We prove that the volume of central hyperplane sections of a unit cube in $\mathbb{R}^n$ orthogonal to a diameter of the cube is a strictly monotonically increasing function of the dimension for $n\geq 3$. Our argument uses an integral formula that goes back to Pólya \cite{P} (see also \cite{H} and \cite{B86}) for the volume of central sections of the cube, and Laplace's method to estimate the asymptotic behaviour of the integral. First we show that monotonicity holds starting from some specific $n_0$. Then, using interval arithmetic (IA) and automatic differentiation (AD), we compute an explicit bound for $n_0$, and check the remaining cases between $3$ and $n_0$ by direct computation.

The most challenging problem of modern physics is how to reconcile quantum theory and general relativity, namely, to find a consistent quantum theory in which gravity is quantized. This Progress Report focuses on such a tentative theory called the information-complete quantum theory (ICQT), in which (1) spacetime (gravity) as a physical quantum system plays a central role for formulating the theory, and (2) there are no any classical systems and concepts. Here universal spacetime-matter entanglement "glues" spacetime and matter (matter fermions and their gauge fields) as an indivisible trinity, encodes information-complete physical predictions of the world, and is as universal as universal gravitation. After summarizing the basic theoretic structure of the ICQT, conceptual advances achieved so far and some new issues within the ICQT are considered. While such a theory integrating quantum gravity is of fundamental interest to a wider audience, its relevance to quantum information technologies is discussed, with emphasis on its potential impacts on quantum computing and quantum communication.

General $L_p$ dual curvature measures have recently been introduced by Lutwak, Yang and Zhang. These new measures unify several other geometric measures of the Brunn-Minkowski theory and the dual Brunn-Minkowski theory. $L_p$ dual curvature measures arise from $q$th dual inrinsic volumes by means of Alexandrov-type variational formulas. Lutwak, Yang and Zhang formulated the $L_p$ dual Minkowski problem, which concerns the characterization of $L_p$ dual curvature measures. In this paper, we solve the existence part of the $L_{p}$ dual Minkowski problem for $p>1$ and $q>0$, and we also discuss the regularity of the solution.

We present a brief history of the construction of models of the universe, followed by calculations of quantitative characteristics of basic geometric and kinematic properties of the Standard Cosmological Model ($Λ$CDM). Using the Friedmann equations of uniform space, we derive equations characterizing a $Λ$CDM model that describes a universe corresponding to current observational data. The equations take into account the effects of radiation and ultra-relativistic neutrinos. It is shown that the universe at very early and late stages can be described to sufficient accuracy by simple formulas. Certain important moments of cosmic evolution are determined: the times when densities of the gravitational components of the universe become equal, when they contribute equally to the gravitational force, when the accelerating expansion of space begins, and several others. The dependences of different distances on redshift and the scale factor of space are derived. The distance to the sphere that expands with the speed of light (the Hubble distance), and its current and future acceleration, are found. Concepts of a horizon, second inflation, and second horizon are discussed. We consider the remote future of the universe and the opportunity, in principle, of connection with extraterrestrial civilizations.

We strengthen the volume inequalities for L_p zonoids of even isotropic measures and for their duals, which are due to Ball, Barthe and Lutwak, Yang, Zhang. Along the way, we prove a stronger version of the Brascamp-Lieb inequality for a family of functions that can approximate arbitrary well some Gaussians when equality holds. The special case p=\infty yields a stability version of the reverse isoperimetric inequality for centrally symmetric bodies.

By creating some new concepts and methods: checking tree, long unit path, direct contradiction unit pair, indirect contradiction unit pair, additional contradiction unit pair, 2-unit layer and 3-unit layer, redundant units, and destroying parallel pairs , we successfully transform solving a 3SAT problem to solving 2SAT problems in polynomial time. Thus we proved that NP=P.

