Papers

A basic dynamical model for (clinical) depression is presented to describe the time evolution of two coupled states: a resilience level and a depression symptom. The resilience level can also be interpreted in terms of the memory of past symptoms. The model consists of a system of two coupled first order differential equations without free parameters that qualitatively captures different courses of illness, without the overhead of a derivation from neuroscientific first principles. A comprehensive mathematical analysis of the model is provided, including equilibria, stability properties, and monotonicity. The model can reproduce chronic, delayed, recovery, and resilience scenarios that are prominent in the literature, as well as scenarios such as improvement from pre-existing conditions, burnout from low-grade adversity, or isolated and recurrent depressive episodes. Supplementary to the manuscript are computational tools to replicate the figures and, without prior mathematical or programming skills, to experiment with the model interactively in the web browser (at https://rsmodel.org/).

Newton's iteration is a fundamental tool for root-finding and numerical solutions of systems of equations. The iteration rapidly refines the initial approximation to the exact root, and in general the convergence is quadratic. Since the method requires finding the function value and its derivative at each iteration, in some cases, it may not converge. This is because the value of the derivative gets close to zero. In this paper, we introduce a modified and stable algorithm of Newton's iteration method that addresses this issue, reduces computational cost, and improves efficiency. In addition. we analyze the convergence properties of the modified method to demonstrate its effectiveness.

We prove seven conjectures involving binomial coefficients. These conjectures appear in The On-Line Encyclopedia of Integer Sequences (OEIS). Among these is an application of Jacobi's residue formula. We also apply MacMahon's master theorem to give a new and independently obtained proof of a recently established result.

We show that for every uniform family of affine algebraic sets, there is a formula in the parameters of the family and in $n$, expressing the cardinality of the set in the ring $\mathbb Z/n \mathbb Z$. In particular, we construct such a formula for the elliptic curves in Weierstrass normal form. These formulas are arithmetic terms: fixed finite compositions of additions, multiplications, subtractions, divisions with remainder and integer exponentiations.

Along the prey nullcline $y=g(x)$ of a planar system, the Jacobian entry satisfies $J_{11}=P\,g'$, where $P=-f_{1y}|_{\gamma}$. Under $P>0$ and predator self-damping, the Hopf trace condition forces $g'(x^\ast)>0$: critical points of $g$ are spectral barriers, so the Hopf locus lies on ascending branches. This geometric localization determines where oscillatory instability can occur; we determine how the bifurcation unfolds there. Let $D=f_{1x}+f_{2y}$ and $\tau=D|_{\gamma}$. Using nullcline coordinates $(u=x,v=y-g(x))$ and Hadamard's lemma gives $\dot u=-vH$, $\dot v=W$, with $H(u,0)=P$. The mixed jet of $W$ is determined by the jets of $\tau$ and $\nu=f_2|_{\gamma}$, yielding a closed five-term formula for the first Lyapunov coefficient $\ell_1$ in trace-field form, valid for predator-dependent functional responses. Three distinct reductions of the cubic jet are established. In general planar Hopf bifurcations, eigenvector pairing gives $\partial\ell_1/\partial f_{1yyy}=\partial\ell_1/\partial f_{2xxx}=0$. Straightening the nullcline removes two further slots, while within the Gause class the reparametrization $\varphi\mapsto g$ removes $p'''$. They have different scopes. For predator-dependent responses, comparison with the $y$-affine truncation of the $3$-jet at fixed linear part gives $\ell_1=\ell_1^{\mathrm{aff}}+\Delta$, where $\Delta=\Lambda\bigl(f_{1xyy}+\omega_0^{-2}\langle\partial_x f,\nabla D\rangle f_{1yy}\bigr)$ and $\Lambda>0$. On the Hopf locus, $\langle\partial_x f,\nabla D\rangle=J_{11}\tau'+(\omega_0^2/P)D_y$. Thus the same quantity $J_{11}=Pg'$ that localizes the bifurcation at first order reappears at third order as the weight of the tangential trace derivative. The bridge identity is independent of eigenvector normalization, while $\ell_1,\ell_1^{\mathrm{aff}},\Delta,\Lambda$ scale by $|c|^2$ under $q\mapsto cq$; their signs and ratios are unchanged.

