Papers

We describe several new algorithms for finding all prime numbers up to a given bound $N$, achieving the first ever speedup by a positive power of $\log N$ over the ancient sieve of Eratosthenes. The fastest version, which is not fully rigorous, runs in \[ N (\log \log N)^{1+o(1)} \] bit operations when analysed in the multitape Turing model. This improves on the best existing algorithms due to Pritchard (1981), Atkin–Bernstein (2004) and Sergeev (2016) by a factor of almost $\log N$. We also present a rigorous randomised (Las Vegas) variant that is slower by a factor of $(\log \log N)^{1+o(1)}$, and a rigorous deterministic variant that is slower by a factor of $(\log N)^{1/2+o(1)}$. The new algorithms make heavy use of fast polynomial arithmetic over finite fields, and also involve ideas from the theory of error-correcting codes.

We use the deformation methods to obtain the strictly log concavity of solution of a class Hessian equation in bounded convex domain in $\mathbb{R}^{n}$, as an application we get the Brunn–Minkowski inequality for the Hessian eigenvalue and characterize the equality case in bounded strictly convex domain in $\mathbb{R}^{n}$.

Let $\pi$ be a cuspidal representation of $GL(n,F)$ over a finite field $F$. Let $P=MN$ be the Levi decomposition of a maximal parabolic subgroup corresponding to the partition $(k,n-k)$ of $n$. Given a rank $r$ character $\psi_r$ of the unipotent radical $N$, the twisted Jacquet module $\pi_{N, \psi_r}$ is a representation of the subgroup $M_r$ of $M$ which stabilizes $\psi_r$. The problem we solve in this work is to determine the structure of $\pi_{N, \psi_r}$ as a $M_r$-module. This problem was first studied by D. Prasad, who solved it for the case $r=k=n/2$ by calculating the character of $\pi_{N, \psi_r}$ and matching it to a known representation of $M_r$. In this work, we solve the problem for all values of $(r,k,n)$ directly without calculating the character of $\pi_{N, \psi_r}$. Our solution depends on two other key conceptual advances: (i) We generalize the Bernstein-Zelevinsky framework for studying representations of the Mirabolic subgroup of $GL(n,F)$, to maximal parabolic subgroups $P$. In particular, we show that the twisted Jacquet functor which takes a representation of $P$ to its twisted Jacquet modules, gives an equivalence of categories between Rep$(P)$ and the direct sum $\oplus_r \text{Rep}(M_r)$. (ii) Using this, we construct a pair of recursively defined representations $\Pi_{k,n}, \Pi_{n-k,n}^\dagger$ of $P$, which generalizes to $P$, the representation of the Mirabolic subgroup obtained from the trivial representation by recursively applying the Bernstein-Zelevinsky $\Phi^+$ functor. Like the representation $(\Phi^+)^{n-1}(1)$ of the Mirabolic subgroup, the representation $\Pi_{n-k,n}^\dagger$ satisfies a universal property with respect to restrictions to $P$ of cuspidal representations of $GL(n,F)$. Our solution of the main problem is a simple consequence of this universal property.

We study a level analog of Euler's totient function for integer mosaics. Let $P_i(n)$ be the set of primes appearing in the first $i$ levels of the mosaic of $n$, and let $\varphi_i(n)$ count the integers $m\leq n$ for which $P_i(m)\cap P_i(n)=\varnothing$. We prove a Möbius divisor-sum formula for $\varphi_i(n)$ and reduce it to a sum over a set $\mathcal{V}_{i,S}$ of powerful integers. If $S\neq\varnothing$, $i\geq 2$, and $q=\min S$, then \[ |\mathcal{V}_{i,S}\cap[1,x]|\sim C_{i,S}x^{1/q}, \] with $C_{i,S}$ an explicit positive Euler-product constant. For fixed $S$, the density $\delta_i(S)$ of integers whose first $i$ levels avoid $S$ exists and has an Euler product; for nonempty $S$, $i\ge2$, and $q=\min S$, the number of such integers up to $N$ is $\delta_i(S)N+O_{i,S}(N^{1/q})$. Taking $S=P_i(n)$ gives \[ \frac{\varphi_i(n)}{n}=\delta_i(P_i(n))+O_\varepsilon(n^{-1/2+\varepsilon}) \] uniformly in $n$.

