One of the fundamental results in graph minor theory is that for every planar graph~$H$, there is a minimum integer~$f(H)$ such that graphs with no minor isomorphic to~$H$ have treewidth at most~$f(H)$. The best known bound for an arbitrary planar $H$ is ${O(|V(H)|^9\operatorname{poly~log} |V(H)|)}$. We show that if $H$ is the disjoint union of cycles, then $f(H)$ is $O(|V(H)|\log^2 |V(H)|)$, which is a $\log|V(H)|$ factor away being optimal.

Given a smooth, bounded domain $Ω\subset\mathbb{R}^N$, we establish the existence of two non-trivial, non-negative solutions to the semilinear degenerate elliptic equation \begin{align*} \left. \begin{array}{l} -Δ_λu=μg(z)|u|^{r-1}u+h(z)|u|^{s-1}u \;\text{in}\; Ω u\in H^{1,λ}_0(Ω) \end{array}\right\} \end{align*} where $Δ_λ=Δ_x+|x|^{2λ}Δ_y$ denotes the Grushin Laplacian Operator, $z=(x,y)\inΩ$, $N=n+m;\, n,\, m\geq 1$, $λ>0$, $0\leq r<1<s<2^*_λ-1$ and $μ$ is a positive parameter. The functions $g$ and $h$ may change sign and $2^*_λ=\frac{2Q}{Q-2}$ is the critical Sobolev exponent associated with the homogeneous dimension $Q=n+(1+λ)m$ of $Δ_λ$. In the critical case $s=2^*_λ-1$, we further show that the problem admits at least two non-trivial, non-negative solutions under the additional assumptions $g\geq 0$ and $h\equiv 1$.

It is a fundamental question why quantum mechanics employs complex numbers rather than solely real numbers. In this work, we conduct the first analysis of imaginarity quantification in neutrino flavor and spin-flavor oscillations. As quantum systems in coherent superposition, neutrinos are ideal candidates for quantifying imaginarity within the resource theoretic framework, using measures such as the $\ell_1$-norm and the relative entropy of imaginarity. We show that in the case of two-flavor mixing, these measures of imaginarity are nonzero. The measures of imaginarity reach their extreme values when the probabilistic features of quantum theory are fully maximized, i.e., both the transitional and survival probabilities are approximately equal. Our study reveals that the imaginarity, as a resource, can be harnessed not solely from the presence of a complex phase in the mixing matrix but also from the intrinsic quantum dynamics of time evolution itself. We further extend our analysis to explore the dynamics of three-flavor neutrino mixing, incorporating the effects of a nonzero $CP$ phase.

Ongoing advancements in microbiome profiling have provided unprecedented insights into the molecular dynamics of microbial communities, sparking a surge of interest in uncovering the microbiome's critical role in human health. Identifying microbial features linked to clinical outcomes, however, remains challenging due to the high-dimensional, sparse, and compositional nature of microbiome data. Additionally, many microbial taxa, although classified as distinct, may share functional roles, complicating traditional variable selection methods. To overcome these obstacles, we introduce Bayesian Regression with Agglomerated Compositional Effects (BRACE), a novel approach using a spike-and-cluster prior combining Bernoulli activity indicators, an Ewens exchangeable partition prior on the finite active set, and a projection-based constrained Gaussian prior on cluster effects to perform data-adaptive clustering and variable selection. The methodological innovation of our work lies in how we combine the Ewens partition prior with a projection-based constrained Gaussian on the cluster atoms to enforce the sum-to-zero constraint. BRACE groups microbial taxa with similar effects on the outcome, yielding more interpretable models while enabling effective dimension reduction. Through comprehensive simulations and a real-world application examining the influence of oral microbiome composition on insulin resistance, we demonstrate BRACE's superior performance over existing methods, particularly in identifying key features with shared effects on outcomes.

