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Many problems in Physics and Chemistry are formulated as the minimization of a functional. Therefore, methods for solving these problems typically require differentiating maps whose input and/or output are functions -- commonly referred to…
One of the fundamental tools of undergraduate calculus is the chain rule. The notion of higher order directional derivatives was developed by Huang, Marcantognini, and Young, along with a corresponding higher order chain rule. When Johnson…
A well-ordering principle is a principle of the form: If $X$ is well-ordered then $F(X)$ is well-ordered, where $F$ is some natural operator transforming linear orders into linear orders. Many important subsystems of Second-order Arithmetic…
In the paper, by induction, the Fa\`a di Bruno formula, and some techniques in the theory of complex functions, the author finds explicit formulas for higher order derivatives of the tangent and cotangent functions as well as powers of the…
We revisit several partition-theoretic generating functions, including the theta quotients from Ramanujan's lost notebook, MacMahon's partition functions, and reciprocal sums of parts in partitions, through the lens of the classical Fa\`{a}…
In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded)…
We introduce a mathematical framework for retrosynthetic analysis, an important research method in synthetic chemistry. Our approach represents molecules and their interaction using string diagrams in layered props - a recently introduced…
Analysis often splits change into components. For example, how much of the observed variance is caused by genes or environment? In many cases, the split is ultimately made by the logic of the chain rule, which divides the difference of a…
Recent theoretical work on automatic differentiation (autodiff) has focused on characteristics such as correctness and efficiency while assuming that all derivatives are automatically generated by autodiff using program transformation, with…
An integro-differential Dirac system with an integral term in the form of convolution is considered. We suppose that the convolution kernel is known a priori on a part of the interval, and recover it on the remaining part, using a part of…
In this paper, we propose a new concept of derivative with respect to an arbitrary kernel-function. Several properties related to this new operator, like inversion rules, integration by parts, etc. are studied. In particular, we introduce…
Guarded recursion is a framework allowing for a formalisation of streams in classical programming languages. The latter take their semantics in cartesian closed categories. However, some programming paradigms do not take their semantics in…
An element of a group is said to be reversible if it is conjugate to its inverse. We characterise the reversible elements in the group of diffeomorphisms of the real line, and in the subgroup of order preserving diffeomorphisms.
We generalize the classical mean value theorem of differential calculus by allowing the use of a Caputo-type fractional derivative instead of the commonly used first-order derivative. Similarly, we generalize the classical mean value…
Automatic differentiation plays a prominent role in scientific computing and in modern machine learning, often in the context of powerful programming systems. The relation of the various embodiments of automatic differentiation to the…
Inverse spectral problems are studied for first-order integro-differential operators on a finite interval. These problems consist in recovering some components of the kernel from one or multiple spectra. Uniqueness theorems are proved for…
Interactions between derivatives and fixpoints have many important applications in both computer science and mathematics. In this paper, we provide a categorical framework to combine fixpoints with derivatives by studying Cartesian…
Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…
We extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional equations, including some convex fully…
In the theory of conditional sets, many classical theorems from areas such as functional analysis, probability theory or measure theory are lifted to a conditional framework, often to be applied in areas such as mathematical economics or…