In 2003 Cohn and Umans introduced a new group-theoretic framework for doing fast matrix multiplications, with several conjectures that would imply the matrix multiplication exponent ω is 2. Their methods have been used to match one of the fastest known algorithms by Coppersmith and Winograd, which runs in O(n2.376) time and implies that ω ≤ 2.376. This thesis discusses the framework that Cohn a...

A ring Λ is called a tiled order if Λ is a prime Noetherian semi-prefect semi-distributive ring with non-zero Jacobson radical. Any tiled order can be constructed from a (non-necessarily commutative) discrete valuation ring and an exponent matrix. The latter means a square integer matrix A = (αps), whose diagonal entries are equal to zero, and for all possible indices i, j, k, the following ine...

We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov-Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asympto...

We derive explicit tail-estimates for the Jacobian of the solution flow for stochastic differential equations driven by Gaussian rough paths. In particular, we deduce that the Jacobian has finite moments of all order for a wide class of Gaussian process including fractional Brownian motion with Hurst parameter H > 1/4. We remark on the relevance of such estimates to a number of significant open...

‎by considering a degenerate $p(x)-$laplacian equation‎, ‎a generalized compact embedding in weighted variable‎ ‎exponent sobolev space is presented‎. ‎multiplicity of positive solutions are discussed by applying fibering map approach for the‎ ‎corresponding nehari manifold‎.

Without an ultraviolet cut-off, the time evolution of the classical Yang-Mills equations give rise to a never ending cascading of the modes towards the ultraviolet, and ergodic measures and dynamical averages, such as the spectrum of characteristic Lyapunov exponents (measures of temporal chaos) or spatial correlation functions, are ill defined. A lattice regularization (in space) provides an u...

We show that two important quantities from two disparate areas of complexity theory — Strassen’s exponent of matrix multiplication ω and Grothendieck’s constant KG — are intimately related. They are different measures of size for the same underlying object — the matrix multiplication tensor, i.e., the 3-tensor or bilinear operator μl,m,n : Fl×m × Fm×n → Fl×n, (A,B) 7→ AB defined by matrix-matri...

Let X1, . . . , Xm denote smooth projective curves of genus gi ≥ 2 over an algebraically closed field of characteristic 0 and let n denote any integer at least equal to 1 + max i=1 gi. We show that the product JX1 × · · · × JXm of the corresponding Jacobian varieties admits the structure of a Prym-Tyurin variety of exponent n. This exponent is considerably smaller than the exponent of the struc...

We study free massive fermionic ghosts, in the presence of an extended line of impurities, relying on the Lagrangian formalism. We propose two distinct defect interactions, respectively, of relevant and marginal nature. The corresponding scattering theories reveal the occurrence of resonances and instabilities in the former case and the presence of poles with imaginary residues in the latter. C...

Subdiffusive fractional equations are not structurally stable with respect to spatial perturbations to the anomalous exponent (Phys. Rev. E 85, 031132 (2012)). The question arises of applicability of these fractional equations to model real world phenomena. To rectify this problem we propose the inclusion of the random death process into the random walk scheme from which we arrive at the modifi...