Papers

Molecular mixing models for the conditional scalar diffusion (CSD) term in the filtered mass density function (FMDF) transport equation used in Large Eddy Simulation (LES-FMDF) approach are evaluated against direct numerical simulation (DNS) of spatially evolving jets. A database of DNS of turbulent mixing slot jets of $C_{7}H_{16} / O_2$ and $C_{7}H_{16} / Air$ is developed. The formulation includes the compressible form of the governing equations, a generalized multicomponent diffusion model, a cubic real gas equation of state and realistic property models. Simulations are conducted over a wide range of initial pressures ($1 atm < P_0 < 100 atm$) and jet width based Reynolds numbers of 850 and 1300. FMDF of mixture fraction at various filter widths are obtained from the simulation at several spatial locations within the flow. The CSD term in the exact transport equation of FMDF is calculated from the DNS. An a priori analysis of Interaction by Exchange of Mean (IEM), Modified Curl (MC) and Mapping Closure (MAPPING) mixing models for CSD is conducted. At high pressures, significant deviations are observed between the CSD measured from the DNS and those predicted by the models. The source of the deviations is determined to be the mixing frequency used in the models. The significance of the mixing frequency and its variation with change in pressure and diffusion models is studied. The primary objective of this work is to assess the applicability of aforementioned mixing models to flows at large pressures and determine the causes for deviation in prediction.

This work introduces innovation-based sampling of continuous stochastic processes, in which a sample is generated only when the process reveals “sufficiently new” information relative to all previous samples. The resulting innovation process admits a lattice structure and a three-dimensional state representation. For a Wiener process, we characterize the direction, timing, and frequency of innovations. We show that direction reversals become increasingly rare and establish limit theorems for their number. Innovations also become progressively sparser: the expected number of innovations grows only as the square root of time, and hence the sampling rate vanishes asymptotically. We then study remote estimation from sparsely received innovations. Notably, the absence of an innovation carries information: the minimum mean-square error (MMSE) estimate evolves with the age of information (AoI) and achieves a substantial MSE reduction over the conventional silence-ignorant zero-order hold (ZOH) estimator. Finally, we develop tractable affine-age and exponential-age approximations for practical use. Overall, information is conveyed not only by the content of innovations, but also by their direction and timing.

For integers $m\geq 2$ and $m\mid N$, let $G(N;m)=\gcd\{\binom{N}{k}:0<k<N,\ m\mid k\}$. We prove a complete $p$-adic valuation formula for $G(N;m)$ at primes $p\equiv -1\pmod m$, under the hypotheses $m\geq 3$, $m\mid N$, and $m<N$. Writing $N=\sum_i d_i p^i$, put $A=\sum_{i\text{ even}}d_i$ and $B=\sum_{i\text{ odd}}d_i$. Then $v_p(G(N;m))$ is $2$ in the exceptional mixed case $(A,B)=(1,1)$ with $p=m-1$, is $1$ in the mixed case $(A,B)=(1,1)$ with $m<p$, is $1$ in the one-parity cases $(A,B)=(m,0)$ and $(A,B)=(0,m)$, and is $0$ otherwise. The proof uses Kummer's theorem to translate the problem into digitwise borrow counts and a minimal signed zero-sum classification. A source-level literature audit through 25 September 2026 located no equivalent prior theorem for the full minus-one classification: McTague's corrected same-residue extension covers the one-parity $p>m$ subcases, but not the mixed-parity branch or the $p=m-1$ regime. The theorem, its plus-one companion, a scaling reduction, and the $m=3,4,6$ specializations have been formalized in Lean 4 / Mathlib.

We discuss some contexts in which the topological dynamics of certain Anosov subgroups of higher-rank Lie groups on "bad" subsets of flag manifolds can be analyzed using results from homogeneous dynamics in the infinite-covolume rank-one setting. For example, we show that a group of projective transformations dividing a strictly convex domain in projective space and intersecting Zariski-densely the stabilizer of an ellipsoid has a dense orbit in the complement of the domain, and acts minimally on the space of full projective flags tangent to the domain. This accounts for all known examples of divisible strictly convex domains in sufficiently high dimensions. We also provide examples of Zariski-dense groups that are Anosov in a partial flag manifold but such that the equivariant projection from the Benoist-Guivarc'h limit set in the Furstenberg boundary to the Anosov limit set is not a fibration.

We study weak and strong forbidden subposet problems in the linear lattice $L_n(q)$. We show that the $q$-analogues of the Bukh–Griggs–Lu conjecture fail for every prime power $q$. If $q\ge 3$ and $n$ and $d$ have opposite parity, then $La_q(n,L_d(q))=La_q^*(n,L_d(q))=\Sigma_q(n,d)$. We also show that $La_q(n,D_s)=La_q^*(n,D_s)=\Sigma_q(n,2)$ for $2\le s\le q$ and all $n$, and for $s=q+1$ when $n$ is odd. For $n$ even, a maximum strong $D_{q+1}$-free family can be contained from the three middle layers. The case $s=2$ settles the $q$-analog of the diamond conjecture in the affirmative.

We study the finite element approximation of the Dirichlet problem for the Bessel $(p,s)$-Laplacian, $\operatorname{div}^{s} \left(|\nabla^{s} u|^{p-2}\nabla^{s} u\right)$, built from the Riesz fractional gradient $\nabla^s$ and posed on the Bessel potential space $X^{s,p}$, obtained by complex interpolation between $L^p$ and $W^{1,p}$. For $p = 2$ this operator reduces to the fractional Laplacian, while for $p \neq 2$ it differs from the fractional $p$-Laplacian arising from real interpolation. Since a direct Galerkin discretization requires reassembling a dense stiffness matrix at every nonlinear iteration, we propose an augmented Lagrangian formulation based on the projected fractional gradient $B_h = \mathbf{\Pi}_h \nabla^s$, in which the only dense matrix is assembled once and the nonlinearity decouples elementwise. We analyze the resulting consistency error, establish a priori convergence rates in $X^{s,p}$, conditional on a discrete inf-sup condition for the augmented Lagrangian scheme, and present numerical experiments that illustrate the convergence of the method and its ability to deal with degenerate and strongly nonlinear problems.

In 1977, Erdős posed the problem of determining the maximum number $f_r(n)$ of edges in an $n$-vertex $r$-uniform hypergraph in which all disjoint pairs of edges have distinct unions. Füredi later conjectured that, for every fixed $r\ge 4$ and all sufficiently large $n$, $f_r(n)=\binom{n-1}{r-1}+\lfloor \frac{n-1}{r}\rfloor$. In this paper, we prove this conjecture and determine all extremal configurations. Our proof combines a stability theorem for such dense hypergraphs with a delicate deletion argument applied to an associated bipartite $3$-graph. The stability theorem also resolves a conjecture of Mubayi.

In this paper, we obtain a classification of isoparametric submanifolds with arbitrary codimensions in products of simply-connected space forms, except for isoparametric hypersurfaces in $\mathbb{S}^n \times \mathbb{S}^m$ and $\mathbb{H}^n \times\mathbb{H}^m$.

Recently derived exact expressions for the distributions of scattering matrix elements show that all combinations of ingoing and outgoing spin orientations are equally distributed. We propose a model suitable for experimental realization that breaks this degeneracy by introducing a spin preference in the channels. With the help of supersymmetry we derive a corresponding nonlinear sigma model. We exactly calculate the distributions of real and imaginary part in first order of the parameter measuring the preference. Quite remarkably, while breaking the spin degeneracy leads to different distributions for the favored and unfavored spin orientation, the real and imaginary parts of these altered distributions are the same as in the case of unbroken spin degeneracy.

The 290-Theorem of Bhargava-Hanke completely classifies all universal quadratic forms over the rational integers in terms of a finite criterion set: a positive definite quadratic form is universal if and only if it represents every integer in this finite set. In this article, we study diagonal forms of higher degree over a totally real number field and prove the existence of a unique finite criterion set, minimal with respect to inclusion, for universal diagonal forms of higher degree. As part of the proof, we establish an analog of the asymptotic local-global principle for such forms and develop a local theory for the representation of integers by diagonal forms of higher degree.

Scattering theory is a key tool for the investigation of quantum systems. Universal approaches are of interest, particularly in the framework of Random Matrix Theory. Dyson's Threefold way classifies systems according to their behavior under time reversal. In these three cases the distribution of off-diagonal scattering matrix elements and cross sections were recently calculated. However, in many applications systems neither display full nor fully broken time reversal invariance, such as systems in tuneable external magnetic fields. The study of these systems has a long history, also in Random Matrix Theory. We investigate the influence of time reversal invariance breaking on the universal behavior of the mentioned distributions. Using supersymmetry we succeed in deriving exact expressions for the distribution of off-diagonal scattering matrix elements and cross sections in systems with half-integer or integer/no spin in the absence of further symmetries. We find that the distributions are symmetric while the scattering matrix elements are not. Furthermore, our results might considerably facilitate the data analysis of time reversal invariance breaking.

We intend to prove that for defocusing nonlinear Schrödinger equations, solutions with finite pseudo conformal energy must be global and scatter. This fact is first proved by Bourgain when $0<s_c<1$: the equation is locally well-posed in $H_x^{s_c}$, and if one further assumes that the initial data satisfies $xu_0\in L_x^2$, then the solution is global and scatters. The $s_c<0$ case remains not studied, and the best result is given by Beceanu, Deng, Soffer, and Wu: If the initial data is in $H_x^{s_c}$, radial, and compactly supported, then the solution is globally well-posed. In this paper, we extend Bourgain's result to the $s_c<0$ case. We prove that if the initial data satisfies $|x|^{-s_c}u_0\in L_x^2$ and $xu_0\in L_x^2$, or if the radial initial data satisfies $u_0\in \dot{H}_x^{s_c}$ and $xu_0\in L_x^2$, then the solution is global and scatters. Bourgain's result is based on the subcritical a priori control on $L_x^{p+2}$, while our argument relies on the supercritical bound on $\mathcal{F}\dot H_x^1$.

Let $K$ be an algebraically closed field of odd characteristic $p$, let $G= {\rm SL}_2(K)$, and let $V$ be the natural representation of $G$. Let $b_k$ denote the number of $G$-indecomposable factors of $V^{\otimes k}$, counted with multiplicity, and let $\delta_p=1-\log_{p^2}\!\bigl(\tfrac{p+1}{2}\bigr)$. Then there exists a smooth, strictly positive, multiplicatively $p^2$-periodic function $\omega(t)$ such that $b_k$ is asymptotic to $\omega(k)k^{-\delta_p}2^k$. We also show that $t^{-\delta}\omega(t)$ arises as the limiting density of renormalized convolutions of rescaled copies of a positive weight $3/2$ theta function, obtained from the boundary heat flux of the Dirichlet heat kernel on $(0,p)$.

Goodness-of-fit testing is a basic tool for assessing whether a fitted procedure has captured the systematic information contained in the covariates. While traditional theory has largely focused on parametric regression models, modern data analysis increasingly relies on flexible black-box learners, whose predictive success alone is insufficient to assess model accuracy. In this paper, we propose SPARK, a general framework for goodness-of-fit testing that applies to traditional statistical models and general black-box learning procedures, continuous and binary responses, and low- and high-dimensional predictors. Based on a debiasing strategy, the residuals from an initial fit of a learning procedure are projected onto nearly orthogonal directions to extract any remaining signal. To capture information across all projection directions, we propose a kernel-based projection method and establish both its asymptotic properties and the consistency of a bootstrap procedure. Comprehensive simulations and real data analyses illustrate the effectiveness and flexibility of our proposed method.

