Papers

We consider beta-Dyson Brownian motion, with $\beta >= 1$, started from a deterministic configuration with uniformly bounded support. Let $\mu_t$ be the semicircular free-convolution flow issued from the initial empirical measure, and set $S_t = supp(\mu_t)$. For every fixed $T, \epsilon > 0$, with probability at least $1 - C \exp(-(log n)^2)$, every particle remains within an epsilon-neighborhood of $S_t$ for all $0 <= t <= T$. The result holds from time zero, requires no regularity assumption at the initial spectral edges, and applies to multi-cut supports with macroscopic interior gaps. A key ingredient is a deterministic local resolvent exclusion principle: an $o((n \eta)^(-1))$ comparison of Stieltjes transforms on a complex disc above a real point separated from the reference support excludes eigenvalues from the corresponding real interval. This gives a model-independent mechanism for converting local resolvent estimates into spectral confinement.

We construct an infinite walk in $\mathbb{Z}^3$ whose steps come from a fixed set of sixteen vectors and no three of whose vertices are collinear, answering a problem of Gerver and Ramsey popularized as Erdős Problem 193.

We survey biLipschitz (BL) and bounded displacement (BD) equivalence of Delone sets, with an emphasis on examples arising in aperiodic order. After recalling the classical results of Burago-Kleiner, McMullen, and Laczkovich, we discuss criteria for rectifiability, uniform spreadness, and BD equivalence, and how these relate to discrepancy and point-counting estimates. We then focus on Delone sets associated with substitution tilings, where these questions can often be studied through the combinatorial and spectral properties of the underlying substitution rules. We also consider selected classes of cut-and-project sets and briefly discuss extensions beyond BL and BD equivalence.

In 1966, McKean asked whether the entropy production of the Boltzmann equation must be monotone decreasing in time. We show that this is not the case even in the space-homogeneous setting for the Boltzmann collision operator with a constant angular cross section and the kinetic parameter $\gamma \in [0,1]$. This recovers the classical case of hard spheres and the simplest case of Maxwell molecules. Our examples are radially symmetric mixtures of Maxwellians.

We study orthogonal polynomials of $d = d_1+d_2$ variables with respect to a wrapped product weight function ${\bm W}({\bm x},{\bm y}) = W_1({\bm x}/\rho({\bm y})) W_2(\bm y)$ for $(\bm x, \bm y) \in \mathbb{R}^{d_1} \times \mathbb{R}^{d_2}$, where $\rho$ is either linear or the square root of a nonnegative quadratic polynomial, and identify all such polynomials that are eigenfunctions of a second-order linear differential operator. For $d =2$, it is known that there are primarily, up to affine transformations, five families of such polynomials, which are products or wrapped products of classical orthogonal polynomials of one variable; all five families have their counterparts in higher dimensions, but no characterization is known in dimension three or higher. Our study explores viable wrapped product families, finds explicit second-order differential operators for two new families of orthogonal polynomials in $d= d_1+d_2$ variables that have not been studied before if either $d_1>1$ or $d_2 > 1$, and provides, in particular, a complete list of such operators among all wrapped product orthogonal polynomials when $d = 3$. The list also includes four families that are eigenfunctions of a fourth-order differential operator, whereas no second-order operator is available. Moreover, orthogonal polynomials on the wrapped quadratic surfaces that are eigenvalues of a second-order differential operator on the surface are also studied.

We study minimal residual space-time finite element discretizations of linear parabolic initial value problems in canonical space-time variational form. To deal with the arising dual norm, we introduce the Riesz lift of the residual as an additional variable. Quasi-optimality of the primal variable of the mixed system follows from a uniform inf-sup condition. This condition is known to be satisfied for finite element spaces w.r.t. prismatic partitions of the space-time cylinder that allow for a decomposition into time-slabs. We prove that this condition cannot be expected to hold otherwise. To recover stability for general partitions and the data at hand, we derive an a posteriori condition on the error between the exact Riesz lift of the residual and its Galerkin approximation – being the secondary variable of our system – under which the primal variable is quasi-optimal. We derive a posteriori error estimators for both variables, and use them in a double-adaptive loop that alternates test-space with trial-space enrichment. We illustrate our findings with numerical experiments in $1+1$ and $2+1$ dimensions.

