Proof of Theorem 3.1 for the mixed class
A Simple Proof of Toda's Theorem
... We start with the following three results, all of which have proofs that easily relativize. Theorem 3.1 (Valiant-Vazirani [8]). There is a probabilistic polynomial-time procedure that, given a Boolean formula φ, ...
6
3 The proof of Theorem 1
... Thus, if n > 1 and φ(n)σ(n) = m 2 for some positive integer m, then m < n, so we can write m = n − a for some positive integer a. In this paper, we look at the positive integers a arising in this way. First, ...
10
3 Proof of theorem
... than 1, then M n is isometric to the Riemannian product S k (r) × H n−k (−1/(r 2 + 1)), for some r > ...≥ 3) in Euclidean space R n+1 with constant scalar curvature and with two ...
9
3 Proof of Theorem 3
... ≥ 1 and a graph G , we use kG to denote k independent graphs isomorphic to G ...G 1 , . . . , G r are r graphs and k 1 , ...k 1 G 1 · · · k r G r to denote a set of k 1 + · ...
11
3 Proof of the Theorem 3
... In this paper we obtain existence results for the positive solution of a singular elliptic boundary value problem.. Our study is motivated by the works of Shu [17], Arcoya, Carmona, Leon[r] ...
10
A. Proof of Theorem 1
... We test out the effect of different matrix properties. For vanilla, we sample the matrix entries from a normal distribution. Next, we apply a ReLU operation after each multiplication. For ReLU learned, we used the ...
9
3 Proof of Theorem 1.8
... a 1 , a 2 , . . . , a k and n are positive integers such that n = a 1 +a 2 +· · ·+a k , then the sum a 1 + a 2 + · · · + a k is said to be a partition of n of length k, and a 1 , a 2 , ...+ ...
11
3 Proof of Theorem 6
... from 3 to ...f 1 -heavy if for every pair of vertices u, v ∈ V (H), d H (u, v) = 2 implies max{d(u), d(v)} ≥ (n + ...R-f 1 -heavy if every induced subgraph of G isomorphic to R is f 1 ...P ...
10
3 Proof of Theorem 2.7
... (4) Extension theorems are often used for optimal linear codes problem, especially to prove the nonexistence of linear codes with certain parameters.. Moreover, the extended matrix of G [r] ...
15
3 Proof of the main theorem
... Claim 3 G is the union of stars and at most one double star. Moreover, if H has no double stars, then G also has no double stars. We may assume that k ≥ 4 since otherwise H ∈ F ∞ ∪ {P 4 }. Assume that G has a path ...
10
3 Proof of Theorem 1.2
... A vector ~y ∈ S is called an optimal vector of λ(G) if λ(G, ~y) = λ(G). The following fact is easily implied by the definition of the Lagrange function. Fact 2.1 Let G 1 , G 2 be r-uniform graphs and G 1 ⊂ ...
11
3. Proof of Theorem 2.11
... Both categories Top of topological spaces and SSets of simplicial sets are right (and left) proper [Hi, Theorems 13.1.11, 13.1.13]. Therefore, in both categories the fibrations are h-fibrations, and thus the h-fibrations ...
28
2 Proof of Theorem 1
... combinatorial proof of the rank-unimodality of the poset of order ideals of a product of chains of lengths 2, n, and m, and find a symmetric chain decomposition in the case where n = ...
8
2 Proof of Theorem 1
... n). In Section 4, we describe a somewhat surprising better embedding, with distortion only O(n 5/12 ). This is already optimal according to Proposition 1. Several interesting problems remain open. The obvious ones ...
11
3 The Compactness Theorem (1st proof)
... S 1 ✓ ⌃ # and S 2 ✓ ⌃ # such that neither { } [ S 1 nor {¬ } [ S 2 is ...S 1 [ S 2 ✓ ⌃ # , finite, so has a model, A ...S 1 , or A ✏ ¬ , so A ✏ {¬ } [ S 2 , a ...
6
4. Proof of the theorem
... 22 (1, 1) rr r ! = ∞ whenever p 6= r. The reasoning in the previous lemmas will not help us now, since in Lemma 4.2 we needed c < ∞. As we shall see in Proposition 5.1, the proof for c = ∞ must ...
36
A Proof of Theorem 1.2
... Let S be the set of robustness slack vectors that correspond to feasible solutions. Observe that S is compact. The existence of a lexicographically minimal element in S then follows because any compact subset of R y ...
5
M PROOF OF THE DIVERGENCE THEOREM AND STOKES THEOREM
... Divergence Theorem, we use the same approach as we used for Green’s Theorem; first prove the theorem for rectangular regions, then use the change of variables formula to prove it for regions ...
7
2 Proof of Theorem 1.15
... ≥ 1 and graphs H 1 , . . . , H k , the Gallai- Ramsey number GR(H 1 , ...H 1 = · · · = H k , we simply write GR k ...≥ 3 and k ≥ 2, let G i = P 2i+3 be a path on 2i + 3 ...
13
2 Proof of the main theorem
... dlog 3 ne and B(n) > k, then N (k) is an unbounded function of k. 1 Introduction Coin-weighing puzzles have been abundantly discussed in the mathematical literature over the past 60 years (see, ...
5