Papers

This paper investigates conditional stability for inverse source problems for the heat equation with a known temporal factor and an unknown spatial component in a bounded $C^{1,1}$ domain. We focus on observations supported on sets of low Hausdorff dimension and establish conditional stability in this setting. For boundary observations on compact sets of positive $q$-dimensional Hausdorff content, we establish logarithmic stability from full-time boundary flux observations and double-logarithmic stability from delayed-time boundary flux observations. The admissible dimensional ranges are $q>d-2$ when the observation set is contained in a flat boundary patch and $q>d-1-c_{d+1}$ on a general $C^{1,1}$ boundary, where $c_{d+1}>0$ depends only on the dimension. A key ingredient in deriving these results is a new boundary spectral inequality for the Dirichlet Laplacian, which controls a finite Dirichlet spectral sum through observations of the normal derivative of its elliptic extension on such a boundary set. Our results also cover inverse heat source problems with interior observations on sets of positive $q$-dimensional Hausdorff content for some $q>d-1$, yielding logarithmic stability from full-time observations for general sources in $H_0^1(\Omega)$ and Hölder stability from terminal-time observations for sources in a suitable spectral Gevrey class.

This paper investigates the quantitative stability of Edelstein hidden-variable fractal interpolation functions (EHVFIFs) and the upper box-counting dimension of their graphs under perturbations of interpolation nodes, ordinate data, hidden-variable data, and vertical scaling functions.

In planar soft-stick percolation, each vertex of the square lattice independently opens one uniformly chosen outgoing arrow and, with probability $\epsilon$, the opposite arrow as well. Motivated by the question on its critical parameter raised by Bäumler et al., we prove that $\epsilon_{c}\le0.99<1$, establishing percolation below the two-arrow endpoint. More precisely, at $\epsilon=0.99$ the origin has an infinite forward cluster with probability greater than 11/20. The proof uses a Peierls argument in which boundary turns are encoded by a three-state transfer matrix to bound the contour sum.

Let $g$ be a nondecreasing left-continuous function and let $\mu_g$ be its Lebesgue–Stieltjes measure. We establish a Banach-valued fundamental theorem of calculus and an Aubin–Lions compactness theorem for evolution measured by $\mu_g$, allowing absolutely continuous, singular continuous and atomic components. If the range space has the Radon–Nikodým property, a curve is $g$-absolutely continuous if and only if it is an indefinite Bochner integral; its strong $g$-derivative is the Bochner density and the variation measure has density equal to its norm. A terminal atom may make the derivative invisible from the Bochner state class, so the natural evolution space is a graph of state–derivative pairs. For $B_0\Subset B\hookrightarrow B_1$, with $B_0$ and $B_1$ reflexive and $1<p_0,p_1<\infty$, the state projection is a compact linear operator into $L_g^{p_0}([a,b);B)$. The proof combines bounded evaluation at atoms, local $\mu_g$-averages at nonatomic points and Ehrling's inequality. The result recovers the classical theorem and yields compactness principles for bounded time scales, weighted sequences and mixed Stieltjes measures.

We study an irredundant covering of an $m \times m$ square grid by $2 \times 2$ cards, where cards may overlap and every card must lie entirely inside the grid. Every square of the grid must be covered, and every card must contain at least one square that is covered only by itself. Let \(F(m)\) denote the maximum number of cards in such an irredundant cover. We develop a general framework for bounding the maximum possible size of such an irredundant cover based on square multiplicities, double counting, and local geometric restrictions arising around highly covered squares. These arguments yield a general upper bound whose leading term is $\frac{1}{2}m^2$, accompanied by a negative linear term. In the opposite direction, we construct a family of irredundant covers based on a density-$\frac{1}{2}$ covering pattern, referred to as the period-four staircase pattern, together with a boundary-repair construction. Thus, we obtain a lower bound with the same leading term and a linear-order error, proving that for every \(m\ge6\), \[ \frac{2m}{9} \le \frac12m^2-F(m) \le 2m-3, \] and consequently that the maximum density of an irredundant cover tends to $\frac{1}{2}$ as $m\to\infty$. The known $10\times 10$ case is used as motivation and as a finite benchmark for the general theory.

