Классическое решение в четверти плоскости смешанной задачи для волнового уравнения

This article presents the classical solution with mixed boundary conditions in the quarter of the plane for the wave equation in the analytical form. The boundary of the region consists of two perpendicular half-straight lines. On one of them, Cauchy’s boundary conditions are assigned. The second half-straight line is divided into two parts. Dirichlet’s condition is assigned on the straight line and Neumann’s conditions – on the half-straight line. The classical solution of the considered problem is defined in the class of double continuous differentiable functions in the quarter of the plane. To build this solution, the partial solution of the initial wave equation is written. For the assigned functions of the problem, the matching conditions are written, which are necessary and enough so that the solution of the problem would be classical and unique.В данной работе в аналитическом виде представлено классическое решение со смешанными граничными условиями в четверти плоскости для волнового уравнения. Граница области состоит из двух перпендикулярных полупрямых. На одной из них задаются условия Коши. Вторая полупрямая разделена на две части: конечный отрезок и оставшаяся часть в виде полупрямой. На отрезке задается условие Дирихле, на полупрямой – условие Неймана. В классе дважды непрерывно дифференцируемых функций в четверти плоскости определяется классическое решение рассматриваемой задачи. Для построения этого решения выписывается частное решение исходного волнового уравнения. Для заданных функций задачи выписываются условия согласования, которые являются необходимыми и достаточными, чтобы решение задачи было классическим и единственным

Pешение произвольной гладкости одномерного волнового уравнения для задачи со смешанными условиями

In this paper, we represented an analytical form of a classical solution of the wave equation in the class of continuously differentiable functions of arbitrary order with mixed boundary conditions in a quarter of the plane. The boundary of the area consists of two perpendicular half-lines. On one of them, the Cauchy conditions are specified. The second half-line is separated into two parts, namely, the limited segment and the remaining part in the form of a half-line. The Dirichlet condition is specified on the segment, as well as the Neumann condition is fulfilled on the second part in the form of a half-line. In a quarter of the plane, the classical solution of the problem under consideration is determined. To construct this solution, a particular solution of the original wave equation is established. For the given functions of the problem, the concordance conditions are written, which are necessary and sufficient for the solution of the problem to be classical of high order of smoothness and unique.В аналитическом виде представлено классическое решение в классе непрерывно дифференцируемых функций произвольного порядка со смешанными граничными условиями в четверти плоскости для волнового уравнения. Граница области состоит из двух перпендикулярных полупрямых. На одной из них задаются условия Коши. Вторая полупрямая разделена на две части: конечный отрезок и оставшаяся часть в виде полупрямой. На отрезке задается условие Дирихле, на второй части в виде полупрямой – условие Неймана. В четверти плоскости определяется классическое решение рассматриваемой задачи при построении которого выписывается частное решение исходного волнового уравнения. Для заданных функций задачи выписываются условия согласования, которые являются необходимыми и достаточными, чтобы решение задачи было классическим высокого порядка гладкости и единственным

Current issues in the management of soft tissues of the oral vestibule before orthodontic treatment

Orthodontic treatment tasks include requirements not only forproper teeth alignment and optimal jaw relationships in thesagittal and vertical planes with the maximum of occlusal contactsbut also for healthy periodontal tissues upon completionof orthodontic treatment. Often, certain occlusal anomaliesaffect the surrounding soft tissues. In this situation, the orthodontistrealizes that not until the periodontal problem isresolved can the orthodontic treatment begin. Soft tissueanomalies include epithelial cords that make the periostealattachments extremely thin, may prevent tooth eruption, orcause multiple gingival recessions. The shallow oral vestibulealso adversely affects the position of the incisors, especiallyin the lower jaw. In such cases, incisors should not be movedusing fixed orthodontic appliances, since this will cause thinningof the attached gingiva in the cervical margins of theteeth, or exacerbate the existing gingival recessions, whichin turn will lead to root sensitivity. If a patient already exhibitsmultiple or severely expressed single gingival recessions, wesuggest their closure by surgery as a primary intervention beforeorthodontic treatment with fixed appliances or aligners,which will continuously traumatize the thin edge of attachedgingiva. In this article, we will address the necessary periodontalprocedures that orthodontists should consider beforeinitiating orthodontic correction to avoid irreversible degradationof the soft tissues of the oral cavity, periodontal tissues

The predictive function of rates of matrix metalloproteinases/inhibitors system when assessing reparative changes in the lung tissue in those with infiltrate pulmonary tuberculosis

The objective: to study the rates of matrix metalloproteinases/inhibitors system in infiltrate pulmonary tuberculosis during changes occurring in the intensive phase of treatment in order to develop a statistic model aimed to predict reparative changes.Subjects and methods. The test of ELISA (Bender MedSystems, USA) was used to measure the levels of MMP-1, MMP-8,-9. TIMM and α2-macroglobulin in blood serum of 35 patients with verified infiltrate pulmonary tuberculosis and various reparative changes during the intensive phase of treatment. The extension of infiltrative changes in the lung tissue was calculated (Vitrea, Nodule Analysis). The software of Statistica 10, ANOVA Repeated Measures and multiple discriminant analysis were used.Results. Infiltrate pulmonary tuberculosis disrupts the balance in MMP/inhibitors system regardless of the character of reparative changes: MMP-9 goes higher and the reference level of TIMM-1 and α2-macroglobulin persists. The level of MMP-9 significantly goes down but does not become normal in all variants of the disease involution. Persisting high levels of MMP-8 reduces the chances of cavity healing, while the increasing level of MMP-1 promotes resolution of cavities. To predict the character of reparative changes upon the completion of the intensive phase of treatment of infiltrate pulmonary tuberculosis, the combination of levels of MMP/inhibitors with lung tissue changes during the intensive phase of treatment is the most informative

Current issues in the management of soft tissues of the oral vestibule before orthodontic treatment

Orthodontic treatment tasks include requirements not only forproper teeth alignment and optimal jaw relationships in thesagittal and vertical planes with the maximum of occlusal contactsbut also for healthy periodontal tissues upon completionof orthodontic treatment. Often, certain occlusal anomaliesaffect the surrounding soft tissues. In this situation, the orthodontistrealizes that not until the periodontal problem isresolved can the orthodontic treatment begin. Soft tissueanomalies include epithelial cords that make the periostealattachments extremely thin, may prevent tooth eruption, orcause multiple gingival recessions. The shallow oral vestibulealso adversely affects the position of the incisors, especiallyin the lower jaw. In such cases, incisors should not be movedusing fixed orthodontic appliances, since this will cause thinningof the attached gingiva in the cervical margins of theteeth, or exacerbate the existing gingival recessions, whichin turn will lead to root sensitivity. If a patient already exhibitsmultiple or severely expressed single gingival recessions, wesuggest their closure by surgery as a primary intervention beforeorthodontic treatment with fixed appliances or aligners,which will continuously traumatize the thin edge of attachedgingiva. In this article, we will address the necessary periodontalprocedures that orthodontists should consider beforeinitiating orthodontic correction to avoid irreversible degradationof the soft tissues of the oral cavity, periodontal tissues