We resolve the ENO--TV conjecture, a discrete coercivity problem in compactness theory for entropy-stable approximations of hyperbolic conservation laws. For order-$k$ essentially non-oscillatory (ENO) reconstruction from compactly supported cell averages, it asks whether the nonnegative ENO source times the $(k-1)$st power of the amplitude uniformly controls the $(k+1)$st absolute-jump moment. We prove a parity dichotomy: the estimate holds for odd $k\ge3$ and fails for even $k\ge4$; the known second-order case completes the classification. Localization gives a selection-independent finite-difference functional uniformly comparable to the source and reduces the conjecture to discrete interpolation. For odd orders, summation by parts reveals a hidden square; a discrete Gagliardo--Nirenberg inequality yields coercivity. For even orders, Euler-polynomial blocks from the functional's polynomial kernel yield counterexamples that persist under arbitrarily small perturbations making all affected ENO comparisons strict. We also prove two coercive estimates for every $k\ge2$: control of jumps larger than a fixed fraction of the amplitude and of local blocks modulo sampled polynomials of degree at most $k-2$. Via the Cayley--Sylvester decomposition, we compute the dimensions of homogeneous first-cohomology spaces for the lattice shift on polynomial jump profiles. At fourth order, for a cubic flux and a globally strictly convex entropy, a total-degree-seven component of a reduced entropy-flux mismatch represents a nonzero class on profiles of degree at most two and hence has no translation-invariant finite-stencil $C^7$ local primitive at the zero constant state. Odd-order coercivity persists on globally quasi-uniform meshes, whereas for each $k\ge2$ it fails on a fixed irregular mesh even though every interface contribution remains nonnegative. This failure is due to the mesh geometry.
The Schiffer conjecture states that if a smooth domain $\Omega \subset \mathbb{R}^n$ admits a Neumann eigenfunction of the Laplacian which is constant at the boundary, then the domain is a ball. It is intimately related to Pompeiu's problem, stating that if a nonzero function integrates zero over any rigid motion of $\Omega$, then $\Omega$ is a ball. We disprove both conjectures in $\mathbb{R}^2$, constructing infinitely many planar domains $\Omega$ which are not balls and satisfy the conditions above. Our domains are $N$-fold symmetric, with $N$ sufficiently large. Our approach is based on a novel strategy of considering a relaxed problem where $N$ can be any real number (which corresponds to the Schiffer problem only when $N$ is a natural number). We then apply bifurcation theory to this relaxed problem, showing that the size of the local bifurcation branch can be taken independently of $N$. This result allows us to conclude that branches starting with $N$ sufficiently close to an integer reach integer values of $N$.
Let $k\ge1$ be an integer and let $\lambda$ be the Liouville function. In 1965, Chowla gave a conjecture that the values of $\lambda(n+h_1),\dots, \lambda(n+h_k)$ are asymptotically unrelated for any distinct natural numbers $h_1, \dots, h_k$. In this article, motivated by the recent work of Bergelson and Richter on the dynamical generalizations of the prime number theorem, we will show a dynamical generalization of Chowla's conjecture on average. In the proof, we follow an approach of Qi and Zheng who established a variant of Bergelson and Richter's theorem over irreducible binary cubic forms. Moreover, we will use this approach to show an analogue of the dynamical Chowla's conjecture along the primes on average. In 2016, Tao proved that the two-point logarithmic Chowla's conjecture holds. Recently, Charamaras and Richter generalized Tao's theorem to bounded arithmetic functions and proposed a conjecture that generalizes Chowla's conjecture to bounded multi-variable arithmetic functions. In the end of this article, we will show an averaged form of this conjecture and a dynamical generalization of Tao's theorem.
We settle two conjectures of Gautschi on the degree dependence of the zeros of the Jacobi polynomials $P_n^{(\alpha,\beta)}$, $\alpha,\beta>-1$, and obtain results substantially stronger than those conjectured. The conjectures stem from a line of questions originating in spherical cubature and hyperinterpolation. A Liouville transformation and Sturm comparison yield affine comparison principles with exact thresholds for the pointwise monotonicity of the rescaled potential. We prove that an increasing affine ordering with a degree-independent shift exists if and only if $|\beta|\leq1/2$. For the spectral scale $n+(\alpha+\beta+1)/2$, we determine the exact parameter regions for the two opposite orderings and show that no uniform spectral ordering is possible outside them. We also characterise all equality cases and derive finite-degree bounds in terms of Bessel zeros. The resulting classifications are exact and cannot be enlarged: outside the stated parameter regions the corresponding uniform zero orderings necessarily fail.
