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The Standard Model of Elementary Particle Physics has global Family flavor symmetries corresponding to the 3 families in the Standard Model. It has been shown that the breaking of these symmetries at low energy produces Dark Energy which is the dominant component of the energy density of the Universe. It has also been shown that this model of Dark Energy not only explains the accelerated expansion of the universe but has additional observational consequences and makes verifiable predictions. This model allows a space-time dependent Dark Energy as indicated by recent observations. Further, the collapse of space dependent Dark Energy configurations leads to the formation of Dark Energy Black Holes resulting in verifiable predictions. We examine the implications of these symmetries at high energies and in the Early Universe. We consider what might happen if these global symmetries get gauged and that the global symmetries today are just a consequence of the gauge fields acquiring a constant vacuum expectation value below some energy scale v. Since, we have not yet seen signatures of such gauge fields at any of our particle accelerators, this would imply that v is greater than the electroweak scale. In the early Universe, when these gauge fields would be dynamical, the Higgs would not yet have acquired a non-zero vacuum expectation value and all Standard Model particles would be massless. The gauging of these global symmetries is desirable from a high energy and quantum gravity perspective. The chiral dynamics has a natural interpretation in terms of D Branes and our 4-dimensional Universe embedded in higher dimensions. Anomaly cancellation in this scenario results from the sum of the cubic terms vanishing. Finally, the analysis in this article can provide an explanation for one of the long standing mysteries of the natural world: the matter - antimatter asymmetry observed in our Universe.

The study of word maps on groups has been of deep interest in recent years. This survey focuses on the case of power maps on groups; $viz.$ the map $x\mapsto x^M$ for a group $G$, and an integer $M\geq 2$. Here, we accumulate various results on the subject and pose some questions., Comment: Survey article; Preliminary version

In this article, we determine the non-real elements--the ones that are not conjugate to their inverses--in the group $G = G_2(q)$ when $char(F_q)\neq 2,3$. We use this to show that this group is chiral; that is, there is a word w such that $w(G)\neq w(G)^{-1}$. We also show that most classical finite simple groups are achiral, Comment: 13 pages. Keywords: Chirality, word maps, non-real elements, an exceptional group of Lie type G_2(q)

To a finite group $G$, one can associate several notions of dimensions (or degrees). In this survey, we attempt to bring together some of the notions of dimensions or degrees defined using representations of the group in General Linear Groups and permutation groups. These are embedding degree, minimal faithful irreducible character degree, minimal faithful permutation representation degree, minimal faithful quasi-permutation representation degree and essential dimension. We briefly present the progress in understanding these notions and the related problems.

Let $K$ be an algebraically closed field and $\mathrm{M}(2,K)$ be the $2\times 2$ matrix algebra over $K$ and $\mathrm{GL}(2,K)$ be the invertible elements in $\mathrm{M}(2,K)$. We explore the image of polynomials with constants, namely from the free algebra $\mathrm{M}(2,K)\langle x, y\rangle$. In this article, we compute the images of the polynomial maps given by (a) generalized sum of powers $Ax^{k_1} + By^{k_2}$ and (b) generalized commutator map $Axy -Byx$, where $A$, $B$ are non-zero elements of $\mathrm{M}(2,K)$. We compute this in the first case by fixing a simultaneous conjugate pair for $A, B$ and it turns out that it is surjective in most of the cases. In the second case, we show that the image of the map is always a vector space., Comment: Preliminary version; 34 pp

The Matrix Waring problem is if we can write every matrix as a sum of $k$-th powers. Here, we look at the same problem for triangular matrix algebra $T_n(\mathbb{F}_q)$ consisting of upper triangular matrices over a finite field. We prove that for all integers $k, n \geq 1$, there exists a constant $\mathcal C(k, n)$, such that for all $q> \mathcal C(k,n)$, every matrix in $T_n(\mathbb{F}_q)$ is a sum of three $k$-th powers. Moreover, if $-1$ is $k$-th power in $\mathbb{F}_q$, then for all $q>\mathcal C(k,n)$, every matrix in $T_n(\mathbb{F}_q)$ is a sum of two $k$-th powers. We make use of Lang-Weil estimates about the number of solutions of equations over finite fields to achieve the desired results.

