Papers

Preserving metrological information under noise is central to quantum sensing, yet finite detectors and memories impose an unavoidable limit on how finely noise trajectories can be resolved. Using exact symmetric-logarithmic-derivative geometry, we determine when a monitored trajectory can be compressed without losing quantum Fisher information. For an explicit faithful monitored qubit family, the complete joint signal-and-noise model is recoverable from only polynomially many type records, requiring $O(\log n)$ terminal memory, whereas deferred recovery of arbitrary $n$-qubit states requires exponentially many trajectories, or $O(n)$ memory. Online correction replaces this terminal storage by an irreducible per-use readout and feedback alphabet. These results establish syndrome information as a task- and timing-dependent resource connecting quantum sensing, statistical sufficiency, and quantum error correction.

This paper is concerned with the Cauchy problem for the 3D incompressible magnetohydrodynamic (MHD) equations in supercritical Sobolev spaces. It is well known that the system is locally well-posed in subcritical Sobolev spaces, whereas the supercritical regime remains largely open. In this work, we establish norm inflation for the incompressible MHD equations, both with and without Laplacian dissipation, in supercritical Sobolev spaces, thereby revealing strong ill-posedness of the system at this regularity level. A distinctive feature of our approach is the introduction of a novel geometric construction, termed the “Magnetic-solo ansatz”, through which, for the ideal MHD system, norm inflation occurs exclusively in the magnetic field $b$ in $H^s$ with $0<s<\frac{5}{2}$, while the $H^s$-norm of the velocity field $u$ remains uniformly bounded. This asymmetric behavior shows that supercritical ill-posedness can be driven exclusively by the magnetic field, highlighting its essential role in the breakdown of well-posedness. Our findings fill a significant gap in the supercritical regularity theory for incompressible MHD and shed light on the distinct mechanisms governing the fluid and magnetic dynamics.

We construct two explicit countable discrete groups $\Gamma_1$ and $\Gamma_2$ that are both ICC and have Kazhdan's property (T). Although $\Gamma_1$ and $\Gamma_2$ are not isomorphic as groups, their group von Neumann algebras are isomorphic: $L(\Gamma_1)\cong L(\Gamma_2)$. This provides a counterexample to Connes' rigidity conjecture for ICC property (T) groups. This result was obtained with assistance from GPT-5.6 Sol, independently of and concurrently with work by OpenAI.

We study slow-fast stochastic systems in which a fast process, evolving within several ergodic components, drives the transition rates of a slow jump process switching between these components. Considering a differential equation whose vector field depends on both processes, we show that, under suitable ergodicity and regularity conditions, its solutions converge in law in $D([0,T],\mathbb R^d)$ to an averaged system depending only on the slow process. The averaged drift is obtained by averaging the original vector field over the invariant measure of the fast process within each component. Our results extend classical averaging theory to systems where the vector field is explicitly influenced by a fast random environment with multi-component dynamics.

We develop a least-squares weak Galerkin (LS-WG) finite element method for the Cauchy problem of the biharmonic equation. The proposed approach reformulates the fourth-order equation as a coupled system of two second-order equations, which are discretized using discrete weak Laplacian operators on weak finite element spaces. The resulting least-squares formulation yields a symmetric positive definite linear system, thereby eliminating the discrete inf-sup condition required by mixed finite element methods while avoiding the construction of globally $C^1$-conforming finite element spaces. Furthermore, the weak Galerkin framework naturally accommodates general polygonal meshes, offering considerable flexibility in mesh generation and approximation. Under the assumption that the continuous biharmonic Cauchy problem admits a unique solution, we establish the uniqueness of the discrete LS-WG solution and derive optimal-order error estimates in a discrete energy norm. Numerical experiments confirm the theoretical convergence rates and demonstrate the accuracy, robustness, and effectiveness of the proposed method.

