The singular value decomposition is widely used in signal processing and data mining. Since the data often arrives in a stream, the problem of updating matrix decompositions under low-rank modification has been widely studied. Brand developed a technique in 2006 that has many advantages. However, the technique does not directly approximate the updated matrix, but rather its previous low-rank ap...

We exploit the critical locus structure on Quot scheme $\mathrm{Quot}_{\mathbb A^3}(\mathscr O^{\oplus r},n)$, in particular associated symmetric obstruction theory, order to define rank $r$ K-theoretic Donaldson-Thomas invariants of Calabi-Yau $3$-fold $\mathbb A^3$. compute partition function as a plethystic exponential, proving conjecture proposed string theory by Awata-Kanno and Benini-Bone...

Let L be a finite geometric lattice of rank r, and for i =0, I, •• •» r, let W. denote the number of elements of L with rank i. F or 1 < k < r 2, we have W, + W. + • • • + V, < W ,+•••+ W ., + W , 12 fe r—k r—l r—1 with equality if and only if the lattice L is modular. We give two further results concerning matchings of lattice elements of rank < k into those of rank > r — k, and observe that a...

We consider relative error low rank approximation of tensors with respect to the Frobenius norm. Namely, given an order-q tensor A ∈ R ∏q i=1 ni , output a rank-k tensor B for which ‖A − B‖F ≤ (1 + ) OPT, where OPT = infrank-k A′ ‖A − A‖F . Despite much success on obtaining relative error low rank approximations for matrices, no such results were known for tensors. One structural issue is that ...

Starting from Katsuno and Mendelzon postulates, a new set of postulates which is minimal and complete has been deened, these postulates characterize an update operator. The update operator class which is obtained is larger than Katsuno and Mendelzon's one, since it includes updating operators which are not inert, and which allow for the existence of unreachable states. The main property is that...