For quadratic postcritically finite polynomials defined over a number field $K$, we establish lower bounds for the Call-Silverman height of wandering algebraic integers lying in cyclotomic extensions of $K$ and provide partial results for the nonintegral case. Our proof combines potential theory and algebraic number theory with equidistribution techniques.

In the localization game on a graph $G$, a team of cops searches for an invisible, mobile robber on $G$ by "probing" vertices; each probe tells the cops the distance from the probed vertex to the robber. The cops win if they can uniquely determine the robber's location. In this paper, we introduce a related game: the directional localization game. In this game, instead of probes returning distances, they return directions: when the cops probe a vertex $v$, the robber must respond with one or more neighbors of $v$ that lie on a shortest path from $v$ to the robber's location. The minimum number of cops needed to win this game on $G$ is the directional localization number of $G$. We study the directional localization game on several classes of graphs, including chordal graphs, Cartesian products, and incidence graphs of projective planes. We also bound the directional localization number of a graph $G$ in terms of the degeneracy and the treewidth of $G$.

Many quantum algorithms for classically difficult optimization tasks must return high-quality bitstrings from finitely many circuit executions, whereas most quantum error-mitigation methods target expectation values. We study sample-level recovery when measured probability mass is distributed around multiple latent bitstrings, called centers. Each component of the measured probability mass is called a source and we assume that each center is associated with one source. We identify dominance-at every coordinate, more than half of a retained region's probability mass comes from one source and agrees with its center-as a sufficient condition under which majority voting recovers that center with exponentially decreasing error probability. We show that nearest-center assignment, as used in clustering algorithms such as the $k$-modes algorithm, can fail to produce dominated regions even when the true centers are known. This failure motivates responsibility thresholding and a local dominance screen, whose combination we call dominance-aware (DA) refinement. Synthetic and simulated MaxCut-QAOA experiments show that DA refinement favors precision, while $k$-modes with DA refinement improves overall center recovery. All procedures are classical post-processing and require no additional quantum-circuit executions.

The rapid growth of AI computing has driven increasing demands for flexible and high-capacity data-center interconnections. Owing to its ultra-wide bandwidth and high spatial reuse capability, terahertz (THz) communication has emerged as a promising solution for future wireless data centers, while digital twins (DTs) enable efficient wireless planning and real-time optimization. In this work, a measurement-driven multi-layer DT framework is proposed for THz wireless data centers, where the physical, channel, evaluation, and manipulation layers are progressively constructed from bottom to top. First, extensive channel measurements are conducted at 140, 220, and 300 GHz to characterize frequency-dependent propagation behaviors. Based on the tri-band measurements, a measurement-calibrated physical twin is established by jointly optimizing the geometry, material, antenna, and hybrid propagation models. On top of the physical twin, a line-of-sight (LoS)-aware implicit neural field is developed to construct an AI channel twin for efficient channel reconstruction. The proposed AI twin learns location-dependent channel statistics from the calibrated twin, enabling real-time prediction of received power and LoS probability. Building upon the reconstructed channel field, a system-level evaluation layer is derived to analyze coverage and interference for both AP-to-rack and rack-to-rack communications. Experimental results show that the proposed AI twin achieves lower power reconstruction error than existing neural-field baselines while maintaining real-time inference capability. Moreover, the ceiling-mounted AP deployment achieves over 90% coverage under a 10 dB signal-to-interference-plus-noise ratio (SINR) threshold, demonstrating the effectiveness of the proposed DT framework for THz wireless data-center planning and optimization.

We consider the linear cocycles generated by the linearized vorticity equation and by passive scalar advection diffusion, both driven by the two dimensional stochastic Navier-Stokes flow on the torus with non degenerate additive forcing, and we prove uniqueness of the stationary measures for the projective process associated to such linear dynamics. The proof relies on a localized asymptotic strong coupling construction, and on the non-degeneracy of the Malliavin matrix of the projective process. Establishing this non-degeneracy for passive scalar advection poses additional challenges, and requires a proof (based on Cameron-Martin analyticity arguments) that generically the passive scalar is nowhere one-dimensional.

