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A canonical formalism and constraint analysis for discrete systems subject to a variational action principle are devised. The formalism is equivalent to the covariant formulation, encompasses global and local discrete time evolution moves…

For integrable systems in the sense of multidimensional consistency (MDC) we can consider the Lagrangian as a form, which is closed on solutions of the equations of motion. For 2-dimensional systems, described by partial difference…

Systems of ordinary differential equations (or dynamical forms in Lagrangian mechanics), induced by embeddings of smooth fibered manifolds over one-dimensional basis, are considered in the class of variational equations. For a given…

In this paper, we consider an infinite derivative scalar field action with infinite derivative kinetic and interaction terms. We establish that the theory is unitary if the correlation functions are formulated in Euclidean space and then…

The mathematics of K-conserving functional differentiation, with K being the integral of some invertible function of the functional variable, is clarified. The most general form for constrained functional derivatives is derived from the…

Discrete choice models are commonly used by applied statisticians in numerous fields, such as marketing, economics, finance, and operations research. When agents in discrete choice models are assumed to have differing preferences, exact…

An Independent Parallelism Theorem is proven in the theory of adhesive HLR categories. It shows the bijective correspondence between sequential independent and parallel independent direct derivations in the Weak Double-Pushout framework,…

Diffusion models have become the de facto standard for modern visual generation, including well-established frameworks such as latent diffusion and flow matching. Recently, modeling high-order dynamics has emerged as a promising frontier in…

We examine the fractional derivative of composite functions and present a generalization of the product and chain rules for the Caputo fractional derivative. These results are especially important for physical and biological systems that…

This paper proposes new derivations of three well-known sorting algorithms, in their functional formulation. The approach we use is based on three main ingredients: first, the algorithms are derived from a simpler algorithm, i.e. the…

We give two proofs to the following theorem and its generalization: if a finite dimensional algebra $A$ is derived equivalent to a smooth projective scheme, then any derived equivalence between $A$ and another algebra $B$ is standard, that…

Variance reduction is a family of powerful mechanisms for stochastic optimization that appears to be helpful in many machine learning tasks. It is based on estimating the exact gradient with some recursive sequences. Previously, many papers…

Coherence in a monoidal category asserts that all morphisms built from structural isomorphisms with a fixed source and target coincide. These structural isomorphisms include, in particular, the associators. Linearly distributive categories…

A folklore result in category theory is that a (weakly) Cartesian closed category with finite co-products is distributive. Usually, the proof of this small result is carried on using the fact that the exponential functor is right adjoint to…

We formulate explicitly the necessary and sufficient conditions for the local invertibility of a field transformation involving derivative terms. Our approach is to apply the method of characteristics of differential equations, by treating…

Invertible neural networks (INNs) are neural network architectures with invertibility by design. Thanks to their invertibility and the tractability of Jacobian, INNs have various machine learning applications such as probabilistic modeling,…

Classifier chains are an effective technique for modeling label dependencies in multi-label classification. However, the method requires a fixed, static order of the labels. While in theory, any order is sufficient, in practice, this order…

Rice's theorem shows that nontrivial extensional properties of partial recursive functions are undecidable. For finite weighted Boolean optimization/CSP-style slices, a Rice-style structural analogue holds for tractability classification:…

Interatomic potentials are essential to go beyond ab initio size limitations, but simulation results depend sensitively on potential parameters. Forward propagation of parameter variation is key for uncertainty quantification, whilst…

In this paper, we introduce two new non-singular kernel fractional derivatives and present a class of other fractional derivatives derived from the new formulations. We present some important results of uniformly convergent sequences of…