A decorated surface S is an oriented topological surface with marked points on the boundary considered modulo the isotopy. We consider the moduli space of hyperbolic structures on S with geodesic boundary, such that the hyperbolic structure near each marked point is a cusp, equipped with a horocycle. This space carries a volume form. Let us fix the set K of distances between the horocycles at the adjacent cusps, and the set L of lengths of boundary circles without cusps. We get a subspace M(S; K,L) with the induced volume form Vol(K,L). However, if the cusps are present, the volume of the space M(S; K,L) is infinite. We introduce the exponential volume form exp(-W)Vol(K,L), where W is a positive function on the moduli space, given by the sum over cusps of the hyperbolic areas enclosed between the cusp and the horocycle at the cusp. We prove that the exponential volume, defined as the integral of the exponential volume form over the moduli space M(S; K,L), is always finite. We suggest that the moduli spaces M(S; K,L) with the exponential volume forms are the true analogs of the classical moduli spaces of Riemann surfaces, with the Weil-Petersson volume forms. In particular, they should be relevant to the open string theory. We support this by proving an unfolding formula for the integrals of measurable functions multiplied by the exponential volume form. It expresses them as finite sums of similar integrals over moduli spaces for simpler surfaces. They generalise Mirzakhani's recursions for the volumes of moduli spaces of hyperbolic surfaces. We show that exponential volumes for elementary decorated surfaces give rise to a commutative algebra E, which we call the positive Hecke-Whittaker algebra for PGL(2,R). Exponential volumes for all decorated surfaces and unfolding formulas extend the algebra E to all decorated surfaces.

Sizeable primordial curvature perturbations needed to seed a population of primordial black holes (PBHs) will be accompanied by a scalar-induced gravitational wave signal that can be detectable by pulsar timing arrays (PTA). We derive conservative bounds on the amplitude of the scalar power spectrum at the PTA frequencies and estimate the implied constraints on the PBH abundance. We show that only a small fraction of dark matter can consist of stellar mass PBHs. The strength and the shape of the constraint depend on the shape of the power spectrum and the nature of the non-Gaussianities. We find that constraints on the PBH abundance arise in the mass range $0.1-10^3\, M_{\odot}$, with the sub-solar mass range being constrained only for narrow curvature power spectra. These constraints are softened when positive non-Gaussianity is introduced and can be eliminated when $f_{\rm NL} \gtrsim 5$. On the other hand, if the PBH abundance is computed via the theory of peaks, the PTA constraints on PBHs are relaxed, signalling once more the theoretical uncertainties in assessing the PBH abundance. We further discuss how strong positive non-Gaussianites can allow for heavy PBHs to potentially seed supermassive BHs.

Computing Persistent Homology for large point clouds remains a bottleneck for the wider adoption of persistent homology by the scientific community. We present an algorithm which can compute the degree-1 Vietoris-Rips Persistent Homology of point clouds in low dimensional Euclidean Space for larger point clouds.

Major players in the global aerospace industry are shifting their focus toward achieving net carbon-neutral operations by 2050. A considerable portion of the overall carbon emission reduction is expected to come from new aircraft technologies, such as flight path optimization. In pursuing these sustainability objectives, we delve into the capacity of quantum computing to tackle computational challenges associated with flight path optimization, an essential operation within the aerospace engineering domain with important ecological and economic considerations. In recent years, the quantum computing field has made significant strides, paving the way for improved performance over classical algorithms. In order to effectively apply quantum algorithms in real-world scenarios, it is crucial to thoroughly examine and tackle the intrinsic overheads and constraints that exist in the present implementations of these algorithms. Our study delves into the application of quantum computers in flight path optimization problems and introduces a customizable modular framework designed to accommodate specific simulation requirements. We examine the running time of a hybrid quantum-classical algorithm across various quantum architectures and their simulations on CPUs and GPUs. A temporal comparison between the conventional classical algorithm and its quantum-improved counterpart indicates that achieving the theoretical speedup in practice may necessitate further innovation. We present our results from running the quantum algorithms on IBM hardware and discuss potential approaches to accelerate the incorporation of quantum algorithms within the problem domain.

Take a closed monotone symplectic manifold containing a smooth anticanonical divisor. The quantum connection on its cohomology has singularities at zero and infinity (in the quantum parameter). At zero it has a regular singular point, by definition. We show that the singularity at infinity is of unramified exponential type. The argument involves: realizing cohomology as a deformation of the symplectic cohomology of the divisor complement; the corresponding deformation of the wrapped Fukaya category; a new categorical interpretation of the Fourier-Laplace transform of D-modules; and the regularity theorem of Petrov-Vaintrob-Vologodsky in noncommutative geometry.

