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In this paper we give a new formula for characters of finite dimensional irreducible $\frak{gl}(m,n)$ modules. We use two main ingredients: Su-Zhang formula and Brion's theorem., Comment: 19 pages. Typos corrected

Textured silicon surface structures, in particular black silicon (b-Si), open up possibilities for Si-based solar cells and photodetectors to be extremely thin and highly sensitive owing to perfect light-trapping and anti-reflection properties. However, near-infrared (NIR) performance of bare b-Si is limited by Si band gap of 1.12 eV or 1100 nm. This work reports a simple method to increase NIR absorption of b-Si by $in$ $vacuo$ silicidation with magnesium. Obtained Mg$_2$Si/b-Si heterostructure has a complex geometry where b-Si nanocones are covered by Mg$_2$Si shells and crowned with flake-like Mg$_2$Si hexagons. Mg$_2$Si formation atop b-Si resulted in 5-fold lower reflectivity and optical absorption to be no lower than 88\% over 200-1800 nm spectral range. More importantly, Mg$_2$Si/b-Si heterostructure is more adjusted to match AM-1.5 solar spectrum with theoretically higher photogenerated current density. The maximal advantage is demonstrated in the NIR region compared to bare b-Si in full accordance with one's expectations about NIR sensitive narrow band gap ($\sim$0.75 eV) semiconductor with high absorption coefficient, which is Mg$_2$Si. Results of optical simulation confirmed the superiority of Mg$_2$Si/b-Si NIR performance. Therefore, this new wide-band optical absorber called black silicide proved rather competitive alongside state-of-the-art approaches to extend b-Si spectral blackness., Comment: 8 pages, 4 figures

Neural machine translation - using neural networks to translate human language - is an area of active research exploring new neuron types and network topologies with the goal of dramatically improving machine translation performance. Current state-of-the-art approaches, such as the multi-head attention-based transformer, require very large translation corpuses and many epochs to produce models of reasonable quality. Recent attempts to parallelize the official TensorFlow "Transformer" model across multiple nodes have hit roadblocks due to excessive memory use and resulting out of memory errors when performing MPI collectives. This paper describes modifications made to the Horovod MPI-based distributed training framework to reduce memory usage for transformer models by converting assumed-sparse tensors to dense tensors, and subsequently replacing sparse gradient gather with dense gradient reduction. The result is a dramatic increase in scale-out capability, with CPU-only scaling tests achieving 91% weak scaling efficiency up to 1200 MPI processes (300 nodes), and up to 65% strong scaling efficiency up to 400 MPI processes (200 nodes) using the Stampede2 supercomputer., Comment: 18 pages, 10 figures, accepted at the 2019 International Supercomputing Conference

Training modern deep learning models requires large amounts of computation, often provided by GPUs. Scaling computation from one GPU to many can enable much faster training and research progress but entails two complications. First, the training library must support inter-GPU communication. Depending on the particular methods employed, this communication may entail anywhere from negligible to significant overhead. Second, the user must modify his or her training code to take advantage of inter-GPU communication. Depending on the training library's API, the modification required may be either significant or minimal. Existing methods for enabling multi-GPU training under the TensorFlow library entail non-negligible communication overhead and require users to heavily modify their model-building code, leading many researchers to avoid the whole mess and stick with slower single-GPU training. In this paper we introduce Horovod, an open source library that improves on both obstructions to scaling: it employs efficient inter-GPU communication via ring reduction and requires only a few lines of modification to user code, enabling faster, easier distributed training in TensorFlow. Horovod is available under the Apache 2.0 license at https://github.com/uber/horovod

Triggering vacuum breakdown at the upcoming laser facilities can provide rapid electron-positron pair production for studies in laboratory astrophysics and fundamental physics. However, the density of the emerging plasma should seemingly stop rising at the relativistic critical density, when the plasma becomes opaque. Here we identify the opportunity of breaking this limit using optimal beam configuration of petawatt-class lasers. Tightly focused laser fields allow plasma generation in a small focal volume much less than ${\lambda}^3$, and creating extreme plasma states in terms of density and produced currents. These states can be regarded as a new object of nonlinear plasma physics. Using 3D QED-PIC simulations we demonstrate the possibility of reaching densities of more than $10^{25}$ cm$^{-3}$, which is an order of magnitude higher than previously expected. Controlling the process via the initial target parameters gives the opportunity to reach the discovered plasma states at the upcoming laser facilities.

