We introduce $p$-uniformity to characterize the scaling of density fluctuations in spatial random systems in $\mathbb{R}^d$, ranging from hyperfluctuation to stealthy hyperuniformity. Our central theorem establishes sufficient conditions to preserve $p$-uniformity under transport. The first condition, a finite $(d+p)$-th moment of the transport distance, allows for a Taylor expansion of the transport. The second condition controls the corresponding terms. We thus solve a previously stated open problem; indeed we extend it, since our result applies to a general $p$-uniform source in any dimension, and the source and transport may be dependent. As an application, we construct new classes of point processes that are isotropic and $p$-uniform with arbitrarily high $p$, and that can be simulated in linear time. We conclude with an outlook on a converse statement.

We develop the differential theory of complex spinorial forms associated with irreducible complex spinors across all dimensions and signatures. This framework enables the study of constrained parallelicity conditions for irreducible complex spinors by reformulating them as equivalent differential systems for exterior forms within a prescribed semi-algebraic body of the Kähler-Atiyah bundle. To illustrate this approach, we first apply it to the spin-c Killing spinor equation in low dimensions, refining existing results by relaxing standard assumptions of simply connectedness and completeness. Then, we proceed to apply our framework to supersymmetry conditions in supergravity, and we prove that every quasi-supersymmetric solution of Freedman's gauged supergravity belongs to an explicit four-parameter family of geodesically complete, globally hyperbolic gyratonic Brinkmann waves with spherical wave fronts. Finally, we study the quasi-supersymmetric solutions of six-dimensional minimal supergravity, defined by a system that couples a self-dual curving on a bundle gerbe to a Lorentzian metric with an irreducible chiral spinor parallel under a metric connection with totally skew-symmetric torsion given by the curvature of the aforementioned curving. Along the way, we prove that a Lorentzian six-manifold admits a skew-torsion parallel spinor with an integrable screen bundle only if it admits a foliation whose leaves are locally conformally Kähler complex surfaces.

We study the equivalence classes of slice-regular functions $f:Ω\to\mathbb{H}$ on a symmetric slice domain $Ω$, and of their subclass made of polynomial slice-regular functions, with respect to the natural action of $\mathrm{PGL}(2,\mathbb{H})$ and its subgroups, by employing the twistor construction. In particular, we characterize slice--regular functions whose twistor lift is planar and belongs to a given orbit, and we find normal classes of slice-regular polynomials with respect to the action of a parabolic subgroup of $\mathrm{GL}(2,\mathbb{H})$.

We prove a complete characterization of the extremal invariant measures for half-space geometric last-passage percolation with an arbitrary boundary parameter. This is the first result of its kind for a model in the KPZ universality class that has boundary effects and an unbounded domain. A description of a class of invariant measures was previously given in a work of Barraquand and Corwin, where it was conjectured that these should comprise all extremal invariant measures. To complete the classification, we prove a one force--one solution principle: when started in the distant past from an arbitrary initial condition with a given asymptotic slope at $\infty$, the recentered solution at time $0$ converges to a process which is distributed as the associated invariant measure with the specified slope. This limiting process is called the Busemann process, the first of its kind constructed for a half-space model. The Busemann process across all slopes is distributed as the joint invariant measure for geometric half-space LPP, recently constructed by Dauvergne and Zhang. There, it was conjectured that the constructed family of jointly invariant measures comprises all extremal jointly invariant measures; our analysis also confirms this conjecture. When the model has a strong (attractive) boundary, the collection of slopes for the invariant measures has a discontinuity, which does not arise in the full-space case. To handle this difficulty, we combine the control of the directions of semi-infinite geodesics with techniques from the theory of half-space Gibbsian line ensembles. Along the way, we classify the set of directions of semi-infinite geodesics for half-space geometric LPP, confirming a recent conjecture of Dauvergne and Zhang.

We present a unified algebraic framework utilizing the formal Bell transform to bridge the Dirichlet convolution of arithmetic functions with the combinatorial structure of infinite Euler-type products. By analyzing the logarithmic derivative of exponential generating functions, we establish explicit mappings between Bell exponents and Möbius inversions. We apply this framework to derive exact vanishing properties and congruence inheritances for classical sequences, including Ramanujan's tau function and prime-colored partitions. Furthermore, we demonstrate that the inverse Bell transform seamlessly recovers classical partition recurrences and provides a discrete combinatorial engine for generating special polynomial families, including classical Appell and Sheffer sequences.

We show that the total Cartier index of varieties with rational singularities in a bounded family is bounded. This solves a problem of Han and Jiang. The overall structure of the proof, which treats the surface case and the higher-dimensional case separately, was originated by generative AI, particularly the Rethlas system, and was substantially corrected and elaborated by hand.

The Flexible Job Shop Scheduling Problem (FJSP) is the optimal allocation of a set of jobs to machines. Two primary challenges persist in FJSP: the unpredictable arrival of future jobs and the combinatorial complexity of the problem, rendering it intractable for conventional mixed-integer linear programming solvers. This paper proposes an event-based \gls{DRL} approach to solve FJSP with random job arrivals. Specifically, we employ the Proximal Policy Optimization algorithm and use lightweight Multi-Layer Perceptrons to train the \gls{DRL} agent for minimizing the total completion time of all jobs. We design the state representation to be directly accessible from the environment, and limit the learning agent to selecting from among a set of well-established dispatching rules. Simulations show that our \gls{DRL} approach outperforms any of the individual dispatching rules on datasets with varying heterogeneity and job arrival rates. We benchmark our \gls{DRL} against an arrival-triggered mixed-integer linear programming solution and show that our method achieves good performance especially when the datasets are heterogeneous.

Every transitive cellular automaton (CA) is sensitive to initial conditions. We study this implication in the more general context of non-uniform cellular automata (NUCA) with finitely many different local update rules assigned to cells. We construct a two-dimensional NUCA that is minimal -- and hence transitive -- but that is not sensitive to initial conditions. The construction is based on an odometer NUCA on $\{0,1,2\}^\mathbb{N}$ which is nearly uniform in the sense that only the first cell uses a different local rule. Then we show that if the assignment of local rules in the cells is recurrent then transitivity implies sensitivity.

We study the high excursion probability of a centered Gaussian field on a square. Writing \(σ\) and \(r\) for its standard deviation and correlation function, we assume that \(σ\) has a unique maximum at the corner \(\boldsymbol{0}=(0,0)\) and \[ 1-σ(\boldsymbol{t}) \sim t_1^β+t_2^β+t_1^a t_2^a , \qquad \boldsymbol{t}=(t_1,t_2)\to\boldsymbol{0} \] in \(\mathbb R_+^2\). The local correlation is assumed to satisfy \[ 1-r(\boldsymbol{t},\boldsymbol{s})\sim |t_1-s_1|^α+|t_2-s_2|^α, \qquad 0<α<β. \] This product form of the standard-deviation loss is not covered by the usual locally additive assumptions. In the range \(a<β/2\), the classical essential rectangle at the variance-loss scale no longer captures the leading contribution; the relevant localization becomes side-attached and, in one regime, effectively one-dimensional. We determine the corresponding high-level asymptotics, including the logarithmic and side-dominated regimes which do not arise in the locally additive case.

We define the grove polynomials, a set-valued extension of forest polynomials. We show that they are $K$-theoretically dual to the quasisymmetric Schubert cells which pave the quasisymmetric flag variety, in the same way that Grothendieck polynomials are dual to Schubert cells in the complete flag variety. As a consequence, the finite truncations of the multi-fundamental quasisymmetric functions of Lam-Pylyavskyy acquire a geometric interpretation as $K$-theoretic representatives of quasisymmetric Schubert cells.

