Topological methods for strong local minimizers and extremals of multiple integrals in the calculus of variations

Let Ω ⊂ Rn be a bounded Lipschitz domain and consider the energy functional F[u, Ω] := ∫ Ω F(∇u(x)) dx, over the space Ap(Ω) := {u ∈ W 1,p(Ω, Rn): u|∂Ω = x, det ∇u> 0 a.e. in Ω}, where the integrand F : Mn×n → R is quasiconvex, sufficiently regular and satisfies a p-coercivity and p-growth for some exponent p ∈ [1, ∞[. A motivation for the study of above energy functional comes from nonlinear elasticity where F represents the elastic energy of a homogeneous hyperelastic material and Ap(Ω) represents the space of orientation preserving deformations of Ω fixing the boundary pointwise. The aim of this thesis is to discuss the question of multiplicity versus uniqueness for extremals and strong local minimizers of F and the relation it bares to the domain topology. Our work, building upon previous works of others, explicitly and quantitatively confirms the significant role of domain topology, and provides explicit and new examples as well as methods for constructing such maps. Our approach for constructing strong local minimizers is topological in nature and is based on defining suitable homotopy classes in Ap(Ω) (for p ≥ n), whereby minimizing F on each class results in, modulo technicalities, a strong local minimizer. Here we work on a prototypical example of a topologically non-trivial domain, namely, a generalised annulus, Ω= {x ∈ Rn : a< |x| <b}, with 0 <a<b< ∞. Then the associated homotopy classes of Ap(Ω) are infinitely many when n =2 and two when n ≥ 3. In contrast, for constructing explicitly and directly solutions to the system of Euler-Lagrange equations associated to F we introduce a topological class of maps referred to as generalised twists and relate the problem to extremising an associated energy on the compact Lie group SO(n). The main result is a surprising discrepancy between even and odd dimensions. In even dimensions the latter system of equations admits infinitely many smooth solutions, modulo isometries, amongst such maps whereas in odd dimensions this number reduces to one. Even more surprising is the fact that in odd dimensions the functional F admits strong local minimizers yet no solution of the Euler-Lagrange equations can be in the form of a generalised twist. Thus the strong local minimizers here do not have the symmetry one intuitively expects

Generalised twists as elastic energy extremals on annuli, Squaternions and lifting twist loops to the spinor groups

Let X = {x ∈ Rⁿ : a < |x| < b} be a generalized annulus and consider the Dirichlet energy functional F[u; X] = 1/2∫x |∇u(x)|²dx, over the space of admissible maps Aϕ(X) = {u ∈ W¹,²(X, Rⁿ) : det ∇u =1 a.e. in X, u|∂X = ϕ}, where ϕ is the identity map. In this paper we consider a class of maps referred to as generalized twists and examine them in connection with the Euler–Lagrange equation associated with F[·, X] on Aϕ(X). The approach is novel and is based on lifting twist loops from SO(n) to its double cover Spin(n) and reformulating the equations accordingly. We restrict our attention to low dimensions and prove that for n = 4 the system admits infinitely many smooth solutions in the form of twists while for n = 3 this number sharply reduces to one. We discuss some qualitative features of these solutions in view of their remarkable explicit representation through the exponential map of Spin(n)

The effects of cyclooxygenase inhibitors on the gastric emptying and small intestine transit in the male rats following traumatic brain injury

Objective(s): This study was carried out to investigate the effects of COX-2 selective inhibitor (Celecoxib) or non-selective COX inhibitor (Ibuprofen) on gastrointestinal motility. Materials and Methods: The rats were randomly divided into five groups including: intact, sham, traumatic brain injury (TBI) group (intact rats under TBI), Celecoxib group (10 mg/kg), Ibuprofen group (10 mg/kg). Rats of the treatment groups received gavages at 1 hr before the TBI induction. The TBI was moderate and diffused using the Marmarou method. The gastric emptying and small intestine transit were measured by phenol red method. Results: The gastric emptying didn&apos;t change following TBI induction compared to intact group. The consumption of ibuprofen or celecoxib didn&apos;t have any effect on gastric emptying compared to sham group. TBI induction didn&apos;t have any effect on the intestinal transit. Also, there was no significant difference between ibuprofen or celecoxib consumption vs. sham group (P&gt;0.05). Conclusion: The COX-2 selective inhibitor (celecoxib) or non-selective COX inhibitor (ibuprofen) have no effects on gastric or small bowel transit. Further work is necessary to investigate the effects of non-selective COX inhibitors and their impact on gastrointestinal motility disorders