A non-intrusive, discretization and PDE agnostic limiting technique is introduced. The method is conservative and can be applied to high-order approximations including spectral methods. It can be applied to time-dependent and time-independent problems. It can be used for elliptic, parabolic and hyperbolic problems.

For every integer $k \ge 1$ and sufficiently large $n$, we show that the extremal construction for the Turán number of the $4$-uniform tight cycle of length $4k+2$ minus one edge is a complete odd-bipartite $4$-graph. In particular, since $C_{6}^{4-}$ contains the $4$-uniform expanded triangle as a subgraph, our result extends that of Frankl and Keevash–Sudakov on the Turán density and the Turán number of the $4$-uniform expanded triangle. We also show that the Turán density of $C_{4k+2}^{4}$ is $1/2$ for all integers $k \ge 2$, and establish the corresponding stability result. This strengthens the result of Sankar on the Turán density of $C_{4k+2}^{4}$ which holds only for all sufficiently large $k$.

In this paper, we study the signless Laplacian spectral radius of pure simplicial complexes under local homological restrictions on links. Let $K$ be a pure $r$-dimensional complex on $n$ vertices, ${\mathfrak q}_{r-1}(K)$ be the spectral radius of the $(r-1)$-up signless Laplacian of $K$, and ${\operatorname{lk}}_K(\sigma)$ be the link of a face $\sigma$ in $K$. We prove that if the homology $\widetilde H_t({\operatorname{lk}}_K(\sigma), {\mathbb R})=0$ for every face $\sigma\in K$ with $|\sigma|=r-t$, then \[ {\mathfrak q}_{r-1}(K)\le tn-(t-1)(r+1).\] Moreover, if $K$ is $r$-down path connected and $n\ge r+2+\binom{r+1}{t}\binom{r}{t}$, equality holds if and only if $K \cong \Delta_{r+1-t} \star \Delta_{n-r-1+t}^{t}$, where $\Delta_n$ denotes a simplex on $n$ vertices, $\Delta_n^{p}$ denotes the $(p-1)$-skeleton of $\Delta_n$, and $\star$ denotes the join of two complexes.

This paper focuses on the free boundary problem of the three-dimensional compressible Navier-Stokes-Poisson equations with degenerate viscosities for self-gravitating viscous gaseous stars. For spherically symmetric barotropic motion, we establish the local well-posedness of classical solutions. The solutions obtained here are smooth all the way up to the moving boundary and capture the physical vacuum boundary behavior of the Lane-Emden star configuration for all adiabatic exponents $\gamma>\frac{4}{3}$.

We introduce a generalized first Ueda obstruction class for a line bundle along a closed analytic subvariety of a complex manifold, allowing singular subvarieties and restrictions that are not unitary flat. Using a Dolbeault resolution, the Chern curvature appears naturally, and non-vanishing of the class obstructs the existence of a smooth Hermitian metric with semi-positive curvature. As an application, we recover several classical examples within a unified framework and provide new examples of nef but not semi-positive line bundles. We also introduce higher degree analogues of these obstruction classes.

It is well-known that deleting or contracting any element of a connected matroid always yields at least one connected minor. However, for a connected polymatroid, only two such elements can be guaranteed, proved by Hall in 2013. This note investigates the connectivity properties of slice-projections of connected polymatroids, which includes deletion and contraction. We establish that for any element of a connected polymatroid, at least one of its two consecutive slice-projections is connected. We also obtain that the $j$-th slice-projection from the top of a polymatroid and $j$-th slice-projection from the bottom of its dual have the same connectedness. These results both extend existing connectedness theorems for graphs and matroids.

This paper investigates the dynamical behaviors of solutions to the Yamabe flow via the modified potential well method. We first establish the local existence and regularity of weak solutions for the flow. Several new results concerning global existence and blowup are obtained by classifying initial data into stable and unstable sets. Specifically, solutions with initial data in the stable set exist globally and extinguish in finite time, whereas those originating from unstable initial data blow up in infinite time. For certain high-energy initial data, we show that the solution decays to zero as time tends to infinity and undergoes finite-time blowup. In addition, we analyze Palais-Smale sequences to reveal the intrinsic relationship between the long-time asymptotic behavior of solutions and steady states. Finally, we derive the Pohozaev identity for the equation and prove the corresponding nonexistence theorem.