For integers $h,k\geq 1$, let $hA$ be the $h$-fold sumset of $A$ and put $\mathcal{R}(h,k)=\{|hA|:A\subset\mathbb{Z}, |A|=k\}$. Previously, the fixed-cardinality exponent law was known only for $k\leq 3$; every fixed $k\geq 4$ remained open. We settle the problem in full by determining the complete fixed-cardinality exponent law: $|\mathcal{R}(h,k)|=\begin{cases}1,&k\leq 2,\\ h,&k=3,\\ h^{k-1+o_k(1)},&k\geq 4\end{cases}$. Here $o_k(1)\to 0$ as $h\to\infty$ with $k$ fixed. More sharply, for fixed $k\geq 4$, an interval of length $\Theta_k(h^{k-1})$ contains at least $h^{k-1-o_k(1)}$ attainable values. At $k=4$ we prove $|\mathcal{R}(h,4)|=\Theta(h^3)$ with positive lower density in its ambient interval, disproving Nathanson's proposed $o(h^3)$ and $O(h^2)$ bounds. One bounded addition-table geometry drives these results, coupling Hilbert-energy amplification to optimal finite-observation compression. Every ordered real $k$-set ($k\geq 2$) has an integer model in $[0,O_k(h^{k-2})]$ preserving every sum equality and strict comparison through degree $h$; the exponent $k-2$ is sharp. The universal label-realization length is therefore $\Theta_k(h^{k-2})$, one power sharper than Nathanson's $O_k(h^{k-1})$ bound. For $h\geq 2$ and $k\geq 3$, minimum active rank equals realization-frequency codimension, exponent-shape codimension, and sampling-rarity exponent; a full-exponent family has maximal-rank witnesses with Cohen-Macaulay toric coordinate rings. At rank zero, for $h\geq 2$, it proves the conjectural OEIS A227589 formula $\binom{h+2}{2}+\mathbf{1}_{\{2\nmid h\}}$ for the least normalized diameter of a four-point $B_h$-set. It also gives exact fixed-$(h,k)$ popularity laws for $k$-subsets of $\{1,\ldots,q\}$ as $q\to\infty$, resolving Nathanson's Problems 9 and 10.

In many inverse problems, the aim is to recover a probability distribution on a latent (or object state) space from indirect, noisy observations. When the observations can be modelled as noisy samples from the mixing latent distribution, the recovery problem is a deconvolution problem on the space of probability measures. A common strategy is to fix a finite set of candidate points and estimate a weight for each, turning the problem into a finite-dimensional concave maximum likelihood problem on the simplex. We study the combined effect of this discretization and of the noise in the context of cryo-electron microscopy (cryo-EM), where the candidate points are biomolecular conformations and the weights describe the relative frequency of each conformation. Our results pertain to both statistical and algorithmic aspects of the estimator. We analyze the weight-recovery problem in which the candidate states and their likelihoods are known. A nearby pair of candidates forces a near-null direction. More generally, the grid and the noise level impose a uniform lower bound on the achievable Kullback–Leibler divergence between observation densities, even with infinite data. The finite-grid estimator is asymptotically normal when its population target is in the interior of the simplex; at a boundary target, its limit is a cone-projected Gaussian. Finally, the exact proximal form of Expectation–Maximization leads to a global high-noise comparison between an early iterate and a KL-penalized likelihood, without a basin assumption or a linearization of the recursion. Tests on synthetic images of the Hsp90 molecule illustrate the theoretical findings and translate them into practical guidelines for interpreting reweighted ensembles.

The hypothalamic pituitary adrenal (HPA) axis is a key regulatory system coordinating endocrine responses to physiological and psychological stress. While the ACTH stimulation test remains a cornerstone of adrenal function assessment, its interpretation is complicated by the dynamic and adaptive nature of the HPA axis under chronic stress exposure. In particular, prolonged stress induces glandular remodeling, glucocorticoid receptor (GR) resistance and delayed feedback recovery, all of which may alter test outcomes without indicating primary adrenal failure. In this study, we present a mechanistic modeling framework that integrates hormonal kinetics, feedback inhibition and functional mass adaptation of the corticotroph and adrenal compartments. We simulate the HPA axis over $180$ days encompassing three phases - baseline, chronic stress and recovery, while introducing a time varying GR resistance function to mimic feedback desensitization and its resolution. Using this framework, we evaluate both low dose ($1 \mu g)$ and high dose ($250 \mu g)$ ACTH stimulation tests across physiological phases. Our simulations show that cortisol responses are highly sensitive to both the magnitude and timing of stress exposure and that ACTH responsiveness is phase dependent and often blunted during recovery due to persistent feedback resistance. Low dose ACTH testing more reliably reflects partial adrenal adaptation, while high dose tests risks masking dysfunction due to supraphysiological drive. These results highlight the limitations of static testing paradigms and suggest that accounting for glandular plasticity and GR feedback dynamics is essential for effective endocrine diagnosis particularly in stress related or treatment induced adrenal disorders.