Motion simulators allow researchers to safely investigate the interaction of drivers with a vehicle. However, many studies that use driving simulator data to predict cognitive load only employ two levels of workload, leaving a gap in research on employing deep learning methodologies to analyze cognitive load, especially in challenging low-light conditions. Often, studies overlook or solely focus on scenarios in bright daylight. To address this gap and understand the correlation between performance and cognitive load, this study employs functional near-infrared spectroscopy (fNIRS) and eye-tracking data, including fixation duration and gaze direction, during simulated driving tasks in low visibility conditions, inducing various mental workloads. The first stage involves the statistical estimation of useful features from fNIRS and eye-tracking data. ANOVA will be applied to the signals to identify significant channels from fNIRS signals. Optimal features from fNIRS, eye-tracking and vehicle dynamics are then combined in one chunk as input to the CNN and LSTM model to predict workload variations. The proposed CNN-LSTM model achieved 99% accuracy with neurological data and 89% with vehicle dynamics to predict cognitive load, indicating potential for real-time assessment of driver mental state and guide designers for the development of safe adaptive systems.

In this paper, we study the representation theory of affine Nappi-Witten Lie algebra $\widehat{H_4}$ corresponding to the Nappi-Witten Lie algebra $H_4$. We completely classify all Cartan-free modules of rank one for the Nappi-Witten Lie algebra $H_4$. With the help of Cartan free $H_4$ modules we classify all Cartan-free modules of rank one over affine Nappi Witten Lie algebra. We also give a necessary and sufficient condition for these modules to be irreducible. Finally as an application we classify Cartan free modules of rank one for affine-Virasoro Nappi-Witten Lie algebras.

A double cover $Y$ of $\mathbb{P}^1 \times \mathbb{P}^2$ ramified over a general $(2,2)$-divisor will have the structure of a geometrically standard conic bundle ramified over a smooth plane quartic $Δ\subset \mathbb{P}^2$ via the second projection. These threefolds are rational over algebraically closed fields; however, over nonclosed fields, including $\mathbb{R}$, their rationality is an open problem. In this paper, we characterize rationality over $\mathbb{R}$ when $Δ(\mathbb{R})$ has at least two connected components (extending work of M. Ji and the second author) and over local fields when all odd degree fibers of the first projection have nonsquare discriminant. We obtain these applications by proving general results comparing the conic bundle structure on $Y$ with the conic bundle structure on a well-chosen intersection of two quadrics. The difference between these two conic bundles is encoded by a constant Brauer class, and we prove that this class encodes the obstruction to the existence of a section of the first projection $Y\to\mathbb{P}^1$.

We study the systole of a model of random hyperbolic 3-manifolds introduced by Petri and Raimbault, answering a question posed in that same article. These are compact manifolds with boundary constructed by randomly gluing truncated tetrahedra along their faces. We prove that the limit, as the volume tends to infinity, of the expected value of their systole exists and we give a closed formula of it. Moreover, we compute a numerical approximation of this value.

In this paper, we characterize the vanishing of twisted central $L$-values attached to newforms of square-free level in terms of certain polynomials of quadratic forms introduced by Zagier and the action of finitely many Hecke operators thereon. To be more precise, we establish that a twisted central $L$-value attached to a newform vanishes if and only if a certain explicitly computable polynomial is constant. We describe these constants explicitly in two different ways. One of the descriptions involves the generalized Hurwitz class numbers, which were introduced by Pei and Wang in $2003$. We provide some numerical examples and conclude by offering some questions for future work.

