$\eta$-RICCI SOLITONS AND GRADIENT RICCI SOLITONS ON $\delta$- LORENTZIAN TRANS-SASAKIAN MANIFOLDS

The objective of the present research article is to study the $\delta$-Lorentzian trans-Sasakian manifolds conceding the $\eta$-Ricci solitons and gradient Ricci soliton. We shown that a symmetric second order covariant tensor in a $\delta$-Lorentzian trans-Sasakian manifold is a constant multiple of metric tensor. Also, we furnish an example of $\eta$-Ricci soliton on 3-diemsional $\delta$-Lorentzian trans-Sasakian manifold is provide in the region where $\delta$-Lorentzian trans-Sasakian manifold is expanding. Furthermore, we discuss some results based on gradient Ricci solitons on $3$-dimensional $\delta$- Lorentzian trans-Sasakian manifold.

Entropy as a Topological Operad Derivation

We share a small connection between information theory, algebra, and topology—namely, a correspondence between Shannon entropy and derivations of the operad of topological simplices. We begin with a brief review of operads and their representations with topological simplices and the real line as the main example. We then give a general definition for a derivation of an operad in any category with values in an abelian bimodule over the operad. The main result is that Shannon entropy defines a derivation of the operad of topological simplices, and that for every derivation of this operad there exists a point at which it is given by a constant multiple of Shannon entropy. We show this is compatible with, and relies heavily on, a well-known characterization of entropy given by Faddeev in 1956 and a recent variation given by Leinster.

Structure and Colour in Triangle-Free Graphs

Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number $\chi$ contains a rainbow independent set of size $\lceil\frac12\chi\rceil$. This is sharp up to a factor $2$. This result and its short proof have implications for the related notion of chromatic discrepancy. Drawing inspiration from both structural and extremal graph theory, we conjecture that every triangle-free graph of chromatic number $\chi$ contains an induced cycle of length $\Omega(\chi\log\chi)$ as $\chi\to\infty$. Even if one only demands an induced path of length $\Omega(\chi\log\chi)$, the conclusion would be sharp up to a constant multiple. We prove it for regular girth $5$ graphs and for girth $21$ graphs. As a common strengthening of the induced paths form of this conjecture and of Johansson's theorem (1996), we posit the existence of some $c >0$ such that for every forest $H$ on $D$ vertices, every triangle-free and induced $H$-free graph has chromatic number at most $c D/\log D$. We prove this assertion with 'triangle-free' replaced by 'regular girth 5'.

Novel Application to Recognize a Breakdown Pressure Event on Time Series Frac Data Vs. an Artificial Intelligence Approach

Breakdown pressure is the peak pressure attained when fluid is injected into a borehole until fracturing occurs. Hydraulic fracturing operations are conducted above the breakdown pressure, at which the rock formation fractures and allows fluids to flow inside. This value is essential to obtain formation stress measurements. The objective of this study is to automate the selection of breakdown pressure flags on time series fracture data using a novel algorithm in lieu of an artificial neural network. This study is based on high-frequency treatment data collected from a cloud-based software. The comma separated (.csv) files include treating pressure (TP), slurry rate (SR), and bottomhole proppant concentration (BHPC) with defined start and end time flags. Using feature engineering, the model calculates the rate of change of treating pressure (dtp_1st) slurry rate (dsr_1st), and bottomhole proppant concentration (dbhpc_1st). An algorithm isolates the initial area of the treatment plot before proppant reaches the perforations, the slurry rate is constant, and the pressure increases. The first approach uses a neural network trained with 872 stages to isolate the breakdown pressure area. The expert rule-based approach finds the highest pressure spikes where SR is constant. Then, a refining function finds the maximum treating pressure value and returns its job time as the predicted breakdown pressure flag. Due to the complexity of unconventional reservoirs, the treatment plots may show pressure changes while the slurry rate is constant multiple times during the same stage. The diverse behavior of the breakdown pressure inhibits an artificial neural network's ability to find one "consistent pattern" across the stage. The multiple patterns found through the stage makes it difficult to select an area to find the breakdown pressure value. Testing this complex model worked moderately well, but it made the computational time too high for deployment. On the other hand, the automation algorithm uses rules to find the breakdown pressure value with its location within the stage. The breakdown flag model was validated with 102 stages and tested with 775 stages, returning the location and values corresponding to the highest pressure point. Results show that 86% of the predicted breakdown pressures are within 65 psi of manually picked values. Breakdown pressure recognition automation is important because it saves time and allows engineers to focus on analytical tasks instead of repetitive data-structuring tasks. Automating this process brings consistency to the data across service providers and basins. In some cases, due to its ability to zoom-in, the algorithm recognized breakdown pressures with higher accuracy than subject matter experts. Comparing the results from two different approaches allowed us to conclude that similar or better results with lower running times can be achieved without using complex algorithms.

