In 2023, the author presented the first computable minimal Euclidean function for a non-trivial number field. Along with a formula for $ϕ_{\mathbb{Z}[i]}$, the minimal Euclidean function on the Gaussian inteers, the same paper introduced a geometric description for $ϕ_{\mathbb{Z}[i]}^{-1}([0,n])$. This paper uses that construction to prove formulas for the size of the function's pre-images, or $|ϕ_{\mathbb{Z}[i]}^{-1}([0,n])|$.

We analyse divergent diagrams of \(k\)-fold map-germs on \((\mathbb{C}^n,0)\), for $k, n \geq 2$, associated with reflections, adapting to the complex setting the theory of folds associated with involutions on \((\mathbb{R}^n,0)\). In the complex case, a \(k\)-fold is naturally related to a cyclic group generated by a reflection, which guides the analytic classification of singularities. Under the conditions of transversality and linearity of the associated reflections, certain conditions related to the nontrivial eigenvalues appear as invariants by simultaneous conjugacy. We also provide a complete classification of pairs of transversal linear reflections and the corresponding divergent diagrams.

In this work, we introduce and study the $p$-$α$-closest-center problem ($pα$CCP), which generalizes the $p$-second-center problem, a recently emerged variant of the classical $p$-center problem. In the $pα$CCP, we are given sets of customers and potential facility locations, distances between each customer and potential facility location as well as two integers $p$ and $α$. The goal is to open facilities at $p$ of the potential facility locations, such that the maximum $α$-distance between each customer and the open facilities is minimized. The $α$-distance of a customer is defined as the sum of distances from the customer to its $α$ closest open facilities. If $α$ is one, the $pα$CCP is the $p$-center problem, and for $α$ being two, the $p$-second-center problem is obtained, for which the only existing algorithm in literature is a variable neighborhood search (VNS). We present four mixed-integer programming (MIP) formulations for the $pα$CCP, strengthen them by adding valid and optimality-preserving inequalities and conduct a polyhedral study to prove relationships between their linear programming relaxations. Moreover, we present iterative procedures for lifting some valid inequalities to improve initial lower bounds on the optimal objective function value of the $pα$CCP and characterize the best lower bounds obtainable by this iterative lifting approach. Based on our theoretical findings, we develop a branch-and-cut algorithm (B&C) to solve the $pα$CCP exactly. We improve its performance by a starting and a primal heuristic, variable fixings and separating inequalities. In our computational study, we investigate the effect of the various ingredients of our B&C on benchmark instances from related literature. Our B&C is able to prove optimality for 17 of the 40 instances from the work on the VNS heuristic.

We prove the existence of spontaneous symmetry breaking in suitably low-energy eigenstates of certain gapless and frustrated many-body quantum systems, namely symmetric quantum perturbations to classical models which exhibit spontaneous symmetry breaking of a finite group at some positive temperature. Additionally, the classical model need not be local in space, as long as it satisfies a quantum analogue of the Peierls condition. As an example of our technique, we establish robust ferromagnetism in random-bond Ising models in $d= 2$ dimensions with sufficiently biased random couplings, with weak transverse field. Our mathematical technique is based on establishing quantum bottlenecks, similar to a "many-body WKB" method for evaluating tunneling rates. Using these same methods, we provide new proofs of metastability and the slow decay of the false vacuum, applicable to gapless metastable states. Our work represents a first step towards a rigorous classification of stable gapless quantum phases.

