Four quotient set gems

Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the Monthly, despite its intense coverage of quotient sets over the years.Comment: 9 pages, to appear in the American Mathematical Monthl

Four Quotient Set Gems

Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the MONTHLY, despite its intense coverage of quotient sets over the years

An Exhibition of Exponential Sums: Visualizing Supercharacters

We discuss a simple mathematical mechanism that produces a variety of striking images of great complexity and subtlety. We briefly explain this approach and present a selection of attractive images obtained using this technique

Two remarks about nilpotent operators of order two

We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator.Comment: 7 pages. To appear in Proceedings of the AM

1976: Abilene Christian College Bible Lectures - Full Text

FREEDOM IN CHRIST Being the Abilene Christian College Annual Bible Lectures 1976 Published by ABILENE CHRISTIAN COLLEGE Book Store ACC Station Abilene, Texas 7960

The Graphic Nature of Gaussian Periods

Recent work has shown that the study of supercharacters on abelian groups provides a natural framework with which to study the properties of certain exponential sums of interest in number theory. Our aim here is to initiate the study of Gaussian periods from this novel perspective. Among other things, this approach reveals that these classical objects display a dazzling array of visual patterns of great complexity and remarkable subtlety

Der ökonomische Determinismus – neu betrachtet

This article revisits the claim that the social relations of production determine, at least in the last instance, the overall nature and dynamic of social formations. The claim in its original form is often misunderstood, its later justifications are generally weak, and efforts to improve it are mostly unconvincing. An alternative interpretation is presented that identifies four complementary meanings of economic determinism that are all consistent with historical materialism. The fourth meaning draws on the relatively new concept of ‘ecological dominance’ but it was anticipated, the author argues, in certain arguments of Marx and Engels themselves

Loss-of-function mutations in the ATP13A2/PARK9 gene cause complicated hereditary spastic paraplegia (SPG78)

Hereditary spastic paraplegias are heterogeneous neurodegenerative disorders characterized by progressive spasticity of the lower limbs due to degeneration of the corticospinal motor neurons. In a Bulgarian family with three siblings affected by complicated hereditary spastic paraplegia, we performed whole exome sequencing and homozygosity mapping and identified a homozygous p.Thr512Ile (c.1535C>T) mutation in ATP13A2. Molecular defects in this gene have been causally associated with Kufor-Rakeb syndrome (#606693), an autosomal recessive form of juvenile-onset parkinsonism, and neuronal ceroid lipofuscinosis (#606693), a neurodegenerative disorder characterized by the intracellular accumulation of autofluorescent lipopigments. Further analysis of 795 index cases with hereditary spastic paraplegia and related disorders revealed two additional families carrying truncating biallelic mutations in ATP13A2. ATP13A2 is a lysosomal P5-type transport ATPase, the activity of which critically depends on catalytic autophosphorylation. Our biochemical and immunocytochemical experiments in COS-1 and HeLa cells and patient-derived fibroblasts demonstrated that the hereditary spastic paraplegia-associated mutations, similarly to the ones causing Kufor-Rakeb syndrome and neuronal ceroid lipofuscinosis, cause loss of ATP13A2 function due to transcript or protein instability and abnormal intracellular localization of the mutant proteins, ultimately impairing the lysosomal and mitochondrial function. Moreover, we provide the first biochemical evidence that disease-causing mutations can affect the catalytic autophosphorylation activity of ATP13A2. Our study adds complicated hereditary spastic paraplegia (SPG78) to the clinical continuum of ATP13A2-associated neurological disorders, which are commonly hallmarked by lysosomal and mitochondrial dysfunction. The disease presentation in our patients with hereditary spastic paraplegia was dominated by an adult-onset lower-limb predominant spastic paraparesis. Cognitive impairment was present in most of the cases and ranged from very mild deficits to advanced dementia with frontotemporal characteristics. Nerve conduction studies revealed involvement of the peripheral motor and sensory nerves. Only one of five patients with hereditary spastic paraplegia showed clinical indication of extrapyramidal involvement in the form of subtle bradykinesia and slight resting tremor. Neuroimaging cranial investigations revealed pronounced vermian and hemispheric cerebellar atrophy. Notably, reduced striatal dopamine was apparent in the brain of one of the patients, who had no clinical signs or symptoms of extrapyramidal involvement