The first aim of this article is to extend the notion of the Gromov product to the class of partial metric spaces. A relation between $\delta$-hyperbolicity of a partial metric space and $\delta$-hyperbolicity of metric spaces endowed with the metrics defined via that partial metric is also established. The second one is to analyze the Gromov product in the wide class of metric spaces in which metrics are defined via Chebyshev sets. Midpoint mappings in this class of spaces will be also examined.

For some discrete parameters, multivariate (Dunkl-)Bessel processes on Weyl chambers $C$ are projections of Brownian motions on Euclidean spaces $V$; the most prominent examples are Dyson Brownian motions. More generally, the projections of Brownian motions on $V$ with drift are Bessel processes with drifts on $C$, where again the associated transition densities can be described in terms of multivariate Bessel functions. Moreover, Dunkl processes with drift can be defined in an analogous way. In this paper we prove central limit theorems for these Bessel and Dunkl processes with drift for regular drift vectors, and arbitrary root systems and multiplicities. These results improve recent results of the author.

We consider an additive white Gaussian noise channel subject to a peak-amplitude constraint and study how the support of the capacity-achieving input distribution (CAID) evolves as the amplitude constraint $A$ varies. Although the CAID is known to be unique, symmetric, discrete, and finitely supported, the structure of its support transitions has remained largely unresolved. We show that the origin is the only possible degenerate support point and the only possible inactive contact point of the KKT function. We then establish continuity properties of the optimal distribution and the KKT function and prove that every nonzero support point is locally stable: under small changes in $A$, it persists as a unique nearby atom whose location and probability mass vary continuously. Combining these results, we prove that, locally, the support cardinality at a nearby amplitude is either unchanged or larger by one and that any such increase can occur only at the origin. Consequently, all local changes in support cardinality are confined to the origin: locally, the only possible transition mechanisms are the appearance or disappearance of the point at the origin and the splitting or merging of the origin into a symmetric pair.

Parametric bootstrap and fiducial generalized methods are proposed to construct simultaneous confidence intervals (SCIs) for all pairwise mean differences of several two-parameter exponential distributions, based on doubly censored samples. Both methods have been shown to have correct asymptotic coverage probabilities. Simulation studies are conducted to compare the performance of the two proposed methods. The usefulness of our proposed procedure is illustrated with two examples.

Let $V$ be a vector space over $\mathbb F=\mathbb R$ or $\mathbb C$. We develop a basis-free umbral calculus over $V$. We define the vector space of polynomials over $V$, and polynomial sequences in it. We discuss shift-invariant operators acting in polynomials over $V$. We define polynomial sequences of binomial type and Sheffer sequences over $V$. We provide equivalent characterizations of these polynomial sequences. We prove two recurrence formulas for Sheffer sequences. With each Sheffer sequence, we associate a linear operator acting in polynomials over $V$, which we call a Sheffer operator. We prove that the set of Sheffer operators is a group for the usual product of linear operators, which is isomorphic to the Riordan group of pairs of formal tensor power series in a variable from $V$. Under the assumption that $V$ is an algebra, we lift every Sheffer sequence over $\mathbb F$ to a Sheffer sequence over $V$. We provide examples of such lifting.

In this paper we introduce a generalization of Helly's theorem closely connected to Bárány-Gromov overlap theorems (also called selection lemmas). Our main result implies both the topological colorful Helly of Kalai and Meschulam and Karasev's topological centerpoint theorem. We further investigate the topological fractional Helly theorem from this overlap perspective, and show an overlap theorem for dense complexes (a continuous second selection lemma for tame maps).

Relative K-stability of toric varieties can be tested through explicit Donaldson-Futaki computations, yet finding unstable examples remains challenging. We develop a reproducible symbolic-numeric framework to investigate all fifteen families of Delzant octagons corresponding to smooth toric surfaces of Picard rank six. While numerical optimization identifies several apparently destabilizing configurations, exact rational reconstruction systematically removes the observed instability. We explain why Wang and Zhou's 2014 tentative construction of a relatively K-unstable octagon does not allow to find an explicit example.

We study reconstruction and profile geometry for divisor schemes of binary linear systems and their sparse Krylov charts. Over the integers, the first nonzero equations of the complete embedded divisor scheme recover the defining linear system functorially under arbitrary base change, yielding a closed immersion from the Grassmannian of linear systems to the corresponding Hilbert scheme. For monomial systems in characteristic zero, arithmetic profiles classify the reduced factorization branches, determine their image dimensions and generic multiplicities, and control geometric reducedness. For complete progressions, the normalizations of the relation branches and their images are explicit products of projective spaces equipped with two natural polarizations. We give an affine normality criterion in terms of the associated Fourier data and a projective criterion obtained by adjoining an endpoint-allocation condition. These results place the closed sparse Krylov rank loci in a uniform reconstruction–normalization framework and yield explicit mixed-degree and formal-profile consequences. They also clarify the limit of normalization data alone: recovering a possibly nonnormal image algebra requires additional information not addressed here.

Let $\alpha \in \mathbb{R} \setminus \mathbb{Q}$, $\beta \in \R$, $N \in \mathbb{R}_{\ge 1}$ and $ \Delta \in (0, 1/2)$. For any real $y$, let $\|y\|$ denote the distance from $y$ to the nearest integer. In the first part of this paper, we show that given two coprime integers $u, v \ge 1$, there are infinitely many primes $\ell \equiv u \bmod v$ such that $ \|\alpha \ell - \beta\| \ll_v \ell^{-1/4} \log^{8} \ell. $ In order to prove this result we first prove the following general theorem and then deduce the above as a corollary. Before stating the result, let us define a function on $f_{\Delta}(\theta)$ on $\R$ such that $f_{\Delta}(\theta)$ is $ 1 \text{ if } \| \theta \| < \Delta$ and $ 0 $ otherwise. Further, suppose that $u, v \in \mathbb{Z}_{\ge 1}$ are coprime and $a$, $q \in \Z$ are coprime with $q> N^{1/4}$ and $|\alpha v -a/q| \le 1/q^2$. Then, for every $\epsilon \in \mathbb{R}_{>0}$ we have \begin{equation*} \sum_{\substack{n=1 \\ n \equiv u \bmod v}}^N \Lambda(n) (f_{\Delta}(\alpha n - \beta) - 2\Delta) \ll_v (Nq^{-1/2} + N^{3/4} + N^{5/6}\Delta^{1/2} + (\Delta Nq)^{1/2} + N^{\epsilon}q \Delta^{1-\epsilon}) \mathcal{L}^8 \end{equation*} where $\mathcal{L}=\log (Nq/\Delta)$. This generalises a well known result of Vaughan from 1977 proving a similar bound for the sum \begin{equation*} \sum_{\substack{n=1}}^N \Lambda(n) (f_{\Delta}(\alpha n - \beta) - 2\Delta). \end{equation*} In the second part of this paper, we explicitly construct an uncountable set $S \subset \R\setminus \Q$ such that for every $\gamma \in S$ there are infinitely many primes $\ell \equiv u \bmod v$ satisfying $\| \gamma \ell \| < \ell^{-1}$. Further we prove unconditionally that not all the elements of $S$ are Liouville numbers. This addresses a question of Erdös and Mahler from 1939.

We address the plasma-vacuum interface problem in the two dimensional space. The plasma region is governed by the ideal incompressible magnetohydrodynamic equations. The vacuum region is described by the pre-Maxwell dynamics, with the unknowns set to zero for simplicity. This is a free boundary problem of ideal Euler type equations. Under the strong horizontal magnetic field in the plasma region, we prove the global in time nonlinear stability of the two dimensional plasma-vacuum interface. The proof relies on understanding the interplay between the dynamics of the fluids inside the domain and those on the free interface, as well as the inherent structures of the problem.

We study the geodesic convex hull of a stationary Poisson point process restricted to a horoball in $d$-dimensional hyperbolic space. The resulting random set is an unbounded hyperbolic polyhedron with a distinguished ideal direction. Projecting its boundary facets to the bounding horosphere yields a stationary Euclidean tessellation of $\mathbb{R}^{d-1}$, which we identify as a dual Poisson–Laguerre tessellation with an explicit height density. We derive an exact formula for its cell intensity and, in the critical regime where the intensity of the Poisson point process is matched with the height of the horoball, we prove local convergence of the projected tessellation to the classical Poisson–Delaunay tessellation in $\mathbb{R}^{d-1}$. As consequences, the typical cell converges in distribution and the intensities of all $k$-dimensional faces converge to their Poisson–Delaunay counterparts. We also study a localised volume functional of the hyperbolic convex hull, determine its limiting expectation in the same regime, and use an Efron-type identity to obtain a second-order asymptotic expansion for the vertex intensity. In addition, we derive an exact formula for the expected localised surface area and determine its critical asymptotics. In dimensions $d\ge3$ the expected unnormalised local surface area converges to a finite limit, whereas in dimension $d=2$ it exhibits logarithmic growth.

We consider the derivative nonlinear Schrödinger equation on the circle, and prove that it is globally well-posed in the Sobolev space $H^1(\TT)$.

Wang and Sitters' rounding scheme for graph balancing (each job allowed on at most two machines) keeps the makespan, the busiest machine's load, within 11/6 of the optimum, and a companion note shows that no smaller constant holds. One suspected cause is Step 3's freedom to match leftover jobs to slots in any valid way. We show that removing that freedom does not improve the constant. An idealized oracle that always picks the valid matching with the smallest makespan still has worst-case constant exactly 11/6 against the optimum. Both natural 7/4 bounds for it, against the relaxation's target and against the optimum, are false. Requiring Step 1 to return a vertex (corner) of the relaxation does not help when that vertex is chosen adversarially. The loss therefore does not come from Step 3's freedom alone. We also prove a reduction confining every overload above 7/4 to two shapes, give a seven-machine instance on which the natural two-phase repair fails, and prove a strict 7/4 bound on the generalized three-path family.

For a graph $F$, the Turán number $\operatorname{ex}(n,F)$ is the maximum number of edges in an $n$-vertex graph containing no copy of $F$. Determining the Turán numbers of even cycles is a central problem in extremal graph theory and remains open in general. For $C_6$, the best previous upper bound was due to Füredi, Naor, and Verstraëte [Advances in Mathematics, 2006], who proved that, for sufficiently large positive integer $n$, $$ \operatorname{ex}(n,C_6) \leq \lambda n^{4/3}+O(n)<0.6272 n^{4/3}, $$ where $\lambda$ is the real root of $ 16\lambda^3-4\lambda^2+\lambda-3=0$. We improve this bound by showing that, for sufficiently large positive integer $n$, $$ \operatorname{ex}(n,C_6) \leq \alpha n^{4/3}+O(n)<0.6144 n^{4/3}, $$ where $\alpha$ is the unique real root of $ 4 \alpha^{3} (3/2)^{1-1/(2\alpha)} =1$ in the interval $(1/2,2/3)$.

Numerically solving McKean-Vlasov stochastic differential equations is computationally challenging due to the compounding costs of discretizing in time and in the distribution of the solution. Multilevel ideas have been proposed to provide speed-ups. In this work, we study the multilevel Monte Carlo method proposed by Ricketson (2015) for equations whose drift and diffusion terms depend on the law of the solution $X_t$ through the expectation $E[R(X_t)]$. The scheme follows the single-ensemble paradigm, where particles interact across levels at each timestep. While cross-level feedback makes this scheme attractive in practice, the correlations it introduces have so far confined its cost-error analysis to a model problem with linear drift, deterministic diffusion, and $R$ the identity. We use additional coarse particles to enforce geometrically decaying coupling errors towards the coarser levels. This allows us to prove our main contribution, an $L^p$-error of $O(\epsilon)$ at cost $O(\epsilon^{-2-\delta})$ for any $p\ge2$ and $\delta>0$ (with a constant that grows as $\delta\to0$), assuming only global Lipschitz bounds on the drift, the diffusion, and $R$. An exploratory experiment is consistent with the derived rates and suggests that in practice the constant does not grow significantly for small $\delta$. Our methodology and proof strategy may also be useful for other single-ensemble multilevel schemes, such as multilevel ensemble Kalman filters.

We consider covariant algebras attached to certain hybrid qudit spaces. We describe these algebras in terms of generators and relations, and we show they are flat deformations of certain explicitly described semigroup algebras.

We continue our studies on parametric norm forms $F_t({\bf x})$, with ${\bf x}=(x_0,x_1,\ldots,x_{d-1})$ lying in some parametric linear subvariety $W_t$ and integers $t$ sufficiently large. In a previous paper [Am-Ma-Za2] we proved some effective specialization results for integer solutions $\bf x$ of $F_t({\bf x})=1$. Here we modify our techniques to treat $F_t({\bf x})=q$ for an arbitrary integer $q$. Under mild conditions (not however including the crucial index assumption in [Am-Ma-Za2]) we show that all $\bf x$ are polynomially bounded in terms of $|q|$ and $t$. As in [Am-Ma-Za2] we use the methods of our paper[Am-Ma-Za] based on diophantine approximation techniques to bound certain heights. In particular we do not use linear forms in logarithms and indeed it seems unlikely that those can lead to such polynomial bounds, even for Thue equations in two variables with $x_2=\cdots=x_{d-1}=0$. We present an example with eight variables.

This paper focuses on the boundary approximate controllability of two classes of linear parabolic systems, namely a system of $n$ heat equations coupled through constant terms and a $2 \times 2$ cascade system coupled by means of a first order partial differential operator with space-dependent coefficients. For each system we prove a sufficient condition in any space dimension and we show that this condition turns out to be also necessary in one dimension with only one control. For the system of coupled heat equations we also study the problem on rectangle, and we give characterizations depending on the position of the control domain. Finally, we exhibit a cascade system for which the distributed controllability holds whereas the boundary controllability does not. The method relies on a general characterization due to H.O. Fattorini.

We explore the homogeneous structures of Albanese morphisms and maximal rationally chain-connected fibrations of algebraic varieties with nef anticanonical divisors. Furthermore, we show that there exists a natural group homomorphism associated with a maximal rationally chain-connected fibration, which induces the Chevalley decomposition of $\operatorname{Aut}^0$ up to an isogeny. This result is analogous to a similar theorem of Nishi and Matsumura for Albanese morphisms. In addition, we reprove Nishi and Matsumura's theorem using the modern language of algebraic geometry.

The contact process on the regular tree $\mathbb{T}_d$ when $d\geq 3$ has the two phase transitions of global and of local survival, found by Pemantle and Liggett at values $\lambda_1$ and $\lambda_2$. We start the system with every vertex infected and let it relax to what is known as the upper invariant infection. In this stationary state, $\lambda_p$ is the critical value beyond which the infected vertices can percolate through $\mathbb{T}_d$, and $\lambda_{p^\complement}$ is the parameter before which the healthy vertices can percolate. We find these are both distinct phase transitions \[0<\lambda_1<\lambda_p<\lambda_2<\lambda_{p^\complement}<+\infty\] on $\mathbb{T}_d$ when $d\geq 7$. The most interesting of these comparisons is $\lambda_1<\lambda_p$, which we find for all $d\geq 3$. This comparison $\lambda_1<\lambda_p$ is a long-standing open question on $\mathbb{Z}^d$ with $d\geq 2$ and was not yet found on any other graphs except where $\lambda_p$ is infinite.

