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In this paper, we first establish an improved Bohr inequality for the class of operator-valued holomorphic functions $f$ on a simply connected domain $\Omega$ in $\mathbb{C}$. Next, we establish a generalization of refined version of the Bohr inequality and the Bohr-Rogosinski inequality with the help of the sequence $\varphi=\{\varphi_n(r) \}^{\infty}_{n=0}$ of non-negative continuous functions in $[0,1)$ such that the series $\sum_{n=0}^{\infty}\varphi_n(r)$ converges locally uniformly on the interval $[0,1)$. All the results are proved to be sharp. Moreover, We establish the Bohr inequality and the Bohr-Rogosinski inequality for the class of operator-valued $\nu$-Bloch functions defined in two different simply connected domains, $\Omega$ and $\Omega_{\gamma}$, in $\mathbb{C}$.

We investigate the astrophysical consequences of black holes in asymptotically safe quantum gravity, characterized by the parameters $S_0$, $S_2$, $m_0$, and $m_2$, in addition to the black hole mass $M$. To evaluate the physical validity of the fundamental asymptotically safe theory of quantum gravity black hole solutions, we analyze their gravitational lensing properties in the strong field regime. Specifically, we examine the shadow cast by the asymptotically safe quantum gravity black hole and constrain its parameters using observational data from the M87* and Sgr A* supermassive black holes. Our analysis reveals that, within the $1\sigma$ confidence level, a significant portion of the parameter space for asymptotically safe quantum gravity black holes is consistent with the Event Horizon Telescope (EHT) observations of M87* and Sgr A*. This suggests these black holes are plausible candidates for describing astrophysical black holes. As an additional observational test, we perform a detailed investigation of the strong gravitational lensing properties of these black holes. We explore the fundamental strong lensing observables in detail, including the angular positions and separations of the lensed images, the relative magnifications, the radius of the outermost Einstein ring, and the relativistic time delay between images. We compare the predictions of the asymptotically safe quantum gravity black hole for each observable with those of the classical Schwarzschild solution using realistic astrophysical data. Our findings provide a pathway for testing asymptotically safe quantum gravity at galactic and extragalactic scales, offering new insights into the observational properties of black hole solutions within this framework., Comment: 26 pages, 38 figures

Dispersion of mobile robots over the nodes of an anonymous graph is an important problem and turns out to be a crucial subroutine for designing efficient algorithms for many fundamental graph problems via mobile robots. In this problem, starting from an arbitrary initial distribution of $n$ robots across the $n$ nodes, the goal is to achieve a final configuration where each node holds at most one robot. This paper investigates the dispersion problem on an oriented grid, considering the possibility of robot failures (crashes) at any time during the algorithm's execution. We present a crash-tolerant dispersion algorithm that solves the dispersion problem on an anonymous oriented grid in $O(\sqrt{n})$ time and using $O(\log n)$ bits of memory per robot. The algorithm is optimal in terms of both time and memory per robot. We further extend this algorithm to deal with weak Byzantine robots. The weak Byzantine fault dispersion algorithm takes optimal $O(\sqrt{n})$ rounds but requires $O(n\log n)$ bits of memory per robot., Comment: arXiv admin note: substantial text overlap with arXiv:2405.02002

A Galactic core-collapse supernova (CCSN) is likely to be observed in neutrino detectors around the world minutes to hours before the electromagnetic radiation arrives. The SNEWS2.0 network of neutrino and dark matter detectors aims to use the relative arrival times of the neutrinos at the different experiments to point back to the supernova. One of the simplest methods to calculate the CCSN direction is to use the first neutrino events detected through the inverse beta decay (IBD) process, $\overline{\nu}_e p\rightarrow e^+n$. We will consider neutrino detectors sensitive to IBD interactions with low backgrounds. The difference in signal arrival times between a large and a small detector will be biased, however, with the first event at the smaller detector, on average, arriving later than that at the larger detector. This bias can be mitigated by using these first events in a data-driven approach without recourse to simulations or models. In this article, we demonstrate this method and its uncertainty estimate using pairs of detectors of different sizes and with different supernova distances. Finally, we use this method to calculate probability skymaps using four detectors (Super-Kamiokande, JUNO, LVD, and SNO+) and show that the calculated probabilities yield appropriate confidence intervals. The resulting skymaps should be useful for the multi-messenger community to follow up on the SNEWS2.0 Galactic CCSN neutrino alert., Comment: 9 pages, 9 figures

Drug-releasing implants are gaining momentum in the treatment of glaucoma. Implants present however several limitations. Among these limitations, there is the inability to provide an even distribution of the drug in the trabecular meshwork. CFD simulations were used in this work to study the interplay between ocular fluid dynamics and drug diffusion to explore the options to achieve a homogeneous drug distribution. The analysis finds that the balance between convection and diffusion flux hinders mixing within the eye's anterior chamber. This results in highly localized drug delivery in the trabecular meshwork when using a gravity-driven location implant. The results also show that varying the size of the implant can help to solve the issue. Given the natural variability of the trabecular meshwork size amongst patients, implants with personalized size may become a potential solution. The location of the implant within the eye is also key to effective drug delivery. Natural laminar flow of aqueous humor within the eye's anterior chamber prevents achieving an even drug distribution at the target tissue for gravity-driven location implants. However, the iris-lens gap can be utilized as a natural mixer when placing the implant in the eye's posterior chamber, thus helping with effective delivery.

In this paper, we first establish refined versions of the Bohr inequalities for the class of holomorphic functions from the unit ball $B_X$ of a complex Banach space $X$ into $\mathbb{C}$. As applications, we will establish refined Bohr inequalities of functional type or of norm type for holomorphic mappings with lacunary series on the unit ball $B_X$ with values in higher dimensional spaces. Next, we obtain the Bohr-Rogosinski inequality for the class of holomorphic functions on $B_X.$ In addition, we establish an improved version of the Bohr inequality for holomorphic functions on $B_X$. All the results are proved to be sharp.