Albertson's conjecture states that every graph of chromatic number r has crossing number at least that of the complete graph K_r. We prove the conjecture for all r. The proof combines new clique-sensitive edge bounds for critical graphs, maximum-clique sampling, and a reserved immersion construction. For graphs of order close to r, we construct edge-disjoint routes while preserving a complete bipartite subgraph whose forced crossings compensate for those introduced when the immersion is smoothed into a drawing. For larger orders, we derive explicit edge bounds using independent-triple deletion and structural properties of critical graphs, then combine them with sampling inequalities that account for crossings within a maximum clique. The remaining numerical inequalities are certified using exact rational arithmetic, with a reproducible certificate and standalone Python verifier.

We prove that, for every integer $k\ge 2$, there exists a constant $c_k>0$ such that every graph on $n\ge R(k,k)$ vertices with maximum degree $\Delta$ contains an induced subgraph on at least $n-c_k\sqrt{\Delta}$ vertices whose maximum degree is attained by at least $k$ vertices. This confirms a conjecture of Caro and Yuster in strong form.

Motivated by the recent study of strongly $\Delta$-clean rings, we introduce and study strongly $m$-$\Delta$-clean rings. We establish their fundamental properties and characterize strongly $m$-$\Delta$-clean rings via lifting $m$-potent elements modulo $\Delta(R)$. We prove that every $m$-$\Delta$-clean ring is clean and that the factor ring of a strongly $m$-$\Delta$-clean ring modulo its Jacobson radical is reduced. Finally, we investigate corner rings, Morita context rings, and the relationships of these rings with $\Delta$-clean, local, semipotent, and strongly $m$-nil clean rings.

An electrical network can be summarized at its boundary by a response matrix: prescribed boundary voltages determine boundary currents. A superport network adds a constraint by grouping boundary terminals into superports, requiring the total current in each group to be zero and making voltage differences inside the groups the natural coordinates. Earlier work determined forest formulas for a single response entry and for the determinant of the whole response matrix. The missing case was an arbitrary subdeterminant, or minor: one needs to know not only which spanning forests contribute, but also the sign carried by each forest. This paper supplies that sign rule. After choosing one reference vertex in each superport, the response is \[ L=\left(D^{T}K^{-1}D\right)^{-1}, \] with \(K\) the grounded weighted Laplacian and \(D\) recording the selected voltage differences. Contracting the components of a physical spanning forest \(F\) produces a much smaller quotient port graph \(H_F\). Its reduced incidence matrix \[ B_F=Q_FD \] has columns only of the forms \[ 0,\qquad \pm e_a,\qquad e_a-e_b. \] Hence every square incidence minor is exactly \(0\) or \(\pm1\). For a \(k\)-set of response coordinates \(I\), append to \(B_F\) the selector rows \(E_I^{T}\) and define the completed quotient-incidence determinant \[ \widehat{\chi}_F(I) = \det \begin{pmatrix} B_F\\ E_I^{T} \end{pmatrix}. \] For coordinate sets \(I,J\) of the same size, the arbitrary response minor is a weighted spanning-forest sum whose coefficient is simply \[ \widehat{\chi}_F(I)\widehat{\chi}_F(J). \] Thus the Jacobi complementary-minor factors used in the derivation disappear from the final theorem. Direct block-triangular reduction gives \[ \widehat{\chi}_F(I)\in\{0,\pm1\}, \] with nonvanishing occurring exactly when the complementary quotient edges \(N\setminus I\) form a spanning tree of \(H_F\).

Interval translation maps are piecewise translations for which the images of distinct continuity intervals may overlap. We study when their measured dynamics can be represented by finite or countable interval exchange transformations. First, we give a direct entropy-based proof of the known fact that an interval translation map is invertible almost everywhere with respect to every nonatomic invariant probability measure. The proof uses a polynomial upper bound for the complexity of the natural branch coding. We then use a consequence of a theorem of Arnoux, Ornstein, and Weiss: every nonatomic measure-preserving automorphism of a standard probability space admits a countable interval exchange transformation model. Applied to the almost-everywhere invertible core, this gives an abstract countable interval exchange transformation model for every measured interval translation map. We next study the canonical order-preserving coordinate defined by the distribution function of the invariant measure. For each branch, we introduce a positive defect measure recording the excess measure carried by its direct image. Its push-forward to the distribution coordinate gives a canonical cut set such that, away from this set, the induced map is locally a translation. Finite defect support yields a finite interval exchange transformation model, while a Lebesgue-null defect cut set yields a countable model. Under branchwise nonsingularity, the defect support is the closure of the cuts arising from active gaps in the support, giving an equivalent geometric characterization of the finite case. Finally, for a self-similar infinite-type Bruin-Troubetzkoy map, we compute the canonical defect cut set explicitly. It is countably infinite and Lebesgue-null, yielding a genuinely countable, non-finite interval exchange transformation model.

The Solomonoff distribution $M$ assigns an a priori probability to a finite binary string $z$ by summing over all programs whose output begins with $z$, weighting a program of length $\ell$ by $2^{-\ell}$. Thus, likely strings are those with many short explanations. This appears to be a purely discrete notion of complexity. However, Riemann sums are also countable. We show that the sum defining $M$, suitably reorganised, contains Riemann sums approximating the Bayesian evidence of any computable statistical model. These evidence integrals are singular integrals whose asymptotics are governed by Singular Learning Theory and, through it, by invariants of algebraic geometry. Concretely, for every computable Bayesian model, we construct a single monotone Turing machine whose induced semimeasure agrees with the evidence $Z_n$ up to a uniform multiplicative constant. If the model also satisfies the hypotheses of Watanabe's free-energy asymptotics, then a sample $X^n=X_1\cdots X_n$ drawn i.i.d. from the true distribution satisfies $-\log M(X^n) \le nL_n(w_0)+\lambda\log n-(m-1)\log\log n+O_{\mathbb{P}}(1)$, where $L_n$ is the empirical loss, $w_0$ is an optimal parameter, $\lambda$ is the learning coefficient, and $m$ is its multiplicity. Thus the learning coefficient, a geometric measure of model simplicity, appears within the Solomonoff distribution as the coefficient of $\log n$ in an upper bound on code length.

We construct a constant-degree and constant-gap quantum expander on $n$ qubits where each unitary can be implemented by a depth-$1$ and 1D circuit of Pauli or CNOT gates. We provide two applications of this expander. First, we use it to construct a family of frustration-free 1D Hamiltonians whose ground states obey the entanglement-gap relation $S = \Theta(\Delta^{-1/2})$; this is believed to be optimal, but achieving it had been open. Second, we use it to provide a streaming protocol that tests for closeness to a class of 1D volume-law entangled states. Moreover, we extend our quantum expander to a constant-degree and constant-gap expander on the unitary group where each unitary is a single $T$ gate, a single $T^{\dagger}$ gate, or a depth-$1$ Clifford circuit. This implies that a random sequence of unitaries from the expander yields a gapped walk on a dense subgroup of the unitary group. This improves upon previous work by Bourgain and Gamburd which did not control the dependence of the gap on the dimension.

Hilbert's Tenth Problem over the integers was solved negatively by Y. Matiyasevich in 1970. In this paper, we prove that there is no algorithm to determine for any polynomial equation $P(z_1,\ldots,z_{10})=0$ (with integer coefficients and ten unknowns) whether it has integer solutions. This improves the previous 11 unknowns theorem.

We study the warm-ion Euler–Poisson system with Maxwell–Boltzmann electrons on the cylinder $\R\times\T$ in the quasineutral regime. Taking a smooth expanding planar simple wave of the effective Euler system as the reference state, we construct an even Debye expansion through arbitrary finite order $M$ with a residual of $O(\eps^{2M+2})$. We rigorously establish nonlinear stability on every fixed interval $[t_0,T]$ ($t_0\ge0$), achieving a lifespan and energy constants strictly independent of the Debye length $0<\eps\le\eps_0$. To overcome the singular scaling of the electric field, we develop a novel compensated energy topology that couples the warm-ion symmetrizer to the time-differentiated nonlinear Poisson constraint. By integrating the top-order electric work directly into the time derivative of a weighted energy functional, this mechanism controls the potential in $H^s$ and its gradient in $\eps H^s$, completely eliminating the $\eps^{-1}$ loss typically encountered in the momentum equation. This framework successfully governs distinct neutral end states, captures genuinely two-dimensional rotational perturbations, and yields an arbitrary-order quasineutral asymptotic expansion for prepared data, providing a critical analytical foundation for the geometric theory of multidimensional quasineutral rarefactions.

