Curves of fixed points of trace maps

We study curves of fixed points for certain diffeomorphisms of ${\mathbb{R}}^3$ that are induced by automorphisms of a trace algebra. We classify these curves. There is a function $E$ which is invariant under all such trace maps and the level surfaces $E_t: E=t$ are invariant; a point of $E_t$ will be said to have level $t$. The surface $E_1$ is significant. Then most fixed points on $E_1$ are actually on a curve $\gamma$ of fixed points interior to $E_1$. We describe the possibilities for the other end of $\gamma$ on $E_1$

Trace identities from identities for determinants

We present new identities for determinants of matrices $(A_{i,j})$ with entries $A_{i,j}$ equal to $a_{i,j}$ or $a_{i,0}a_{0,j}-a_{i,j}$, where the $a_{i,j}$'s are indeterminates. We show that these identities are behind trace identities for $SL(2,\Bbb C)$ matrices found earlier by Magnus in his study of trace algebras.Comment: 14 pages, AmS-Te

Commutative Schur Rings Over Symmetric Groups II: The Case n=6

We determine the commutative Schur rings over $S_6$ that contain the sum of all the transpositions in $S_6$. There are eight such types (up to conjugacy), of which four have the set of all the transpositions as a principal set of the Schur ring

Braid groups, infinite Lie algebras of Cartan type and rings of invariants

AbstractIn this paper we show that each element α of the pure braid group Pn or the pure symmetric automorphism group H(n) of the free group Fn of rank n can be represented as α=exp(D)=id+D+(D2/2!)+(D3/3!)+⋯, where D=D(α) is an element of an infinite-dimensional Lie algebra h(n). Each such D is a derivation of the power series ring C[[a1,…,ar]], r=n2−n, which fixes the volume form a1∧⋯∧ar and so h(n) is a subalgebra of Sn2−n, the special Lie algebra of Cartan type. There is a corresponding action of these groups on C[[a1,…,ar]] and C[a1,…,ar]. We use the representation α=exp(D) to prove results about the ring of invariants for this action of the pure braid group. The Lie algebra h(n) is a subalgebra of a graded Lie algebra h(n); we also calculate the Poincaré series of the Lie algebra l(n) and of certain of its subalgebras, and show that these Poincaré series are rational

Cogrowth of groups and a matrix of redheffer

AbstractWe describe a relationship between the cogrowth function (and other similarly defined functions) of a presentation of a torsion-free group and the Riemann hypothesis. This relationship is determined using a certain matrix of Redheffer

Arctic Standards: Recommendations on Oil Spill Prevention, Response, and Safety in the U.S. Arctic Ocean

Oil spilled in Arctic waters would be particularly difficult to remove. Current technology has not been proved to effectively clean up oil when mixed with ice or when trapped under ice. An oil spill would have a profoundly adverse impact on the rich and complex ecosystem found nowhere else in the United States. The Arctic Ocean is home to bowhead, beluga, and gray whales; walruses; polar bears; and other magnificent marine mammals, as well as millions of migratory birds. A healthy ocean is important for these species and integral to the continuation of hunting and fishing traditions practiced by Alaska Native communities for thousands of years.To aid the United States in its efforts to modernize Arctic technology and equipment standards, this report examines the fierce Arctic conditions in which offshore oil and gas operations could take place and then offers a summary of key recommendations for the Interior Department to consider as it develops world-class, Arctic-specific regulatory standards for these activities. Pew's recommendations call for improved technology,equipment, and procedural requirements that match the challenging conditions in the Arctic and for full public participation and transparency throughout the decision-making process. Pew is not opposed to offshore drilling, but a balance must be achieved between responsible energy development and protection of the environment.It is essential that appropriate standards be in place for safety and for oil spill prevention and response in this extreme, remote, and vulnerable ecosystem. This report recommends updating regulations to include Arctic specific requirements and codifying temporary guidance into regulation. The appendixes to this report provide substantially more detail on the report's recommendations, including technical background documentation and additional referenced materials. Please refer to the full set of appendixes for a complete set of recommendations. This report and its appendixes offer guidelines for responsible hydrocarbon development in the U.S. Arctic Ocean

The genes for mouse globin and the post-transcriptional control of their expression

In this work I have investigated the possible role of gene reiteration, amplification or deletion, as a means of regulating the expression of the globin gene during tissue differentiation in the mouse. This has employed a probe of complementary DNA (CDNA) prepared by copying reticulocyte 9S RNA using avian myeloblastosis virus reverse transcriptase. Since this RNA contains the messenger RNAs for alpha and beta globin, the cDNA produced is complementary to the alpha and beta globin genes. The fidelity of this cDNA said its use as a probe for globin sequences in DNA and RNA are rigorously justified. The rates of hybridisation of this probe to whole mouse embryo DNA, mouse erythroid DNA from 14 day foetal livers, and mouse germ line DNA from sperm are compared. The small corrections required for the size differences of the hybridising molecules, said the slight mismatching in the cDNA are estimated. The study demonstrates that there are 1-2 copies of the globin alpha and beta genes in all tissues studied, and thus mechstnisms other than gene amplification or deletion are required to explain the control of globin gene expression. The existence of post-transcriptional mechanisms by which this regulation may occur was then investigated. The first of these is the translational role of the poly(A) segment found on the 3' of eukaryotic mRNAs. This segment was removed from reticulocyte 9S RNA, using polynucleotide phosphorylase . Several experiments were performed to demonstrate the absence of the poly(A) sequence from the mRNA. The ability of this mRNA to code for alpha and beta globin chain synthesis in several cell-free protein synthesising systems was compared with intact mRNA. The results demonstrate that the poly(A) segment is not required for efficient initiation, elongation, termination or reinitiation of protein synthesis in a cell-free system. The role of the poly (A) sequence in determining the nuclear metabolism of globin RNA was then investigated. Low levels of globin RNA sequences were detected using cDNA in the nucleus of several non-erythroid tissues, adult mouse brain and liver, and a cultured lymphoma cell line, L5178Y. Very much lower levels of globin RNA sequences were detected in the cytoplasm of these cells. In contrast, in the erythroid 14 day foetal mouse liver, the amounts of globin RNA sequences found in the cytoplasm were very much higher than that in the nucleus. This indicates the operation of a post-transcriptional regulatory mechanism, whereby the nuclear said cytoplasmic levels of globin RNA sequences are regulated independently. However, this mechanism appears not to be mediated through association of nuclear globin RNA sequences with poly(A) since in both erythroid and non-erythroid tissues, where the fraction of total cell globin RNA sequences in the cytoplasm varies considerably, the proportion of polyadenylated nuclear globin RNA sequences is the same. At the present time, the possible functions of this low-level transcription of the globin gene in non-erythroid tissues are unclear

Malentangled: Function Redacting Tape

The neologism entanglement proposes that all things are connected through super-complex meshworks of mutable interdependencies. This entanglement of interdependencies is often obscured through forgetness, a radically reductive process by which things are taken to be isolated and interdependencies are forgotten. In some instances – for example when objects break – people are again reminded of the interdependentness of things. Malentanglement theory proposes that forgetness may also encounter a remindness through humour, and not only through catastrophe (depunctualisation). The ‘Function Redacting Tape’ project takes redaction as a method for doctoring documents, but it deploys this method in the material context of design. Project Participants are provided with black PVC adhesive tape and invited to consider the functions of designed objects. They are then asked to redact these functions (using the tape) and in doing so to make documented interventions that draw back the metaphorical veil of forgetness for reasons of design enquiry. The project functions as a sort of rudimentary cultural probe that might shed some light on entanglement, humour, and design, whilst simultaneously testing the employment of humour to aid participation in design research