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How small can a set of vertices in the $n$-dimensional hypercube $Q_n$ be if it meets every copy of $Q_d$? The asymptotic density of such a set (for $d$ fixed and $n$ large) is denoted by $\gamma_d$. It is easy to see that $\gamma_d \leq 1/(d+1)$, and it is known that $\gamma_d=1/(d+1)$ for $d \leq 2$, but it was recently shown that $\gamma_d < 1/(d+1)$ for $d \geq 8$. In this paper we show that the latter phenomenon also holds for $d=7$ and $d=6$., Comment: 8 pages

We show that if $\mathcal{A} \subset {[n] \choose n/2}$ with measure bounded away from zero and from one, then the $\Omega(\sqrt{n})$-iterated upper shadows of $\mathcal{A}$ and $\mathcal{A}^c$ intersect in a set of positive measure. This confirms (in a strong form) a conjecture of Friedgut. It can be seen as a stability result for the Kruskal--Katona theorem., Comment: Minor corrections. 8 pages

Let $G$ be a nontrivial finite permutation group of degree $n$. If $G$ is transitive, then a theorem of Jordan states that $G$ has a derangement. Equivalently, a finite group is never the union of conjugates of a proper subgroup. If $G$ is intransitive, then $G$ may fail to have a derangement, and this can happen even if $G$ has only two orbits, both of which have size $(1/2+o(1))n$. However, we conjecture that if $G$ has two orbits of size exactly $n/2$ then $G$ does have a derangement, and we prove this conjecture when $G$ acts primitively on at least one of the orbits. Equivalently, we conjecture that a finite group is never the union of conjugates of two proper subgroups of the same order, and we prove this conjecture when at least one of the subgroups is maximal. The same conjecture was implicitly made by Feldman. We prove other cases of the conjecture, and we highlight connections our results have with intersecting families of permutations and roots of polynomials modulo primes. Along the way, we also prove a linear variant on Isbell's conjecture regarding derangements of prime-power order., Comment: 26 pages; references added

An $r$-daisy is an $r$-uniform hypergraph consisting of the six $r$-sets formed by taking the union of an $(r-2)$-set with each of the 2-sets of a disjoint 4-set. Bollob\'as, Leader and Malvenuto, and also Bukh, conjectured that the Tur\'an density of the $r$-daisy tends to zero as $r \to \infty$. In this paper we disprove this conjecture. Adapting our construction, we are also able to disprove a folklore conjecture about Tur\'an densities of hypercubes. For fixed $d$ and large $n$, we show that the smallest set of vertices of the $n$-dimensional hypercube $Q_n$ that meets every copy of $Q_d$ has asymptotic density strictly below $1/(d+1)$, for all $d \geq 8$. In fact, we show that this asymptotic density is at most $c^d$, for some constant $c<1$. As a consequence, we obtain similar bounds for the edge-Tur\'an densities of hypercubes. We also answer some related questions of Johnson and Talbot, and disprove a conjecture made by Bukh and by Griggs and Lu on poset densities., Comment: 14 pages. Minor corrections made since last version

If $G$ is a group, we say a subset $S$ of $G$ is product-free if the equation $xy=z$ has no solutions with $x,y,z \in S$. For $D \in \mathbb{N}$, a group $G$ is said to be $D$-quasirandom if the minimal dimension of a nontrivial complex irreducible representation of $G$ is at least $D$. Gowers showed that in a $D$-quasirandom finite group $G$, the maximal size of a product-free set is at most $|G|/D^{1/3}$. This disproved a longstanding conjecture of Babai and S\'os from 1985. For the special unitary group, $G=SU(n)$, Gowers observed that his argument yields an upper bound of $n^{-1/3}$ on the measure of a measurable product-free subset. In this paper, we improve Gowers' upper bound to $\exp(-cn^{1/3})$, where $c>0$ is an absolute constant. In fact, we establish something stronger, namely, product-mixing for measurable subsets of $SU(n)$ with measure at least $\exp(-cn^{1/3})$; for this product-mixing result, the $n^{1/3}$ in the exponent is sharp. Our approach involves introducing novel hypercontractive inequalities, which imply that the non-Abelian Fourier spectrum of the indicator function of a small set concentrates on high-dimensional irreducible representations. Our hypercontractive inequalities are obtained via methods from representation theory, harmonic analysis, random matrix theory and differential geometry. We generalize our hypercontractive inequalities from $SU(n)$ to an arbitrary $D$-quasirandom compact connected Lie group for $D$ at least an absolute constant, thereby extending our results on product-free sets to such groups. We also demonstrate various other applications of our inequalities to geometry (viz., non-Abelian Brunn-Minkowski type inequalities), mixing times, and the theory of growth in compact Lie groups., Comment: References updated