The conformal boundary condition for gravity fixes the boundary conformal class and the mean curvature, leaving the trace-free extrinsic curvature free as the conjugate response. It is the boundary-value form of York's conformal decomposition of gravitational data, shown to be well posed for Einstein metrics by Anderson. Witten identified it as the elliptic replacement for the ill-posed Dirichlet condition in the finite-boundary perturbative Euclidean gravitational path integral. We show that this perturbative construction admits no half-supersymmetric extension in linearized minimal supergravity. For fixed conformal bosonic data, no half-dimensional gravitino boundary condition (local or pseudodifferential, APS-type included, with any compatible ghost condition at highest-derivative order) closes the full preserved chiral supersymmetry. Supersymmetry first selects the natural local chiral gravitino datum. Acting back on this datum then produces the trace-free extrinsic curvature, precisely the response that the conformal prescription leaves unfixed. The obstruction is therefore not the failure of a particular elliptic ansatz: even the chiral/Robin completion that is LS-elliptic and BRST-compatible at highest-derivative order would impose Dirichlet control on a Neumann response. The obstruction is pointwise in tangential momentum and survives compensating gauge transformations. It is a linearized, highest-derivative obstruction, not a global or nonlinear no-go; nonlinear supercovariant boundary terms may evade it by tying the trace-free extrinsic curvature to gravitino bilinears.

Fix an integer $m\geq 0$. We study the one-dimensional problem of deciding when a system of linear inequalities \begin{equation*} \sum_{j=1}^M A_{ij}(x)F_j(x)\leq f_i(x),\qquad i=1,\ldots,N, \end{equation*} with fixed semialgebraic coefficients $A_{ij}:\mathbb R\to\mathbb R$ and given functions \begin{equation*} f_i\in C^\infty(\mathbb R),\qquad i=1,\ldots,N, \end{equation*} admits a solution \begin{equation*} F=(F_1,\ldots,F_M)\in C^m(\mathbb R,\mathbb R^M). \end{equation*} The analogous problem for systems of linear equations admits a finite linear differential criterion, whereas for systems of inequalities such a criterion fails in general in dimensions at least two, already for continuous solutions. This paper gives a complete answer in one dimension for a system of linear inequalities. For $m=0,1,2$, the existence of a $C^m$ solution is characterized by a finite differential criterion. In contrast, for every $m\geq 3$, no finite criterion of this local differential type exists in general, even for a fixed constant-coefficient system. Thus, $m=2$ is the sharp regularity threshold for finite differential characterization of the one-dimensional inequality problem.

Consider a distributed storage system consisting of $N$ non-colluding servers that collectively store a database of $M$ files encoded using an $[N,K]$ maximum distance separable(MDS) code. A user wishes to retrieve one file privately by accessing the servers without revealing the identity of the requested file. A scheme designed for this purpose is called a joint MDS-coded private information retrieval(PIR) scheme, which was first introduced by Sun and Tian in 2019 to break the capacity $\frac{1-K/N}{1-(K/N)^M}$ of the separate MDS-coded PIR schemes established by Banawan and Ulukus. However, the capacity of joint MDS-coded PIR remains largely unexplored. In this paper, we study the capacity of joint MDS-coded PIR with systematic MDS array storage codes under prescribed storage patterns. Specifically, we first derive upper bounds on the capacity of joint MDS-coded PIR for $K=Mt$ and $K=Mt+1$, respectively. We then construct three joint MDS-coded PIR schemes for the cases $N\le K+t, K=Mt$, $N>K+t, K=Mt$ and $N\le K+t, K=Mt+1$. The proposed schemes require small file sizes and achieve higher retrieval rates: the first and third schemes exceed the capacity of separate MDS-coded PIR schemes, while the second scheme does so when the storage rate $\frac{K}{N}>r_M$ for some $0<r_M<\frac{M}{M+1}$. In particular, for $K=Mt$ and $N\leq K+t$, the proposed scheme achieves the derived upper bound, thereby establishing that the optimal joint MDS-coded PIR capacity under the considered storage pattern is $1-(1-\frac{1}{M})\frac{K}{N}$. Compared with capacity-achieving separate MDS-coded PIR schemes at the same storage-code rate, the proposed schemes may achieve a substantial relative retrieval-rate improvement: the maximum improvement can exceed $15\%$ when $M\geq 4$, exceed $20\%$ when $M\geq 9$, and asymptotically approach $1-2/e\approx 26.42\%$ as M increases.

