Unitary equivalence to a truncated Toeplitz operator: analytic symbols

Unlike Toeplitz operators on $H^2$, truncated Toeplitz operators do not have a natural matricial characterization. Consequently, these operators are difficult to study numerically. In this note we provide criteria for a matrix with distinct eigenvalues to be unitarily equivalent to a truncated Toeplitz operator having an analytic symbol. This test is constructive and we illustrate it with several examples. As a byproduct, we also prove that every complex symmetric operator on a Hilbert space of dimension $\leq 3$ is unitarily equivalent to a direct sum of truncated Toeplitz operators.Comment: 15 page

On the closure of the complex symmetric operators: compact operators and weighted shifts

We study the closure $\bar{CSO}$ of the set $CSO$ of all complex symmetric operators on a separable, infinite-dimensional, complex Hilbert space. Among other things, we prove that every compact operator in $\bar{CSO}$ is complex symmetric. Using a construction of Kakutani as motivation, we also describe many properties of weighted shifts in $\bar{CSO} \backslash CSO$. In particular, we show that weighted shifts which demonstrate a type of approximate self-similarity belong to $\bar{CSO}\backslash CSO$. As a byproduct of our treatment of weighted shifts, we explain several ways in which our result on compact operators is optimal.Comment: 19 pages, to appear in J. Funct. Ana

Unitary Equivalence to a Complex Symmetric Matrix: Low Dimensions

A matrix T∈Mn(C) is UECSM if it is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We develop several techniques for studying this property in dimensions three and four. Among other things, we completely characterize 4×4 nilpotent matrices which are UECSM and we settle an open problem which has lingered in the 3×3 case. We conclude with a discussion concerning a crucial difference which makes dimension three so different from dimensions four and above

Unitary Equivalence to a Complex Symmetric Matrix: A Modulus Criterion

We develop a procedure for determining whether a square complex matrix is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We compare our approach to several existing methods [1, 19, 20] and present a number of examples

Quotient Sets and Diophantine Equations

Quotient sets U / U = {u / u¹: u, u¹ ϵ U} have been considered several times before in the MONTHLY. We consider more general quotient sets U / V and we apply our results to certain simultaneous Diophantine equations with side constraints

Adult-Onset Obesity Reveals Prenatal Programming of Glucose-Insulin Sensitivity in Male Sheep Nutrient Restricted during Late Gestation

BACKGROUND: Obesity invokes a range of metabolic disturbances, but the transition from a poor to excessive nutritional environment may exacerbate adult metabolic dysfunction. The current study investigated global maternal nutrient restriction during early or late gestation on glucose tolerance and insulin sensitivity in the adult offspring when lean and obese. METHODS/PRINCIPAL FINDINGS: Pregnant sheep received adequate (1.0M; CE, n = 6) or energy restricted (0.7M) diet during early (1-65 days; LEE, n = 6) or late (65-128 days; LEL, n = 7) gestation (term approximately 147 days). Subsequent offspring remained on pasture until 1.5 years when all received glucose and insulin tolerance tests (GTT & ITT) and body composition determination by dual energy x-ray absorptiometry (DXA). All animals were then exposed to an obesogenic environment for 6-7 months and all protocols repeated. Prenatal dietary treatment had no effect on birth weight or on metabolic endpoints when animals were 'lean' (1.5 years). Obesity revealed generalised metabolic 'inflexibility' and insulin resistance; characterised by blunted excursions of plasma NEFA and increased insulin(AUC) (from 133 to 341 [s.e.d. 26] ng.ml(-1).120 mins) during a GTT, respectively. For LEL vs. CE, the peak in plasma insulin when obese was greater (7.8 vs. 4.7 [s.e.d. 1.1] ng.ml(-1)) and was exacerbated by offspring sex (i.e. 9.8 vs. 4.4 [s.e.d. 1.16] ng.ml(-1); LEL male vs. CE male, respectively). Acquisition of obesity also significantly influenced the plasma lipid and protein profile to suggest, overall, greater net lipogenesis and reduced protein metabolism. CONCLUSIONS: This study indicates generalised metabolic dysfunction with adult-onset obesity which also exacerbates and 'reveals' programming of glucose-insulin sensitivity in male offspring prenatally exposed to maternal undernutrition during late gestation. Taken together, the data suggest that metabolic function appears little compromised in young prenatally 'programmed' animals so long as weight is adequately controlled. Nutritional excess in adulthood exacerbates any programmed phenotype, indicating greater vigilance over weight control is required for those individuals exposed to nutritional thrift during gestation

Reduced Cortisol and Metabolic Responses of Thin Ewes to an Acute Cold Challenge in Mid-Pregnancy: Implications for Animal Physiology and Welfare

