Papers

We study the fully parabolic Keller–Segel system \[ u_t=\Delta u-\chi\nabla\!\cdot(u\nabla v)+\lambda u-\mu u^2, \qquad \tau v_t=\Delta v-v+u \] in a bounded smooth convex domain. For every fixed $\tau>0$, we prove that \[ \mu>\frac{N\chi}{4} \] guarantees global existence and uniform-in-time boundedness. This coefficient-explicit sufficient condition is independent of $\tau$ and involves no embedding or maximal-regularity constants. To the best of our knowledge, it is the first coefficient-explicit boundedness criterion that remains unchanged for all $\tau>0$ in arbitrary space dimension. The proof is built on a new auxiliary comparison function \[ Y_\tau =u+\frac{\chi\tau}{2}|\nabla v|^2-(\tau-1)\Delta v, \] which satisfies a closed scalar parabolic inequality for every $\tau>0$. When $\tau\ge1$, this inequality yields a direct pointwise comparison and an explicit bound for $u$. When $0<\tau<1$, it instead provides a uniform upper bound for $v$. Applying a parabolic squeezing argument to the transform $z=e^{-\chi v/2}$ then yields a uniform Hölder bound for $v$. Hölder–Sobolev interpolation and weighted maximal $L^p$-regularity subsequently give an $L^p$-bound for $u$ with sufficiently large $p$, and standard parabolic smoothing closes the argument.

We give a new characterisation of Auslander-Gorenstein finite dimensional algebras by showing that they are exactly the finite dimensional algebras with a well-defined Auslander-Reiten bijection. This proves a conjecture of Marczinzik. We use this to give a new proof that a finite lattice with an Auslander-Gorenstein incidence algebra has to be distributive.

Empirical work often removes fixed effects, latent factors, or high-dimensional controls before estimating structural relationships. These transformations reduce confounding but may also remove identifying variation. We study linear panel IV after one equation-compatible nuisance projection under two-way dependence. The projected Jacobian determines which structural directions remain visible; the projected-score law determines their precision; and, on Gaussian fixed-rank strata, they combine in a Projected Information Matrix. We derive weak-identification limits with dimension-specific information accumulation, feasible factor-transfer conditions, identification-robust tests, bootstrap procedures for non-Gaussian interaction limits, and inference for the projected spectrum, rank, subspaces, and information matrix. Simulations show that a raw first-stage statistic above 500 can support the wrong sign while projected diagnostics reveal weak valid information. In an international monetary application, common projection substantially attenuates apparent foreign-output persistence, while Gaussian-reference Anderson–Rubin sets remain unbounded. Identification should therefore be assessed after nuisance removal.

We study convex-concave minimax optimization problems with bilinear coupling of the form $\min_{x\in \mathcal X}\max_{y\in \mathcal Y} \; f(x)+\langle y,\mathbf{B}x\rangle-g(y),$ where the functions $f$ and $g$ have Hölder continuous (sub)gradients. This setting covers a broad range of regimes, from nonsmooth problems with bounded subgradient variation to smooth problems with Lipschitz continuous gradients; for a smooth component used in the coupling-induced regularizer, its Lipschitz-gradient constant is assumed to hold in the ambient space. We propose a sliding method that exploits the composite structure of the problem by querying the oracles associated with $f$, $g$, and the bilinear coupling operator at prescribed frequencies determined by their individual properties. The method is based on a recursive sliding scheme for monotone variational inequalities. We establish convergence guarantees under Hölder continuity and show how the resulting complexity bounds depend explicitly on the Hölder exponents, Hölder constants, strong convexity parameters, and spectral properties of the coupling matrix. Our analysis covers nonstrongly convex and partially strongly convex regimes. For stochastic function oracles, we prove componentwise Hölder guarantees with separate variance costs: a uniform expected-gap bound on bounded domains and, when both functions are strongly convex on their feasible sets, a restarted Lyapunov guarantee without an anchor noise floor. Numerical experiments reproduce the predicted Hölder exponents and confirm that the number of gradient evaluations required for each function separates according to its own smoothness level rather than the worse of the two. A tomographic benchmark shows runtime gains when gradient evaluations are more expensive than the additional matrix-vector products.

