In this work, first we examine the energy conditions in the context of the generalized metric-Palatini hybrid gravity, known as $f(R, {\cal R})$ gravity. We show that for the proposed model in this study, {\it i.e.} $f(R, {\cal R})= R +\alpha{\cal R}^{n}$, one of the four fundamental energy conditions, specifically the strong energy condition, does not hold for some values of $n$. Therefore, it seems that hybrid gravity can provide a model for the accelerated expansion of the universe. In continuation of completing our study in this work, we try to analyze the impact of hybrid metric-Palatini gravity on the gravitational baryogenesis process. The hybrid metric-Palatini model combines two gravitational theories that allow for a more detailed examination of the behavior of space-time and its interaction with matter. This combination is critical in the early radiation-dominant universe, where unusual gravitational effects may play a key role in generating baryonic asymmetry and the production of baryons and anti-baryons.

Developed by OpenAI, ChatGPT (Conditional Generative Pre-trained Transformer) is an artificial intelligence technology that is fine-tuned using supervised machine learning and reinforcement learning techniques, allowing a computer to generate natural language conversation fully autonomously. ChatGPT is built on the transformer architecture and trained on millions of conversations from various sources. The system combines the power of pre-trained deep learning models with a programmability layer to provide a strong base for generating natural language conversations. In this study, after reviewing the existing literature, we examine the applications, opportunities, and threats of ChatGPT in 10 main domains, providing detailed examples for the business and industry as well as education. We also conducted an experimental study, checking the effectiveness and comparing the performances of GPT-3.5 and GPT-4, and found that the latter performs significantly better. Despite its exceptional ability to generate natural-sounding responses, the authors believe that ChatGPT does not possess the same level of understanding, empathy, and creativity as a human and cannot fully replace them in most situations.

We perform a Lie symmetry analysis on the tempered-fractional Keller Segel (TFKS) system, a chemo- taxis model incorporating anomalous diffusion. A novel approach is used to handle the nonlocal nature of tempered fractional operators. By deriving the full set of Lie point symmetries and identifying the optimal one-dimensional subalgebras, we reduce the TFKS PDEs to ordinary differential equations (ODEs), yielding new exact solutions. These results offer insights into the long-term behavior and aggregation dynamics of the TFKS model and present a methodology applicable to other tempered fractional differential equations.

We study Euler Poincare dynamics on Lie groupoids with a focus on optimal control. Extending the classical Lie group formulation, we derive reduced equations on trivial Lie groupoids and interpret the result as a generalized rigid body on the sphere. This model couples internal rotational dynamics with translational motion on the Sphere. We illustrate the framework with an example from collective cell migration, showing how groupoid-based dynamics capture the trade-off between energy and migration time. The results demonstrate how Lie groupoid methods broaden the applicability of geometric control beyond standard rigid body systems

It is shown that for every multidimensional metric in the multiply warped product form $\bar{M} = K\times_{f_1} M_1\times_{f_2}M_2$ with warp functions $f_1$, $f_2$, associated to the submanifolds $M_1$, $M_2$ of dimensions $n_1$, $n_2$ respectively, one can find the corresponding Einstein equations $\bar{G}_{AB}=-\bar{\Lambda}\bar{g}_{AB}$, with cosmological constant $\bar{\Lambda}$, which are reducible to the Einstein equations $G_{\alpha\beta} = -\Lambda_1 g_{\alpha\beta}$ and $G_{ij} =-\Lambda_2 h_{ij}$ on the submanifolds $M_1$, $M_2$, with cosmological constants ${\Lambda_1}$ and ${\Lambda_2}$, respectively, where $\bar{\Lambda}$, ${\Lambda_1}$ and ${\Lambda_2}$ are functions of ${f_1}$, ${f_2}$ and $n_1$, $n_2$.

In this paper, we will analyze a time dependent geometry in a massive theory of gravity. This will be done by analyzing Vaidya space-time in such a massive theory of gravity. As gravitational collapse is a time dependent system, we will analyze it using the Vaidya space-time in massive gravity. The Vainshtein and dRGT mechanisms are used to obtain a ghost free massive gravity, and construct such time dependent solutions. Singularities formed, their nature and strength will be studied in detail. We will also study the thermodynamical aspects of such a geometry by calculating the important thermodynamical quantities for such a system, and analyzing the thermodynamical behavior of such quantities.

Recent advances in language models (LMs), have demonstrated significant efficacy in tasks related to the arts and humanities. While LMs have exhibited exceptional performance across a wide range of natural language processing tasks, there are notable challenges associated with their utilization on small datasets and their ability to replicate more creative human capacities. In this study, we aim to address these challenges by training a Persian classical poetry generation model using a transformer architecture on a specialized dataset with no pretraining. Additionally, we propose a novel decoding method to enhance coherence and meaningfulness in the generated poetry, effectively managing the tradeoff between diversity and quality. Furthermore, the results of our training approach and the proposed decoding method are evaluated through comprehensive set of automatic and human evaluations and showed its superior capability to generate coherent and meaningful poetry in compare to other decoding methods and an existing Persian large language model (LLM).

