On factorization of the shift semigroup

Let $\mathscr{S}^\mathcal{E}=(S_t^\mathcal{E})_{t\ge 0}$ on $L^2(\mathbb{R}_+,\mathcal{E})$ be the right shift semigroup for a separable Hilbert space $\mathcal{E}$. Let $\mathscr{V}_1 = (V_{1,t})_{t\ge 0}$ and $\mathscr{V}_2 = (V_{2,t})_{t\ge 0}$ be a pair of semigroups of contractions which satisfy $V_{1,t}V_{2,s} = V_{2,s}V_{1,t}$ and $S^\mathcal{E}_t=V_{1,t} V_{2,t}$ for every $s,t \ge 0$. Such a pair is called a factorization of $\mathscr{S}^\mathcal{E}$. The main result of this note completely describes all factorizations of $\mathscr{S}^\mathcal{E}$ when $\mathcal{E}$ is finite dimensional. Using the known fact that $\mathscr{S}^\mathcal{E}$ is unitarily equivalent to a semigroup of multiplication operators on the vector valued Hardy space $H^2_\mathbb{D}(\mathcal{E})$, we employ novel function theoretic methods and classical convex analysis to arrive at the factorization

Frederic Andrews Gibbs and the assassination of John Fitzgerald Kennedy

Frederic Gibbs, the peerless expert on electroencephalogrphy was summoned to provide opinion on the EEG tracing of Jack Ruby, who murdered Lee Harvey Oswald, the assassin of John Fitzgerald Kennedy, the American President, in 1963. Gibbs pleaded that the tracing suggested features indicative of psychomotor epilepsy and Ruby killed Oswald in a state of fugue. His view was not agreed upon but Gibbs stood his ground unflinchingly. Subsequent appeals to the higher court spared Ruby from imminent execution and finally he died a natural death from metastatic complications of carcinoma of the lung in 1967

The stretch reflex and the contributions of C David Marsden

The stretch reflex or myotatic reflex refers to the contraction of a muscle in response to its passive stretching by increasing its contractility as long as the stretch is within physiological limits. For ages, it was thought that the stretch reflex was of short latency and it was synonymous with the tendon reflex, subserving the same spinal reflex arc. However, disparities in the status of the two reflexes in certain clinical situations led Marsden and his collaborators to carry out a series of experiments that helped to establish that the two reflexes had different pathways. That the two reflexes are dissociated has been proved by the fact that the stretch reflex and the tendon reflex, elicited by stimulation of the same muscle, have different latencies, that of the stretch reflex being considerably longer. They hypothesized that the stretch reflex had a transcortical course before it reached the spinal motor neurons for final firing. Additionally, the phenomenon of stimulus-sensitive cortical myoclonus lent further evidence to the presence of the transcortical loop where the EEG correlate preceded the EMG discharge. This concept has been worked out by later neurologists in great detail , and the general consensus is that indeed, the stretch reflex is endowed with a conspicuous transcortical component

Johann bernhard aloys von Gudden and the Mad King of Bavaria

Bernhard von Gudden was a psychiatrist in Prussia and he was summoned in March 1886 to examine King Ludwig II for his apparently insane activities like, profligate spending and erratic behaviour. A team of four estimable psychiatrists pronounced that he was not capable ruling. Consequently, he was dethroned and kept in a castle under supervision of von Gudden. Gudden championed the idea of 'no restraint' and advocated free movement of insane persons and one evening in June, he accompanied the King during an evening stroll to a lake. A few hours later, the corpus of both of them were recovered under mysterious circumstances. Autopsy suggested that the King was drowned but no post-mortem examination was performed on von Gudden. There are plenty of controversies regarding their death like, murder, accidental death or even natural death from cardiac arrest following immersion in cold water, but no incontrovertible conclusion could be arrived at, even after scrupulous analysis by historians and even the diagnosis of insanity of the King has been doubted. Some even suggested that the opinion of psychiatrists were sought as a pretense in order to depose the King

Edgar adrian and patrick merton: Names blurred with the passage of time

Edgar Douglas Adrian and Patrick Anthony Merton are two supreme neurophysiologists from England in the last century whose names are almost forgotten these days. Adrian's work on all-or-none phenomenon in nerve and muscle excitability ushered in a new era and Merton's servo theory of muscular movement and muscle fatigue added a new dimension to the understanding of stretch reflex and deep tendon reflexes. Both of them trained and worked at Trinity College, Cambridge and both were elected as Fellow of the Royal Society and Adrian in addition, was awarded the Nobel Prize in 1932 along with Charles Scott Sherrington

Adolf Hitler and His Parkinsonism

Research works have suggested almost incontrovertibly, that Adolf Hitler suffered from Parkinsonism. However, the precise nature of his illness had always been controversial and post-encephalitic and idiopathic varieties were the ones which were most commonly thought as the possible etiology. He displayed features like oculogyric crisis, palilalia, and autonomic symptoms which strongly implicate post-encephalitic etiology in the genesis of his illness. Others on the contrary, observed premorbid personality traits like non-flinching mental rigidity, extreme inflexibility, and awesome pedantry; which are often observed in idiopathic Parkinson′s disease. Moreover, nonmotor symptoms like disturbed sleep, proneness to temper tantrums, phases of depression, suspiciousness, and lack of trust on colleagues have also been described by various authors. Additionally, he was prescribed methamphetamine by his personal doctor and that might have led to the development of some of the later traits in his personality

Induction of a high affinity binding site for auxin in Avena root membrane

A membrane preparation from Avena sativa root has been found to contain only one low-affinity IAA-binding site having a Kd value of 8.4 ×

Joseph Godwin Greenfield: The father of neuropathology (1884-1958)

The systematic study of neuropathology was initiated by Godwin Greenfield in the early part of the 20 th century. He worked at the National Hospital, Queen Square, London for the major period of his life and worked on various subjects like cerebrospinal fluid, intracranial tumours, cerebellar ataxias, dystrophia myotonica, disseminated sclerosis, subacute combined degeneration of the spinal cord, and the like. After his retirement he visited the National Institute of Neurological Diseases and Blindness in Bethesda, Maryland, USA, from time to time and there he died suddenly from myocardial infarction