Papers

We study the long-wave spectral instability of the two-dimensional compressible Euler equations around smooth monotone shear layers. We construct decaying half-line solutions of the compressible Rayleigh equation through a long-wave expansion and derive a second-order expansion of the matching Wronskian. For every fixed Mach number $m>0$, we prove the existence of unstable modes for sufficiently small wavenumbers. For $m<\sqrt2$, this holds for a class of profiles, with $c_i$ tending to a positive constant as $\alpha\to0$. At $m=\sqrt{2}$, the profile $U_s(Y)=\tanh Y$ admits an unstable mode with $c\to0$ and $c_i$ of order $\alpha^{1/3}$. For $m>\sqrt2$, the same profile remains unstable, with $c\to c_*(m)\in(0,1)$ and $c_i>0$ of order $\alpha$. The corresponding temporal growth rates are of order $\alpha$, $\alpha^{4/3}$ and $\alpha^2$, respectively, showing a change in the long-wave instability scaling at $m=\sqrt2$. In the zero-thickness limit, the supercritical unstable eigenvalue approaches the real axis, consistently with the stability results for supersonic compressible vortex sheets in \cite{CS1,CS2}.

The frequency $K_i$s ($i\in[4,n]$) are studied for symmetric traveling salesman problem ($TSP$) to characterize the structure properties of the edges inside and outside the optimal Hamiltonian cycle ($OHC$). Given a $K_i$ in $K_n$ where $i\in [4,n]$, the frequency $K_i$ is computed with the set of ${{i}\choose{2}}$ optimal $i$-vertex paths with fixed endpoints (optimal $i$-vertex paths) in the $K_i$. Given an $OHC$ edge in a $K_i$, it has a frequency bigger than $\frac{1}{2}{{i}\choose{2}}$ in the frequency $K_i$, and that of an ordinary edge outside the $OHC$ is smaller than $\frac{1}{2}{{i}\choose{2}}$. As the frequency of an edge is computed with the frequency $K_i$s, an $OHC$ edge of $K_n$ has an average frequency bigger than $\frac{1}{2}{{i}\choose{2}}$. It indicates an $OHC$ edge of $K_n$ is also one $OHC$ edge of a $K_i$ containing it. It also found that the probability that an $OHC$ edge has the frequency bigger than $\frac{1}{2}{{i}\choose{2}}$ increases according to $i\in [4, n]$ based on the frequency $K_i$s. For an ordinary edge outside the $OHC$, the probability that it has a frequency smaller than $\frac{1}{2}{{i}\choose{2}}$ increases according to $i$. Based on the findings, a method is given to identify the ordinary edges for $TSP$.

Using different approaches, Simon Donaldson and Dashen Yan recently constructed $\mathbb Z_2$-harmonic functions on $\mathbb R^3$ and on $\mathbb R^n$ for $n\geqslant 3$, respectively. Their examples have quadratic growth at infinity and $\mathcal{O}(r^{3/2})$ local growth near a smooth codimension-two branching set $\Sigma$. We show that these properties uniquely characterise such $\mathbb Z_2$-harmonic functions up to rigid motions of $\mathbb R^n$.

Motivated by the conjectured monotonicity of the magnitude functions of metric spaces of negative type, we investigate the monotonicity problem for finite semimetric spaces, namely, “metric spaces without the triangle inequality”. We completely resolve this problem by proving that the monotonicity holds for negative type semimetric spaces with at most three points, while constructing negative type semimetric spaces with at least four points whose magnitude functions are not monotonically increasing. We also prove monotonicity for all 4-point metric spaces and construct 12-point metric spaces of negative type whose magnitude functions are not monotonically increasing.

This paper presents a numerical finite volume method for conservation law systems that are adapted to the principle of maximal dissipation. The general assumptions are the existence of a strictly convex entropy functional and the finite propagation speed property of a given system. The procedure is based on a numerical flux construction obtained by the minimization of the entropy functional in each time step. The scheme satisfies assumptions of the Lax-Wendroff theorem. A limiting solution obtained by this scheme is compared with the classical weak solutions obtained by the Glimm or the Wave Front Tracking algorithm for one-dimensional systems.

