polynomial

A polynomial is a mathematical expression consisting of a sum of terms, each term including a variable or variables raised to a power and multiplied by a coefficient. The simplest polynomials have one variable. A one-variable (univariate) polynomial of degree n has the following form:

a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x¹ + a₀x⁰

where the a's represent the coefficients and x represents the variable. Because x¹ = x and x⁰ = 1 for all complex numbers x, the above expression can be simplified to:

a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀

When an nth-degree univariate polynomial is equal to zero, the result is a univariate polynomial equation of degree n:

a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀ = 0

There may be several different values of x, called roots, that satisfy a univariate polynomial equation. In general, the higher the order of the equation (that is, the larger the value of n), the more roots there are.

A univariate polynomial equation of degree 1 (n = 1) constitutes a linear equation. When n = 2, it is a quadratic equation; when n = 3, it is a cubic equation; when n = 4, it is a quartic equation; when n = 5, it is a quintic equation. The larger the value of n, the more difficult it is to find all the roots of a univariate polynomial equation.

Some polynomials have two, three, or more variables. A two-variable polynomial is called bivariate; a three-variable polynomial is called trivariate.

cyclic redundancy checking

Cyclic redundancy checking is a method of checking for errors in data that has been transmitted on a communications link. A sending device applies a 16- or 32-bit polynomial to a block of data that is to be transmitted and appends the resulting cyclic redundancy code (CRC) to the block. The receiving end applies the same polynomial to the data and compares its result with the result appended by the sender. If they agree, the data has been received successfully. If not, the sender can be notified to resend the block of data.

The ITU-TS (CCITT) has a standard for a 16-bit polynomial to be used to obtain the cyclic redundancy code (CRC) that is appended. IBM's Synchronous Data Link Control and other protocols use CRC-16, another 16-bit polynomial. A 16-bit cyclic redundancy code detects all single and double-bit errors and ensures detection of 99.998% of all possible errors. This level of detection assurance is considered sufficient for data transmission blocks of 4 kilobytes or less. For larger transmissions, a 32-bit CRC is used. The Ethernet and Token Ring local area network protocols both used a 32-bit CRC.

In Europe, CRC-4 is a multiframe system of cyclic redundancy checking that is required for switches on E-1 lines.

A less complicated but less capable error detection method is the checksum method. See modem error-correcting protocols for a list of protocols that use either of these methods.

ohm

The ohm is the standard unit of electrical resistance in the International System of Units (SI). Ohms are also used, when multiplied by imaginary numbers, to denote reactance in alternating-current (AC) and radio-frequency (RF) applications. Reduced to base SI units, one ohm is the equivalent of one kilogram meter squared per second cubed per ampere squared (1 kg times m² · s^-3 · A^-2. The ohm is also the equivalent of a volt per ampere (V/A).

In a direct-current (DC) circuit, a component has a resistance of one ohm when a potential difference of one volt produces a current of one ampere through the component. In AC and RF circuits, resistive ohms behave the same as they do in DC circuits, provided the root-mean-square (rms) AC voltage is specified. In AC and RF circuits, reactance exists only when there is a net capacitance or inductance. Capacitive reactances have negative imaginary ohmic values; inductive reactances have positive imaginary ohmic values. The reactance of a particular capacitor or inductor depends on the frequency.

Resistances and reactances are sometimes expressed in units representing power-of-10 multiples of one ohm. A kilohm is equal to one thousand (10³) ohms. A megohm is equal to one million (10⁶) ohms. Fractional prefix multipliers are seldom used for resistance or reactances; rarely will you hear or read about a milliohm or a microhm. Extremely small resistances and reactances are usually referred to in terms of conductance. Also see conductance, Ohm's Law, prefix multipliers, resistance, reactance, siemens, and International System of Units (SI).

coefficient

1) In a mathematical equation, a coefficient is a constant by which a variable is multiplied. Consider the following equations:

3x² + 5x - 3 = 0
ax³ + by² = z

The values 3 and 5 in the first equation are coefficients of x, a variable. In the second equation, if a and b are constants, then a is a coefficient of x³, and b is a coefficient of y². It is customary to use lower-case, italic letters from the first half of the alphabet to represent constants whose values are not specified numerically. Lower-case, italic letters from the second half of the alphabet generally represent variables.

2) In physics and engineering, a coefficient is a quantitative expression of a specific property of matter, or of a phenomenon. Consider an electronic component whose value changes with temperature. It is tested and found to have a resistance of 100 ohms at a temperature of +20 degrees Celsius (°C), and a resistance of 101 ohms at a temperature of +70 °C. This is the equivalent of a change in resistance of +1 ohm for a temperature change of +50 °C, or +0.02 ohm per degree Celsius. Between temperatures of +20 °C and +70 °C, therefore, this component has a temperature coefficient of +0.02 ohm per degree Celsius, assuming the resistance-versus-temperature function is linear over that range of temperatures.