For integers $n \geq k \geq 1$, the {\em Kneser graph} $K(n, k)$ is the graph with vertex-set consisting of all the $k$-element subsets of $\{1,2,\ldots,n\}$, where two $k$-element sets are adjacent in $K(n,k)$ if they are disjoint. We show that if $(n,k,s) \in \mathbb{N}^3$ with $n > 10000 k s^5$ and $\mathcal{F}$ is set of vertices of $K(n,k)$ of size larger than $\{A \subset \{1,2,\ldots,n\}:\ |A|=k,\ A \cap \{1,2,\ldots,s\} \neq \varnothing\}$, then the subgraph of $K(n,k)$ induced by $\mathcal{F}$ has maximum degree at least \[ \left(1 - O\left(\sqrt{s^3 k/n}\right)\right)\frac{s}{s+1} \cdot {n-k \choose k} \cdot \frac{|\mathcal{F}|}{\binom{n}{k}}.\] This is sharp up to the behaviour of the error term $O(\sqrt{s^3 k/n})$. In particular, if the triple of integers $(n, k, s)$ satisfies the condition above, then the minimum maximum degree does not increase `continuously' with $|\mathcal{F}|$. Instead, it has $s$ jumps, one at each time when $|\mathcal{F}|$ becomes just larger than the union of $i$ stars, for $i = 1, 2, \ldots, s$. An appealing special case of the above result is that if $\mathcal{F}$ is a family of $k$-element subsets of $\{1,2,\ldots,n\}$ with $|\mathcal{F}| = {n-1 \choose k-1}+1$, then there exists $A \in \mathcal{F}$ such that $\mathcal{F}$ is disjoint from at least $$\left(1/2-O\left(\sqrt{k/n}\right)\right){n-k-1 \choose k-1}$$ of the other sets in $\mathcal{F}$; this is asymptotically sharp if $k=o(n)$. Frankl and Kupavskii, using different methods, have recently proven closely related results under the hypothesis that $n$ is at least quadratic in $k$., Comment: 31 pages. Clarifications in response to comments of two anonymous referees

One of the great pleasures of working with Imre Leader is to experience his infectious delight on encountering a compelling combinatorial problem. This collection of open problems in combinatorics has been put together by a subset of his former PhD students and students-of-students for the occasion of his 60th birthday. All of the contributors have been influenced (directly or indirectly) by Imre: his personality, enthusiasm and his approach to mathematics. The problems included cover many of the areas of combinatorial mathematics that Imre is most associated with: including extremal problems on graphs, set systems and permutations, and Ramsey theory. This is a personal selection of problems which we find intriguing and deserving of being better known. It is not intended to be systematic, or to consist of the most significant or difficult questions in any area. Rather, our main aim is to celebrate Imre and his mathematics and to hope that these problems will make him smile. We also hope this collection will be a useful resource for researchers in combinatorics and will stimulate some enjoyable collaborations and beautiful mathematics.

Prior to the deep learning era, shape was commonly used to describe the objects. Nowadays, state-of-the-art (SOTA) algorithms in medical imaging are predominantly diverging from computer vision, where voxel grids, meshes, point clouds, and implicit surface models are used. This is seen from numerous shape-related publications in premier vision conferences as well as the growing popularity of ShapeNet (about 51,300 models) and Princeton ModelNet (127,915 models). For the medical domain, we present a large collection of anatomical shapes (e.g., bones, organs, vessels) and 3D models of surgical instrument, called MedShapeNet, created to facilitate the translation of data-driven vision algorithms to medical applications and to adapt SOTA vision algorithms to medical problems. As a unique feature, we directly model the majority of shapes on the imaging data of real patients. As of today, MedShapeNet includes 23 dataset with more than 100,000 shapes that are paired with annotations (ground truth). Our data is freely accessible via a web interface and a Python application programming interface (API) and can be used for discriminative, reconstructive, and variational benchmarks as well as various applications in virtual, augmented, or mixed reality, and 3D printing. Exemplary, we present use cases in the fields of classification of brain tumors, facial and skull reconstructions, multi-class anatomy completion, education, and 3D printing. In future, we will extend the data and improve the interfaces. The project pages are: https://medshapenet.ikim.nrw/ and https://github.com/Jianningli/medshapenet-feedback, Comment: 16 pages

