T. Berry Brazelton
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View article: Monodromy in the space of symmetric cubic surfaces with a line
Monodromy in the space of symmetric cubic surfaces with a line Open
We explore the enumerative problem of finding lines on cubic surfaces defined by symmetric polynomials. We prove that the moduli space of symmetric cubic surfaces is an arithmetic quotient of the complex hyperbolic line, and determine cons…
View article: There is no Cazanave's Theorem for punctured affine space
There is no Cazanave's Theorem for punctured affine space Open
In his thesis, Cazanave proved that the set of naive $\mathbb{A}^1$-homotopy classes of endomorphisms of the projective line admits a monoid structure whose group completion is genuine $\mathbb{A}^1$-homotopy classes of endomorphisms of th…
View article: Residue sums of Dickson polynomials over finite fields
Residue sums of Dickson polynomials over finite fields Open
Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime p. The sum over the distinct residues can sometimes be computed independent of the prime p; for example, Gau…
View article: $\mathbb{A}^1$-Brouwer degrees in Macaulay2
$\mathbb{A}^1$-Brouwer degrees in Macaulay2 Open
We describe the Macaulay2 package "A1BrouwerDegrees" for computing local and global $\mathbb{A}^1$-Brouwer degrees and studying symmetric bilinear forms over the complex numbers, the real numbers, the rational numbers, and finite fields of…
View article: Bézoutians and the 𝔸1-degree
Bézoutians and the 𝔸1-degree Open
We prove that both the local and global ށ 1 -degree of an endomorphism of affine space can be computed in terms of the multivariate Bézoutian.In particular, we show that the Bézoutian bilinear form, the Scheja-Storch form, and the ށ 1 …
View article: Management of Critically Ill Patients With Spinal Muscular Atrophy Admitted With Acute Respiratory Failure: The Authors' Reply
Management of Critically Ill Patients With Spinal Muscular Atrophy Admitted With Acute Respiratory Failure: The Authors' Reply Open
The authors wish to thank the editors for the opportunity to respond to Bach et al. The authors are unclear why there would be a lack of clarity regarding our noninvasive ventilation (NIV) approach to our subjects with spinal muscular atro…
View article: Lifts, transfers, and degrees of univariate maps
Lifts, transfers, and degrees of univariate maps Open
One can compute the local $\mathbb{A}^1$-degree at points with separable residue field by base changing, working rationally, and post-composing with the field trace. We show that for endomorphisms of the affine line, one can compute the lo…
View article: Management of Critically Ill Patients With Spinal Muscular Atrophy Admitted With Acute Respiratory Failure
Management of Critically Ill Patients With Spinal Muscular Atrophy Admitted With Acute Respiratory Failure Open
The subjects with SMA types I or II can be treated successfully with NIV and aggressive airway clearance during acute respiratory failure. Similarly, when intubation is required, successful extubation can be achieved with NIV transitional …
View article: Equivariant enumerative geometry
Equivariant enumerative geometry Open
We formulate an equivariant conservation of number, which proves that a generalized Euler number of a complex equivariant vector bundle can be computed as a sum of local indices of an arbitrary section. This involves an expansion of the Po…
View article: Managing multiple perspectives in the collaborative design process of a team health information technology
Managing multiple perspectives in the collaborative design process of a team health information technology Open
We need to design technologies that support the work of health care teams; designing such solutions should integrate different clinical roles. However, we know little about the actual collaboration that occurs in the design process for a t…
View article: An enriched degree of the Wronski
An enriched degree of the Wronski Open
Given $mp$ different $p$-planes in general position in $(m+p)$-dimensional space, a classical problem is to ask how many $p$-planes intersect all of them. For example when $m = p = 2$, this is precisely the question of "lines meeting four …
View article: Bézoutians and the $\mathbb{A}^1$-degree
Bézoutians and the $\mathbb{A}^1$-degree Open
We prove that both the local and global $\mathbb{A}^1$-degree of an endomorphism of affine space can be computed in terms of the multivariate Bézoutian. In particular, we show that the Bézoutian bilinear form, the Scheja--Storch form, and …
View article: Residue sums of Dickson polynomials over finite fields
Residue sums of Dickson polynomials over finite fields Open
Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime $p$. The sum over the distinct residues can sometimes be computed independent of the prime $p$; for example,…
View article: Distinct residues of Lucas polynomials over $\mathbb{F}_p$
Distinct residues of Lucas polynomials over $\mathbb{F}_p$ Open
Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime $p$. The sum over these residues can sometimes be computed independently of the choice of prime $p$; for exa…
View article: A note on twisted group rings and semilinearization
A note on twisted group rings and semilinearization Open
In this short note, we construct a right adjoint to the functor which associates to a ring $R$ equipped with a group action its twisted group ring. This right adjoint admits an interpretation as semilinearization, in that it sends an $R$-m…
View article: Homotopy Mackey functors of equivariant algebraic $K$-theory
Homotopy Mackey functors of equivariant algebraic $K$-theory Open
Given a finite group $G$ acting on a ring $R$, Merling constructed an equivariant algebraic $K$-theory $G$-spectrum, and work of Malkiewich and Merling, as well as work of Barwick, provides an interpretation of this construction as a spect…
View article: Early infant behavioural correlates of social skills in adolescents
Early infant behavioural correlates of social skills in adolescents Open
Background Autism Spectrum Disorder (ASD) is a neurodevelopmental disorder typically diagnosed after the second year of life; however, differences in brain structure and function associated with ASD have been ascertained in early infancy. …
View article: An Introduction to $\mathbf{A}^1$-Enumerative Geometry
An Introduction to $\mathbf{A}^1$-Enumerative Geometry Open
We provide an expository introduction to $\mathbb{A}^1$-enumerative geometry, which uses the machinery of $\mathbb{A}^1$-homotopy theory to enrich classical enumerative geometry questions over a broader range of fields. Included is a discu…
View article: Physician Perceptions of the Electronic Problem List in Pediatric Trauma Care
Physician Perceptions of the Electronic Problem List in Pediatric Trauma Care Open
Objective To describe physician perceptions of the potential goals, characteristics, and content of the electronic problem list (PL) in pediatric trauma. Methods We conducted 12 semistructured interviews with physicians involved in the ped…
View article: Constructing the Unstable Motivic Homotopy Category Using $(\infty,1)$-Categories
Constructing the Unstable Motivic Homotopy Category Using $(\infty,1)$-Categories Open
This is an expository paper providing an overview of the unstable motivic homotopy category using the theory of $(\infty,1)$-categories. In this paper, we examine two constructions in the literature and discuss their equivalence.
View article: Zeros of Newform Eisenstein Series on $\Gamma_0(N)$
Zeros of Newform Eisenstein Series on $\Gamma_0(N)$ Open
We examine the zeros of newform Eisenstein series $E_{\\chi_1,\\chi_2,k}(z)$ of\nweight $k$ on $\\Gamma_0(q_1 q_2)$, where $\\chi_1$ and $\\chi_2$ are primitive\ncharacters modulo $q_1$ and $q_2$, respectively. We determine the location an…
View article: Zeros of Newform Eisenstein Series on $Γ_0(N)$
Zeros of Newform Eisenstein Series on $Γ_0(N)$ Open
We examine the zeros of newform Eisenstein series $E_{χ_1,χ_2,k}(z)$ of weight $k$ on $Γ_0(q_1 q_2)$, where $χ_1$ and $χ_2$ are primitive characters modulo $q_1$ and $q_2$, respectively. We determine the location and distribution of a sign…