Racah coefficients and extended HOMFLY polynomials for all 5-, 6- and 7-strand braids

Basing on evaluation of the Racah coefficients for SU_q(3) (which supported the earlier conjecture of their universal form) we derive explicit formulas for all the 5-, 6- and 7-strand Wilson averages in the fundamental representation of arbitrary SU(N) group (the HOMFLY polynomials). As an application, we list the answers for all 5-strand knots with 9 crossings. In fact, the 7-strand formulas are sufficient to reproduce all the HOMFLY polynomials from the katlas.org: they are all described at once by a simple explicit formula with a very transparent structure. Moreover, would the formulas for the relevant SU_q(3) Racah coefficients remain true for all other quantum groups, the paper provides a complete description of the fundamental HOMFLY polynomials for all braids with any number of strands.Comment: 16 pages + Tables and Appendice

Matrix model version of AGT conjecture and generalized Selberg integrals

Operator product expansion (OPE) of two operators in two-dimensional conformal field theory includes a sum over Virasoro descendants of other operator with universal coefficients, dictated exclusively by properties of the Virasoro algebra and independent of choice of the particular conformal model. In the free field model, these coefficients arise only with a special "conservation" relation imposed on the three dimensions of the operators involved in OPE. We demonstrate that the coefficients for the three unconstrained dimensions arise in the free field formalism when additional Dotsenko-Fateev integrals are inserted between the positions of the two original operators in the product. If such coefficients are combined to form an $n$-point conformal block on Riemann sphere, one reproduces the earlier conjectured $\beta$-ensemble representation of conformal blocks, thus proving this (matrix model) version of the celebrated AGT relation. The statement can also be regarded as a relation between the $3j$-symbols of the Virasoro algebra and the slightly generalized Selberg integrals $I_Y$, associated with arbitrary Young diagrams. The conformal blocks are multilinear combinations of such integrals and the remaining part of the original AGT conjecture relates them to the Nekrasov functions which have exactly the same structure.Comment: 19 page

Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation

We continue the program of systematic study of extended HOMFLY polynomials. Extended polynomials depend on infinitely many time variables, are close relatives of integrable tau-functions, and depend on the choice of the braid representation of the knot. They possess natural character decompositions, with coefficients which can be defined by exhaustively general formula for any particular number m of strands in the braid and any particular representation R of the Lie algebra GL(\infty). Being restricted to "the topological locus" in the space of time variables, the extended HOMFLY polynomials reproduce the ordinary knot invariants. We derive such a general formula, for m=3, when the braid is parameterized by a sequence of integers (a_1,b_1,a_2,b_2,...), and for the first non-fundamental representation R=[2]. Instead of calculating the mixing matrices directly, we deduce them from comparison with the known answers for torus and composite knots. A simple reflection symmetry converts the answer for the symmetric representation [2] into that for the antisymmetric one [1,1]. The result applies, in particular, to the figure eight knot 4_1, and was further extended to superpolynomials in arbitrary symmetric and antisymmetric representations in arXiv:1203.5978.Comment: 22 pages + Tables of knot polynomial

HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations

Explicit answer is given for the HOMFLY polynomial of the figure eight knot $4_1$ in arbitrary symmetric representation R=[p]. It generalizes the old answers for p=1 and 2 and the recently derived results for p=3,4, which are fully consistent with the Ooguri-Vafa conjecture. The answer can be considered as a quantization of the \sigma_R = \sigma_{[1]}^{|R|} identity for the "special" polynomials (they define the leading asymptotics of HOMFLY at q=1), and arises in a form, convenient for comparison with the representation of the Jones polynomials as sums of dilogarithm ratios. In particular, we construct a difference equation ("non-commutative A-polynomial") in the representation variable p. Simple symmetry transformation provides also a formula for arbitrary antisymmetric (fundamental) representation R=[1^p], which also passes some obvious checks. Also straightforward is a deformation from HOMFLY to superpolynomials. Further generalizations seem possible to arbitrary Young diagrams R, but these expressions are harder to test because of the lack of alternative results, even partial.Comment: 14 page

Heights integrated model as instrument for simulation of hydHeights Integrated Model as Instrument for Simulation of Hydrodynamic, Radiation Transport, and Heat Conduction Phenomena of Laser-Produced Plasma in EUV Applications.

The HEIGHTS integrated model has been developed as an instrument for simulation and optimization of laser-produced plasma (LPP) sources relevant to extreme ultraviolet (EUV) lithography. The model combines three general parts: hydrodynamics, radiation transport, and heat conduction. The first part employs a total variation diminishing scheme in the Lax-Friedrich formulation (TVD-LF); the second part, a Monte Carlo model; and the third part, implicit schemes with sparse matrix technology. All model parts consider physical processes in three-dimensional geometry. The influence of a generated magnetic field on laser plasma behavior was estimated, and it was found that this effect could be neglected for laser intensities relevant to EUV (up to {approx}10{sup 12} W/cm{sup 2}). All applied schemes were tested on analytical problems separately. Benchmark modeling of the full EUV source problem with a planar tin target showed good correspondence with experimental and theoretical data. Preliminary results are presented for tin droplet- and planar-target LPP devices. The influence of three-dimensional effects on EUV properties of source is discussed

Problem-solving for problem-solving: Data analytics to identify families for service intervention

The article draws on Bacchi’s ideas about problematisation (2020) and links to technological solutionism as governing logics of our age, to explore the double-faceted problem-solving logic operating in the UK family policy and early intervention field. Families with certain characteristics are identified as problematic, and local authorities are tasked with intervening to fix that social problem. Local authorities thus need to identify these families for problem-solving intervention, and data analytics companies will solve that problem for them. In the article, we identify discourses of transmitted deprivation and anti-social behaviour in families and the accompanying costly public sector burden as characteristics that produce families as social problems, and discursive themes around delivering powerful knowledge, timeliness and economic efficiently in data analytic companies’ problem solving claims for their data linkage and predictive analytics systems. These discursive rationales undergird the double-faceted problem-solving for problem-solving logic that directs attention away from complex structural causes

Tubular cell and keratinocyte single-cell transcriptomics applied to lupus nephritis reveal type I IFN and fibrosis relevant pathways.

The molecular and cellular processes that lead to renal damage and to the heterogeneity of lupus nephritis (LN) are not well understood. We applied single-cell RNA sequencing (scRNA-seq) to renal biopsies from patients with LN and evaluated skin biopsies as a potential source of diagnostic and prognostic markers of renal disease. Type I interferon (IFN)-response signatures in tubular cells and keratinocytes distinguished patients with LN from healthy control subjects. Moreover, a high IFN-response signature and fibrotic signature in tubular cells were each associated with failure to respond to treatment. Analysis of tubular cells from patients with proliferative, membranous and mixed LN indicated pathways relevant to inflammation and fibrosis, which offer insight into their histologic differences. In summary, we applied scRNA-seq to LN to deconstruct its heterogeneity and identify novel targets for personalized approaches to therapy