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Lattice models parameterized using first-principles calculations constitute an effective framework to simulate the thermodynamic behavior of physical systems. The cluster expansion method is a flexible lattice-based method used extensively…
Let $\bx_j = \btheta +\bep_j, j=1,...,n$, be observations of an unknown parameter $\btheta$ in a Euclidean or separable Hilbert space $\scrH$, where $\bep_j$ are noises as random elements in $\scrH$ from a general distribution. We study the…
This paper focuses on a wide class of Collatz-type arithmetic dynamics, and presents a systematic derivation of recursive formulas and functional equations satisfied by the associated generating functions. The main tools belong to complex…
Estimating individual-level treatment effect from observational data is a fundamental problem in causal inference and has attracted increasing attention in the fields of education, healthcare, and public policy.In this work, we concentrate…
Classical functional linear regression models the relationship between a scalar response and a functional covariate, where the coefficient function is assumed to be identical for all subjects. In this paper, the classical model is extended…
This paper is devoted to the construction of order reduced method of fourth order problems. A framework is presented such that a problem on a high-regularity space can be deduced in a constructive way to an equivalent problem on three…
Traditional analytical reflectance models, while compact and interpretable, lack the capacity to accurately represent physical measurements. Recent neural models, which closely fit input data, are less generalizable and often more expensive…
In this paper, we study harmonic analysis on finite homogeneous spaces whose associated permutation representation decomposes with multiplicity. After a careful look at Frobenius reciprocity and transitivity of induction, and the…
We analyze the problem of global reconstruction of functions as accurately as possible, based on partial information in the form of a truncated power series at some point, and additional analyticity properties. This situation occurs…
We introduce a generalization of the Heisenberg algebra which is written in terms of a functional of one generator of the algebra, $f(J_0)$, that can be any analytical function. When $f$ is linear with slope $\theta$, we show that the…
Multivariate information decompositions hold promise to yield insight into complex systems, and stand out for their ability to identify synergistic phenomena. However, the adoption of these approaches has been hindered by there being…
This paper is dedicated to the study of an estimator of the generalized Hoeffding decomposition. We build such an estimator using an empirical Gram-Schmidt approach and derive a consistency rate in a large dimensional settings. Then, we…
The affine Hilbert function is a classical algebraic object that has been central, among other tools, to the development of the polynomial method in combinatorics. Owing to its concrete connections with Gr\"obner basis theory, as well as…
We extend the shell and kernel reductions for hyperexponential functions over the field of rational functions to a monomial extension. Both of the reductions are incorporated into one algorithm. As an application, we present an additive…
In this paper we consider Idempotent Functional Analysis, an `abstract' version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a review of the basic ideas of Idempotent Analysis. The correspondence between…
Classical functional calculus is primarily spectral, capturing eigenvalue information through resolvent methods while largely ignoring nilpotent structure. Building on the projector-nilpotent characterization developed in our companion…
A generally intelligent learner should generalize to more complex tasks than it has previously encountered, but the two common paradigms in machine learning -- either training a separate learner per task or training a single learner for all…
A new notion in frame theory has been introduced recently that called woven frames. %From the perspective of others, Woven and weaving frames are powerful tools for pre-processing signals and distributed data processing. The purpose of…
Constructing complex computation from simpler building blocks is a defining problem of computer science. In algebraic automata theory, we represent computing devices as semigroups. Accordingly, we use mathematical tools like products and…
Researchers have proposed a wide variety of model explanation approaches, but it remains unclear how most methods are related or when one method is preferable to another. We describe a new unified class of methods, removal-based…