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Complex phenomena can be better understood when broken down into a limited number of simpler "components". Linear statistical methods such as the principal component analysis and its variants are widely used across various fields of applied science to identify and rank these components based on the variance they represent in the data. These methods can be seen as factorizations of the matrix collecting all the data, which are assumed to be a collection of time series sampled from fixed points in space. However, when data sampling locations vary over time, as with mobile monitoring stations in meteorology and oceanography or with particle tracking velocimetry in experimental fluid dynamics, advanced interpolation techniques are required to project the data onto a fixed grid before carrying out the factorization. This interpolation is often expensive and inaccurate. This work proposes a method to decompose scattered data without interpolating. The approach is based on physics-constrained radial basis function regression to compute inner products in space and time. The method provides an analytical and mesh-independent decomposition in space and time, demonstrating higher accuracy than the traditional approach. Our results show that it is possible to distill the most relevant "components" even for measurements whose natural output is a distribution of data scattered in space and time, maintaining high accuracy and mesh independence.

Given a homomorphism $\tau$ from a suitable finite group $\mathsf{\Gamma}$ to $\mathsf{SU}(4)$ with image $\mathsf{\Gamma}^\tau$, we construct a cohomological gauge theory on a noncommutative resolution of the quotient singularity $\mathbb{C}^4/\mathsf{\Gamma}^\tau$ whose BRST fixed points are $\mathsf{\Gamma}$-invariant tetrahedron instantons on a generally non-effective orbifold. The partition function computes the expectation values of complex codimension one defect operators in rank $r$ cohomological Donaldson-Thomas theory on a flat gerbe over the quotient stack $[\mathbb{C}^4/\,\mathsf{\Gamma}^\tau]$. We describe the generalized ADHM parametrization of the tetrahedron instanton moduli space, and evaluate the orbifold partition functions through virtual torus localization. If $\mathsf{\Gamma}$ is an abelian group the partition function is expressed as a combinatorial series over arrays of $\mathsf{\Gamma}$-coloured plane partitions, while if $\mathsf{\Gamma}$ is non-abelian the partition function localizes onto a sum over torus-invariant connected components of the moduli space labelled by lower-dimensional partitions. When $\mathsf{\Gamma}=\mathbb{Z}_n$ is a finite abelian subgroup of $\mathsf{SL}(2,\mathbb{C})$, we exhibit the reduction of Donaldson-Thomas theory on the toric Calabi-Yau four-orbifold $\mathbb{C}^2/\,\mathsf{\Gamma}\times\mathbb{C}^2$ to the cohomological field theory of tetrahedron instantons, from which we express the partition function as a closed infinite product formula. We also use the crepant resolution correpondence to derive a closed formula for the partition function on any polyhedral singularity., Comment: 79 pages; v2: minor changes and corrections

Coarse-Grain Reconfigurable Arrays (CGRAs) represent emerging low-power architectures designed to accelerate Compute-Intensive Loops (CILs). The effectiveness of CGRAs in providing acceleration relies on the quality of mapping: how efficiently the CIL is compiled onto the platform. State of the Art (SoA) compilation techniques utilize modulo scheduling to minimize the Iteration Interval (II) and use graph algorithms like Max-Clique Enumeration to address mapping challenges. Our work approaches the mapping problem through a satisfiability (SAT) formulation. We introduce the Kernel Mobility Schedule (KMS), an ad-hoc schedule used with the Data Flow Graph and CGRA architectural information to generate Boolean statements that, when satisfied, yield a valid mapping. Experimental results demonstrate SAT-MapIt outperforming SoA alternatives in almost 50\% of explored benchmarks. Additionally, we evaluated the mapping results in a synthesizable CGRA design and emphasized the run-time metrics trends, i.e. energy efficiency and latency, across different CILs and CGRA sizes. We show that a hardware-agnostic analysis performed on compiler-level metrics can optimally prune the architectural design space, while still retaining Pareto-optimal configurations. Moreover, by exploring how implementation details impact cost and performance on real hardware, we highlight the importance of holistic software-to-hardware mapping flows, as the one presented herein.