When estimating causal effects from observational studies, researchers often need to adjust for many covariates to deconfound the non-causal relationship between exposure and outcome, among which many covariates are discrete. The behavior of commonly used estimators in the presence of many discrete covariates is not well understood since their properties are often analyzed under structural assumptions including sparsity and smoothness, which do not apply in discrete settings. In this work, we study the estimation of causal effects in a model where the covariates required for confounding adjustment are discrete but high-dimensional, meaning the number of categories $d$ is comparable with or even larger than sample size $n$. Specifically, we show the mean squared error of commonly used regression, weighting and doubly robust estimators is bounded by $\frac{d^2}{n^2}+\frac{1}{n}$. We then prove the minimax lower bound for the average treatment effect is of order $\frac{d^2}{n^2 \log^2 n}+\frac{1}{n}$, which characterizes the fundamental difficulty of causal effect estimation in the high-dimensional discrete setting, and shows the estimators mentioned above are rate-optimal up to log-factors. We further consider additional structures that can be exploited, namely effect homogeneity and prior knowledge of the covariate distribution, and propose new estimators that enjoy faster convergence rates of order $\frac{d}{n^2} + \frac{1}{n}$, which achieve consistency in a broader regime. The results are illustrated empirically via simulation studies.

Motivated by Hadwiger's conjecture and related problems for list-coloring, we study graphs $H$ for which every graph with minimum degree at least $|V(H)|-1$ contains $H$ as a minor. We prove that a large class of apex-outerplanar graphs satisfies this property. Our result gives the first examples of such graphs whose vertex cover numbers are significantly larger than half of the number of its vertices, which breaks a barrier for attacking related coloring problems via extremal functions, and recovers all known such graphs that have arbitrarily large maximum degree. Our proof can be adapted to directed graphs to show that if $\vec H$ is the digraph obtained from a directed cycle or an in-arborescence by adding an apex source, then every digraph with minimum out-degree $|V(\vec H)|-1$ contains $\vec H$ as a subdivision or a butterfly minor respectively. These results provide the optimal upper bound for the chromatic number and dichromatic number of graphs and digraphs that do not contain the aforementioned graphs or digraphs as a minor, butterfly minor and a subdivision, respectively. Special cases of our results solve an open problem of Aboulker, Cohen, Havet, Lochet, Moura and Thomassé and strengthen results of Gishboliner, Steiner and Szabó.

Given any pair of quantum channels $Φ_1,Φ_2$ such that at least one of them has Kraus rank one, as well as any respective Stinespring isometries $V_1,V_2$, we prove that there exists a unitary $U$ on the environment such that $\|V_1-({\bf1}\otimes U)V_2\|_\infty\leq\sqrt{2\|Φ_1-Φ_2\|_\diamond}$. Moreover, we provide a simple example which shows that the factor $\sqrt2$ on the right-hand side is optimal, and we conjecture that this inequality holds for every pair of channels.

We prove the dynamical Manin-Mumford conjecture for regular polynomial maps of A^2 and irreducible curves avoiding super-attracting orbits at infinity, over any field of characteristic 0.

We develop a nilpotent approximation theory for Smith ideals, extending adic completion for commutative rings to monoid objects in locally presentable symmetric monoidal abelian categories and to $\mathbb{E}_\infty$-algebra objects in stable symmetric monoidal model categories. The main result is a formal completeness theorem: finite generation of a Smith ideal forces completeness of its nilpotent approximation. This gives a categorical analogue of the finite generation completeness phenomenon in classical adic completion, while remaining distinct from ordinary adic completion of quotient rings. As applications, we construct an almost mathematics version of nilpotent approximation and prove a homotopical completeness theorem for weakly compact Smith ideals. We then apply the general theory to motivic spectra. For the canonical morphism from algebraic cobordism to algebraic K-theory, we construct the corresponding K-theoretic nilpotent approximation of algebraic cobordism, prove its homotopical completeness and Bott periodicity, and establish a mod-$\ell$ Gabber rigidity theorem for the analogous approximation of $\mathbf{MGL}/\ell$ by $\mathbb{K}/l$.

We present general reduction procedures for Courant, Dirac and generalized complex structures, in particular when a group of symmetries is acting. We do so by taking the graded symplectic viewpoint on Courant algebroids and carrying out graded symplectic reduction, both in the coisotropic and hamiltonian settings. Specializing the latter to the exact case, we recover in a systematic way the reduction schemes of Bursztyn-Cavalcanti-Gualtieri.