We consider the problem of estimating the missing mass, partition function or evidence and its probability distribution in the case that for each sample point in the discrete sample space its (unnormalized) probability mass is revealed. Estimating the missing mass or partition function (evidence) is a well-studied problem for which, in different contexts, the harmonic mean estimator and the Good-Turing (and related) estimators are available. For sampling on a discrete set with revealed probability masses these estimators can be Rao-Blackwellized, leading to self-consistent estimators not involving an auxiliary distribution with known total mass. For the case of sampling from a mixture distribution this offers the perspective of anchoring the estimator at both ends: at the diffuse end (high temperature in statistical physics) via an explicit expression for the total probability mass and at the peaked end (low temperature) via the feature of repeated entries in the sample. Estimation is model-free, but to provide a probability distribution for the missing mass or partition function a model is needed for the distribution of mass. We present one such model, identify sufficient reduced statistics, and analyze the model in various ways -- Bayesian, profile likelihood, maximum likelihood and moment matching -- with the objective of eliminating the mathematical (nuisance) parameters for a final expression in terms of the observed data. The most satisfactory (explicit and transparent) result is obtained by a mixed method that combines Bayesian marginalization or profile likelihood optimization for all but one of the parameters with plain maximum likelihood optimization of the final parameter.

We introduce jacobian graphs, which are explicit families of regular graphs that are spectrally indistinguishable from random graphs, but whose local structure is very different from that of random graphs. The construction relies on the geometric properties of generalized jacobians of curves and on general equidistribution theorems for character sums over finite fields.

Cops and Robbers is a pursuit-evasion game played on graphs, of which many variants have been developed and studied. We introduce a variant of this game, "Sneaky-Active Cops and Robbers", where all cops and robber must move on their turn, and where the robber is allowed to move onto a cop position without being captured. We show that for reflexive graphs, this game is equivalent to the classical cops and robbers and that the cop number for a graph is invariant under $\times$-homotopy equivalence. We then develop further properties of this game, computing cop numbers for a number of graph families and developing results about the behavior of categorical and box products of graphs.

The farthest-first traversal of Gonzalez is a classical $2$-approximation algorithm for solving the $k$-center problem, but its sequential nature makes it difficult to scale to very large datasets. In this work we study the effect of running farthest-first on a $δ$-cover of the dataset rather than on the full set of points. A $δ$-cover provides a compact summary of the data in which every point lies within distance $δ$ of some selected center. We prove that if farthest-first is applied to a $δ$-cover, the resulting $k$-center radius is at most twice the optimal radius plus $δ$. In our experiments on large high-dimensional datasets, we show that restricting the input to a $δ$-cover dramatically reduces the running time of the farthest-first traversal while only modestly increasing the $k$-center radius.

In this paper, we consider a wave equation with strong damping and logarithmic nonlinearity. This paper aims to study the local and global existence, uniqueness and the uniform energy decay rate of a weak solution under some sufficient conditions on the initial data. Unlike previous literature restricted to the lower subcritical range $2 < γ< \frac{2(n-1)}{n-2}$, we successfully extend the validity of the well-posedness and stabilization results to the upper subcritical range $\frac{2(n-1)}{n-2} \leq γ< \frac{2n}{n-2}$.

A total graph is an ordered triple $(V_0, V_1, E)$, where $V_0, V_1$ are the sets of empty and full vertices, respectively, $V_0 \cap V_1 = \emptyset$, and the set of edges $E$ is a subset of \(\binom{V_0 \cup V_1}{2}\) $(E\cap(V_0 \cup V_1)=\emptyset)$. A simple graph is a total graph in which all vertices are full. We say that a total graph $G$ is locally irregular if every two adjacent vertices have different total degrees, where by the total degree of a vertex $v$ in $G$ we mean the number of edges in $G$ that contain $v$ plus 1 if $v$ is full, or plus 0 if $v$ is empty. A total coloring of a graph $G$ whose colors induce locally irregular total subgraphs is called locally irregular total coloring, and the minimum number of colors required in such a coloring of $G$ is denoted by ${\rm tlir}(G)$. In 2015, Baudon, Bensmail, Przybyło, and Woźniak conjectured that ${\rm tlir}(G)\leq 2$ for every graph $G$. In this paper, we prove this conjecture for cacti, subcubic graphs, and split graphs. We also provide a general upper bound for ${\rm tlir}(G)$ depending on the chromatic number of $G$, and a constant upper bound if $G$ is planar or outerplanar. In our proofs, we utilize special decompositions of graphs and the connection between acyclic vertex coloring and locally irregular total coloring.

