background: we conducted a retrospective analysis to evaluate the impact of omission of supraclavicular radiotherapy on supraclavicular failure rate and treatment outcomes in n1 breast cancer patients with evaluation of prognostic factors that affected supraclavicular recurrence free survival. methods: this study analyzed the medical records of 109 patients with n1 breast cancer. all patients u...

In this paper, we investigate the global behavior of the difference equation of order three xn+1 = α+ xn−2 xn , n = 0, 1, . . . where the parameters α, k ∈ (0,∞) and the initial values x−2, x−1 and x0 are arbitrary positive real numbers.

In this paper, existence criteria for single and multiple positive solutions of periodic boundary value problems for first order difference equations of the form

and Applied Analysis 3 After k steps we obtain the following formula xn A ⎛ ⎝ A x r/p n−k ⎛ ⎝ A x r/p2 n−k x r/p n−k−1 ⎛ ⎝ A x r/p3 n−k x r/p2 n−k−1x r/p n−k−2

The paper investigates relationships between algebraic expressions and graphs. Using the decomposition method we generate special simultaneous systems of linear recurrences for sizes of graph expressions. We propose techniques which provide closed-form solutions for these systems.