We give a uniform interpretation of the classical continuous Chebyshev's and Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie algebra gl(N), where N is any complex number. One can similarly interpret Chebyshev's and Hahn's q-polynomials and introduce orthogonal polynomials corresponding to Lie superlagebras. We also describe the real forms of gl(N), quasi-finite modules over gl(N), and conditions for unitarity of the quasi-finite modules. Analogs of tensors over gl(N) are also introduced., Comment: 25 pages, LaTeX

Consider the complex matrix Lie superalgebra $gl_{N|N}$ with the standard generators $E_{ij}$ where $i,j=-N,...,-1,1,...,N$. Define an involutive automorphism $\eta$ of $gl_{N|N}$ by $\eta(E_{ij})=E_{-i,-j}$. The queer Lie superalgebra $q_N$ is the fixed point subalgebra in $gl_{N|N}$ relative to the automorphism $\eta$. Consider the twisted polynomial current Lie superalgebra $g=\{X(t)\in\gl_{N|N}[t]:\eta(X(t))=X(-t)\}$. The enveloping algebra $U(g)$ of the Lie superalgebra $g$ has a deformation, called the Yangian of $q_N$. For each $M=1,2,...$ denote by $A_N^M$ the centralizer of $q_M\subset q_{N+M}$ in the superalgebra $U(q_{N+M})$. We describe the projective limit of the sequence of centralizer algebras $A_N^1,A_N^2,...$ in terms of the Yangian of $q_N$., Comment: 24 pages

Let A be an associative complex algebra and L an invariant linear functional on it (trace). Let i be an involutive antiautomorphism of A such that L(i(a))=L(a) for any a in A. Then A admits a symmetric invariant bilinear form (a, b)=L(a i(b)). For A=U(sl(2))/m, where m is any maximal ideal of U(sl(2)), Leites and I have constructed orthogonal basis whose elements turned out to be, essentially, Chebyshev and Hahn polynomials in one discrete variable. Here I take A=U(gl(3))/m for the maximal ideals m which annihilate irreducible highest weight gl(3)-modules of particular form (generalizations of symmetric powers of the identity representation). In this way we obtain multivariable analogs of Hahn polynomials., Comment: 12p., Latex

Casimir operators -- the generators of the center of the enveloping algebra -- are described for simple or close to them ``classical'' finite dimensional Lie superalgebras with nondegenerate symmetric even bilinear form in Sergeev A., The invariant polynomials on simple Lie superalgebras. Represent. Theory 3 (1999), 250--280; math-RT/9810111 and for the ``queer'' series in Sergeev A., The centre of enveloping algebra for Lie superalgebra Q(n, C). Lett. Math. Phys. 7, no. 3, 1983, 177--179. Here we consider the remaining cases, and state conjectures proved for small values of parameter. Under deformation (quantization) the Poisson Lie superalgebra po(0|2n) on purely odd superspace turns into gl(2^{n-1}|2^{n-1}) and, conjecturally, the lowest terms of the Taylor series expansion with respect to the deformation parameter (Planck's constant) of the Casimir operators for gl(2^{n-1}|2^{n-1}) are the Casimir operators for po(0|2n). Similarly, quantization sends po(0|2n-1) into q(2^{n-1}) and the above procedure makes Casimir operators for q(2^{n-1}) into same for po(0|2n-1). Casimir operators for the Lie superalgebra vect(0|m) of vector fields on purely odd superspace are only constants for m>2. Conjecturally, same is true for the Lie superalgebra svect(0|m) of divergence free vector fields, and its deform, for m>3. Invariant polynomials on po(0|2n-1) are also described. They do not correspond to Casimir operators., Comment: 4 p., Latex

I show that the projective Schur functions may be interpreted as bispherical functions of either the triple (q(n),gl(n,n),q(n)), where q(n) is the "odd" (queer) analog of the general liner Lie algebra, or the triple (p(n),gl(n,n),p(n)), where p(n) is the periplectic Lie superalgebra which preserves the nondegenerate odd, bilinear form (either symmetric or skew symmetric). Making use of this interpretation I characterize projective Schur functions as common eigenfunctions of an algebra differential operators., Comment: Latex, 18 pages

A depending on a complex parameter $k$ superanalog ${\mathcal S}{\mathcal L}$ of Calogero operator is constructed; it is related with the root system of the Lie superalgebra ${\mathfrak{gl}}(n|m)$. For $m=0$ we obtain the usual Calogero operator; for $m=1$ we obtain, up to a change of indeterminates and parameter $k$ the operator constructed by Veselov, Chalykh and Feigin [2,3]. For $k=1, \frac12$ the operator ${\mathcal S}{\mathcal L}$ is the radial part of the 2nd order Laplace operator for the symmetric superspaces corresponding to pairs $(GL(V)\times GL(V), GL(V))$ and $(GL(V), OSp(V))$, respectively. We will show that for the generic $m$ and $n$ the superanalogs of the Jack polynomials constructed by Kerov, Okunkov and Olshanskii [5] are eigenfunctions of ${\mathcal S}{\mathcal L}$; for $k=1, \frac12$ they coinside with the spherical functions corresponding to the above mentioned symmetric superspaces. We also study the inner product induced by Berezin's integral on these superspaces.