相关论文: Fun with "Analysis I": basic theorems in calculus …

A new continuity for set-valued functions is introduced, and an existence theorem is proved for such continuous set-valued functions.

The paper continues the intriguing theme that many key facts of (single-variable) Real Analysis are not only crucially dependent on the completeness of the real numbers, but are actually equivalent to it. The list of these characterizations…

We axiomatize and generalize Markov's approach to the continuity problem for Type 1 computable functions, i.e. the problem of finding sufficient conditions on a computable topological space to obtain a theorem of the form "computable…

We initiate the study of parallel quantum programming by defining the operational and denotational semantics of parallel quantum programs. The technical contributions of this paper include: (1) find a series of useful proof rules for…

It has been shown that a functional interpretation of proofs in mathematical analysis can be given by the product of selection functions, a mode of recursion that has an intuitive reading in terms of the computation of optimal strategies in…

In this paper, we study the consequences of the fundamental theorem of calculus from an algebraic point of view. For functions with singularities, this leads to a generalized notion of evaluation. We investigate properties of such…

The optimal value function is one of the basic objects in the field of mathematical optimization, as it allows the evaluation of the variations in the cost/revenue generated while minimizing/maximizing a given function under some…

Encodings or the proof of their absence are the main way to compare process calculi. To analyse the quality of encodings and to rule out trivial or meaningless encodings, they are augmented with encodability criteria. There exists a bunch…

This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits…

How can complexity theory and algorithms benefit from practical advances in computing? We give a short overview of some prior work using practical computing to attack problems in computational complexity and algorithms, informally describe…

Logic-based approaches to AI have the advantage that their behavior can in principle be explained to a user. If, for instance, a Description Logic reasoner derives a consequence that triggers some action of the overall system, then one can…

Starting from the Colombeau's full generalized functions, the sharp topologies and the notion of generalized points, we introduce a new kind differential calculus (for functions between totally disconnected spaces). We study generalized…

Models of computation operating over the real numbers and computing a larger class of functions compared to the class of general recursive functions invariably introduce a non-finite element of infinite information encoded in an arbitrary…

This chapter provides a tutorial overview of first principles methods to describe the properties of matter at the ground state or equilibrium. It begins with a brief introduction to quantum and statistical mechanics for predicting the…

We consider the space of convex functions defined in the Euclidean $n$-dimensional space, which are lower semi-continuous and tend to infinity at infinity. We study real-valued valuations defined on this space of functions, which are…

There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been…

Metric regularity is among the central concepts of nonlinear and variational analysis, constrained optimization, and their numerous applications. However, metric regularity can be elusive for some important ill-posed classes of problems…

Fair termination is the property of programs that may diverge "in principle" but that terminate "in practice", i.e. under suitable fairness assumptions concerning the resolution of non-deterministic choices. We study a conservative…

Hilbert's epsilon calculus is an extension of elementary or predicate calculus by a term-forming operator $\varepsilon$ and initial formulas involving such terms. The fundamental results about the epsilon calculus are so-called epsilon…

We introduce a general technique to create an extended formulation of a mixed-integer program. We classify the integer variables into blocks, each of which generates a finite set of vector values. The extended formulation is constructed by…