We present Tuna-step, a novel microfluidic module based on step emulsification that allows for reliable generation of droplets of different sizes. Until now, sizes of droplets generated with step emulsification were hard-wired into the geometry of the step emulsification nozzle. To overcome this, we incorporate a thin membrane underneath the step nozzle that can be actuated by pressure, enabling the tuning of the nozzle size on-demand. By controllably reducing the height of the nozzle, we successfully achieved a three-order-of-magnitude variation in droplet volume without any adjustment of the flow rates of the two phases. We further enhanced our system by developing and applying a new hydrophilic surface modification, ensuring long-term stability and preventing swelling of the device when generating Oil-in-Water droplets. We used our system in the manufacturing of functionally graded soft materials with adjustable porosity and material content. By combining our microfluidic device with a custom 3D printer, we generated and extruded Oil-in-Water emulsions in an agarose gel bath, allowing for the creation of unique self-standing 3D hydrogel structures with porosity decoupled from flow rate and with composition gradients of external phases. We upscaled Tuna-step by setting 14 actuatable nozzles in parallel, offering a step-emulsification-based single chip solution that can accommodate various requirements in terms of throughput, droplet volumes, flow rates, and surface chemistry.

Particle Image Velocimetry (PIV) estimates velocities through correlations of particle images within interrogation windows, leading to a spatial modulation of the velocity field. Although in principle Particle Tracking Velocimetry (PTV) estimates locally a non-modulated particle displacement, to exploit the scattered data from PTV it is necessary to interpolate these data on a structured grid, which implies a spatial modulation effect that biases the resulting velocity field. This systematic error due to finite spatial resolution inevitably depends on the interrogation window size and on the interparticle spacing. It must be observed that all these operations (cross-correlation, direct interpolation or averaging in windows) induce modulation on both the mean and the fluctuating part. We introduce a simple trick to reduce this systematic error source of PIV/PTV measurements exploiting ensemble statistics. Ensemble Particle Tracking Velocimetry (EPTV) can be leveraged to obtain the high-resolution mean flow by merging the different instantaneous realisations. The mean flow can be estimated with EPTV, and the fluctuating part can be measured from PIV/PTV. The high-resolution mean can then be superposed to the instantaneous fluctuating part to obtain velocity fields with lower systematic error. The methodology is validated against datasets with a progressively increasing level of complexity: two virtual experiments based on direct numerical simulations (DNS) of the wake of a fluidic pinball and a channel flow and the experimental data of a turbulent boundary layer. For all the cases both PTV and PIV are analysed.

We study rank $r$ cohomological Donaldson-Thomas theory on a toric Calabi-Yau orbifold of $\mathbb{C}^4$ by a finite abelian subgroup $\mathsf\Gamma$ of $\mathsf{SU}(4)$, from the perspective of instanton counting in cohomological gauge theory on a noncommutative crepant resolution of the quotient singularity. We describe the moduli space of noncommutative instantons on $\mathbb{C}^4/\mathsf{\Gamma}$ and its generalized ADHM parametrization. Using toric localization, we compute the orbifold instanton partition function as a combinatorial series over $r$-vectors of $\mathsf\Gamma$-coloured solid partitions. When the $\mathsf\Gamma$-action fixes an affine line in $\mathbb{C}^4$, we exhibit the dimensional reduction to rank $r$ Donaldson-Thomas theory on the toric Kahler three-orbifold $\mathbb{C}^3/\mathsf{\Gamma}$. Based on this reduction and explicit calculations, we conjecture closed infinite product formulas, in terms of generalized MacMahon functions, for the instanton partition functions on the orbifolds $\mathbb{C}^2/\mathbb{Z}_n\times\mathbb{C}^2$ and $\mathbb{C}^3/(\mathbb{Z}_2\times\mathbb{Z}_2)\times\mathbb{C}$, finding perfect agreement with new mathematical results of Cao, Kool and Monavari., Comment: 66 pages; v2: typos fixed, comments added; v3: exposition improved, appendix on generalised ADHM construction added; Final version to appear in Advances in Theoretical and Mathematical Physics