We show that the scaling limit of the random walk loop soup on suitable planar graphs is the Brownian loop soup, under a topology on multisets of unrooted, unparameterized, and macroscopic loops. The result holds assuming only convergence of simple random walk to Brownian motion, a Russo-Seymour-Welsh type crossing estimate, and the bounded density of the graphs. The proof relies on Wilson's algorithm and Schramm's finiteness theorem. Precisely, we approximate the random walk loop soup by the set of loops erased in a greedy variant of Wilson's algorithm, thereby establishing convergence. The resulting limit is identified using the result of Lawler and Ferreras arXiv:math/0409291.

We consider complex-valued functions on the complex plane and the task of computing their zeros from samples taken along a finite grid. We introduce PhaseJumps, an algorithm based on comparing changes in the complex phase and local oscillations among grid neighboring points. The algorithm is applicable to possibly non-analytic input functions, and also computes the direction of phase winding around zeros. PhaseJumps provides a first effective means to compute the zeros of the short-time Fourier transform of an analog signal with respect to a general analyzing window, and makes certain recent signal processing insights more widely applicable, overcoming previous constraints to analytic transformations. We study the performance of (a variant of) PhaseJumps under a stochastic input model motivated by signal processing applications and show that the input instances that may cause the algorithm to fail are fragile, in the sense that they are regularized by additive noise (smoothed analysis). Precisely, given samples of a function on a grid with spacing $δ$, we show that our algorithm computes zeros with accuracy $\sqrtδ$ in the Wasserstein metric with failure probability $O\big(\log^2(\tfrac{1}δ) δ\big)$, while numerical experiments suggests even better performance.

This paper develops a quantized Q-learning algorithm for the optimal control of controlled diffusion processes on $\mathbb{R}^d$ under both discounted and ergodic (average) cost criteria. We first establish near-optimality of finite-state MDP approximations to discrete-time discretizations of the diffusion, then introduce a quantized Q-learning scheme and prove its almost-sure convergence to near-optimal policies for the finite MDP. These policies, when interpolated to continuous time, are shown to be near-optimal for the original diffusion model under discounted costs and -- via a vanishing-discount argument -- also under ergodic costs for sufficiently small discount factors. The analysis applies under mild conditions (Lipschitz dynamics, non-degeneracy, bounded continuous costs, and Lyapunov stability for ergodic case) without requiring prior knowledge of the system dynamics or restrictions on control policies (beyond admissibility). Our results complement recent work on continuous-time reinforcement learning for diffusions by providing explicit near-optimality rates and extending rigorous guarantees both for discounted cost and ergodic cost criteria for diffusions with unbounded state space.

The Mayer-Vietoris theorem is known for its wide applications, especially in determining homology. In fact, this theorem provides us with a long exact sequence, where the underlying homology groups fit in. However, this theorem does not provide an explicit way to compute homology. In this paper we prove an ``effective" version of the Mayer-Vietoris theorem using discrete Morse theory. Suppose, we have a Mayer-Vietoris type setup, i.e., let $X$ be a simplicial complex and $A$ and $B$ be two subcomplexes of $X$, such that $A \cup B=X$. Moreover, let $\mathcal{W}_A$, $\mathcal{W}_{B}$ and $\mathcal{W}_{A \cap B}$ be gradient vector fields on $A$, $B$ and $A \cap B$ respectively (which need not be ``coherent", i.e., they do not need to coincide on their intersection). Then, the main theorem of our paper provides an explicit way to compute the homology groups of $X$, using the combinatorial information regarding the trajectories of the aforementioned gradient vector fields, we do not even need to know the individual homology groups $H_{*}(A)$, $H_{*}(B)$ and $H_{*}(A \cap B)$. In principle, the homology of $X$ can always be computed explicitly using our theorem irrespective of the choice of the gradient vector fields. Further, if we choose the subcomplexes $A$ and $B$ wisely so that each of $A$, $B$ and $A \cap B$ admits an efficient gradient vector field, then the computation of the homology groups is considerably reduced.

