Papers

Data assimilation for time-dependent partial differential equations (PDEs) requires Bayesian updates of an evolving field from streaming, sparse, and noisy observations, while keeping the filtering state finite dimensional. We introduce a random-feature Kalman filtering framework for linear PDE data assimilation. Once the random features are frozen and the linear PDE is Galerkin discretized, the coefficient vector satisfies a finite-dimensional linear-Gaussian state-space model, so the Kalman recursion gives the exact posterior for the chosen coefficient model. For non-orthogonal random-feature draws, we construct a mass-whitened effective-rank coordinate system that removes near-null mass directions and identifies the posterior dimension $r$. For the heat equation with implicit-Euler time stepping, we prove a high-probability posterior-contraction and PDE-consistency theorem in these mass-whitened coordinates. The mean-square $L^2$ reconstruction error separates into an effective-rank feature approximation term, a deterministic time-consistency term, and a Bayesian estimation term. In the high-information regime, the leading posterior contribution scales as $r\sigma^2/N_o$, where $\sigma^2$ is the observation-noise variance and $N_o$ is the number of observations per analysis time. Thus the analysis distinguishes the exact coefficient-space posterior from deterministic PDE approximation errors, and gives a checkable uncertainty-quantification guarantee for random-feature filtering of a representative parabolic PDE.

Let A be a 1-domestic Brauer graph algebra. By covering theory and the characterization of simple-minded systems of 2-domestic Brauer graph algebras, we construct a family of objects in A-stmod to be a simple-minded system and our construction provides all simple-minded systems in A-stmod.

We study small-noise asymptotics for a class of reversible stochastic evolution equations in infinite dimensions. The dynamics are of the form \[ dX_t=-A\nabla F(X_t)\,dt+\sqrt{2\beta^{-1}A}\,dW_t, \] where $F$ is a regular multi-well potential, $A$ is a selfadjoint mobility operator, $W$ is a cylindrical Brownian motion and $\beta\gg 1$ is the inverse noise strength. The invariant measure is a Gibbs perturbation of a Gaussian reference measure, and the resulting framework covers, in particular, the stochastic Allen-Cahn and stochastic Cahn-Hilliard equations on bounded intervals. In the double-well case, we derive a sharp asymptotic formula for the first nonzero eigenvalue of the generator. This gives an infinite-dimensional Eyring-Kramers law for the spectral gap, with exponential rate determined by the communication height and leading prefactor determined by the local quadratic behavior at the relevant minima and saddle points. Our approach provides a general strategy for lifting finite-dimensional Eyring-Kramers analysis to infinite-dimensional stochastic gradient systems.

Let A be a self-injective algebra over an algebraically closed field. We study reduction of simple-minded systems over stable module category A-stmod. We present a recollement and study gluing technique of simple-minded systems through a subset of a simple-minded system in A-stmod. As a byproduct, we also study the extendible property of simple-minded systems.

A soft set $(F,E)$ determines the selection space $\SE(F)=\prod_{e\in E}F(e)$. This paper studies two natural structures on that space. For at most countable $E$, the series metric $\dPi$ induces the product topology. For arbitrary $E$, the sup metric $\dsup$ induces uniform convergence. We prove that these metric spaces are complete exactly when all fibres are complete. We then compare global contractions with coordinatewise contractions and give counterexamples when coordinate separability or a uniform contractive bound is absent. Standard fixed-point theorems for complete metric spaces are recorded as direct consequences, without repeating their classical proofs. For uncountable $E$, the product topology may fail to be metrizable, so we work with its product uniformity. A parameterwise contraction theorem gives a unique fixed point and convergence in the product topology; a common bound below one gives convergence in $\dsup$.

This study systematically compares Turbo-Quant and SpectralQuant KV-cache compression, evaluating non-dominated schemes, including WHT rotation with Beta Lloyd-Max and QJL, through a statistical validation methodology that separates systematic codec differences from implementation variance. Key findings reveal that while eigenbasis-based methods fail on heavy-tailed data due to covariance instability, they excel in structured regimes, with the effective semantic dimension ($d_{eff}$) adapting to calibration budgets rather than true data rank. (this is an abstract of the abstract thank you )

In this paper, we establish commutativity criteria for the quotient ring $R/P$ through pairs of $\ast$-reverse $P$-derivations satisfying suitable commutator and anti-commutator differential identities. As applications, a number of results concerning $\ast$-reverse derivations on prime rings are recovered.