This paper investigates the non-resistive compressible magnetohydrodynamic (MHD) equations in $\mathbb{R}^2$. We establish the global existence and stability of classical solutions for initial data sufficiently close to a constant equilibrium state. A distinguishing feature of our result is that global stability is derived solely from pure $H^s$ energy estimate and intrinsic $L^2$ time-decay mechanism, thereby bypassing the traditional initial data requirement of $L^1$ integrability or negative-order Sobolev norm regularity. To achieve this goal, firstly we introduce some quantities motivated by effective viscous flux, which intrinsically couples density and magnetic field perturbation. Secondly, to overcome the critical time-decay obstacle arising from the absence of negative-index regularity, we develop a novel pseudo-negative-derivative technique. Moreover, we regard the wildest nonlinear term as a whole and abandon obtaining time decay estimate for each item. These approaches enable us to close the higher-order energy estimate entirely within standard Sobolev spaces.

Optimization over the intersection of two manifolds arises in a broad range of applications, but is hindered by the coupled geometry of the feasible region. In this paper, we prove that the regularities -- clean intersection and intrinsic transversality -- are equivalent, which yields a tractable projection onto the tangent space of the intersection. Therefore, we propose a geometric method that employs a retraction on only one manifold and updates the iterate along two orthogonal directions. Specifically, the iterates stay on one manifold, and the two directions are responsible for asymptotically approaching the other manifold and decreasing the objective function, respectively. Under intrinsic transversality, we derive the convergence rate for both the feasibility and optimality measures, and show that every accumulation point is first-order stationary. Numerical experiments on problems stemming from sparse and low-rank optimization, including fitting spherical data, approximating hyperbolic embeddings on real data, and computing compressed modes, demonstrate the effectiveness of the proposed method.

We prove Shokurov's global index conjecture for foliations in dimension at most three. This answers a question of the first author, Meng, and Xie in dimension three. The main result of this paper is partially obtained by generative AI, particularly the Rethlas system.

Let $G$ be a simple connected graph on $n$ vertices, and let $λ_1(G),λ_2(G),\ldots,λ_n(G)$ be the eigenvalues of its adjacency matrix $A(G)$. For $p>0$, define the $p$-energy of $G$ by $\mathcal E_p(G)=\sum_{i=1}^n |λ_i(G)|^p$. We prove that, for every real number $p\ge 2$ and every simple connected graph $G$ on $n$ vertices, $$ \mathcal E_p(G)\ge \mathcal E_p(P_n), $$ where $P_n$ denotes the path on $n$ vertices. Moreover, for each fixed $p>2$, equality holds if and only if $G\cong P_n$. Together with the previously known star-minimality results, this completes the solution of two questions of Nikiforov. The proof combines two different comparison principles. For $2<p<4$, we use a bipartite reduction, a Mellin representation of fractional powers, and a determinant comparison involving matching generating polynomials and tree shifts. For $p\ge4$, we prove a second-order stop-loss comparison for the squared singular values of bipartite graphs. This comparison is established by rank-one spectral-shift estimates, deletion-minimal counterexamples, and a finite certified analysis of the terminal sparse-sun configurations. As applications, we obtain sharp path-minimality results for positive $p$-energies in several cases, and for Laplacian and signless Laplacian power sums and related indices.

In this paper, we devise a $\operatorname{prox}$-based semi-smooth Newton method for the non-differentiable TV-minimization problem. To this end, the primal-dual optimality conditions are reformulated as a nonlinear operator equation with Newton-(type-)differentiable structure. We investigate the question of well-posedness of the resulting semi-smooth Newton scheme in the infinite-dimensional setting and identify structural properties of the associated Newton-type derivatives. For a conforming finite element discretization, we prove that the resulting semi-smooth Newton method is globally well-posed and locally super-linearly convergent. The approach extends to a large class of convex minimization problems, coincides with established semi-smooth Newton methods for obstacle problems, satisfies a primal-dual invariance, and, under suitable additional assumptions, is well-posed in the infinite-dimensional setting. Numerical experiments indicate a robust practical performance of the proposed method, including reliable reduction of the discrete primal-dual gap estimator to machine precision, robustness with respect to the choice of proximity parameters, an improved convergence basin compared to a canonical primal semi-smooth Newton method, and effective performance even for quadratically graded meshes using only a mesh-independent initialization criterion.