For a finite group $G$, the representation dimension is the smallest integer realizable as the degree of a complex faithful representation of $G$. In this article, we compute representation dimension for some $p$-groups, their direct products, and groups with certain conditions on nonlinear irreducible characters. We also make similar computations for the smallest integer realizable as the degree of an irreducible complex faithful representation of $G$, if one exists. In the appendix, we present GAP codes to compute these numbers.

For $n\geq 2$, we consider the map on $M_n(\mathbb K)$ given by evaluation of a polynomial $f(X_1, \ldots, X_m)$ over the field $\mathbb K$. In this article, we explore the image of the diagonal map given by $f=\delta_1 X_1^{k_1} + \delta_2 X_2^{k_2} + \cdots +\delta_m X_m^{k_m}$ in terms of the solution of certain equations over $\mathbb K$. In particular, we show that for $m\geq 2$, the diagonal map is surjective when (a) $\mathbb K= \mathbb C$, (b) $\mathbb K= \mathbb F_q$ for large enough $q$. Moreover, when $\mathbb K= \mathbb R$ and $m=2$ it is surjective except when $n$ is odd, $k_1, k_2$ are both even, and $\delta _1\delta_2>0$ (in that case the image misses negative scalars), and the map is surjective for $m\geq 3$. We further show that on $M_n(\mathbb H)$ the diagonal map is surjective for $m\geq 2$, where $\mathbb H$ is the algebra of Hamiltonian quaternions.

Let $\text{U}(n,\mathbb{F}_{q^2})$ denote the subgroup of unitary matrices of the general linear group $\text{GL}(n,\mathbb{F}_{q^2})$ which fixes a Hermitian form and $M\geq 2$ an integer. This is a companion paper to the previous works where the elements of the groups $\text{GL}(n,\mathbb{F}_{q})$, $\text{Sp}(2n,\mathbb{F}_{q})$, $\text{O}^{\pm}(2n,\mathbb{F}_{q})$ and $\text{O}(2n+1,\mathbb{F}_{q})$ which has an $M$-th root in the concerned group, have been described. Here we will describe the $M$-th powers in unitary groups for the regular semisimple, semisimple and cyclic elements. Our methods are parallel to those of the Memoir ``A generating function approach to the enumeration of matrices in classical groups over finite fields" by Fulman, Neumann and Praeger., Comment: 11 pages; preliminary version; comments are welcome

Differential central simple algebras are the main object of study in this survey article. We recall some crucial notions such as differential subfields, differential splitting fields, tensor products etc. Our main focus is on differential splitting fields which connects these objects to the classical differential Galois theory. We mention several known results and raise some questions along the line.

Automation of on-call customer support relies heavily on accurate and efficient speech-to-intent (S2I) systems. Building such systems using multi-component pipelines can pose various challenges because they require large annotated datasets, have higher latency, and have complex deployment. These pipelines are also prone to compounding errors. To overcome these challenges, we discuss an end-to-end (E2E) S2I model for customer support voicebot task in a bilingual setting. We show how we can solve E2E intent classification by leveraging a pre-trained automatic speech recognition (ASR) model with slight modification and fine-tuning on small annotated datasets. Experimental results show that our best E2E model outperforms a conventional pipeline by a relative ~27% on the F1 score., Comment: Accepted at EMNLP 2022 Industry Track

We study differential splitting fields of quaternion algebras with derivations. A quaternion algebra over a field $k$ is always split by a quadratic extension of $k$. However, a differential quaternion algebra need not be split over any algebraic extension of $k$. We use solutions of certain Riccati equations to provide bounds on the transcendence degree of splitting fields of a differential quaternion algebra.