Let $S_g$ be a closed oriented surface of genus $g\ge2$. For a reductive representation $\rho:\pi_1(S_g)\to\PU(2,1)$, let $E_\rho$ be the energy function on Teichmüller space associated to equivariant harmonic maps into $\CH^2$. For every positive integer $d$ with $3\nmid d$, all sufficiently large $h$, and every $g>h$, we construct an irreducible reductive representation \[ \rho_{g,h,d}:\pi_1(S_g)\to\PU(2,1) \] with \[ \tau(\rho_{g,h,d})=2h-2-\frac{2d}{3}\notin\mathbb Z, \qquad \operatorname{Crit}(E_{\rho_{g,h,d}})=\varnothing. \] Consequently, the associated branched-minimal-surface forgetful map is not surjective in these nonintegral Toledo components.

Weak shock reflection at nearly glancing incidence is governed, after the transonic weak-shock scaling, by a self-similar unsteady transonic small-disturbance (UTSD) free-boundary problem. We derive and differentiate the first finite-strength perturbation of the corresponding isentropic potential-flow problem along paths of fixed canonical incidence $a=\alpha/\delta$, where $\mu=\delta^2=2(M^2-1)$. A second-order refluxed adaptive finite-volume method, exact discrete tangents and adjoints, and a Rankine–Hugoniot-constrained fitted principal front give the fixed-$a$ canonical sensitivity $H_{2,a}^{\PF}(0.5;1.4)=-0.217\pm0.012$; a fully differentiated physical back-map gives the diagnostic fixed-$a$ coefficient $K_{2,a}^{\PF}\simeq-0.266$. We derive the exact chain rule that converts these quantities to the distinguished fixed-$\lambda$ path $\lambda=(M-1)/\alpha^2$, showing explicitly that the conversion requires the independent incidence derivative of the leading UTSD branch and therefore cannot be inferred from the fixed-$a$ calculation alone. A matched-boundary self-similar Euler study with strength-dependent refinement contains 21 qualified nonlinear states and 252 evaluations of a common front-functional family. Coupled extrapolation gives $g_0^{\Eul}=0.510\pm0.006$ and the physical leading-angle coefficient $G_0^{\Eul}=0.256\pm0.004$, consistent with the shock-fitted UTSD limits. A separate same-strength phase-bracket audit using 18 Euler states shows that the captured-shock subcell phase is comparable to the desired cubic signal. The resulting finite-resolution Euler secants are compatible with the potential-flow correction, but their $\rho=h_\eta/\sqrt\mu\to0$ extrapolation is not model-stable. Thus leading-order Euler–UTSD correspondence is numerically verified, whereas cubic-order Euler correspondence remains unresolved.

Arboricity and degeneracy are two fundamental and closely related graph parameters that measure the sparsity of a graph. Every $k$-degenerate graph is $k$-arboric, but some $k$-arboric graphs are only $(2k-1)$-degenerate. However, every maximal $k$-arboric multigraph with $n$ vertices and every maximal $k$-degenerate multigraph with $n$ vertices has exactly $k(n-1)$ edges. These basic observations lead to a natural structural question: How far are $k$-arboric graphs from being $k$-degenerate? We answer this question by showing that: By at most a $(k-1)$-bounded-degree graph apart. More specifically, we prove that a $k$-arboric multigraph admits a $(k,k-1)$-decomposition, that is, its edges can be partitioned into two multisets such that one spans a $k$-degenerate multigraph and the other spans a multigraph with every vertex having degree at most $k-1$. Moreover, we provide a complete characterisation of all possible such decomposition types. Namely, for any integers $k \ge 1$ and $d,h \ge 0$ we show that every $k$-arboric multigraph admits a $(d,h)$-decomposition if and only if $d\geq k$ and $d+h\geq 2k-1$. Our proofs are constructive and we present a polynomial time algorithm that produces such decompositions. By contrast, we show that related decision problems for general graphs (without constraints on the arboricity) are NP-complete.