We prove that all automorphisms of twisted Chevalley groups of type ${}^2D_\ell$, $\ell\ge4$, and their elementary subgroups over commutative local rings with $1/2$ are standard. More precisely, for elementary groups, every automorphism is a composition of strictly inner, diagonal, and ring automorphisms; for full groups, a central factor may also occur. In the adjoint case, diagonal automorphisms are strictly inner and the centre is trivial, so only strictly inner and ring factors are needed for both the elementary and full groups.

We consider a diffusion-relaxation system and investigate the conditions on parameters leading to equilibration versus localization. When the diffusion is dominant, solutions converge toward homogeneous equilibria. By contrast, when the effective diffusion is weak, localization emerges. Such behaviors have been studied for various models through formal asymptotic arguments and linearized stability analysis, but rigorous understanding of the associated nonlinear phenomena remains limited, particularly in higher dimensions. In the equilibration regime, we establish convergence toward constant equilibria by exploiting an energy dissipation structure and invariant-region estimates. In the localization regime, we study self-similar solutions and transform the problem of their existence into an autonomous dynamical system. The existence of self-similar profiles associated to localizing solutions is reduced to the construction of a heteroclinic orbit for an autonomous dynamical system. Their existence is obtained through an application of geometric singular perturbation theory. Our analysis provides a rigorous characterization of the transition from equilibration to localization.

We introduce a mathematical framework for shape comparison based on mapping functions from the shape domain to a common reference domain. This Push-Forward Transform enables invariant and robust comparison of shapes, preserving intrinsic geometric information. Quantitatively comparing shapes and their temporal evolution is a fundamental challenge in image analysis. Meaningful shape comparison requires representations that are invariant to transformations that do not alter shape itself, such as translation, rotation, reflection, re-parametrization, and uniform scaling, while remaining sensitive to intrinsic geometric variation. Existing approaches often rely on sensitive parameterizations, landmark correspondence, or learned representations that are difficult to interpret and reproduce. We show that the Push-Forward Transform (PF-T) applied to Signed Distance Functions (SDFs) yields a continuous representation that captures both boundary and interior geometry. We derive an interpretable morphometric that quantifies shape similarity and reveals features such as skeletal topology and rotational symmetries. The push-forward transform applies consistently to two- and three-dimensional shapes, extends to time-evolving geometries, and supports the joint analysis of shape and additional scalar fields defined over shapes, such as intensity or molecular signals. We present the mathematical formulation, describe an efficient algorithm, and benchmark the approach on 2D, 3D, and temporal data sets.

Let $T_n=K_{1,1,n}$, and let $P_n(x)$ denote the Kazhdan–Lusztig polynomial of its graphic matroid. We prove that, whenever $n\ge2$ and $0\le\lambda\le n/2$, the polynomial $P_n(x)+\lambda x$ has exactly $\lfloor n/2\rfloor$ zeros, all of which are negative and simple. In particular, the Kazhdan–Lusztig polynomials of the graphic matroids of $T_n$ and $K_{2,n}$ are real-rooted. We also prove that, for $n\ge2$, the common polynomial $Z_{T_n}(x)=Z_{K_{2,n}}(x)$ has $n+1$ distinct negative zeros. The proofs use a common rational transformation, reducing the Kazhdan–Lusztig case to alternating sign evaluations at the zeros of a Chebyshev polynomial and the $Z$-polynomial case to a unit-circle criterion for self-inversive polynomials.

Let \(\cG_1,\ldots,\cG_k\) be fixed vertex-ordered graphs, each containing at least one edge. The ordered Ramsey number \(\oR(\cG_1,\ldots,\cG_k)\) is the least integer \(N\) such that every \(k\)-edge-coloring of the ordered complete graph \(\cK_N\) contains an order-preserving copy of \(\cG_i\) in color \(i\) for some \(i\in[k]\). For positive weights \(\blambda=(\lambda_1,\ldots,\lambda_k)\), let \(\oM_{\blambda}(n;\cG_1,\ldots,\cG_k)\) denote the minimum weighted number of correctly colored, order-preserving copies of the target graphs over all \(k\)-edge-colorings of \(\cK_n\). When \(\blambda=\bf{1}\), \(\oM_{\bf{1}}(n;\cG_1,\ldots,\cG_k)=\oM(n;\cG_1,\ldots,\cG_k)\) is called the ordered Ramsey multiplicity. In this paper, we first establish the amplification inequality \[ \oM_{\blambda}(n;\cG_1,\ldots,\cG_k) \ge \oM_{\blambda}(t;\cG_1,\ldots,\cG_k) \frac{\binom{n}{\hmin}}{\binom{t}{\hmin}}, \] where $h_i=v(\cG_i),\hmin=\min_{i\in[k]}h_i$, and $n\ge t\ge\oR(\cG_1,\ldots,\cG_k)$. Let $\cS_{r,s}$ be the ordered star whose center has $r-1$ leaves to its left and $s-1$ leaves to its right, and let $\bB_m$ be the family of all ordered perfect matchings on $[2m]$ containing the edge $\{1,2m\}$. We apply the amplification inequality to obtain the multiplicity lower bounds for ordered stars and ordered perfect matchings. We then obtain the upper bound $\oM_{\boldsymbol\lambda} (n;\cS_{r_1,s_1},\cS_{r_2,s_2}) \le \min\{\lambda_1 B_{h_1}(n),\lambda_2 B_{h_2}(n)\}$ by constructions, where $B_{h_i}(n):= \binom{\lfloor n/2\rfloor}{h_i} + \binom{\lceil n/2\rceil}{h_i}$ and $h_i=r_i+s_i-1$ for $i\in [2]$. We also derive a random-coloring upper bound for ordered stars and prove \[\oM(n; \bB_m,\bB_m) \le \binom{n}{2m} \frac{(2m-2)!}{2^{2m-2}(m-1)!}.\] Finally, we establish a regularity-based lifting theorem for ordered colorings.

Creator economy platforms face a strategic dilemma: allocating traffic to established stars for immediate ad revenue versus nurturing emerging creators to build a follower base for future monetization. We develop a continuous-time dynamic optimization model to characterize the optimal traffic allocation policy for a platform managing heterogeneous creators with dual revenue streams (advertising and direct follower contributions). We characterize the optimal policy analytically, revealing a “most-valuable-creator-first" rule driven by a forward-looking activation set. Under Bass diffusion dynamics, this policy exhibits a sophisticated “conditional reversal" strategy, where the platform temporarily prioritizes lagging creators to capitalize on word-of-mouth effects. Regarding the ecosystem structure, we find the optimal policy acts as a selective gatekeeper. Unlike myopic policies that lead to a harsh “winner-take-all" market, or naive fairness-driven heuristics that foster inefficient “indiscriminate growth," the optimal policy imposes a strict capability threshold that screens which creators receive platform traffic. Furthermore, it enforces disciplined growth for successful entrants, capping their follower bases at an optimal ceiling to prevent over-investment. Finally, we demonstrate that simple heuristics can lead to significant revenue losses (up to 25%) and propose a practical “follower-growth adjusted" heuristic that achieves near-optimal performance by leveraging observed growth momentum.

We prove a new Carnot-scale $n$-step nilpotent $\pi_1$-de Rham theorem: along every convergent scale sequence, the asymptotic homotopy class of a long trajectory is identified with the nilpotent development of its macroscopic horizontal path. The key analytic mechanism is an asymptotic-development theorem showing that uniform convergence of rescaled horizontal paths, together with a uniform bound on variation, forces uniform, layer-by-layer convergence of their full Carnot-rescaled nilpotent developments. The theorem recovers Schwartzman's theory of asymptotic cycles in step one and the 2-step asymptotic homotopy theory of Benardete and Mitchell in step two, while extending the correspondence to arbitrary nilpotent step.

We study a reconstruction problem of planar domains from non-local integral-geometric invariants. We show that a generic polygonal domain, not necessarily convex, is uniquely determined, up to Euclidean isometry, by the interpoint distance distribution (IDD), which, for convex domains, is equivalent to the chord length distribution. Using a boundary representation of the Riesz energy function, we replace the IDD of the domain by an equivalent boundary IDD weighted by the scalar product of the outer unit normals, which we call $\nu$-weighted IDD of the boundary. It enables us to reduce the problem to a one-dimensional problem. By analyzing jumps and blow-up terms of the second and third derivatives of the $\nu$-weighted IDD of the boundary, we recover the side lengths, their cyclic incidence, and the exterior angles of the polygon. This can be considered as extending Waksman's classical generic reconstruction theorem for convex polygons to non-convex setting, as well as extending generic polygonal reconstruction from directional covariogram data to a one-dimensional invariant in which directional information has been integrated out.

Massive connectivity, a key building block of 5G, is expected to play an important role in the next generation of wireless systems, and its expected requirements are being revolutionized through the modeling of the information dynamics related to the vast numbers of Internet of things (IoT) devices. Motivated by this, the present paper introduces a model that captures the spatio-temporal nature of freshness of information sent via random access channel policies from an extremely large set of IoT devices via simple scalar parameters, i.e., the probability of success and accuracy of received updates. There are many information freshness metrics, starting from the age of information (AoI), all of which are proxies for the actual application performance, characterized over the temporal dimension. Our model adds the spatial dimension to this picture, observing that sensors distributed over the same area may have a strong correlation, and information from multiple close-by sensors may improve the overall accuracy of the receiver. We focus on characterizing the uncertainty of the receiver, expressed through the conditional entropy, considering a network of partially reliable, spatially distributed sensors observing the same process and reporting their measurements over a slotted ALOHA channel. We consider a simple forgetful receiver and a more complete model which accounts for the full history of past observations, deriving their performance, and optimizing the transmission probability of nodes to minimize the expected uncertainty.

We study uniform rigidity and topological mild mixing through continuous observables. For a fixed sequence of times, the observables rigid along that sequence form a closed unital $T^{\pm1}$-invariant algebra and determine the maximal factor uniformly rigid along the prescribed sequence. We then give functional forms of the classical ${\rm SIP}^{*}$- and ${\rm IP}^{*}$-return-time criteria: a topological dynamical system is mildly mixing exactly when it has no nonconstant locally SIP-rigid observable, and in the minimal category the same property is equivalent to the absence of nonconstant locally IP-rigid observables. Finally, a locally IP-rigid observable yields a canonical orbit-name factor carrying marked local data. For fixed local data, the $T^{\pm 1}$-invariant core of the local rigidity algebra determines a uniformly rigid factor whenever the core is nontrivial.

For a group $\Omega$, the associated power graph $P(\Omega)$ is defined as the graph whose vertices are the elements of $\Omega$, with two distinct vertices $u,v\in \Omega$ being adjacent if either $u=v^m$ or $v=u^n$ for some $m,n \in \mathbb{N}$. In this paper, we completely characterise the structure of the power graph associated with the class of metacyclic groups. Building on this structural description, we derive explicit expressions for the characteristic polynomials of the adjacency, Laplacian, and signless Laplacian matrices. Moreover, we obtain lower and upper bounds for the spectral radii of the adjacency and signless Laplacian matrices.

We prove that for all fixed $k\geq 4$, any $N$ vertex graph with no independent set of size $n$ and $N\geq \Omega(n^{k-1}/\log^{k-2}n)$ contains at least $$ \Omega\bigg(\binom Nk \Big(\frac{\log n}{n}\Big)^{\binom k2}/\log n\bigg) $$ cliques of order $k$, and for $k\geq 5$ this is best possible conditional on the known upper bounds for $r(k,n)$. This is also true and tight for $k=2$ by Turán's Theorem and for $k=3$ by a result of Bohman and Mubayi. We show the bound is also tight for $k=4$. We obtain other supersaturation results using the same methods.