A result of Barnea and Isaacs states that if $L$ is a finite dimensional nilpotent Lie algebra with exactly two distinct centralizer dimensions, then nilpotency class of $L$ is either $2$ or $3$. In this article, we classify all such finite dimensional $3$-step nilpotent Lie algebras over a finite field.

Dark Energy is the largest fraction of the energy density of our Universe - yet it remains one of the enduring enigmas of our times. Here we show that Dark Energy can be used to solve 2 tantalizing mysteries of the observable universe. We build on existing models of Dark Energy linked to neutrino masses. In these models Dark Energy can undergo Phase Transitions and form Black Holes. Here we look at the implications of the family structure of neutrinos for the phase transitions in dark energy and associated peaks in black hole formation. It has been previously shown that one of these peaks in Black Hole formation is associated with the observed peak in Quasar formation. Here, we predict that there will also be an earlier peak in the Dark Energy Black Holes at high redshifts. These Dark Energy Black Holes formed at high redshifts are Intermediate Mass Black Holes (IMBHs).These Dark Energy Black Holes at large redshift can help explain both the EDGES observations and the observations of large Supermassive Black Holes (SMBHs) at redshifts $z \sim 7$. The existence of an earlier phase of Dark Energy Black Holes take care of some current challenges to theory implied by existing astronomical data and also helps us look for these Dark Energy Black Holes at high redshifts as predicted here through targeted searches for these Black Holes at the redshifts $z \sim 18$. There is a slight dependence of the location of the peak on the lightest neutrino mass - so the peak may be located at a slightly lower value of the redshift. This may actually enable a measurement of the lightest neutrino mass. Finding these Dark Energy Black Holes of Intermediate Mass should be within the reach of upcoming observations - particularly with the James Webb Space Telescope - but perhaps also through the use of other innovative techniques focusing specifically on the redshifts around $z \sim 18$., Comment: As embargo period is over - this version corresponds to the final published version

Automatic Speech Recognition(ASR) has been dominated by deep learning-based end-to-end speech recognition models. These approaches require large amounts of labeled data in the form of audio-text pairs. Moreover, these models are more susceptible to domain shift as compared to traditional models. It is common practice to train generic ASR models and then adapt them to target domains using comparatively smaller data sets. We consider a more extreme case of domain adaptation where text-only corpus is available. In this work, we propose a simple baseline technique for domain adaptation in end-to-end speech recognition models. We convert the text-only corpus to audio data using single speaker Text to Speech (TTS) engine. The parallel data in the target domain is then used to fine-tune the final dense layer of generic ASR models. We show that single speaker synthetic TTS data coupled with final dense layer only fine-tuning provides reasonable improvements in word error rates. We use text data from address and e-commerce search domains to show the effectiveness of our low-cost baseline approach on CTC and attention-based models., Comment: Accepted at ECNLP @ACL 2022

We prove that for all integers $k \geq 1$, there exists a constant $C_k$ depending only on $k$ such that for all $q > C_k$ and for all $n \geq 1$ every matrix in $M_n(\mathbb F_q)$ is a sum of two $k$th powers.

Magnetic skyrmions are vortex-like spin textures, which can be manipulated by external stress or pressure via magnetoelastic effects. Here, we present the observation of isostructural phase transition in a biskyrmion host hexagonal MnNiGa at a pressure around 4 GPa using pressure-dependent synchrotron x-ray powder diffraction (XRD) data analysis. Our XRD data reveals anisotropic compression behavior with pressure with different compression rates of the a-axis in the basal plane and the c-axis in the prismatic plane. However, the hexagonal symmetry remains unchanged for pressure up to 14 GPa. Fitting of unit cell volume with pressure using a second-order Birch-Murnagan equation of state reveals that the data to fall into two distinct curves for those above and below 4 GPa. The present study contributes to the understanding of crystal structure with the application of hydrostatic pressure in the biskyrmion host MnNiGa, wherein the skyrmion textures can be manipulated by pressure due to their magnetoelastic character.