We present a data-driven reduced-order modeling (ROM) framework for the zeroth angular moment of the solution to the radiative transfer equation (RTE) rather than the full phase-space solution. Our construction is based on the Peierls integral formulation of the angularly averaged density. For media with isotropic scattering, the density satisfies a closed second-kind Fredholm equation with a globally attenuated, weakly singular kernel. We project this equation directly. For media with anisotropic scattering, we utilize the average-fluctuation decomposition to derive a closed system for a projection-based ROM. Numerical simulations are presented to illustrate the effectiveness of the ROMs we implemented.

We study boundary-weighted barycenter spaces, in which an interior support point has cost (or weight) two and a boundary support point has cost one. These spaces are finite-dimensional topological models for the concentration patterns produced by noncompact boundary Euler–Lagrange functionals: an interior bubble carries twice the quantized mass of a boundary bubble, and very negative sublevels are therefore modeled by boundary-weighted rather than ordinary barycenters. The paper gives a systematic algebraic-topological treatment of these spaces. We construct the mixed strata, the closed-stratum poset, and the triangular boundary-weighted colimit filtration; compute the Euler characteristic; and prove a homology decomposition in terms of ordinary barycenter spaces of \(\partial M\) and \(M/\partial M\). We then specialize the formulas to compact connected orientable surfaces with boundary and to hemispheres, obtaining explicit rational Betti polynomials and, for surfaces, the mod-two polynomials required by the mean-field application. Finally, we explain how these polynomials measure the topology at infinity in the resonant Neumann mean-field equation on a compact surface with boundary.

We study quantitative mean-field limits for classical particles with Yukawa (screened Coulomb) interactions in every fixed dimension $d\ge2$, uniformly as the screening parameter $\kappa$ tends to zero. Our main result is a first-order commutator estimate in the natural Yukawa modulated energy, with an additive error of order $N^{-2/d}$ for $d\ge3$ and $(1+\log N)/N$ for $d=2$. It requires only a bounded reference density and a Lipschitz transport field, with no uniform lower bound on interparticle distances and no negative power of $\kappa$. The modified Helmholtz operator $-\Delta+\kappa^2$ creates the main new difficulty. Truncating the potential to a constant inside each truncation ball produces a surface charge and a positive volume charge whose total mass is strictly less than one. We keep the reference density unchanged and control this loss of mass through an exact Green function representation and renormalized energy identities. A stress-energy identity with interface terms and averaging over the truncation radii then give the uniform commutator estimate. Combined with the modulated energy dissipation identity and a normalized quadratic transport cost, this estimate yields weak–strong stability, propagation of chaos, and time-integrated control of the mean-square difference between empirical and mean-field forces. We also prove a quantitative Yukawa-to-Coulomb limit. For smooth product data, $N\to\infty$ and $\kappa\downarrow0$ may be taken simultaneously with no relation between their rates; in dimension three, a direct comparison at the particle level also holds for general symmetric initial laws with finite initial error.

We study uniform $L^2$ non-localization of Dirichlet Laplacian eigenfunctions on planar integrable polygons, with particular emphasis on spectral degeneracy and quantitative dependence on the observation set. For a measurable set $V\subset\Omega$ of positive measure, define \[ C_2(V;\Omega) := \inf_{\lambda\in\sigma(-\Delta_\Omega)} \inf_{0\neq u\in E_\lambda(\Omega)} \frac{\|u\|_{L^2(V)}}{\|u\|_{L^2(\Omega)}}. \] We prove that $C_2(V;\Omega)>0$ for rectangles, isosceles right triangles, equilateral triangles, and hemi-equilateral triangles, uniformly over the complete eigenspaces and hence independently of spectral multiplicity. For rectangles, we obtain a quantitative refinement. If the reflected extension of $V$ has finite perimeter and $\alpha=|V|/|\Omega|$, we derive an explicit sufficient threshold $\lambda_*(\Omega,V)$ such that every eigenfunction with $\lambda\geq\lambda_*(\Omega,V)$ satisfies \[ \frac{\|u\|_{L^2(V)}}{\|u\|_{L^2(\Omega)}} \geq \left[ \frac{\alpha}{2} \left( 1-\frac{\sin(\pi\alpha)}{\pi\alpha} \right) \right]^{1/2}. \] We further establish stability under bounded real-valued potentials on the rectangular branch and under controlled spectral defects, yielding corresponding non-localization results for sufficiently accurate quasimodes and narrow spectral clusters.