Background: Low food availability leading to reductions in Body Condition Score (BCS; 0 indicates emaciation and 5 obesity) in sheep often coincides with low temperatures associated with the onset of winter in New Zealand. The ability to adapt to reductions in environmental temperature may be impaired in animals with low BCS, in particular during pregnancy when metabolic demand is higher. Here we assess whether BCS affects a pregnant animal’s ability to cope with cold challenges. Methods: Eighteen pregnant ewes with a BCS of 2.760.1 were fed to attain low (LBC: BCS2.360.1), medium (MBC: BCS3.260.2) or high BCS (HBC: BCS3.660.2). Shorn ewes were exposed to a 6-h acute cold challenge in a climate-controlled room (wet and windy conditions, 4.460.1uC) in mid-pregnancy. Blood samples were collected during the BCS change phase, acute cold challenge and recovery phase. Results: During the BCS change phase, plasma glucose and leptin concentrations declined while free fatty acids (FFA) increased in LBC compared to MBC (P,0.01, P,0.01 and P,0.05, respectively) and HBC ewes (P,0.05, P,0.01 and P,0.01, respectively). During the cold challenge, plasma cortisol concentrations were lower in LBC than MBC (P,0.05) and HBC ewes (P,0.05), and FFA and insulin concentrations were lower in LBC than HBC ewes (P,0.05 and P,0.001, respectively). Leptin concentrations declined in MBC and HBC ewes while remaining unchanged in LBC ewes (P,0.01). Glucose concentrations and internal body temperature (Tcore) increased in all treatments, although peak Tcore tended to be higher in HBC ewes (P,0.1). During the recovery phase, T4 concentrations were lower in LBC ewes (P,0.05). Conclusion: Even though all ewes were able to increase Tcore and mobilize glucose, low BCS animals had considerably reduced cortisol and metabolic responses to a cold challenge in mid-pregnancy, suggesting that their ability to adapt to cold challenges through some of the expected pathways was reduced

Case Reports1. A Late Presentation of Loeys-Dietz Syndrome: Beware of TGFβ Receptor Mutations in Benign Joint Hypermobility

Background: Thoracic aortic aneurysms (TAA) and dissections are not uncommon causes of sudden death in young adults. Loeys-Dietz syndrome (LDS) is a rare, recently described, autosomal dominant, connective tissue disease characterized by aggressive arterial aneurysms, resulting from mutations in the transforming growth factor beta (TGFβ) receptor genes TGFBR1 and TGFBR2. Mean age at death is 26.1 years, most often due to aortic dissection. We report an unusually late presentation of LDS, diagnosed following elective surgery in a female with a long history of joint hypermobility. Methods: A 51-year-old Caucasian lady complained of chest pain and headache following a dural leak from spinal anaesthesia for an elective ankle arthroscopy. CT scan and echocardiography demonstrated a dilated aortic root and significant aortic regurgitation. MRA demonstrated aortic tortuosity, an infrarenal aortic aneurysm and aneurysms in the left renal and right internal mammary arteries. She underwent aortic root repair and aortic valve replacement. She had a background of long-standing joint pains secondary to hypermobility, easy bruising, unusual fracture susceptibility and mild bronchiectasis. She had one healthy child age 32, after which she suffered a uterine prolapse. Examination revealed mild Marfanoid features. Uvula, skin and ophthalmological examination was normal. Results: Fibrillin-1 testing for Marfan syndrome (MFS) was negative. Detection of a c.1270G > C (p.Gly424Arg) TGFBR2 mutation confirmed the diagnosis of LDS. Losartan was started for vascular protection. Conclusions: LDS is a severe inherited vasculopathy that usually presents in childhood. It is characterized by aortic root dilatation and ascending aneurysms. There is a higher risk of aortic dissection compared with MFS. Clinical features overlap with MFS and Ehlers Danlos syndrome Type IV, but differentiating dysmorphogenic features include ocular hypertelorism, bifid uvula and cleft palate. Echocardiography and MRA or CT scanning from head to pelvis is recommended to establish the extent of vascular involvement. Management involves early surgical intervention, including early valve-sparing aortic root replacement, genetic counselling and close monitoring in pregnancy. Despite being caused by loss of function mutations in either TGFβ receptor, paradoxical activation of TGFβ signalling is seen, suggesting that TGFβ antagonism may confer disease modifying effects similar to those observed in MFS. TGFβ antagonism can be achieved with angiotensin antagonists, such as Losartan, which is able to delay aortic aneurysm development in preclinical models and in patients with MFS. Our case emphasizes the importance of timely recognition of vasculopathy syndromes in patients with hypermobility and the need for early surgical intervention. It also highlights their heterogeneity and the potential for late presentation. Disclosures: The authors have declared no conflicts of interes