We consider the magnetic Neumann Laplacian on rectangles, subject to non-homogeneous fields. We prove that the square is a local minimiser of the lowest eigenvalue among all rectangles of a fixed area for weak magnetic fields . We conjecture that it is a global minimiser both under the area or perimeter constraints and partially prove the conjecture by establishing lower and upper bounds to the principal eigenvalue.

We provide a proof of Crouzeix's conjecture, which combines the tools developed previously for weaker estimates with a simple perturbation lemma for $2$-dilations. Applying the lemma to the iterates $f^{n}$ in the double-layer potential representation yields the conjectured bound.

We organize and generalize several dualities involving continuous posets. The main theorem reads $\mathbf{St}_\alpha\mathbf{Inf}_{\alpha'}\mathbf{Cont}_{\beta'}\mathbf{Sup}_\beta \simeq (\mathbf{St}_\beta\mathbf{Inf}_{\beta'}\mathbf{Cont}_{\alpha'}\mathbf{Sup}_\alpha)^{\mathrm{op}}$, where $\mathbf{St}$, $\mathbf{Inf}$, $\mathbf{Cont}$ and $\mathbf{Sup}$ refer to stability, completeness, continuity and cocompleteness. The indices are "ladders", i.e., classes of sets $\lambda$ stable under dependent sums and quotients, with associated notions of $\lambda$-small infima and ${\lambda}$-filtered suprema. In the second half of the paper, we discuss algebraicity, proximity lattices and perfect maps.

Suda introduced the notion of a $Q$-polynomial coherent configuration, which provides a natural and important concept. Subsequently, Lato introduced a notion of a $P$-polynomial coherent configuration and proved that every such configuration satisfying the definition has at most two fibers. Although Lato's definition is interesting, particularly because it characterizes distance-biregular graphs, we argue that an alternative definition is desirable. In this paper, we propose an alternative notion of $P$-polynomial coherent configurations that is naturally aligned with Suda's $Q$-polynomial framework. We show that every two-fiber coherent configuration that is $P$-polynomial in Lato's sense is also $P$-polynomial in our sense, whereas the converse does not hold. We further prove that every coherent configuration of type $(2,2;3)$, $(3,2;3)$ or $(3,3;3)$ is $P$-polynomial in our sense. In addition, we present three families of $P$-polynomial coherent configurations with an arbitrary number of fibers: those arising from tight Euclidean $t$-designs in $\mathbb R^2$, the Terwilliger algebra of $H(n,2)$, and the set of all subspaces of $\mathbb F_q^n$. Finally, we give an equivalent condition for the cross-block intersection matrices to be tridiagonal and verify that all three families satisfy this condition.

This study investigates a virtual multiuser multiple-input multiple-output (MU-MIMO) system with PSK modulation, realized with a reconfigurable intelligent surface (RIS)-based transmitter. The study focuses on minimizing transmit power under quality-of-service (QoS) constraints while addressing the associated computational complexity. A discrete phase-shift RIS model is considered, and the power minimization problem is formulated in two scenarios. First, for QPSK user data, the symbol-error probability (SEP) is adopted as the QoS criterion. Second, for general $M$-PSK modulation, the union-bound SEP (UBSEP) is used to define the QoS constraints. Based on the considered formulations, a partial branch-and-bound (PBB) approach is developed, which improves on full branch-and-bound (FBB) methods in the sense of allowing for favorable complexity performance trade-offs. For the special case of high-resolution RIS, the discrete phase-shift set is approximated by its continuous counterpart, enabling the reformulation of the original problems as constrained optimizations on an oblique manifold, which are solved with reduced computational complexity with the proposed bisection method. Numerical results demonstrate the effectiveness of the proposed approaches in minimizing the transmit power for different SEP requirements and showcase the balance between power efficiency and computational complexity

Mutually unbiased measurements are commonly expected to expose independent facets of a quantum state: a correlation that is classical in one basis should disappear in a complementary basis. In higher dimensions, however, this intuition becomes particularly subtle because correlations recovered in different settings need not represent different information. To expose this loophole, we propose a two-branch classical null test: before the setting is chosen, a shared bit selects one of two orthogonal product preparations, producing a rank-two classical–classical state, and the candidate protocol then runs unchanged. Different settings can read the same bit through different outcome patterns. This two-branch classical architecture disproves the complementary-quantum correlation (CQC) conjecture in every dimension $d\geq3$. A distinct rank-two classical–classical state disproves its complete-basis extension (ECQC) at $d=7$, with an overrun that grows without bound along prime dimensions. Its qutrit CQC instance also gives classical false positives for a proposed quantum-correlation measure and a proposed one-sided semi-device-independent steering criterion, and refutes a conditional-probability conjecture. The failures identify the missing requirement: information read in different settings must be nonredundant. In experiments and applications, the same low-overhead architecture can serve as a calibration test before a multibasis score is assigned quantum meaning.