In this manuscript, we determine the critical exponent for the three-dimensional semilinear wave equation with strong damping and thereby resolve an open problem posed in 2014. The threshold for the power nonlinearity $|u|^p$ is \begin{align*} p=p_{\mathrm{crit}}=\frac{7}{3}. \end{align*} Sufficiently small data generate global in-time solutions for $p>\frac{7}{3}$, whereas there exist arbitrarily small smooth compactly supported data whose solutions blow up in finite time for $1<p\leqslant\frac{7}{3}$. Positivity of the full velocity fundamental solution, combined with a dimension-descent formula, yields a positive half-line kernel and replaces the finite propagation property unavailable for strongly damped waves. This leads to a nonlinear lower-bound parabolic iteration without radial symmetry or pointwise sign assumptions on the initial data. At the critical power, a refined slicing argument on moving shells converts the borderline logarithmic gain into the growth required for blow-up.

In this paper, we introduce a new symmetric monoidal structure on $\mathbf{Cat}$, called the graph tensor product, with unit given by the terminal category. This tensor product falls in the middle of a factorization between the funny tensor product and the Cartesian product, giving a factorization connecting these two classical monoidal structures. We extend this construction to a symmetric monoidal structure on $2\mathbf{Cat}$, again with unit $D^0$, which provides an analogous factorization between the funny tensor product and the Cartesian product of $2$-categories. Using the $(\mathrm{bo},\mathrm{lff})$ factorization system on $2\mathbf{Cat}$, we construct a new symmetric monoidal closed model structure on $2\mathbf{Cat}$ whose tensor product restricts to the Cartesian product on the subcategory of flexible $2$-categories. Finally, we prove that this symmetric monoidal model structure fits into a square of weak symmetric monoidal Quillen equivalences relating the Gray tensor product and the flexible tensor product.

We investigate $t$-intersecting families in direct-product set systems obtained by prescribing the number of selected elements in each part of a partitioned ground set. For a finite union of layers, we prove an Erdős–Ko–Rado result under coordinatewise linear part-size conditions. As an application, we establish a new range of parameters for which a conjecture of Frankl et al.\ [J. Combin. Theory Ser. A 155 (2018), 493–502] holds. Under an explicit polynomial large-part hypothesis, we also characterize the maximum nontrivial $t$-intersecting families for the single-layer setting. In particular, for $t=1$, this answers the problem of Kwan et al.\ [J. Combin. Theory Ser. A 156 (2018), 44–60] asking for a classification of all extremal families, including the possible non-shifted maximizers.

In [Publ. Mat. 67 (2023), no. 1, 173-198], G. Navarro asked whether the character field of an odd-degree character of a finite group is generated by a single character value. We show by explicit examples that the answer is negative in general.

This note archives the status of Samart's Table-6 conjecture $n_4(81)=40M_7$, $M_7:=L'(g_7,0)$, where $g_7(\tau)=\eta(\tau)^3\eta(7\tau)^3$ is the newform of $S_3(\Gamma_0(7),\chi_{-7})$ and $n_4(s):=4m(x^4+y^4+z^4+1+s^{1/4}xyz)$. The conjecture is the cleanest of Samart's open interior-point entries (discriminant $-7$, class number $1$, a single $L$-value), and it was dropped as a theorem target in the companion paper, where the two $n_2$-family conjectures were proved. We record what is proved and precisely where those methods fail. First, a complete proof of the $L$-value side (P1): at the CM point $\tau_2=(7+\sqrt{-7})/4$ the Eisenstein–Kronecker expression underlying Samart's formula evaluates exactly to $\mathrm{EK}_4(\tau_2)=40M_7$, via lattice sums over the ring of integers of $\mathbb{Q}(\sqrt{-7})$ and the principal ideal $(\bar\varpi)$, with an exact cancellation of the parasitic $\zeta_K(2)$-terms. Second, two quantitative obstructions to the remaining half $n_4(81)=\mathrm{EK}_4(\tau_2)$: the critical image of the $n_4$-family is a two-dimensional astroid disc containing the parameter $c=3$ in its interior (in contrast to the one-dimensional slit $[0,64]$ of the $n_2$-family), so no continuation path can approach the CM point; and Samart's $U$-series converges on all of the upper half-plane but leaves the geometric sheet of the holomorphic Mahler measure everywhere below $\mathrm{Im}\,\tau=1/\sqrt{2}$, so the premise of the differential-comparison continuation fails. A 20-digit direct torus integration then decides the conjecture numerically: $n_4(81)-40M_7=+0.0586706795972872...$, five orders of magnitude above the integration error floor, so the identity as literally stated is refuted; a closed form for the true value $n_4(81)$ remains open and appears to require regulator/monodromy machinery.