We study steady states of evolutionary games on networks in which some players adopt pure strategies while others play mixed strategies. We refer to these configurations as boundary mixed steady states. Such states arise naturally in structured populations and have no counterpart in the classical well-mixed setting. We introduce a relaxed equilibrium notion, called boundary Nash equilibrium, in which the Nash condition is imposed only on non-pure players. In two-strategy systems, this notion characterizes boundary mixed steady states, while this correspondence breaks down in higher dimensions. The stability of these states is governed by the interaction structure among mixed players. When mixed players do not interact, the system exhibits continua of equilibria. In contrast, any nontrivial interaction generically produces instability. In particular, boundary mixed steady states that are not fully degenerate are never asymptotically stable. Degeneracies are further linked to the rank properties of the underlying interaction. These results reveal a structural instability mechanism specific to networked replicator dynamics, highlighting a qualitative gap with respect to the classical well-mixed case and showing how network topology influences the local behavior of equilibria.

This paper develops a delayed generalized Lotka–Volterra model for systems in which structural change and recognition of change are not synchronized. The starting point is the idea that a system may accumulate internal transformations while remaining locally stable at the level of interpretation, governance, or perception. Instability becomes effective only when accumulated structural stress crosses a threshold and is recognized after a delay. We introduce a vector of civilizational state variables including systemic complexity, informational entropy, systemic pressure, resilience, and legitimacy. Their interaction is modeled through a delayed generalized Lotka–Volterra system. The delays represent the temporal separation between structural variation and its cognitive, institutional, or social effects. A composite instability functional is then defined, and threshold crossing is interpreted as the transition from latent instability to recognized instability. The paper studies positivity, equilibria, linearization, and local stability of the delayed system. Particular attention is given to delay-induced instability: even when the corresponding non-delayed system is locally stable, sufficiently large delays may destabilize the equilibrium and generate oscillatory or nonlinear behavior. The model provides a mathematical framework for delayed recognition, threshold instability, and the emergence of crises in complex adaptive systems.

A symmetric function is called Schur-positive if it admits an expansion in the Schur basis with nonnegative coefficients. In this paper, we study the Schur positivity of symmetric functions naturally associated with set partitions, with respect to two different notions of descent. In the first case, the Schur expansion involves hook-shaped Young diagrams, and the corresponding coefficients are given by Touchard-Riordan polynomials, which enumerate matchings by their number of crossings. In the second case, the Schur functions correspond to two-rows Young diagrams, and the coefficients are partial sums of associated Bell numbers. A key ingredient of our approach in the second case is the notion of a removable singleton, defined algebraically and shown to admit an equivalent combinatorial interpretation via jeu-de-taquin rectification of skew tableaux. As an application, we establish Schur positivity for various classes of symmetric functions indexed by non-crossing partitions and partitions with a given number of parts. We provide an explicit combinatorial description of the tableaux that contribute to the Schur expansion, and we connects the obtained coefficients to some known integer sequences.

This paper develops a generalized transform framework for a logistic–Allee tumor-growth model. The method combines a generalized Laplace transform, Adomian decomposition, Chebyshev–Padé rational reconstruction, and a \(\mu\)-scaled generalized transform to obtain admissible semi-analytical approximations. Comparisons with experimental tumor-growth data show that the resulting compact representations are stable, admissible, and comparable in error to standard numerical reference solutions.

We describe two self-replicating mechanisms of energy accumulation in the seismic focus, which modify the Gutenberg-Richter law in the region of the big earthquakes. The first mechanism acts for magnitudes smaller than a narrow region of large critical magnitudes; it slows down the energy accumulation and may produce precursors. The second mechanism acts above that region, and accelerates the energy accumulation; the precursors may be absent. Both mechanisms reduce the Gutenberg-Richter excedence distribution. On the left of the critical region the Gutenberg-Richter magnitude probability density is unchanged, while on the right the probability density is reduced. The procedure described in this paper introduces a critical-magnitude region (range) as an additional fitting parameter. The results may bear relevance upon the recent concepts of "self-arresting" and "dragon-king" earthquakes. The two self-replicating mechanisms may introduce a magnitude gap between the two types of big earthquakes, and two branches in the excedence law, in the vicinity of the critical region.

