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Articles 1 - 11 of 11
Full-Text Articles in Algebraic Geometry
Spacetime Geometry Of Acoustics And Electromagnetism, Lucas Burns, Tatsuya Daniel, Stephon Alexander, Justin Dressel
Spacetime Geometry Of Acoustics And Electromagnetism, Lucas Burns, Tatsuya Daniel, Stephon Alexander, Justin Dressel
Mathematics, Physics, and Computer Science Faculty Articles and Research
Both acoustics and electromagnetism represent measurable fields in terms of dynamical potential fields. Electromagnetic force-fields form a spacetime bivector that is represented by a dynamical energy–momentum 4-vector potential field. Acoustic pressure and velocity fields form an energy–momentum density 4-vector field that is represented by a dynamical action scalar potential field. Surprisingly, standard field theory analyses of spin angular momentum based on these traditional potential representations contradict recent experiments, which motivates a careful reassessment of both theories. We analyze extensions of both theories that use the full geometric structure of spacetime to respect essential symmetries enforced by vacuum wave propagation. The …
A Strong-Type Furstenberg–Sárközy Theorem For Sets Of Positive Measure, Polona Durcik, Vjekoslav Kovač, Mario Stipčić
A Strong-Type Furstenberg–Sárközy Theorem For Sets Of Positive Measure, Polona Durcik, Vjekoslav Kovač, Mario Stipčić
Mathematics, Physics, and Computer Science Faculty Articles and Research
For every β ∈ (0,∞), β ≠ 1, we prove that a positive measure subset A of the unit square contains a point (x0, y0) such that A nontrivially intersects curves y − y0 = a(x −x0)β for a whole interval I ⊆ (0,∞) of parameters a ∈ I . A classical Nikodym set counterexample prevents one to take β = 1, which is the case of straight lines. Moreover, for a planar set A of positive density, we show that the interval I can be arbitrarily large on the logarithmic scale. These results can …
Infinite-Order Differential Operators Acting On Entire Hyperholomorphic Functions, Daniel Alpay, Fabrizio Colombo, Stefano Pinton, Irene Sabadini, Daniele C. Struppa
Infinite-Order Differential Operators Acting On Entire Hyperholomorphic Functions, Daniel Alpay, Fabrizio Colombo, Stefano Pinton, Irene Sabadini, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
Infinite-order differential operators appear in different fields of mathematics and physics and in the past decade they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial datum for Schrödinger equation. Inspired by the operators arising in quantum mechanics, in this paper, we investigate the continuity of a class of infinite-order differential operators acting on spaces of entire hyperholomorphic functions. We will consider two classes of hyperholomorphic functions, both being natural extensions of holomorphic functions of one complex variable. We show that, even though these two notions of hyperholomorphic functions are quite different …
Diagrams In Intra-Configurational Analysis, Marco Panza, Gianluca Longa
Diagrams In Intra-Configurational Analysis, Marco Panza, Gianluca Longa
MPP Published Research
In this paper we would like to attempt to shed some light on the way in which diagrams enter into the practice of ancient Greek geometrical analysis. To this end, we will first distinguish two main forms of this practice, i.e., trans-configurational and intra-configurational. We will then argue that, while in the former diagrams enter in the proof essentially in the same way (mutatis mutandis) they enter in canonical synthetic demonstrations, in the latter, they take part in the analytic argument in a specific way, which has no correlation in other aspects of classical geometry. In intra-configurational analysis, diagrams represent …
Analysis, Constructions And Diagrams In Classical Geometry, Marco Panza
Analysis, Constructions And Diagrams In Classical Geometry, Marco Panza
MPP Published Research
Greek ancient and early modern geometry necessarily uses diagrams. Among other things, these enter geometrical analysis. The paper distinguishes two sorts of geometrical analysis and shows that in one of them, dubbed “intra-confgurational” analysis, some diagrams necessarily enter as outcomes of a purely material gesture, namely not as result of a codifed constructive procedure, but as result of a free-hand drawing.