Single-particle cryo-electron microscopy (cryo-EM) pose estimation is traditionally solved anew for each dataset, where iterative refinement is done from scratch while the estimator learns to store the molecule in its weights. In this work, we show that pose inference is a generalizable, specimen-agnostic operation when conditioned explicitly on a reference volume. We introduce ARCHER, an amortized contrastive classifier that models the pose posterior over a discrete rotation grid. Trained across a variety of protein structures, it operates zero-shot without retraining per structure. This transferability is grounded in Fourier-space information mechanics, where all specimen dependence is captured by the reference structure's power spectrum and spatial extent. ARCHER achieves a median angular error of 5.0{\deg} on 100 held-out test structures and 2.5{\deg} on experimental particles, matching dedicated estimators within 0.16 Å in 3D reconstruction. Crucially, downstream conformational signal is preserved. The leading conformational coordinate correlates at 0.97 with deposited benchmarks, faithfully reconstructing free-energy basins and mobile domains. These results overall demonstrate that cryo-EM pose estimation can be generalized across different structures.

We derive an exact and novel expression for an averaged two-point correlation function in the statistically stationary, forced-dissipative two-dimensional Navier-Stokes equations subject to the Coriolis force under the beta-plane approximation. This identity is related to the so-called geostrophic balance: it connects the effect of the Coriolis force to the pressure gradient through a two-point correlation function. Additionally, we provide sufficient conditions under which the asymptotics of the averaged third-order structure function at large spatial scales follow the universal third-order law of two-dimensional turbulence in the absence of the Coriolis force. This complements our previous results on small spatial scales. Together, our results provide a clear picture of the role of the Coriolis force in beta-plane turbulence. On the one hand, the spherically averaged rates of enstrophy and energy transfer are not affected by the Coriolis force. On the other hand, the Coriolis force contributes to anisotropic large-scale organization by altering the spatial distribution of energy and promoting the formation of zonal structures. The proof relies on a new formulation of the Karman-Howarth-Monin relation. For the geostrophic balance, we use a novel antisymmetric projection of the KHM relation under which only the pressure and Coriolis terms survive. For the cascade laws, we show that the Coriolis contribution to the averaged classical KHM relation vanishes identically at any scale.

In this paper, numerical solutions of a class of nonlinear ordinary differential equations are obtained using a collocation method based on the shifted Horadam polynomials. The proposed problem, which is of the fourth-order, satisfies a class of two-point boundary conditions. We first discuss definitions and basic properties of the Horadam polynomials before presenting new and useful differentiation formulae for them. Novel interesting identities are deduced from the differentiation properties. The collocation method under consideration assumes that the solution of the proposed problem can be expressed as a shifted Horadam polynomial series. To determine the expansion coefficients of the series solution, one collocates at the zeros of the shifted Horadam polynomials, and the proposed boundary value problem is subsequently reduced to a set of nonlinear algebraic equations. These algebraic equations are then solved using Newton's iterative method to obtain the numerical values of the expansion coefficients of the shifted Horadam polynomial series solution. Several examples of the proposed nonlinear boundary value problem are considered to demonstrate the accuracy, efficiency, and reliability of the proposed method. Numerical solutions and errors obtained are compared with existing solutions. Comparisons of results, which are shown in tables and graphs, clearly reveal that the shifted Horadam collocation method outperforms the existing methods.

We construct Khovanov–Lauda–Rouquier-like generators for certain modified forms of non-degenerate affine Hecke algebras associated with arbitrary Weyl groups after localization. These generators yield non-graded KLR-like presentations of the modified algebras. As an application, we establish isomorphisms between direct sums of blocks of cyclotomic Hecke algebras of arbitrary Weyl groups and cyclotomic quotients of the resulting non-graded KLR-like algebras. We also explain how these isomorphisms identify the generalized weight-space decomposition of cyclotomic Hecke modules with the idempotent decomposition of modules over the KLR-like presentation.