Highly accurate stability boundary values for the ADER-DG method are obtained for arbitrary degrees $N$ of basis polynomials. In the linear case, stability is violated precisely when one of the matrix eigenvalues reaches $\lambda = -1$, regardless of the phase $\theta$. A rigorous mathematical framework for the stability is developed. The stability condition is significantly simplified, reducing it to the problem of calculating the roots of polynomials in the Courant number $\mathrm{CFL}$. The maximum of the Courant numbers $\mathrm{CFL}_{\rm max}(N)$ are calculated. These results are new and very convenient for practical use. A comparison of the obtained results with existing results reveals differences that may be significant for the selection of calculation parameters, especially for high degrees $N$. It is shown that widely used existing estimates $\mathrm{CFL}_{\rm max}(N) \propto 1/(2N+1)$ are overestimated. An interesting qualitative asymptotic $\mathrm{CFL}_{\rm max}(N) \propto 1/(N+1)^{2}$ is obtained. A rigorous direct proof of the approximation is presented. Approximation orders $p = N+1$ for arbitrary degrees $N$ are rigorously derived. A set of numerical experiments is carried out to apply the ADER-DG method to solving both a linear advection equation and an Euler system of equations. The results obtained in these calculations confirm the theoretical results well. In particular, an excess of the Courant number over the $\mathrm{CFL}_{\rm max}(N)$ by even 1% in the linear case immediately leads to significant instability of the numerical solution. The obtained estimates of the boundary Courant number in the nonlinear case are somewhat underestimated – by no more than 5%, which is due to the diffusivity and stability of the approximate Riemann solver. Empirical convergence orders are obtained, which are in good agreement with the theoretical results.

We establish sufficient conditions for robust positive invariance under state- and input-dependent disturbances with anisotropic covariance structure. The proposed ansatz maps a fixed ellipsoidal template through a GP-derived positive-definite matrix field, subsuming scalar homothetic scaling while retaining finite graph-based verification. The resulting LMI conditions couple the learned field to Schur-stable dynamics; an isotropic fallback with inflation factor $r=1/(1-\gamma_{\mathrm{cl}})$ proves admissibility. During each learning epoch the field is frozen, so online tube evaluation is one GP covariance query and a small matrix square root, with no online set iteration or LMI solve. Quadrotor simulations show a $195\times$ reduction in 3D velocity-tube volume and a $2.1{\times}10^5$ reduction in the joint 7D velocity-control subspace relative to a non-adaptive homothetic baseline. This extended version adds full proofs, a separated offline/online complexity analysis, and controller-sweep, contraction, and projection-area studies.

Using cohomological Mackey functors, we give an explicit Brauer group classification of torsors under tori of generalized del Pezzo varieties. This identifies the image of Lamarche's Brauer invariants and extends the Brauer-theoretic description in Blunk's classification of degree-six del Pezzo surfaces. We also determine which finite field extensions can appear in other Brauer group presentations of these torsors. The same tori act on Losev-Manin spaces, so the computation applies to these toric varieties as well.

This paper proves that every $2$-connected regular matroid of sufficiently large rank has a $2$-connected graphic flat of large rank. Furthermore, we explicitly determine a list of such unavoidable flats.

We investigate massive one-dimensional Dirac operators perturbed by diagonal matrix potentials of the form $i V$ where the function $V$ is real-valued and unbounded at infinity. For such operators we find an $L^2$-realization with non-empty resolvent set using generalized coercivity and Schur complement dominance techniques. In the prototypical Airy-Dirac case $V(x)=x$, $x \in \mathbb{R}$, we derive the precise asymptotic behavior of the resolvent as the spectral parameter tends to infinity and also as the mass $m$ tends to $0$. Finally, we find the asymptotics of the resolvent norm for general potentials $V$ in terms of the Airy-Dirac resolvent, which in particular yields an asymptotic shape of $\varepsilon$-pseudospectral curves and establishes the optimality of the pseudospectral region found in [32].

Let $f:M\to M$ be a diffeomorphism on a compact connected smooth manifold of dimension at least $5$. It is known that the Nielsen number $N(f)$ of $f$ vanishes if and only if $f$ is isotopic to a fixed point free map. For any $n\ge 5$, there exists a compact $n$-dimensional nilmanifold $M$ such that every self homeomorphism on $M$ is isotopic to a fixed point free map. The proof depends on the fact that nilmanifolds are of Jiang-type and the fundamental group $\pi_1(M)$ has the property $R_{\infty}$ for certain nilmanifolds $M$. In this paper we construct the first known examples (an infinite family) of non-aspherical closed manifolds whose fundamental groups have property $R_{\infty}$ and that are also of Jiang-type. In particular, every self homeomorphism on such a manifold is isotopic to be fixed point free. The main objective of this work is to construct non-aspherical manifolds whose fundamental groups have property $R_{\infty}$ and that are also of Jiang-type.