In this note, we show various minimality results for a geometric theory of fields $T$: $T$ is stable if and only if it is strongly minimal, $T$ is simple if and only if it has SU-rank 1, and $T$ is rosy if and only if $T$ is surgical. Combining the first equivalence with an earlier result of Hrushovski, we deduce that algebraically bounded stable fields are precisely expansions of algebraically closed fields by constants. We then consider algebraically bounded and o-minimal expansions of fields with generic derivations. We show that if $\mathbb{M}$ is a simple algebraically bounded structure and $Δ$ is a generic tuple of derivations on $\mathbb{M}$, then $(\mathbb{M};Δ)$ is supersimple if and only if the derivations commute. Similarly, if $\mathbb{M}$ is an o-minimal structure and $Δ$ is a generic tuple of $T$-derivations on $\mathbb{M}$, then $(\mathbb{M};Δ)$ is superrosy if and only if the derivations commute. We obtain explicit bounds on ranks using the Kolchin polynomial.

We study the approximation and statistical complexity of learning collections of operators in a shared multi-task setting, with a focus on the Multiple Neural Operators (MNO) architecture. For broad classes of Lipschitz multiple operator maps, we derive near-optimal upper bounds for approximation and statistical generalization. On the lower-bound side, we establish a curse of parametric complexity and prove corresponding minimax rates. Together, these results show that shared representations across tasks do not increase the overall cost: multi-task operator learning follows the same scaling laws as single operator learning. We also compare MNO with a multi-task extension of DeepONet based on concatenated task inputs and show that, from a worst-case approximation-complexity perspective, both architectures satisfy essentially the same asymptotic rates.

A central error measure in Gaussian DDPMs is the path-space KL divergence between the exact reverse chain and the learned Gaussian reverse process. This quantity is especially relevant for procedures such as classifier guidance, which perturb the entire reverse trajectory rather than only the terminal sample. Prior analyses show that standard isotropic reverse covariances suffer an unavoidable $Ω(1/T)$ path-KL error as the number of denoising steps $T$ grows. We show that matching the full posterior covariance breaks this barrier, yielding an order-wise improvement that reduces the path KL to $O(1/T^2)$. To make full covariance matching practical, we introduce the Lanczos Gaussian sampler (LGS), a training-free, matrix-free method for sampling from the optimal reverse covariance using only covariance-vector products, which are available through Jacobian-vector products of the posterior mean. LGS avoids dense covariance storage and auxiliary covariance models. We prove that LGS approximation error decays exponentially in the number of Lanczos steps, where each Lanczos step requires a single Jacobian-vector product. Empirically, using only just three such steps improves sample quality over strong diagonal-covariance baselines, including OCM-DDPM, across standard image benchmarks. This identifies full covariance matching as both theoretically valuable and practically accessible for fast DDPM sampling.

We consider bipartite exact entanglement embezzlement with a catalyst state vector $ψ$ in a Hilbert space $\mathcal{H}$ using unitaries (or more generally, contractions). If $\mathcal{M} \subseteq \mathcal{B}(\mathcal{H})$ is a von Neumann algebra and $U \in M_d \otimes \mathcal{M}$ and $V \in \mathcal{M}' \otimes M_d$ are unitaries (or more generally contractions), then such a protocol is of the form $(U \otimes I_d)(I_d \otimes V)(e_0 \otimes ψ\otimes e_0)=\sum_{i=0}^{d-1} α_i e_i \otimes ψ\otimes e_i$, where each $α_i>0$ and $\sum_{i=0}^{d-1} α_i^2=1$. We show that any such protocol must arise from a unique state on the tensor product $\mathcal{O}_d \otimes \mathcal{O}_d$ of the Cuntz algebra with itself. As a result, we prove that exact entanglement embezzlement is a self-test for a collection of $d$ Cuntz isometries for each party and a unique quasi-free state on the Cuntz algebra $\mathcal{O}_d$ in the sense of \cite{Iz93}. Moreover, we use modular theory to show that the von Neumann algebra generated by the copy of $\mathcal{O}_d$ is the unique separable approximately finite-dimensional Type $\text{III}_λ$ factor for some $0<λ\leq 1$, where $λ$ can be determined by an algebraic condition on the Schmidt coefficients of the state $φ=\sum_{i=0}^{d-1} α_i e_i \otimes e_i$.

