THE RAMSEY NUMBERS OF LARGE CYCLES VERSUS SMALL
... F : |NX (aj )| ≥ 2 and |NX (aj+1 )| ≥ 2, and without loss of generality we may assume aj is adjacent to x1 and x3 , and aj+1 is adjacent to x2 and x4 . This implies x1 x3 , x2 x4 , x0 aj+2 , x0 aj−1 ∈ E(F ) since otherwise F ⊇ W5 (Note that F 6⊇ Ck implies neither of aj−1 and aj+2 is adjacent to a v ...
... F : |NX (aj )| ≥ 2 and |NX (aj+1 )| ≥ 2, and without loss of generality we may assume aj is adjacent to x1 and x3 , and aj+1 is adjacent to x2 and x4 . This implies x1 x3 , x2 x4 , x0 aj+2 , x0 aj−1 ∈ E(F ) since otherwise F ⊇ W5 (Note that F 6⊇ Ck implies neither of aj−1 and aj+2 is adjacent to a v ...
MTH304 - National Open University of Nigeria
... of confusion: 2 + 3 = 5 stands for (2,1)+(3,1)=(5,1). The equation 3x = 8 that started this all may then be interpreted as shorthand for the equation (3,1)(u, v)=(8,1), and one easily verifies that x =(u, v)=(8,3) is a solution. Now, if someone runs at you in the night and hands you a note with 5 wr ...
... of confusion: 2 + 3 = 5 stands for (2,1)+(3,1)=(5,1). The equation 3x = 8 that started this all may then be interpreted as shorthand for the equation (3,1)(u, v)=(8,1), and one easily verifies that x =(u, v)=(8,3) is a solution. Now, if someone runs at you in the night and hands you a note with 5 wr ...
ON THE LIMIT POINTS OF THE FRACTIONAL PARTS OF
... and so on. All Sn , n = 2, 3, . . . , are divisible by 4. Hence Yn = Sn + Sn+1 modulo 4 is equal to 2 for n = 1 and to zero for all non-negative n 6= 1. Suppose that θ > 1 solves z 3 − z − 1 = 0. Then θ is a Pisot number having a pair complex conjugates inside the unit circle. Clearly, P (1) = −1. T ...
... and so on. All Sn , n = 2, 3, . . . , are divisible by 4. Hence Yn = Sn + Sn+1 modulo 4 is equal to 2 for n = 1 and to zero for all non-negative n 6= 1. Suppose that θ > 1 solves z 3 − z − 1 = 0. Then θ is a Pisot number having a pair complex conjugates inside the unit circle. Clearly, P (1) = −1. T ...
SINGULAR CONTINUOUS SPECTRUM OF HALF
... 3. SUFFICIENT CONDITIONS FOR σsc 6= ∅ AND FOR σ = σsc = R+ In this section we give sufficient conditions on X and α for the operators Hδ,X,α and Hδ0 ,X,α to have non-empty singular continuous spectra and to have even purely singular continuous spectra. Finally, we give the proof of Theorem 1.1 formu ...
... 3. SUFFICIENT CONDITIONS FOR σsc 6= ∅ AND FOR σ = σsc = R+ In this section we give sufficient conditions on X and α for the operators Hδ,X,α and Hδ0 ,X,α to have non-empty singular continuous spectra and to have even purely singular continuous spectra. Finally, we give the proof of Theorem 1.1 formu ...
Chapter 9 Infinite Series
... Chapter Comments You may want to think of this chapter as two parts. Part I (Sections 9.1 through 9.6) covers sequences and series of constant terms and Part II (Sections 9.7 through 9.10) covers series with variable terms. Part I should be covered quickly so that most of your time in this chapter i ...
... Chapter Comments You may want to think of this chapter as two parts. Part I (Sections 9.1 through 9.6) covers sequences and series of constant terms and Part II (Sections 9.7 through 9.10) covers series with variable terms. Part I should be covered quickly so that most of your time in this chapter i ...
ON TOPOLOGICAL NUMBERS OF GRAPHS 1. Introduction
... conditions for a graph to have set-graceful and set-sequential labeling. In 1999 Acharya and Hegde putforward many problems regarding set-valuation of graphs in [6]. A new momentum to this area of study was triggered by Acharya [3] in 2001. Many authors [4, 9, 13] later investigated various aspects ...
... conditions for a graph to have set-graceful and set-sequential labeling. In 1999 Acharya and Hegde putforward many problems regarding set-valuation of graphs in [6]. A new momentum to this area of study was triggered by Acharya [3] in 2001. Many authors [4, 9, 13] later investigated various aspects ...
Here - Dartmouth Math Home
... In how many ways... • can we pick 6 numbers from 1 to 15 so that no two are consecutive? • can we climb a ladder if we move up either one or two steps at a time? • can 7 balls be placed in 4 boxes if no box is to be left empty? • can we give change of a pound? As suggested by the title of the course ...
... In how many ways... • can we pick 6 numbers from 1 to 15 so that no two are consecutive? • can we climb a ladder if we move up either one or two steps at a time? • can 7 balls be placed in 4 boxes if no box is to be left empty? • can we give change of a pound? As suggested by the title of the course ...
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals.