Papers

We present a nonabelian complex self-dual antisymmetric tensor model, with gauged internal symmetry, comprising a complex multiplet of massless self-dual antisymmetric tensor fields and its conjugate (interacting with massless Yang-Mills gauge fields). We build the model at the classical level, via a first order formulation, assigning local fields to a component expansion over the Dirac algebra and a basis of internal matrices, with Lorentz and gauge invariance ensured by tracing, and applying an appropriate bilinear form. The full expansion requires auxiliary place-holder fields, which are not potentials, but become curvatures (field strengths) after elimination via their algebraic equations of motion. The component expansion over the Dirac algebra is closed under multiplication by additional internal matrices, which are not gauge group generators, but mix internal components, and link auxiliary curvatures with potentials. The model is shown to be invariant under such transformations, and thus possesses a type of hidden internal symmetry. We analyze the specific case of gauge group $SU(2)\! \times \! U(1)$, and as higher symmetry candidates, we consider both the Lie group $SU(3)$ and the Lie superalgebra $SU(2/1)$ (with the additional matrices acting via Lie brackets, or graded Lie brackets, respectively).

We revisit the problem of approximating a linear time-invariant LTI model by a reduced-order model (ROM) in an $H_2$-optimal sense. A widely used method for computing such ROMs is the iterative rational Krylov algorithm (IRKA). However, in practice, the stability of the resulting ROM is not generally guaranteed. In this work, we develop a novel framework that can be used to obtain $H_2$-optimal models of single-input single-output LTI systems while enforcing stability of the ROM. We use Malmquist-Takenaka orthogonal expansions in the corresponding $H_2$-spaces and their connection to hyperbolic geometry to develop an equilibrium function. We show that the poles of an $H_2$-optimal ROM transfer function are defined by the zeros of this equilibrium function. The resulting nonlinear root-finding problem is solved using a Newton-type method formulated on the products of Poincaré disk manifolds, where updates are performed along hyperbolic geodesics ensuring that the stability constraint is satisfied at every iteration. We present numerical experiments on benchmark systems for both discrete- and continuous-time LTI MOR problems, including examples where IRKA produces unstable reduced models while the proposed method preserves stability throughout the iteration.

We study sampling from $\pi(x)\propto\mathrm{e}^{-V(x)}$ on $\mathbb{R}^d$, where $V$ is $\alpha$-strongly convex and $\beta$-smooth with condition number $\kappa=\beta/\alpha$. For this problem, the proximal sampler preserves the target exactly and recovers overdamped Langevin dynamics in the small-step limit, but its Gaussian resampling step discards directional information. Underdamped Langevin dynamics retains such information through momentum to accelerate exploration, motivating a proximal construction that combines target preservation with directional persistence. We introduce reflected proximal dynamics, which combines proximal sampling with an exact reflection of the auxiliary variable, conditional kinetic motion, and controlled refreshment. We develop two realizations: a flux-resampled underdamped half-turn and a persistent-momentum quarter-turn. From a warm start with constant $\chi^2$ divergence, both methods achieve total variation error $\varepsilon$ using $\widetilde O(\sqrt{\kappa}d^{1/4})$ expected oracle queries. We further remove the supplied warm-start assumption by combining either sampler with existing posterior samplers in a proximal initialization stage with an optimized coarse step size. Accounting for both initialization and sampling yields $\widetilde O(\kappa^{6/7}d^{1/4})$ expected oracle queries. In both regimes, the dependence on the target accuracy $\varepsilon$ is $\mathrm{polylog}(1/\varepsilon)$, making reflected proximal dynamics a promising framework for accelerated high-accuracy log-concave sampling.

The object of interest is the maximum number, $h(n)$, of Hamilton cycles in an $n$-tournament. By considering the expected number of Hamilton cycles in various classes of random tournaments, we obtain new asymptotic lower bounds on $h(n)$. The best result so far is approximately $ 2.85584...$ times the expected number $g(n)$ of Hamilton cycles in a random $n$-tournament, and it is conjectured that $h(n)\sim cg(n)$ where $c\approx 2.855958$. The same statements hold for Hamilton paths.

