zero-sum

Examples of zero-sum in the following topics:

• The Loop Rule

  • Applied to circuitry, it is implicit that the directed sum of the electrical potential differences (voltages) around any closed network is equal to zero.
  • Mathematically, Kirchhoff's loop rule can be represented as the sum of voltages in a circuit, which is equated with zero:
  • In this example, the sum of v1, v2, v3, and v4 (and v5 if it is included), is zero.
  • Kirchhoff's loop rule states that the sum of all the voltages around the loop is equal to zero: v1 + v2 + v3 - v4 = 0.
  • An example of Kirchhoff's second rule where the sum of the changes in potential around a closed loop must be zero.

• Infinite Geometric Series

  • If the terms of a geometric series approach zero, the sum of its terms will be finite.
  • As the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series being infinite.
  • A geometric series with a finite sum is said to converge.
  • What follows in an example of an infinite series with a finite sum.
  • We will calculate the sum $s$ of the following series:

• First Condition

  • The first condition of equilibrium is that the net force in all directions must be zero.
  • This means that both the net force and the net torque on the object must be zero.
  • Here we will discuss the first condition, that of zero net force.
  • In order to achieve this conditon, the forces acting along each axis of motion must sum to zero.
  • The forces acting on him add up to zero.

• Static Equilibrium

  • It is qualitatively described by an object at rest and by the sum of all forces, with the sum of all torques acting on that object being equal to zero.
  • Therefore, the sum of the forces and torques at any point within the static liquid or gas must be zero.
  • Similarly, the sum of the forces and torques of an object at rest within a static fluid medium must also be zero.
  • For static equilibrium to be achieved, the sum of these forces must be zero, as shown in .
  • In the case on an object at stationary equilibrium within a static fluid, the sum of the forces acting on that object must be zero.

• Two-Component Forces

  • An object with constant velocity has zero acceleration.
  • A motionless object still has constant (zero) velocity, so motionless objects also have zero acceleration.
  • so objects with constant velocity also have zero net external force.
  • In static systems, in which motion does not occur, the sum of the forces in all directions always equals zero.
  • A moving car for which the net x and y force components are zero

• Zero-Coupon Bonds

  • Examples of zero-coupon bonds include U.S.
  • Treasury bills, U.S. savings bonds, and long-term zero-coupon bonds.
  • This creates a supply of new zero coupon bonds.
  • Each of these investments then pays a single lump sum.
  • Zero coupon bonds may be long- or short-term investments.

• Summing an Infinite Series

  • A series is the sum of the terms of a sequence.
  • The sequence of partial sums ${S_k}$ associated to a series $\sum_{n=0}^\infty a_n$ is defined for each k as the sum of the sequence ${a_n}$ from $a_0$ to $a_k$:
  • By definition the series $\sum_{n=0}^\infty a_n$ converges to a limit $L$ if and only if the associated sequence of partial sums  converges to $L$.
  • An easy way that an infinite series can converge is if all the $a_{n}$ are zero for sufficiently large $n$s.
  • Working out the properties of the series that converge even if infinitely many terms are non-zero is, therefore, the essence of the study of series.

• The Junction Rule

  • Kirchhoff's junction rule states that at any circuit junction, the sum of the currents flowing into and out of that junction are equal.
  • Kirchhoff's junction rule states that at any junction (node) in an electrical circuit, the sum of the currents flowing into that junction is equal to the sum of the currents flowing out of that junction.
  • In other words, given that a current will be positive or negative depending on whether it is flowing towards or away from a junction, the algebraic sum of currents in a network of conductors meeting at a point is equal to zero.
  • Thus, Kirchoff's junction rule can be stated mathematically as a sum of currents (I):

• Partitioning the Sums of Squares

  • Partition sum of squares Y into sum of squares predicted and sum of squares error
  • Define r2 in terms of sum of squares explained and sum of squares Y
  • The sum of squares error is the sum of the squared errors of prediction.
  • First, notice that the sum of y and the sum of y' are both zero.
  • This indicates that although some Y values are higher than their respective predicted Y values and some are lower, the average difference is zero.

• Series Solutions

  • $\displaystyle{f= \sum_{k=0}^\infty A_kz^k \ f'= \sum_{k=0}^\infty kA_kz^{k-1} \ f''= \sum_{k=0}^\infty k(k-1)A_kz^{k-2}}$
  • $\begin{aligned} & {} \sum_{k=0}^\infty k(k-1)A_kz^{k-2}-2z \sum_{k=0}^\infty kA_kz^{k-1}+ \sum_{k=0}^\infty A_kz^k=0 \\ & = \sum_{k=0}^\infty k(k-1)A_kz^{k-2}- \sum_{k=0}^\infty 2kA_kz^k+ \sum_{k=0}^\infty A_kz^k \end{aligned}$
  • $\begin{aligned} & = \sum_{k+2=0}^\infty (k+2)((k+2)-1)A_{k+2}z^{(k+2)-2}- \sum_{k=0}^\infty 2kA_kz^k+ \sum_{k=0}^\infty A_kz^k \\ & = \sum_{k=0}^\infty (k+2)(k+1)A_{k+2}z^k- \sum_{k=0}^\infty 2kA_kz^k+ \sum_{k=0}^\infty A_kz^k \\ & = \sum_{k=0}^\infty \left((k+2)(k+1)A_{k+2}+(-2k+1)A_k \right)z^k \end{aligned}$
  • If this series is a solution, then all these coefficients must be zero, so:
  • The exponential function (in blue), and the sum of the first $n+1$ terms of its Maclaurin power series (in red).