We formulate the loop equations of pure $4d, N=1$ super Yang–Mills (SYM) theory in a finite geometric form. The usual equal-point loop derivatives are singular because the order of gauge-field insertions is lost when contour points coincide. Our path-ordered operator calculus (POOC) keeps the insertions in distinct ordered slots, forms the graded commutators and Jacobi combinations, and only then takes the coincidence limit. This removes the spurious kinematical singularities without introducing a cutoff; the genuine ultraviolet contact remains a separate physical distribution. We apply POOC to the exact Itoyama–Takashino superloop hierarchy and construct a Lorentzian supersymmetric Hodge-dual (SHD) surface functional. Its Hodge-resolved area derivative is annihilated by the local chiral and anti-chiral loop operators. Exact additivity under surface-geodesic sewing shows that the exponential of this zero mode multiplies any solution of the complete finite-$N$ hierarchy without changing the equations. For planar contours the SHD functional is the geometric area. In particular, for a long rectangular Wilson loop, $W[C_{T,L}]\sim\exp[-i\sigma_kLT]$ and $E_k(L)=\sigma_kL>0$. Thus the construction gives an exact nonperturbative confining area-law factor in $ N=1$ SYM. After Euclidean continuation, the same SHD factor is an exact fixed point of the deterministic zero-noise SYM gradient flow. Its interpretation as a dynamically selected equilibrium still requires stability in the long-flow-time, large-volume, and zero-noise limits, but this does not affect the exact zero-mode and finite-$N$ dressing theorems. The complete planar Wilson-loop solution, including the undressed fluctuation factor and excitation spectrum, will be developed in the next papers of this series.

Motivated by the use of modern high-capacity models in real-time control problems, this paper studies the adaptive control of high-dimensional discrete-time nonlinear systems with an unknown matrix-valued parameter. We focus on regimes where the number of unknown parameter entries is large, but the parameter matrix possesses exploitable structure, such as entrywise sparsity, group sparsity, low rank, and row-stochastic or density-matrix structure. To quantify transient performance, we consider a regret criterion relative to a nominal controller with full knowledge of the true parameter. We show that standard Euclidean update schemes, including recursive least squares and gradient descent, are ill-suited to this setting: their transient performance deteriorates as the dimension increases, even when the true parameter has low intrinsic complexity. To address this limitation, we propose a novel class of mirror-descent-type adaptive laws equipped with a non-Euclidean Polyak-type step size that exploit the geometry induced by the parameter structure. For the proposed update laws, we establish asymptotic state convergence and derive regret bounds with at most logarithmic dependence on the ambient dimension. Numerical experiments demonstrate the effectiveness of the proposed schemes.

We prove two conjectures of Samart as identities of genuine Mahler measures. The first is the case $k=1$ of his evaluations for the family $(x+x^{-1})(y+y^{-1})(z+z^{-1})+k^{1/2}$: $m((x+x^{-1})(y+y^{-1})(z+z^{-1})+1)=4L'(g_7,0)$, where $g_7(\tau)=\eta(\tau)^3\eta(7\tau)^3$ is the unique newform of $S_3(\Gamma_0(7),\chi_{-7})$ (LMFDB label 7.3.b.a). The second is the conjugate pair of quadratic CM entries of his 2015 table: $n_2((47\pm 45\sqrt{-7})/2)=(4/7)(54L'(g_7,0)+L'(\chi_{-7},-1))$, where $n_2(s):=2m((x+x^{-1})(y+y^{-1})(z+z^{-1})+\sqrt{s})$. In both cases the parameter lies inside (or on the boundary of) the critical locus, so the passage from the holomorphic (modified) Mahler measure—where Samart computed the $L$-value side conditionally, and Fei the real-part level—to the true Mahler measure was open. For the first theorem we close the gap by a differential-comparison continuation of the Mahler differential along an explicitly certified path, followed by a continuity argument at the interior point. For the second theorem no continuation is needed: the second preimage of the parameter under the modular parametrization is the Fricke partner of Samart's CM point and lies in his proved region, and the evaluation is an exact CM lattice-sum computation. The analytic and arithmetic inputs not proved here are stated explicitly with references; the finite numerical inequalities used in the continuation argument are certified by interval arithmetic, and all identities are confirmed numerically to 41–60 digits.

