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It was shown in 1989 by Delandtsheer and Doyen that, for a $2$-design with $v$ points and block size $k$, a block-transitive group of automorphisms can be point-imprimitive (that is, leave invariant a nontrivial partition of the point set) only if $v$ is small enough relative to $k$. Recently, exploiting a construction of block-transitive point-imprimitive $2$-designs given by Cameron and the last author, four of the authors studied $2$-designs admitting a block-transitive group that preserves a two-dimensional grid structure on the point set. Here we consider the case where there a block-transitive group preserves a multidimensional grid structure on points. We provide necessary and sufficient conditions for such $2$-designs to exist in terms of the parameters of the grid, and certain `array parameters' which describe a subset of points (which will be a block of the design). Using this criterion, we construct explicit examples of $2$-designs for grids of dimensions three and four, and pose several open questions.

In this article, we prove that if $\mathcal{D}$ is a $2$-design with $k=7$ admitting flag-transitive almost simple automorphism group with socle an alternating group, then $\mathcal{D}$ is $PG_{2}(3,2)$ with parameter set $(15,7,3)$ and $G=A_7$, or $\mathcal{D}$ is the $2$-design with parameter set $(55, 7, 1680)$ and $G=A_{11}$ or $S_{11}$.

In this article, we study symmetric designs admitting flag-transitive, point-imprimitive almost simple automorphism groups with socle sporadic simple groups. As a corollary, we present a classification of symmetric designs admitting flag-transitive automorphism group whose socle is a sporadic simple group, and in conclusion, there are exactly seven such designs, one of which admits a point-imprimitive automorphism group and the remaining are point-primitive.

The main aim of this article is to study the quantitative structure of projective symplectic groups $PSp_{4}(q)$ with $q>2$ even. Indeed, we prove that the groups $PSp_{4}(q)$ with $q>2$ even are uniquely determined by their orders and the set of the number of elements of the same order. This result links to the well-known J. G. Thompson's problem (1987) for finite simple groups.

In this paper, we present a classification of $2$-designs with $\gcd(r,\lambda)=1$ admitting flag-transitive automorphism groups. If $G$ is a flag-transitive automorphism group of a non-trivial $2$-design $\mathcal{D}$ with $\gcd(r,\lambda)=1$, then either $(\mathcal{D},G)$ is one of the known examples described in this paper, or $\mathcal{D}$ has $q = p^{d}$ points with $p$ prime and $G$ is a subgroup of $A\Gamma L_{1}(q)$.

We study point-block incidence structures $(\mathcal{P},\mathcal{B})$ for which the point set $\mathcal{P}$ is an $m\times n$ grid. Cameron and the fourth author showed that each block $B$ may be viewed as a subgraph of a complete bipartite graph $\mathbf{K}_{m,n}$ with bipartite parts (biparts) of sizes $m, n$. In the case where $\mathcal{B}$ consists of all the subgraphs isomorphic to $B$, under automorphisms of $\mathbf{K}_{m,n}$ fixing the two biparts, they obtained necessary and sufficient conditions for $(\mathcal{P},\mathcal{B})$ to be a $2$-design, and to be a $3$-design. We first re-interpret these conditions more graph theoretically, and then focus on square grids, and designs admitting the full automorphism group of $\mathbf{K}_{m,m}$. We find necessary and sufficient conditions, again in terms of graph theoretic parameters, for these incidence structures to be $t$-designs, for $t=2, 3$, and give infinite families of examples illustrating that block-transitive, point-primitive $2$-designs based on grids exist for all values of $m$, and flag-transitive, point-primitive examples occur for all even $m$. This approach also allows us to construct a small number of block-transitive $3$-designs based on grids.

This article investigates the large deflection and post-buckling of composite plates by employing the Carrera Unified Formulation (CUF). As a consequence, the geometrically nonlinear governing equations and the relevant incremental equations are derived in terms of fundamental nuclei, which are invariant of the theory approximation order. By using the Lagrange expansion functions across the laminate thickness and the classical finite element (FE) approximation, layer-wise (LW) refined plate models are implemented. The Newton-Raphson linearization scheme with the path-following method based on the arc-length constraint is employed to solve geometrically non-linear composite plate problems. In this study, different composite plates subjected to large deflections/rotations and post-buckling are analyzed, and the corresponding equilibrium curves are compared with the results in the available literature or the traditional FEM-based solutions. The effects of various parameters, such as stacking sequence, number of layers, loading conditions, and edge conditions are demonstrated. The accuracy and reliability of the proposed method for solving the composite plates' geometrically nonlinear problems are verified.

A computational method based on the non-linear Gaussian process (GP), known as deep Gaussian processes (deep GPs) for uncertainty quantification & propagation in modelling of flow through heterogeneous porous media is presented. The method is also used for reducing dimensionality of model output and consequently emulating highly complex relationship between hydrogeological properties and reduced order fluid velocity field in a tractable manner. Deep GPs are multi-layer hierarchical generalisations of GPs with multiple, infinitely wide hidden layers that are very efficient models for deep learning and modelling of high-dimensional complex systems by tackling the complexity through several hidden layers connected with non-linear mappings. According to this approach, the hydrogeological data is modelled as the output of a multivariate GP whose inputs are governed by another GP such that each single layer is either a standard GP or the Gaussian process latent variable model. A variational approximation framework is used so that the posterior distribution of the model outputs associated to given inputs can be analytically approximated. In contrast to the other dimensionality reduction, methods that do not provide any information about the dimensionality of each hidden layer, the proposed method automatically selects the dimensionality of each hidden layer and it can be used to propagate uncertainty obtained in each layer across the hierarchy. Using this, dimensionality of the full input space consists of both geometrical parameters of modelling domain and stochastic hydrogeological parameters can be simultaneously reduced without the need for any simplifications generally being assumed for stochastic modelling of subsurface flow problems. It allows estimation of the flow statistics with greatly reduced computational efforts compared to other stochastic approaches such as Monte Carlo method., Comment: 27 pages, 11 figures, presented in SIAM Conference on Uncertainty Quantification (UQ16)