We give negative answers to two questions of Farjoun about localizations of groups. For every odd prime $p$, we construct a finite $p$-group $G$, an infinite non-nilpotent group $H$, and a local embedding $f\colon G\hookrightarrow H$. The associated localization satisfies $L_f(G)\cong H$, showing that localizations need not preserve nilpotence and that the localization map $G\to L(G)$ need not be surjective when $G$ is a finite $p$-group.

The Recursive Projection Aggregation (RPA) decoder for Reed-Muller codes uses, in its default implementation, all one-dimensional subspaces at each level of recursion. We restrict our attention to the case where the number of errors is $t<d_{\mathrm{min}}/2$. It is known that if the number of errors is $t<d_{\mathrm{min}}/2$, then $2t+1$ subspaces at each level of recursion are sufficient to correct these errors. We show that, for structured error patterns, the number of subspaces required for correction can be reduced further by appropriately selecting the subspaces used for projection. In particular, when errors occur as a single burst of length $t < d_{\mathrm{min}}/2$, three subspaces at each level of recursion suffice for correction. We obtain similar results for multiple bursts and other structured error patterns as well.

In this paper, we study perfect codes in bi-Cayley graphs, focusing on when two subgroups $H$ and $K$ of a group G can form a perfect code of the form $H_0 \cup K_1$. We first consider the problem for a general group G and then examine what happens for special classes of groups, particularly abelian groups. Along the way, we obtain several results, including a necessary and sufficient condition for normal subgroups to form a perfect code, as well as an equivalent condition for abelian groups in terms of their Sylow 2-subgroups. Finally, we study the same problem for dihedral groups, providing results for this important class of non-abelian groups.

Understanding the relationship between generalization and hallucination requires clarifying the role of memorization. We study this relationship from an information-theoretic perspective through a Selector-Density Conditional Mutual Information (CMI) framework. In the sparse random-fact limit, a uniformly bounded information budget per fact forces a nonvanishing false-positive rate at recall bounded away from zero, even when all facts are observed during training. In a rules-and-facts setting, we decompose model-data mutual information into rule learning, atomic-fact memory, and excess memorization, revealing competition between rule learning and factual memory even without excess memorization. A joint budget bound quantifies the resulting constraint on atomic hallucination under perfect atomic recall. Moreover, an additive Gaussian model shows that maintaining high recall for atomic and composite facts incurs false positives even under optimal data mixing. Finally, controlled relational experiments hint how training exposure, rule complexity, and model capacity shape these trade-offs.

This work studies the relationship between the multigrading in the cohomology of moment-angle complexes, introduced by V. Buchstaber and T. Panov in their computation of these cohomology groups, and the multigrading in the cohomology of the corresponding complements of coordinate subspace arrangements, arising from the mixed Hodge structure. It is shown that these multigradings coincide. This result refines a theorem from Yu. Eliyashev's 2013 dissertation. Furthermore, results on the mixed Hodge structure in the cohomology of complements of coordinate subspace arrangements are applied to the study of the same varieties over finite fields. In particular, various formulas are obtained for the number of $\mathbb{F}_q$-rational points and for the associated global $\zeta$-functions.

We study the incompressible Oldroyd-B system on $\mathbb R^d$, $d\ge2$, in the regime in which the velocity is the only dissipated quantity: the stress tensor undergoes neither relaxation nor diffusion. In this regime the operator $\mathbb P\operatorname{div}$ acting on symmetric stresses has an infinite-dimensional kernel consisting of exact stationary states, so no decay estimate can hold for the full stress tensor and the existing sharp-decay theory for dissipative systems does not apply. We identify the active pair $U=(u,\Lambda^{-1}\mathbb P\operatorname{div}\tau)$ as the correct decaying observable and establish, in a critical hybrid Besov framework, a complete characterization of its large-time behavior. For $-d/2<\sigma_1<\sigma\le d/2-1$ we prove that $\|U(t)\|_{\mathcal H^{\sigma}}=O(t^{-(\sigma-\sigma_1)/2})$ holds if and only if the low-frequency part of $U_0$ belongs to $\dot B^{\sigma_1}_{2,\infty}$, and, for $\sigma<d/2-1$, that the two-sided estimate $\|U(t)\|_{\mathcal H^{\sigma}}\simeq t^{-(\sigma-\sigma_1)/2}$ holds if and only if the low-frequency part of $U_0$ belongs to a nondegenerate subclass $\mathfrak B^{\sigma_1}_{2,\infty}$ of $\dot B^{\sigma_1}_{2,\infty}$.