Let $\mathcal{A}$ be the class of all analytic functions $f$ defined on the open unit disk $\mathbb{D}$ with the normalization $f(0)=0=f^{\prime}(0)-1$. This paper examines the radius of concavity for various subclasses of $\mathcal{A}$, namely $\mathcal{S}_0^{(n)}$, $\mathcal{K(\alpha,\beta)}$, $\mathcal{\tilde{S^*}(\beta)}$, and $\mathcal{S}^*(\alpha)$. It also presents results for various classes of analytic functions on the unit disk. All the radii are best possible., Comment: 14 pages, 0 figures

The Bohr radius for an arbitrary class $\mathcal{F}$ of analytic functions of the form $f(z)=\sum_{n=0}^{\infty}a_nz^n$ on the unit disk $\mathbb{D}=\{z\in\mathbb{C} : |z|<1\}$ is the largest radius $R_{\mathcal{F}}$ such that every function $f\in\mathcal{F}$ satisfies the inequality \begin{align*} d\left(\sum_{n=0}^{\infty}|a_nz^n|, |f(0)|\right)=\sum_{n=1}^{\infty}|a_nz^n|\leq d(f(0), \partial f(\mathbb{D})), \end{align*} for all $|z|=r\leq R_{\mathcal{F}}$ , where $d(0, \partial f(\mathbb{D}))$ is the Euclidean distance. In this paper, our aim is to determine the sharp improved Bohr radius for the classes of analytic functions $f$ satisfying differential subordination relation $zf^{\prime}(z)/f(z)\prec h(z)$ and $f(z)+\beta zf^{\prime}(z)+\gamma z^2f^{\prime\prime}(z)\prec h(z)$, where $h$ is the Janowski function. We show that improved Bohr radius can be obtained for Janowski functions as root of an equation involving Bessel function of first kind. Analogues results are obtained in this paper for $\alpha$-convex functions and typically real functions, respectively. All obtained results in the paper are sharp and are improved version of [{Bull. Malays. Math. Sci. Soc.} (2021) 44:1771-1785]., Comment: 31 pages, 0 figures

A sock ordering is a finite string of elements, each of which has a color. Foot-sorting is a non-deterministic sorting algorithm with the aim of grouping all socks of the same color together, under the constraints of a stack structure; a sock ordering is foot-sortable if there is some way to apply the foot-sorting algorithm such that this aim is realizable. In introducing sock orderings and the foot-sorting algorithm, Defant and Kravitz ask whether there is some list of patterns which all foot-sortable sock orderings must avoid, thus characterizing which sock orderings are foot-sortable. We enumerate this list, the basis of foot-sortable sock orderings, and show that this basis is infinite. We also describe a deterministic polynomial-time algorithm for sorting socks., Comment: 20 pages

Let $G = (V,E)$ be an $n$-vertex graph and let $c: E \to \mathbb{N}$ be a coloring of its edges. Let $d^c(v)$ be the number of distinct colors on the edges at $v \in V$ and let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$. H. Li proved that $\delta^c(G) > n/2$ guarantees a rainbow triangle in $G$. We give extensions of Li's result to cliques $K_r$ for $r \ge 4$., Comment: 16 pages

Let $ \mathcal{S}(p) $ be the class of all meromorphic univalent functions defined in the unit disc $ \mathbb{D} $ of the complex plane with a simple pole at $ z=p $ and normalized by the conditions $ f(0)=0 $ and $ f^{\prime}(0)=1 $. In this paper, we find radius of concavity and compute the same for functions in $ \mathcal{S}(p) $ and for some other well-known classes of functions on unit disk. We explore general linear combinations $F(z):=\lambda_1f_1(z)+\cdots+\lambda_{2n} f_{2n}(z),\; \lambda_j\in\mathbb{C} $, $ n\in\mathbb{N} $, of functions belonging to the class $\mathcal{S}(p)$ and some other classes of functions of analytic univalent functions and investigate their radii of univalence, convexity and concavity.

The neutrino signal from the next galactic core-collapse supernova will provide an invaluable early warning of the explosion. By combining the burst trigger from several neutrino detectors, the location of the explosion can be triangulated minutes to hours before the optical emission becomes visible, while also reducing the rate of false-positive triggers. To enable multi-messenger follow-up of nearby supernovae, the SuperNova Early Warning System 2.0 (SNEWS 2.0) will produce a combined alert using a global network of neutrino detectors. This paper describes the trigger publishing and alert formation framework of the SNEWS 2.0 network. The framework is built on the HOPSKOTCH publish-subscribe system to easily incorporate new detectors into the network, and it implements a coincidence system to form alerts and estimate a false-positive rate for the combined triggers. The paper outlines the structure of the SNEWS 2.0 software and the initial testing of coincident signals.

This study deals with the new class of embedded wormhole solutions in the background of general relativity. Two newly calculated wormhole solutions satisfy all the required properties. All the energy conditions are discussed through their validity regions for the different ranges of involved parameters. In maximum regions, all energy conditions are violated. We investigate the shadow and strong gravitational lensing by the wormhole throat for the two new wormhole models, namely Model-I and Model-II. The present paper considers the wormhole throat to act as a photon sphere. We first derive null geodesics using the Hamilton-Jacobi separation method to investigate the shadow and strong gravitational lensing caused by the wormhole throat. We then numerically obtain the radius of wormhole shadow, strong deflection angle, and various lensing observables by taking the example of supermassive black M87* and Sgr A* in the context of both Model-I and Model-II. Keeping all other parameters fixed, it is observed that the parameters $\zeta_1$ and $\zeta_2$ for Model-I; and $\chi_1$ and $\chi_2$ for Model-II have significant effects on the wormhole shadow and strong gravitational lensing phenomena. Our conclusion is that it is possible to detect relativistic images, such as Einstein rings, produced by wormholes with throat radii of $r_{th}=3M$. Additionally, current technology enables us to test hypotheses related to astrophysical wormholes., Comment: 22 figures

We explore the bound neutrons decay into invisible particles (e.g., $n\rightarrow 3 \nu$ or $nn \rightarrow 2 \nu$) in the JUNO liquid scintillator detector. The invisible decay includes two decay modes: $ n \rightarrow { inv} $ and $ nn \rightarrow { inv} $. The invisible decays of $s$-shell neutrons in $^{12}{\rm C}$ will leave a highly excited residual nucleus. Subsequently, some de-excitation modes of the excited residual nuclei can produce a time- and space-correlated triple coincidence signal in the JUNO detector. Based on a full Monte Carlo simulation informed with the latest available data, we estimate all backgrounds, including inverse beta decay events of the reactor antineutrino $\bar{\nu}_e$, natural radioactivity, cosmogenic isotopes and neutral current interactions of atmospheric neutrinos. Pulse shape discrimination and multivariate analysis techniques are employed to further suppress backgrounds. With two years of exposure, JUNO is expected to give an order of magnitude improvement compared to the current best limits. After 10 years of data taking, the JUNO expected sensitivities at a 90% confidence level are $\tau/B( n \rightarrow { inv} ) > 5.0 \times 10^{31} \, {\rm yr}$ and $\tau/B( nn \rightarrow { inv} ) > 1.4 \times 10^{32} \, {\rm yr}$., Comment: 28 pages, 7 figures, 4 tables