Formulating a consistent integration measure for quantum geometric fluctuations without violating diffeomorphism invariance remains a theoretical challenge. In this work, a framework rooted in the statistics of discrete Poisson point processes is proposed. The formulation yields a double-exponential probability functional characterized by a capacity limit, which acts as an amplitude regularizer suppressing ultraviolet singularities. To evaluate this model at macroscopic scales, a statistical bifurcation of the stochastic action is identified. First, the macroscopic mean condenses to define the classical continuous spacetime background and its matter distribution. Second, at macroscopic scales, the Law of Large Numbers dictates that the residual ultraviolet noise maps into an infrared continuous zero-mean Gaussian martingale within the bulk. Third, this zero-mean Gaussian noise linearly generates standard quantum kinematic effects. Fourth, evaluating the non-linear exponential action separates the variance of this Gaussian noise from the linear cancellation, rectifying it into a macroscopic drift that manifests as the dark energy density. Diluted by the Bekenstein-Hawking entropy of the observable universe, this bulk variance dictates a continuous field cutoff at 6 TeV. Building upon this framework, broad phenomenological applications are demonstrated: (1) establishing a UV-finite effective field theory preserving gauge symmetries in 4D; (2) constructing a topological model of particles deriving Standard Model hierarchies; (3) formulating a non-singular cosmological model predicting observed large-scale power suppression in the cosmic microwave background; and (4) deriving foundational axioms of quantum mechanics as emergent statistical phenomenologies. Collectively, this framework provides a falsifiable synthesis bridging discrete quantum geometry and continuous macroscopic physics.

Semiparametric efficiency theory provides the mathematical foundation for influence-function-based estimation, including one-step estimators, targeted minimum loss estimators, and many modern inferential methods used in causal inference and missing data analysis. Despite its widespread use, the theory is often presented through a collection of technical constructions whose geometric meaning remains opaque. As a result, influence functions are often derived and applied without an intuitive understanding of the principles connecting scores, tangent spaces, nuisance tangent spaces, and efficient influence functions. This tutorial develops a geometric exposition of semiparametric efficiency theory as a form of differential calculus on a space of probability distributions. Drawing systematic parallels with ordinary multivariable calculus, we show that paths of distributions play the role of curves, scores play the role of velocity vectors, influence functions play the role of gradients, and efficient influence functions arise as projected gradients. This perspective provides a unified explanation for several foundational questions, including why perturbation directions are represented by functions, why tangent spaces depend only on the statistical model whereas nuisance tangent spaces depend on the parameter of interest, and why efficient influence functions arise through orthogonal projection. The resulting framework offers a geometric perspective on semiparametric efficiency theory and influence-function-based inference.

We show that an integral non-degenerate fusion category $\mathcal{C}$ is group-theoretical if the Frobenius-Perron dimensions of its simple objects are either 1 or powers of a prime $p$.

We introduce a new four-point comparison principle, called the $(\varepsilon,\delta)$-weak quadruple condition, for non-Riemannian spaces with synthetic non-negative curvature. This condition holds not only for classical Alexandrov spaces with non-negative curvature, but also for many genuinely non-Riemannian spaces. In particular, we show that this condition is intrinsic to spaces satisfying Ohta's $S$-concavity in the full parameter range. Using this comparison principle, we develop a non-symmetric strainer framework and establish a Burago–Gromov–Perelman-type structure theory for finite-dimensional $S$-concave Busemann concave spaces. We prove that these spaces have constant integer Hausdorff dimension, satisfy the measure contraction property, are rectifiable, and admit unique Banach tangent cones almost everywhere. We further show that each such space contains an open dense topological manifold part of top dimension and full measure. Finally, we establish Hausdorff dimension estimates for singular strata and construct natural measure-theoretic stratifications of these spaces. Our framework includes Alexandrov spaces with non-negative curvature as a special case, and provides tools for studying Finslerian metric spaces whose tangent cones need not be metric cones and angles need not be symmetric.

