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The nature of "Little Red Dots" and their relation to other forms of accreting supermassive black holes remain an open question. Here we report the discovery of a Little Red Dot at $z=7.3$. It is attenuated by moderate amounts of dust, $A_V = {2.79}\,\textrm{mag}$, with an intrinsic bolometric luminosity of $10^{46.6}\,\textrm{erg}\,\textrm{s}^{-1}$ and a SMBH mass of $5\times10^8\,\textrm{M}_\odot$. Most notably, this object is embedded in an overdensity of eight nearby galaxies, allowing us to calculate a spectroscopic estimate of the clustering of galaxies around Little Red Dots. We find a Little Red Dot-galaxy cross-correlation length of $r_0\!=\!8\pm2\,\textrm{h}^{-1}\,\textrm{cMpc}$, comparable to that of $z\!\sim\!6$ UV-luminous quasars. The resulting estimate of their minimum dark matter halo mass of $\log_{10}(M_{\textrm{halo, min}}/\textrm{M}_{\odot})= 12.0_{-1.0}^{+0.8}$ indicates that nearly all halos above this mass must host actively accreting SMBHs at $z\approx7$, in strong contrast with the far smaller duty cycle of luminous quasars ($<1\%$). Our results, taken at face value, motivate a picture in which SMBHs in Little Red Dot phases could serve as the obscured p

For the isospectral Darboux transformations of the discrete quantum mechanics with real shifts, there are two methods: type I and type II constructions. Based on the type I construction, the type I multi-indexed little $q$-Jacobi and little $q$-Laguerre orthogonal polynomials were presented in J. Phys. {\bf A50} (2017) 165204. Based on the type II construction, we present the type II multi-indexed little $q$-Jacobi and little $q$-Laguerre orthogonal polynomials.

In this paper, we generalize the Dunn-Brinkmeier~additivity theorem, which establishes a weak equivalence $\mathcal{C}_n \otimes \mathcal{C}_m \simeq \mathcal{C}_{n+m}$ for the little cubes operad $\mathcal{C}_n$. We introduce equivariant framed little disk operads, a new class of operads that simultaneously generalize the framed little disk operads and the little $V$-disk operads associated with a $G$-representation $V$. We prove that these operads satisfy an analogous additivity property, extending the classical theorem to settings involving group actions and framings.

We construct a model for the (non-unital) S^1-framed little 2d-dimensional disks operad for any positive integer d using logarithmic geometry. We also show that the unframed little 2d-dimensional disks operad has a model which can be constructed using log schemes with virtual morphisms.

The little $n$-disks operad is $SO(n)$ and $O(n)$-equivariantly formal over the rationals. Equivalently, the oriented and unoriented framed little disks operads are rationally formal as $\infty$-operads.

Little Higgs models address the hierarchy problem by identifying the SM Higgs doublet as pseudo-Nambu--Goldstone bosons (pNGB) arising from global symmetries with collective breakings. These models are designed to address the little hierarchy problem up to a scale of $Λ\!\sim\! {\cal O}(10)$ TeV. Consequently, these models necessitate an ultraviolet (UV) completion above this scale. On the other hand, conformal extensions of the Standard Model are intriguing because scales emerge as a consequence of dimensional transmutation. In this study, we present a unified framework in which the electroweak hierarchy problem is tackled through a conformal symmetry collectively broken around the TeV scale, offering an appealing UV completion for little Higgs models. Notably, this framework automatically ensures the presence of the required UV fixed points, eliminating the need for careful adjustments to the particle content of the theory. Moreover, this framework naturally addresses the flavor puzzles associated with composite or little Higgs models. Furthermore, we suggest that in this framework all known little Higgs models can be UV-completed through conformal dynamics above the scale $Λ$ up