Relative periodic orbits (RPOs) are solutions of the three-body problem that are periodic in a uniformly rotating reference frame and, in general, quasi-periodic in inertial coordinates. We present a numerical procedure for computing and continuing one-parameter families of RPOs of the planar Newtonian three-body problem. The method exploits consecutive syzygies, understood here as configurations in which the three bodies are aligned and their velocities satisfy the corresponding symmetry conditions. Matching the positions and momenta at two consecutive alignments reduces the computation of RPOs to a low-dimensional nonlinear problem. Its solutions are then numerically continued, and linear stability is determined from the nontrivial eigenvalues of the rotated monodromy matrix after removing the neutral directions associated with conserved quantities and continuous symmetries. The procedure is applied to several mass distributions and initial configurations, producing families of Poincaré, Hill, and binary-type solutions. These families exhibit transitions from nearly circular to highly eccentric motion, changes of stability near resonances and turning points, and absolute periodic solutions when the rotation angle is a rational multiple of 2{\pi}. In the Hill families, the continuation connects satellite configurations with circumstellar motion as the smallest body loses its gravitational binding to the intermediate body. Circumbinary and circumstellar configurations are also obtained in the binary regime. The results illustrate the dynamical diversity of RPOs and provide coherent three-body motions that can be used as prescribed trajectories in restricted four-body models.

Additive functionals $\overline{J}_t=\frac{1}{t}\int_0^tU(X_s)\circ dX_s$ of Stratonovich-type recently attracted much attention in the context of inference of thermodynamic properties of complex systems from observations $U$ of individual fluctuating paths $(X_s)_{0\le s\le t}$, whereby $X_0$ is initiated from some general measure. Concentration results on $\overline{J}_t$, albeit desirable, are virtually nonexistent. They turn out to be significantly more challenging to prove than for classical Lebesgue-type functionals $\overline{\rho}_t=\frac{1}{t}\int_0^t V(X_s)ds$ because the tilt deforms the full second-order structure of the Feynman-Kac generator instead of contributing an additive potential. This renders the generator generally non-self-adjoint even under detailed balance. We overcome this by working with a symmetrized Dirichlet form with a new effective potential that now couples the observable to the non-equilibrium character of the dynamics. We prove concentration inequalities for $\overline{J}_t$ for any bounded, sufficiently smooth vector-valued function $U$ of a general geometrically ergodic diffusion process $X_s$, including explicit sub-gamma and Bernstein-type inequalities, and we obtain explicit upper bounds on ${\rm Var}(\overline{J}_t)$. Strikingly, under detailed balance the concentration of $\overline{J}_t$ is distinctively sub-Gaussian at all times and all deviations, with a variance proxy fixed by the noise alone and independent of the spectral gap, which has no analog for $\overline{\rho}_t$.

We study optimal sampling recovery in reproducing kernel Hilbert spaces (RKHS) in the uniform norm. For every RKHS with bounded kernel, we establish new comparisons between linear sampling widths and Gelfand widths that overcome the known square-root gap, without requiring a measure or a Christoffel-type condition. Our bounds rely on nested sampling designs obtained by kernel interpolation at (weak) P-greedy points. Under additional (polynomial) decay assumptions the decay rate of the Gelfand widths directly transfers to the sampling widths. With either a logarithmic oversampling or passing to the square root of the Gelfand widths we obtain a direct comparison (requiring no decay assumption) between them. This is particularly effective for super-polynomial decay, such as in Paley-Wiener spaces. Our results follow from representations of both widths in terms of kernel translates and yield, in the opposite direction, a new existence result for a sharp reduced basis selection. Numerical experiments for Legendre, mixed-Sobolev, and Paley-Wiener kernels illustrate our findings.