Semi-analytical evolution models of galaxies are a useful and computationally inexpensive tool for fast assessment of individual properties and their evolution. In this work, specifically the influence of a metallicity and star-formation rate (SFR) dependent galaxy-wide stellar initial mass function (IGIMF) on the self-regulation of star-formation in a galaxy is of interest. All models -- both non-varying gwIMFs and the IGIMF -- reproduce reasonable gas fractions, gas depletion timescales and the main sequence of star-forming galaxies. However, only the IGIMF model accurately predicts the mass-metallicity relation and provides a more comprehensive description of quenched elliptical galaxies. For massive ellipticals all models suggest the need for an additional gas heating source to reach a quenched state. Using a different stellar yield table in the IGIMF model does not significantly affect the results. In all models, the galaxies evolve self-regulated, determined by the accretion rate. The self-regulated constancy of the SFR reflects the constant SFRs of nearby star-forming galaxies. The specific gas-accretion rate of all galaxies appears to be comparable to the Hubble constant. The inclusion of outflows improves the results for the canonical gwIMF model, but not significantly, while for the IGIMF model it has no significant impact.

We describe novel local singularity models for $\mathbb Z/2$ harmonic 1-forms, self-dual 2-forms and spinors in dimension 4. These models are homogeneous versions on $\R^4$ whose singular sets are cones on the 1-skeletal of certain regular 4-dimensional polytopes.

We apply harmonic analysis to study the $T\bar{T}$-deformed torus partition function. We first express the CFT partition functions in terms of Maass waveforms, including the Eisenstein series and cusp forms. These basis functions turn out to deform in a very simple way under the $T\bar{T}$-deformation. The spectral decomposition provides a numerically stable and efficient method to compute the partition function at finite values of the deformation parameter $λ$, allowing us to clearly resolve the analytic structure of the partition function as a function of $λ$. The resulting deformed partition function exhibits a Hagedorn singularity. Building on harmonic analysis approach, we propose a natural analytic continuation beyond the Hagedorn singularity, which enables us to compute the full partition function for any value of $λ$.

Optical frequency combs have revolutionized precision science and technology, yet their nanophotonic implementations have failed to simultaneously achieve high efficiency, power, and coherence. Optically driven microcombs provide broad and stable spectra but low usable power, whereas active comb generators, including mode-locked lasers, can be efficient yet offer less control over coherence. We introduce the laser electro-optic (LEO) frequency comb, a comb-generation mechanism in which coherent continuous-wave injection drives a phase-modulated laser cavity above threshold into a distinct operating regime. Unlike other coherently driven integrated comb sources, the LEO comb generates comb powers that exceed the injected continuous-wave power by an order of magnitude. We realize the LEO comb in a hybrid lithium niobate/III-V nanophotonic circuit and demonstrate milliwatt-level power per comb line, 1.76-ps pulses, a 4.7-nm background-free spectrum, and linewidths as narrow as 19.6 kHz. By unifying high efficiency, power, and coherence, this architecture establishes a definitive route to chip-scale frequency comb sources that deliver on the promise of scalable, high-performance coherent optical technologies.

This paper proves that a wide class of local search algorithms extend as is to the fully dynamic setting with an adaptive adversary, achieving an amortized $\tilde{O}(1)$ number of local-search steps per update. A breakthrough by Moser (2009) introduced the witness-tree and entropy compression techniques for analyzing local resampling processes for the Lovász Local Lemma. These methods have since been generalized and expanded to analyze a wide variety of local search algorithms that can efficiently find solutions to many important local constraint satisfaction problems. These algorithms either extend a partial valid assignment and backtrack by unassigning variables when constraints become violated, or they iteratively fix violated constraints by resampling their variables. These local resampling or backtracking procedures are incredibly flexible, practical, and simple to specify and implement. Yet, they can be shown to be extremely efficient on static instances, typically performing only (sub)-linear number of fixing steps. The main technical challenge lies in proving conditions that guarantee such rapid convergence. This paper extends these convergence results to fully dynamic settings, where an adaptive adversary may add or remove constraints. We prove that applying the same simple local search procedures to fix old or newly introduced violations leads to a total number of resampling steps near-linear in the number of adversarial updates. Our result is very general and yields several immediate corollaries. For example, letting $Δ$ denote the maximum degree, for a constant $ε$ and $Δ= \text{poly}(\log n)$, we can maintain a $(1+ε) Δ$-edge coloring in $\text{poly}(\log n)$ amortized update time against an adaptive adversary. The prior work for this regime has exponential running time in $\sqrt{\log n}$ [Christiansen, SODA '26].

We consider minimax (saddle-point) problems of the form max_{c \in C} min_{β\in S} g(c; β), where C and S are compact convex sets, and g is concave-convex. Applying the Alternating Direction Method of Multipliers (ADMM) requires evaluating a proximal operator that is, in general, as hard as the original problem. We show that when the outcome function g is bilinear, i.e. g(c; β) = c^T A β, the proximal operator reduces to a generalized projection onto the confidence region S. This reduction is exact -- it involves no approximation or linearization. The resulting ADMM algorithm alternates between (i) a generalized projection onto S and (ii) a Euclidean projection onto C. We describe the derivation, state the algorithm, and discuss convergence.

We study the ground-state phase diagram of a spin-1 Heisenberg chain with staggered long-range (LR) interactions decaying as $\propto r^{-α}$ using a quantum Monte Carlo approach based on the split-spin representation. This formulation enables efficient large-scale simulations by mapping the spin-1 model onto spin-$1/2$ degrees of freedom with local projection constraints. We resolve the continuous quantum phase transition between the gapped Haldane phase at large $α$ (short-range regime) and a gapless antiferromagnetically ordered Néel phase at small $α$ (LR regime), where the continuous SU(2) symmetry is broken. From finite-size scaling and crossing point analyses, we determine the critical point to be at $α_c = 2.48(2)$ and extract the associated critical exponents, which indicate unconventional criticality. In particular, the transition is found to be nonconformal, characterized by a dynamic exponent $z \neq 1$. We further analyze the scaling of entanglement entropy and bipartite fluctuations across the transition, and determine the corresponding universal scalings in both phases and at criticality.

Semi-discrete (differential-difference) matrix Lax representations (Lax pairs) play an essential role in the theory of integrable differential-difference equations. Fix a (1+1)-dimensional evolutionary differential-difference (semi-discrete) equation and consider matrix Lax representations (MLRs) of this equation. Two MLRs are said to be gauge equivalent if one of them can be obtained from the other by applying a (local) matrix gauge transformation. Gauge transformations (GTs) form an infinite-dimensional group, which acts on the set of MLRs of a given equation. Two MLRs are gauge equivalent iff they belong to the same orbit of this action. When one tries to establish integrability (in the sense of soliton theory) for a given equation, one is interested in MLRs which depend on a parameter (usually called the spectral parameter) such that the parameter cannot be removed by any GT. We introduce and study explicit invariants with respect to the action of GTs on the set of MLRs for a given (1+1)-dimensional evolutionary differential-difference equation with any number of components. Using these invariants, we obtain the following results: - Consider a MLR with a parameter $λ$. If at least one of the invariants computed for this MLR depends nontrivially on $λ$, then the parameter cannot be removed by any GT. - When we have two different MLRs for a given equation, we present necessary conditions for these two MLRs to be gauge equivalent. Our results on semi-discrete MLRs of differential-difference equations are inspired by results of S.Yu. Sakovich and M. Marvan on (continuous) zero-curvature representations of partial differential equations. A comparison with some of the results of S.Yu. Sakovich and M. Marvan is presented.

We study exploration properties of a random walk on a network. For a fully connected network we find that the problem can be mapped to the well known coupon collector problem, thus allowing us to estimate form of $P(S,t)$: the distribution of number of distinct nodes $S$ visited by the random walk upto time $t$. From a practical point of view, however, both the fully connected network and hops taking place after fixed intervals are an idealization. We solve this problem by introducing the formalism of continuous time random walks wherein the random walk spends a random amount of time a node before hopping to its neighboring node. The formalism allows us to study the large deviation limit of $P(S,t)$ under very mild conditions that the distribution of waiting times $ψ(τ)$ exhibits analyticity at small times. Furthermore, we find that at small times, the properties of $P(S,t)$ are largely independent of the network topology, and are governed solely by the waiting time characteristics.

