A derivative for complex Lipschitz maps with generalised Cauchy–Riemann equations

AbstractWe introduce the Lipschitz derivative or the L-derivative of a locally Lipschitz complex map: it is a Scott continuous, compact and convex set-valued map that extends the classical derivative to the bigger class of locally Lipschitz maps and allows an extension of the fundamental theorem of calculus and a new generalisation of Cauchy–Riemann equations to these maps, which form a continuous Scott domain. We show that a complex Lipschitz map is analytic in an open set if and only if its L-derivative is a singleton at all points in the open set. The calculus of the L-derivative for sum, product and composition of maps is derived. The notion of contour integration is extended to Scott continuous, non-empty compact, convex valued functions on the complex plane, and by using the L-derivative, the fundamental theorem of contour integration is extended to these functions

The Lean mathematical library

This paper describes mathlib, a community-driven effort to build a unified library of mathematics formalized in the Lean proof assistant. Among proof assistant libraries, it is distinguished by its dependently typed foundations, focus on classical mathematics, extensive hierarchy of structures, use of large- and small-scale automation, and distributed organization. We explain the architecture and design decisions of the library and the social organization that has led us here

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Biometric Systems

Because of the accelerating progress in biometrics research and the latest nation-state threats to security, this book's publication is not only timely but also much needed. This volume contains seventeen peer-reviewed chapters reporting the state of the art in biometrics research: security issues, signature verification, fingerprint identification, wrist vascular biometrics, ear detection, face detection and identification (including a new survey of face recognition), person re-identification, electrocardiogram (ECT) recognition, and several multi-modal systems. This book will be a valuable resource for graduate students, engineers, and researchers interested in understanding and investigating this important field of study

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This collection of essay explore the different facets of the contingency/necessity debate in the philosophy of scienc