We present a study of the principal deuterium Hugoniot for pressures up to $150$ GPa, using Machine Learning potentials (MLPs) trained with Quantum Monte Carlo (QMC) energies, forces and pressures. In particular, we adopted a recently proposed workflow based on the combination of Gaussian kernel regression and $\Delta$-learning. By fully taking advantage of this method, we explicitly considered finite-temperature electrons in the dynamics, whose effects are highly relevant for temperatures above $10$ kK. The Hugoniot curve obtained by our MLPs shows a good agreement with the most recent experiments, particularly in the region below 60 GPa. At larger pressures, our Hugoniot curve is slightly more compressible than the one yielded by experiments, whose uncertainties generally increase, however, with pressure. Our work demonstrates that QMC can be successfully combined with $\Delta$-learning to deploy reliable MLPs for complex extended systems across different thermodynamic conditions, by keeping the QMC precision at the computational cost of a mean-field calculation., Comment: 7 + 10 pages; new version with improved QMC dataset. Hugoniot curve and discussion updated accordingly, main physical outcomes unchanged

Topological data analysis (TDA) is a new emerging and powerful tool to understand the medium range structure ordering of multi-scale data. This study investigates the density anomalies observed during cooling of liquid silica from topological point of view using TDA. The density of liquid silica does not monotonically increase during cooling; it instead shows a maximum and minimum. Despite tremendous efforts, the structural origin of these density anomalies is not clearly understood. Our approach reveals that the one-dimensional topology of the -Si-Si- network changes at the temperatures at which the maximum and minimum densities are observed in our MD simulations, while those of the -O-O- and -Si-O- networks change at lower temperatures. These results are also supported by conventional ring analysis. Our work demonstrates the value of new topological techniques in understanding the transitions in glassy materials and sheds light on the characterization of glass-liquid transitions., Comment: 25 pages, 5 figures, in the main text

This is a mini-review about generalized instantons of noncommutative gauge theories in dimensions 4, 6 and 8, with emphasis on their realizations in type II string theory, their geometric interpretations, and their applications to the enumerative geometry of non-compact toric varieties., Comment: 27 pages, 4 figures; v2: typos fixed, minor improvements; v3: references updated; Contribution to the special issue of the European Physical Journal on "Noncommutativity and Physics"

We introduce a novel end-to-end approach to improving the resolution of PIV measurements. The method blends information from different snapshots without the need for time-resolved measurements on grounds of similarity of flow regions in different snapshots. The main hypothesis is that, with a sufficiently large ensemble of statistically-independent snapshots, the identification of flow structures that are morphologically similar but occurring at different time instants is feasible. Measured individual vectors from different snapshots with similar flow organisation can thus be merged, resulting in an artificially increased particle concentration. This allows to refine the interrogation region and, consequently, increase the spatial resolution. The measurement domain is split in subdomains. The similarity is enforced only on a local scale, i.e. morphologically-similar regions are sought only among subdomains corresponding to the same flow region. The identification of locally-similar snapshots is based on unsupervised K-nearest neighbours search in a space of significant flow features. Such features are defined in terms of a Proper Orthogonal Decomposition, performed in subdomains on the original low-resolution data, obtained either with standard cross-correlation or with binning of Particle Tracking Velocimetry data with a relatively large bin size. A refined bin size is then selected according to the number of "sufficiently close" snapshots identified. The statistical dispersion of the velocity vectors within the bin is then used to estimate the uncertainty and to select the optimal K which minimises it. The method is tested and validated against datasets with a progressively increasing level of complexity: two virtual experiments based on direct simulations of the wake of a fluidic pinball and a channel flow and the experimental data collected in a turbulent boundary layer.