In this paper, we first study biplanes $\mathcal{D}$ with parameters $(v,k,2)$, where the block size $k\in\{13,16\}$. These are the smallest parameter values for which a classification is not available. We show that if $k=13$, then either $\mathcal{D}$ is the Aschbacher biplane or its dual, or $Aut(\mathcal{D})$ is a subgroup of the cyclic group of order $3$. In the case where $k=16$, we prove that $|Aut(\mathcal{D})|$ divides $2^{7}\cdot 3^{2}\cdot 5\cdot 7\cdot 11\cdot 13$. We also provide an example of a biplane with parameters $(16,6,2)$ with a flag-transitive and point-primitive subgroup of automorphisms preserving a homogeneous cartesian decomposition. This motivated us to study biplanes with point-primitive automorphism groups preserving a cartesian decomposition. We prove that such an automorphism group is either of affine type (as in the example), or twisted wreath type., Comment: 24 pages

In this article, we investigate symmetric $(v,k,\lambda)$ designs $\mathcal{D}$ with $\lambda$ prime admitting flag-transitive and point-primitive automorphism groups $G$. We prove that if $G$ is an almost simple group with socle a finite simple group of Lie type, then $\mathcal{D}$ is either the point-hyperplane design of a projective space $\mathrm{PG}_{n-1}(q)$, or it is of parameters $(7,4,2)$, $(11,5,2)$, $(11,6,2)$ or $(45,12,3)$.

In this article, we study symmetric $(v, k, \lambda)$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle is a projective special unitary group of dimension at most five. We, in particular, determine all such possible parameters $(v, k, \lambda)$ and show that there exist eight non-isomorphic of such designs for which $\lambda\in\{3,6,12, 16, 18\}$ and $G$ is $PSU_{3}(3)$, $PSU_{3}(3):2$, $PSU_{4}(2)$ or $PSU_{4}(2):2$.

In this article, we investigate $2$-$(v,k,\lambda)$ designs with $\gcd(r,\lambda)=1$ admitting flag-transitive automorphism groups $G$. We prove that if $G$ is an almost simple group, then such a design belongs to one of the seven infinite families of $2$-designs or it is one of the eleven well-known examples. We describe all these examples of designs. We, in particular, prove that if $\mathcal{D}$ is a symmetric $(v,k,\lambda)$ design with $\gcd(k,\lambda)=1$ admitting a flag-transitive automorphism group $G$, then either $G\leq A\Gamma L_{1}(q)$ for some odd prime power $q$, or $\mathcal{D}$ is a projective space or the unique Hadamard design with parameters $(11,5,2)$., Comment: arXiv admin note: text overlap with arXiv:1904.10518

In this article, we study $2$-designs with $\gcd(r, \lambda)=1$ admitting a flag-transitive automorphism group. The automorphism groups of these designs are point-primitive of almost simple or affine type. We determine all pairs $(\mathcal{D}, G)$, where $\mathcal{D}$ is a $2$-design with $\gcd(r, \lambda)=1$ and $G$ is a flag-transitive almost simple automorphism group of $\mathcal{D}$ whose socle is $X=\mathrm{PSU}(n, q)$ with $(n, q)\neq (3, 2)$ and prove that such a design belongs to one of the two infinite families of Hermitian unitals and Witt-Bose-Shrikhande spaces, or it is isomorphic to a design with parameters $(6, 3, 2)$, $(7, 3, 1)$, $(8, 4, 3)$, $(10, 6, 5)$, $(11, 5, 2)$ or $(28, 7, 2)$., Comment: arXiv admin note: text overlap with arXiv:1904.10518

Safety is an essential aspect in the facilitation of automated vehicle deployment. Current testing practices are not enough, and going beyond them leads to infeasible testing requirements, such as needing to drive billions of kilometres on public roads. Automated vehicles are exposed to an indefinite number of scenarios. Handling of the most challenging scenarios should be tested, which leads to the question of how such corner cases can be determined. We propose an approach to identify the performance boundary, where these corner cases are located, using Gaussian Process Classification. We also demonstrate the classification on an exemplary traffic jam approach scenario, showing that it is feasible and would lead to more efficient testing practices., Comment: 6 pages, 5 figures, accepted at 2019 IEEE Intelligent Transportation Systems Conference - ITSC 2019, Auckland, New Zealand, October 2019

In this article, we study $2$-designs with prime replication number admitting a flag-transitive automorphism group. The automorphism groups of these designs are point-primitive of almost simple or affine type. We determine $2$-designs with prime replication number admitting an almost simple automorphism group.

In this article, we study symmetric $(v, k, \lambda)$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle is $PSU_{4}(q)$. We prove that there exist eight non-isomorphic such designs for which $\lambda\in\{3,6,18\}$ and $G$ is either $PSU_{4}(2)$, or $PSU_{4}(2):2$.

In this article, we study flag-transitive automorphism groups of non-trivial symmetric $(v, k, \lambda)$ designs, where $\lambda$ divides $k$ and $k\geq \lambda^2$. We show that such an automorphism group is either point-primitive of affine or almost simple type, or point-imprimitive with parameters $v=\lambda^{2}(\lambda+2)$ and $k=\lambda(\lambda+1)$, for some positive integer $\lambda$. We also provide some examples in both possibilities.