International benchmarking competitions have become fundamental for the comparative performance assessment of image analysis methods. However, little attention has been given to investigating what can be learnt from these competitions. Do they really generate scientific progress? What are common and successful participation strategies? What makes a solution superior to a competing method? To address this gap in the literature, we performed a multi-center study with all 80 competitions that were conducted in the scope of IEEE ISBI 2021 and MICCAI 2021. Statistical analyses performed based on comprehensive descriptions of the submitted algorithms linked to their rank as well as the underlying participation strategies revealed common characteristics of winning solutions. These typically include the use of multi-task learning (63%) and/or multi-stage pipelines (61%), and a focus on augmentation (100%), image preprocessing (97%), data curation (79%), and postprocessing (66%). The "typical" lead of a winning team is a computer scientist with a doctoral degree, five years of experience in biomedical image analysis, and four years of experience in deep learning. Two core general development strategies stood out for highly-ranked teams: the reflection of the metrics in the method design and the focus on analyzing and handling failure cases. According to the organizers, 43% of the winning algorithms exceeded the state of the art but only 11% completely solved the respective domain problem. The insights of our study could help researchers (1) improve algorithm development strategies when approaching new problems, and (2) focus on open research questions revealed by this work., Comment: accepted to CVPR 2023

In this very short note, we give a counterexample to a recent conjecture of Gilmer which would have implied the union-closed conjecture., Comment: 3 pages

We prove an analogue of Bonami's (hypercontractive) lemma for complex-valued functions on $\mathcal{L}(V,W)$, where $V$ and $W$ are vector spaces over a finite field. This inequality is useful for functions on $\mathcal{L}(V,W)$ whose `generalised influences' are small, in an appropriate sense. It leads to a significant shortening of the proof of a recent seminal result by Khot, Minzer and Safra that pseudorandom sets in Grassmann graphs have near-perfect expansion, which (in combination with the work of Dinur, Khot, Kindler, Minzer and Safra) implies the 2-2 Games conjecture (the variant, that is, with imperfect completeness)., Comment: 46 pages

We study an analogue of the Erd\H{o}s-S\'os forbidden intersection problem, for families of linear maps. If $V$ and $W$ are vector spaces over the same field, we say a family $\mathcal{F}$ of linear maps from $V$ to $W$ is \emph{$(t-1)$-intersection-free} if for any two linear maps $\sigma_1,\sigma_2 \in \mathcal{F}$, $\dim(\{v \in V:\ \sigma_1(v)=\sigma_2(v)\}) \neq t-1$. We prove that if $n$ is sufficiently large depending on $t$, $q$ is any prime power, $V$ is an $n$-dimensional vector space over $\mathbb{F}_q$, and $\mathcal{F} \subset \textrm{GL}(V)$ is $(t-1)$-intersection-free, then $|\mathcal{F}| \leq \prod_{i=1}^{n-t}(q^n - q^{i+t-1})$. Equality holds only if there exists a $t$-dimensional subspace of $V$ on which all elements of $\mathcal{F}$ agree, or a $t$-dimensional subspace of $V^*$ on which all elements of $\{\sigma^*:\ \sigma \in \mathcal{F}\}$ agree. Our main tool is a `junta approximation' result for families of linear maps with a forbidden intersection: namely, that if $V$ and $W$ are finite-dimensional vector spaces over the same finite field, then any $(t-1)$-intersection-free family of linear maps from $V$ to $W$ is essentially contained in a $t$-intersecting \emph{junta} (meaning, a family $\mathcal{J}$ of linear maps from $V$ to $W$ such that the membership of $\sigma$ in $\mathcal{J}$ is determined by $\sigma(v_1),\ldots,\sigma(v_M),\sigma^*(a_1),\ldots,\sigma^*(a_N)$, where $v_1,\ldots,v_M \in V$, $a_1,\ldots,a_N \in W^*$ and $M+N$ is bounded). The proof of this in turn relies on a variant of the `junta method' (originally introduced by Dinur and Friedgut, and powefully extended by Keller and the last author), together with spectral techniques and a hypercontractive inequality., Comment: 23 pages

Semiconductor quantum dots embedded in hybrid circular Bragg gratings are a promising platform for the efficient generation of nonclassical light. The scalable fabrication of multiple devices with similar performance is highly desirable for their practical use as sources of single and entangled photons, while the ability to operate at telecom wavelength is essential for their integration with the existing fiber infrastructure. In this work we combine the promising properties of broadband hybrid circular Bragg gratings with a membrane-transfer process performed on 3" wafer scale. We develop and study single-photon sources based on InAs/GaAs quantum dots emitting in the telecom O-band, demonstrating bright single-photon emission with Purcell factor > 5 and count rates up to 10 MHz. Furthermore, we address the question of reproducibility by benchmarking the performance of 10 devices covering a wide spectral range of 50 nm within the O-band.

The Peak-Patch algorithm is used to identify the densest minicluster seeds in the initial axion density field simulated from string decay. The fate of these dense seeds is found by tracking the subsequent gravitational collapse in cosmological $N$-body simulations. We find that miniclusters at late times are well described by NFW profiles, although for around 80\% of simulated miniclusters a single power-law density profile of $r^{-2.9}$ is an equally good fit due to the unresolved scale radius. Under the assumption that all miniclusters with an unresolved scale radius are described by a power-law plus axion star density profile, we identify a significant number of miniclusters that might be dense enough to give rise to gravitational microlensing if the axion mass is $0.2 \,\mathrm{meV}\lesssim m_a \lesssim 3\,\mathrm{meV}$. Higher resolution simulations resolving the inner structure and axion star formation are necessary to explore this possibility further., Comment: 10+8 pages, 15 figures, v2 further comments on density profile fits, matches version accepted in Phys Rev D

Existing work on personality traits in software development excludes game developers as a discrete group. Whilst games are software, game development has unique considerations, so game developers may exhibit different personality traits from other software professionals. We assessed responses from 123 game developers on an International Personality Item Pool Five Factor Model scale and demographic questionnaire using factor analysis. Programmers reported lower Extraversion than designers, artists and production team members; lower Openness than designers and production, and reported higher Neuroticism than production -- potentially linked to burnout and crunch time. Compared to published norms of software developers, game developers reported lower Openness, Conscientiousness, Extraversion and Agreeableness, but higher Neuroticism. These personality differences have many practical implications: differences in Extraversion among roles may precipitate communication breakdowns; differences in Openness may induce conflict between programmers and designers. Understanding the relationship between personality traits and roles can help recruiters steer new employees into appropriate roles, and help managers apply appropriate stress management techniques. To realise these benefits, individuals must be distinguished from roles: just because an individual occupies a role does not mean they possess personality traits associated with that role., Comment: 10 pages, 2 figures, 4 tables

For integers $r \geq 3$ and $t \geq 2$, an $r$-uniform $t$-daisy $\mathcal{D}^t_r$ is a family of $\binom{2t}{t}$ $r$-element sets of the form $$\{S \cup T \ : T\subset U, \ |T|=t \}$$ for some sets $S,U$ with $|S|=r-t$, $|U|=2t$ and $S \cap U = \emptyset$. It was conjectured by Bollob\'as, Leader and Malvenuto (and independently Bukh) that the Tur\'an densities of $t$-daisies satisfy $\lim\limits_{r \to \infty} \pi(\mathcal{D}_r^t) = 0$ for all $t \geq 2$; this has become a well-known problem, and it is still open for all values of $t$. In this paper, we give lower bounds for the Tur\'an densities of $r$-uniform $t$-daisies. To do so, we introduce (and make some progress on) the following natural problem in additive combinatorics: for integers $m \geq 2t \geq 4$, what is the maximum cardinality $g(m,t)$ of a subset $R$ of $\mathbb{Z}/m\mathbb{Z}$ such that for any $x \in \mathbb{Z}/m\mathbb{Z}$ and any $2t$-element subset $X$ of $\mathbb{Z}/m\mathbb{Z}$, there are $t$ distinct elements of $X$ whose sum is not in the translate $x+R$? This is a slice-analogue of the extremal Hilbert cube problem considered by Gunderson and R\"odl and its generalization studied by Cilleruelo and Tesoro., Comment: 11 pages. Minor changes made in response to comments of two anonymous referees

