This article offers a simple but rigorous proof of Brouwer’s fixed point theorem using Sperner’s Lemma.The general method I have used so far in the proof is mainly to convert the n-dimensional shapes to the correspo...This article offers a simple but rigorous proof of Brouwer’s fixed point theorem using Sperner’s Lemma.The general method I have used so far in the proof is mainly to convert the n-dimensional shapes to the corresponding case under the Sperner’s Labeling and apply the Sperner’s Lemma to solve the question.展开更多
We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlinearity is superlinear at both the origin and +infinity. Our method of proof combines the Kneser’s theorem wit...We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlinearity is superlinear at both the origin and +infinity. Our method of proof combines the Kneser’s theorem with the well-known from combinatorial topology Sperner’s lemma. We also notice that our geometric approach is strongly based on the associated vector field.展开更多
Let m, n, S_1, S_2, …, S_n, be non-negative integers with 0≤m≤n. Assume μ(S_1, S_2, …, S_n)={(a_1, a_2, …, a_n)|0≤a_i≤S_i for each i} is a poser, Where (a_1, a_2, …, a_n)<(b_1, b_2, …, b_n) if and only if...Let m, n, S_1, S_2, …, S_n, be non-negative integers with 0≤m≤n. Assume μ(S_1, S_2, …, S_n)={(a_1, a_2, …, a_n)|0≤a_i≤S_i for each i} is a poser, Where (a_1, a_2, …, a_n)<(b_1, b_2, …, b_n) if and only if a_i<b_i for all i. A subset of μ(s_1, s_2, …, S_n) is called a two-part Sperner family in μ(s_1, s_2, …, s_n) if for any a=(a_1, a_2, …, an), b=(b_1, b_2, …, b_n) ∈μ(s_1, s_2, …, s_n), (i) a_i=b_i(1≤i≤m) and a_i≤b_i(m+1≤i≤n) imply a_i=b_i for all i, and (ⅱ) a_i≤b_i(1≤i≤m) and a_i=b_i(m+1≤i≤n) imply a_i=b_i for all i.In this paper, we prove that if is a two-part Sperner family in μ(s_1, s_2,…, s_n), then展开更多
Letn andk be arbitrary positive integers,p a prime number and L(k n)(p) the subgroup lattice of the Abelianp-group (Z/p k ) n . Then there is a positive integerN(n,k) such that whenp N(n,k),L (k N )(p) has the strong ...Letn andk be arbitrary positive integers,p a prime number and L(k n)(p) the subgroup lattice of the Abelianp-group (Z/p k ) n . Then there is a positive integerN(n,k) such that whenp N(n,k),L (k N )(p) has the strong Sperner property.展开更多
基金by Dr Kemp from National Mathematics and Science College.
文摘This article offers a simple but rigorous proof of Brouwer’s fixed point theorem using Sperner’s Lemma.The general method I have used so far in the proof is mainly to convert the n-dimensional shapes to the corresponding case under the Sperner’s Labeling and apply the Sperner’s Lemma to solve the question.
文摘We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlinearity is superlinear at both the origin and +infinity. Our method of proof combines the Kneser’s theorem with the well-known from combinatorial topology Sperner’s lemma. We also notice that our geometric approach is strongly based on the associated vector field.
文摘Let m, n, S_1, S_2, …, S_n, be non-negative integers with 0≤m≤n. Assume μ(S_1, S_2, …, S_n)={(a_1, a_2, …, a_n)|0≤a_i≤S_i for each i} is a poser, Where (a_1, a_2, …, a_n)<(b_1, b_2, …, b_n) if and only if a_i<b_i for all i. A subset of μ(s_1, s_2, …, S_n) is called a two-part Sperner family in μ(s_1, s_2, …, s_n) if for any a=(a_1, a_2, …, an), b=(b_1, b_2, …, b_n) ∈μ(s_1, s_2, …, s_n), (i) a_i=b_i(1≤i≤m) and a_i≤b_i(m+1≤i≤n) imply a_i=b_i for all i, and (ⅱ) a_i≤b_i(1≤i≤m) and a_i=b_i(m+1≤i≤n) imply a_i=b_i for all i.In this paper, we prove that if is a two-part Sperner family in μ(s_1, s_2,…, s_n), then
文摘Letn andk be arbitrary positive integers,p a prime number and L(k n)(p) the subgroup lattice of the Abelianp-group (Z/p k ) n . Then there is a positive integerN(n,k) such that whenp N(n,k),L (k N )(p) has the strong Sperner property.