Touching and dividing cell nuclei may appear as connected structures in microscopy images, making it difficult to distinguish neighboring nuclei during segmentation. We introduce and analyze a regularized subjective-surface model designed for this setting. On a smooth bounded domain $\Omega\subset\mathbb{R}^n$ with homogeneous Dirichlet boundary conditions, the model is \[ u_t = \nu\Delta u + \left(\varepsilon^2+|\nabla u|^2\right)^{1/2} \operatorname{div}\!\left( G(x)\frac{\nabla u} {\left(\varepsilon^2+|\nabla u|^2\right)^{1/2}} \right) - \mu\Lambda(x)H_\eta(u-q). \] Here, $\varepsilon>0$ and $\nu>0$ are fixed regularization parameters, $G$ is a strictly positive smooth edge coefficient, and the nonnegative interaction weight $\Lambda$ incorporates fixed information about neighboring nucleus candidates. The principal objective of this work is to establish an existence theory for the proposed model. For compatible $C^{2+\alpha}$ initial data taking values in $[0,1]$, we prove the existence and uniqueness of a global classical solution whose restriction to every finite time interval is Schauder-classical, together with preservation of the physical range, finite-time Schauder estimates, and $L^\infty$-nonexpansive dependence on the initial data. The main analytical step is a global spatial-gradient bound, obtained by combining gradient estimates near $\partial\Omega$ with interior gradient estimates. These results provide a mathematical foundation for applying the model to the analysis of touching and dividing nuclei in 3D and 3D+time microscopy image data.

In this paper, we study $L^p$-estimates for maximal averaging operators $\mathcal M$ along hypersurfaces $S$ in $\mathbb{R}^{3}$ which are the graph of a mixed homogeneous function $\Phi$ which is analytic away from the origin. The closure of such a surface will pass through the origin, so that the usual transversality condition that had been imposed in many previous works on maximal averages along hypersurfaces will not hold even when $\Phi$ is analytic at the origin. As our main result, under mild assumptions which are satisfied for instance for every mixed homogeneous polynomial $\Phi,$ we determine the critical Lebesgue exponent $p_c$ for which $\mathcal M$ is $L^p$-bounded for every $p>p_c,$ but unbounded for $p<p_c,$ in terms of multiplicities of the real roots of the Hessian determinant of $\Phi.$ It turns out that the study of the contributions by neighborhoods of a certain type of roots is closely related to recent work by Dendrinos, Ikromov and the first and third author on sharp estimates for a maximal averaging operator along a transversal hypersurface of an “exceptional” class, whose $L^p$-boundedness had been an open problem for a long time and which has recently been established by means of their new theory of FIO-cone multipliers.

Let $A$ be a finite-dimensional basic hereditary algebra and let $T(A)=A\ltimes D(A)$ be its trivial extension. Building on the classification of indecomposable $\tau$-rigid $T(A)$-modules, we give explicit Hom-vanishing conditions characterizing arbitrary basic $\tau$-rigid $T(A)$-modules. For such a module $M$, we also determine its maximal projective complement in terms of the support of the underlying $A$-module $U(M)$, and hence obtain an explicit criterion for $M$ to be support $\tau$-tilting. As an application, for the linearly oriented quiver of type $A_n$, we classify all basic rank-two $\tau$-rigid modules over $T(\Bbbk A_n)$ and prove that their number is \[ \binom{n}{2}\binom{n+1}{2}. \] We also show that every basic $\tau$-tilting $T(\Bbbk A_n)$-module contains an indecomposable projective direct summand. Finally, for two orientations of a quiver of type $D_4$, we determine the corresponding support $\tau$-tilting compatibility graphs and their face distributions, from which we obtain and compare the associated F-triangles.

A generalized Steinhaus $s$-triangle is obtained from a binary sequence by repeatedly replacing each block of $s$ consecutive entries by its sum modulo $2$. We study the triangles generated by canonical basis vectors. Expressing their entries through bi$^{s}$nomial coefficients, we prove that the truncated coefficient profiles of the successive rows are purely periodic and determine their exact period. This periodicity yields a power-of-two derivative identity and a decomposition into identical finite blocks. We consequently obtain an exact recurrence for the number of ones and show that, on every residue class, this weight is an affine function of the length. We also derive a rational generating function, determine the asymptotic growth rate, and prove that these canonical triangles have zero density of ones. Explicit formulas are obtained for the first two canonical vectors and for the block weight in the classical case. These results provide a unified description of the periodic and enumerative structure of generalized Steinhaus triangles generated by canonical basis vectors.

