The precision era of cosmology demands accurate theoretical predictions from inflationary models. In quantitative reheating analyses, inflationary observables depend sensitively on the number of e-folds between horizon exit and the end of inflation, $N_k$, whose determination relies on slow-roll approximations near the end of inflation. Since inflation ends when the first slow-roll parameter reaches unity, even modest inaccuracies in this approximation can shift the end of inflation and thereby alter $N_k$, leading to modifications in predicted observables -- including those evaluated at leading-order. While such effects are implicit in standard treatments, their quantitative impact on observable constraints has not been systematically assessed. In this work, we first re-evaluate leading-order slow-roll predictions using an improved determination of $N_k$ within a simple quantitative reheating framework, and then incorporate higher-order slow-roll corrections consistently with the revised background evolution. Applying this framework to the Starobinsky model, we find that improved end-of-inflation dynamics alone can induce shifts of order $Δn_s \sim 10^{-3}$, while higher-order slow-roll corrections provide additional refinements at the $\sim 4 \times 10^{-4}$ level. The cumulative effect yields a maximum shift of $Δn_s \sim 1.2 \times 10^{-3}$ within the allowed reheating range. To our knowledge, this is the first systematic decomposition of end-of-inflation corrections and their individual contributions to $n_s$ in the Starobinsky model, with implications for model discrimination in next-generation CMB surveys. These results demonstrate that an accurate determination of the end of inflation is essential for precision tests of inflationary models.

Fractons are exotic quasiparticles whose mobility in space is restricted by symmetries. In potential real-world realisations, fractons are likely lodged to a physical material rather than absolute space. Motivated by this, we propose and explore a new symmetry principle that restricts the motion of fractons relative to a physical solid. Unlike models with restricted mobility in absolute space, these fractonic solids admit gauge-invariant momentum density, are compatible with boost symmetry, and can consistently be coupled to gravity. We also propose a holographic model for fractonic solids.

We generalize Iarrobino's symmetric decomposition for the associated graded algebra of an Artinian Gorenstein algebra to a symmetric decomposition of finite-length self-dual modules over a local algebra, and we deduce consequences for the Hilbert functions of such self-dual modules. We classify the local Hilbert functions for small degree modules. We generalize Kunte's criterion for self-duality in terms of Macaulay's inverse systems.

We continue a series of papers studying the parity of families of eta-quotients, which provide implications for the parity of the partition function as well as an overarching conjecture on related $q$-series. The present article focuses on two classes. One consists of eta-quotients of the form $f_t^3/f_1$, a distinguished case of Andrews' singular overpartitions that has recently attracted attention among researchers. In addition, we investigate the parity of certain pure eta-powers $f_1^t$, appending new results to known density theorems.

Measurement-induced quantum computation with continuous-variable cluster states utilizes teleportation to transmit and alter quantum states via measurement-and-feedforward control. One of the key challenges of this approach is the deterioration of quantum states caused by the noise added due to imperfect entanglement of the cluster. We analyze the propagation of a quantum non-Gaussian state with nonlinear squeezing through a small cluster state. We show that a nonlinear feedforward in the deterministic teleportation protocol reduces the added noise and improves the nonlinear squeezing transferred. In a probabilistic regime, the improvement can be manifested even with current experimental resources. Better processing of non-Gaussian states can bring us closer to the necessary interplay between cluster states and non-Gaussianity required by quantum computing.

We revisit a classical problem in dynamic storage allocation. Items arrive in a linear storage medium, modeled as a half-axis, at a Poisson rate $r$ and depart after an independent exponentially distributed unit mean service time. The arriving item sizes (lengths) are assumed to be independent and identically distributed (i.i.d.) from a common distribution $H$. A widely employed algorithm for allocating the items is the "first-fit" discipline, namely, each arriving item is placed in the left-most vacant interval large enough to accommodate it. In a seminal 1985 paper, Coffman, Kadota, and Shepp ([6]) proved that in the special case of unit length items (i.e. degenerate $H$), as $r$ tends towards infinity, the first-fit algorithm is asymptotically optimal in the following sense: the steady-state ratio of expected "empty space" (gaps between items) to expected occupied space tends towards $0$. In a sequel to [6], Coffman, Kadota, and Shepp ([5]) conjectured that the first-fit discipline is also asymptotically optimal for non-degenerate $H$. In this paper we provide the first proof of first-fit asymptotic optimality for non-degenerate distributions $H$ of item sizes. Our main result is for the case when $H$ is concentrated on countably many positive real sizes forming an increasing sequence that is either finite or goes to infinity, with the average item size being finite. We prove that under the first-fit discipline, as $r$ tends towards infinity, the steady-state packing configuration (scaled down by $r$) converges in distribution to the limiting packing configuration with smaller items on the left, larger items on the right, and with no gaps between. In particular, this proves asymptotic optimality of first-fit in the sense that in steady-state the empty space (scaled down by $r$) vanishes.