We establish that Rudin's diagonal restriction operator is contractive from the Hardy space $H^p$ of the bidisc to the Bergman space $A^p$ of the disc for $2\leq p<\infty$, and characterize all cases of equality. We extend the result to polydiscs and obtain applications to polynomial point evaluation among others. The proof relies on an explicit formula, deduced from certain Green's formulas, for the defect between the $p$th powers of the two $L^p$ norms.

Let $p\geq11$ be a prime, and let $\bar\rho:\Gal(\bar\QQ_p/\QQ_p)\rightarrow\GL_2(\bar\FF_p)$ be a reducible and very generic representation. In this paper, we use ghost series \cite{Lghost2} to study the irreducible components of Kisin's potentially crystalline deformation rings of $\bar\rho$. For wildly ramified principal series inertial types and arbitrary regular Hodge–Tate weights, we distinguish the components by the crystabelline slopes on their generic fibers and determine the total number of irreducible components. Combining this with an explicit formula for the total Hilbert–Samuel multiplicity of the special fiber obtained from the Breuil–Mézard conjecture, we compute the Hilbert–Samuel multiplicity of the special fiber of each component. Every such multiplicity is one when $\bar\rho$ is non-split, and is either one or two when $\bar\rho$ is split, according to whether the generic fiber of the corresponding component parametrizes reducible or irreducible Galois representations.

We study the exponential map on the group of real two-by-two matrices with determinant one, endowed with a left-invariant Riemannian metric. Using the Euler-Poincaré equation and its linearization, we determine the conjugate and Maxwell times and identify a domain on which the exponential map is a global diffeomorphism onto the complement of the Maxwell stratum. This provides an alternative proof that the first Maxwell time coincides with the cut time. We obtain an explicit matrix description of the cut locus and characterize the globally minimizing geodesics. Furthermore, we analyze the first Maxwell-time function, determine its unique global minimum, and show that the injectivity radius is $\frac{\pi \sqrt{14}}{4}$. Finally, we characterize the intersection of the cut and conjugate loci and prove that every geodesic is equi-optimal.

We establish a family of sharp higher order isoperimetric type inequalities for smooth compact surfaces in Euclidean spaces of arbitrary codimension. These inequalities involve arbitrarily high derivatives of the boundary curve, together with the mean curvature vector of the surface. For example, the first and second order inequalities yield the sharp two-sided estimate for the isoperimetric deficit: \begin{align*} & \frac{\pi}{R}\int_{\partial M}|x-R\nu|^2\, ds +2\pi\int_M x\cdot H\, dA \le L^2-4\pi A\\ \le& \frac{\pi}{R}\int_{\partial M}|x-R\nu|^2\, ds +2\pi\int_M x\cdot H\, dA +\frac{\pi}{3R}\int_{\partial M}|x+R^2\kappa|^2\, ds, \end{align*} where $R=\frac{L}{2\pi}$ and the origin is chosen to be the centroid of the boundary. For minimal surfaces, equality in the lower bound holds precisely for flat disks, whereas the second inequality admits non-round equality cases. Our results extend that of Carleman, Sachs, Chakerian, and Li-Schoen-Yau. A distinctive feature of our work is the unexpected rigidity structure arising at higher orders. At the first and second orders, equality curves admit decompositions into circular Fourier modes supported in mutually orthogonal planes. Remarkably, this phenomenon fails at higher orders. We establish a multiplicative factorization of the unit tangent map into elementary planar rotations acting on a fixed unit vector, providing a characterization of equality cases at arbitrary order. As a result, we construct a third order equality curve in $\mathbb R^3$ whose Fourier modes cannot be decomposed into mutually orthogonal circular modes.

We study the limiting distribution of the largest real part of the eigenvalues of truncated unitary matrices. Let \(U_m\) be an \(m\times m\) Haar-distributed unitary matrix, and let \(Z_1,\ldots,Z_n\) be the eigenvalues of its upper-left \(n\times n\) principal submatrix. We determine the limiting distribution of \(\max_{1\le i\le n}\Re Z_i\) under a unified normalization: the limiting law is Gumbel when $q:=m-n\to\infty$ and is of Weibull type when $q$ is fixed. The fixed-\(q\) limiting laws converge to the Gumbel law as \(q\to\infty\). We also give a unified formulation of the normalizing constants for the spectral radius that recovers the previously known limiting laws.

