a) What are the possible values for the rank of A? Why? Solution By the Rank-Nullity Theorem 0 ≤ rank (A) ≤ min{6, 4} = 4. b) What are the possible values for the dimension of the kernel of A? Why? Solution Since dim(image(A)) ≤ 4, by the Rank-Nullity Theorem 2 ≤ dim ker(A) ≤ 6. c) Suppose the rank of A is as large as possible. What is the dimension of ker(A)? Explain. Solution Since then dim(i...

This paper initiates the study of sets in Euclidean space R (n ≥ 2) that are defined in terms of the dimensions of their elements. Specifically, given an interval I ⊆ [0, 1], we are interested in the connectivity properties of the set DIM consisting of all points in R whose (constructive Hausdorff) dimensions lie in the interval I. It is easy to see that the sets DIM and DIM(n−1,n] are totally ...

In this paper, for any two bipartite ordered sets P and Q; we de ne the convolution P Q of P and Q: For dim(P ) = s and dim(Q) = t; we prove that s+ t (U + V ) 2 dim(P Q) s+t (U+V )+2; where U+V is the max-min integer of the certain realizers. In particular, we also prove that dim(Pn;k) = n+k b n+k 3 c for 2 k n < 2k and dim(Pn;k) = n for n 2k; where Pn;k = Sn Sk is the convolution of two stand...

The antitumor effects of 3,3'-diindolylmethane (DIM) are exhibited in a number of human cancer cells. However, there have been few studies performed concerning the effect of DIM on nasopharyngeal cancer (NPC) cells. In the present study, we examined the in vitro antitumor activity of DIM on the poorly differentiated NPC cell line CNE-2. The potential molecular mechanisms of the activity were al...

In this paper, we determine the dimension of the rectangular product of certain nite lattices. In fact, if L1 and L2 be nite lattices which satisfy the some conditions, then we have dim(L1 L2) = dim(L1) + dim(L2) 1:

It is shown that if $\gamma: [a,b] \to S^2$ $C^3$ with $\det(\gamma, \gamma', \gamma'') \neq 0$, and $A \subseteq \mathbb{R}^3$ a Borel set, then $\dim \pi_{\theta} (A) \geq \min\left\{ 2,\dim A, \frac{ \dim A}{2} + \frac{3}{4} \right\}$ for a.e. $\theta \in [a,b]$, where $\pi_{\theta}$ denotes projection onto the orthogonal complement of $\gamma(\theta)$ ``$\dim$'' refers to Hausdorff dimensio...

Algebraically closed fields, real closed fields, and pseudofinite fields have, for every natural number n, have a finite number of extensions of degree n; we show they share this property with all fields which, like them, satisfy a very basic property of preservation of a type of model-theoretic dimension. This result is attained by showing that a certain action of the group GLn on such a field...

Comparability graphs are graphs which have transitive orientations. The dimension of a poset is the least number of linear orders whose intersection gives this poset. The dimension dim(X) of a comparability graph X is the dimension of any transitive orientation of X, and by k-DIM we denote the class of comparability graphs X with dim(X) ≤ k. It is known that the complements of comparability gra...

Although 3,3'-diindolylmethane (DIM) has been suggested to reduce the risk of colorectal cancer, the underlying biological mechanism is not clearly understood. In the present study, we investigated the effect of DIM on the migratory and invasive activities of the human colorectal cancer cell lines DLD-1 and HCT116. DIM significantly inhibited the migration and invasion of colorectal cancer cell...