The (Local) Lifting Property ((L)LP) is introduced by Kirchberg and deals with lifting completely positive maps. We give a characterization of the (L)LP in terms of lifting $\ast$-homomorphisms. We use it to prove that if $A$ and $B$ have the LP and $F$ is their finite-dimensional C*-subalgebra, then $A\ast_F B$ has the LP. This answers a question of Ozawa. We prove that Exel's soft tori have the LP. As a consequence we obtain that $C^*(F_n\times F_n)$ is inductive limit of RFD C*-algebras with the LP. We prove that for a class of C*-algebras including $C^*(F_n\times F_n)$, all contractible C*-algebras and all suspensions, the LLP is equivalent to Ext being a group. As byproduct of methods developed in the paper we generalize Kirchberg's theorem about extensions with the WEP, give short proofs of several, old and new, facts about soft tori, new unified proofs of Li and Shen's characterization of RFD property of free products amalgamated over a finite-dimensional subalgebra and Blackadar's characterization of semiprojectivity of them.

We extend generalized functional linear models under independence to a situation in which a functional covariate is related to a scalar response variable that exhibits spatial dependence-a complex yet prevalent phenomenon. For estimation, we apply basis expansion and truncation for dimension reduction of the covariate process followed by a composite likelihood estimating equation to handle the spatial dependency. We establish asymptotic results for the proposed model under a repeating lattice asymptotic context, allowing us to construct a confidence interval for the spatial dependence parameter and a confidence band for the regression parameter function. A binary conditionals model with functional covariates is presented as a concrete illustration and is used in simulation studies to verify the applicability of the asymptotic inferential results. We apply the proposed model to a problem in which the objective is to relate annual corn yield in counties of states in the Midwestern United States to daily maximum temperatures from April to September in those same geographic regions. The extension to an expanding lattice context is further discussed in the supplement.

Given two real symmetric matrices, their eigenvalue configuration is the relative arrangement of their eigenvalues on the real line. In this paper, we consider the following problem: given two parametric real symmetric matrices and an eigenvalue configuration, find a simple condition on the parameters such that their eigenvalues have the given configuration. We give an algorithm which expresses the eigenvalue configuration problem as a real root counting problem of certain symmetric polynomials, whose roots can be counted using the Fundamental Theorem of Symmetric Polynomials and Descartes' rule of signs.

We demonstrate remnant structural universality in a putative S-wave $2n$-halo-bound $D^0nn$ system in the $J=0, T=3/2$ channel by invoking the zero-coupling limit (ZCL), which eliminates sub-threshold decay channels. Within this framework, we evaluate the one- and two-body matter density form factors, their associated root mean-square radii, and the $n$-$D^0$-$n$ opening angle. Our analysis is carried out at leading order using a quantum mechanical Faddeev technique in the momentum representation. Employing Jacobi momenta, we construct a complete partial-wave basis to expand the full three-body $D^0nn$ wave function across distinct rearrangement channels. Projection onto this basis yields a coupled set of Faddeev integral equations that govern the multiple-scattering dynamics of the constituent coupled spin-isospin subsystems. By introducing short-range separable interactions and expressing the two-body scattering amplitudes via spectator functions, we establish a direct correspondence with the familiar Skornyakov-Ter-Martirosyan equations from halo-EFT approach at leading order. A regulator-dependent analysis highlights the Efimov-like character of the three-body observables, with ground state properties exhibiting marked sensitivity to cutoff variations. However, the inclusion of a three-body force suppresses this dependence, as expected from renormalization-group invariance. We thereby conclude that, for sufficiently shallow three-body binding, the $D^0nn$ system in the ZCL exhibits a universal halo-bound structure. The subtle implications of range-like corrections at LO are addressed at a qualitative level in this analysis.