The K-energy functional is extended to complexified Kähler classes, providing a variational approach to study the scalar curvature equation with B-field introduced by Schlitzer and Stoppa. The extended K-energy is convex along geodesics in the space of almost calibrated representatives of the complexified Kähler class. This fact is used to show that, in some situations, solutions of the scalar curvature equation with B-field are unique in their class, up to pullbacks by reduced automorphisms of the manifold.

Processing-in-Memory (PIM) architectures are evolving to minimize data movement by leveraging the same physical devices for both memory and logic functionalities. While analog PIM harnesses crossbar arrays for efficient approximate matrix-vector multiplication, digital PIM architectures facilitate massively-parallel bitwise operations for more general workloads. Recent works have extended digital PIM towards the full-precision acceleration of convolutional neural networks (CNNs), yet a comprehensive comparison with GPUs remains a gap in the literature that may illuminate the limitations of digital PIM. This paper aims to fill this void by conducting a thorough examination of CNN acceleration through an updated quantitative comparison with GPUs. Our approach begins with a theoretical investigation into various PIM architectures, shedding light on their performance characteristics and constraints. Subsequently, through a series of benchmarks spanning memory-bound vectored arithmetic to CNN acceleration, we provide insights into digital PIM performance that may guide the acceleration of applications in the future.

Environmental problems are receiving increasing attention in socio-economic and health studies, fostering advances in recording and data collection of related real-life processes. However, traditional tools for data processing are often found too restrictive as they do not account for the rich nature of such data sets. In this paper, we propose a simple statistical perspective on forecasting environmental data collected sequentially over time across some predefined geographic region. We treat such data set as a surface (or functional) time series with a possibly complicated geographical domain. Using techniques from functional data analysis, we develop a forecasting methodology that allows to account for both geographic and temporal dependencies. This methodology allows integration of traditional multivariate techniques to provide forecasts surfaces. We demonstrate the practical value of our approach with a forecasting example of ground-level ozone concentration across Germany, showcasing its effectiveness and potential for broad application.

We extend discrete Morse-Bott theory to the setting of loop-free (or acyclic) categories. First of all, we state a homological version of Quillen's Theorem A in this context and introduce the notion of cellular categories. Second, we present a notion of vector field for loop-free categories. Third, we prove a homological collapsing theorem in the absence of critical objects in order to obtain the Morse inequalities. Examples are provided through the exposition. This answers partially a question by T. John: whether there is a Morse theory for loop-free (or acyclic) categories? [14].

We define the eñe product for the multiplicative group of polynomials and formal power series with coefficients on a commutative ring and unitary constant coefficient. This defines a commutative ring structure where multiplication is the additive structure and the eñe product is the multiplicative one. For polynomials with complex coefficients, the eñe product acts as a multiplicative convolution of their divisor. We study its algebraic properties, its relation to symmetric functions on an infinite number of variables, to tensor products, and Hecke operators. The exponential linearizes also the eñe product. The eñe product extends to rational functions and formal meromorphic functions. We also study the analytic properties over the complex numbers, and for entire functions. The eñe product respects Hadamard-Weierstrass factorization and is related to the Hadamard product. The eñe product plays a central role in predicting the phenomenon of the "statistics on Riemann zeros" for Riemann zeta function and general Dirichlet $L$-functions discovered by the author. It also gives reasons to believe in the Riemann Hypothesis as explained in the survey "Notes on the Riemann Hypothesis".

In this paper, we give an axiomatic characterization of the cap product in the Hochschild theory of associative unital algebras which are projective over a commutative unital ring. We also give an interpretation of the cap product with coefficients in the algebra via chain maps. We illustrate these results by computing the cap product for truncated polynomial algebras $k[x]/(x^N)$ and for polynomial algebras, where it is identified with the contraction of differential forms by polyvector fields.