Asymptotic normality approximations often fail to hold for extremum estimators when the true value of the parameter is at or close to the boundary of a parameter space. I analyze and develop tests using a quasi-unconstrained estimator, which is asymptotically normal even when the true parameter vector is near or at the boundary. These results generalize previous work with this estimator by allowing for more types of constraints and showing how the method can naturally be modified when a nuisance parameter is also high-dimensional. I show that variations of Wald, Likelihood Ratio, and Lagrange Multiplier tests can control size in a uniform sense, provided the initial constrained estimator is sufficiently accurate. Lastly, I apply the method to an application involving network estimation with panel data.

The Stone-von Neumann theorem makes canonical quantization unique only after one imposes strong continuity of the Weyl translation groups. We show that this continuity assumption cannot be certified by any finite-resolution interrogation of the Weyl relations. A finite protocol probes only finitely many phase-space displacements and bounded expectation values, thereby determining a protocol algebra \(C_S\) generated by a finitely generated subgroup of Weyl translations, but not the regularity of an extension to the full Weyl algebra. Using the polymer representation as a canonical non-regular representation, we prove that for every regular Weyl representation, every normal state, and every finite family of bounded protocol observables, a polymer state reproduces the same expectation values to arbitrary prescribed accuracy. Thus finite agreement with Schrödinger quantum mechanics does not operationally certify the regularity hypothesis entering uniqueness. The result does not assert physical equivalence between regular and polymer quantizations, it identifies where additional input, such as Hamiltonian dynamics, energy regularity, semiclassicality, or a continuum limit, is required to distinguish or select representations. In this precise sense, non-regular polymer kinematics are not excluded by finite Weyl data alone, a point of direct relevance to quantum-gravity motivated quantizations.

We construct an IFS consisting of three contracting similarities on $\mathbb{R}^{9}$ with a countable compact sub-self-similar set $E$ with distinct Hausdorff and lower box dimensions. This provides a concrete counterexample to the question raised by Falconer concerning the equality of Hausdorff and box dimensions for sub-self-similar sets without the open set condition.

Many imaging problems seek to reconstruct underlying configurations from partial observable measurements. While reconstruction algorithms operate on these measurements, the observable organization induced by the measurement process is rarely represented explicitly. We introduce TENSKEL, a combinatorial observable framework for structured measurement and reconstruction based on tensor representations defined over discrete domains. Starting from a binary latent ensemble, the framework constructs a hierarchy of tensors coupling measurement contexts to a latent Pascal organization through successive aggregation and folding operations. Each measurement context induces an observable partition of the same latent ensemble, and the resulting tensor formulation makes explicit the associated combinatorial multiplicities, shell organization, degeneracies, and induced reconstruction geometry. Rather than introducing a new reconstruction algorithm, this framework provides a mathematical representation of how latent configurations become organized under observation. The induced tensor kernel characterizes similarities between latent coordinates through their measurement-context responses, while regularized inversion provides a structured reconstruction of the latent representation from observable measurements. The binary construction further admits a natural multinomial extension to discrete simplex-supported latent representations. Connections to Pascal cellular automata and structured discrete color mappings illustrate respectively compressed and multinomial realizations of the framework. More generally, TENSKEL provides a combinatorial basis for reasoning about the organization induced by observation, with potential relevance to computational imaging, inverse problems, structured sensing, and quantum-inspired measurement formulations.

