Schwartz regularity of differential operators on the cylinder

This article presents an investigation of global properties of a class of differential operators on \T^1\times\R. Our approach involves the utilization of a mixed Fourier transform, incorporating both partial Fourier series on the torus and partial Fourier transform in Euclidean space. By examining the behavior of the mixed Fourier coefficients, we obtain necessary and sufficient conditions for the Schwartz global hypoellipticity of this class of differential operators, as well as conditions for the Schwartz global solvability of said operators.Comment: arXiv admin note: text overlap with arXiv:2306.1557

Fourier Analysis on Tm×Rn\mathbb{T}^m\times\mathbb{R}^n and Applications to Global Hypoellipticity

This article presents a convenient approach to Fourier analysis for the investigation of functions and distributions defined in $\mathbb{T}^m \times \mathbb{R}^n$. Our approach involves the utilization of a mixed Fourier transform, incorporating both partial Fourier series on the torus for the initial variables and partial Fourier transform in Euclidean space for the remaining variables. By examining the behaviour of the mixed Fourier coefficients, we achieve a comprehensive characterization of the spaces of smooth functions and distributions in this context. Additionally, we apply our results to derive necessary and sufficient conditions for the global hypoellipticity of a class first order differential operators defined on $\mathbb{T}^m \times \mathbb{R}^n$, including all constant coefficient first order differential operators

Global hypoellipticity of GG-invariant operators on homogeneous vector bundles

We establish necessary and sufficient conditions for the global hypoellipticity of $G$-invariant operators on homogeneous vector bundles. These criteria are established in terms of the corresponding matrix-valued symbols as developed by Ruzhansky and Turunen and extended in [7] to homogeneous vector-bundles.Comment: 15 page

Grupos de lIE

Orientador: Professor Hudson do Nascimento LimaTrabalho de Conclusão de Curso (graduação) - Universidade Federal do Paraná, Setor de Ciências Exatas, Curso de Graduação em MatemáticaInclui referênciasResumo : O trabalho reúne definições e resultados básicos da teoria de Grupos de Lie, entre eles o Teorema do Subgrupo Fechado e os 3 Teoremas de Lie. Mostra-se que é possível definir um funtor entre a categoria dos Grupos de Lie e a das Álgebras de Lie de dimensão finita, que, quando restrito a subcategoria de grupos simplesmente conexos, é fiel. Também se classificam os Grupos de Lie abelianos e conexos, e são estudadas algumas propriedades de Grupos e Álgebras compactas. Por fim, reúnem-se exemplos dos principais grupos de LieAbstract: The paper compiles basic results and definitions from the thoery of Lie Groups, among them the Closed Subgroup Theorem and the Lie’s 3 Theorems. It is shown that its possible to define the a functor between the category of Lie Groups e the category of Lie Algebras of finite dimension and, when this functor is restricted to simply connected Groups, this is a faithful functor. The abelian connected subgroups are classified and some properities of compact Lie Groups and Algebras are also exibihite

A relevancia do lobo temporal como area irritativa e sintomatogenica na epilepsia associada a calcificações intracerebrais anormais