The $\sigma_k$-Loewner–Nirenberg problem is a fully nonlinear generalization of the classical Loewner–Nirenberg problem of constructing complete conformal metrics with constant negative scalar curvature in the interior of domains. For the $\sigma_k$-version, once $k > 1$, one must impose a negative admissibility condition to ensure ellipticity of the equation. In this note, we exhibit topological obstructions to admissibility and illustrate this with examples.

Modern deep learning architectures are increasingly multi-task and multi-modal, using a pretrained foundation model combined with task-specific, fine-tuned models. Empirically, exploiting similarity across different problems, instead of solving them individually, can significantly improve overall performance. While the generalization and sample complexity properties of multitask learning have been widely studied, the parametric complexity of joint approximation in comparison to separate approximation remains less well understood. The question is particularly relevant in modern deep learning, where models are increasingly required to satisfy structural constraints such as equivariance, conservation laws, or orthogonality. We prove lower and upper bounds on the description-length for separate and joint approximation classes, respectively, in uniform norm. We build a class of orthogonal functions by composing a shared hard feature, realized by a Rademacher-Haar wavelet series, with Sawtooth-Walsh readouts to enforce orthogonality of output coordinates. The dyadic tree structure of the Rademacher-Haar wavelet concentrates the approximation hardness in the common feature component, while the readouts act as task-specific heads. Using an information-theoretic framework, we obtain a sharp gap between the optimal approximation rates achievable by joint and separate coding. Finally, we realize this separation in a neural network model using Heaviside activations via reduction to triangle-wave approximation. Our results show that even under an orthogonality constraint joint approximation requires strictly fewer bits in compositional architectures, provided the tasks share a latent hard feature. This provides theoretical insight into the description-length-efficiency of compositional multi-output architectures and clarifies how neural networks can retain expressivity under geometric constraints.

By comparing simultaneous multigraded blow-ups with suitable iterated blow-up construction, we establish a birational correspondence between the associated exceptional divisors. We further investigate blow-ups arising from rational maps to multiprojective spaces and derive intersection-theoretic formulas for the Chern classes of the resulting exceptional divisors and pullback of tautological line bundles. These constructions have two main applications. First, they yield a proof of the product formula for Segre classes without any pure-dimensionality hypothesis. Second, they lead to a degree formula for arbitrary closed subschemes of multiprojective spaces, recovering and extending the classical formula of van der Waerden.

Stochastic search processes in which agents are continuously inserted into and removed from a target search domain are fundamental to a wide class of physical and artificial systems. The theory of such non-conservative search processes is, however, much less developed than for search processes with a fixed number of agents. Recent work on the non-conservative case has characterized the fastest first-passage time (FPT) to a target, after which the search process is terminated. Here we consider the complementary problem of continuous arrival statistics to a target. In particular, we derive the full time-dependent distribution of target arrivals, which we use to show that the inter-arrival time distribution is exactly exponential in steady-state regardless of the underlying search process. Thus, counterintuitively, the arrival statistics of non-conservative search are both universal and much simpler than sequential search-and-capture processes involving a fixed number of agents. Our methods also yield an exact expression for the fastest FPT distribution with time-dependent insertion, thereby generalizing previous work on fixed-rate insertion for non-conservative search. Altogether these results have major implications for target resource accumulation, wherein the delivery of resources is counter-balanced by their downstream consumption.

In this note, we prove a non-asymptotic version of a theorem by Rácz and Richey, showing that a Wishart matrix is close in total variation to an affine transformation of a GOE matrix. The proof mirrors a proof in a paper by Bubeck, Ding, Eldan, and Rácz, with some changes made to make it non-asymptotic.

The pressureless Euler equations in dimension 1 describe the evolution of a distribution of mass over the real line. We are interested in sticky particles solutions: distributional solutions in which mass that collides sticks together for all time. Given initial data, a projection formula is known to give a sticky particles solution. We obtain a necessary and sufficient condition for this solution to go to equilibrium based only on the initial condition. Furthermore, we give a precise characterization of the equilibrium, as well as a formula for the time of collapse.

We develop an algebraic approach to matrix discrepancy based on the representation theory of finite-dimensional C$^*$-algebras. As an application, we resolve a substantial structured special case of the Matrix Spencer conjecture. In particular, we show that for every family of contractions $A_1,\ldots,A_n$ that are contained in a finite-dimensional $C^*$-algebra $\mathcal A$ with $\text{dim}_{\mathbb C} (\mathcal A) \lesssim n$, there exists signs $x\in\{\pm1\}^n$ such that $\|\sum_{i=1}^n x_i A_i\| \le O(\sqrt n)$. As a noteworthy special case, our main result also resolves the Group Spencer conjecture of (Bandeira'24). We furthermore prove that Matrix Spencer continues to hold for low-rank perturbations of matrix families coming from an $C^*$-algebra of small dimension.