We present a novel dynamical dark energy model within the frameworks of both Einstein gravity and Horava-Lifshitz gravity. Utilizing a new parametrization of the dark energy equation of state $\omega(z)$, we derive solutions to the field equations. By employing recent cosmological datasets, such as cosmic chronometer datasets, Type Ia Supernovae datasets, Baryonic Oscillation datasets, and the recent Hubble constant value measured by the Hubble Space Telescope and the SH0ES Team as an additional prior. We validate our model and determine optimal parameter values. Furthermore, we analyze the evolution of the Universe by showing the redshift dependence plots of key cosmological parameters through graphical representations. We also perform diagnostic analyses to compare our model with the standard model. Using the Akaike Information Criterion (AIC), we compare the three models and find that all of them are supported by the current data, making it impossible to discard any of them. Our model aligns well with recent observations and unveils intriguing features of the Universe, particularly the late-time behavior of the Universe., Comment: 19 pages, 5 figures

In this paper, we first obtain a refined Bohr radius for invariant families of bounded analytic functions on unit disk $ \mathbb{D} $. Then, we obtain Bohr inequality for certain integral transforms, namely Fourier (discrete) and Laplace (discrete) transforms of bounded analytic functions $ f(z)=\sum_{n=0}^{\infty}a_nz^n $, in a simply connected domain \begin{align*} \Omega_\gamma=\biggl\{z\in\mathbb{C}: \bigg|z+\dfrac{\gamma}{1-\gamma}\bigg|<\dfrac{1}{1-\gamma}\;\mbox{for}\; 0\leq \gamma<1\biggr\}, \end{align*} where $ \Omega_0=\mathbb{D} $. These results generalize some existing results. We also show that a better estimate can be obtained in radius and inequality can be shown sharp for Laplace transform of $ f $., Comment: 18 pages, 1 figure

The introduction and study of dispersing mobile robots across the nodes of an anonymous graph have recently gained traction and have been explored within various graph classes and settings. While optimal dispersion solution was established for {\em oriented} grids [Kshemkalyani et al., WALCOM 2020], a significant unresolved question pertains to whether achieving optimal dispersion is feasible on an {\em unoriented} grid. This paper investigates the dispersion problem on unoriented grids, considering both non-faulty and faulty robots. The challenge posed by unoriented grids lies in the absence of a clear sense of direction for a single robot moving between nodes, as opposed to the straightforward navigation of oriented grids. We present three deterministic algorithms tailored to our robot model. The first and second algorithms deal with the dispersion of faulty and non-faulty robots, ensuring both time and memory optimization in oriented and unoriented grids, respectively. Faulty robots that are prone to crashing at any time, causing permanent failure. In both settings, we achieve dispersion in $O(\sqrt{n})$ rounds while requiring $O(\log n)$ bits of memory per robot. The third algorithm tackles faulty robots prone to crash faults in an unoriented grid. In this scenario, our algorithm operates within $O(\sqrt{n} \log n)$ time and uses $O(\sqrt{n} \log n)$ bits of memory per robot. The robots need to know the value of $n$ for termination.

In this article, we first establish a generalized Bohr inequality and examine its sharpness for a class of analytic functions $f$ in a simply connected domain $\Omega_\gamma,$ where $0\leq \gamma<1$ with a sequence $\{\varphi_n(r) \}^{\infty}_{n=0}$ of non-negative continuous functions defined on $[0,1)$ such that the series $\sum_{n=0}^{\infty}\varphi_n(r)$ converges locally uniformly on $[0,1)$. Our results represent twofold generalizations corresponding to those obtained for the classes $\mathcal{B}(\mathbb{D})$ and $\mathcal{B}(\Omega_{\gamma})$, where \begin{align*} \Omega_{\gamma}:=\biggl\{z\in \mathbb{C}: \bigg|z+\dfrac{\gamma}{1-\gamma}\bigg|<\dfrac{1}{1-\gamma}\biggr\}. \end{align*} As a convolution counterpart, we determine the Bohr radius for hypergeometric function on $ \Omega_{\gamma} $. Lastly, we establish a generalized Bohr inequality and its sharpness for the class of $ K $-quasiconformal, sense-preserving harmonic maps of the form $f=h+\overline{g}$ in $\Omega_{\gamma}.$

In this article, we introduce a new method for solving general quadratic functional equations in $\mathbb{C}^n$. By utilizing Nevanlinna theory in $\mathbb{C}^n$, we explore the existence and form of solutions for the several systems of general quadratic difference or partial differential-difference equations of the form $af^2 + 2\alpha fg + bg^2 + 2\beta f + 2\gamma g + C=0$, where $f$ and $g$ are non-constant meromorphic functions in $\mathbb{C}^n$. The results in this paper are some improvements and generalizations of several existing results on the existence and forms of solutions for these systems. Specifically, the existing results were related to Circular-type $ f^2+g^2=1 $ and quadratic trinomial functional equations $ f^2+2\alpha fg+g^2=1 $. We provide examples to illustrate the validity of our conclusions regarding the existence and forms of solutions for such system of equations., Comment: 22 pages, 0 figure

Of a given bipartite graph $G = (V, E)$, it is elementary to construct a bipartition in time $O(|E|)$. For a given $k$-graph $H = H^{(k)}$ with $k \geq 3$ fixed, Lov\'asz proved that deciding whether $H$ is bipartite is NP-complete. Let $\mathcal{B}_n$ denote the collection of all $[n]$-vertex bipartite $k$-graphs. We construct, of a given $H \in \mathcal{B}_n$, a bipartition in time averaging $O(n^k)$ over the class $\mathcal{B}_n$. (When $k=3$, our result expedites one of Person and Schacht.) We also consider an application to the class of all $[n]$-vertex 3-graphs $H$ forbidding the Fano plane as a subhypergraph., Comment: 11 pages

