Abstract This paper presents a conjecture concerning the minimum possible size of a pair of maximal orthogonal partial Latin squares of a given order n . We show that in the balanced case the optimal structure is formed from a pair of partial Latin squares, each comprising three subsquares whose orders are as close as possible to one another and sum to n . Further results are obtained in unbalanced cases. The problem can be recast in terms of finding the minimum number of blocks in a maximal partial transversal design TD(4, n ), and as finding the minimum number of codewords in an n -ary code of length 4 having minimum distance 3 and covering radius 2. The conjecture is extended to sets of k maximal mutually orthogonal partial Latin squares and hence to n -ary codes of length $$k+2$$ k + 2 , minimum distance $$k+1$$ k + 1 and covering radius k .

Multiple myeloma (MM) is a plasma cell cancer that occurs in the bone marrow. A leading treatment for MM is the monoclonal antibody Daratumumab, targeting the CD38 receptor, which is highly overexpressed in myeloma cells. In this work we model drug resistance via loss of CD38 expression, which is a proposed mechanism of resistance to Daratumumab treatment. We develop an ODE model that includes drug resistance via two mechanisms: a direct effect in which CD38 expression is lost without cell death in response to Daratumumab, and an indirect effect in which CD38 expression switches on and off in the cancer cells; myeloma cells that do not express CD38 have lower fitness but are shielded from the drug action. The model also incorporates competition with healthy cells, death of healthy cells due to off-target drug effects, and a Michaelis-Menten type immune response. Using optimal control theory, we study the effect of the drug resistance mechanisms and the off-target drug effect on the optimal treatment regime. We identify a general increase in the duration and costs of optimal treatment, as a result of these added mechanisms. Several distinct optimal treatment regimes are identified within the parameter space.

Abstract A lower bound is presented for the minimal number of filled cells in a maximal partial Latin hypercube of dimension d and order n. The result generalises and extends previous results for $$d=2$$ d = 2 (Latin squares) and $$d=3$$ d = 3 (Latin cubes). Explicit constructions show that this bound is near-optimal for large $$n> d$$ n > d . For $$d>n$$ d > n , a connection with Hamming codes shows that this lower bound gives a related upper bound for the same quantity. The results can be interpreted in terms of independent dominating sets in certain graphs, and in terms of codes that have covering radius 1 and minimum distance at least 2.

In 1980 Alltop produced a family of cubic phase sequences that nearly meet the Welch bound for maximum non-peak correlation magnitude. This family of sequences were shown by Wooters and Fields to be useful for quantum state tomography. Alltop's construction used a function that is not planar, but whose difference function is planar. In this paper we show that Alltop type functions cannot exist in fields of characteristic 3 and that for a known class of planar functions, x3is the only Alltop type function.