We develop a rigorous framework for the kinematics of a material point whose trajectory contains a jump discontinuity, modeled as a smooth curve on an interval punctured at one instant, with finite one-sided limits of position and velocity. A position jump is called an instantaneous teleportation. The event is treated as exogenous to Newtonian dynamics, imposed only before and after the removed instant. Separating the position and velocity discontinuities, we prove that the net linear impulse equals the momentum jump $\mathbf J=m\,\Delta\mathbf v$, carried entirely by the velocity discontinuity, while the position jump contributes none. This yields a momentum-compatibility condition: an instantaneous displacement transfers no net impulse if and only if the velocity is preserved. For rigid bodies it splits into preservation of translational and angular velocity. For the rotating, revolving Earth, a kinematic fact governs the estimates: for two points of the same Earth at the same instant the orbital velocity cancels, bounding the jump by $2\omega R_{\oplus}\approx 9.3\times10^{2}\ \mathrm{m\,s^{-1}}$; the larger scale $6.0\times10^{4}\ \mathrm{m\,s^{-1}}$ arises only between orbital epochs or frames. The internal stress depends on how momentum is delivered, and the implied strains lie outside linear elasticity, so the figures are indicative.

We consider polyconvex functionals defined on vector valued functions, the model case being the graph area functional, and we analyze the relaxed energy with respect to the strict convergence in $BV$. In several cases where the relaxed area is known, by exploiting a continuity result by Reshetnyak we are able to find an explicit formula for wide classes of integrands with linear growth in the minors of the gradient. We preliminarily extend to the $BV$ setting a continuity property observed by Acerbi–Dal Maso in the Sobolev case. Finally, partial results concerning the relaxation of the vortex map with respect to the $L^1$ convergence are obtained.

A b-coloring is a proper vertex coloring such that every color class contains a vertex, a so-called b-vertex, which sees all colors in its closed neighborhood. This type of coloring has been intensively studied from both structural and algorithmic point of view. Recently, Zaker [DAM 2025] introduced the notion of a b*-coloring, which is a b-coloring in which there is a vertex that sees a b-vertex of every color in its closed neighborhood. The b*-chromatic number is the maximum integer k such that there is a b*-coloring with k colors. We partially answer a question posed by Zaker and prove that graphs of girth at least 7 are b*-monotonic, which means that the b*-chromatic number does not increase by taking an induced subgraph. In addition, we discover a class of d-regular graphs of girth at least 5 with b*-chromatic number d+1, which strengthens a result about b-colorings by Dettlaff, Furmańczyk, Peterin, Roux, and Ziemann [AMC 2024]. We also study the parameterized complexity of finding b*-colorings, and show that for many structural parameters, the complexity coincides with that of finding b-colorings. In particular, the b*-chromatic number can be computed in polynomial time on any class of bounded clique-width. For most parameters, the translation from b-colorings is straightforward but for the feedback edge number, the FPT algorithm for b*-colorings is actually much simpler than that for b-colorings by Balabán [MFCS 2026].

For every admissible pair of Brieskorn parameters in the Plotnick–Suciu construction, we obtain a pair of oriented $2$-knots whose knot groups are isomorphic, whose second homotopy modules are semilinearly isomorphic under a suitable group isomorphism, and whose fundamental quandles are isomorphic, while no compatible group and module isomorphisms carry one first Postnikov invariant to the other. Consequently, their exteriors are not homotopy equivalent. Thus, the knot group, the second homotopy module up to semilinear equivalence, and the fundamental quandle do not determine the homotopy type of an oriented $2$-knot exterior. Using an alternative construction from Suciu's thesis based on punctured lens spaces, the same peripheral argument yields, for every $N\geq2$, a family of $N$ oriented $2$-knots with these properties. We additionally show that the Tanaka–Taniguchi examples with isomorphic knot groups and distinct fundamental quandles have pairwise inequivalent second homotopy modules: no isomorphism between two of the knot groups makes the corresponding second homotopy modules semilinearly isomorphic.