We study the signal percolation through heart-like biological system. Starting from an initial distribution of waiting and inactive cells with probabilities $p$ and $(1-p)$ respectively, the signal propagation is observed in terms of active cells. As the signal enters the system from one end, the number of arrival of active sites at the other end is studied and analysis of the system behaviour is made by varying a few important parameters of the system like $p_{switch}$ (switching probability from inactive to waiting) and $p_{act}$ (switching probability from waiting to active). In this connection, the non-regular heart rhythms are discussed. Fraction of paths percolating through the system shows a transition from $0$ to $1$ near $p=p_c$. Some other important quantities like tortuosity and cluster distribution are discussed. Several critical exponents have been obtained and compared the exponents of standard percolation., Comment: 11 pages, 10 figures, 4 tables, revtex4-2

Leader election is one of the fundamental and well-studied problems in distributed computing. In this paper, we initiate the study of leader election using mobile agents. Suppose $n$ agents are positioned initially arbitrarily on the nodes of an arbitrary, anonymous, $n$-node, $m$-edge graph $G$. The agents relocate themselves autonomously on the nodes of $G$ and elect an agent as a leader such that the leader agent knows it is a leader and the other agents know they are not leaders. The objective is to minimize time and memory requirements. Following the literature, we consider the synchronous setting in which each agent performs its operations synchronously with others and hence the time complexity can be measured in rounds. The quest in this paper is to provide solutions without agents knowing any graph parameter, such as $n$, a priori. We first establish that, without agents knowing any graph parameter a priori, there exists a deterministic algorithm to elect an agent as a leader in $O(m)$ rounds with $O(n\log n)$ bits at each agent. Using this leader election result, we develop a deterministic algorithm for agents to construct a minimum spanning tree of $G$ in $O(m+n\log n)$ rounds using $O(n \log n)$ bits memory at each agent, without agents knowing any graph parameter a priori. Finally, using the same leader election result, we provide improved time/memory results for other fundamental distributed graph problems, namely, gathering, maximal independent set, and minimal dominating sets, removing the assumptions on agents knowing graph parameters a priori., Comment: 25 pages

Heavy neutrinos with masses in the MeV range can in principle simultaneously explain the light neutrino masses and the origin of baryonic matter in the universe. The strongest constraints on their properties come from their potential impact on the formation of light elements in the early universe. Since these constraints rely on assumptions about the cosmic history, independent checks in the laboratory are highly desirable. In this paper, we discuss the opportunity to search for heavy neutrinos within the MeV mass range in short and medium baseline reactor neutrino experiments, using the SoLid, JUNO and TAO experiments as examples. This kind of experiments can give the currently strongest upper bound on the mixing between the light electron neutrinos and the heavy neutrino in the 2-9 MeV mass range., Comment: submitted to JHEP

In this paper, we study some improved and refined versions of the classical Bohr inequality applicable to the class $\mathcal{B}$, which consists of self-analytic mappings defined on the unit disk $\mathbb{D}$. First, we improve the Bohr inequality for the class $\mathcal{B}$ of analytic self-maps, incorporating the area measurements of sub-disks $\mathbb{D}_r$ of $\mathbb{D}$. Secondly, we establish a sharp inequality with suitable setting as an improved version of the classic Bohr inequality. Then we obtain a sharp refined Bohr inequality in which the coefficients $|a_k|$ $(k=0, 1, 2, 3)$ in the majorant series $M_f(r)$ of $f$ are replaced by $|f^{(k)}(z)|/k!$. Finally, for a certain class $\mathcal{P}^0_{\mathcal{H}}(M)$ of harmonic mappings of the form $f=h+\overline{g}$, we generalize the Bohr inequality incorporating a sequence $\{\varphi_n(r)\}_{n=0}^{\infty}$ of continuous functions of $r$ in $[0, 1)$ and give some applications., Comment: 25 pages, 0 figures

This study investigates the accelerated cosmic expansion within the Ho\v{r}ava-Lifshitz Model. To constrain the cosmological parameters of this model, we incorporate 17 Baryon Acoustic Oscillation points, 31 Cosmic Chronometer points, 40 Type Ia Supernovae points, 24 quasar Hubble diagram points, and 162 Gamma Ray Bursts points, along with the latest Hubble constant measurement (R22). We treat $r_{d}$ as a free parameter to extract $H_{0}$ and $r_{d}$ using late-time datasets, aiming for optimal fitting values in each model. Treating $r_{d}$ as free improves precision, reduces bias, and enhances dataset compatibility. The obtained values of $H_{0}$ and $r_{d}$ are compared to the $\Lambda$CDM model, showing consistency with previous estimates from Planck and SDSS studies. The Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) favor the Ho\v{r}ava-Lifshitz model, with the $\Lambda$CDM model having the lowest AIC. Additionally, we conduct $\Delta$AIC and $\Delta$BIC analyses to assess model preference.} Validation using the reduced $\chi_{red}^{2}$ statistic indicates satisfactory fits for the Ho\v{r}ava-Lifshitz model, while recognizing $\Lambda$CDM as the preferred model. Extensions of the analysis warrant further investigation., Comment: 9 pages, 2 figures

This study focuses on the implementation of modern and intelligent logistics vehicles equipped with advanced tracking and security features. In response to the evolving landscape of logistics management, the proposed system integrates cutting edge technologies to enhance efficiency and ensure the security of the entire logistics process. The core component of this implementation is the incorporation of state-of-the art tracking mechanisms, enabling real-time monitoring of vehicle locations and movements. Furthermore, the system addresses the paramount concern of security by introducing advanced security measures. Through the utilization of sophisticated tracking technologies and security protocols, the proposed logistics vehicles aim to safeguard both customer and provider data. The implementation includes the integration of QR code concepts, creating a binary image system that conceals sensitive information and ensures access only to authorized users. In addition to tracking and security, the study delves into the realm of information mining, employing techniques such as classification, clustering, and recommendation to extract meaningful patterns from vast datasets. Collaborative filtering techniques are incorporated to enhance customer experience by recommending services based on user preferences and historical data. This abstract encapsulates the comprehensive approach of deploying modern logistics vehicles, emphasizing their intelligence through advanced tracking, robust security measures, and data-driven insights. The proposed system aims to revolutionize logistics management, providing a seamless and secure experience for both customers and service providers in the dynamic logistics landscape., Comment: 8 pages

