A new submodule clustering method via sparse and lowrank representation for multi-way data is proposed in this paper. Instead of reshaping multi-way data into vectors, this method maintains their natural orders to preserve data intrinsic structures, e.g., image data kept as matrices. To implement clustering, the multi-way data, viewed as tensors, are represented by the proposed tensor sparse an...

The Normalized Cut is a widely used measure of separation between clusters in a graph. In this paper we provide a novel probabilistic perspective on this measure. We show that for a partition of a graph into two weakly connected sets, V = A ⊎ B, in fact Ncut(V ) = 1/τA→B + 1/τB→A, where τA→B is the uni-directional characteristic exit time of a random walk from subset A to subset B. Using matrix...

‎Let $R$ be a domain with quotiont field $K$‎, ‎and‎ ‎let $N$ be a submodule of an $R$-module $M$‎. ‎We say that $N$ is‎ ‎powerful (strongly primary) if $x,yin K$ and‎ ‎$xyMsubseteq N$‎, ‎then $xin R$ or $yin R$ ($xMsubseteq N$‎ ‎or $y^nMsubseteq N$ for some $ngeq1$)‎. ‎We show that a submodule‎ ‎with either of these properties is comparable to every prime‎ ‎submodule of $M$‎, ‎also we show tha...

Jiang and Su and (independently) Elliott discovered a simple, nuclear, infinite-dimensional C-algebra Z having the same Elliott invariant as the complex numbers. For a nuclear C-algebra A with weakly unperforated K∗-group the Elliott invariant of A ⊗ Z is isomorphic to that of A. Thus, any simple nuclear C-algebra A having a weakly unperforated K∗-group which does not absorb Z provides a counte...

Let M be a right module over a ring R. In this manuscript, we shall study on a special case of F-inverse split modules where F is a fully invariant submodule of M introduced in [12]. We say M is Z 2(M)-inverse split provided f^(-1)(Z2(M)) is a direct summand of M for each endomorphism f of M. We prove that M is Z2(M)-inverse split if and only if M is a direct...

Normalized cut is a widely used measure of separation between clusters in a graph. In this paper we provide a novel probabilistic perspective on this measure. We show that for a partition of a graph into two weakly connected sets, V = A B, the multiway normalized cut is approximately MNcut ≈ 1/τA→B + 1/τB→A, where τA→B is the unidirectional characteristic exit time of a random walk from subset ...

We describe the structure, including composition factors and submodule lattices, of cross-characteristic permutation modules for the natural actions of the orthogonal groups O± m(3) with m ≥ 6 on nonsingular points of their standard modules. These actions together with those studied in [2] are all examples of primitive rank 3 actions of finite classical groups on nonsingular points.

Given a Young diagram λ and the set Hλ of partitions of {1, 2, . . . , |λ|} of shape λ, we analyze a particular S|λ|-module homomorphism QH λ → QHλ ′ to show that QHλ is a submodule of QHλ ′ whenever λ is a hook (n, 1, 1, . . . , 1) with m rows, n ≥ m, or any diagram with two rows.

In the present paper we consider recursive games that satisfy an absorbing property defined by Vieille. We give two sufficient conditions for existence of an equilibrium payoff in such games, and prove that if the game has at most two non-absorbing states, then at least one of the conditions is satisfied. Using a reduction of Vieille, we conclude that every stochastic game which has at most two...