The discriminants associated to isotropy representations of symmetric spaces

We consider a generalized discriminant associated to a symmetric space which generalizes the discriminant of real symmetric matrices, and note that it can be written as a sum of squares of real polynomials. A method to estimate the minimum number of squares required to represent the discrimininant is developed and applied in examples.

The effects of catecholamine depletion on the neural response to fearful faces in remitted depression

Recent evidence suggests that increased psychophysiological response to negatively valenced emotional stimuli found in major depressive disorder (MDD) may be associated with reduced catecholaminergic neurotransmission. Fourteen unmedicated, remitted subjects with MDD (RMDD) and 13 healthy control subjects underwent catecholamine depletion with oral α-methyl-para-tyrosine (AMPT) in a randomized, placebo-controlled, double-blind crossover trial. Subjects were exposed to fearful (FF) and neutral faces (NF) during a scan with [15O]H2O positron emission tomography to assess the brain-catecholamine interaction in brain regions previously associated with emotional face processing. Treatment with AMPT resulted in significantly increased, normalized cerebral blood flow (CBF) in the left inferior temporal gyrus (ITG) and significantly decreased CBF in the right cerebellum across conditions and groups. In RMDD, flow in the left posterior cingulate cortex (PCC) increased significantly in the FF compared to the NF condition after AMPT, but remained unchanged after placebo, whereas healthy controls showed a significant increase under placebo and a significant decrease under AMPT in this brain region. In the left dorsolateral prefrontal cortex (DLPFC), flow decreased significantly in the FF compared to the NF condition under AMPT, and increased significantly under placebo in RMDD, whereas healthy controls showed no significant differences. Differences between AMPT and placebo of within-session changes in worry-symptoms were positively correlated with the corresponding changes in CBF in the right subgenual prefrontal cortex in RMDD. In conclusion, this study provided evidence for a catecholamine-related modulation of the neural responses to FF expressions in the left PCC and the left DLPFC in subjects with RMDD that might constitute a persistent, trait-like abnormality in MDD.

Applicability of the GP Device to the Circle of Willis Arteries by Using a Mathematical Model

According to the World Health Organization, 15 million people suffer stroke worldwide each year, of these, 5 million die and 5 million are permanently disabled. Stroke is therefore a major cause of mortality world-wide. The majority of strokes are caused by a blood clot that occludes an artery in the brain, and although thrombolytic agents such as Alteplase are used to dissolve clots that arise in the arteries of the brain, there are limitations on the use of these thrombolytic agents. However over the past decade, other methods of treatment have been developed which include Thrombectomy Devices e.g. the 'GP' Thrombus Aspiration Device ('GP' TAD). Such devices may be used as an alternative to thrombolytics or in conjunction with them to extract blood clots in arteries such as the middle cerebral artery of the midbrain brain, and the posterior inferior cerebellar artery (PICA) of the posterior aspect of the brain. In this paper, we mathematically model the removal of blood clots using the 'GP' TAD from selected arteries of the brain where blood clots may arise taking into account factors such as the resistances, compliances and inertances effects. Such mathematical modelling may have potential uses in predicting the pressures necessary to extract blood clots of given lengths, and masses from arteries in the Circle of Willis - posterior circulation of the brain

A Catalogue of estates in Cambridge belonging to Harvard College, 1799 January 11

Two folio-sized leaves containing a handwritten list of Cambridge real estate belonging to Harvard and the amount of related rents, signed by President Willard.

Letter from Samuel Willard, Cambridge, Massachusetts, to William Willard, Petersham, Massachusetts, September 6, 1799

This letter written to his father describes his arrival at Harvard, book expenses, and present financial situation; he also asks his father to build him a writing desk. Willard discusses the family of his uncle, Harvard president Joseph Willard, and his uncle’s health and issues with jaundice.

Letter from Samuel Willard, Cambridge, Massachusetts, to Catherine Willard, Petersham, Massachusetts, September 6, 1799

Willard complains about not having received requested items sooner, explains that his aunt gave him some furniture, discusses friends in Lancaster, mentions a blue coat that he would like to have made, and describes the house that he lives in, which was called “the Den or College House.”

Letter from Samuel Willard, Cambridge, Massachusetts, to William Willard, Petersham, Massachusetts, November 22, 1799

Willard details his various expenses for food, wood, tuition, and books, and explains his bill payment schedule. He also mentions President Willard and sends best wishes to his mother, brother, and sister.

Letter from Samuel Willard, Cambridge, Massachusetts, to Catherine Willard, Petersham, Massachusetts, November 22, 1799

Willard discusses tailoring his coat, asks to be sent another towel and a Bible, and describes the latest Harvard fashion: “Deep blue is the Colour most in Vogue in this Place.” He also mentions that the “President and his family are pretty well,” and that he is invited to dine with them about once a fortnight.

Letter from Samuel Willard, Harvard College, Cambridge, Massachusetts, to William Willard and Katherine Willard, Petersham, Massachusetts, April 4, 1801

Willard informs his parents of the death of Deacon Fairbank. He assures them that on his return to Cambridge, he was “received with great apparent cordiality both by the government + the president’s family.” He also reports on the health of President Willard, whose health has improved. Willard concludes the letter by asking for money to pay his expenses.

Letter from Samuel Willard, Harvard College, Cambridge, Massachusetts, to Catherine Willard, Petersham, Massachusetts, May 1, 1801

Willard alludes to a situation regarding his father and praises him profusely.

Letter from Samuel Willard, Harvard College, Cambridge, Massachusetts, to Abigail Willard, Petersham, Massachusetts, May 1, 1801

Willard thanks his sister for writing to him and asks her to write as often as possible. He also mentions his cousin Sophia Chadwick, who has been living with President Willard.

Letter from Samuel Willard, Harvard College, Cambridge, Massachusetts, to Catherine Willard, Petersham, Massachusetts, June 26, 1801

Willard discusses various articles of clothing.

Letter from Samuel Willard, Harvard College, Cambridge, Massachusetts, to William Willard, Petersham, Massachusetts, October 16, 1801

Willard reports on President Willard’s travels and general well-being and asks his father to apologize to his sister for not writing to her often enough. He also tells his father that he is in debt, details his purchases, and asks for money.

Letter from Samuel Willard, Harvard College, Cambridge, Massachusetts, to William Willard, Petersham, Massachusetts, February 28, 1802

Willard describes his trip back to Harvard from Petersham, reports that President Willard and family were doing well, although “he has had an ill turn since, but has recovered,” and concludes the letter by asking for more money.