We study {\em right-invariant (resp., left-invariant) Poisson quasi-Nijenhuis structures} on a Lie group $G$ and introduce their infinitesimal counterpart, the so-called {\em r-qn structures} on the corresponding Lie algebra $\mathfrak g$. We investigate the procedure of the classification of such structures on the Lie algebras and then for clarity of our results we classify, up to a natural equivalence, all $r$-$qn$ structures on two types of four-dimensional real Lie algebras. We mention some remarks on the relation between $r$-$qn$ structures and the generalized complex structures on the Lie algebras $\mathfrak g$ and also the solutions of modified Yang-Baxter equation on the double of Lie bialgebra $\mathfrak g\oplus\mathfrak g^*$. The results are applied to some relevant examples.

We study the thermodynamical features and dynamical evolutions of various apparent horizons associated with the Vaidya evaporating black hole surrounded by the cosmological fields of dust, radiation, quintessence, cosmological constant-like and phantom. In this regard, we address in detail how do these surrounding fields contribute to the characteristic features of a surrounded dynamical black hole in comparison to a dynamical black hole in an empty background.

In superdimension $(2|2)$ there are only three non-Abelian Lie superalgebras admitting non-degenerate ad-invariant supersymmetric metric, the well-known Lie superalgebra $gl(1|1)$, and two more, $({\C}^3 + \A)$ and $({\C}_0^5 +{\A})$. After a brief review of the construction of the Wess-Zumino-Witten (WZW) models based on the $GL(1|1)$ and $(C^3 + A)$ Lie supergroups, we proceed to construct the WZW model on the $({C}_0^5 +{A})$ Lie supergroup. Unfortunately, this model does not include the super Poisson-Lie symmetry. In the following, three new exact conformal field theories of the WZW type are constructed by gauging an anomaly-free subgroup SO(2) of the Lie supergroups mentioned above. The most interesting indication of this work is that the gauged WZW model on the supercoset $(C^3 + A)/$SO(2) has super Poisson-Lie symmetry; most importantly, its dual model is conformally invariant at the one-loop order, and this is presented here for the first time. Finally, in order to study the Yang-Baxter (YB) deformations of the $({C}_0^5 +{A})$ WZW model we obtain the inequivalent solutions of the (modified) graded classical Yang-Baxter equation ((m)GCYBE) for the $({\C}_0^5 +{\A})$ Lie superalgebra. Then, we classify all possible YB deformations for the $({C}_0^5 +{A})$ and settle also the issue of an one-loop conformality of the deformed backgrounds. The classification results are important, in particular in the Lie supergroup case they are rare, much hard technical work was needed to obtain them.

We study the thermodynamical features and dynamical evolutions of various apparent horizons associated with the Vaidya evaporating black hole surrounded by the cosmological fields of dust, radiation, quintessence, cosmological constant-like and phantom. In this regard, we address in detail how do these surrounding fields contribute to the characteristic features of a surrounded dynamical black hole in comparison to a dynamical black hole in an empty background.

The associated Legendre functions $P_{l}^{(m)}(x)$ for a given $l-m$, may be taken into account as the increasing infinite sequences with respect to both indices $l$ and $m$. This allows us to construct the exponential generating functions for them in two different methods by using Rodrigues formula. As an application then we present a scheme to construct generalized coherent states corresponding to the spherical harmonics $Y_{m}^{m}(\theta,\phi)$.

This paper focuses on the introduction of right-invariant Poisson-Nijenhuis structures on Lie groupoids and their infinitesimal counterparts, also known as structures. A Poisson-Nijenhuis structure refers to a combination of a Poisson structure and a Nijenhuis structure. The paper also presents a mutual correspondence between Poisson-Nijenhuis structures on Lie algebroids and Poisson-Nijenhuis structures on their corresponding Lie groupoids, under certain conditions. This correspondence demonstrates a connection between the infinitesimal and global structures. Finally, the paper concludes with an illustrative example to enhance the understanding of the concepts introduced.

By Poissonization of Jacobi structures on real three-dimensional Lie groups $\mathbf{G}$ and using the realizations of their Lie algebras, we obtain integrable bi-Hamiltonian systems on $\mathbf{G}\otimes \mathbb{R}$.

The static and time-dependent behaviours of adhesively bonded polyethylene Double-Strap (DS) joints were investigated to assess the viability of this joint configuration relative to the Single-Lap (SL) joints. Both experiments and finite element simulations are conducted. First, we individually characterise the tensile and creep behaviour of the adhesive and adherent materials; an epoxy-based adhesive and polyethylene, respectively. This information is used to develop suitable constitutive models that are then implemented in the commercial finite element package ABAQUS by means of user material subroutines, UMATs. The numerical models are used to design the creep tests on the adhesive joints. Afterwards, an extensive experimental campaign is conducted where we characterise the static and creep behaviour of two joint configurations, SL and DS joints, and three selected values of the overlap length. In regard to the static case, results reveal an increase in the failure load with increasing overlap length, of up to 10% for an overlap length of 39 mm. Also, slightly better performance is observed for the SL joint configuration. For the creep experiments, we show that the DS adhesive joint configuration leads to much shorter elongations, relative to the SL joints. These differences diminish with increasing overlap length but remain substantial in all cases. In both joint configurations, the elongation increases with decreasing overlap length. For instance, increasing the overlap length to 39 mm led to a 50% and a 30% reduction in elongation for SL and DS joints, respectively. Moreover, the numerical predictions show a good agreement with the experiments. The stress redistribution is investigated and it is found that the shear stress is highly sensitive to the testing time, with differences being more noticeable for the DS joint system.