The Künneth Formula And Applications, Melissa Sugimoto
The Künneth Formula And Applications, Melissa Sugimoto
SURF Posters and Papers
The de Rham cohomology of a manifold is a homotopy invariant that expresses basic topological information about smooth manifolds. The q-th de Rham cohomology of the n-dimensional Euclidean space is the vector space defined by the closed q-forms over the exact q-forms. Furthermore, the support of a continuous function f on a topological space X is the closure of the set on which f is nonzero. The result of restricting the definition of the de Rham cohomology to functions with compact support is called the de Rham cohomology with compact support, or the compact cohomology. The concept of cohomology can …
Enthymemathical Proofs And Canonical Proofs In Euclid’S Plane Geometry, Abel Lassalle, Marco Panza
Enthymemathical Proofs And Canonical Proofs In Euclid’S Plane Geometry, Abel Lassalle, Marco Panza
MPP Published Research
Since the application of Postulate I.2 in Euclid’s Elements is not uniform, one could wonder in what way should it be applied in Euclid’s plane geometry. Besides legitimizing questions like this from the perspective of a philosophy of mathematical practice, we sketch a general perspective of conceptual analysis of mathematical texts, which involves an extended notion of mathematical theory as system of authorizations, and an audience-dependent notion of proof.
Beurling-Lax Type Theorems In The Complex And Quaternionic Setting, Daniel Alpay, Irene Sabadini
Beurling-Lax Type Theorems In The Complex And Quaternionic Setting, Daniel Alpay, Irene Sabadini
Mathematics, Physics, and Computer Science Faculty Articles and Research
We give a generalization of the Beurling–Lax theorem both in the complex and quaternionic settings. We consider in the first case functions meromorphic in the right complex half-plane, and functions slice hypermeromorphic in the right quaternionic half-space in the second case. In both settings we also discuss a unified framework, which includes both the disk and the half-plane for the complex case and the open unit ball and the half-space in the quaternionic setting.
Quaternionic Hermitian Spinor Systems And Compatibility Conditions, Alberto Damiano, David Eelbode, Irene Sabadini
Quaternionic Hermitian Spinor Systems And Compatibility Conditions, Alberto Damiano, David Eelbode, Irene Sabadini
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we show that the systems introduced in [12] and [22] are equivalent, both giving the notion of quaternionic Hermitian monogenic functions. This makes it possible to prove that the free resolution associated to the system is linear in any dimension, and that the first cohomology module is nontrivial, thus generalizing the results in [22]. Furthermore, exploiting the decomposition of the spinor space into sp(m)-irreducibles, we find a certain number of "algebraic" compatibility conditions for the system, suggesting that the usual spinor reduction is not applicable.
Rethinking Geometrical Exactness, Marco Panza
Rethinking Geometrical Exactness, Marco Panza
MPP Published Research
A crucial concern of early modern geometry was fixing appropriate norms for deciding whether some objects, procedures, or arguments should or should not be allowed into it. According to Bos, this is the exactness concern. I argue that Descartes’s way of responding to this concern was to suggest an appropriate conservative extension of Euclid’s plane geometry (EPG). In Section 2, I outline the exactness concern as, I think, it appeared to Descartes. In Section 3, I account for Descartes’s views on exactness and for his attitude towards the most common sorts of constructions in classical geometry. I also explain in …
Hartogs' Phenomenon For Polyregular Functions And Projective Dimension Of Related Modules Over A Polynomial Ring, W. W. Adams, P. Loustaunau, V. P. Palamadov, Daniele C. Struppa
Hartogs' Phenomenon For Polyregular Functions And Projective Dimension Of Related Modules Over A Polynomial Ring, W. W. Adams, P. Loustaunau, V. P. Palamadov, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we prove that the projective dimension of Mn = R^4/(An) is 2n -1, where R is the ring of polynomials in 4n variables with complex coefficients and (An) is the module generated by the columns of a 4x4n matrix which arises as the Fourier transform of the matrix of differential operators associated with the regularity condition for a function of n quaternionic variables. As a corollary we show that the sheaf R of regular functions has flabby dimension 2n -1, and we prove a cohomology vanishing theorem for open sets in the space Hn of quaternions. We …