This paper proposes a unified decision making and local trajectory planning framework based on Time-Varying Artificial Potential Fields (TVAPFs). The TVAPF explicitly models the predicted motion via bounded uncertainty of dynamic obstacles over the planning horizon, using information from perception and V2X sources when available. TVAPFs are embedded into a finite horizon optimal control problem that jointly selects the driving maneuver and computes a feasible, collision free trajectory. The effectiveness and real-time suitability of the approach are demonstrated through a simulation test in a multi-actor scenario with real road topology, highlighting the advantages of the unified TVAPF-based formulation.

We establish a variant of the log-Sobolev and transport-information inequalities for mixture distributions. If a probability measure $π$ can be decomposed into components that individually satisfy such inequalities, then any measure $μ$ close to $π$ in relative Fisher information is close in relative entropy or transport distance to a reweighted version of $π$ with the same mixture components but possibly different weights. This provides a user-friendly interpretation of Fisher information bounds for non-log-concave measures and explains phenomena observed in the analysis of Langevin Monte Carlo for multimodal distributions.

We continue the program initiated in \cite{SVGS}. In this paper, we focus on the infinite $d-$regular tree, and prove the monotonicity of a weighted Dirichlet energy, a Weiss-type monotonicity formula, and a generalization of the Almgren monotonicity formula of \cite{SVGS} for $p\ge 1$. We also compute examples in the infinite $2-$ and $3-$regular trees.

We develop two penalty based difference of convex (DC) algorithms for solving chance constrained programs. First, leveraging a rank-based DC decomposition of the chance constraint, we propose a proximal penalty based DC algorithm in the primal space that does not require a feasible initialization. Second, to improve numerical stability in the general nonlinear settings, we derive an equivalent lifted formulation with complementary constraints and show that, after minimizing primal variables, the penalized lifted problem admits a tractable DC structure in the dual space over a simple polyhedron. We then develop a penalty based DC algorithm in the lifted space with a finite termination guarantee. We establish exact penalty and stationarity guarantees under mild constraint qualifications and identify the relationship of the local minimizers between the two formulations. Numerical experiments demonstrate the efficiency and effectiveness of our proposed methods compared with state-of-the-art benchmarks.

We consider vacuum anisotropic spatially homogeneous models in certain modified gravity theories (such as Hořava-Lifshitz, $λ$-$R$ or $f(R)$ gravity), which are expected to describe generic spacelike singularities for these theories. These models perturb the well-known Bianchi models in general relativity (GR) by a parameter $v\in (0,1)$ with GR recovered at $v=1/2$. We prove an analogue of the well-known Ringström attractor theorem in GR to the supercritical theories: for any $v\in (1/2,1)$, all solutions of Bianchi type $\mathrm{IX}$ converge to an analogue of the Mixmaster attractor, consisting of Bianchi type I solutions (Kasner states) and heteroclinic chains of Bianchi type II solutions. In contrast to GR, there are no solutions that converge to a different set other than the Mixmaster (such as the locally rotationally symmetric solutions in GR).

Let $\mathcal{C}$ be a proper minor-closed class of graphs. Given the minors excluded in $\mathcal{C}$, we determine the maximum $q$-centered chromatic number and the maximum $q$th weak coloring number of graphs in $\mathcal{C}$ within an $\mathcal{O}(q)$-factor. Moreover, when $\mathcal{C}$ excludes a planar graph, we determine it within a constant factor. Our results imply that the $q$-centered chromatic number of $K_t$-minor-free graphs is in $\mathcal{O}(q^{t-1})$, improving on the previously known $\mathcal{O}(q^{h(t)})$ bound with a large and non-explicit function $h$. We include similar bounds for another family of parameters, the fractional treedepth fragility rates. All our bounds are proved via the same general framework.