This article reviews the Little Higgs models of electroweak symmetry breaking and their phenomenology. Little Higgs models incorporate a light composite Higgs boson and remain perturbative until a scale of order 10 TeV, as required by precision electroweak data. The collective symmetry breaking mechanism, which forms the basis of Little Higgs models, is introduced. An explicit, fully realistic implementation of this mechanism, the Littlest Higgs model, is then discussed in some detail. Several other implementations, including simple group models and models with T parity, are also reviewed. Precision electroweak constraints on a variety of Little Higgs models are summarized. If a Little Higgs model is realized in nature, the predicted new particles should be observable at the Large Hadron Collider (LHC). The expected signatures, as well as the experimental sensitivities and the possible strategies for confirming the Little Higgs origin of new particles, are discussed. Finally, several other related topics are briefly reviewed, including the ultraviolet completions of Little Higgs models, as well as the implications of these models for flavor physics and cosmology.

We examine in more detail specific models which yield a little rip cosmology, i.e., a universe in which the dark energy density increases without bound but the universe never reaches a finite-time singularity. We derive the conditions for the little rip in terms of the inertial force in the expanding universe and present two representative models to illustrate in more detail the difference between little rip models and those which are asymptotically de Sitter. We derive conditions on the equation of state parameter of the dark energy to distinguish between the two types of models. We show that coupling between dark matter and dark energy with a little rip equation of state can alter the evolution, changing the little rip into an asymptotic de Sitter expansion. We give conditions on minimally-coupled phantom scalar field models and on scalar-tensor models that indicate whether or not they correspond to a little rip expansion. We show that, counterintuitively, despite local instability, a little-rip cosmology has an infinite lifetime.

Little Higgs models predict new gauge bosons, fermions and scalars at the TeV scale that stabilize the Higgs mass against quadratically divergent one-loop radiative corrections. We categorize the many little Higgs models into two classes based on the structure of the extended electroweak gauge group and examine the experimental signatures that identify the little Higgs mechanism in addition to those that identify the particular little Higgs model. We find that by examining the properties of the new heavy fermion(s) at the LHC, one can distinguish the structure of the top quark mass generation mechanism and test the little Higgs mechanism in the top sector. Similarly, by studying the couplings of the new gauge bosons to the light Higgs boson and to the Standard Model fermions, one can confirm the little Higgs mechanism and determine the structure of the extended electroweak gauge group.

Little Higgs theories are an attempt to address the little hierarchy problem, i.e., the tension between the naturalness of the electroweak scale and the precision measurements showing no evidence for new physics up to 5-10 TeV. In little Higgs theories, the Higgs mass-squareds are protected to the one-loop order from the quadratic divergence. This allows the cutoff to be raised up to \~10 TeV, beyond the scales probed by the precision data. However, strong constraints can still arise from the contributions of the new TeV scale particles and hence re-introduces the fine-tuning problem. In this paper we show that a new symmetry, denoted as T-parity, under which all heavy gauge bosons and scalar triplets are odd, can remove all the tree-level contributions to the electroweak observables and therefore makes the little Higgs theories completely natural. The T-parity can be manifestly implemented in a majority of little Higgs models by following the most general construction of the low energy effective theory a la Callan, Coleman, Wess and Zumino. In particular, we discuss in detail how to implement the T-parity in the littlest Higgs model based on SU(5)/SO(5). The symmetry breaking scal

The Hubble constant, H0, or its dimensionless equivalent, "little h", is a fundamental cosmological property that is now known to an accuracy better than a few percent. Despite its cosmological nature, little h commonly appears in the measured properties of individual galaxies. This can pose unique challenges for users of such data, particularly with survey data. In this paper we show how little h arises in the measurement of galaxies, how to compare like-properties from different datasets that have assumed different little h cosmologies, and how to fairly compare theoretical data with observed data, where little h can manifest in vastly different ways. This last point is particularly important when observations are used to calibrate galaxy formation models, as calibrating with the wrong (or no) little h can lead to disastrous results when the model is later converted to the correct h cosmology. We argue that in this modern age little h is an anachronism, being one of least uncertain parameters in astrophysics, and we propose that observers and theorists instead treat this uncertainty like any other. We conclude with a "cheat sheet" of nine points that should be followed when deali