The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{\alpha_1}\times\mathbb{Z}_4^{\alpha_2}\times\mathbb{Z}_8^{\alpha_3}$, and a $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is the Gray map image of such a code. A recursive construction of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive Hadamard codes $\mathcal H^{t_1,t_2,t_3}$, with all $\alpha_i\neq0$, $t_1\geq1$, $t_2\geq0$, and $t_3\geq1$, is known, as are the linearity, kernel dimension, and rank of the corresponding codes $H^{t_1,t_2,t_3}$ of length $2^t$, where $t+1=3t_1+2t_2+t_3$. Yet these invariants do not completely classify the family. Two infinite families of pairs of distinct types share the length, rank and kernel dimension, and for classified lengths $2^t$, $3\leq t\leq11$, such pairs were separated only by computer equivalence tests. In this paper, we introduce an equivalence invariant that resolves these cases. The kernel partitions the binary coordinates into blocks, two coordinates lying in the same block when every kernel word takes the same value in both; the kernel-block rank profile is the multiset of the dimensions of the linear span punctured on these blocks. Unlike rank and kernel dimension, which are global, this invariant records how much of the span survives on each block. We compute it for the whole family: there are $2^{t_1+t_2+t_3-1}$ blocks, all of size $2^{2t_1+t_2}$, and the profile takes at most two values $t_2+\binom{t_1+2}{2}$ and $t_2+2+\binom{t_1+1}{2}$, whose difference is $t_1-1$; it is constant precisely when $t_1=1$. Hence, the profile recovers $t_1$, and the length and kernel dimension recover $t_2$ and $t_3$. Two codes of the family with the same length are therefore equivalent if and only if their types coincide, and the number of pairwise nonequivalent such codes of length $2^t$ is $\lfloor(t^2+6)/12\rfloor$ for every $t\geq3$.

A recent program tries to find which functions appear as conjugator length functions. In this note, we show that any (computable) increasing function larger than $n$ appears as $\mathrm{Cl}_G$ for some finitely generated (recursively presented) group. On the other hand, we demonstrate that either $\mathrm{Cl}_G$ must be constant or $\mathrm{Cl}_G(n)\succ n$. Combining these, we obtain a complete description of which functions appear as conjugator length functions of finitely generated groups.

We classify the multi-solitons, in the positive sense of time, of the focusing energy-critical wave equation in space dimension $5$, more precisely the solutions $u(t,x)$ of the equation \[ \partial_t^2 u - \Delta u - |u|^{\frac 4{3}} u = 0, \quad (t,x)\in [T_0,\infty)\times {\mathbb R}^5, \] which satisfy the estimate \[ \left\|\nabla_{t,x} \left(u(t) - \sum_{k} W_k(t) \right) \right\|_{L^2} \lesssim t^{-\frac12^+} \quad \hbox{for all $t\gg 1$}. \] Here, $\{W_k\}_k$ is any given finite family of traveling waves, based on the explicit ground state solution \[ W(x) = \left(1+ \frac{|x|^2}{15}\right)^{-\frac32} \] with collinear two-by-two different velocities.

We prove the $C^{p'}$-regularity conjecture in every dimension when $p>2$ is sufficiently close to $2$. To this end, we establish improved Hölder estimates for the gradients of $p$-harmonic functions. These estimates also determine the first-order asymptotics of the optimal gradient Hölder exponent in every dimension. The proof combines compactness, harmonic rigidity of the limiting profiles, and a sharp uniform gap estimate for the first variation of the gradient excess.

In this work we study the null-controllability properties of linear parabolic systems with constant coefficients in the case where several controls are acting on different distributed subdomains and/or on the boundary. We prove a Kalman rank condition in the one-dimensional case. In the case where only distributed controls are considered we also establish related results such as a Carleman estimate.