On the Displacement of Eigenvalues when Removing a Twin Vertex

Twin vertices of a graph have the same open neighbourhood. If they are not adjacent, then they are called duplicates and contribute the eigenvalue zero to the adjacency matrix. Otherwise they are termed co-duplicates, when they contribute $-1$ as an eigenvalue of the adjacency matrix. On removing a twin vertex from a graph, the spectrum of the adjacency matrix does not only lose the eigenvalue $0$ or $-1$. The perturbation sends a rippling effect to the spectrum. The simple eigenvalues are displaced. We obtain a closed formula for the characteristic polynomial of a graph with twin vertices in terms of two polynomials associated with the perturbed graph. These are used to obtain estimates of the displacements in the spectrum caused by the perturbation

The Vector-like Twin Higgs

We present a version of the twin Higgs mechanism with vector-like top partners. In this setup all gauge anomalies automatically cancel, even without twin leptons. The matter content of the most minimal twin sector is therefore just two twin tops and one twin bottom. The LHC phenomenology, illustrated with two example models, is dominated by twin glueball decays, possibly in association with Higgs bosons. We further construct an explicit four-dimensional UV completion and discuss a variety of UV completions relevant for both vector-like and fraternal twin Higgs models.Comment: 39 pages; v2 published versio

Strong Cospectrality and Twin Vertices in Weighted Graphs

We explore algebraic and spectral properties of weighted graphs containing twin vertices that are useful in quantum state transfer. We extend the notion of adjacency strong cospectrality to arbitrary Hermitian matrices, with focus on the generalized adjacency matrix and the generalized normalized adjacency matrix. We then determine necessary and sufficient conditions such that a pair of twin vertices in a weighted graph exhibits strong cospectrality with respect to the above-mentioned matrices. We also generalize known results about equitable and almost equitable partitions, and use these to determine which joins of the form $X\vee H$, where $X$ is either the complete or empty graph, exhibit strong cospectrality.Comment: 25 pages, 6 figure

Gluing two affine Yangians of gl1\mathfrak{gl}_1

We construct a four-parameter family of affine Yangian algebras by gluing two copies of the affine Yangian of $\mathfrak{gl}_1$. Our construction allows for gluing operators with arbitrary (integer or half integer) conformal dimension and arbitrary (bosonic or fermionic) statistics, which is related to the relative framing. The resulting family of algebras is a two-parameter generalization of the $\mathcal{N}=2$ affine Yangian, which is isomorphic to the universal enveloping algebra of $\mathfrak{u}(1)\oplus \mathcal{W}^{\mathcal{N}=2}_{\infty}[\lambda]$. All algebras that we construct have natural representations in terms of "twin plane partitions", a pair of plane partitions appropriately joined along one common leg. We observe that the geometry of twin plane partitions, which determines the algebra, bears striking similarities to the geometry of certain toric Calabi-Yau threefolds.Comment: 88 pages, 12 figure

Growth, coalescence and equilibration of metallic nanoparticles and nanoalloys studied by computational methods

Among nanoscale systems, metallic nanoparticles (NPs) certainly play a primary role, due to their highly tunable properties and to the wide variety of their applications. The properties of NPs are known to strongly depend on their size and geometric shape. In the case of bimetallic nanoparticles, also known as nanoalloys, further parameters can be exploited, i.e. the NP composition and the spatial arrangement of the two atomic species within the NP volume, here referred to as chemical ordering. Within this framework, the fine control of the NP configuration (here intended as the interplay between size, shape, composition and chemical ordering) is essential in sight of the possible technological applications. To this aim, a deep understanding of the NP formation process is highly desirable: one has to clearly know what are the different stages of such process, and what are the physical forces and the chemical effects involved. Moreover, a clear knowledge of the thermodynamic stability of the produced phases under the operating conditions is desirable as well. Computer simulations can be of great help in this sense, as they can provide clear information on both the equilibrium properties and the kinetic behaviour of the NPs. Specifically, the most thermodynamically favourable configurations of a given system can be determined, and the evolution pathways can be simulated and analysed at the atomic level, therefore allowing to rationalize the experimental findings. This Ph.D. thesis is devoted to the computational study of mono- and bi-metallic NPs, with particular attention to some of the nonequilibrium phenomena undergone by them. Different examples are presented and discussed; specifically, different metallic systems are treated, all of which are of great interest due to their practical applications, and different phenomena are analysed

A Geometric Tension Dynamics Model of Epithelial Convergent Extension

Epithelial tissue elongation by convergent extension is a key motif of animal morphogenesis. On a coarse scale, cell motion resembles laminar fluid flow; yet in contrast to a fluid, epithelial cells adhere to each other and maintain the tissue layer under actively generated internal tension. To resolve this apparent paradox, we formulate a model in which tissue flow occurs through adiabatic remodelling of the cellular force balance causing local cell rearrangement. We propose that the gradual shifting of the force balance is caused by positive feedback on myosin-generated cytoskeletal tension. Shifting force balance within a tension network causes active T1s oriented by the global anisotropy of tension. Rigidity of cells against shape changes converts the oriented internal rearrangements into net tissue deformation. Strikingly, we find that the total amount of tissue extension depends on the initial magnitude of anisotropy and on cellular packing order. T1s degrade this order so that tissue flow is self-limiting. We explain these findings by showing that coordination of T1s depends on coherence in local tension configurations, quantified by a certain order parameter in tension space. Our model reproduces the salient tissue- and cell-scale features of germ band elongation during Drosophila gastrulation, in particular the slowdown of tissue flow after approximately twofold extension concomitant with a loss of order in tension configurations. This suggests local cell geometry contains morphogenetic information and yields predictions testable in future experiments. Furthermore, our focus on defining biologically controlled active tension dynamics on the manifold of force-balanced states may provide a general approach to the description of morphogenetic flow.Comment: 44 pages, 19 figure

Diffusion-adapted spatial filtering of fMRI data for improved activation mapping in white matter

Brain activation mapping using fMRI data has been mostly focused on finding detections in gray matter. Activations in white matter are harder to detect due to anatomical differences between both tissue types, which are rarely acknowledged in experimental design. However, recent publications have started to show evidence for the possibility of detecting meaningful activations in white matter. The shape of the activations arising from the BOLD signal is fundamentally different between white matter and gray matter, a fact which is not taken into account when applying isotropic Gaussian filtering in the preprocessing of fMRI data. We explore a graph-based description of the white matter developed from diffusion MRI data, which is capable of encoding the anisotropic domain. Based on this representation, two approaches to white matter filtering are tested, and their performance is evaluated on both semi-synthetic phantoms and real fMRI data. The first approach relies on heat kernel filtering in the graph spectral domain, and produced a clear increase in both sensitivity and specificity over isotropic Gaussian filtering. The second approach is based on spectral decomposition for the denosing of the signal, and showed increased specificity at the cost of a lower sensitivity.Novel approach to white matter filtering We introduced new advanced methods for filtering brain scans. Using them, we managed to improve the detection of activity in the white matter of the brain