Protective effects of an interaction between vagus nerve and melatonin on gastric ischemia/reperfusion: the role of oxidative stress

Objectives: Vagal pathways in gastrointestinal tract are the most important pathways that regulate ischemia/reperfusion (I/R). Gastrointestinal tract is one of the important sources of melatonin production. The aim of this study was to investigate probable protective effect of the interaction between vagus nerve and melatonin after I/R. Materials and methods: This study was performed in male rats that were divided into six groups. Cervical vagus nerve was cut bilaterally after induction of I/R and the right one was stimulated by stimulator. Melatonin or vehicle was injected intraperitoneally. The stomach was removed for histopathological and biochemical investigations. Results: A significant decrease in infiltration of gastric neutrophils and malondialdehyde (MDA) level after I/R was induced by melatonin and was disappeared after vagotomy. The stimulation of vagus nerve significantly enhanced these effects of melatonin. However, a stimulation of vagus nerve alone increased neutrophils infiltration and MDA level. Melatonin significantly increased the activities of catalase, glutathione peroxidase (GPx), superoxide dismutases (SOD). Unlike stimulation of vagus nerve, vagotomy decreased these effects of melatonin. Conclusion: According to these results, it is probable that protective effects of melatonin after I/R may be mediated by vagus nerve. Therefore, there is an interaction between melatonin and vagus nerve in their protective effects

Protective effect of Mumiju against acetic acid-induced ulcerative colitis in rats

Objective: In this study, we elucidated the ameliorative effect of aqueous extract of leaves of Mumiju against acetic acid-induced experimental colitis in male rats. Materials and Methods: The animals were randomly divided into four groups (n=7) including I: control group, II: vehicle group (injected with 2 ml acetic acid (4%) intra rectally), III and IV: treatment groups which received Mumiju (250 mg/kg) orally or intraperitoneally for 4 consecutive days after ulcer induction. Ulcer index, severity of inflammation, colonic levels of superoxide dismutase (SOD), glutathione (GSH), and malondialdehyde (MDA), and histological changes were recorded after the treatment regimen of 4 days. Results: The ulcer index, severity of inflammation and colonic MDA levels were increased following intrarectal instillation of acetic acid. Also, acetic acid significantly decreased the SOD and GSH levels. Treatment with Mumiju for 4 days exhibited significantly lowered oxidative stress, while elevated of SOD and GSH levels. Regenerative-healing patterns also was seen by histopathological findings after treatment with Mumiju. Conclusion: The present investigation demonstrates that Mumiju could be regarded as a herb with potent therapeutic value in the amelioration of experimental colitis in laboratory animals by modulation of oxidant- antioxidant system

Laparoscopic repair of iatrogenic long ureteral injury

Introduction: Transection of the ureter in laparoscopic retroperitoneal lymph node dissection (RPLND) is a rare iatrogenic complication. Its repair is possible by end-to-end anastomosis when the defect is short, but if there is a long defect, laparoscopic repair is a challenge. Methods: A 30-year-old man underwent modified RPLND of a clinically stage-A mixed germ cell of the left testis. While cutting the gonadal vein, 1.5 cm of the ureter was transected 2 cm below the renal pelvis and clipped by one of our residents. The injury was diagnosed immediately. After the completion of the modified RPLND, the clips were removed and a 20-mm defect of the ureter was seen precluding anastomosis. The kidney was dissected and pulled down 35 mm. The lower pole of the kidney was then fixed to the psoas muscle by a 2-0 vicryl, making anastomosis possible after an 8-mm spatulation of both sides. Anastomosis was done over a stent by interrupted sutures using 4-0 vicryl. Results: Leakage was 400 mL at the first postoperative day and ceased at the 4th day. On the control excretion urography, the kidney function was normal and a fullness of the caliceal system was seen. Conclusion: Long defect of the ureter that may occur in laparoscopic urological surgeries could be manageable laparoscopically using methods for shortening of the interval such as relocation of the kidney. Thus, a same approach as open surgeries can yield acceptable results in laparoscopic surgeries