We prove a Nakai-Moishezon-type criterion for complex Hessian-type equations on projective manifolds whose associated degree-$n$ polynomials are strongly strictly right-Noetherian. For strictly right-Noetherian polynomials of arbitrary degree, we prove a uniform Nakai-Moishezon-type criterion. This class includes the complex Hessian and Hessian quotient equations.

A remarkable result of Guba shows that the Dehn function of Thompson's group $F$ is quadratic (Inv. Math. 163:313-342). In another paper (Alg. prob. in Grps. and Semigrps, 91-120), Guba showed that Thompson's group $T$ and $V$ have Dehn functions bounded above by $n^7$ and $n^{11}$, respectively. Recently, Migliorini showed that the Dehn function of Thompson's group $T$ is also quadratic (For. of Math. Sigma, 13:e109). In this paper, we improve the upper bound of the Dehn function of Thompson's group $V$ to $n^6$ and explore the intricacies of its relations via explicit van Kampen diagrams.

We prove that the Chow polynomial of the noncrossing partition lattice is equal to the descent generating function of tieless parking functions. As a consequence, we obtain new results on the descent statistic for parking functions.

Kazhdan–Lusztig $W$-graphs encode the cell structure of Hecke algebras, while their bidirected connected components are called molecules. In finite type~$A$, the Robinson–Schensted correspondence describes cells and molecules, and Beissinger's row insertion constructs the common tableau associated with an involution. In affine type~$A$, the affine matrix-ball construction (AMBC) assigns an affine permutation a pair of tabloids together with a dominant weight, and Marberg introduced affine fixed-point-free (FPF) $W$-graphs indexed by affine FPF involutions. We classify and enumerate the molecules of Marberg's $\m$-type affine FPF $W$-graph $\Gamma_n^{\m}$ and show that molecules with the same AMBC shape have isomorphic underlying simple bidirected graphs. We also prove that labelled complete-cycle row-Beissinger truncations recover the full AMBC datum of an affine FPF involution.

Alienation-based abstention makes participation endogenous because the platforms chosen by parties determine who votes. We study how voter tolerance affects the existence, location, and persistence of pure-strategy equilibria in two versions of a two-party spatial model. In the first, office-seeking parties maximize voter plurality; in the second, they care about the policy implemented. We characterize all office-seeking equilibria through a condition called window dominance, which gives a distributional characterisation of profitable deviations both near to and far from a party's platform. For general continuous voter distributions, pure strategy equilibrium in the office seeking game may fail to exist, may be non-unique, and may appear and disappear as tolerance increases. To resolve this, we identify a centripetal structure in the set of positions that can not lose to each rival. When this structure holds, a single anchor defeats every other platform and determines the unique office-seeking equilibrium. The classical Downs and Llavador results are special cases. We show that this structure persists at all larger tolerance radii once it holds at a radius of at least half the policy space and at least twice the median's distance from the center. The centripetal structure emerges before full participation for all voter distributions. Finally, the same structure yields a complete classification of equilibrium when parties care about the policy implemented rather than electoral victory alone.

The cone $M \times \mathbb{R}_+$ over a Sasakian manifold $M$ is equipped with a canonical Kähler structure with specific homogeneity properties with respect to the $\mathbb{R}_+$ coordinate. This Kähler structure completely encodes the underlying Sasakian structure. Recently, Grabowski, Grabowska and Mohseni provided a broader conceptual framework for this phenomenon via homogeneous Kähler structures, i.e. Kähler structures on a principal $\mathbb{R}^\times$-bundle $P$ satisfying similar homogeneity properties. This approach successfully extends Sasakian geometry from cooriented contact manifolds (where $P$ is a trivial principal bundle) to non-necessarily coorientable contact structures (where $P$ is non-necessarily trivial). Homogeneous Kähler structures are genuinely more general than Sasakian structures and this note precisely characterizes the extent of this generalization. This is achieved through a detailed analysis of all the involved compatibilities in terms of the line bundle tautologically associated to $P$. We also show that modifying the homogeneity condition on the Kähler structure allows this framework to encompass co-Kähler structures and a natural generalization of those as well.

