A common approach for solving clustering problems is to design algorithms to approximately optimize various objective functions (e.g., k-means or min-sum) defined in terms of some given pairwise distance or similarity information. However, in many learning motivated clustering applications (such as clustering proteins by function) there is some unknown target clustering; in such cases the pairw...

We prove a result on the length of a longest cycle in a graph on n vertices that contains a 2-factor and satisfies d(u)+ d(v)+d(w)~> n + 2 for every triple u, v, w of independent vertices. As a corollary we obtain the following improvement of a conjecture of H/iggkvist (1992): Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least n-k and ass...

A vector x E Rn is weakly k-majorized by a vector q 6 R^ if the sum of r largest components of x is less than or equal to the sum of r largest components of q for r = 1,2,. . . , k and k < n. In this paper we extend the components of x to their absolute values in the above description and generalize some results in [2] and [3] by G. Dahl and F. Margot.

Let K be a totally real number field with Galois closure L. We prove that if f ∈ Q[x1, . . . , xn] is a sum of m squares in K[x1, . . . , xn], then f is a sum of 4m · 2[L:Q]+1 ([L : Q] + 1 2 ) squares in Q[x1, . . . , xn]. Moreover, our argument is constructive and generalizes to the case of commutative K-algebras. This result gives a partial resolution to a question of Sturmfels on the algebra...

In this paper, we consider the problem of the interference alignment for the K-user SISO interference channel (IC) with blind channel state information (CSI) at transmitters. Our achievement contrary to the traditional K−user interference alignment (IA) scheme has more practical notions. In this case, every receiver is equipped with one reconfigurable antenna which tries to place its desired si...

Abstract. We show that the polynomial Sm,k(A, B), that is the sum of all words in noncommuting variables A and B having length m and exactly k letters equal to B, is not equal to a sum of commutators and Hermitian squares in the algebra R〈X, Y 〉, where X = A and Y 2 = B, for all even values of m and k with 6 ≤ k ≤ m − 10, and also for (m, k) = (12, 6). This leaves only the case (m, k) = (16, 8)...

Let Ak*(n) be the number of positive integers a coprime to n such that the equation a n=1 m1+ } } } +1 mk admits a solution in positive integers (m1 , ..., mk). We prove that the sum of A2*(n) over n x is both >>x log 3 x and also <<x log x. For the corresponding sum where the a's are counted with multiplicity of the number of solutions we obtain the asymptotic formula. We also show that Ak*(n)...

Theorem 2 is the main result in this note. Lemma 1 is key to our main result. It is different from the result in [2] because it does not restrict the number of terms in the partial sum to an even one. Also, our inequalities (2) should be compared with the corresponding result in [2] for k = 2. Lemma 1. Let sn(p) be the nth partial sum of the p-series ∑∞ i=1(1/i), and let k be any integer greate...

A sequence in an additively written abelian group is called a minimal zero-sum sequence if its sum is the zero element of the group and none of its proper subsequences has sum zero. The structure of the longest minimal zero-sum sequences in the group C2 ⊕C2k is known. Their length is equal to 2k + 1. We characterize the minimal zero-sum sequences in C2 ⊕ C2k (k ≥ 3) with lengths at least 2�k/2�...

A subset S of a finite Abelian group G is said to be a sum cover of G if every element of G can be expressed as the sum of two not necessarily distinct elements in S, a strict sum cover of G if every element of G can be expressed as the sum of two distinct elements in S, and a difference cover of G if every element of G can be expressed as the difference of two elements in S. For each type of c...