We determine the representation type and Schurian finiteness of the Tits algebra of a finite central real hyperplane arrangement over an algebraically closed field. The algebra is representation-finite when the arrangement has at most two hyperplanes, tame when it has three, and wild when it has at least four. Moreover, it is Schurian-finite if and only if the arrangement has at most two hyperplanes. Both classifications hold over algebraically closed fields of arbitrary characteristic. The proof combines Saliola's bound-quiver presentation with explicit module constructions, convex subquivers, essentialization, and deletion. In the $\mathbb{R}^2$ case, we describe all eleven indecomposable modules for two lines and construct a full exact embedding of modules over a free algebra for at least four lines.

We study two related questions for nonlocal equations of Bellman type: the regularity of solutions when the set of controls depends on the state variable, and the periodic homogenization of stationary nonlocal Hamilton-Jacobi equations driven by uniformly elliptic integro-differential operators of order $\sigma\in(0,2)$, whose kernels depend simultaneously on the slow variable, the fast periodic variable and the control. The effective equation is again a nonlocal Bellman equation, but its supremum runs over a set of viscosity Mather measures that varies with the space variable, in a merely Hölder continuous way. We first develop a $C^{\sigma+\alpha}$ regularity theory for nonlocal Bellman equations with Hölder continuous, state-dependent index sets, extending the Schauder estimates of Caffarelli-Silvestre and Serra, in a general framework independent of homogenization. We then use these estimates as a source of compactness to construct a classical solution of the effective problem, unique among bounded classical solutions. Finally, combining this regularity with discounted approximate correctors, we make the perturbed test function method quantitative, and prove that the whole family of solutions converges to the effective solution, with the explicit rate $\epsilon^{\sigma\alpha/(2+\alpha)}$, where $\alpha$ is the Hölder exponent of the data.

Let $P$ be a short-range projector on $\ell^2(\mathbb{Z}^{2n},\mathbb{C}^m)$ and $U=(U_1,\dots,U_{2n})$ be a transverse family of open subsets of $\mathbb{R}^{2n}$. We give an elementary proof of a recent result of Martinian-Stoiber, which extends a two-dimensional result of Drouot-Zhu to all even dimensions. Specifically, using only Stokes' theorem and the Poincaré lemma, we show that the geometric Kubo formula $\operatorname{Tr}\big(P[P,\mathbf{1}_{U_1}]\wedge\cdots\wedge[P,\mathbf{1}_{U_{2n}}]\big)$ is the product of a topological integer $\mathcal{X}_U$, which depends only on the relative positions of $U_1, \dots, U_{2n}$, and of the bulk conductance of $P$. Our proof writes the geometric bulk conductance as the pairing of $P$ with a cochain on $\mathbb{Z}^{2n}$ - an idea originally due to Kitaev - and deforms the cochain rather than the sets $(U_1, \dots, U_{2n})$. The proof presented in this paper is due to Claude Opus 5.5.

We show that the Stein degree of horizontal divisors on generalised log Calabi-Yau fibrations or certain vertical divisors of Fano type fibrations over arbitrary fields of characteristic zero is bounded by constants that are independent on the ground field.

We resolve a conjecture of Łuczak, Mieczkowska and Šileikis on the maximal tail probability of sums of independent and identically distributed nonnegative random variables. Specifically, for every integer $k\ge1$ and every $0\le x\le1/k$, we prove \[ \sup_{\mathbb E X_1\le x}\mathbb{P}(X_1+\cdots+X_k\ge1) =\max\{1-(1-x)^k,(kx)^k\}, \] where the supremum is over nonnegative i.i.d.\ random variables $X_1,\ldots,X_k$. We also characterize all maximizing distributions. Using this result, we prove the Erdős Matching Conjecture for every fixed $k\ge3$, all sufficiently large $s$, and every $n\ge k(s+1)$. More specifically, we show that in this range, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $s+1$ pairwise disjoint edges is \[ \max\left\{ \binom{k(s+1)-1}{k}, \binom nk-\binom{n-s}{k} \right\}. \] Moreover, every extremal family consists either of all $k$-sets meeting a fixed $s$-set or of all $k$-sets contained in a fixed $(k(s+1)-1)$-set.

