Embedding Non-planar Graphs: Storage and Representation

In this paper, we propose a convention for repre-senting non-planar graphs and their least-crossing embeddings in a canonical way. We achieve this by using state-of-the-art tools such as canonical labelling of graphs, Nauty’s Graph6 string and combinatorial representations for planar graphs. To the best of our knowledge, this has not been done before. Besides, we implement the men-tioned procedure in a SageMath language and compute embeddings for certain classes of cubic, vertex-transitive and general graphs. Our main contribution is an extension of one of the graph data sets hosted on MathDataHub, and towards extending the SageMath codebase.

The Drosophila Mushroom Body: From Architecture to Algorithm in a Learning Circuit

The Drosophila brain contains a relatively simple circuit for forming Pavlovian associations, yet it achieves many operations common across memory systems. Recent advances have established a clear framework for Drosophila learning and revealed the following key operations: a) pattern separation, whereby dense combinatorial representations of odors are preprocessed to generate highly specific, nonoverlapping odor patterns used for learning; b) convergence, in which sensory information is funneled to a small set of output neurons that guide behavioral actions; c) plasticity, where changing the mapping of sensory input to behavioral output requires a strong reinforcement signal, which is also modulated by internal state and environmental context; and d) modularization, in which a memory consists of multiple parallel traces, which are distinct in stability and flexibility and exist in anatomically well-defined modules within the network. Cross-module interactions allow for higher-order effects where past experience influences future learning. Many of these operations have parallels with processes of memory formation and action selection in more complex brains.

Slow presynaptic mechanisms that mediate adaptation in the olfactory pathway of Drosophila

The olfactory system encodes odor stimuli as combinatorial activity of populations of neurons whose response depends on stimulus history. How and on which timescales previous stimuli affect these combinatorial representations remains unclear. We use in vivo optical imaging in Drosophila to analyze sensory adaptation at the first synaptic step along the olfactory pathway. We show that calcium signals in the axon terminals of olfactory receptor neurons (ORNs) do not follow the same adaptive properties as the firing activity measured at the antenna. While ORNs calcium responses are sustained on long timescales, calcium signals in the postsynaptic projection neurons (PNs) adapt within tens of seconds. We propose that this slow component of the postsynaptic response is mediated by a slow presynaptic depression of vesicle release and enables the combinatorial population activity of PNs to adjust to the mean and variance of fluctuating odor stimuli.

Combinatorial representations for the scheme of allocations of distinguishable particles into indistinguishable cells

For a scheme of allocation of distinguishable particles into indistinguishable cells we describe types of representation, numbering and enumerating of its outcomes in terms of the transition graph; this graph allows, in particular, to find the distribution on the set of outcomes. Several methods of statistical simulation of scheme outcomes are described.

On Offer Shai’s Contribution to Mechanical Engineering and Design

This paper presents the contribution of Offer Shai to mechanical engineering and design. Over a period of three decades Shai has created an impressive research program that is founded on solid mathematical grounds — combinatorial representations of systems. On this foundation he made contributions that ranged from inventing new concepts in mechanics (e.g., face force), new ways to characterize systems (e.g., singularity positions), new ways to create building blocks to model discrete systems (e.g., Assur graphs and their synthesis), and new methods in design (e.g., infused design). This paper summarizes some of these contributions in an attempt to describe the breadth and depth and attract researchers to continue develop his ideas.

A LOWER BOUND FOR THE NUMBER OF FORBIDDEN MOVES TO UNKNOT A LONG VIRTUAL KNOT

Nelson and Kanenobu showed that forbidden moves unknot any virtual knot. Similarly a long virtual knot can be unknotted by a finite sequence of forbidden moves. Goussarov, Polyak and Viro introduced finite type invariants of virtual knots and long virtual knots and gave combinatorial representations of finite type invariants. We introduce Fn-moves which generalize the forbidden moves. Assume that two long virtual knots K and K′ are related by a finite sequence of Fn-moves. We show that the values of the finite type invariants of degree 2 of K and K′ are congruent modulo n and give a lower bound for the number of Fn-moves needed to transform K to K′.