In QFT on curved spacetime, the Hadamard condition is often presented as a necessary condition on 'physically reasonable' states of the quantum field, and plays a central role in many theoretical and foundational applications - ranging from proofs of the renormalizability of Wick polynomials to derivations of the Hawking temperature. Yet despite this, the philosophical and foundational underpinnings of the Hadamard condition remain murky. I critically discuss existing motivations for the Hadamard condition in the literature, before arguing in favour of an alternative motivation for the Hadamard condition, according to which it is best understood as a necessary and sufficient condition for the existence of a well-defined operator product of a certain form on a sufficiently large space of observables of the quantum field, satisfying a variety of further conditions. This clarifies the role and status of the Hadamard condition, including its relationship to the equivalence principle, to well-definedness of physical quantities such as Wick polynomials and the expectation value of the stress-energy operator, and the sense in which Hadamard states are 'vacuum-like'.

The system coupled by the Schrödinger equation and the Intermediate Long Wave (ILW) equation is a particular model describing the interaction of long and short waves. We prove local and global well-posedness of the initial value problem associated with this system in Sobolev spaces of low regularity. Our approach relies on the use of energy estimates, Bourgain spaces, and Tao's gauge transformation.

This paper studies the long-time dynamics of a coupled hyperbolic system for magnetizable piezoelectric beams with infinite viscoelastic memory. Memory dissipation is characterized by $A^\alpha$, the fractional power of a positive self-adjoint operator $A$ with $\alpha\in[0,1)$. An asymmetric fractional magnetoelectric coupling is adopted: the mechanical-to-magnetic coupling uses integer-order operator $A$, while the magnetic feedback to mechanics is governed by fractional operator $A^\beta$ ($\beta\in[0,1)$). This model bridges the gap between the well-studied integer coupling case ($\beta=1$) and the unsolved fully fractional symmetric coupling problem, offering a universal framework for related coupled systems. Under mild assumptions on memory kernels and system parameters, we prove well-posedness via semigroup theory and derive an explicit polynomial decay estimate for smooth initial data: \[ \|X(t)\|_{\mathcal H} \le C t^{-\frac{1}{4-2\beta-2\alpha}}\|X_0\|_{D(\mathcal A)},\quad \forall\,t\ge 1, \] where the decay exponent is explicitly determined by $\alpha$ and $\beta$. For exponentially decaying memory kernels, the decay rate is sharp if stiffness coefficients satisfy $\alpha_1\ne\alpha_2$. When $\alpha_1=\alpha_2$, we only obtain an upper bound $\delta\le \frac{1}{3-\beta-2\alpha}$ for decay index $\delta$, leaving the optimal rate open. Comparisons with integer feedback coupling ($\beta=1$) show fractional feedback ($\beta<1$) slows energy decay. It demonstrates that $\beta$ weakens indirect damping and degrades structural stabilization. The results reveal the intrinsic interaction between fractional memory dissipation and fractional coupling in strongly coupled dissipative systems.

We analyze the stability of the Hagstrom-Warburton nonreflecting boundary conditions (HW-NRBCs) for the first-order time-dependent Maxwell equations. The HW-NRBCs enjoy very small reflection coefficients and do not use high-order derivatives or nonlocal boundary operators, which makes them well-suited for high-order accurate numerical discretizations. The main result of this paper is an $L^2$ a-priori bound on the solution in terms of initial and boundary data and volume sources in a half-space. To obtain this result, we first derive a mapping that takes outgoing components to ingoing components of the solution of the Maxwell equations with HW-NRBCs, which shows that the Kreiss condition does not hold uniformly. Next, to prove stability, several symmetrizers are constructed, which establishes well-posedness in a generalized sense.