Let $ \mathcal{H}(\Omega) $ be the class of complex-valued functions harmonic in $ \Omega\subset\mathbb{C} $ and each $f=h+\overline{g}\in \mathcal{H}(\Omega)$, where $ h $ and $ g $ are analytic. In the study of Bohr phenomenon for certain class of harmonic mappings, it is to find a constant $ r_f\in (0, 1) $ such that the inequality \begin{align*} M_f(r):=r+\sum_{n=2}^{\infty}\left(|a_n|+|b_n|\right)r^n\leq d\left(f(0), \partial\Omega\right) \;\mbox{for}\;|z|=r\leq r_f, \end{align*} where $ d\left(f(0), \partial\Omega\right) $ is the Euclidean distance between $ f(0) $ and the boundary of $ \Omega:=f(\mathbb{D}) $. The largest such radius $ r_f $ is called the Bohr radius and the inequality $ M_f(r)\leq d\left(f(0), \partial\Omega\right) $ is called the Bohr inequality for the class $ \mathcal{H}(\Omega) $. In this paper, we study Bohr phenomenon for the class of close-to-convex harmonic mappings establishing several inequalities. All the results are proved to be sharp., Comment: 26 pages, 9 figures

Triangle counting in a graph is a fundamental problem and has a wide range of applications in various domains. It is crucial in understanding the structural properties of a graph and is often used as a building block for more complex graph analytics. In this paper, we solve the triangle counting problem in an anonymous graph in a distributed setting using mobile agents and subsequently use this as a subroutine to tackle the truss decomposition and triangle centrality problem. The paper employs mobile agents, placed on the nodes of the graph to coordinate among themselves to solve the triangle enumeration problem for the graph. Following the literature, we consider the synchronous systems where each robot executes its tasks concurrently with all others and hence time complexity can be measured as the number of rounds needed to complete the task. The graph is anonymous, i.e., without any node labels or IDs, but the agents are autonomous with distinct IDs and have limited memory. Agents can only communicate with other agents locally i.e., if and only if they are at the same node. The goal is to devise algorithms that minimise both the time required for triangle counting and the memory usage at each agent. We further demonstrate how the triangle count obtained through the mobile agent approach can be leveraged to address the truss decomposition, triangle centrality and local clustering coefficient problems, which involves finding maximal sub-graphs with strong interconnections. Truss decomposition helps in identifying maximal, highly interconnected sub-graphs, or trusses, within a network, thus, revealing the structural cohesion and tight-knit communities in complex graphs, facilitating the analysis of relationships and information flow in various fields, such as social networks, biology, and recommendation systems.

Linear programming (LP) problems are pervasive in real-life applications. However, despite their apparent simplicity, an untrained user may find it difficult to determine the linear model of their specific problem. We envisage the creation of a goal-oriented conversational agent that will engage in conversation with the user to elicit all information required so that a subsequent agent can generate the linear model. In this paper, we present an approach for the generation of sample dialogues that can be used to develop and train such a conversational agent. Using prompt engineering, we develop two agents that "talk" to each other, one acting as the conversational agent, and the other acting as the user. Using a set of text descriptions of linear problems from NL4Opt available to the user only, the agent and the user engage in conversation until the agent has retrieved all key information from the original problem description. We also propose an extrinsic evaluation of the dialogues by assessing how well the summaries generated by the dialogues match the original problem descriptions. We conduct human and automatic evaluations, including an evaluation approach that uses GPT-4 to mimic the human evaluation metrics. The evaluation results show an overall good quality of the dialogues, though research is still needed to improve the quality of the GPT-4 evaluation metrics. The resulting dialogues, including the human annotations of a subset, are available to the research community. The conversational agent used for the generation of the dialogues can be used as a baseline., Comment: GEM Workshop @ EMNLP 2023

In this paper, a significant improvement has been achieved in the classical Bohr's inequality for the class $ \mathcal{B} $ of analytic self maps defined on the unit disk $ \mathbb{D} $. More precisely, we generalize and improve several Bohr-type inequalities by combining appropriate improved and refined versions of the classical Bohr's inequality with some methods concerning the area measure of bounded analytic functions in $ \mathcal{B} $. In addition, we obtain Bohr-type and Bohr-Rogosinski-type inequalities for the subordination class and also for the class of $ K $-quasiconformal harmonic mappings. All the results are proved to be sharp.

Wurtzite ferroelectrics possess transformative potential for next-generation microelectronics. A comprehensive understanding of their ferroelectric properties and domain energetics is crucial for tailoring their ferroelectric characteristics and exploiting their functional properties in practical devices. Despite burgeoning interest, the exact configurations, and electronic structures of the domain walls in wurtzite ferroelectrics remain elusive. In this work, we elucidate the atomic configurations and electronic properties of electric-field-induced domain walls in ferroelectric ScGaN. By combining transmission electron microscopy and theoretical calculations, a novel charged domain wall with a buckled two-dimensional hexagonal phase is revealed. The dangling bonds associated with these domain walls give rise to unprecedented metallic-like mid-gap states within the forbidden band. Quantitative analysis further unveils a universal charge-compensation mechanism stabilizing antipolar domain walls in ferroelectric materials, wherein the polarization discontinuity at the 180{\deg} domain wall is compensated by the dangling bond electrons. Furthermore, the reconfigurable conductivity of these domain walls is experimentally demonstrated, showcasing their potential for ultra-scaled device applications. Our findings represent a pivotal advancement in understanding the structural and electronic properties of wurtzite ferroelectric domain walls and lay the groundwork for fundamental physics studies and device applications.

In this paper, we study Bohr's inequality and refined versions of Bohr-Rogosinski inequalities involving Schwarz functions. Moreover, we establish a version of multidimensional analogue of Bohr inequality and Bohr-Rogosinski inequalities involving Schwarz functions. Finally, we establish a multidimensional analogue of the refined version of Bohr inequalities with the initial coefficient being zero. All the results are proved to be sharp.

Matom\"aki proved that if $\alpha\in \mathbb{R}$ is irrational, then there are infinitely many primes $p$ such that $|\alpha-a/p|\le p^{-4/3+\varepsilon}$ for a suitable integer a. In this paper, we extend this result to all quadratic number fields under the condition that the Grand Riemann Hypothesis holds for their Hecke $L$-functions., Comment: 31 pages

In this paper, we explore the complex symmetrical characteristics of weighted composition operators $W_{u, v}$ and weighted composition-differentiation operators $W_{u, v, k_1, k_2, \ldots, k_n}$ on the Hardy space $H^2(\mathbb{D}^n)$ over the Polydisk $\mathbb{D}^n$, with respect to the standard conjugation $\mathcal{J}$. We specify explicit conditions that confirm the Hermitian characteristics of the operator $W_{u, v, k_1, k_2, \ldots, k_n}$ and describe the conditions necessary for it to exhibit normal behavior. Additionally, we identify the kernels of the generalized weighted composition-differentiation operators and their corresponding adjoint operators.