At the critical density, the classical density theory of the Bargmann–Fock space no longer determines exact zero sets. We show that this failure is substantial: the complete radial counting function together with any prescribed finite collection of angular moments does not determine exact realizability, and the finite-dimensional relation between the dimension of the vanishing space and the zero-insertion capacity breaks down in infinite dimension. For arbitrary locally finite multisets, we establish a variational characterization of exact zero sets in terms of a renormalized logarithmic potential and relative entropy. This characterization yields sharp criteria for several classes of critical configurations.

This work constructs explicit closed-form Green's functions for the Helmholtz equation with Robin boundary conditions on balls in all dimensions $d\ge 2$. Despite the canonical geometry, such kernels have remained unavailable because the Robin condition couples the field and its normal derivative, preventing the method of images and obstructing standard eigenfunction expansions. By a decomposition–expansion method, the free-space fundamental solution is expanded via Graf's addition theorem in two dimensions and the hyperspherical addition theorem in higher dimensions; the regular correction is then determined algebraically by matching Robin boundary data mode by mode. The resulting series converge absolutely on compact interior subsets and are computable to machine precision. These explicit kernels yield a complete characterisation of the resonance spectra: each spectral branch increases strictly with the impedance parameter, interpolating between Neumann and Dirichlet eigenvalues. We establish a low-frequency spectral gap with an explicit cutoff estimate, and a universal high-frequency mode spacing with a second-order correction that distinguishes the Robin condition from the Dirichlet and Neumann extremes. The closed-form expressions furnish exact benchmark solutions for wave simulations in impedance-matched cavities, eliminating geometric discretisation error and providing reference data for the validation of finite-element and boundary-element algorithms in acoustics, electromagnetism and quantum mechanics.

We determine the cotangent operator space structure at the origin of the tetrablock (denoted by $\mathbb E$). In particular we prove that $\operatorname{COT}_0(\overline{\mathbb E}) =\operatorname{MIN}(\ell_1^2\oplus_\infty\mathbb C)$ completely isometrically. This shows the existence of commuting tuple of operators having $\overline{\mathbb E}$ as a spectral set but not as a complete spectral set.

Scattering theory is at the core of gaining information about most quantum systems. In general, analytic progress is difficult and the investigation of universal features using statistical methods is of interest. Commonly this is carried out within the Heidelberg approach which models the Hamiltonian of the target by a random matrix. Hitherto, such investigations were performed for systems that are effectively spinless or have total angular momentum conservation. In light of recent experimental progress it has become relevant to also investigate systems with half-integer spin. Using supersymmetry, we derive the distribution for the real and imaginary parts of the scattering matrix elements and the corresponding cross sections in the case of symplectic symmetry. These results complete the derivations for Dysons threefold way.

We prove that every codimension-one holomorphic Anosov diffeomorphism of a compact connected complex manifold is biholomorphically conjugate to a hyperbolic automorphism of a complex torus. This verifies a conjecture of Ghys in the codimension-one case and, in particular, in complex dimension three.

This is the first in a series of papers extending Roe algebras and coarse index theory to the groupoid-equivariant setting. We introduce some techniques to develop a framework of Roe algebras and their K-theory for spaces equipped with proper actions of étale groupoids. For a fixed étale groupoid G and G-C* -algebra A, we construct a functor KC(-; G, A) from the category of locally compact Hausdorff proper G-spaces and equivariant proper continuous maps to the category of graded abelian groups, which provides a natural receptacle for an equivariant coarse index map. We study the existence of universal modules, an analog of ample modules in non-equivariant setting, and develop a decomposition technique to establish their existence. As an application, we prove that groupoid simplicial complexes satisfying suitable hypotheses admit universal modules.

For prosolvable groups, we provide conditions under which two locally conjugate supplements of a normal pronilpotent subgroup are conjugate, analogous to those given by Losey and Stonehewer for finite groups. Our proof relies on a primary-type decomposition of the first cohomology set which may be of independent interest. We also give local conditions for subgroup inclusion and a fixed-point result in the style of Glauberman.

