Search results
Results from the WOW.Com Content Network
The purpose of this page is to catalog new, interesting, and useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number , or equivalently the Dirichlet convolution of an arithmetic function () with one:
The multiplicative identity is the unit function ε defined by ε(n) = 1 if n = 1 and ε(n) = 0 if n > 1. The units (invertible elements) of this ring are the arithmetic functions f with f(1) ≠ 0. Specifically, [1] Dirichlet convolution is associative, = (), distributive over addition
Horologium Oscillatorium: Sive de Motu Pendulorum ad Horologia Aptato Demonstrationes Geometricae (English: The Pendulum Clock: or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks) is a book published by Dutch mathematician and physicist Christiaan Huygens in 1673 and his major work on pendula and horology.
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
Lemma 1. ′ =, where ′ is the differential of . This equation means that the differential of det {\displaystyle \det } , evaluated at the identity matrix, is equal to the trace. The differential det ′ ( I ) {\displaystyle \det '(I)} is a linear operator that maps an n × n matrix to a real number.
The star graphs K 1,3, K 1,4, K 1,5, and K 1,6. A complete bipartite graph of K 4,7 showing that Turán's brick factory problem with 4 storage sites (yellow spots) and 7 kilns (blue spots) requires 18 crossings (red dots) For any k, K 1,k is called a star. [2] All complete bipartite graphs which are trees are stars. The graph K 1,3 is called a ...
This number can be computed from C = {c k, ..., c 2, c 1} with c k > ... > c 2 > c 1 as follows. From the definition of the ordering it follows that for each k-combination S strictly less than C, there is a unique index i such that c i is absent from S, while c k, ..., c i+1 are present in S, and no other value larger than c i is.
Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class. [note 3] For this reason, a small group of authors have adopted conditions 4-6 to define a Dynkin system. An important fact is that any Dynkin system that is also a π-system (that is, closed under finite intersections) is a 𝜎-algebra. This ...