The acceleration mechanisms of cosmic rays (CR) in supernova remnants (SNRs) and their associated compact central objects (CCOs) remain an open question in high-energy astrophysics. In this study, we perform a modeling of CR transport and gamma-ray emission from SNR G296.5+10.0 and its CCO 1E 1207.4-5209, using the latest public release of the GALPROP code (v57) and focusing, in particular, on the contribution from the CCO. Our simulations predict the contribution of CR from this source to the Galactic flux, accounting for energy losses and particle interaction processes. We find that, under time-evolving scenarios, the environment around SNR G296.5+10.0 and 1E 1207.4-5209 is suitable for CR acceleration and gamma-ray production. The analysis distinguishes between gamma rays produced by hadronic interactions in SNR G296.5+10.0 and by leptonic processes in CCO 1E 1207.4-5209, revealing that each mechanism dominates in different energy bands. We show that the Cherenkov Telescope Array Observatory (CTAO) can detect this emission with a significance of 5σ after 50 h of exposure, providing the first constraints on particle acceleration in this unique CCO-SNR system. These findings suggest that CCOs may be efficient electron accelerators, even in the absence of pulsar wind nebulae, and emphasize the critical role of next-generation observatories such as CTAO in unraveling CR acceleration processes in low-luminosity SNR-CCO systems.

This paper presents a joint theoretical and numerical study of a stochastic version of the compressible Navier-Stokes equations within the location uncertainty (LU) framework, applied to problems related to upper ocean vertical mixing. This approach builds on an extended stochastic form of the Reynolds transport theorem, incorporating stochastic source terms. As in the deterministic case, this conservation theorem is applied to mass, mass of species (such as salinity), momentum, and total energy, leading to transport equations for the primitive variables: density, mass fraction of species, velocity, and temperature. We subsequently apply the Boussinesq approximations to this general system, and recover existing formulations of the incompressible stochastic Navier-Stokes and stochastic Boussinesq equations. We employ this new framework in a Boussinesq large-eddy simulation of temperature-driven free convection event, and highlight the potential of stochastic transport to reproduce penetrative convection effects at the base of the mixed layer under the Boussinesq approximation. Compression effects identified in our stochastic Boussinesq hydrostatic model are found to be negligible in the temperature equation when expressed in terms of internal energy, in agreement with Boussinesq approximation. However, when expressed in terms of potential energy, compression effects become significant, and reveal interesting properties of the stochastic pressure terms within the mixed layer. We believe this later results open new physical modelling perspective enabling to represent oceanic dynamics within a stochastic framework that more fully accounts for physical uncertainties and approximations, while also providing a basis for improving the energetic consistency of subgrid-scale vertical models used in ocean general circulation models.

Every Model of High-Level Computation (MHC) has an underlying composition mechanism for combining simple computing devices into more complex ones. Composition can be done by (explicitly or implicitly) defining control flow, data flow or any combination thereof. Control flow specifies the order in which individual computations are activated, whereas data flow defines how data is exchanged among them. Unfortunately, traditional MHCs either mix data and control or only consider one dimension explicitly, which makes it difficult to reason about data flow and control flow separately. Reasoning about these dimensions orthogonally is a crucial desideratum for optimisation, maintainability and verification purposes. In this paper, we introduce a novel MHC that explicitly treats data flow and control flow as separate dimensions, while providing modularity. As the model is rooted in category theory, it provides category-theoretic operations for compositionally constructing sequential, parallel, branching or iterative composites. Compositionality entails that a composite exhibits the same properties as its respective constituents, including separation of concerns and modularity. We conclude the paper by demonstrating how our proposal can be used to model high-level computations in two different application domains: software engineering and artificial intelligence.

Let $\mathbb{F}_q[t]$ be the polynomial ring over the finite field $\mathbb{F}_q$ of $q$ elements. A polynomial in $\mathbb{F}_q[t]$ is called $m$-smooth (or $m$-friable) if all its irreducible factors are of degree at most $m$. In this paper, we investigate the distribution of $m$-smooth (or $m$-friable) polynomials with prescribed coefficients. Our technique is based on character sum estimates on smooth (friable) polynomials, Bourgains's argument (2015) applied for polynomials by Ha (2016) and on double character sums on smooth (friable) polynomials.

The CDF II detector at the Tevatron collider reported significant tension between the measurement of the W boson mass and the Standard Model prediction, assuming that 125 GeV scalar discovered at the LHC is the Higgs boson. We calculate one loop corrections to the W boson mass within the theory of noncontractible space without the Higgs boson. It turns out that our theory provides better agreement with the CDF II detector result than the Standard Model.

With probability 1, we assess the average behaviour of various arithmetic functions at the values of degree d polynomials f that are ordered by height. This allows us to establish averaged versions of the Bateman-Horn conjecture, the polynomial Chowla conjecture and to address a basic question about the integral Hasse principle for norm form equations. Moreover, we are able to quantify the error term in the asymptotics and the size of the exceptional set of f, both with arbitrary logarithmic power savings.

