Related papers: Improving Cauchy's Theorem in Constructive Analysi…
We develop a linear theory of discrete complex analysis on general quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on…
Structured and decorated cospans are broadly applicable frameworks for building bicategories or double categories of open systems. We streamline and generalize these frameworks using central concepts of double category theory. We show that,…
We use the Cauchy-Crofton formula to show that every definable cell (bounded by a ball with rational radius) in an O-minimal expansion of a field extension of the real numbers satisfies the Whitney arc property.
We study different notions of connected constructive metric spaces. They differ the types of connected components and how different components relate to each other. These notions are equivalent in classical point set topology but they give…
We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of (valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The…
We give necessary and sufficient conditions for the functor that forgets the $(C, \gamma)$-coaction to be separable. This leads to a generalized notion of integrals. Finally, the applications of our results are considered.
Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory and classical reasoning is typically…
Given a compact of ${\bf R}^n$, there is always a doubling measure having it as its support. We use this fact to construct an integral operator that extends differentiable functions defined on any compact set of ${\bf R}^n$ to the whole of…
Stieltjes integral theorem is more commonly known by the phrase 'integration by parts' and enables rearrangement of an otherwise intractable integral to a more amenable form; often permitting completion of an integral in closed form.…
Sharing of notations and theories across an inheritance hierarchy of mathematical structures, e.g., groups and rings, is important for productivity when formalizing mathematics in proof assistants. The packed classes methodology is a…
We present Nonstandard Analysis by three axioms: the {\em Extension, Transfer and Saturation Principles} in the framework of the superstructure of a given infinite set. We also present several applications of this axiomatic approach to…
We explore a function theory connected with the principal series representation of SL(2,R) in contrast to standard complex analysis connected with the discrete series. We construct counterparts for the Cauchy integral formula, the Hardy…
A trichotomy theorem for countable, stable, unsuperstable theories is offered. We develop the notion of a `regular ideal' of formulas and study types that are minimal with respect to such an ideal.
Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in constructive locale theory. We show that…
Homotopy type theory is an interpretation of Martin-L\"of's constructive type theory into abstract homotopy theory. There results a link between constructive mathematics and algebraic topology, providing topological semantics for…
We generalize the notion of harmonic conjugate functions and Hilbert transforms to higher dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These conjugate functions are in general far from being…
Variable selection for models including interactions between explanatory variables often needs to obey certain hierarchical constraints. The weak or strong structural hierarchy requires that the existence of an interaction term implies at…
We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in…
We extend Bishop's one-fourth three-fourths principle for constructing peak functions belonging to a uniform algebra to a situation where the ``approximate barriers'' associated with the Bishop construction are not uniformly bounded.
We investigate bounds in Ramsey's theorem for relations definable in NIP structures. Applying model-theoretic methods to finitary combinatorics, we generalize a theorem of Bukh and Matousek [B. Bukh, J. Matou\v{s}ek.…