In previous studies [1, 2, 3] the possible existence dark matter admixed pulsar have been discussed based on three different dark matter density profiles, Singular Isothermal Sphere density profile, Universal Rotational Curve density profile and Navarro-Frenk-White density profile. They have been used to discuss the pulsars present in our Milky Way galaxy as well as some satellite dwarf galaxies of Milky Way. In this article we use the Universal Rotational Curve (URC) dark matter density profile to observe similar effects on galaxies M31 and M87. These study hold significant importance, as now it can be concluded that there is a fair possibility of presence of dark matter admixed pulser in M31 and M87 galaxies as well., Comment: 17 pages, 16 figures, 2 tables

This paper reports on the use of prompt engineering and GPT-3.5 for biomedical query-focused multi-document summarisation. Using GPT-3.5 and appropriate prompts, our system achieves top ROUGE-F1 results in the task of obtaining short-paragraph-sized answers to biomedical questions in the 2023 BioASQ Challenge (BioASQ 11b). This paper confirms what has been observed in other domains: 1) Prompts that incorporated few-shot samples generally improved on their counterpart zero-shot variants; 2) The largest improvement was achieved by retrieval augmented generation. The fact that these prompts allow our top runs to rank within the top two runs of BioASQ 11b demonstrate the power of using adequate prompts for Large Language Models in general, and GPT-3.5 in particular, for query-focused summarisation., Comment: 9 pages, 7 figures, 4 tables

The role of type Ia supernovae (SN Ia), mainly the Delay Time Distributions (DTDs) determined by the binary systems, and the yields of elements created by different explosion mechanisms, are studied by using the MulChem chemical evolution model applied to our Galaxy. We explored 15 DTDs and 12 tables of elemental yields produced by different SN Ia explosion mechanisms, doing a total of 180 models. Chemical abundances for $\alpha$-elements (O, Mg, Si, Ca) and Fe derived from these models are compared with recent solar region observational data of $\alpha$-elements over Fe relative abundances, [X/Fe], as a function of [Fe/H] and age. A multidimensional maximum likelihood analysis shows that 52 models are able to fit all these datasets simultaneously, considering the 1$\sigma$ confidence level. The combination of STROLG1 DTD and LN20181 SN Ia yields provides the best fit. The exponential model with very prompt events is a possible DTD, but a combination of several channels is more probable. The SN Ia yields that include MCh or Near MCh correspond to 39 (75%) of the 52 best models. Regarding the DTD, 31 (60%) of the 52 most probable models correspond to the SD scenario, while the remaining 21 (40%) are based on the DD scenario. Our results also show that the relatively large dispersion of the observational data may be explained by the stellar migration from other radial regions, and/or perhaps a combination of DTDs and explosion channels., Comment: 26 pages, 28 figures. Accepted for publication in MNRAS

We investigate the strong gravitational lensing phenomena caused by a black hole with a dark matter halo. In this study, we examine strong gravitational lensing with two significant dark matter models: the universal rotation curve model and the cold dark matter model. To do this, we first numerically estimate the strong lensing coefficients and strong deflection angles for both the universal rotation curve and cold dark matter models. It is observed that the deflection angle, denoted as $\alpha_D$, increases with the parameter $\alpha$ while holding the value of $\gamma \cdot 2M$ constant. Additionally, it increases with the parameter $\gamma \cdot 2M$ while keeping the value of $\alpha$ constant. The strong deflection angle, $\alpha_D$, for the black hole with a dark matter halo, with parameters $\alpha=0$ and $\gamma \cdot 2M$, greatly enhances the gravitational bending effect and surpasses the corresponding case of the standard Schwarzschild black hole ($A=0, B=0, \alpha=0, \gamma \cdot 2M=0$). Furthermore, we investigate the astrophysical consequences through strong gravitational lensing observations, using examples of various supermassive black holes such as $M87^{*}$ and $SgrA^{*}$ located at the centers of several galaxies. It is observed that black holes with dark matter halos can be quantitatively distinguished and characterized from the standard Schwarzschild black hole ($A=0, B=0, \alpha=0, \gamma \cdot 2M=0$). The findings in our analysis suggest that observational tests for black holes influenced by dark matter halos are indeed feasible and viable., Comment: 21 pages, 24 figures

In this work, we assume the FRLW Universe which is filled with dark matter along with dark energy in the framework of Horava-Lifshitz (HL) gravity. The dark energy is considered as the Linear (Model I) and CPL (Model II) parametrizations of the equation of state parameter. For both models, we express the Hubble parameter $H(z)$ in terms of the model parameters and redshift $z$. To rigorously constrain the model, we have employed a comprehensive set of recent observational datasets including cosmic chronometers (CC), Type Ia Supernovae (SNIa), Baryon Acoustic Oscillations (BAO), Gamma-ray Burst (GRB), Quasar (Q) and Cosmic Microwave Background Radiation (CMB). Through the joint analysis of this diverse collection of datasets, we have achieved tighter constraints on the model's parameters. This, in turn, allows us to delve into both the physical and geometrical aspects of the model with greater precision. Furthermore, our analysis has enabled us to determine the present values of crucial cosmological parameters, including $H_{0}$, $\Omega_{m0}$, $\Omega_{k0}$ and $\Omega_{\Lambda0}$. It's noteworthy that our results are consistent with recent findings from Planck 2018, underscoring the reliability and relevance of our models in the current cosmological context. We also conduct analysis of cosmographic parameters and apply statefinder and diagnostic tests to explore the evolution of the Universe. In addition, the AIC and BIC suggest that the $\Lambda$CDM model is the preferred model among all our considered models. Our investigation into the models has unveiled intriguing features of the late universe, Comment: 24 pages, 20 figures, accepted for publication in European Physical Journal C (EPJC)