A novel theoretical framework, the inverse problem approach, is proposed to calculate non-perturbative quantities in quantum chromodynamics (QCD). Based on the dispersion relation of quantum field theory, this approach determines unknown low-energy non-perturbative quantities from known high-energy perturbative inputs via solving an inverse problem. The resulting inverse problem is rigorously proven to be ill-posed, with the solutions being unique but unstable. To address this instability, the well-established Tikhonov regularization is employed, yielding stable approximate solutions that converge to the true values as input errors vanish. The key features of this approach are illustrated through three toy models, demonstrating that solution precision can be systematically improved through reduced input errors and optimized regularization strategies.

The formulation of combinatorial differential forms, proposed by Forman for analysis of topological properties of discrete complexes, is extended by defining the operators required for analysis of physical processes dependent on scalar variables. The resulting description is intrinsic, different from the approach known as Discrete Exterior Calculus, because it does not assume the existence of smooth vector fields and forms extrinsic to the discrete complex. In addition, the proposed formulation provides a significant new modelling capability: physical processes may be set to operate differently on cells with different dimensions within a complex. An application of the new method to the heat/diffusion equation is presented to demonstrate how it captures the effect of changing properties of microstructural elements on the macroscopic behavior. The proposed method is applicable to a range of physical problems, including heat, mass and charge diffusion, and flow through porous media.

With the dual variational principle and the saddle point reduction we use the abstract bifurcation theory recently developed by author in previous work to prove many new bifurcation results for solutions of four types of Hamiltonian boundary value problems nonlinearly depending on parameters. The most interesting and important among them are those alternative results which can only be proved with our generalized versions of the famous Rabinowitz's alternative bifurcation theorem.

This work is a probabilistic study of the 'primes' of the Cramér model. We prove that there exists a set of integers $\mathcal S$ of density 1 such that \begin{equation}\liminf_{ \mathcal S\ni n\to\infty} (\log n)\mathbb{P} \{S_n\ \hbox{prime} \} \ge \frac{1}{\sqrt{2πe}\, }, \end{equation} and that for $b>\frac12$, the formula \begin{equation} \mathbb{P} \{S_n\ \text{prime}\, \} \, =\, \frac{ (1+ o( 1) )}{ \sqrt{2πB_n } } \int_{m_n-\sqrt{ 2bB_n\log n}}^{m_n+\sqrt{ 2bB_n\log n}} \, e^{-\frac{(t - m_n)^2}{ 2 B_n } }\, {\rm d}π(t), \end{equation} in which $m_n=\mathbb{E} S_n,B_n={\rm Var }\,S_n$, holds true for all $n\in \mathcal S$, $n\to \infty$. Further we prove that for any $0<η<1$, and all $n$ large enough and $ ζ_0\le ζ\le \exp\big\{ \frac{c\log n}{\log\log n}\big\}$, letting $S'_n= \sum_{j= 8}^n ξ_j$, \begin{eqnarray*} \mathbb{P}\big\{ S'_n\hbox{\ $ζ$-quasiprime}\big\} \,\ge \, (1-η) \frac{ e^{-γ} }{ \log ζ}, \end{eqnarray*} according to Pintz's terminology, where $c>0$ and $γ$ is Euler's constant. We also test which infinite sequences of primes are ultimately avoided by the 'primes' of the Cramér model, with probability 1. Moreover we show that the Cramér model has incidences on the Prime Number Theorem, since it predicts that the error term is sensitive to subsequences. We obtain sharp results on the length and the number of occurrences of intervals $I$ such as for some $z>0$, \begin{equation}\sup_{n\in I} \frac{|S_n-m_n|}{ \sqrt{B_n}}\le z, \end{equation} which are tied with the spectrum of the Sturm-Liouville equation.

A wide range of natural and industrial processes involve heat and mass transport in porous media. In some important cases the transported substance may undergo phase change, e.g. from liquid to solid and vice versa in the case of freezing and thawing of soils. The predictive modelling of such phenomena faces physical (multiple physical processes taking place) and mathematical (evolving interface with step change of properties) challenges. In this work, we develop and test a non-local approach based on bond-based peridynamics which addresses the challenges successfully. Our formulation allows for predicting the location of the interface between phases, and for calculating the temperature and pressure distributions within the saturated porous medium under the conditions of pressure driven water flow. The formulation is verified against existing analytical solutions for 1D problems, as well as finite element transient solutions for 2D problems. The agreement found by the verification exercise demonstrates the accuracy of the proposed methodology. The detailed coupled description of heat and hydraulic processes can be considered as a critical step towards a thermo-hydro-mechanical model, which will allow, for example, description of the hydrological behaviour of permafrost soils and the frost heave phenomenon.