We explore two classes of 6d $\mathcal{N}=(1,0)$ little string theories obtained from type IIA/IIB NS5-branes probing $D_n$ singularities. Their tensor branches are described by effective gauge theories whose instanton solitons are macroscopic little strings. We specifically study two families of 2d $\mathcal{N}=(0,4)$ gauge theories which describe at low energy the worldsheet dynamics of the type IIA/IIB little strings. These gauge theories are useful to calculate the supersymmetric partition functions of the little string theories on $\mathbf{R}^4 \times T^2$. We establish the T-duality of the little string theories by utilizing their BPS spectra as a probe.

In this manuscript the theory and phenomena associated with the Little Effect are introduced as the spin induced orbital dynamics of confined fermions under strong magnetic and thermal environments. This Little Effect is considered in details for the electron transfer reactions associated with redox processes of Cu-Ag alloy within deionized water and for the orbital dynamics during the iron catalyzed covalent bond rearrangements associated with amorphous carbon conversion to diamond. Furthermore, prolong extreme conditions of 74,000 amps, 403 V, strong Lorentz compression, and thermal stresses upon this Cu-Ag- H2O system on the basis of the Little Effect of high spin, thermally induced orbital dynamics are predicted and demonstrated to cause the magnetically organized reverse beta, electron capture, proton capture and neutron capture processes for various infrequent pycnonuclear transmutations within the Cu-Ag coil. The general experimental verification and the broad implications of this Little Effect on chemistry are demonstrated within these two ideal systems: an ionic case and a molecular case. The Little Effect is contrasted with the Hedvall Effect as a dynamical phenomenon cau

Dark energy of phantom or quintessence nature with an equation of state parameter $w$ almost equal to -1 often leads the universe evolution to a finite-time future singularity. An elegant solution to this problem has been recently proposed \cite{frampton11} under the form of the so-called Little Rip cosmology which appears to be a realistic alternative to the $Λ$CDM model. A viscous Little Rip cosmology is here proposed. Whereas generically bulk viscosity tends to promote the Big Rip, we find that there are a number of situations where this is not the case and where the formalism nicely adjusts itself to the Little Rip scenario. We prove, in particular, that a viscous fluid (or, equivalently, one with an inhomogeneous (imperfect) equation of state) is perfectly able to produce a Little Rip cosmology as a purely viscosity effect. The possibility of its induction as a combined result of viscosity and a general (power-like) equation of state is also investigated in detail. To finish, a physical, inertial force interpretation of the dissolution of bound structures in the Little Rip cosmology is presented.

In \cite{DFW} and \cite{Fu07}, little $q$-Schur algebras were introduced as homomorphic images of the infinitesimal quantum groups. In this paper, we will investigate representations of these algebras. We will classify simple modules for little $q$-Schur algebras and classify semisimple little $q$-Schur algebras. Moreover, through the classification of the blocks of little $q$-Schur algebras for $n=2$, we will determine little $q$-Schur algebras of finite representation type in the odd roots of unity case.

This paper presents a little reflection about the Sleeping Beauty Problem, maybe contributing to shed light on it and perhaps helping to find a simple and elegant solution that could definitively resolve the controversies about it.

A model for particles based on preons in chiral, vector and tensor/graviton supermultiplets of unbroken global supersymmetry is engineered. The framework of the model is little string theory. Phenomenological predictions are discussed.

The little Higgs mechanism provides an alternative solution to the hierarchy problem, arguably fitting better into the phenomenological hint of the "little hierarchy" which may cause some fine-tuning for the case of supersymmetry. We discuss an aspect of little Higgs physics lacking proper attention -- the construction of an interesting and consistent chiral fermionic sector and its phenomenological implications. At least for the kind of example models to be discussed, the gauge and top sector structure of a model largely dictates, through gauge anomaly cancellation conditions, a specific chiral fermion spectrum. The spectrum has interesting, family non-universal, flavor structure. The implications for flavor physics are specially interesting. We also add a brief comment of little Higgs versus supersymmetry.