Designing implants for large and complex cranial defects is a challenging task, even for professional designers. Current efforts on automating the design process focused mainly on convolutional neural networks (CNN), which have produced state-of-the-art results on reconstructing synthetic defects. However, existing CNN-based methods have been difficult to translate to clinical practice in cranioplasty, as their performance on complex and irregular cranial defects remains unsatisfactory. In this paper, a statistical shape model (SSM) built directly on the segmentation masks of the skulls is presented. We evaluate the SSM on several cranial implant design tasks, and the results show that, while the SSM performs suboptimally on synthetic defects compared to CNN-based approaches, it is capable of reconstructing large and complex defects with only minor manual corrections. The quality of the resulting implants is examined and assured by experienced neurosurgeons. In contrast, CNN-based approaches, even with massive data augmentation, fail or produce less-than-satisfactory implants for these cases. Codes are publicly available at https://github.com/Jianningli/ssm, Comment: 9 pages

Resonant Inelastic X-Ray Scattering (RIXS) spectra of hematite were measured at the Fe L3-edge for heteroepitaxial thin films which were undoped and doped with 1% Ti, Sn or Zn, in the energy loss range in excess of 1 eV to study electronic transitions. The spectra were measured for several momentum transfers (q), conducted at both low temperature (T=14K) and room temperature. While we can not rule out dispersive features possibly owing to propagating excitations, the coarse envelopes of the general spectra did not appreciably change shape with q, implying that the bulk of the observed L-edge RIXS intensity originates from (mostly) non-dispersive ligand field (LF) excitations. Summing the RIXS spectra over q and comparing the results at T=14 K to those at T=300 K, revealed pronounced temperature effects, including an intensity change and energy shift of the 1.4 eV peak, a broadband intensity increase of the 3-4 eV range, and higher energy features. The q-summed spectra and their temperature dependences are virtually identical for nearly all of the samples with different dopants, save for the temperature dependence of the Ti-doped sample's spectrum, which we attribute to being affected by a large number of free charge carriers. Comparing with magnetization measurements for different temperatures and dopings likewise did not show a clear correlation between the RIXS spectra and the magnetic ordering states. To clarify the excited states, we performed spin multiplet calculations which were in excellent agreement with the RIXS spectra over a wide energy range and provide detailed electronic descriptions of the excited states. The implications of these findings to the photoconversion efficiency of hematite photoanodes is discussed., Comment: 11 pages, 5 figures, and .pdf supplementary file

Our aim in this note is to show that, for any $\epsilon>0$, there exists a union-closed family $\mathcal F$ with (unique) smallest set $S$ such that no element of $S$ belongs to more than a fraction $\epsilon$ of the sets in $\mathcal F$. More precisely, we give an example of a union-closed family with smallest set of size $k$ such that no element of this set belongs to more than a fraction $(1+o(1))\frac{\log_2 k}{2k}$ of the sets in $\mathcal F$. We also give explicit examples of union-closed families containing `small' sets for which we have been unable to verify the Union-Closed Conjecture., Comment: 5 pages