We establish a quantitative stability estimate for codimension-one immersions into a round sphere. Let $(M,g)$ be an oriented Riemannian manifold and suppose that a prescribed shape operator is realised by a smooth isometric immersion $\theta:(M,g)\to\mathbb S^{n+1}$. We prove that, on every relatively compact strongly Lipschitz domain and for every $1<p<\infty$, any Sobolev immersion $\phi$ is close to $\theta$ modulo an ambient rotation, with the $W^{1,p}$-distances between both the immersions and their Gauss maps controlled by the $L^p$ stretching-plus-bending energies of $\phi$. No a priori bounds on the fundamental forms of $\phi$ are required. The proof proceeds by extending the immersions along normal geodesics and reducing the problem to an equidimensional geometric rigidity estimate on the sphere. A finite localisation and patching argument handles the possible non-injectivity of the normal extension of the reference immersion.

We study the decay rate for a quasilinear heat equation of the form $ \dot u-\Delta u = \operatorname{div B}(\nabla u)$. The term on the right-hand-side is interpreted as a nonlinear perturbation of the ordinary heat equation and we will prove an optimal decay rate for the solution. More precisely, under natural conditions on the operator $B \colon \R^d \to \R^d$, we show that for sufficiently regular initial data the solution decays as $t^{-d/4}$ and its gradient as $t^{-d/4 - 1/2}$ for large times, that is at the same rates as for the free heat equation. Furthermore, we prove that the difference between the quasilinear and the linear solution decays strictly faster than the solution of either equation generically does, that is that there is asymptotic simplification. All estimates are completely explicit and rely on variations of Fourier splitting techniques, originally introduced by Schonbek.

The Runge phenomenon in polynomial interpolation is often considered a classical analogue of the double descent phenomenon in machine learning. In this note, we explore overparameterized polynomial interpolation in three popular polynomial bases: Monomial, Chebyshev and Legendre basis with coefficients that are minimal in the $\ell^2$-norm (and, for the monomial basis, also those minimal in the $\ell^1$-norm). We present our results primarily for equidistant and Chebyshev data points, but many results are independent of the exact form of sampling.

This paper is devoted to studying asymptotic spreading speeds of a road-field reaction-diffusion predator-prey system, in which predators diffuse faster on the straight line $\R\times \{0\}$ than in the half plane $\R\times (0,+\infty)$. We give a sharp characterization of the spreading properties of solutions of the system with compactly supported initial values by appealing to the invasion speed of the prey when predators are absent, and the invasion speed of predators when the prey is saturated. Notably, since the line may accelerate the expansion of predators in a cone of directions, the invasion speed of predators varies with the direction. To overcome the difficulty arising from the coupling between the predator-prey interaction and the road-field interaction, we first derive pointwise comparisons between predators and the prey and then reduce the components of predators in the system to a truncated field-road system depending on the invasion speed of the prey. By using known results for the single-species field-road model, we further show the propagation properties of the truncated system. This step is very crucial. Once the propagation properties of the truncated system are established, we can obtain the expected spreading properties of solutions to the original system by applying the comparison principle and contradiction arguments, and constructing appropriate Lyapunov functionals.

We argue that the exponential quantum advantage in Shor's algorithm is broken under one-layer depolarizing noise of arbitrary small error rate, by analyzing a resonance breaking phenomenon in noisy quantum circuits. First, we express the distribution of measurements on $n$-qubit strings as the superposition of the $4^n$ wave functions in the Pauli path integral. In the noiseless case, the bit-strings achieving resonant peaks guaranteed a constant rate of successful measurements to factor a large number. Secondly, when the middle layer of the quantum circuit has independent depolarizing noise of rate $\lambda$, for any Hamming weight $d$, we obtain corresponding measurement rate bounded by $2(1-\lambda)^d$ for higher frequency terms and by $O(n^d)/2^{n/2}$ for low frequency terms. The rate approaches to zero as $n$ and $d$ approaches to infinity. Resonance breaking destroys the exponential quantum advantage. Furthermore, we design a classical factoring algorithm in polynomial time $O(n^d)$ to substitute the low frequency contribution.