In this paper we obtain explicit linear upper bounds for the virtually cyclic dimension of normally poly-surface and normally poly-free groups. Our approach is based on a structural study of the balanced property (Lück's Condition~C), which provides structural control over commensurators of virtually cyclic subgroups. We prove general stability results showing that the balanced property is preserved under suitable short exact sequences, direct limits, and acylindrical graph of groups decompositions. As applications, we establish that normally poly-hyperbolic groups, normally poly-free groups, and normally poly-surface groups are balanced. These classes include, in particular, pure braid groups of surfaces with non-empty boundary, Artin groups of FC-type, right-angled Artin groups, and fundamental groups of mapping tori of surface homeomorphisms.

We consider a quantum gas consisting of $N$ hard spheres with radius $\frak{a} > 0$, obeying bosonic statistics and moving in the box $Λ= [0;L]^3$ with periodic boundary conditions. We are interested in the ground state energy per unit volume in the thermodynamic limit, with $N, L \to \infty$ at fixed density $ρ= N / L^3$. We derive an upper bound for the ground state energy density, matching the famous Lee-Huang-Yang formula, up to lower order terms, in the dilute limit $ρ\frak{a}^3 \ll 1$.

The Cauchy problem for the two-dimensional incompressible Euler equation is globally well-posed for smooth initial data. In this paper, we show that for a dense $G_δ$ set of initial data, the solutions lose regularity in infinite time, thereby confirming a long-standing conjecture of Yudovich in the smooth setting.

What is a diffusion model actually doing when it turns noise into a photograph? We show that the deterministic DDIM reverse chain operates as a Partitioned Iterated Function System (PIFS) and that this framework serves as a unified design language for denoising diffusion model schedules, architectures, and training objectives. From the PIFS structure we derive three computable geometric quantities: a per-step contraction threshold $L^*_t$, a diagonal expansion function $f_t(λ)$ and a global expansion threshold $λ^{**}$. These quantities require no model evaluation and fully characterize the denoising dynamics. They structurally explain the two-regime behavior of diffusion models: global context assembly at high noise via diffuse cross-patch attention and fine-detail synthesis at low noise via patch-by-patch suppression release in strict variance order. Self-attention emerges as the natural primitive for PIFS contraction. The Kaplan-Yorke dimension of the PIFS attractor is determined analytically through a discrete Moran equation on the Lyapunov spectrum. Through the study of the fractal geometry of the PIFS, we derive three optimal design criteria and show that four prominent empirical design choices (the cosine schedule offset, resolution-dependent logSNR shift, Min-SNR loss weighting, and Align Your Steps sampling) each arise as approximate solutions to our explicit geometric optimization problems tuning theory into practice.

We investigate the algebraic structure of the two-time physics introduced some time ago by I. Bars and his co-authors, clarifying its relations with quadratic and cubic Jordan algebras, as well as with reduced Freudenthal triple systems (FTS) based on them. In particular, the `extended' phase space introduced by Bars can be endowed with the structure of a reduced FTS constructed over a semi-simple cubic Jordan algebra (named Lorentzian spin factor), characterized by a primitive, invariant symmetric tensor of rank $4$. The $Sp(2,\mathbb{R})$-gauge fixing procedure typical of two-time physics yields algebraic-differential constraints on the quartic polynomial associated to such a tensor, implying that only two (isomorphic) nilpotent orbits of the non-transitive action of the automorphism group of the Lorentzian spin factor are spanned by the conjugated variables which coordinatize the `extended' phase space. We illustrate our results in relativistic, manifestly Lorentz-covariant physical systems, as well as in non-relativistic systems (such as the non-relativistic massive particle, the hydrogen atom, and the Carroll particle with non-vanishing energy).

