The Decomposition of Lie Powers

Let G be a group, F a field of prime characteristic p and V a finite-dimensional FG-module. Let L(V) denote the free Lie algebra on V regarded as an FG-submodule of the free associative algebra (or tensor algebra) T(V). For each positive integer r, let L^r(V) and T^r(V) be the rth homogeneous components of L(V) and T(V), respectively. Here L^r(V) is called the rth Lie power of V. Our main result is that there are submodules B_1, B_2, ... of L(V) such that, for all r, B_r is a direct summand of T^r(V) and, whenever m \geq 0 and k is not divisible by p, $$L^{p^mk}(V) = L^{p^m}(B_k) \oplus L^{p^{m-1}}(B_{pk}) \oplus ... \oplus L^p(B_{p^{m-1}k}) \oplus L^1(B_{p^mk}).$$ Thus every Lie power is a direct sum of Lie powers of p-power degree. The approach builds on an analysis of T^r(V) as a bimodule for G and the Solomon descent algebra.Comment: 32 page

Combinatorial Markov chains on linear extensions

We consider generalizations of Schuetzenberger's promotion operator on the set L of linear extensions of a finite poset of size n. This gives rise to a strongly connected graph on L. By assigning weights to the edges of the graph in two different ways, we study two Markov chains, both of which are irreducible. The stationary state of one gives rise to the uniform distribution, whereas the weights of the stationary state of the other has a nice product formula. This generalizes results by Hendricks on the Tsetlin library, which corresponds to the case when the poset is the anti-chain and hence L=S_n is the full symmetric group. We also provide explicit eigenvalues of the transition matrix in general when the poset is a rooted forest. This is shown by proving that the associated monoid is R-trivial and then using Steinberg's extension of Brown's theory for Markov chains on left regular bands to R-trivial monoids.Comment: 35 pages, more examples of promotion, rephrased the main theorems in terms of discrete time Markov chain

Comparison of prognostic scores and surgical approaches to treat spinal metastatic tumors: A review of 57 cases

Surgical treatment of metastatic spinal cord compression with or without neural deficit is controversial. Karnofsky and Tokuhashi scores have been proposed for prognosis of spinal metastasis. Here, we conducted a retrospective analysis of Karnofsky and modified Tokuhashi scores in 57 consecutive patients undergoing surgery for secondary spinal metastases to evaluate the value of these scores in aiding decision making for surgery. Comparison of preoperative Karnofsky and modified Tokuhashi scores with the type of the surgical approach for each patient revealed that both scores not only reliably estimate life expectancy, but also objectively improved surgical decisions. When the general status of the patient is poor (i.e., Karnofsky score less than 40% or modified Tokuhashi score of 5 or greater), palliative treatments and radiotherapy, rather than surgery, should be considered

Allergenic Lipid Transfer Proteins from Plant-Derived Foods Do Not Immunologically and Clinically Behave Homogeneously: The Kiwifruit LTP as a Model

BACKGROUND: Food allergy is increasingly common worldwide. Tools for allergy diagnosis measuring IgE improved much since allergenic molecules and microarrays started to be used. IgE response toward allergens belonging to the same group of molecules has not been comprehensively explored using such approach yet. OBJECTIVE: Using the model of lipid transfer proteins (LTPs) from plants as allergens, including two new structures, we sought to define how heterogeneous is the behavior of homologous proteins. METHODS: Two new allergenic LTPs, Act d 10 and Act c 10, have been identified in green (Actinidia deliciosa) and gold (Actinidia chinensis) kiwifruit (KF), respectively, using clinically characterized allergic patients, and their biochemical features comparatively evaluated by means of amino acid sequence alignments. Along with other five LTPs from peach, mulberry, hazelnut, peanut, mugwort, KF LTPs, preliminary tested positive for IgE, have been immobilized on a microarray, used for IgE testing 1,003 allergic subjects. Comparative analysis has been carried out. RESULTS: Alignment of Act d 10 primary structure with the other allergenic LTPs shows amino acid identities to be in a narrow range between 40 and 55%, with a number of substitutions making the sequences quite different from each other. Although peach LTP dominates the IgE immune response in terms of prevalence, epitope recognition driven by sequence heterogeneity has been recorded to be distributed in a wide range of behaviors. KF LTPs IgE positive results were obtained in a patient subset IgE positive for the peach LTP. Anyhow, the negative results on homologous molecules allowed us to reintroduce KF in patients' diet. CONCLUSION: The biochemical nature of allergenic molecule belonging to a group of homologous ones should not be taken as proof of immunological recognition as well. The availability of panels of homologous molecules to be tested using microarrays is valuable to address the therapeutic intervention