Maximizing the number of nonnegative subsets, SIAM J. Discrete
... partitions A = A1 ∪· · ·∪Ak and B = B1 ∪· · ·∪Bl such that, for every i ∈ [k], j ∈ [l], in the induced bipartite graph G[Ai , Bj ] all the vertices in Ai have equal degrees and all the vertices in Bj have equal degrees too. Define an auxiliary bipartite graph H on the same vertex set, and replace ev ...
... partitions A = A1 ∪· · ·∪Ak and B = B1 ∪· · ·∪Bl such that, for every i ∈ [k], j ∈ [l], in the induced bipartite graph G[Ai , Bj ] all the vertices in Ai have equal degrees and all the vertices in Bj have equal degrees too. Define an auxiliary bipartite graph H on the same vertex set, and replace ev ...
ABSOLUTE VALUE INEQUALITIES Chapter 1 Section 6
... • NO! Absolute value is always positive. • Cases: ...
... • NO! Absolute value is always positive. • Cases: ...
Prime Factors
... remainders) by itself and the number 1. Examples: Let’s look at the factors of several numbers. ...
... remainders) by itself and the number 1. Examples: Let’s look at the factors of several numbers. ...
Comparing and Ordering Fractions - Mendenhall-Jr-PLC
... than (<). To do this, multiply the numerator in the first fraction by the denominator in the second fraction (2 x 4). Write the product above or below the first fraction. Then multiply the numerator in the second fraction by the denominator of the first fraction (3 x 3). Write the product above or b ...
... than (<). To do this, multiply the numerator in the first fraction by the denominator in the second fraction (2 x 4). Write the product above or below the first fraction. Then multiply the numerator in the second fraction by the denominator of the first fraction (3 x 3). Write the product above or b ...
Chapter 8 Introduction To Number Theory Prime
... • If smallest is m= ø(n) then a is called a primitive root. • If p is prime, then successive powers of a "generate" the group mod p, e.g. for prime number 19; its primitive root are 2,3,10,13,14 and 15. • These are useful but relatively hard to find. ...
... • If smallest is m= ø(n) then a is called a primitive root. • If p is prime, then successive powers of a "generate" the group mod p, e.g. for prime number 19; its primitive root are 2,3,10,13,14 and 15. • These are useful but relatively hard to find. ...
Fractions and Decimals - MakingMathsMarvellous
... Work with ratio may be linked with the Golden Rectangle. Many windows are Golden Rectangles, as are some of the buildings in Athens such as the Parthenon. The ratio of the dimensions of the Golden Rectangle was known to the ancient Greeks: ...
... Work with ratio may be linked with the Golden Rectangle. Many windows are Golden Rectangles, as are some of the buildings in Athens such as the Parthenon. The ratio of the dimensions of the Golden Rectangle was known to the ancient Greeks: ...
Addition
Addition (often signified by the plus symbol ""+"") is one of the four elementary, mathematical operations of arithmetic, with the others being subtraction, multiplication and division.The addition of two whole numbers is the total amount of those quantities combined. For example, in the picture on the right, there is a combination of three apples and two apples together; making a total of 5 apples. This observation is equivalent to the mathematical expression ""3 + 2 = 5"" i.e., ""3 add 2 is equal to 5"".Besides counting fruits, addition can also represent combining other physical objects. Using systematic generalizations, addition can also be defined on more abstract quantities, such as integers, rational numbers, real numbers and complex numbers and other abstract objects such as vectors and matrices.In arithmetic, rules for addition involving fractions and negative numbers have been devised amongst others. In algebra, addition is studied more abstractly.Addition has several important properties. It is commutative, meaning that order does not matter, and it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter (see Summation). Repeated addition of 1 is the same as counting; addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.Performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers; the most basic task, 1 + 1, can be performed by infants as young as five months and even some non-human animals. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day.