遠距離幾何学における普遍性と量子的複雑性

In 1905, Albert Einstein published his special theory of relativity. According to this theory, no matter in the universe can attain a speed greater than the speed of light in a vacuum, which thus serves as the speed limit of the universe.
This has been verified in many experiments. However, space and time are no longer absolute. Two observers moving relative to one another do not agree on the length of objects or the passage of time. The mechanics of objects based on Newton's laws of...

Consider a region consisting of several individual conductors with a definite charge density in the region between these conductors. The second uniqueness theorem states that if the total charge on each conductor and the charge density in the in-between region are known, then the electric field can be uniquely determined.
In contrast, consider that the electric field is non-unique and apply Gauss's law in divergence form in the region between the conductors and the integral form to the...

For transition metal complexes, the coordination number determines the geometry around the central metal ion. Table 1 compares coordination numbers to molecular geometry. The most common structures of the complexes in coordination compounds are octahedral, tetrahedral, and square planar.

In the macroscopic world, objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a table will behave like a particle; it will continue traveling in a straight line unless it collides with another ball, or it is acted on by some other force, such as friction. The ball has a well-defined position and velocity or well-defined momentum, p = mv, which is defined by mass m and velocity v at any given moment. This is the typical...

The parallel-axis theorem provides a convenient and quick method of finding the moment of inertia of an object about an axis parallel to the axis passing through its center of mass. Consider a thin rod as an example. There is a striking similarity between the process of finding the moment of inertia of a thin rod about an axis through its middle, where the center of mass lies, and about an axis through its end using the conventional method. In the conventional method, the concept of linear mass...

A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...