$-tuple mSGD, for the setting in which data is missing in a block-wise, tuple pattern. We prove that our proposed method uses unbiased estimates of the gradient of the least squares objective in the presence of tuple missing data. We also draw connections between $\ell$-tuple mSGD and previously established SGD-type methods for missing data. Furthermore, we prove our algorithm converges when using updating step sizes and empirically demonstrate the convergence of $\ell$-tuple mSGD on synthetic data. Lastly, we evaluate $\ell$-tuple mSGD applied to real-world continuous glucose monitoring (CGM) device data.
We propose a CJ-FEAST GSVDsolver to compute a partial generalized singular value decomposition (GSVD) of a large matrix pair $(A,B)$ with the generalized singular values in a given interval. The solver is a highly nontrivial extension of the FEAST eigensolver for the (generalized) eigenvalue problem and CJ-FEAST SVDsolver for the SVD problem. For a partial GSVD problem, given three left and right searching subspaces, we propose a general projection method that works on $(A,B)$ {\em directly}, and computes approximations to the desired GSVD components. For the concerning GSVD problem, we exploit the Chebyshev--Jackson (CJ) series to construct an approximate spectral projector of the generalized eigenvalue problem of the matrix pair $(A^TA,B^TB)$ associated with the generalized singular values of interest, and use subspace iteration on it to generate a right subspace. Premultiplying it with $A$ and $B$ constructs two left subspaces. Applying the general projection method to the subspaces constructed leads to the CJ-FEAST GSVDsolver. We derive accuracy estimates for the approximate spectral projector and its eigenvalues, and establish a number of convergence results on the underlying subspaces and the approximate GSVD components obtained by the CJ-FEAST GSVDsolver. We propose general-purpose choice strategies for the series degree and subspace dimension. Numerical experiments illustrate the efficiency of the CJ-FEAST GSVDsolver.
I propose an alternative algorithm to compute the MMS voting rule. Instead of using linear programming, in this new algorithm the maximin support value of a committee is computed using a sequence of maximum flow problems.
The paper introduces a geometrically unfitted finite element method for the numerical solution of the tangential Navier--Stokes equations posed on a passively evolving smooth closed surface embedded in $\mathbb{R}^3$. The discrete formulation employs finite difference and finite elements methods to handle evolution in time and variation in space, respectively. A complete numerical analysis of the method is presented, including stability, optimal order convergence, and quantification of the geometric errors. Results of numerical experiments are also provided.
Stiff systems of ordinary differential equations (ODEs) and sparse training data are common in scientific problems. This paper describes efficient, implicit, vectorized methods for integrating stiff systems of ordinary differential equations through time and calculating parameter gradients with the adjoint method. The main innovation is to vectorize the problem both over the number of independent times series and over a batch or "chunk" of sequential time steps, effectively vectorizing the assembly of the implicit system of ODEs. The block-bidiagonal structure of the linearized implicit system for the backward Euler method allows for further vectorization using parallel cyclic reduction (PCR). Vectorizing over both axes of the input data provides a higher bandwidth of calculations to the computing device, allowing even problems with comparatively sparse data to fully utilize modern GPUs and achieving speed ups of greater than 100x, compared to standard, sequential time integration. We demonstrate the advantages of implicit, vectorized time integration with several example problems, drawn from both analytical stiff and non-stiff ODE models as well as neural ODE models. We also describe and provide a freely available open-source implementation of the methods developed here.
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