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These notes aim to provide a classical approach to solving some conformable differential equations based on prior knowledge of how to solve ordinary differential equations. That is, using the methods of separation of variables, homogeneous…
Employing a recently proposed separability criterion we develop analytical lower bounds for the concurrence and for the entanglement of formation of bipartite quantum systems. The separability criterion is based on a nondecomposable…
Linear causal disentanglement is a recent method in causal representation learning to describe a collection of observed variables via latent variables with causal dependencies between them. It can be viewed as a generalization of both…
We introduce and explain key relations between a posteriori error estimates and subspace correction methods viewed as preconditioners for problems in infinite dimensional Hilbert spaces. We set the stage using the Finite Element Exterior…
The momentous objective of this work is to discuss some qualitative properties of solutions such as the estimate on the solutions, the continuous dependence of the solutions on initial conditions as well as the existence and uniqueness of…
In this paper, we study the weighted difference substitutions from geometrical views. First, we give the geometric meanings of the weighted difference substitutions, and introduce the concept of convergence of the sequence of substitution…
Causal structure discovery from observational data is fundamental to the causal understanding of autonomous systems such as medical decision support systems, advertising campaigns and self-driving cars. This is essential to solve well-known…
Conformal prediction methods provide statistically rigorous marginal coverage guarantees for machine learning models, but such guarantees fail to account for algorithmic biases, thereby undermining fairness and trust. This paper introduces…
A general sufficient condition for the convergence of subsequences of solutions of non-autonomous, nonlinear difference equations and systems is obtained. For higher order equations the delay sizes and patterns play essential roles in…
The paper considers causal smoothing of the real sequences, i.e.,discrete time processes in a deterministic setting. A family of causal linear time-invariant filters is suggested. These filters approximate the gain decay for some non-causal…
Many standard structural quantities, such as order parameters and correlation functions, exist for common condensed matter systems, such as spherical and rod-like particles. However, these structural quantities are often insufficient for…
Inference of causality in time series has been principally based on the prediction paradigm. Nonetheless, the predictive causality approach may overlook the simultaneous and reciprocal nature of causal interactions observed in real world…
One can theoretically conceive of processes where the causal order between quantum operations is no longer well-defined. Certain such causally indefinite processes have an operational interpretation in terms of quantum operations on…
Determinantal point process have recently been used as models in machine learning and this has raised questions regarding the characterizations of conditional independence. In this paper we investigate characterizations of conditional…
We define and prove existence of fractional $P(\phi)_1$-processes as random processes generated by fractional Schr\"odinger semigroups with Kato-decomposable potentials. Also, we show that the measure of such a process is a Gibbs measure…
We recently characterized the separated determinantal point processes $\Lambda_\phi$ associated with Fock spaces $\mathcal F_\phi$ in the plane with doubling weight $\phi$. We also showed that, as expected, a more restrictive condition is…
Determinantal point processes (DPPs for short) are a class of repulsive point processes. They have found some statistical applications to model spatial point pattern datasets with repulsion between close points. In the case of DPPs on…
To capture the relationship between samples and labels, conditional generative models often inherit spurious correlations from the training dataset. This can result in label-conditional distributions that are imbalanced with respect to…
The main purpose of this paper is to exhibit a simple variational setting for finding fully nontrivial solutions to the weakly coupled elliptic system (1.1). We show that such solutions correspond to critical points of a…
A recent framework of quantum theory with no global causal order predicts the existence of "causally nonseparable" processes. Some of these processes produce correlations incompatible with any causal order (they violate so-called "causal…