Regional climate risk assessment is critical for applications such as infrastructure design, disaster forecasting, and insurance resource allocation. However, estimating regional (i.e., high-spatial-resolution) risk with global climate models (GCMs) remains computationally prohibitive, which has driven the development of downscaling methods for coarse GCM outputs. Downscaling is vital for rare events, since quantifying their extreme properties requires high spatial resolution and very long GCM simulations. These methods non-intrusively increase GCM resolution while correcting statistical biases from unresolved fine-scale processes, thereby improving the accuracy of extreme event statistics with long return periods. A key challenge is preserving dynamical consistency, as freely evolving GCM trajectories are not expected to track the observational dataset used for training the correction operator. This is critical for causal extreme event analyses, where storyline-based risk assessment, i.e., extreme event catalogs, is necessary for effective planning. We address this challenge by introducing Dynamically and Statistically Consistent downscaling (DySCo), a non-intrusive framework yielding high-resolution climate projections consistent with coarse GCM dynamics. DySCo relies on a data-driven reformulation of nudging to create dynamically paired training trajectories without intrusive GCM modifications. Using these paired trajectories, we train a dynamically and statistically consistent, two-stage operator. We evaluate the method by downscaling the Community Earth System Model v2 Large Ensemble (LENS2) in time and space towards historical reanalysis. Results show DySCo achieves superior dynamical consistency with the coarse GCM trajectories, essentially applying a minimal, causal correction to the GCM, preserving top statistical performance comparable to state-of-the-art unsupervised models.

I provide a detailed exposition of the fundamental group calculations in the “reverse engineering" construction of an exotic $\CP ^2\#3\overline{\CP }^2$ and of an exotic $\CP ^2\#2\overline{\CP }^2$.

We classify the compactified universal Jacobian spaces that are projective over the moduli stack of stable pointed curves.

We characterize infinitesimal finite forcibility for bounded symmetric real kernels. We prove that the graph-density gradients at a kernel span a finite-dimensional space if and only if the kernel is a step kernel. Combined with known finite-forcing results for step kernels, this gives a positive answer to a question of Lovász and Szegedy on whether every infinitesimally finitely forcible kernel is finitely forcible. The proof combines spectral methods with a compression argument based on book graphs.

We determine the exact local second-order rate region of the finite-alphabet Wyner–Ahlswede–Körner problem under full support, at informative boundary points exposed by a finite supporting slope. The characterization permits nonunique optimizing test channels with unequal information variances and requires no neighborhood smoothness of the optimized first-order value. The optimal success function is a conditional Gaussian envelope averaged over the fluctuation of the helper's observed source type. For each type, the helper selects the minimum or maximum conditional variance according to the remaining description budget, while preserving both first-order rates. The converse follows from a uniform Gaussian bound for ordinary channel images. An exact posterior score decomposition separates a common source score, a nonnegative optimality gap, a negligible source residual on low-cost histories, and a conditionally independent channel fluctuation. Conditional-type covering with two endpoint optimizers and one common binning map attains the bound. Under a strict first-order benefit from the helper, a separate limiting-rate argument also gives the unrestricted weighted second-order optimum. A strictly positive finite source exhibits a strict gain over every fixed-optimizer Gaussian expression, and an exact binary image calculation illustrates the limiting coefficient.

We prove that if $Z$ is a tropical cycle in a tropical variety $Y$ which is algebraically trivial, then for $k\gg1$ sufficiently large, $k!\cdot Z^k\subset Y^k$ is rationally trivial. The proof involves explicitly constructing a rational equivalence $k!\cdot(p-q)^k\sim0$ in the product $C^k$, where $p$ and $q$ are any points on $C$. We use this to formulate a tropical version of Voevodsky's conjecture, provide explicit examples, and comment on potential applications to symplectic geometry.

We provide a criterion for determining when elements of the Bernstein center of a totally disconnected locally compact group are admissible invariant distributions in the sense of Harish-Chandra \cite{HC99}. As a consequence, we deduce the local integrability results for elements of bounded depth or Bernstein supports in the Bernstein centers of reductive $p$-adic groups. Our methods apply uniformly to both complex and mod-$\ell$ coefficients, generalizing results of Moy and Tadić \cite{MT02} in the complex case.

We investigate how the complex analytic Morse Splitting Lemma relates to the generic Lê numbers of a function, and what the relation tells us about the function. As special cases, we consider functions for which the 0-dimensional generic Lê number is at most 3 and functions for which the critical locus has dimension 1.

We prove that the torsion three-form in the preferred splitting of a left-invariant generalized nilsoliton on a nilpotent Lie group is harmonic. This gives a positive answer to Question 7.12 proposed by Fusi-Lafuente-Stanfield in [3].