A direct formulation of linear elasticity of cell complexes based on discrete exterior calculus is presented. The primary unknown are displacements, represented by primal vector-valued 0-cochain. Displacement differences and internal forces are represented by primal vector-valued 1-cochain and dual vector-valued 2-cochain, respectively. The macroscopic constitutive relation is enforced at primal 0-cells with the help of musical isomorphisms mapping cochains to smooth fields and vice versa. The balance of linear momentum is established at primal 0-cells. The governing equations are solved as a Laplace equation with a non-local and non-diagonal material Hodge star. Numerical simulations of several classical problems with analytic solutions are presented to validate the formulation. Good agreement with known solutions is obtained. The formulation provides a method to calculate the relations between displacement differences and internal forces for any lattice structure, when the structure is required to follow a prescribed macroscopic elastic behaviour. This is also the first and critical step in developing formulations for dissipative processes in cell complexes.

A formulation of elliptic boundary value problems is used to develop the first discrete exterior calculus (DEC) library for massively parallel computations with 3D domains. This can be used for steady-state analysis of any physical process driven by the gradient of a scalar quantity, e.g. temperature, concentration, pressure or electric potential, and is easily extendable to transient analysis. In addition to offering this library to the community, we demonstrate one important benefit from the DEC formulation: effortless introduction of strong heterogeneities and discontinuities. These are typical for real materials, but challenging for widely used domain discretization schemes, such as finite elements. Specifically, we demonstrate the efficiency of the method for calculating the evolution of thermal conductivity of a solid with a growing crack population. Future development of the library will deal with transient problems, and more importantly with processes driven by gradients of vector quantities.

We prove that QCSP$(\mathbb{N};x=y\rightarrow y=z)$ is PSpace-complete, settling a question open for more than ten years. This completes the complexity classification for the QCSP over equality languages as a trichotomy between Logspace, NP-complete and PSpace-complete. We additionally settle the classification for bounded alternation QCSP$(Γ)$, for $Γ$ an equality language. Such problems are either in Logspace, NP-complete, co-NP-complete or rise in complexity in the Polynomial Hierarchy.

We construct a coherent Hopf 2-algebra in terms of Hopf coquasigroups, which relax the coassociativity condition and generalize the results in \cite{XH2023}. We also study quasi coassociative Hopf coquasigroups, and show that they give rise to coherent Hopf 2-algebras with nontrivial coassociators. As an example, we investigate the algebra of functions on a Cayley algebra basis.

This paper presents an experimental performance study of implementations of three symbolic algorithms for solving band matrix systems of linear algebraic equations with heptadiagonal, pentadiagonal, and tridiagonal coefficient matrices. The only assumption on the coefficient matrix in order for the algorithms to be stable is nonsingularity. These algorithms are implemented using the GiNaC library of C++ and the SymPy library of Python, considering five different data storing classes. Performance analysis of the implementations is done using the high-performance computing (HPC) platforms "HybriLIT" and "Avitohol". The experimental setup and the results from the conducted computations on the individual computer systems are presented and discussed. An analysis of the three algorithms is performed.

We study asymptotic distribution of zeros of random holomorphic sections of high powers of positive line bundles defined over projective homogenous manifolds. We work with a wide class of distributions that includes real and complex Gaussians. As a special case, we obtain asymptotic zero distribution of multivariate complex polynomials given by linear combinations of orthogonal polynomials with i.i.d. random coefficients. Namely, we prove that normalized zero measures of m i.i.d random polynomials, orthonormalized on a regular compact set $K\subset \Bbb{C}^m,$ are almost surely asymptotic to the equilibrium measure of $K$.

We study ergodic properties of compositions of holomorphic endomorphisms of the complex projective space chosen independently at random according to some probability distribution. Along the way, we construct positive closed currents which have good invariance and convergence properties. We provide a sufficient condition for these currents to have Hölder continuous quasi-potentials. We also prove central limit theorem for d.s.h and Hölder continuous observables.

In this note, we prove, for instance, that the automorphism group of a rational manifold X which is obtained from CP^k by a finite sequence of blow-ups along smooth centers of dimension at most r with k>2r+2 has finite image in GL(H^*(X,Z)). In particular, every holomorphic automorphism $f:X\to X$ has zero topological entropy.

We study limiting distribution of the sequence of pull-backs of smooth $(1,1)$ forms and positive closed currents by meromorphic self-maps of compact Kähler manifolds.