The material conditional has long been charged with paradox. Defined truth-functionally, it renders true any conditional whose antecedent is false or consequent true -- hence, seemingly absurd statements such as `If unicorns exist, then $2+2=4$'. This has been taken as proof that the connective cannot capture the meaning of ordinary if-then sentences, which appear to imply a causal or evidential link. I argue, by contrast, that the paradoxes arise from a confusion of what it expresses caused by cognitive biases. The material conditional properly belongs to the class of indicative, not subjunctive, conditionals -- those that register patterns of co-variation rather than counterfactual dependence. When understood as a formal device marking entailment under a background theory, it faithfully represents a mode of reasoning essential to science itself: correlation without causation. The faults ascribed to it, therefore, are not flaws in meaning or standard use, but as misapplication and misreading.

We provide an adelic relative trace formula proof to the Petersson/Bruggeman-Kuznetsov (PBK) formulas, specifically in the holomorphic case for $κ=2$ and the non-holomorphic case for $m_1m_2<0$. Given two sets of hypothesis on the non archimedean test function $f$, called the geometric and spectral assumptions, this approach allows us to obtain refined PBK formulas.

This survey reviews a collection of parallel phenomena between free boundary submanifolds in the Euclidean unit ball and closed submanifolds in the sphere, with particular emphasis on rigidity mechanisms, pinching thresholds, and canonical models. We do not regard the two theories as a unified system in one-to-one correspondence. Rather, we emphasize that in several typical settings -- including low topology, strong pinching, spectral extremality, and symmetry reduction -- the free boundary condition often forces stronger rigidity in the unit ball than in the closed setting. The exposition is organized around six interconnected themes. We first contrast the failure of the spherical Bernstein problem in high dimensions with the dimension-independent rigidity of free boundary minimal disks in the unit ball. We then discuss the parallel roles played by the Clifford torus and the critical catenoid in uniqueness, Morse index, and eigenvalue characterizations. Next, we review the transition from the Lawson--Simons stable currents method to the Bochner--Hardy techniques developed for free boundary problems, summarize pinching and gap theorems driven by the second fundamental form and its traceless part, and outline the linear comparison framework between Morse index and topology in the minimal, constant mean curvature, and weighted settings. Finally, we survey existence results obtained from group actions, isoparametric foliations, and recent equivariant eigenvalue optimization, thereby illustrating both the striking analogies and the essential boundary-driven differences between the closed spherical theory and the free boundary theory in the ball.

We consider a stochastic optimization problem involving two random variables: a context variable $X$ and a dependent variable $Y$. The objective is to minimize the expected value of a nonlinear loss functional applied to the conditional expectation $\mathbb{E}[f(X, Y,β) \mid X]$, where $f$ is a nonlinear function and $β$ represents the decision variables. We focus on the practically important setting in which direct sampling from the conditional distribution of $Y \mid X$ is infeasible, and only a stream of i.i.d.\ observation pairs $\{(X^k, Y^k)\}_{k=0,1,2,\ldots}$ is available. In our approach, the conditional expectation is approximated within a prespecified parametric function class. We analyze a simultaneous learning-and-optimization algorithm that jointly estimates the conditional expectation and optimizes the outer objective, and establish that the method achieves a convergence rate of order $\mathcal{O}\big(1/\sqrt{N}\big)$, where $N$ denotes the number of observed pairs.

We study an embedded Trefftz discontinuous Galerkin method for the Helmholtz equation. The method starts from a polynomial DG space and enforces the Trefftz property through local constraints, avoiding an explicit construction of Trefftz basis functions. For the global coupling we use a simple symmetric interior penalty DG bilinear form. Since the resulting formulation is not coercive, stability is proved by a $T$-coercivity argument combined with a Schatz-type duality technique. This yields wavenumber-explicit stability, quasi-optimality, and convergence estimates in standard DG norms under an explicit mesh resolution condition.