A decomposition formula for the wall heat flux of a compressible boundary layer

Understanding the generation mechanism of the heat flux is essential for the design of hypersonic vehicles. We proposed a novel formula to decompose the heat flux coefficient into the contributions of different terms by integrating the conservative equation of the total energy. The reliability of the formula is well demonstrated by the direct numerical simulation results of a hypersonic transitional boundary layer. Through this formula, the exact process of the energy transport in the boundary layer can be explained and the dominant contributors to the heat flux can be explored, which are beneficial for the prediction of the heat and design of the thermal protection devices.

Decomposition of degenerate Gromov–Witten invariants

We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family $X \longrightarrow B$ with singular fibre over $b_0\in B$ yields a family $\mathscr {M}(X/B,\beta ) \longrightarrow B$ of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over $b_0$ in terms of rigid tropical maps to the tropicalization of $X/B$. This generalizes one aspect of known results in the case that the fibre $X_{b_0}$ is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.

Lifting modules with finite internal exchange property and direct sums of hollow modules

A module [Formula: see text] is said to be lifting if, for any submodule [Formula: see text] of [Formula: see text], there exists a decomposition [Formula: see text] such that [Formula: see text] and [Formula: see text] is a small submodule of [Formula: see text]. A lifting module is defined as a dual concept of the extending module. A module [Formula: see text] is said to have the finite internal exchange property if, for any direct summand [Formula: see text] of [Formula: see text] and any finite direct sum decomposition [Formula: see text], there exists a direct summand [Formula: see text] of [Formula: see text] [Formula: see text] such that [Formula: see text]. This paper is concerned with the following two fundamental unsolved problems of lifting modules: “Classify those rings all of whose lifting modules have the finite internal exchange property” and “When is a direct sum of indecomposable lifting modules lifting?”. In this paper, we prove that any [Formula: see text]-square-free lifting module over a right perfect ring satisfies the finite internal exchange property. In addition, we give some necessary and sufficient conditions for a direct sum of hollow modules over a right perfect ring to be lifting with the finite internal exchange property.

Performances Management When Modelling Internal Structure of a Production Process

Performance management is a central point for both public and private organizations. In the data envelopment analysis (DEA) method, performance management takes the form of measuring relative efficiency. Furthermore, considering each organization and or production process as a black box,  inputs are transformed into outputs. In reality, production organizations or processes are composed of different parts that carry out different related activities. For this reason, modeling the internal structure of a production process like a system of interconnected parts makes it possible to measure its performance at the sub-process level. In this paper, we hypothesized a production process, made up of three interconnected parts. It is a new strategy to acquire relative efficiency consisting of building a block inside the system with at least two sub-processes. This step refers to a basic model of relational Network Data Envelopment Analysis (NDEA). Also, we used the additive decomposition formula to measure the efficiency of the whole process. We highlighted the differences in the measurement, between the direct application of the relational NDEA model and the measurement with the block approach model.We compared the cumulative empirical distribution functions of the efficiency scores of a sub-process with the decomposition formula multiplicative and our  approach. In conclusion, the paper proposes, a new strategy to measure the relative performances of a production process model as a network system of three subprocesses, which combines the NDEA and the DEA. This allows us to reevaluate, the indications of policy at the individual sub-process level (block). Moreover, it is a versatile approach which allows aggregation of the sub-processes in blocks, according to the particular policy requirements, legislative technological constraints, etc.

Edge geodetic self-decomposition in graphs

Let [Formula: see text] be a simple connected graph of order [Formula: see text] and size [Formula: see text]. A decomposition of a graph [Formula: see text] is a collection of edge-disjoint subgraphs [Formula: see text] of [Formula: see text] such that every edge of [Formula: see text] belongs to exactly one [Formula: see text]. The decomposition [Formula: see text] of a connected graph [Formula: see text] is said to be an edge geodetic self-decomposition if [Formula: see text] for all [Formula: see text]. Some general properties satisfied by this concept are studied.

The Weitzenböck Type Curvature Operator for Singular Distributions

We study geometry of a Riemannian manifold endowed with a singular (or regular) distribution, determined as an image of the tangent bundle under smooth endomorphisms. Following construction of an almost Lie algebroid on a vector bundle, we define the modified covariant and exterior derivatives and their L 2 adjoint operators on tensors. Then, we introduce the Weitzenböck type curvature operator on tensors, prove the Weitzenböck type decomposition formula, and derive the Bochner–Weitzenböck type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian. The assumptions used in the results are reasonable, as illustrated by examples with f-manifolds, including almost Hermitian and almost contact ones.

Fundamental Solutions for a Class of Multidimensional Elliptic Equations with Several Singular Coefficients

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. These fundamental solutions are directly connected with multiple hypergeometric functions and the decomposition formula is required for their investigation which would express the multivariable hypergeometric function in terms of products of several simpler hypergeometric functions involving fewer variables. In this paper, such a formula is proved instead of a previously existing recurrence formula.The order of singularity and other properties of the fundamental solutions that are necessary for solving boundary value problems for degenerate second-order elliptic equations are determined