In the first version of Chang, Slote, Volberg, and Zhang's paper \cite{BSA_of_PTF}, the authors modify a nice recursive approach due to Kane in \cite{Correct_exponent_for_AS} where he bounded the average sensitivity of polynomial threshold functions. In \cite{BSA_of_PTF} Kane's argument was adopted to estimate the boolean surface area of polynomial threshold function. The bridge is a combinatorial averaging lemma considering all balanced partitions. The lemma serves as a substitute for an additive property of average sensitivity. With the lemma, one can apply a Kane-type algorithm to derive a recurrence. Solving the recurrence then gives an upper bound of $e^{C_d \sqrt{\log n}}$ for the boolean surface area. In the second version of the same paper, the authors derive a polylog upper bound for BSA of PTFs. The difference is that they use a tail estimate for the sensitivity function. With the help of a polynomial restriction lemma in \cite{poly_restriction} they sharpen the upper bound. It is noteworthy that when applying the polynomial restriction, each coordinate is put into each part independently with equal probability. As a result, a partition does not necessarily have equal-size blocks. In other words, it may not be balanced. In this note, we first investigate the effect of different partitioning. Second, we use the recursive method in the first version to derive a polylog upper bound for $\mathbb E[s(x)^{\eta}]$ where $\eta < 1/2$. It is interesting to note the phase transition that happens at $\eta=1/2$ in both versions of the proof (but in a completely different form). Section 2.5 treats that.

One way to construct Difference Distance Magic Oriented Graphs (DDMOGs) is via a recently introduced technique called weighted sum. We explore the quality of said construction further by introducing the notion of an ornament. An ornament is an oriented graph that, when used in a weighted sum with an existing DDMOG, creates a new DDMOG. We provide results on the construction of a specific type of ornaments, called s-nodes ornaments. We conclude the paper with a list of open questions related to DDMOGs and ornaments.

In this work we construct multi-slit chordal Loewner chains with multiple attraction points on the real line. We build a model domain with the use of close-to-convex functions, in a way that allow us to prescribe the attraction points. These families of Loewner chains resemble the classical model for semigroups of holomorphic maps of the upper half-plane. Our work is to choose an appropriate set of arbitrarily many parameters and to study the geometric behavior of the model domain, as it evolves with respect to time, in order to derive the aforementioned Loewner chain. Finally, with the use of harmonic measure, we calculate the angles of convergence of each slit of the chain.

Given a resolution $\widetilde{\mathrm{Gr}}^{\underline{\lambda}} \rightarrow \overline{\mathrm{Gr}}^\lambda$ of an affine Schubert variety for $GL_n$, we define its noncommutative version – a sheaf of algebras on $\overline{\mathrm{Gr}}^\lambda$, derived equivalent to $\widetilde{\mathrm{Gr}}^{\underline{\lambda}}$ as well as its Steinberg versions in both zero and positive characteristics. This, in particular, allows us to define the perversely-exotic t-structure on the derived category of equivariant coherent sheaves on $\widetilde{\mathrm{Gr}}^{\underline{\lambda}}$, analogously to Bezrukavnikov–Mirković in the case of Springer resolution. We study the basis of classes of irreducible objects in the equivariant K-theory, and explicitly identify it with the (parabolic) Kazhdan–Lusztig canonical basis in a certain cell quotient. This allows us to relate it to the canonical basis for the quantum affine group. In the course of the proof, we establish some properties of coherent-constructible equivalences.

We prove uniqueness for a finite-dimensional anisotropic Calderón problem in dimensions $n\geq 3$. The unknown conductivity is a symmetric uniformly elliptic matrix field whose restriction to each cell of a known finite partition is polynomial; jumps across cell interfaces are allowed. We assume that the cells can be ordered so that each successive cell is accessible through sufficiently many flat faces. Under this assumption, the local Dirichlet-to-Neumann map on an exterior boundary patch determines every polynomial piece. The proof first recovers one polynomial matrix from its boundary invariants on several flat faces and then propagates the required local data through the recovered cells by unique continuation and Runge approximation. A general finite-dimensional analytic stability theorem also gives Hölder stability on compact admissible parameter sets.