We discuss the action of the center in the categorical local Langlands program in the form of [FS24]. We provide a spectral description of the convolution action of a central torus in this situation. We show that the veracity of the $\ell$-adic categorical Langlands conjecture passes to the quotient by a central torus $D$ satisfying a minor cohomological condition in characteristic 0 and we show that under restriction to Langlands-Shahidi type parameters the conjecture passes to such a quotient in all characterstics and even integrally. In that situation we also show that the splitting of the semi-orthogonal decomposition descends. Finally we use these results to extend [Zou25] to $\mathrm{PGL}_n$.

In this paper, we study value selection for fixed-pattern factorized sparse approximate inverse preconditioners. Given a prescribed sparsity pattern for a factor $G,$ we choose its admissible entries by optimizing spectral objectives of the congruent preconditioned operator $P(G)=GAG^T.$ This differs from classical sparse approximate inverse and FSAI constructions, which choose entries through algebraic Frobenius-residual criteria. For symmetric positive definite systems, the spectral target is a cluster near $+1.$ For symmetric indefinite systems, where congruence preserves inertia, we introduce a bimodal loss that drives positive and negative eigenvalues toward separated clusters near $+1$ and $-1,$ while penalizing eigenvalues near zero. To make these objectives practical for large sparse matrices, we derive projected Krylov support-gradients. Lanczos runs provide both a stochastic trace estimate of the spectral objective and a Ritz approximation to the exact gradient. We implement the resulting gradient through a detached Rayleigh surrogate: the Lanczos data are computed without gradient tracking and held fixed, while the backward pass differentiates only recomputed Rayleigh quotients with respect to the admissible entries of $G.$ This avoids differentiating through the Lanczos recurrence while returning a matrix-free gradient on the prescribed support. We also discuss a projected Kernel Polynomial Method rule as a finite polynomial comparison. Experiments on finite-element test problems show that spectral value selection improves fixed-support preconditioners, especially for symmetric indefinite saddle-point systems. We further demonstrate a graph neural network model for predicting admissible factor entries across related matrices.

We prove that a group $G$ is locally finite if and only if, for some (equivalently, every) infinite set $A$, every cellular automaton $A^G\to A^G$ has closed image in the prodiscrete topology. Equivalently, this holds if and only if every linear cellular automaton $V^G\to V^G$ has closed image for some pair $(K,V)$ with $V$ infinite-dimensional over the field $K$ (equivalently, for every such pair). This gives affirmative answers to Open Problems 6 and 7 of Ceccherini-Silberstein and Coornaert. More precisely, if $G$ is not locally finite, then for every infinite set $A$ there is a finite-memory cellular automaton $A^G\to A^G$ with non-closed image, and for every field $K$ and every infinite-dimensional $K$-vector space $V$ there is such a linear cellular automaton $V^G\to V^G$. The common obstruction is constructed on a countable direct-sum alphabet from an infinite ray in a locally finite Cayley graph. A direct-summand argument gives arbitrary vector-space alphabets, while an alphabet-retract principle gives arbitrary infinite set alphabets.

Likelihood is standard for Hawkes-process inference, while less computationally demanding methods have largely developed separately. We show that least squares, Takács–Fiksel, and related moment-based estimators form a single class of compensator-based estimating equations, with the likelihood score as the efficient benchmark. For fixed-dimensional multivariate Hawkes processes with compact memory, nonlinear positive links, and signed kernels allowing inhibition, every suitably regular predictable functional of a fixed lag window yields an unbiased estimating equation when integrated against $\text{d} N-\lambda\,\text{d} t$. Under common regularity, identification, and rank conditions, estimators based on every admissible finite library achieve uniform high-probability and pointwise almost-sure $\mathcal O(\sqrt{\log(T)/T})$ rates, asymptotic normality with Godambe covariance, and admit feasible two-step optimal weighting. A projection identity quantifies each library's exact efficiency loss as the score information outside its predictable span; a two-point bound shows the root-$T$ scale cannot be improved uniformly. Although compact memory localizes the intensity rather than the stationary law, exponential forgetting yields Bernstein-type concentration and transfers the theory to nonstationary starts after a logarithmic burn-in. Within this scope, the compensator class is exhaustive for finite-library comparisons: it contains the score, gives admissible libraries common guarantees, and quantifies their efficiency gaps exactly.