For 24 years, it has been an open problem to obtain improved bounds, for the maximal function over a sparse sequence of discrete spherical averages, going beyond the range for the full discrete spherical maximal function. I formulate a conjecture to characterize the boundedness of such maximal functions and state a theorem in support of it.

We study the cohomological classification of vector bundles on smooth real affine surfaces and threefolds. We show that, as was observed in joint work in A. Asok and J. Fasel and in a coming joint paper with S. Banerjee and J. Fasel, under suitable cohomological assumptions on the real locus of such varieties, this classification mirrors the one obtained on algebraically closed base fields by Mohan Kumar and Murthy and by Asok and Fasel. Using an argument due to Fasel, we also give an efficient proof of a theorem of Kucharz characterising the triples of algebraic cycles that can be realised as the Chern classes of a rank $3$ bundle on a smooth real affine threefold. We further answer the questions left open by Kucharz; to our knowledge, we give the first instance of a projective module over a smooth affine $\mathbb{R}$-algebra of dimension $3$ with trivial Chern classes which is not stably free.

In this paper, we introduce and study a new class of rings calling them {\it weakly $ΔU$-rings}, hereafter abbreviated as {\it $WΔU$-rings} for short. A ring $R$ is said to be $WΔU$ if every unit of $R$ can be expressed as $\pm 1 + d$ for some $d \in Δ(R)$, where $Δ(R)$ is the largest Jacobson radical of $R$ that is closed under multiplication by units. Utilizing the known structure of $Δ(R)$, we investigate the relationships between $WΔU$ rings and certain classical concepts such as $ΔU$-rings, $UJ$-rings, $WUJ$-rings, as well as clean and exchange rings. Among the main results, we show that a matrix ring $M_n(R)$ is never $WΔU$ for any $n \ge 2$. We also provide complete characterizations of local, semi-local, semi-simple and semi-regular rings that are $WΔU$. Furthermore, it is shown for exchange rings that the $WΔU$ property is equivalent to being $WUJ$. Furthermore, the behavior of $WΔU$-rings under various ring extensions, including skew polynomial rings, skew power series rings, triangular matrix rings, trivial extensions and group rings, is thoroughly examined. Several examples are given to illustrate that the class of $WΔU$-rings properly contains the class of $ΔU$-rings. Finally, necessary and sufficient conditions for a group ring $RG$ to be $WΔU$ are established too. Resuming all of the presented above, our results expanded those by Karabaçak et al. published in J. Algebra \& Appl. (2021).

The dynamical systems having both bosonic and fermionic variables play an important role in the theory of supersymmetry. This article addresses the control problems including both bosonic and fermionic variables on Lie supergroup as the configuration space. Here, the control systems are characterized using the normalizer of Lie subsuperalgebra of left-invariant vector fields in the Lie superalgebra of all smooth vector fields of Lie supergroup. Then, the linear control system is studied in detail and its controllability criterion is proposed along with suitable examples.

Extreme events occur across the natural, engineering, and socioeconomic sciences, where rare but high-impact episodes can lead to disproportionate consequences that pose major challenges for prediction and risk management. Existing studies have mainly focused on the statistics, sampling, forecasting, and attribution of extremes from observable variables. In this paper, we develop a mathematical framework for studying the mechanisms and pathways of extreme events in partially-observed stochastic dynamical systems with hidden variables. By integrating data assimilation with information-theoretic and trajectory-based diagnostics, we infer latent precursor dynamics from observations, quantify their uncertainty, and determine how their influence propagates toward observed extreme events. Conditional Gaussian models provide a tractable analytical setting for deriving closed-form diagnostics, while the framework extends through numerical methods. The analysis proceeds from two complementary perspectives. From a trajectory-wise viewpoint, we compare filtering and smoothing distributions to identify the onset of hidden precursors and quantify temporal influence. From a statistical viewpoint, we construct event-conditioned hidden-state distributions to identify sensitive triggering directions, latent pathways, and multiple classes of extreme-event mechanisms through clustering. Three numerical examples illustrate the methodology. In an intermittent stochastic model, hidden damping dynamics emerge before observed bursts, where discrepancies between the filter and smoother provide an onset diagnostic. In a stochastic model with damping and forcing, separate damping-induced, forcing-driven, and mixed pathways to extremes are identified. In a nonlinear topographic-flow model, distinct mechanisms and pathways for blocking and unblocking patterns associated with observed extreme events are revealed.

