Spontaneous pneumomediastinum in limited cutaneous systemic sclerosis and myositis overlap

A spontaneous process is one that occurs naturally under certain conditions. A nonspontaneous process, on the other hand, will not take place unless it is “driven” by the continual input of energy from an external source. Processes have a natural tendency to occur in one direction under a given set of conditions. Water will naturally flow downhill (spontaneous process), but uphill flow (nonspontaneous process) requires outside intervention such as the use of a pump. Iron exposed to...

The relative amounts of reactants and products represented in a balanced chemical equation are often referred to as stoichiometric amounts. However, in reality, the reactants are not always present in the stoichiometric amounts indicated by the balanced equation.

Spin systems where the difference in chemical shifts of the coupled nuclei is greater than ten times J are called first-order spin systems. These nuclei are weakly coupled, and their chemical shifts and coupling constant can generally be estimated from the well-separated signals in the spectrum.
As Δν decreases and the signals move closer, the doublets appear increasingly distorted. The intensities of the inner lines increase at the cost of those of the outer lines as the signals are...

The number e is a fundamental constant in calculus, playing a central role in describing continuous change, particularly exponential growth. It is most naturally defined through its relationship with the natural logarithm, which is the inverse of the exponential function with base e. This relationship allows e to be characterized using basic principles of differentiation rather than as an arbitrary numerical constant.A key property of the natural logarithm function, ln x, is that its derivative...

Limits are a key mathematical concept for understanding how functions behave as their input approaches specific values, particularly when the function is undefined. They help reveal trends and discontinuities by examining the values a function approaches rather than its actual value.One-sided limits focus on the direction from which a value is approached. When a function behaves differently depending on whether the input approaches from the left or the right, the two one-sided limits may not...

Limit laws provide essential tools for analyzing how functions behave as their input approaches a specific value. These laws are particularly useful when dealing with combinations of functions, provided the individual limits exist. The Sum and Difference Laws state that the limit of the sum or difference of two functions equals the sum or difference of their respective limits:The Product Law asserts that the limit of the product of two functions equals the product of their individual limits:A...