The core-collapse supernova (CCSN) is considered one of the most energetic astrophysical events in the universe. The early and prompt detection of neutrinos before (pre-SN) and during the supernova (SN) burst presents a unique opportunity for multi-messenger observations of CCSN events. In this study, we describe the monitoring concept and present the sensitivity of the system to pre-SN and SN neutrinos at the Jiangmen Underground Neutrino Observatory (JUNO), a 20 kton liquid scintillator detector currently under construction in South China. The real-time monitoring system is designed to ensure both prompt alert speed and comprehensive coverage of progenitor stars. It incorporates prompt monitors on the electronic board as well as online monitors at the data acquisition stage. Assuming a false alert rate of 1 per year, this monitoring system exhibits sensitivity to pre-SN neutrinos up to a distance of approximately 1.6 (0.9) kiloparsecs and SN neutrinos up to about 370 (360) kiloparsecs for a progenitor mass of 30 solar masses, considering both normal and inverted mass ordering scenarios. The pointing ability of the CCSN is evaluated by analyzing the accumulated event anisotropy of inverse beta decay interactions from pre-SN or SN neutrinos. This, along with the early alert, can play a crucial role in facilitating follow-up multi-messenger observations of the next galactic or nearby extragalactic CCSN., Comment: 24 pages, 9 figures, accepted for the publication at JCAP

Research involving computing with mobile agents is a fast-growing field, given the advancement of technology in automated systems, e.g., robots, drones, self-driving cars, etc. Therefore, it is pressing to focus on solving classical network problems using mobile agents. In this paper, we study one such problem -- finding small dominating sets of a graph $G$ using mobile agents. Dominating set is interesting in the field of mobile agents as it opens up a way for solving various robotic problems, e.g., guarding, covering, facility location, transport routing, etc. In this paper, we first present two algorithms for computing a {\em minimal dominating set}: (i) an $O(m)$ time algorithm if the robots start from a single node (i.e., gathered initially), (ii) an $O(\ell\Delta\log(\lambda)+n\ell+m)$ time algorithm, if the robots start from multiple nodes (i.e., positioned arbitrarily), where $m$ is the number of edges and $\Delta$ is the maximum degree of $G$, $\ell$ is the number of clusters of the robot initially and $\lambda$ is the maximum ID-length of the robots. Then we present a $\ln (\Delta)$ approximation algorithm for the {\em minimum} dominating set which takes $O(n\Delta\log (\lambda))$ rounds.

The current manuscript deals with the tidal force effects, geodesic deviation, and shadow constraints of the Schwarzschild-like black hole theorised in Starobinsky-Bel-Robinson gravity exhibiting M-theory compactification. In the current analysis, we explore the radial and angular tidal force effects on a radially in-falling particle by the central black hole, which is located in this spacetime. We also numerically solve the geodesic deviation equation and study the variation of the geodesic separation vector with the radial coordinate for two nearby geodesics using suitable initial conditions. All the obtained results are tested for Sag A* and M87* by constraining the value of the stringy gravity parameter $\beta$ using the shadow data from the event horizon telescope observations. All the results are compared with Schwarzschild black hole spacetime. In our study, we found that both the radial and angular tidal forces experienced by a particle switch their initial behaviour and turn compressive and stretching, respectively, before reaching the event horizon. The geodesic deviation shows an oscillating trend as well for the chosen initial condition. For the constrained value of $\beta$, we see that the spacetime geometry generated by Sag A* and M87* is effectively same for both Schwarzschild and Starobinsky-Bel-Robinson black hole. Furthermore, we also calculated the angular diameter of the shadow in Starobinsky-Bel-Robinson black hole and compared with the Schwarzschild black hole. It is observed that the angular diameter of shadow for M87* and Sgr A* in Starobinsky-Bel-Robinson black hole is smaller than the Schwarzschild black hole. The calculated results satisfy the event horizon telescope observational constraints. Finally, we have concluding remarks., Comment: 12 pages, 18 figures, accepted for publication in European Physical Journal C

In this article, we first establish operator-valued analogues of multidimensional refined Bohr inequality. Then we establish operator-valued analogues of multidimensional improved Bohr inequality with a certain power of the norm of the initial coefficient. Finally, we establish operator-valued analogues of a multidimensional sharp version of Bohr inequality with the initial coefficient being replaced by the norm value of the function. In addition, we establish operator-valued analogues of multidimensional improved Bohr inequality using the quantity $S_r/\pi$. All the results prove to be sharp., Comment: 19. arXiv admin note: text overlap with arXiv:2111.11713 by other authors

Let $ \mathcal{H}(\mathbb{D}) $ be the class of all holomorphic functions in the unit disk $ \mathbb{D} $. We aim to explore the complex symmetry exhibited by generalized weighted composition-differentiation operators, denoted as $L_{n, \psi, \phi}$ and is defined by \begin{align*} L_{n, \psi, \phi}:=\sum_{k=1}^{n}c_kD_{k, \psi_k, \phi},\; \mbox{where }\; c_k\in\mathbb{C}\; \mbox{for}\; k=1, 2, \ldots, n, \end{align*} where $ D_{k, \psi, \phi}f(z):=\psi(z)f^{(k)}(\phi(z)),\; f\in \mathcal{A}^2_{\alpha}(\mathbb{D}), $ in the reproducing kernel Hilbert space, labeled as $\mathcal{A}^2_{\alpha}(\mathbb{D})$, which encompasses analytic functions defined on the unit disk $\mathbb{D}$. By deriving a condition that is both necessary and sufficient, we provide insights into the $ C_{\mu, \eta} $-symmetry exhibited by $L_{n, \psi, \phi}$. The explicit conditions for which the operator T is Hermitian and normal are obtained through our investigation. Additionally, we conduct an in-depth analysis of the spectral properties of $ L_{n, \psi, \phi} $ under the assumption of $ C_{\mu, \eta} $-symmetry and thoroughly examine the kernel of the adjoint operator of $L_{n, \psi, \phi}$., Comment: 25

