Interval total colorings of graphs

A total coloring of a graph $G$ is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. An \emph{interval total $t$-coloring} of a graph $G$ is a total coloring of $G$ with colors $1,2,\...,t$ such that at least one vertex or edge of $G$ is colored by $i$, $i=1,2,\...,t$, and the edges incident to each vertex $v$ together with $v$ are colored by $d_{G}(v)+1$ consecutive colors, where $d_{G}(v)$ is the degree of the vertex $v$ in $G$. In this paper we investigate some properties of interval total colorings. We also determine exact values of the least and the greatest possible number of colors in such colorings for some classes of graphs.Comment: 23 pages, 1 figur

All Stable Characteristic Classes of Homological Vector Fields

An odd vector field $Q$ on a supermanifold $M$ is called homological, if $Q^2=0$. The operator of Lie derivative $L_Q$ makes the algebra of smooth tensor fields on $M$ into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called characteristic classes. These take values in the cohomology of the operator $L_Q$ and are represented by $Q$-invariant tensors made up of the homological vector field and a symmetric connection on $M$ by means of tensor operations.Comment: 17 pages, references and comments adde

Graph complexes in deformation quantization

Kontsevich's formality theorem and the consequent star-product formula rely on the construction of an $L_\infty$-morphism between the DGLA of polyvector fields and the DGLA of polydifferential operators. This construction uses a version of graphical calculus. In this article we present the details of this graphical calculus with emphasis on its algebraic features. It is a morphism of differential graded Lie algebras between the Kontsevich DGLA of admissible graphs and the Chevalley-Eilenberg DGLA of linear homomorphisms between polyvector fields and polydifferential operators. Kontsevich's proof of the formality morphism is reexamined in this light and an algebraic framework for discussing the tree-level reduction of Kontsevich's star-product is described.Comment: 39 pages; 3 eps figures; uses Xy-pic. Final version. Details added, mainly concerning the tree-level approximation. Typos corrected. An abridged version will appear in Lett. Math. Phy

A Review of Disintegration Mechanisms and Measurement Techniques

Pharmaceutical solid dosage forms (tablets or capsules) are the predominant form to administer active pharmaceutical ingredients (APIs) to the patient. Tablets are typically powder compacts consisting of several different excipients in addition to the API. Excipients are added to a formulation in order to achieve the desired fill weight of a dosage form, to improve the processability or to affect the drug release behaviour in the body. These complex porous systems undergo different mechanisms when they come in contact with physiological fluids. The performance of a drug is primarily influenced by the disintegration and dissolution behaviour of the powder compact. The disintegration process is specifically critical for immediate-release dosage forms. Its mechanisms and the factors impacting disintegration are discussed and methods used to study the disintegration in-situ are presented. This review further summarises mathematical models used to simulate disintegration phenomena and to predict drug release kinetics.We would like to acknowledge the U.K. Engineering and Physical Sciences Research Council (EPSRC) for funding (EP/L019922/1)

The homotopy theory of simplicial props

The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on "higher props," we show that the category of all small colored simplicial props admits a cofibrantly generated model category structure. With this model structure, the forgetful functor from props to operads is a right Quillen functor.Comment: Final version, to appear in Israel J. Mat

Review of sensing technologies for measuring powder density variations during pharmaceutical solid dosage form manufacturing

Oral solid dosage forms, the most widely used pharmaceutical products, are typically manufactured through a series of processes that transform a blend of drug and excipient particles into a densified product with consistent quality attributes. While the densification of powder during processing is crucial and directly impacts the quality of the drug product, there is still scarcity of non-destructive and fast sensor systems that provide access to the powder density at critical process stages. This review discusses methods for monitoring density variations of particulate matter by describing their principles and presenting application examples. The techniques discussed range from common in-line methods such as near-infrared spectroscopy, acoustic emission and ultrasonic methods as well as techniques with potential to be more frequently applied in a pharmaceutical manufacturing line, i.e., terahertz spectroscopy and imaging, microwave technique, electrical tomography and X-ray based methods. This review also compares these techniques in terms of measurement and data processing time, resolution and its ability to be integrated in a process

DM2 platform II : AI-assisted optimization of oral solid dosage form development

Digital Medicines Manufacturing (DM2) platform II aims to leverage AI to develop an autonomous workflow for drug product manufacturing and testing system by identifying optimal formulation and process parameters that deliver desired critical quality attributes (CQAs). This approach aims to de-risk and accelerate drug product development by reducing experiments, development time, and materials use by 60%. The initial objective is to develop a database of hundreds of historical data and new experiments. This database is then used to develop a hybrid machine for predicting product (tablets and capsules) attributes by utilizing domain knowledge (empirical/mechanistic models) and AI-powered models (where domain knowledge is not available/reliable). The hybrid machine is then integrated into an iterative, model-informed optimization framework, to smartly plan experiments that drive the automated manufacturing and testing system, which will be used to collect multi-scale and –point data and update the model(s) to learn from the experiments. Fit-for-purpose optimization algorithms will be employed to execute this loop based on the objective of experiments (i.e. exploration of the knowledge space or exploitation of the model-based optimization) and it continues until the targets are achieved. The proposed smart experimental planning method has been tested using the Gurnham (empirical) model as a predictor of porosity based on peak compression pressure. The initial analysis of the uncertainty of fit demonstrates that adding only 1 data point results in a 20-fold improvement in the accuracy of prediction while adding 8 more data points leads to a minimal improvement, highlighting the significance of the here-developed smart experimental planning procedure in achieving acceptable prediction accuracy at a minimal experimentation cost

Multi-modal dissolution testing system for pharmaceutical tablets

This project aims to develop a novel multimodal sensor system that is capable of resolving the key processes, as well as how these processes are linked to microstructure, formulation and raw material attributes