Software is everywhere, from mission critical systems such as industrial power stations, pacemakers and even household appliances. This growing dependence on technology and the increasing complexity software has serious security implications as it means we are potentially surrounded by software that contain exploitable vulnerabilities. These challenges have made binary analysis an important area of research in computer science and has emphasized the need for building automated analysis systems that can operate at scale, speed and efficacy; all while performing with the skill of a human expert. Though great progress has been made in this area of research, there remains limitations and open challenges to be addressed. Recognizing this need, DARPA sponsored the Cyber Grand Challenge (CGC), a competition to showcase the current state of the art in systems that perform; automated vulnerability detection, exploit generation and software patching. This paper is a survey of the vulnerability detection and exploit generation techniques, underlying technologies and related works of two of the winning systems Mayhem and Mechanical Phish.

Recently, graphics processors (GPUs) have been increasingly leveraged in a variety of scientific computing applications. However, architectural differences between CPUs and GPUs necessitate the development of algorithms that take advantage of GPU hardware. As sparse matrix vector multiplication (SPMV) operations are commonly used in finite element analysis, a new SPMV algorithm and several variations are developed for unstructured finite element meshes on GPUs. The effective bandwidth of current GPU algorithms and the newly proposed algorithms are measured and analyzed for 15 sparse matrices of varying sizes and varying sparsity structures. The effects of optimization and differences between the new GPU algorithm and its variants are then subsequently studied. Lastly, both new and current SPMV GPU algorithms are utilized in the GPU CG Solver in GPU finite element simulations of the heart. These results are then compared against parallel PETSc finite element implementation results. The effective bandwidth tests indicate that the new algorithms compare very favorably with current algorithms for a wide variety of sparse matrices and can yield very notable benefits. GPU finite element simulation results demonstrate the benefit of using GPUs for finite element analysis, and also show that the proposed algorithms can yield speedup factors up to 12-fold for real finite element applications.

Backside power-delivery-network (BSPDN) schemes require wafer-to-wafer bonding steps that do not leave high order shape changes or localized stresses in the bonded stack to ensure that these don't get transferred to the thinned wafer for further lithographic exposure. However, bonding mechanics involve strong adhesive forces which can inherently create localized distortions that lithography tools must eventually compensate. Some contributors to grid distortion of the target wafer are, method of bond initiation, bond front velocity variations, and lack of symmetry between the wafers. In this work, we evaluate a low-distortion bonding approach in the SUSS XBA tool, where some of these contributors are tackled at the source, namely by initiating the bond without a localized external force, and keeping the wafers compliant & symmetric during bonding. Wafers are bonded with a slight pre-stress due to controlled gas pressure applied over the whole backside of the wafers throughout bonding. Wafer-shape measurements of the bonded stacks are used to perform gradient-based in-plane-displacements (IPD) modelling to estimate bonding-induced grid distortion. Dense scanner metrology is used on patterned-bonded wafers to confirm the location and severity of distortion predictions from patterned wafer geometry (PWG) measurements. On the PWG distortion maps and scanner grid readouts, we perform alignment and CPE modelling with different field layouts. Sub-10 nm levels of residual grid distortion are achievable with relatively low-order correction models, and less than or equal to 3 nm by using advanced CPE models. These results demonstrate that pneumatically warped bonding yields low-distortion bonded stacks, and that simple process tuning can decouple the dominant distortion source from edge-related variability.

Rare $α$-particle-induced charge clusters appear in LSST images as compact, PSF-like sources with a median FWHM of $0.\!\!^{\prime\prime}95$ and median ellipticity consistent with zero, closely resembling unresolved astrophysical point sources. These events are detected in both dark and science exposures at a rate of approximately $10^{-12}\ \mathrm{pixel}^{-1}\ \mathrm{s}^{-1}$. Their collected charge and morphology are consistent with energy deposition from $\sim$5 MeV $α$-particles in silicon CCDs, and their spatial distribution across the focal plane suggests a localized material origin, plausibly associated with trace radioactive contamination in the cryostat aluminum. Despite their deceptive appearance, we demonstrate that a simple broadness statistic based on fourth-order moments cleanly separates these events from stellar PSFs, enabling efficient rejection in coadded images and real-time alert streams. Such charge clusters do not impose an intrinsic bright-end contamination floor for Rubin transient searches, as genuine fast astrophysical events would exhibit characteristically different morphological signatures.