Let $ \mathcal{B}:=\{f(z)=\sum_{n=0}^{\infty}a_nz^n\; \mbox{with}\; |f(z)|<1\;\mbox{for all}\; z\in\mathbb{D}\} $. The improved version of the classical Bohr's inequality \cite{Bohr-1914} states that if $ f\in\mathcal{B} $, then the associated majorant series $ M_f(r):=\sum_{n=0}^{\infty}|a_n|r^n\leq 1 $ holds for $ |z|=r\leq 1/3 $ and the constant $ 1/3 $ cannot be improved. Bohr's original theorem and its subsequent generalizations remain active fields of study, driving investigations in a wide range of function spaces. In this paper, first we establish a generalized Bohr inequality for the class $\mathcal{B}$ by allowing a sequence $\{\varphi_n(r)\}_{n=0}^{\infty}$ of non-negative continuous functions on $[0, 1)$ in the place of $\{r^n\}_{n=0}^{\infty}$ of the majorant series $M_f(r)$ introducing a weighted sequence of non-negative continuous functions $\{\Phi_n(r)\}_{n=0}^{\infty}$ on $[0, 1)$. Secondly, as a generalization, we obtain a refined version of the Bohr inequality for a certain class $\tilde{G}^0_{\mathcal{H}}(\beta)$ of harmonic mappings. All the results are proved to be sharp., Comment: 22

The Hankel and Toeplitz determinants $H_{2,1}(F_{f^{-1}}/2)$ and $T_{2,1}(F_{f^{-1}}/2)$ are defined as: \begin{align*} H_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} \Gamma_1 & \Gamma_2 \Gamma_2 & \Gamma_3 \end{vmatrix} \;\;\mbox{and} \;\; T_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} \Gamma_1 & \Gamma_2 \Gamma_2 & \Gamma_1 \end{vmatrix} \end{align*} where $\Gamma_1, \Gamma_2,$ and $\Gamma_3$ are the first, second and third logarithmic coefficients of inverse functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we establish sharp inequalities $|H_{2,1}(F_{f^{-1}}/2)|\leq 1/4$, $|H_{2,1}(F_{f^{-1}}/2)| \leq 1/36$, $|T_{2,1}(F_{f^{-1}}/2)|\leq 5/16$ and $|T_{2,1}(F_{f^{-1}}/2)|\leq 145/2304$ for the logarithmic coefficients of inverse functions for the classes starlike functions and convex functions with respect to symmetric points. In addition, our findings are substantiated further through the incorporation of illustrative examples, which support the strict inequality and lend credence to our conclusions., Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:2305.12500, arXiv:2307.14365

The Hankel determinant $H_{2,1}(F_{f^{-1}}/2)$ of logarithmic coefficients is defined as: \begin{align*} H_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} \Gamma_1 & \Gamma_2 \Gamma_2 & \Gamma_3 \end{vmatrix}=\Gamma_1\Gamma_3-\Gamma^2_2, \end{align*} where $\Gamma_1, \Gamma_2,$ and $\Gamma_3$ are the first, second and third logarithmic coefficients of inverse functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we establish sharp inequalities $|H_{2,1}(F_{f^{-1}}/2)|\leq 19/288$, $|H_{2,1}(F_{f^{-1}}/2)| \leq 1/144$, and $|H_{2,1}(F_{f^{-1}}/2)| \leq 1/36$ for the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order $1/2$, respectively., Comment: 14. arXiv admin note: substantial text overlap with arXiv:2307.02741

This paper is devoted to studying the observational signatures modified by Bardeen black hole via shadow and strong lensing observations. Influence of the modified Bardeen black hole parameters q, g, and the parameter $\mu$ on the shadow radius of the black hole have been investigated numerically and graphically. Recently, EHT collaboration observed the image and shadow of supermassive black holes $M87^*$ and $SgrA^*$ where the shadow angular diameter $\theta_d=42\pm3$ for $M87^*$ and $\theta_d=51.8\pm2.3$ for $SgrA^*$. The modified black hole parameters q and $\mu$ for the fixed value of g have been constrained by the EHT collaboration data for the angular shadow diameter of $M87^*$ and $SgrA^*$. It has been observed that the constrain ranges of the parameters $\mu$ and $q$ of modified Bardeen black hole as $-0.89\leq \mu/8M^2 \leq 0.4$ and $0\leq |q|\leq 0.185$ for $M87^*$; and $-1.38\leq \mu/8M^2 \leq 0.1$ and $0\leq |q|\leq 0.058$ for $SgrA^*$, keeping the fixed value $g/2M=0.2$. Modified Bardeen black holes with the additional parameters $\mu$,$g$ and $q$ besides the mass M of the black hole as the supermassive black holes $M87^*$ and $SgrA^*$; and it is observed that to be a viable astrophysical black hole candidate. Furthermore, Gravitational lensing in the strong field limit for modified Bardeen black hole has been investigated numerically as well as graphically and compared to the other ordinary astrophysical black hole such as Schwarzschild ($\mu=\&q=0$) and regular Bardeen ($\mu=0$) black hole., Comment: 20 pages, 30 figures

We compile a catalog of 578 highly probable and 62 likely red supergiants (RSGs) of the Milky Way, which represents the largest list of Galactic RSG candidates designed for continuous follow-up to date. We match distances measured by Gaia DR3, 2MASS photometry, and a 3D Galactic dust map to obtain luminous bright late-type stars. Determining the stars' bolometric luminosities and effective temperatures, we compare to Geneva stellar evolution tracks to determine likely RSG candidates, and quantify contamination using a catalog of Galactic AGB in the same luminosity-temperature space. We add details for common or interesting characteristics of RSG, such as multi-star system membership, variability, and classification as a runaway. As potential future core-collapse supernova (SN) progenitors, we study the ability of the catalog to inform the Supernova Early Warning System (SNEWS) coincidence network made to automate pointing, and show that for 3D position estimates made possible by neutrinos, the number of progenitor candidates can be significantly reduced, improving our ability to observe the progenitor pre-explosion and the early phases of the core-collapse supernova., Comment: 22 pages, 10 figures, 5 table. This matches the version published in MNRAS. Data available at https://github.com/SNEWS2/candidate_list

In this paper, utilizing Nevanlinna theory, we study existence and forms of the entire solutions $ f $ of the quadratic trinomial-type partial differential-difference equations in $ \mathbb{C}^n $ \begin{align*} a\left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right)^2 + 2 \omega \left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right) f(z + c) + b f(z + c)^2 = e^{g(z)} \end{align*} and \begin{align*} a\left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right)^2 & + 2 \omega \left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right) \Delta_cf(z) + b [\Delta_cf(z)]^2 = e^{g(z)}, \end{align*} where $ a, \omega, b\in\mathbb{C} $, $ g $ is a polynomial in $ \mathbb{C}^n $ and $ \Delta_cf(z)=f(z+c)-f(z) $. The main results of the paper improve several existence results in $ \mathbb{C}^n $ for integer $ n\geq 2 $ and $ 1\leq i