By Matthias Beck, Sinai Robins
ISBN-10: 0387291393
ISBN-13: 9780387291390
This textbook illuminates the sphere of discrete arithmetic with examples, idea, and functions of the discrete quantity of a polytope. The authors have weaved a unifying thread via easy but deep rules in discrete geometry, combinatorics, and quantity idea.
We come across right here a pleasant invitation to the sphere of "counting integer issues in polytopes", and its quite a few connections to effortless finite Fourier research, producing features, the Frobenius coin-exchange challenge, sturdy angles, magic squares, Dedekind sums, computational geometry, and extra.
With 250 workouts and open difficulties, the reader appears like an lively player.
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Additional info for Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra (Undergraduate Texts in Mathematics)
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41. , in light of the Morales–Denham theorem mentioned in the Notes. , arithmetic sequences. 42. For which 0 ≤ n ≤ b − 1 is sn (a1 , a2 , . . , ad ; b) = 0? 2 A Gallery of Discrete Volumes Few things are harder to put up with than a good example. Mark Twain (1835–1910) A unifying theme of this book is the study of the number of integer points in polytopes, where the polytopes lives in a real Euclidean space Rd . The integer points Zd form a lattice in Rd , and we often call the integer points lattice points.
Md ) ∈ Zd : all mj > 0, m1 a1 + · · · + md ad = n ; that is, p◦A (n) counts the number of partitions of n using only the elements of A as parts, where each part is used at least once. Find formulas for p◦A for A = {a} , A = {a, b} , A = {a, b, c} , A = {a, b, c, d}, where a, b, c, d are pairwise relatively prime positive integers. Observe that in all examples, the counting functions pA and p◦A satisfy the algebraic relation p◦A (−n) = (−1)d−1 pA (n) . 32. Prove that p◦A (n) = pA (n − a1 − a2 − · · · − ad ).
Our pyramid P that started this section is a pyramid over the unit (d − 1)cube, and so EhrP (z) = 1 1−z d−1 k=1 d−1 k=1 A (d − 1, k) z k−1 = (1 − z)d A (d − 1, k) z k−1 . 22). Let’s summarize what we have proved for the pyramid over the unit cube. 5. Let P be the d-pyramid P = (x1 , x2 , . . , xd ) ∈ Rd : 0 ≤ x1 , x2 , . . , xd−1 ≤ 1 − xd ≤ 1 . (a) The lattice-point enumerator of P is the polynomial LP (t) = 1 (Bd (t + 2) − Bd ) . d P (b) Its evaluation at negative integers yields (−1)d LP (−t) = LP ◦ (t).
Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra (Undergraduate Texts in Mathematics) by Matthias Beck, Sinai Robins
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