Atmospheric temperature on Earth results from complex phenomena that climate models must account for. One important module is radiation, which has three sources: infrared radiation emitted by the Earth and the air, and visible light from the Sun. The latter arrives from an almost point-like source in the sky, producing a Dirac singularity in the boundary conditions known as collimated light. Following Siewert and Maiorino (1980), we propose a very fast numerical implementation to handle the singularity.

We establish a mesoscopic local law and quantitative eigenvector-delocalization estimates for the Sachdev–Ye–Kitaev model (SYK) Hamiltonian of $N$ interacting Majorana fermions subject to a $q$-body interaction. For even $q\ll N^{\frac{1}{2}}$, we prove that the normalized Stieltjes transform is approximated, uniformly on bounded energy intervals, by an explicit deterministic profile which we construct as an asymptotic expansion in the proportion $\alpha_{N,q}$ of anticommuting $q$-body interaction terms. The result holds both on the full Hilbert space and in each fermion-parity sector. We derive mesoscopic eigenvalue counting and spectral form factor estimates. For fixed $q$, we further obtain high-probability $\ell^\infty$-delocalization bounds in any deterministic orthonormal basis for individual bulk eigenvectors. These are the first mesoscopic laws and eigenvector delocalization estimates for the SYK Hamiltonian, which we obtain using the resolvent method.

We study the composition of temporal ergodic averaging with spatial averaging over shrinking metric balls, and determine sharp Orlicz endpoints for the corresponding unrestricted joint limit. For consecutive Birkhoff averages normalized by $N\Lambda_q(N)$ ($q\ge 0$), the sharp Orlicz endpoint is $L\log_{q+1}L$. In particular, the ordinary case $q=0$ yields an $L\log L$ local joint convergence theorem under the Lebesgue differentiation property alone, while $L^1$ fails even on the Euclidean interval, answering two questions of Young. The same $L\log_{q+1}L$ endpoint holds for prime averages for every $q\ge 1$. For arbitrary time sequences, $L\log_qL$ always suffices at the same normalization $N\Lambda_q(N)$ ($q\ge 1$), and this endpoint is sharp in the Orlicz sense for fixed-base exponential sequences $k^n$ ($k\ge 2$) and for sequences with polynomial ratio separation, including $n!$. The positive results rest on a local stability principle: under the ball Lebesgue differentiation property, pointwise temporal convergence lifts to the local joint limit whenever the associated temporal maximal function admits an $L^1$ majorant. The additional logarithm in the regular-time case comes from lifting restricted logarithmic maximal estimates from sets to general functions. The arbitrary-sequence theorem uses a dyadic decomposition instead. The lower bounds are local $\infty$-sweeping out constructions, obtained by weighted local sweeping out for polynomial-growth regular times, by residue constructions for polynomially ratio-separated sequences, and by digit constructions for fixed-base exponentials.

We prove that a Noetherian local ring $R$ with canonical module $\omega_R$ is quasi-Gorenstein if and only if $\Fitt_1^R(\omega_R) \cong \omega_R$. This settles the expectation of Eisenbud, Ficarra, Herzog, and Moradi on canonical ideals.

The well-known persistence algorithm summarizes the evolution of homological cycles into what is called a barcode while scanning an input simplicial filtration. We show that this summarization process can be enriched by monitoring other algebraic structures that weave through different dimensions. In particular, we propose an algorithm to monitor the $(p+1)$-chains that make $p$-cycles to be $p$-boundaries and then morph into $(p+1)$-cycles. In effect, we get extra bars called links connecting the bars in dimension $p$ with the bars in dimension $p+1$ in the persistence barcode. The links produce extra barcodes, which we call link barcodes in addition to the usual ones obtained by standard persistence. The link barcodes, as such, are not stable. However, we can make them stable using a fixed “reference” filtration. We apply the link barcodes to the graph isomorphism problem and to the link prediction problem in temporal networks exhibiting its discriminating power through these experiments.