Three-dimensional trans-Sasakian manifolds and solitons

PurposeThe authors set the goal to find the solution of the Eisenhart problem within the framework of three-dimensional trans-Sasakian manifolds. Also, they prove some results of the Ricci solitons, η-Ricci solitons and three-dimensional weakly  symmetric trans-Sasakian manifolds. Finally, they give a nontrivial example of three-dimensional proper trans-Sasakian manifold.Design/methodology/approachThe authors have used the tensorial approach to achieve the goal.FindingsA second-order parallel symmetric tensor on a three-dimensional trans-Sasakian manifold is a constant multiple of the associated Riemannian metric g.Originality/valueThe authors declare that the manuscript is original and it has not been submitted to any other journal for possible publication.

Neural network approach to data-driven estimation of chemotactic sensitivity in the Keller-Segel model

<abstract><p>We consider the mathematical model of chemotaxis introduced by Patlak, Keller, and Segel. Aggregation and progression waves are present everywhere in the population dynamics of chemotactic cells. Aggregation originates from the chemotaxis of mobile cells, where cells are attracted to migrate to higher concentrations of the chemical signal region produced by themselves. The neural net can be used to find the approximate solution of the PDE. We proved that the error, the difference between the actual value and the predicted value, is bound to a constant multiple of the loss we are learning. Also, the Neural Net approximation can be easily applied to the inverse problem. It was confirmed that even when the coefficient of the PDE equation was unknown, prediction with high accuracy was achieved.</p></abstract>

On Volume Functions of Special Flow Polytopes Associated to the Root System of Type $A$

In this paper, we consider the volume of a special kind of flow polytope. We show that its volume satisfies a certain system of differential equations, and conversely, the solution of the system of differential equations is unique up to a constant multiple. In addition, we give an inductive formula for the volume with respect to the rank of the root system of type $A$.

Large complete minors in random subgraphs

Let G be a graph of minimum degree at least k and let G p be the random subgraph of G obtained by keeping each edge independently with probability p. We are interested in the size of the largest complete minor that G p contains when p = (1 + ε)/k with ε > 0. We show that with high probability G p contains a complete minor of order $\tilde{\Omega}(\sqrt{k})$ , where the ~ hides a polylogarithmic factor. Furthermore, in the case where the order of G is also bounded above by a constant multiple of k, we show that this polylogarithmic term can be removed, giving a tight bound.

Fluctuation Bounds for the Max-Weight Policy with Applications to State Space Collapse

We consider a multihop switched network operating under a max-weight scheduling policy and show that the distance between the queue length process and a fluid solution remains bounded by a constant multiple of the deviation of the cumulative arrival process from its average. We then exploit this result to prove matching upper and lower bounds for the time scale over which additive state space collapse (SSC) takes place. This implies, as two special cases, an additive SSC result in diffusion scaling under non-Markovian arrivals and, for the case of independent and identically distributed arrivals, an additive SSC result over an exponential time scale.

An Energy Bound in the Affine Group

We prove a nontrivial energy bound for a finite set of affine transformations over a general field and discuss a number of implications. These include new bounds on growth in the affine group, a quantitative version of a theorem by Elekes about rich lines in grids. We also give a positive answer to a question of Yufei Zhao that for a plane point set $P$ for which no line contains a positive proportion of points from $P$, there may be at most one line, meeting the set of lines defined by $P$ in at most a constant multiple of $|P|$ points.