A compact metric tree is the natural domain of spatial data recorded along a river network, a root system, or the branched processes of a cell, and a suitable distance between two points is the length of the path joining them. That distance admits almost none of the covariance models of spatial statistics, since only the roughest members of each family stay nonnegative definite, and the resulting fields are nowhere mean square differentiable. The repair proposed here is a change of metric. Comparing two points through their distances to the leaves gives a Hausdorff-type metric built from the shortest-path one. The new metric has a closed form, agrees with the shortest-path one inside every edge, and is strictly smaller across branching points. Any correlation function valid in a Euclidean space of dimension equal to the number of leaves may then be composed with it, returning the Matérn, powered exponential, and generalized Cauchy families at every degree of smoothness, and, on trees with few leaves, also correlations that change sign or have compact support. The same construction yields the multivariate Matérn and space-time models of the Gneiting class. In simulations covering a range of smoothness and of spatial scale, maximum likelihood provides estimates without appreciable bias, and on a reconstructed microglial cell the smooth models reduce the cross-validated prediction error by half.

Existing cumulative regret guarantees for Bayesian optimization (BO) do not necessarily characterize the quality of the algorithm's final output. This gap is critical under noisy evaluations, where the goal is to return a single high-quality recommendation. To address it, we introduce efficient exploration strategies that can be incorporated into any BO procedure by occasionally replacing the acquisition query with an exploration query. To validate this approach, we bound the Gaussian process posterior standard deviation over the search space via the fill distance of the queries, which measures the largest region left unexplored. We then use this result to derive upper bounds on the inference regret, i.e., the gap between the global optimum and the function value at the algorithm's returned point. Empirically, these exploration strategies improve the final recommendations of existing BO algorithms on synthetic benchmarks, hyperparameter optimization, and a control task.

We study the price of anarchy (PoA) in nonatomic routing games as a function of the full vector of origin–destination demands. We show that the Beckmann value function is continuously differentiable on the positive demand cone, with gradient equal to the equilibrium origin–destination cost vector; hence the PoA is continuously differentiable on the regular demand region. We prove that the nondifferentiability set has Lebesgue measure zero and, when every edge cost is strictly increasing or constant, is covered by countably many codimension-one submanifolds. Under the same cost structure, all equilibrium objects are cellwise as smooth as the edge costs. For affine costs, the exceptional sets become finite unions of lower-dimensional polyhedral cells linked by an exact scaling law, and the maximum of the PoA over a cell closure is attained on the cell boundary, generalizing the one-dimensional endpoint result. The theory is illustrated on the Sioux Falls benchmark network.

The spectral form factor (SFF), which is the Fourier transform of the two-point spectral correlation, is a fundamental diagnostics of quantum chaos. Nevertheless, its experimental detection in quantum many-body systems was less unexplored until recently. Here, we establish a relationship between the SFF with the quantum Fisher information (QFI) by means of quantum sensing protocols. Furthermore, when the eigenvalues and eigenstates of the disordered Hamiltonian are statistically independent, we prove that QFI is typical both in short and long timescales while numerics suggest that it is also typical at intermediate timescales, which ensures self-averaging of QFI. The self-averaging property renders the experimental measurement of QFI with our protocol less noisy in contrast with the direct measurement of SFF which is known to be non-self-averaging. Our results reveal an intriguing connection between quantum noise and the signatures of quantum chaos.

We prove a regularity lemma for polynomials with fixed degree $d$ and any number of standard Gaussian input variables. Following modern approaches to the Szemerédi regularity lemma, we deduce our results from a general formulation of the regularity lemma using ultraproducts. Along the way, we give a direct combinatorial proof of the celebrated “fourth moment theorem” of Nualart and Peccati, which characterizes the asymptotically Gaussian polynomials inside degree-$d$ Gaussian chaos as those such that $\mathbb{E} p^2 \to 1$ and $\mathbb{E} p^4 \to 3$.