Recently, a new class of inflationary models, so called \emph{gauge-flation} or non-Abelian gauge field inflation has been introduced where the slow-roll inflation is driven by a non-Abelian gauge field $\textbf{A}$ with the field strength $\textbf{F}$. This class of models are based on a gauge field theory having $\textbf{F}^2$ and $\textbf{F}^4$ terms with a non-Abelian gauge group minimally coupled to gravity. Here, we present a new class of such inflationary models based on a gauge field theory having only $\textbf{F}^2$ term with non-Abelian gauge fields non-minimally coupled to gravity. The non-minimal coupling is set up by introducing the Einstein tensor besides the metric tensor within the $\textbf{F}^2$ term, which is called kinetic coupled gravity. A perturbation analysis is performed to confront the inflation under consideration with Planck and BICEP2 results.

In this study, we proceed to investigate the Thurston geometries from the point of view of their Poisson-Lie (PL) T-dualizability. First of all, we find all subalgebras of Killing vectors that generate group of isometries acting freely and transitively on the three-dimensional target manifolds, where the Thurston metrics are defined. It is shown that three-dimensional Lie subalgebras are isomorphic to the Bianchi type algebras. We take the isometry subgroup of the metric as the first subgroup of Drinfeld double. In order to investigate the non-Abelian T-duality, the second subgroup must be chosen to be Abelian. Accordingly, the non-Abelian target space duals of these geometries are found via PL T-duality approach in the absence of $B$-field. We also comment on the conformal invariance conditions of the T-dual $\sigma$-models under consideration.

Silicon-on-insulator (SOI) waveguides with different geometries have been employed to design various integrated optical components. Reducing the bending radius of the SOI waveguides with low bending loss is essential in minimizing the footprint of light-wave circuits. The propagating mode is less confined in the core of the ultra-thin SOI waveguide and penetrates to substrate and cladding, leading to higher bending loss compared with conventional SOI waveguide with the thicker guiding layer. While various bending mechanisms have been utilized to reduce the bending loss of conventional SOI waveguides, the ultra-thin SOI waveguide bends have not been studied in detail. In this paper, we present a 60 nm-thick SOI waveguide bend based on the truncated Eaton lens implemented by varying thickness of the guiding layer. The three-dimensional full-wave simulations reveal that the designed waveguide bend, with a radius of 3.9 $\mu m$, reduces the bending loss from 3.3 to 0.42 dB at the wavelength of 1550 nm. Moreover, the bending loss for the wavelength range of 1260-1675 nm is lower than 0.67 dB while the bending loss in the C-band is lower than 0.45 dB.

In the present paper, quantum speed limit (QSL) time of a bipartite V-type three-level atomic system under the effect of two-qutrit entanglement is investigated. Each party interacts with own independent reservoir. By considering two local unitarily equivalent Wener states and the Horodecki PPT state, as initial states, the QSL time is evaluated for each of them in the respective entangled regions. It is counterintu-itively observe that the effect of entanglement on the QSL time driven from each of the initial Werner states are completely different when the degree of non-Markovianity is considerable. In addition, it is interesting that the effect of entanglement of the non-equivalent Horodecki state on the calculated QSL time displays an intermediate behavior relative to the cases obtained for the Werner states.

We study the propagation of massless fermionic fields, implementing a family of special functions: Heun functions, in solving the wave equation in three three-dimensional backgrounds, including the BTZ black hole in string theory and Lifshitz black hole solutions in conformal gravity and Hu-Sawicki $F(R)$ theory. The main properties of the selected black hole solutions is that their line elements are Weyl related to that of a homogeneous spacetime, whose spatial part possesses Lie symmetry, described by Lobachevsky-type geometry with arbitrary negative Gaussian curvature. Using the Weyl symmetry of massless Dirac action, we consider the perturbation equations of fermionic fields in relation to those of the homogeneous background, which having definite singularities, are transformed into Heun equation. We point out the existence of quasinormal modes labeled by the accessory parameter of the Heun function. The distribution of the quasinormal modes has been clarified to satisfy the boundary conditions that require ingoing and decaying waves at the event horizon and conformal infinity, respectively. It turned out that the procedure based on the Heun function, beside reproducing the previously known results obtained via hypergemetric function for the BTZ and Lifshitz black hole solution in conformal gravity, brings up new families of quasinormal frequencies, which can also contain purely imaginary modes. Also, the analysis of the quasinormal modes shows that with the negative imaginary part of complex frequencies $\omega=\omega_{Re}+i\omega_{Im}$, the fermionic perturbations are stable in this background.