Edinah K. Gnang

Anticipatory smooth eye movements with random-dot kinematograms

Anticipatory smooth eye movements were studied in response to expectations of motion of random-dot kinematograms (RDKs). Dot lifetime was limited (52-208 ms) to prevent selection and tracking of the motion of local elements and to disrupt the perception of an object moving across space. Anticipatory smooth eye movements were found in response to cues signaling the future direction of global RDK motion, either prior to the onset of the RDK or prior to a change in its direction of motion. Cues signaling the lifetime of the dots were not effective. These results show that anticipatory smooth eye movements can be produced by expectations of global motion and do not require a sustained representation of an object or set of objects moving across space. At the same time, certain properties of global motion (direction) were more sensitive to cues than others (dot lifetime), suggesting that the rules by which prediction operates to influence pursuit may go beyond simple associations between cues and the upcoming motion of targets.

A Spectral Theory for Tensors

In this paper we propose a general spectral theory for tensors. Our proposed factorization decomposes a tensor into a product of orthogonal and scaling tensors. At the same time, our factorization yields an expansion of a tensor as a summation of outer products of lower order tensors . Our proposed factorization shows the relationship between the eigen-objects and the generalised characteristic polynomials. Our framework is based on a consistent multilinear algebra which explains how to generalise the notion of matrix hermicity, matrix transpose, and most importantly the notion of orthogonality. Our proposed factorization for a tensor in terms of lower order tensors can be recursively applied so as to naturally induces a spectral hierarchy for tensors.

Zeroless arithmetic: representing integers ONLY using ONE

We use recurrence equations (alias difference equations) to enumerate the number of formula representations of positive integers using only addition and multiplication, and using addition, multiplication and exponentiation, where all the inputs are ones. We also describe efficient algorithms for the random generation of such representations, and use dynamical programming to find a shortest possible formula representing any given positive integer.

Counting arithmetic formulas

An arithmetic formula is an expression involving only the constant 1, and the binary operations of addition and multiplication, with multiplication by 1 not allowed. We obtain an asymptotic formula for the number of arithmetic formulas evaluating to n as n goes to infinity, solving a conjecture of E.K.?Gnang and D.?Zeilberger. We give also an asymptotic formula for the number of arithmetic formulas evaluating to n and using exactly k multiplications. Finally we analyze three specific encodings for producing arithmetic formulas. For almost all integers n , we compare the lengths of the arithmetic formulas for n that each encoding produces with the length of the shortest formula for n (which we estimate from below). We briefly discuss the time-space tradeoff offered by each.