Orientador: Lineu Cesar WerneckDissertação (mestrado) - Universidade Federal do Parana, Setor de Ciencias da SaudePara testar a hipótese de que o lobo temporal tem papel relevante na epilepsia associada a calcificações intracerebrais sugestivas de neurocisticercose, foram comparadas a gravidade da epilepsia (critérios de Pazzaglia, modificados), a sintomatologia ictal, a localização e a quantidade de calcificações intracerebrais e a localização do foco eletrencefalográfico interictal. Foram estudados três grupos de pacientes recrutados do ambulatório de neurologia do Hospital de Clinicas - UFPR, Curitiba, Paraná sendo um grupo de pacientes com epilepsia, calcificações intracerebrais e sem foco eletrencefalográfico intercrítico (grupo A, n=24); outro grupo com epilepsia, calcificações intracerebrais e foco eletrencefalografico intercritico (grupo B, n=23); e um terceiro grupo com epilepsia sem calcificações intracerebrais, porem com foco eletrencefalografico interictal (grupo C, n=21). As anormalidades eletrencefalograficas ocorreram sobre o lobo temporal em 23 dos 24, pacientes do grupo B e nos 23 pacientes do grupo C. Enquanto nos grupos B e C os sintomas ictais iniciais apresentados pelos pacientes foram, mais frequentemente, compatíveis com aqueles descritos para crises do lobo temporal, no grupo A predominaram sintomas iniciais compatíveis com envolvimento do córtex sensório-motor. A epilepsia foi mais grave nos pacientes dos grupos B e C que naqueles do grupo A (p=0,0001 e 0,0054, respectivamente), mas não houve diferença de gravidade entre os grupos B e C. A diferença na gravidade pode ser atribuída a presença de crises parciais de alta frequência (grupo B) e parciais com generalização de alta frequência (B e C). Não houve relação direta entre o numero de calcificações e a gravidade da epilepsia nos grupos A e B. O estudo confirmou a hipótese de que, na epilepsia associada a calcificações intracerebrais sugestivas de neurocisticercose, o envolvimento do lobo temporal no processo epileptogênico e relevante na determinação dos sintomas ictais, na frequência das crises e na gravidade da epilepsia, independentemente do numero e localização das calcificações.Abstract: To test the hypothesis that the temporal lobe is relevant in determining the clinical behavior of epilepsy associated to intracranial calcifications suggestive of neurocysticercosis, severity of epilepsy (Pazzaglia criteria, modified), seizure symptomatology, location and number of intracranial calcifications and location of interjetai EEG foci were compared between three group of epileptic individuals. Patients were serially selected at the outpatient clinic of the Hospital de Clinicas - UFPR, at Curitiba, Brazil: a group of patients with epilepsy and intracranial calcifications, but without interictal EEG foci (group A, n=24); other with epilepsy, intracranial calcifications, and interictal EEG foci (group B, n=23); and a third group with epilepsy and interictal EEG foci, but without intracranial calcifications (group C, n=21). EEG abnormalities located over the temporal lobe in 23 patients of group B, in and 23 patients of group C. While initial seizure symptoms presented by groups B and C patients were suggestive of those related to temporal lobe seizures in most cases, they related to somato-sensory cortex in most of group A patients. Epilepsy was most severe in groups B and C than in group A patients (p=0,0001 and 0,0054, respectively), but there was no difference in severity between groups B and C. Difference in severity could be atributted to the ocurrence of high frequency partial seizures (group B) and secondarily generalized seizures (groups B and C). There was not a direct relationship between the number of calcifications and severity of epilepsy in groups A and B. The study confirmed the hypothesis that in epilepsy associated to intracerebral calcifications suggestive of neurocysticercosis, involvement of the temporal lobe in the epileptogenic process is relevant in determining seizure symptoms and frequency, and severity of the epilepsy, irrespective to the number and location of intracerebral calcifications

Phenytoin-induced isolated chronic, nocturnal dry cough

AbstractWe report a 72-year-old man with a four-year history of dyscognitive seizures (with occasional secondary generalization) who developed isolated, nocturnal dry cough immediately after being started on PO phenytoin. The cough was not accompanied by any other symptom or sign as his physical exam was completely normal. Further investigation with chest CT and spirometry was unremarkable. This symptom persisted for six months and did not resolve until we weaned him off of phenytoin. According to the Naranjo Adverse Drug Reaction Probability Scale, his cough was classified as being probably (score +6) related to the use of this antiepileptic drug. To our knowledge, there has been only one study that reported phenytoin-triggered cough. It described a postoperative patient who developed cough and bronchospasm after receiving IV phenytoin. By reporting our case and discussing the literature on this specific topic, we have essentially two goals. First, we intend to remind clinicians that isolated persistent cough can be an adverse reaction to phenytoin. Second, we hope to encourage further studies that will be able to elucidate the association presented herein

Global Properties for first order differential operators on Tr+1×(S3)s\mathbb{T}^{r+1}\times(\mathbb{S}^{3})^s

In this paper, we study the global properties of a class of evolution-like differential operator with a 0-order perturbation defined on the product of $r+1$ tori and $s$ spheres $\mathbb{T}^{r+1}\times(\mathbb{S}^{3})^s$, with $r$ and $s$ non-negative integers. By varying the values of $r$ and $s$, we show that it is possible to recover results already known in the literature and present new results. The main tool used in this study is Fourier analysis, taken partially with respect to each copy of the torus and sphere. We obtain necessary and sufficient conditions related to Diophantine inequalities, change of sign and connectivity of level sets associated the operator's coefficients

Cerebral Venous Thrombosis with Migraine-Like Headache and the Trigeminovascular System

Cerebral venous thrombosis- (CVT-) associated headache is considered a secondary headache, commonly presenting as intracranial hypertension headache in association with seizures and/or neurological signs. However, it can occasionally mimic migraine. We report a patient presenting with a migraine-like, CVT-related headache refractory to several medications but intravenous dihydroergotamine (DHE). The response to DHE, which is considered to be an antimigraine medication, in addition to the neurovascular nature of migraine, points out to a probable similarity between CVT-headache and migraine. Based on experimental studies, we discuss this similarity and hypothesize a trigeminovascular role in the genesis of CVT-associated headache