Measurement-induced phase transitions (MIPTs) in monitored quantum systems are typically diagnosed using entanglement-based measures. Here, we develop a complementary thermodynamic perspective based on the arrow of time (AoT), which arises from the intrinsic irreversibility of the quantum measurements driving these transitions. We study the AoT - defined as the logarithmic ratio of forward and backward trajectory probabilities - across a family of models exhibiting MIPTs. We find that, like entanglement entropy, the AoT is a nonlinear functional of the averaged density matrix; however, in contrast to entanglement, it is associated with a local operator. To determine whether the AoT exhibits critical behavior, we formulate and exactly solve a model of a random quantum circuit with non-projective measurements. This allows us to analytically demonstrate that the AoT displays nonanalytic behavior and identify its critical exponent. Our results establish the AoT as a novel diagnostic for phase transitions in monitored quantum systems.

Large-deviation upper bounds on compact sets do not, in general, extend to arbitrary closed sets without additional tightness. We show that this obstruction already occurs in static entropic optimal transport. More precisely, we construct a fixed-cost model with continuous cost and nonatomic marginals for which the entropic minimisers converge in total variation to an optimal plan with noncompact support, the known compact-set upper bound remains valid, but the corresponding closed-set upper bound fails on a specific closed subset of the ambient space. For a fixed closed set, we identify the exact tail criterion for passing from compact to closed sets. We show that there does not exist a full large-deviation principle (LDP) on the ambient space at speed $1/\varepsilon$ with an arbitrary lower semicontinuous rate function.

Phishing attacks remain one of the most prevalent threats to online security, with the Anti-Phishing Working Group reporting over 890,000 attacks in Q3 2025 alone. Traditional password-based authentication is particularly vulnerable to such attacks, prompting the development of more secure alternatives. This paper examines passkeys, also known as FIDO2, which claim to provide phishing-resistant authentication through asymmetric cryptography. In this approach, a private key is stored on a user's device, the authenticator, while the server stores the corresponding public key. During authentication, the server generates a challenge that the user signs with the private key; the server then verifies the signature and establishes a session. We present passkey workflows and review state-of-the-art attack vectors from related work alongside newly identified approaches. Two attacks are implemented and evaluated: the Infected Authenticator attack, which generates attacker-known keys on a corrupted authenticator, and the Authenticator Deception attack, which spoofs a target website by modifying the browser's certificate authority store, installing a valid certificate, and intercepting user traffic. An attacker relays a legitimate challenge from the real server to a user, who signs it, allowing the attacker to authenticate as the victim. Our results demonstrate that successful attacks on passkeys require substantial effort and resources. The claim that passkeys are phishing-resistant largely holds true, significantly raising the bar compared to traditional password-based authentication.

The internet folklore of the Cat Distribution System (CDS) humorously suggests that cats are "assigned" to people rather than intentionally sought. Beyond its playful origins, CDS reflects a culturally resonant way people perceive and engage in adoption, and this user context can guide the redesign and improvement of adoption systems. In the Philippines, where an estimated 13.11 million stray cats and dogs place the country sixth worldwide in overpopulation, this framing offers a novel way to rethink adoption platforms. We developed a prototype application inspired by CDS principles, focusing on features such as algorithmic matchmaking, community reporting, and proximity-based discovery. An initial evaluation with potential users (n=35) indicated that the system was positively received for its ease of use and its alignment with users' intuitive expectations, though participants highlighted areas for improvement in transparency of matchmaking and owner-adopter communication. The findings suggest that culturally embedded metaphors like CDS can shape mental models, making adoption processes feel more serendipitous and less transactional.

In this paper, we introduce $S$-prime elements in $V$-lattices, where $S$ is a multiplicatively closed subset of a $V$-lattice $L$. In addition, we introduce the $S$-Prime Element Principle to prove that certain elements in $V$-lattices are $S$-prime elements. This principle leads to a direct and uniform approach to the results on the existence of prime elements in multiplicative lattices when $S=\{1\}$.

