We discuss the definition of the Laplace operator on an open subset in Euclidean space as a self-adjoint operator with Dirichlet or Neumann boundary conditions and we derive its basic spectral properties. Among others, we include the spectral inequalities of Faber–Krahn, Hersch, and Friedlander. Then, using the technique of Dirichlet–Neumann bracketing, we derive Weyl's law for the asymptotic distribution of eigenvalues. We supplement this with a discussion of non-asymptotic bounds, including Pólya's conjecture and its proof for tiling domains and domains of product form. We present the sharp eigenvalue bounds of Berezin and Li–Yau. Finally, using separation of variables in spherical coordinates, we discuss the Laplacian on a ball.

The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras.

The building blocks of the theory are geometric objects associated to a reflection arrangement such as faces, flats, lunes, and their orbits under the action of the Coxeter group. A generalized notion of zeta and Möbius function play a fundamental role in all aspects of the theory, including exp-log correspondences and results such as the Poincarö–Birkhoff–Witt theorem. The Tits algebra and its invariant subalgebra also play key roles.

This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.

The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras.

The building blocks of the theory are geometric objects associated to a reflection arrangement such as faces, flats, lunes, and their orbits under the action of the Coxeter group. A generalized notion of zeta and Möbius function play a fundamental role in all aspects of the theory, including exp-log correspondences and results such as the Poincarö–Birkhoff–Witt theorem. The Tits algebra and its invariant subalgebra also play key roles.

This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.

The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras.

The building blocks of the theory are geometric objects associated to a reflection arrangement such as faces, flats, lunes, and their orbits under the action of the Coxeter group. A generalized notion of zeta and Möbius function play a fundamental role in all aspects of the theory, including exp-log correspondences and results such as the Poincarö–Birkhoff–Witt theorem. The Tits algebra and its invariant subalgebra also play key roles.

This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.

The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras.

The building blocks of the theory are geometric objects associated to a reflection arrangement such as faces, flats, lunes, and their orbits under the action of the Coxeter group. A generalized notion of zeta and Möbius function play a fundamental role in all aspects of the theory, including exp-log correspondences and results such as the Poincarö–Birkhoff–Witt theorem. The Tits algebra and its invariant subalgebra also play key roles.

This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.

We provide a brief, but self-contained, introduction to the theory of Sobolev spaces. We prove some facts from the calculus with weak derivatives, including product and chain rules. We discuss various kinds of Sobolev inequalities, including those by Gagliardo–Nirenberg, Poincaré, Friedrichs, and Hardy, both on the whole space and on domains, and include some information on their sharp constants. Furthermore, we discuss Rellich’s compactness theorem, the Sobolev extension property of a domain, as well as homogeneous Sobolev spaces.

The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras.

The building blocks of the theory are geometric objects associated to a reflection arrangement such as faces, flats, lunes, and their orbits under the action of the Coxeter group. A generalized notion of zeta and Möbius function play a fundamental role in all aspects of the theory, including exp-log correspondences and results such as the Poincarö–Birkhoff–Witt theorem. The Tits algebra and its invariant subalgebra also play key roles.

This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.

The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras.

The building blocks of the theory are geometric objects associated to a reflection arrangement such as faces, flats, lunes, and their orbits under the action of the Coxeter group. A generalized notion of zeta and Möbius function play a fundamental role in all aspects of the theory, including exp-log correspondences and results such as the Poincarö–Birkhoff–Witt theorem. The Tits algebra and its invariant subalgebra also play key roles.

This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.

The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras.

The building blocks of the theory are geometric objects associated to a reflection arrangement such as faces, flats, lunes, and their orbits under the action of the Coxeter group. A generalized notion of zeta and Möbius function play a fundamental role in all aspects of the theory, including exp-log correspondences and results such as the Poincarö–Birkhoff–Witt theorem. The Tits algebra and its invariant subalgebra also play key roles.

This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.

The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras.

The building blocks of the theory are geometric objects associated to a reflection arrangement such as faces, flats, lunes, and their orbits under the action of the Coxeter group. A generalized notion of zeta and Möbius function play a fundamental role in all aspects of the theory, including exp-log correspondences and results such as the Poincarö–Birkhoff–Witt theorem. The Tits algebra and its invariant subalgebra also play key roles.

This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.