The study of intersection problems in Extremal Combinatorics dates back perhaps to 1938, when Paul Erd\H{o}s, Chao Ko and Richard Rado proved the (first) `Erd\H{o}s-Ko-Rado theorem' on the maximum possible size of an intersecting family of $k$-element subsets of a finite set. Since then, a plethora of results of a similar flavour have been proved, for a range of different mathematical structures, using a wide variety of different methods. Structures studied in this context have included families of vector subspaces, families of graphs, subsets of finite groups with given group actions, and of course uniform hypergraphs with stronger or weaker intersection conditions imposed. The methods used have included purely combinatorial ones such as shifting/compressions, algebraic methods (including linear-algebraic, Fourier analytic and representation-theoretic), and more recently, analytic, probabilistic and regularity-type methods. As well as being natural problems in their own right, intersection problems have connections with many other parts of Combinatorics and with Theoretical Computer Science (and indeed with many other parts of Mathematics), both through the results themselves, and the methods used. In this survey paper, we discuss both old and new results (and both old and new methods), in the field of intersection problems. Many interesting open problems remain; we will discuss several. For expositional and pedagogical purposes, we also take this opportunity to give slightly streamlined versions of proofs (due to others) of several classical results in the area. This survey is intended to be useful to PhD students, as well as to more established researchers. It is a personal perspective on the field, and is not intended to be exhaustive; we apologise for any omissions. It is an expanded version of a paper that will appear in the Proceedings of the 29th British Combinatorial Conference., Comment: 60 pages; references added, corrections made following reviewer and editor feedback

When testing for a disease such as COVID-19, the standard method is individual testing: we take a sample from each individual and test these samples separately. An alternative is pooled testing (or "group testing"), where samples are mixed together in different pools, and those pooled samples are tested. When the prevalence of the disease is low and the accuracy of the test is fairly high, pooled testing strategies can be more efficient than individual testing. In this chapter, we discuss the mathematics of pooled testing and its uses during pandemics, in particular the COVID-19 pandemic. We analyse some one- and two-stage pooling strategies under perfect and imperfect tests, and consider the practical issues in the application of such protocols., Comment: Extended version of a book chapter to appear in "Pandemics: Insurance and Social Protection", edited by M. C. Boado-Penas, J. Eisenberg and \c{S}. \c{S}ahin and to be published by Springer. Typos corrected

In recent years, hematite potential as a photoanode material for solar hydrogen production has ignited a renewed interest in its physical and interfacial properties, which continues to be an active field of research. Research on hematite photoanodes provides new insights on the correlations between electronic structure, transport properties, excited state dynamics and charge transfer phenomena, and expands our knowledge on solar cell materials into correlated electron systems. This research news article presents a snapshot of selected theoretical and experimental developments linking the electronic structure to the photoelectrochemical performance, with particular focus on optoelectronic properties and charge carrier dynamics.

The spatial collection efficiency portrays the driving forces and loss mechanisms in photovoltaic and photoelectrochemical devices. It is defined as the fraction of photogenerated charge carriers created at a specific point within the device that contribute to the photocurrent. In stratified planar structures, the spatial collection efficiency can be extracted out of photocurrent action spectra measurements empirically, with few a priori assumptions. Although this method was applied to photovoltaic cells made of well-understood materials, it has never been used to study unconventional materials such as metal-oxide semiconductors that are often employed in photoelectrochemical cells. This perspective shows the opportunities that this method has to offer for investigating new materials and devices with unknown properties. The relative simplicity of the method, and its applicability to operando performance characterization, makes it an important tool for analysis and design of new photovoltaic and photoelectrochemical materials and devices.

The photoelectrochemical behavior of a planar 1 cm2 thick Ti-doped hematite film deposited on F:SnO2 coated glass was studied with both front and back illumination. Despite low quantum efficiency, photocurrent was observed upon back illumination with low wavelengths, indicating that some photogenerated holes are able to traverse at least 700 nm across the hematite film and effectively oxidize water. This cannot be accounted for using the commonly accepted hole collection length of hematite based on fitting to the Gartner model. Furthermore, under back illumination, 450 nm excitation resulted in increased photocurrent as compared to 530 nm excitation despite most of the light being absorbed further away from the surface. These results demonstrate that the photocurrent is strongly dependent on the optical excitation wavelength, and related to both delocalized holes with long lifetime and localized excitations rather than only being dependent on the proximity of the absorption to the surface.

