# residue

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## residue

Law what is left of an estate after the discharge of debts and distribution of specific gifts
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Residue

(1) In the theory of numbers, number a is the residue of number b modulo m if the difference a — b is divisible by m (a, b, and m > 0 are integers). For example, the number 24 is the residue of the number 3 modulo 7, since 24 - 3 is divisible by 7. The set of m integers, each of which is the residue of one and only one of the numbers 0, 1, . . . , m—1, is said to be a complete residue system modulo m. For example, the numbers 1, 6, 11, 16, 21, 26 form the complete residue system modulo 6. The number a is called a residue of order n (n≥ 2—an integer) modulo m if there exists an integer x such that the difference xn - a is divisible by m. If this is not the case, then a is called a nonresidue of order n. For example, 2 and 3 are a residue and a nonresidue of the second order (quadratics) modulo 7.

### REFERENCE

Vinogradov, I. M. Osnovy teorii chisel, 7th ed. Moscow, 1965.

A. A. KARATSUBA

(2) In the theory of analytic functions, the residue of a single-valued analytic function f(z) at an isolated singular point Z0 is the coefficient of (z - Z0)—1 in the expansion of f(z) in a series of powers of z - Z0 (Laurent series) valid in a neighborhood of Z0. The notation used is f(z). If Y is a circle with center at Z0 and sufficiently small radius (a circle so small that within it the function f(z) has no singular points other than Z0), then

The importance of residues stems from the following theorem. Let f(Z) be a single-valued function analytic in a domain D, with the exception of isolated singular points, and let G be a simple closed rectifiable curve belonging to domain D together with its interior and not passing through the singular points of f(z); if Z1, . . . , zn are all the singular points of f(z) lying within G, then

Since it is relatively simple to calculate residues, this theorem is an effective means for evaluating integrals.

A. A. GONCHAR

## residue

[′rez·ə‚dü]
(chemical engineering)
The substance left after distilling off all but the heaviest components from crude oil in petroleum refinery operations. Also known as bottoms; residuum.
Solids deposited onto the filter medium during filtration. Also known as cake; discharged solids.
(geology)
The in-place accumulation of rock debris which remains after weathering has removed all but the least soluble constituent.
(mathematics)
The residue of a complex function ƒ(z) at an isolated singularity z0 is given by (1/2π i) ∫ ƒ(z) dz along a simple closed curve interior to an annulus about z0; equivalently, the coefficient of the term (z-z0)-1 in the Laurent series expansion of ƒ(z) about z0.
In general, a coset of an ideal in a ring.
A residue of m of order n, where m and n are integers, is a remainder that results from raising some integer to the n th power and dividing by m.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
Depending on the bauxite ore used and operational parameters within the Bayer process, residues may be dominated by a coarse-textured fraction (residue sand) or a fine-textured residue (red mud) (Courtney and Timpson 2005b; Wehr et al.
Despite the numerous studies that have been undertaken to understand the properties of residue and identify strategies that allow a vegetation cover to be established in this 'plant-hostile' material, procedures to monitor and evaluate rehabilitation performance are scarce in the published literature.
Dr Masood Arshad, Director Water, Climate and Energy, WWF-Pakistan, concluded the event and thanked speakers and participants in developing concrete steps for future actions to control crop residue burning and smog.
Dr Ali Alzarouni, EGA's executive vice president Midstream, said: "We are now moving into the next stage of this important research on the use of bauxite residue, with detailed work on specific potential applications in construction.
Bauxite residue is a by-product of alumina refining - the process of turning bauxite ore into alumina, which is the feedstock for aluminium smelters.
The emissions from residue burning of other crops have not been estimated due to the fact that since smog usually happens in October and November and these two months only involve burning of rice residue.
In carrying out the study the focus group, in collaboration with agriculture specialist from the Punjab Agriculture Department, held discussions with farmers which helped understand the reasons, and get insights about the trends in residue burning practices.
"Such type of campaign is necessary to sensitize the farmers about management for residue. Several efforts are being made to provide the various residue management implements on subsidy to the farmers.
Underscoring the impact of crop residue burning, he said crop burning leads to air, land and water pollution.
"burning of rice residue is a big problem for Pakistan as well South Asian countries and Pak Seeder is breakthrough for rice-wheat growers of Pakistan, which will not only save the time and resources but also enhance our crop production", official said.
Summary: This research work was designed to estimate the gradually increasing use of insecticides on vegetables to create awareness among the consumers regarding their insecticide residues. The main aim of this study is to highlight the harmful effects of insecticides in comparison with dietary intake assessment of vegetables.
For residues determination of insecticides, 16 samples of multi-floral raw honey of Apis dorsata F.

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