Small resolutions of Schubert varieties in symplectie and orthogonal Grassmannians

This article does not have an abstract

Construction of Permutation Polynomials of Certain Specific Cycle Structure over Finite Fields

For a finite field of odd number of elements we construct families of permutation binomials and permutation trinomials with one fixed-point (namely zero) and remaining elements being permuted as disjoint cycles of same length. Binomials and trinomials providing permutations with cycles of many lengths with certain frequency are also constructed.Comment: 10 pages, 1 table, Comments welcom

Small resolutions of Schubert varieties and Kazhdan-Lusztig polynomials

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Collection of Polynomials over Finite Fields Providing Involutary Permutations

For an odd prime power $q$ satisfying $q\equiv 1\pmod 3$ we construct totally $2(q-1)$ permutation polyomials, all giving involutory permutations with exactly $1+ \frac{q-1}3$ fixed points. Among them $(q-1)$ polynomials are trinomials, and the rest are 6-term polynomials.Comment: 10 pages; Comments welcome

Involutary pemutations over finite fields given by trinomials and quadrinomials

For all finite fields of $q$ elements where $q\equiv1\pmod4$ we have constructed permutation polynomials which have order 2 as permutations, and have 3 terms, or 4 terms as polynomials. Explicit formulas for their coefficients are given in terms of the primitive elements of the field. We also give polynomials providing involutions with larger number of terms but coefficients will be conveniently only two possible values. Our procedure gives at least $(q-1)/4$ trinomials, and $(q-1)/2$ quadrinomials, all yielding involutions with unique fixed points over a field of order $q$. Equal number of involutions with exactly $(q+1)/2$ fixed-points are provided as quadrinomials.Comment: 10 pages; comments welccom

Exceptional Quartics are Ubiquitous

For each real quadratic field we constructively show the existence of infinitely many exceptional quartic number fields containing that quadratic field. On the other hand, another infinite collection of quartic exceptional fields without any quadratic subfields is also provided. Both these families are non-Galois extensions of $\mathbf{Q}$, and their normal closu res have Galois groups $D_4$ and $S_4$ respectively. We also show that an infinite number of these exceptional quartic fields have power integral basis, i.e., monogenic. We also construct large collections of exceptional number fields in all degrees greater than 4.Comment: 13 pages. Conjecture in earlier version is prove

LXR Deficiency Confers Increased Protection against Visceral Leishmania Infection in Mice

Leishmania spp. are protozoan single-cell parasites that are transmitted to humans by the bite of an infected sand fly, and can cause a wide spectrum of disease, ranging from self-healing skin lesions to potentially fatal systemic infections. Certain species of Leishmania that cause visceral (systemic) disease are a source of significant mortality worldwide. Here, we use a mouse model of visceral Leishmania infection to investigate the effect of a host gene called LXR. The LXRs have demonstrated important functions in both cholesterol regulation and inflammation. These processes, in turn, are closely related to lipid metabolism and the development of atherosclerosis. LXRs have also previously been shown to be involved in protection against other intracellular pathogens that infect macrophages, including certain bacteria. We demonstrate here that LXR is involved in susceptibility to Leishmania, as animals deficient in the LXR gene are much more resistant to infection with the parasite. We also demonstrate that macrophages lacking LXR kill parasites more readily, and make higher levels of nitric oxide (an antimicrobial mediator) and IL-1β (an inflammatory cytokine) in response to Leishmania infection. These results could have important implications in designing therapeutics against this deadly pathogen, as well as other intracellular microbial pathogens