Suppose that K and K' are knots inside the homology spheres Y Y', respectively. Let X = (K, K') be 3-manifold obtained by splicing complements of Z three-manifold 0 surgery on K. When Y' is an L-space, we use formula [1] to show rank (X ) bounded below (Y if τ(K 2 rank( (Z)) − (Y)) + 1 τ(K') ≠ 0.

In this paper, we consider the problem of computing a low-rank factorization of an m× n matrix in the general turnstile update model. We consider both the private and non-private setting. 1. In the non-private setting, we give a space-optimal algorithm that computes a low-rank factorization. Our algorithm maintains three sketches of the matrix instead of five as in Boutsidis et al. (STOC 2016)....

Most existing literature focuses on the exterior temporal rhythm of human movement to infer the functional regions in a city, but they neglects the underlying interdependence between the functional regions and human activities which uncovers more detailed characteristics of regions. In this research, we proposed a novel model based on the low rank approximation (LRA) to detect the functional re...

In this study‎, ‎a new polynomial rank transmutation is proposed with the help of‎ ‎ the idea of quadratic rank transmutation mapping (QRTM)‎. ‎This polynomial rank‎ ‎ transmutation is allowed to extend the range of the transmutation parameter from‎ ‎  [-1,1] to [-1,k]‎‎. ‎At this point‎, ‎the generated distributions gain more&lrm...

Tian and Styan have shown many rank equalities for the sum of two and three idempotent matrices and pointed out that rank equalities for the sum P₁ + ⋯+P k with P₁,…, P k be idempotent (k > 3) are still open. In this paper, by using block Gaussian elimination, we obtained rank equalities for the sum of finitely many idempotent matrices and then solved the open problem mentioned above. Extension...

The positive semidefinite (psd) rank of a polytope is the smallest k for which the cone of k × k real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a polytope is at least the dimension of the polytope plus one, and we characterize those polytopes whose psd rank equals this lower bound.

The rank conjecture says that rank of an elliptic curve is one less the arithmetic complexity of the corresponding non-commutative torus. We prove the conjecture for elliptic curves E(K) over a number field K. As a corollary, one gets a simple estimate for the rank in terms of the length of period of a continued fraction attached to the E(K). As an illustration, we consider a family of elliptic...

A Galois correspondence for finitely generated field extensions k/h is presented in the case characteristic h = p ^ 0. A field extension k/h is Galois if it is modular and h is separably algebraically closed in k. Galois groups are the direct limit of groups of higher derivations having rank a power of p. Galois groups are characterized in terms of abelian iterative generating sets in a manner ...

It has been shown that a best rank-R approximation of an order-k tensor may not exist when R ≥ 2 and k ≥ 3. This poses a serious problem to data analysts using tensor decompositions. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and subtracting best rank-1 approximations. The reason for this is that subtracting a best rank-1 approximati...

Let K denote a compact real symmetric subset of C and let AR(K) denote the real Banach algebra of all real symmetric continuous functions on K which are analytic in the interior K of K, endowed with the supremum norm. We characterize all unimodular pairs (f, g) in AR(K) 2 which are reducible. In addition, for an arbitrary compact K in C, we give a new proof (not relying on Banach algebra theory...