We give explicit complex polynomials $P,Q$ in three independent standard real Gaussian variables such that \[ {\mathbb E}(P^m)=0,\qquad {\mathbb E}(QP^m)=m!\neq0 \] for every $m\geq1$. In natural complex linear coordinates, $P$ has five terms and total degree $4$. Hence the Gaussian Moments Conjecture is false in every dimension $n\geq3$. We also give a six-term cubic example in four variables, which was found first and already proves failure for every $n\geq4$. Both examples follow from the same coefficient identity. The search was prompted by Levent Alp\"oge's public announcement of an explicit three-dimensional counterexample to the Jacobian Conjecture. Although the main theorem of Derksen, van den Essen, and Zhao is stated globally in dimension, its proof has fixed-dimensional content: a noninvertible cubic-homogeneous Keller map in $r$ variables forces the failure of ${\mathrm GMC}(2r)$. Tracking a standard Bass--Connell--Wright reduction of the announced map gives a conservative cubic-homogeneous counterexample in $79$ variables, and hence a route-based failure of ${\mathrm GMC}(158)$. That route is nonconstructive at the final Gaussian step and does not furnish explicit polynomials $P,Q$. The much smaller explicit failures in dimensions $4$ and $3$ below were not derived from the announced Jacobian map.
The Sprott--Zeraoulia conjecture, formulated on the basis of numerical experiments, states that three-dimensional quadratic continuous-time systems with symmetric Jacobian matrices cannot exhibit chaotic behavior. We give an analytical resolution of this conjecture in terms of the principal precise notions of chaos used for continuous dynamical systems. The starting point is the classical, but decisive, fact that symmetry of the Jacobian on $\mathbb{R}^3$ is equivalent to a global gradient representation. Hence every system in the conjectured class can be written as $\dot{x}~=~\nabla V(x)$, where $V$ is a polynomial of degree at most three, and the conjecture becomes a problem about cubic polynomial gradient flows. For bounded trajectories, the monotonicity property of gradient systems first confines every omega-limit set to a critical level of $V$. When the equilibria on that level are isolated, convergence follows from the connectedness of the omega-limit set. In the general case, which may include curves or surfaces of equilibria, the \L{}ojasiewicz gradient inequality implies finite length and convergence of every bounded forward trajectory to a single equilibrium. Consequently, compact invariant sets support no nonstationary recurrence, positive-entropy measure, topological transitivity, Smale horseshoe, or Devaney, Auslander--Yorke, Li--Yorke, mean Li--Yorke, and distributional chaos. Solutions escaping to infinity are treated through the Poincar\'e compactification. Finally, we connect the gradient formulation with earlier Darboux-theoretic criteria: invariant algebraic surfaces satisfy $\langle\nabla V,\nabla f\rangle=Kf$, and an invariant affine plane with constant cofactor produces an exact tangential--normal decomposition of the potential and the flow.
Let $1\le m<n$ and let $u$ be an $m$-subharmonic function on a domain in $\mathbb{C}^n$. We study local exponential and polynomial integrability, with particular attention to the sharp polynomial exponent predicted by B{\l}ocki's conjecture. Explicit radial examples show that direct analogues of the Guan--Zhou strong openness theorem and Skoda's integrability criterion formulated in terms of the $m$-Lelong number fail when $m<n$. We classify a family of radial power-logarithmic singularities and determine the exact $L^p$-integrability range for each member, including endpoint behavior. We resolve two problems posed by Benali--Ghiloufi. The normalized limit of the ball maximum always equals the $m$-Lelong number; this follows by combining their spherical-mean formula with the strong uniqueness theorem for tangents. The pointwise integrability exponent is lower semicontinuous in the base point. However, even when restricted to $SH_m$, it is not lower semicontinuous with respect to the $L^1_{\loc}$ topology. We also disprove their polynomial openness conjecture using an explicit power-logarithmic endpoint example. Finally, we introduce a scale of local Hessian-capacity conditions, denoted by $C_{m,\delta}$. The volume-capacity inequality and the layer-cake formula yield $$u\in L^s_{loc}\quad\text{for every}\quad s<\frac{(m+\delta)n}{n-m}.$$ The critical condition $\mathrm C_{m,0}=\mathrm C_m$ holds for negative functions of finite total Hessian mass with relatively compact deep sublevel sets, and for radial functions. More generally, functions in the energy class $\mathcal E_{p,m}$ satisfy $\mathrm C_{m,p}$, recovering the full {\AA}hag--Czy{\.z} Sobolev exponent. These results provide partial progress toward B{\l}ocki's conjecture, which has remained open for more than two decades.