Tridiagonal $k$-Toeplitz operators provide a natural framework for modelling one-dimensional $k$-periodic lattice systems. A fundamental connection is obtained between Coburn's lemma for tridiagonal $k$-Toeplitz operators and the existence of edge modes. We reveal that topological edge modes are characterised by the eigenvalues of the leading principal submatrix of the symbol function. A complete analysis of tridiagonal interface operators satisfying global inversion symmetry is then presented. These results are applied to finite one-dimensional $k$-periodic chains of damped resonators that satisfy both local and global inversion symmetry. Additionally, disordered tight-binding interface operators are shown to support a topologically robust zero-energy interface state. Numerical simulations are conducted to illustrate the theoretical findings.

We study the Zarankiewicz problem for $r$-partite, $r$-uniform intersection hypergraphs arising from $r$ families of axis-parallel boxes in $\mathbb{R}^d$ with prescribed directions $F_1, \dots, F_r \subseteq \{1, \dots, d\}$. This extends the problems studied by Chan and Har-Peled on points and $d$-dimensional boxes in $\mathbb{R}^d$, corresponding to $(F_1,F_2)=(\varnothing,[d])$, as well as by Chan, Keller, and Smorodinsky on $r$ families of $d$-dimensional boxes, corresponding to $(F_1,\dots,F_r)=([d],\dots,[d])$. Our main result establishes a sharp dichotomy for the Zarankiewicz number in this setting: it is either $Θ_r(tn^{r-1})$ or at least $Ω\bigl( tn^{r-1} \cdot \frac{\log n}{\log\log n} \bigr)$, depending only on a simple set-theoretic condition on $(F_1,\dots,F_r)$, which we call $2$-coherence. Informally, $2$-coherence captures whether the configuration contains an underlying two-dimensional incidence structure, which is precisely what gives rise to the extra polylogarithmic factor. Our proof proceeds via a sequence of reductions and a geometric slicing argument that reduces the problem to planar incidence bounds.

Fast radio bursts (FRBs) are millisecond-duration extragalactic radio transients of unknown origin. Rotation measures (RMs) probe their local magneto-ionic environments and provide important clues to their nature. While RM variability has been observed in several repeating FRBs, it is typically gradual or stochastic. Recently, observations of FRB~20220529 revealed an abrupt RM excursion followed by rapid recovery on week-long timescales, termed an ``RM flare'', suggesting a potentially distinct form of RM variability associated with localized magnetized plasma. In this work, we perform a systematic search for RM flare candidates in repeating FRBs with multi-epoch RM measurements. Using a $3σ$ significance threshold, we identify two candidates with multiple observational epochs (FRB~20121102A and FRB~20201124A) and two additional single-epoch candidates (FRB~20180916B), in addition to the event in FRB~20220529A. Our results suggest that RM flares, if confirmed, may not be rare among repeating FRBs and point to highly dynamic magnetized environments local to the sources. Future high-cadence polarimetric observations, particularly following the discovery of RM excursions, will be essential for confirming these candidates and constraining their physical origin.

We develop a method for the rigorous estimation of Hausdorff dimensions of limit sets produced by continued fraction iterated function systems. Our method is based on the approximation of a Perron-Frobenius operator using the finite element method with B-splines as the choice of basis functions. This choice provides key numerical advantages including higher-order convergence and computational flexibility. We prove an analogue of Falk and Nussbaum's result on "hidden positivity" for B-spline quasi-interpolants to give rigorous upper and lower bounds for the Hausdorff dimensions of various limit sets. We provide numerical results to verify both the rigor and higher-order convergence of our method for quadratic B-spline interpolants in one and two dimensions.

Synthetic DNA approaches 227.5 exabytes per gram of storage density with stability over millennial timescales. Realising this capacity requires error-correction codes that recover data from substantial synthesis and sequencing errors. Existing codecs convert noisy sequencer output into discrete base calls before error correction, discarding probabilistic information about which positions are reliable. Here we present a coding scheme that retains the sequencer's per-position posterior distributions through an integrated decoder of profile hidden Markov model alignment, log-product fusion across reads, and ordered-statistics decoding. On the DT4DDS channel simulator, the codec recovers 155.8 and 25.9 exabytes per gram of dsDNA under high- and low-fidelity conditions, exceeding the highest prior-art density on each channel by 11 and 52 percent. Under a single-encode-then-degrade protocol mapped to depurination kinetics at 25 °C in the dry state, the codec projects 282 years of decodable storage at 17.1 exabytes per gram. These results place DNA storage density within reach of the Shannon bound of the underlying channel.