In this very short paper, we point out that the average overlap density of a union-closed family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$ may be as small as $\Theta((\log \log |\mathcal{F}|)/(\log |\mathcal{F}|))$, for infinitely many positive integers $n$., Comment: 4 pages

We show that the Union-Closed Conjecture holds for the union-closed family generated by the cyclic translates of any fixed set., Comment: 3 pages. Updated references

Electrification of transport is a key strategy in reducing carbon emissions. Many countries have adopted policies of complete but gradual transformation to electric vehicles (EVs). However, mass EV adoption also means a spike in load (kW), which in turn can disrupt existing electricity infrastructure. Smart or controlled charging is widely seen as a potential solution to alleviate this stress on existing networks. Learning from the recent EV trials in the UK and elsewhere we take into account two key aspects which are largely ignored in current research: EVs actually charging at any given time and wide range of EV types, especially battery capacity-wise. Taking a minimalistic scenario-based approach, we study forecasting models for mean number of active chargers and mean EV consumption for distinct scenarios. Focusing on residential charging the models we consider range from simple regression models to more advanced machine and deep learning models such as XGBoost and LSTMs. We then use these models to evaluate the impacts of different levels of future EV penetration on a specimen distribution transformer that captures typical real-world scenarios. In doing so, we also initiate the study of different types of controlled charging when fully controlled charging is not possible. This aligns with the outcomes from recent trials which show that a sizeable proportion of EV owners may not prefer fully controlled centralized charging. We study two possible control regimes and show that one is more beneficial from load-on-transformer point of view, while the other may be preferred for other objectives. We show that a minimum of 60% control is required to ensure that transformers are not overloaded during peak hours., Comment: 22 pages (including references and appendix); revised title to include new analysis and results

We use a modified version of the Peak Patch excursion set formalism to compute the mass and size distribution of QCD axion miniclusters from a fully non-Gaussian initial density field obtained from numerical simulations of axion string decay. We find strong agreement with N-Body simulations at a significantly lower computational cost. We employ a spherical collapse model and provide fitting functions for the modified barrier in the radiation era. The halo mass function at $z=629$ has a power-law distribution $M^{-0.6}$ for masses within the range $10^{-15}\lesssim M\lesssim 10^{-10}M_{\odot}$, with all masses scaling as $(m_a/50\mu\mathrm{eV})^{-0.5}$. We construct merger trees to estimate the collapse redshift and concentration mass relation, $C(M)$, which is well described using analytical results from the initial power spectrum and linear growth. Using the calibrated analytic results to extrapolate to $z=0$, our method predicts a mean concentration $C\sim \mathcal{O}(\text{few})\times10^4$. The low computational cost of our method makes future investigation of the statistics of rare, dense miniclusters easy to achieve.

A family of permutations $\mathcal{F} \subset S_{n}$ is said to be $t$-intersecting if any two permutations in $\mathcal{F}$ agree on at least $t$ points. It is said to be $(t-1)$-intersection-free if no two permutations in $\mathcal{F}$ agree on exactly $t-1$ points. If $S,T \subset \{1,2,\ldots,n\}$ with $|S|=|T|$, and $\pi: S \to T$ is a bijection, the $\pi$-star in $S_n$ is the family of all permutations in $S_n$ that agree with $\pi$ on all of $S$. An $s$-star is a $\pi$-star such that $\pi$ is a bijection between sets of size $s$. Friedgut and Pilpel, and independently the first author, showed that if $\mathcal{F} \subset S_n$ is $t$-intersecting, and $n$ is sufficiently large depending on $t$, then $|\mathcal{F}| \leq (n-t)!$; this proved a conjecture of Deza and Frankl from 1977. Equality holds only if $\mathcal{F}$ is a $t$-star. In this paper, we give a more `robust' proof of a strengthening of the Deza-Frankl conjecture, namely that if $n$ is sufficiently large depending on $t$, and $\mathcal{F} \subset S_n$ is $(t-1)$-intersection-free, then $|\mathcal{F} \leq (n-t)!$, with equality only if $\mathcal{F}$ is a $t$-star. The main ingredient of our proof is a `junta approximation' result, namely, that any $(t-1)$-intersection-free family of permutations is essentially contained in a $t$-intersecting {\em junta} (a `junta' being a union of a bounded number of $O(1)$-stars). The proof of our junta approximation result relies, in turn, on a weak regularity lemma for families of permutations, a combinatorial argument that `bootstraps' a weak notion of pseudorandomness into a stronger one, and finally a spectral argument for pairs of highly-pseudorandom fractional families. Our proof employs four different notions of pseudorandomness, three being combinatorial in nature, and one being algebraic., Comment: 28 pages