Largest minimally inversion-complete and pair-complete sets of permutations

We solve two related extremal problems in the theory of permutations. A set Q of permutations of the integers 1 to n is inversion-complete (resp., pair-complete) if for every inversion (j; i), where 1 j), where i 6= j), there exists a permutation in Q where j is before i. It is minimally inversion-complete if in addition no proper subset of Q is inversion-complete; and similarly for pair-completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion- complete set of permutations, and that of a minimal pair-complete set of permutations. The latter problem arises in the determination of the Caratheodory numbers for certain abstract convexity structures on the (n1)-dimensional real and integer vector spaces. Using Mantel's Theorem on the maximum number of edges in a triangle-free graph, we determine these two maximum cardinalities and we present a complete description of the optimal sets of permutations for each problem. Perhaps surprisingly (since there are twice as many pairs to cover as inversions), these two maximum cardinalities coincide when ever n>=4

Largest minimal inversion-complete and pair-complete sets of permutations

We solve two related extremal problems in the theory of permutations. A set $Q$ of permutations of the integers 1 to $n$ is inversion-complete (resp., pair-complete) if for every inversion $(j,i)$, where 1 \le i \textless{} j \le n, (resp., for every pair $(i,j)$, where $i\not= j$) there exists a permutation in~$Q$ where $j$ is before~$i$. It is minimally inversion-complete if in addition no proper subset of~$Q$ is inversion-complete; and similarly for pair-completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion-complete set of permutations, and that of a minimal pair-complete set of permutations. The latter problem arises in the determination of the Carath\'eodory numbers for certain abstract convexity structures on the $(n-1)$-dimensional real and integer vector spaces. Using Mantel's Theorem on the maximum number of edges in a triangle-free graph, we determine these two maximum cardinalities and we present a complete description of the optimal sets of permutations for each problem. Perhaps surprisingly (since there are twice as many pairs to cover as inversions), these two maximum cardinalities coincide whenever $n \ge 4$

Strategies for identifying exact structure of neural circuits with broad light microscopy connectivity probes

Dissecting the structure of neural circuits in the brain is one of the central problems of neuroscience. Until present day, the only way to obtain complete and detailed reconstructions of neural circuits was thought to be the serial section Electron Microscopy, which could take decades to complete a small circuit. In this paper, we develop a mathematical framework that allows performing such reconstructions much faster and cheaper with existing light microscopy and genetic tools. In this framework, a collection of genetically targeted light probes of connectivity is prepared from different animals and then used to systematically deduce the circuit's connectivity. Each measurement is represented as mathematical constraint on the circuit architecture. Such constraints are then computationally combined to identify the detailed connectivity matrix for the probed circuit. Connectivity here is understood broadly, such as that between different identifiable neurons or identifiable classes of neurons, etc. This paradigm may be applied with connectivity probes such as ChR2-assisted circuit mapping, GRASP or transsynaptic viruses, and genetic targeting techniques such as Brainbow, MARCM/MADM or UAS/Gal4, in model organisms such as C. Elegans, Drosophila, zerbafish, mouse, etc. In particular, we demonstrate how, by using this paradigm, the wiring diagram between all neurons in C. Elegans may be reconstructed with GRASP and Brainbow and off-the-shelf light microscopy tools in the time span of one week or less. Described approach allows recovering exact connectivity matrix even if neurons may not be targeted individually in ~Np*log(N) time (Np is the number of nonzero entries and N is the size of the connectivity matrix). For comparison, the minimal time that would be necessary to determine connectivity matrix directly by probing connections between individual neurons when one knows a-priory which pairs should be tested, e.g. with whole-cell patches, is ~Np

Monodromy transform and the integral equation method for solving the string gravity and supergravity equations in four and higher dimensions

The monodromy transform and corresponding integral equation method described here give rise to a general systematic approach for solving integrable reductions of field equations for gravity coupled bosonic dynamics in string gravity and supergravity in four and higher dimensions. For different types of fields in space-times of $D\ge 4$ dimensions with $d=D-2$ commuting isometries -- stationary fields with spatial symmetries, interacting waves or partially inhomogeneous cosmological models, the string gravity equations govern the dynamics of interacting gravitational, dilaton, antisymmetric tensor and any number $n\ge 0$ of Abelian vector gauge fields (all depending only on two coordinates). The equivalent spectral problem constructed earlier allows to parameterize the infinite-dimensional space of local solutions of these equations by two pairs of \cal{arbitrary} coordinate-independent holomorphic $d\times d$- and $d\times n$- matrix functions ${\mathbf{u}_\pm(w), \mathbf{v}_\pm(w)}$ of a spectral parameter $w$ which constitute a complete set of monodromy data for normalized fundamental solution of this spectral problem. The "direct" and "inverse" problems of such monodromy transform --- calculating the monodromy data for any local solution and constructing the field configurations for any chosen monodromy data always admit unique solutions. We construct the linear singular integral equations which solve the inverse problem. For any \emph{rational} and \emph{analytically matched} (i.e. $\mathbf{u}_+(w)\equiv\mathbf{u}_-(w)$ and $\mathbf{v}_+(w)\equiv\mathbf{v}_-(w)$) monodromy data the solution for string gravity equations can be found explicitly. Simple reductions of the space of monodromy data leads to the similar constructions for solving of other integrable symmetry reduced gravity models, e.g. 5D minimal supergravity or vacuum gravity in $D\ge 4$ dimensions.Comment: RevTex 7 pages, 1 figur

On the hardness of switching to a small number of edges

Seidel's switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other one by a sequence of switches. Jel\'inkov\'a et al. [DMTCS 13, no. 2, 2011] presented a proof that it is NP-complete to decide if the input graph can be switched to contain at most a given number of edges. There turns out to be a flaw in their proof. We present a correct proof. Furthermore, we prove that the problem remains NP-complete even when restricted to graphs whose density is bounded from above by an arbitrary fixed constant. This partially answers a question of Matou\v{s}ek and Wagner [Discrete Comput. Geom. 52, no. 1, 2014].Comment: 19 pages, 7 figures. An extended abstract submitted to COCOON 201

The complexity of classification problems for models of arithmetic

We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.Comment: 15 page