Classical chaining controls an indexed stochastic process through a single worst-case bound and can therefore obscure substantial variation across the index set. We develop the first simultaneous pointwise majorization theory for Banach-valued processes with finite-metric mixed-tail increments. Suppose that an anchored process $(Z_t)_{t\in T}$ satisfies, for some integer $m\ge1$, pseudo-metrics $d_1,\ldots,d_m$, and orders $\alpha_1,\ldots,\alpha_m>0$, \begin{align*} \mathbb{P}\{\|Z_t-Z_s\|>\sum_{j=1}^m u^{1/\alpha_j}d_j(s,t)\}\le 2e^{-u},s,t\in T. \end{align*} For ambient priors $\mu_1,\ldots,\mu_m$, let $v_j(t):=d_j(t,t_0), \Phi_j(t):=\int_0^{4v_j(t)}(\log\frac{1}{\mu_j(B_{d_j}(t,r))})^{1/\alpha_j}dr$. We prove that, $\forall \delta\in(0,1)$, with probability at least $1-\delta$, simultaneously for all $t\in T$, \begin{align*} \|Z_t\|\le C_{m,\boldsymbol\alpha}\sum_{j=1}^m\{\Phi_j(t)+v_j(t)(\log(e/\delta))^{1/\alpha_j}\}. \end{align*} Here $\boldsymbol\alpha:=(\alpha_1,\ldots,\alpha_m)$ and $C_{m,\boldsymbol\alpha}$ depend only on $m$ and these tail orders. The result subsumes single-metric sub-Weibull processes of every positive order as the case $m=1$. In the Gaussian setting, it sharpens the pointwise upper bound of \citet{xu2026} by eliminating the logarithmic terms generated by dyadic peeling. The proof retains the index-wise costs of measure-generated admissible chains and synchronizes the regimes through a nested common refinement. Finally, we apply our theorems to stationary diffusion empirical processes and decoupled Gaussian chaos to obtain simultaneous pointwise envelope bounds, which can further be applied to other statistics problems.

Let $X$ be a smooth complex projective variety of dimension $n$, and let $L$ be an ample Cartier divisor. We prove that $K_X+mL$ is globally generated for every integer $m\geq\lceil C_0n\rceil$, where $C_0=1.77629\ldots$ is an explicit constant. In particular, $K_X+2nL$ is globally generated. We also prove that if $Z$ is a normal projective variety of dimension $n$ with $K_Z$ Cartier, nef and big, then $|mK_Z|$ defines a birational map for every integer $m\geq1+\lceil C_0n+3\sqrt{2\mathrm{e}n}\rceil$. Our main input is a new estimate for the multiplicity of a minimal log canonical center. If $(X,\Delta)$ is log canonical near a closed point $x$ but is not klt at $x$, and $W$ is the positive-dimensional minimal log canonical center through $x$, then $2\overline{e}_1(\mathfrak m_{W,x})\leq\bigl(\dim W-\operatorname{lct}_x((X,\Delta);\mathfrak m_x)\bigr)\operatorname{mult}_xW$, where $\overline{e}_1(\mathfrak m_{W,x})$ is the first normal Hilbert coefficient of the maximal ideal of $\mathcal O_{W,x}$. This implies $\operatorname{mult}_xW\leq \frac{(a+c)^{a+c}}{a^ac^c}$, where $a:=\frac{\dim W-\operatorname{lct}_x((X,\Delta);\mathfrak m_{x})}{2}$ and $c:=\operatorname{edim}\mathcal O_{W,x}-\dim W$.

Fraenkel's conjecture asserts that a partition of the integers into at least three Beatty sequences with distinct moduli has the binary densities $1,2,4,\ldots,2^{m-1}$, normalized by $2^m-1$. We prove the conjecture through a dimension-free intermediate statement: every such partition contains a component of density at least 1/3. After reducing the partition to primitive common-period data, Fourier cancellation produces a finite inverse-sine system. We prove that no such system can exist when every density is below 1/3. The proof combines a divisor-concentration identity with uniform analytic estimates and three exact finite verifications, all carried out with integer or rational arithmetic. The component supplied by the density bound has mean spacing at most three. Deleting it preserves balance, and every surviving periodic balanced set is again a rational Beatty set. Induction determines the surviving binary scales, while a two-sequence disjointness criterion forces the deleted density to be the next binary scale. This yields the asserted density pattern.An eventual Beatty representation of balanced binary indicators extends the classification to one-sided balanced sequences on at least three letters with distinct positive densities, proving the balanced-sequence conjecture of Altman, Gaujal, and Hordijk.

In this note, we prove the existence of smooth complete admissible hypersurfaces in hyperbolic space with constant higher order mean curvature and a prescribed asymptotic boundary at infinity. The result holds for a large part of indices in the subcritical range where the concavity inequality loses its effectiveness. We supply with new arguments to reduce the proof to a semi-convex situation in which case the concavity inequality can play a role.