Distinguishing long memory behaviour from nonstationarity can be very difficult as in both cases the sample autocovariance function decays very slowly. Available stationarity tests either do not include long memory or fare poorly in terms of empirical size, especially near the boundary between long memory and nonstationarity. We propose a testing procedure based on evaluating periodograms at different epochs. Limiting distributions established here are easily tractable as sum of weighted independent $\chi^2$ random variables. Moreover, numerical studies are provided to show that the proposed approach seems to outperform existing methods.

Let $k\geq2$ be an integer. A $1\bmod k$ edge-coloring of a graph $G$ is an edge-coloring in which every nonzero degree in each color class is congruent to $1$ modulo $k$. Let $\chi'_k(G)$ denote the minimum number of colors required, and let $\chi'_k$ be the supremum of $\chi'_k(G)$ over all finite simple graphs $G$. Botler, Colucci, and Kohayakawa conjectured that there exists an absolute constant $C$ such that $\chi'_k(G)\leq k+C$ for every $k$ and every $G$. We disprove this conjecture, even within the class of bipartite graphs. More precisely, for all integers $c\geq0$ and $k\geq3c+2$, we construct a finite simple bipartite graph $G_{k,c}$ satisfying $\chi'_k(G_{k,c})=k+c+1$. Consequently, $\chi'_k\geq k+\lfloor(k+1)/3\rfloor$ for every $k\geq2$. For $k_m=2\cdot3^{m-1}$, we give an affine-hyperplane construction of a finite simple bipartite graph $G_m$ satisfying $\Delta(G_m)=\chi'_{k_m}(G_m)=3^m=3k_m/2$. More generally, for every sufficiently large $k$, we construct a finite simple bipartite graph $G_k$ such that $\Delta(G_k)=\chi'_k(G_k)\geq3k/2-10(k\log k)^{1/3}$. Our proofs combine a codegree obstruction with explicit cyclic and affine-geometric constructions and a structured random perturbation.

A group word $w$ is said to be strongly concise in a class $\mathcal{C}$ of profinite groups if, for every group $G \in \mathcal{C}$ such that $w$ takes less than $2^{\aleph_0}$ values in $G$, the verbal subgroup $w(G)$ is finite. Using the notion of polynomial mappings introduced by Passi, we establish that every group word is strongly concise in the class of profinite metabelian groups. With this new approach, we also give an alternative proof for the fact that every group word is strongly concise in the class of profinite nilpotent groups.

Let $H_k$ be an $L^2$-normalized Hecke basis for the space of all holomorphic cusp forms of weight $k$. We show that $\max_{f\in H_k}\Vert F\Vert_6\gg (\log\log k)^{\frac{1}{2}}$ where $F(z)=(\Im z)^{\frac{k}{2}}f(z).$ This confirms that the $L^6$-norm of Hecke eigenforms does not converge uniformly as the weight goes to infinity. We also give some results on the joint mass of degree $6$.

Let $X$ be a normed linear space, $Y$ a Banach space, $G\subset X$ an open set and $f:G \to Y$ a mapping. We show that the Fréchet derivative $f'$ of $f$ is of the first Baire class on the (possibly empty) set $D\subset G$ where it is defined.