In this work, we have considered the Laguerre polynomial. This polynomial has been studied in several branches of theoretical physics and applied Mathematics. J. K. Prajapat at.al derived condition so that Laguerre polynomial satisfy convexity, strong starlikeness, close-to-convexity and strongly convexity. In this article, characteristics properties such as exponential subordination have been studied. Moreover Janowski starlikeness and convexity have been investigated for this polynomial. Several examples and corollaries have been mentioned to validates the result.

Motivated by a question of L. Robert, asking whether $\rm L(T(A)) = Lsc_{C}(T(A))$ for any separable C*-algebra A, we introduce and initiate the study of \emph{tracially reflexive C*-algebras}. We first prove that commutative C*-algebras are tracially reflexive. We also prove that tracial reflexiveness satisfies permanence properties, such as being preserved under inductive limits. Subsequently, we expose two criteria for tracial reflexiveness, using the Cuntz semigroup and a weak version of the Schröder-Simpson theorem, respectively. In particular, separable topological dimension zero C*-algebras are tracially reflexive. We end the manuscript by closing remarks that could lead to further lines of investigation involving tracial reflexiveness.

This note is a follow up on our recent paper with L. Malý (to appear in Rev. Mat. Complut.). We provide a simple example of a compact metric space $\mathcal{P}$ for which $L^\infty(\mathcal{P})$ has the Vitali-Carathéodory property, the Sobolev $C_\infty$-capacity is an outer capacity, but the Newtonian space $N^{1,\infty}(\mathcal{P})$ contains functions which are not weakly quasicontinuous. The novelty here is that the Vitali-Carathéodory property is satified. We also obtain some related results about quasicontinuous functions in $N^{1,\infty}(\mathcal{P})$ and a characterization of when $L^\infty(\mathcal{P})$ has the Vitali-Carathéodory property.

We prove that all lattice path matroids are Ehrhart positive. This unifies and generalizes numerous results on the Ehrhart positivity of matroids developed over the last two decades. We rely on our previous work on the positivity of order polynomials of fences. Our main result supports the conjecture by Ferroni, Jochemko, and Schröter (2022) on the Ehrhart positivity of positroids. Furthermore, our main result implies that all Schubert matroids are Ehrhart positive, which thus settles a conjecture by Fan and Li (2024), and supports a conjecture by Monical, Tokcan, and Yong (2019) on the Ehrhart positivity of Schubitopes.

A new family of local-global conjectures in the representation theory of finite groups has recently been proposed by Moretó. We show that one of the strongest of these conjectures, the strong subnormalizer conjecture, holds for $p$-solvable groups when $p$ is odd, under the condition that the subnormalizer subset is a subgroup. We also prove it in general when $p$ is odd and the $p$-length of the group is 1 and, in the process, obtain new properties related to the Glauberman correspondence.

A fuzzy Boolean function is a map $f:\cube^n\to [0,1]$, where $n\in\mathbb N$. We introduce and compare three ways of saying that such a function has bounded complexity. The first is a sampling property: the value $f(x)$ can be recovered, up to small error and with high probability, from the values of a bounded number of randomly chosen coordinates of $x$. We call this the holographic property. The second is a structural property: $f$ is uniformly close to a bounded-degree polynomial in boundedly many bounded linear coordinate forms. The third is computational: $f$ is uniformly close to the output of a neural network with a bounded number of non-input neurons, bounded Lipschitz activation functions and bounded incoming weights. We prove that these three properties are equivalent up to quantitative changes of the parameters. The implication from holography to polynomial structure uses a variant of a weak version of hypergraph regularity.