Mumijo Protection gainst Acetaminophen-Induced Acute Hepatic Injury: Role of Oxidative Stress

Background: A majority of people widely use acetaminophen as a sedative. Overusing the drug for prolonged periods of time can lead to acute liver damage. Mumijo, as a strong antioxidant and anti-inflammatory drug, could possibly reduce some of the acetaminophen-induced side effects on the liver. Thus, the aim of this study is to evaluate the effect of Mumijo on the liver damage caused by the use of acetaminophen. Methods: 40 male Wistar rats were randomly divided into five groups: sham, acetaminophen, low and high doses of mumijo, and vehicle. All groups except the sham group received a single dose of 500 mg/kg acetaminophen via ip injection. Then the groups that were under treatment received 150 mg/kg (low dose) and 250 mg/kg (high dose) of mumijo, and the vehicle group received distilled water as vehicle. After 24 hours, blood samples were taken for biochemical tests, and a part of the liver was extracted for histopathological examination. Results: acetaminophen increases the activities of functional liver enzymes including alanine amino transferase (ALT), aspartate aminotransferase (AST), and gamma glutamine transferase (GGT). In groups under treatment, values of the mentioned enzymes were significantly reduced in comparison with the acetaminophen and vehicle groups (P <0.05), and on the other hand, malondialdehyde (MDA), nitric oxide (NO), and protein carbonyl (PC) increase caused by acetaminophen were reduced by mumijo. Furthermore, the amount of glutathione (GPX) was increased by mumijo (P <0.05). From a histopathological point of view, necrosis and liver damage caused by acetaminophen was decreased by mumijo. Conclusion: The findings showed that mumijo is salient in preventing liver damage caused by consumption of high doses of acetaminophen probably through reducing oxidant activities and also through increasing anti-inflammatory and antioxidant activities

Effect of antioxidant therapy on the sperm DNA integrity improvement; a longitudinal cohort study

Background:&nbsp;The effect of antioxidant therapy on sperm DNA fragmentation index (DFI)&nbsp;and achieving natural pregnancy were under debate. Very few studies have showed&nbsp;the rate of pregnancy rate after the antioxidant therapy due to ethical and technical&nbsp;limitations. Objective:&nbsp;The aim of this cohort study was to determine the improvement rate of&nbsp;sperm DFI and natural pregnancy rate after the antioxidant therapy in infertile men. Materials and Methods:&nbsp;1645 infertile men were subjected for this study from May&nbsp;2015 to December 2017. The Spermogram and sperm DFI were assessed using World&nbsp;Health Organization (WHO) 2010-based protocols and sperm chromatin structure assay&nbsp;(SCSA), respectively, in sperm samples before and after antioxidant therapy. Results:&nbsp;The total sperm DFI improvement rate was 38.9% in the total population.&nbsp;Sperm DFI improvement had close correlation with total motility (r= 0.731, p= 0.001)&nbsp;and progressive motility improvement (r= 0.885, p= 0.001); 16.8% of individuals who&nbsp;completed antioxidant therapy for nine months achieved natural pregnancy. Conclusion:&nbsp;The results of the current study suggested that SCSA along with&nbsp;spermogram might be a suitable option for the evaluation of fertility potential. In&nbsp;addition, antioxidant therapy may be useful for men with high levels of sperm DFI.&nbsp;However, the rate of pregnancy was still low and other treatment protocols such as&nbsp;assisted reproductive technology may be necessary. Key words: Antioxidant, Reactive oxygen species, DNA fragmentation, Male infertility