The well-known Borde-Guth-Vilenkin Theorem shows that inflationary spacetimes are generically geodesically past-incomplete, necessitating the existence of a pre-inflationary boundary of some sort, possibly singular. I discuss the generalization of the BGV theorem to spacetimes beyond inflation, including inhomogeneous and cyclic models. As an example, I show that the cyclic model proposed by Ijjas and Steinhardt is geodesically incomplete.

We present an ongoing effort to build a framework and a library in Isabelle/HOL for formalising primal-dual arguments for the analysis of algorithms. We discuss a number of example formalisations from the theory of matching algorithms, covering classical algorithms like the Hungarian Method, widely considered the first primal-dual algorithm, and modern algorithms like the Adwords algorithm, which models the assignment of search queries to advertisers in the context of search engines.

We present a method for learning quantum hardware noise from a measurement distribution of a single device experiment. Each noise channel is represented by automatically differentiable Kraus operators obtained from a Stinespring-based parameterization that is completely positive and trace preserving by construction, and circuits are simulated with a matrix product density operator forward model. Independent channels are attached to each native gate type, to each nearest-neighbor crosstalk interaction, and to state preparation and measurement, and all channels are optimized end-to-end against a distance between the simulated and observed measurement distributions. On ibm_fez, a Heron-generation superconducting processor, training on a ripple-carry adder circuit reproduces the device output distribution, and the same learned parameters, applied without retraining, also track the device distribution of an unrelated multiplier circuit, indicating that the method captures intrinsic device characteristics rather than overfitting to the training circuit. A systematic evaluation across a range of benchmark circuits confirms that this generalization is consistent. We further use the learned model to perform an offline feasibility assessment of the quantum approximate optimization algorithm with an error detection scheme, demonstrating the kind of noise-aware prediction the framework is designed to enable.

Introductory Software Engineering (SE) courses face rapidly increasing student enrollment numbers, participants with diverse backgrounds and the influence of Generative AI (GenAI) solutions. High teacher-to-student ratios often challenge providing timely, high-quality, and personalized feedback a significant challenge for educators. To address these challenges, we introduce NAILA, a tool that provides 24/7 autonomous feedback for student exercises. Utilizing GenAI in the form of modern LLMs, NAILA processes student solutions provided in open document formats, evaluating them against teacher-defined model solutions through specialized prompt templates. We conducted an empirical study involving 900+ active students at the University of Duisburg-Essen to assess four main research questions investigating (1) the underlying motivations that drive students to either adopt or reject NAILA, (2) user acceptance by measuring perceived usefulness and ease of use alongside subjective learning progress, (3) how often and how consistently students engage with NAILA, and (4) how using NAILA to receive AI feedback impacts on academic performance compared to human feedback.

The solar interior is probed by the properties of the Sun's acoustic oscillations (p-modes) observed on the solar surface. The frequencies of these p-modes measured in the last three decades show long term variation similar to the 11 year cyclic behaviour exhibited by 10.7 cm radio flux, sunspot numbers and other solar activity indices. It is also now established that the cyclic behavior of some of the solar proxies are connected with geomagnetic activities and have implications for space weather. Hence, in recent years efforts have been made using machine-learning methods to forecast these solar proxies with a view to improve our understanding of space weather. Developing a comparable method for forecasting p-mode frequency shifts is therefore of interest for two reasons. Firstly, it will facilitate future investigations into its potential role in tracing energy drivers from the Sun's interior to the geospace response by improving models of solar interior dynamics to coronal and heliospheric plasma conditions. In other words, it will help establish a more robust and quantitative link between the Sun's interior and its exterior. Secondly, it may provide us with an independent indicator or an early indicator of ascending and descending phase of solar activity which might be useful for space weather forecasting. In this article, we develop and apply the standard time-series analysis and machine-learning based methods to characterise p-mode frequency shifts for the remaining solar cycle 25.