Linked projective spaces are quiver Grassmannians of subspaces of dimension 1 of certain quiver representations. Degenerations of linear series produce these representations, with the limit divisors parameterized by the associated linked projective spaces. It is not known whether all linked projective spaces arise this way. If they do, they are degenerations of the (small) diagonal in a product of projective spaces. In any case, we prove here that they have the (multivariate) Hilbert polynomial of the diagonal. To achieve this, we first extend a Hilbert-polynomial formula for multiplicity-free varieties to (simple) normal-crossings schemes with multiplicity-free strata in products of projective spaces. Then we prove that a linked projective space is normal-crossings, by describing it locally in terms of Mustafin varieties. Finally, we use a relation between intersections of components of the linked projective space and certain polytopes in the tiling of a simplex associated to the linked net to prove that our formula for the Hilbert polynomial applies.

For a bounded linear operator on an infinite-dimensional Hilbert space, we consider the class $\beta(H)$ of operators whose restrictions to all nonzero reducing subspaces attain their norms. We prove that $\beta(H)$ is dense in $\mathcal{B}(H)$ in the operator norm. Since $\beta(H)$ is not stable under arbitrary compact perturbations, we introduce a natural subclass $\beta_0(H)$ and investigate its perturbation properties. We show that $\beta_0(H)$ is stable under finite-rank perturbations and obtain a complete characterization of those compact perturbations that preserve membership in $\beta_0(H)$. We also determine the maximal compactly perturbation-invariant subset of $\beta_0(H)$. These results clarify the spectral structure and stability of norm-attaining operators on reducing subspaces.

In two and three dimensions, we analyze a mixed formulation of the nonlinear Biot equations in terms of the poroelastic stress tensor, the displacement, the pressure, and the rotation tensor. To establish the well-posedness of the problem, we introduce suitable stabilization terms, leading to an augmented mixed formulation in which the associated parameters are chosen appropriately. Owing to the nonlinearity of the problem, we employ a fixed-point strategy to prove the existence and uniqueness of solutions. We propose a finite element scheme for which we establish existence and uniqueness of a discrete solution, together with a priori error estimates. Additionally, we design and analyze an a posteriori error estimator of the residual type which results to be reliable and efficient. Finally, we present a series of numerical experiments in two and three dimensions in order to assess the performance of the proposed method.

Let the theta and Epstein zeta functions be $\Theta(\alpha,L)=\sum_{v\in L}e^{-\pi\alpha|v|^2}$ for $\alpha>0$ and $E(L,s)=\sum_{v\in L\setminus\{0\}}{|v|^{-2s}}$ for $s>2$, respectively. We consider full-rank lattices $L\subset\mathbb R^4$. Let the covolume of $L$ be one, and let $\mathcal D_4=2^{-1/4}D_4$ be the root lattice $D_4$ rescaled to covolume one. We prove that, up to orthogonal transformations, \begin{equation}\nonumber arg\,min_{covol(L)=1}\Theta(\alpha,L)=\mathcal D_4 \qquad\text{if}\quad \alpha>0, \end{equation} and this implies that \begin{equation}\nonumber arg\,min_{covol(L)=1}E(L,s)=\mathcal D_4 \qquad\text{if}\quad s>2. \end{equation} This proves that the root lattice minimizes the theta and Epstein zeta functions among all lattices in dimension four, as conjectured by Sarnak-Strömbergsson (\cite{SS}, 2006), who established the local minimality of $\mathcal D_4$. Thereby, this resolves the Rankin-Sobolev problem in dimension four dating back to Éndibaev (\cite{End1978}, 1978) and Shushbaev (\cite{Shu1978}, 1978).