In this note, we introduce the notion of ranked spreadness, a strengthening of the usual spread condition in which the elements of each member can be ordered so that their one-coordinate marginals decay geometrically with their rank. This additional structure removes the dependence on the maximum set size in random-containment estimates. We prove width-free hitting and weighted-concentration theorems for ranked-spread set systems, together with an elementary kernel-extraction theorem showing that ranked spreadness arises naturally in arbitrary distributions on small sets. Our main application is to the simulation of nonadaptive property testers by sample-based testers. If a one-sided tester has average query complexity $d$ and rejects every far input with probability at least $\delta$, then, for every integer $c>d/\delta$, it admits a one-sided sample-based simulation with expected sample complexity $O_{d,\delta,|\Sigma|}\bigl(n^{1-1/c}\bigr)$. More generally, if positive inputs are rejected with probability at most $\gamma$ and far inputs with probability at least $\delta>\gamma$, the same conclusion holds for every $c>d/(\delta-\gamma)$. In particular, for constant-query nonadaptive testers we obtain an exponent $1-\Theta(1/q)$, matching, up to the dependence on the rejection gap, the exponent conjectured by Fischer, Lachish, and Vasudev.

Remote threshold decisions require more than accurate state estimates: the posterior must support reliable alarm/no-alarm decisions and, when possible, anticipate early critical decisions. We study this problem over short-packet wireless links with outage risk. We derive false-positive/false-negative feasibility conditions that define a decision-feasible region of the estimation and yield a predictive decision-update trigger. To protect predictive updates from outages, we add AoI-controlled resilience updates that both detect disruptions and maintain freshness. A two-state Markov surrogate of the thresholded process, matched to its one-step switching statistics, enables tractable long-term reliability-energy analysis. Then, we jointly optimized transmit power and AoI-controlled resilience update probabilities. Simulations show earlier, reliable decisions at competitive energy with baselines.

Let $\psi(n)$ and $\phi(n)$ denote Dedekind's arithmetic function and Euler's totient function, respectively. We study the distribution of the compositions $\phi(\psi(n))$ and $\psi(\psi(n))$. In particular, we obtain quantitative upper bounds for the exceptional set associated with $\phi(\psi(n))$, thereby refining a density result of Sándor. We also prove that, for every fixed $c>0$, the set of integers $n\leq x$ satisfying $\psi(\psi(n))\leq cn$ has asymptotic density zero. Our method adapts sieve ideas used by Dixit and Bhattacharjee to compositions involving Dedekind's arithmetic function.

Collecting multidimensional user data is essential for extracting rich insights across various applications. Local Differential Privacy (LDP) has emerged as a de facto standard for mitigating privacy risks in such scenarios. A key challenge in privacy-preserving multidimensional data collection lies in inter-attribute dependencies, as they can inadvertently reveal correlated information and increase privacy vulnerabilities. Therefore, accurately measuring correlation-induced privacy leakage (CPL) is essential for privacy analysis and privacy-utility trade-off. However, existing CPL analysis solutions either require accurate prior knowledge or face scalability challenges for large numbers of attributes and high-cardinality attributes. These limit their practical applicability in real data. To address this research gap, we propose a novel metric, “Dependency Triad” (DT), which summarizes the pairwise dependency information relevant to CPL using three parameters and yields a constant-time conservative estimator of pairwise CPL. DT explicitly models uncertainty in prior distributional knowledge through its parameters, delivering robust leakage estimates. Moreover, its robustness to sparse distributions makes it particularly suitable for high-cardinality attributes, while the pairwise formulation serves as a tractable building block for assessing total leakage in multidimensional settings. Extensive experiments on both synthetic and real datasets demonstrate that DT consistently estimates CPL across diverse dependency regimes and prior uncertainties.

A finite density of mutually avoiding directed polymers in a quenched random medium is a minimal model of glassy line matter. The dilute theory, solved by replica Bethe ansatz, predicts an interaction free energy proportional to $\rho^2$ and disorder cumulants with distinct power-law dependences on the density $\rho$, but direct numerical tests have been hindered by the combinatorially large many-polymer transfer matrix. We recast the problem as filling logarithmic eigenvalues of a single-polymer transfer-matrix product, obtaining the quenched free energy, its cumulants, and a disorder-induced linear spectral edge consistent with the replica prediction.