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The authors proved three theorems about the exact solutions of a generalized or interacting Black–Scholes equation that explicitly includes arbitrage bubbles. These arbitrage bubbles can be characterized by an arbitrage number AN. The first theorem states that if AN = 0, then the solution at maturity of the interacting equation is identical to the solution of the free Black–Scholes equation with the same initial interest rate of r. The second theorem states that if AN ≠ 0, then the interacting solution can be expressed in terms of all higher derivatives of the solutions to the free Black–Scholes equation with an initial interest rate of r. The third theorem states that for a given arbitrage number, the interacting solution is a solution to the free Black–Scholes equation but with a variable interest rate of r(τ) = r + (1/τ)AN(τ), where τ = T − t.

Perivascular adipose tissue (PVAT) regulates vascular function due to its capacity to synthesize vasoactive products and its mechanical properties. PVATs most abundant cells are adipocytes, and their populations are maintained by the maturation of adipocyte progenitor cells (APC), which may play a pivotal role in the pathogenesis of cardiovascular diseases. However, the distribution of APC within PVAT depots, their potential variation in spatial location, and the influence of sex and age on their abundance remain unknown. We hypothesize that APC abundance in PVAT is affected by location, age, sex and that APC subtypes have specific spatial distributions. PVAT from thoracic and abdominal aorta, and mesenteric arteries, and AT from interscapular, gonadal, and subcutaneous depots from 13-week and 30-week-old females and males Pdgfrα-CreERT2 x LSLtdTomato mice (n = 28) were analyzed. Abdominal aorta PVAT had fewer progenitors than mesenteric PVAT and gonadal AT. Aging reduced the abundance of APC in the thoracic aorta but increased their numbers in mesenteric PVAT. Females had more APC than males in mesenteric PVAT and gonadal AT depots. APC exhibited unique spatial distribution in the aorta and mesenteric PVAT where they localized neighboring vasa vasorum and arteries. APC subtypes (APC1, APC2, APC3, diff APC) were identified in all PVAT depots. Thoracic aorta PVAT APC3 were located in the adventitia while diff APC were in the parenchyma. This study identified variability in APC populations based on depot, age, and sex. The distinctive spatial distribution and the presence of diverse APC subtypes suggest that they may contribute differently to cardiovascular diseases-induced PVAT remodeling. [ABSTRACT FROM AUTHOR]

Perivascular adipose tissue (PVAT) is increasingly recognized for its function in mechanotransduction. However, major gaps remain in our understanding of the cells present in PVAT, as well as how different cells contribute to mechanotransduction. We hypothesized that snRNA-seq would reveal the expression of mechanotransducers, and test one (PIEZO1) to illustrate the expression and functional agreement between single-nuclei RNA sequencing (snRNA-seq) and physiological measurements. To contrast two brown tissues, subscapular brown adipose tissue (BAT) was also examined. We used snRNA-seq of the thoracic aorta PVAT (taPVAT) and BAT from male Dahl salt-sensitive (Dahl SS) rats to investigate cell-specific expression mechanotransducers. Localization and function of the mechanostransducer PIEZO1 were further examined using immunohistochemistry (IHC) and RNAscope, as well as pharmacological antagonism. Approximately 30,000 nuclei from taPVAT and BAT each were characterized by snRNA-seq, identifying eight major cell types expected and one unexpected (nuclei with oligodendrocyte marker genes). Cell-specific differential gene expression analysis between taPVAT and BAT identified up to 511 genes (adipocytes) with many (≥20%) being unique to individual cell types. Piezo1 was the most highly, widely expressed mechanotransducer. The presence of PIEZO1 in the PVAT but not the adventitia was confirmed by RNAscope and IHC in male and female rats. Importantly, antagonism of PIEZO1 by GsMTX4 impaired the PVAT's ability to hold tension. Collectively, the cell compositions of taPVAT and BAT are highly similar, and PIEZO1 is likely a mechanotransducer in taPVAT. NEW & NOTEWORTHY This study describes the atlas of cells in the thoracic aorta perivascular adipose tissue (taPVAT) of the Dahl-SS rat, an important hypertension model. We show that mechanotransducers are widely expressed in these cells. Moreover, PIEZO1 expression is shown to be restricted to the taPVAT and is functionally implicated in stress relaxation. These data will serve as the foundation for future studies investigating the role of taPVAT in this model of hypertensive disease. [ABSTRACT FROM AUTHOR]

Introduction: Tunica media extracellular matrix (ECM) remodeling is well understood to occur in response to elevated blood pressure, unlike the remodeling of other tunicas. We hypothesize that perivascular adipose tissue (PVAT) is responsive to hypertension and remodels as a protective measure. Methods: The adventitia and PVAT of the thoracic aorta were used in measuring ECM genes from 5 pairs of Dahl SS male rats on 8 or 24 weeks of feeding from weaning on a control (10% Kcal fat) or high-fat (HF; 60%) diet. A PCR array of ECM genes was performed with cDNA from adventitia and PVAT after 8 and 24 weeks. A gene regulatory network of the differentially expressed genes (DEGs) (HF 2-fold > con) was created using Cytoscape. Results: After 8 weeks, 29 adventitia but 0 PVAT DEGs were found. By contrast, at 24 weeks, PVAT possessed 47 DEGs while adventitia had 3. Top DEGs at 8 weeks in adventitia were thrombospondin 1 and collagen 8a1. At 24 weeks, thrombospondin 1 was also a top DEG in PVAT. The transcription factor Adarb1 was identified as a regulator of DEGs in 8-week adventitia and 24-week PVAT. Conclusion: These data support that PVAT responds biologically once blood pressure is elevated. [ABSTRACT FROM AUTHOR]