A note on the almost-Schur lemma on smooth metric measure spaces

A space truss is a three-dimensional counterpart of a planar truss. These structures consist of members connected at their ends, often utilizing ball-and-socket joints to create a stable and versatile framework. The space truss is widely used in various construction projects due to its adaptability and capacity to withstand complex loads.
At the core of a space truss lies the fundamental unit known as the tetrahedron. This structure is composed of six members that form a three-dimensional shape...

The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...

Smooth endoplasmic reticulum or smooth ER is a sub-organelle with specialized functions in animal cells and plant cells. It is often associated with the tubule morphology of the endoplasmic reticulum.
The ER provides optimal conditions for synthesizing steroid hormones and lipids, such as phospholipids and triglycerides. Traditionally, lipid metabolism was considered to be a smooth ER function. However, there is no direct evidence to prove that rough ER is completely excluded from lipid...

Smooth muscles are an important type of muscle tissue that plays a vital role in the involuntary movements of internal organs. For example, they help regulate the movement of food through the gut and the flow of blood through the circulatory system.
Function of visceral smooth muscles
Visceral smooth muscle is found in the walls of all hollow organs, except the heart, and is a key player in the involuntary movements that drive the functioning of these internal organs. This tissue is arranged in...

Smooth muscle contraction is a complex process vital for various bodily functions, from maintaining blood vessel tension to facilitating the movement of food through the digestive tract. Unlike striated muscles, smooth muscle contraction begins more slowly and lasts longer.
The onset of contraction is triggered by an increase in calcium ions within the sarcoplasm, similar to the process in striated muscle. However, smooth muscles have a relatively smaller reservoir of the sarcoplasmic...

A space truss is a three-dimensional counterpart of a planar truss. These structures consist of members connected at their ends, often utilizing ball-and-socket joints to create a stable and versatile framework. Due to its adaptability and capacity to withstand complex loads, the space truss is widely used in various construction projects.
Consider a tripod consisting of a tetrahedral space truss with a ball-and-socket joint at C. Suppose the height and lengths of the horizontal and vertical...