We study the existence of normalized solutions for the nonlinear Dirac equation \[ \begin{cases} -i\sum\limits_{k=1}^3\alpha_k\partial_k u + m\beta u - |x|^{-b}|u|^{p-2}u = \mu u, \quad x\in\mathbb{R}^3, \int_{\mathbb{R}^3}|u|^2 dx = a, \end{cases} \] where $b\in(0,1)$, $p\in(2,3-b)$, $a>0$ is a prescribed mass, and $\mu\in\mathbb{R}$ is a Lagrange multiplier. For any $b\in(0,1)$ and $p\in(2,3-b)$, we establish the existence of a normalized solution for all sufficiently small masses $a>0$, with $\mu\in(0,m)$ and $u\in H^{\frac12}(\mathbb{R}^3;\mathbb{C}^4)$. Our results cover the full range of nonlinearities, including mass-subcritical, mass-critical, and mass-supercritical cases. The main challenges are the strongly indefinite nature of the Dirac operator and the loss of translation invariance caused by the singular weight $|x|^{-b}$. We overcome these difficulties by combining a constrained min-max reduction method with a novel weighted compact embedding in $L^p(\mathbb{R}^3,|x|^{-b}\,\mathrm{d}x;\mathbb{C}^4)$. This approach circumvents the singular potential at the origin and yields a unified existence theory valid in the small-mass regime.

This paper establishes a rigorous connection between Noether's second theorem and the structure of Pareto fronts in multi-criteria variational problems. We prove that for a family of functionals invariant under a common gauge Lie group, the regular part of the Pareto front is contained in the union of solutions to weighted Euler–Lagrange equations, and that on Pareto-optimal sections, combinations of Noether currents are conserved. Gauge symmetries are shown to impose additional constraints on the Lagrange multipliers via Bianchi-type identities, reducing the dimension of the admissible weight space. Two practical approaches are derived from this result: dimensionality reduction via gauge fixing and scalarization with penalty terms. The theory is validated on three classical problems: the linear-quadratic regulator with two criteria, a model predictive control problem with a minimax criterion, and the double integrator with control constraints. In each case, the Pareto front is shown to satisfy an exact conservation law, and the analytical structure of the front is recovered from the symmetries of the problem. The results demonstrate that gauge functions (null Lagrangians) can restore invariants even when the dynamics break the symmetry, and that control constraints do not destroy the underlying conservation laws but merely restrict the admissible set. The proposed approach yields a Pareto front with minimal deviation from the theoretical curve, outperforming known multi-objective algorithms, and provides a new foundation for the analytical treatment of multi-criteria optimization in nonlinear dynamics and control.

Octonions and split-octonions have been used to express the Dirac equation in physics in several conceptually distinct ways. This review organizes them into four groups: the 2-factor, 3-factor, and projection representations, which use (split-)octonion basis elements natively to model a spacetime basis, and the conventional use of octonions to carry Dirac algebra acting on spinors. For each group we identify the originating construction, relate later variants to it, and point to modern applications and to recent methods for native split-octonionic analysis. Because the multiplication table of the (split-)octonions is not unique, forms that look different in print can coincide after a norm-preserving linear isometry and a multiplication-preserving change of basis; we make such relations explicit and tabulate the basis conventions used across select sources. The 2-factor representation is treated in most detail: we document its origin and show that a recently proposed split-octonionic Dirac equation coincides with it after such an isometry and change of basis, while crediting the independent contributions of that work.

In this article, we study Riemannian planes $(M,g)$ which satisfies a bounded geometry condition ( bounded sectional curvature and a non-vanishing injectivity radius of $(M,g)$). We show that such a Riemannian plane is always quasi-isometric to a radially symmetric plane which also satisfies the aforesaid bounded geometry condition. Using that, we study the quasi-redirecting boundary of Riemannian planes with bounded geometry and show that it is either a single point or $\mathbb{S}^1$. We contrast the quasi-redirecting boundary with the Martin boundary and construct Riemannian planes with bounded geometry for which the We also characterise quasi-redirection in Riemannian planes with bounded geometry via their asymptotic cones and their cut points. Lastly, we characterise $\mathbb{E}^2$ and $\mathbb{H}^2$ upto quasi-isometry by their asymptotic cones.