In this paper, introduce a new finiteness condition for commutative rings, called $Z$-Noetherian, under which every ideal contained in the set of zero-divisors $Z(R)$ is finitely generated. This condition lies properly between Noetherian and $Nil_{*}$-Noetherian. We establish intrinsic characterizations of $Z$-Noetherian rings and investigate their stability under nilpotent quotients and localizations. Under the assumption that $Z(R)$ is an ideal, we obtain a primary decomposition theorem for ideals contained in $Z(R)$. For $ZD$-rings, we prove a Cohen-type criterion characterizing $Z$-Noetherian in terms of the finite generation of prime ideals contained in $Z(R)$. We further establish $Z$-Noetherian analogues of the Hilbert Basis Theorem and Krull's Intersection Theorem. Finally, we establish a principal ideal theorem for $Z$-Noetherian rings. Collectively, these results show that several fundamental finiteness and structural properties of Noetherian rings extend to the broader class of $Z$-Noetherian rings.

On a complete noncompact Kähler manifold $M^n$ of complex dimension $n$ with nonnegative Ricci curvature, we prove that polynomial growth holomorphic functions of degree $d$ ($d\geq 0$) have an dimension upper bound $C(n)(1+d)^n$. In particular, $\dim_{\C}\OO_d(M)\leq C'(n)d^n,$ for $d\geq1$, and the estimate is sharp.

Let $\mu_q$ be the distribution of the normalized Kloosterman sums $\mathrm{Kl}(a;q)$, $a\in\mathbb{F}_q^\times$, and let $\mu_{\mathrm{ST}}$ be the Sato–Tate law. We prove $W_s(\mu_q,\mu_{\mathrm{ST}})\ll q^{-1/3}$ for $1\le s\le2$, $W_s(\mu_q,\mu_{\mathrm{ST}})\ll q^{-1/6-1/(3s)}$ for $2\le s<\infty$, and $W_\infty(\mu_q,\mu_{\mathrm{ST}})\ll q^{-1/6}$, answering a question of Kowalski and Untrau. More generally, we bound $W_s(\mu,\mu_{\mathrm{ST}})$ for all $s$ whenever the Chebyshev moments of $\mu$ satisfy bounds of Katz type. For $s=2$ the bound also follows from a smoothing inequality of Borda and Cuenin; for $s>2$ and $s=\infty$ our bounds are stronger. The proof uses the quantile coupling, which arises from Stieltjes sums over two different intervals whose cells are paired so as to balance two measures, and which is optimal for all $s$ simultaneously; an explicit $W_2$ smoothing inequality relative to $\mu_{\mathrm{ST}}$; and a sharp comparison between $W_s$ and the Kolmogorov distance for laws with square-root edges. The last result shows that $q^{-1/6}$ is the best rate for $W_\infty$ obtainable from Niederreiter's discrepancy bound. Numerically, for primes $10^3<p<10^6$, all these distances appear to decay like $p^{-1/2}$.

A first-order theory $T$ has the finite model property if every first-order sentence that is implied by $T$ has a finite model. We give a characterization of when a complete $\omega$-categorical theory $T$ has the finite model property. The characterization makes it possible to transfer stability theoretic results in [29] to certain classes of finite structures. This approach also gives information concerning properties that a sequence of finite “base structures” of ever larger size must have in order for its random expansions to a larger vocabulary to have a chance to satisfy a convergence law for first-order logic.

We construct a counterexample to integral Aleksandrov-Bakelman-Pucci estimates for nonlocal inequalities of the form \begin{equation*} L_au(x):=\int_{\mathbb R^n}\delta_yu(x) \frac{a(x,y)}{|y|^{n+\sigma}}\,dy \ge-f(x)\quad\text{in }B_1, \end{equation*} with $u\le0$ in $\mathbb R^n\setminus B_1$, where $\delta_yu(x)=u(x+y)+u(x-y)-2u(x)$, and the measurable coefficient satisfies \begin{equation}\tag{C} \lambda\le a(x,y)\le\Lambda,\qquad a(x,y)=a(x,-y),\qquad x\in\mathbb R^n,\quad y\ne0. \end{equation} More precisely, for every $n\ge1$, $0<\sigma<2$, and $0<\lambda<\Lambda$, there exist a fixed coefficient $a(x,y)$ satisfying (C), a constant $d>0$, and nonnegative functions $u_j,f_j\in C_c^\infty(B_1)$ such that $L_au_j\ge-f_j$ in $B_1$, $0\le f_j\le1$, and $|\operatorname{supp}f_j|\to0$ as $j\to\infty$, while $u_j\ge d$ on $B_{1/4}$ for every $j$. Consequently, this fixed operator admits no $L^p$ maximum estimate for any $1\le p<\infty$.