Usually, radius problems between Ma–Minda classes are solved one pair at a time, by fitting a disk inside the target region, providing a sufficient but not always the sharp radius. The same class appearing under different names makes comparison of results harder, with mistakes and duplication eluding detection. The Geometric Function Atlas catalogs radius problems using exact generators and a direction (source $\to$ target), ensuring unique identification. For each problem, we then compute the radius numerically, together with the point where the image first meets the target boundary. This helps us identify the sharp estimate which guides the proof we write. With this system, we were able to refine two earlier stated estimates, the disk radius for the sine $\to$ modified sigmoid and sine $\to$ rational cases from $\operatorname{arsinh} c$ to $\arcsin c$, and correct earlier stated values for the crescent and exponential classes: the sharp radius is $\sin 1$ from crescent to exponential, and $\log(1+\sqrt2)$ from exponential to crescent. All four proofs follow a common strategy: an estimate that is exact at the contact point, followed by the standard extremal function. Applying the method, we derive fifteen further sharp radii, four of them known.

Reconstruction problems lie at the very heart of both mathematics and science, posing the fundamental challenge: How does one reconstruct a hidden structure from incomplete, fragmented, or distorted data? In this paper, we introduce a new approach that harnesses the insights of the Vaisman Atiyah–Molino framework. In contrast to conventional methods that depend on persistent homology, our approach exploits the concept of the Vaisman centroid—an intrinsic invariant that encapsulates the averaged geometry of a data set—to resolve the inherent ambiguities of inverse problems. In the present paper, we focus on the theory and applications of the Vaisman centroid, offering a new perspective for Topological Data Analysis that eschews persistent homology in favour of a unified geometric paradigm. A subsequent paper will extend these ideas to a complete reconstruction scheme via the Atiyah–Molino framework. Our method provides a robust and computationally tractable framework for the recovery of hidden structures while opening new avenues for the analysis of high-dimensional and noisy data across the mathematical sciences.

Let $d\geq 1$. Let $K\subset\mathbb{R}^d$ be a non-singleton self-similar set generated by a finite strongly irreducible iterated function system satisfying the open set condition, and let $\delta=\dim_{\mathrm H} K$. For $\tau>1/d$, set \[ W_d(\tau) = \left\{ \mathbf{x}\in\mathbb{R}^d: |q\mathbf{x}-\mathbf{p}|<q^{-\tau} \text{ for infinitely many }(\mathbf{p},q)\in\mathbb{Z}^d\times\mathbb{N} \right\}. \] We prove that there exists $\varepsilon_K>0$ such that, for every $1/d<\tau<1/d+\varepsilon_K$, \[ \mathcal{H}^{s(\tau)}(K\cap W_d(\tau))=\infty, \qquad\text{with } s(\tau):=\delta+\frac{d+1}{1+\tau}-d, \] and consequently \[ \dim_{\mathrm H}(K\cap W_d(\tau)) = \delta+\frac{d+1}{1+\tau}-d. \] In dimension one, specializing to the middle-third Cantor set, this establishes the Bugeaud–Durand conjectural formula for $\tau>1$ sufficiently close to $1$.

We study the dimension and dimensional stability of polynomial spline spaces of bi-degree $(m,m')$ with prescribed smoothness orders over planar T-meshes. The homological dimension formula writes the spline dimension as the sum of an Euler characteristic term and a correction term. For a fixed ordered bi-degree, the Euler characteristic term is determined by the mesh structure and the prescribed smoothness orders. The correction term can be written as a quotient of coefficient spaces attached to maximal interior segments (MISs). We prove a weighted deletion theorem: when the available vertex relations generate the coefficient space of an MIS, its summand can be removed from the quotient without changing the correction term. This operation changes neither the T-mesh nor its chain complexes. Repeating the deletion leaves a weighted completely non-diagonalizable component (CNDC). We prove that the weighted CNDC is independent of the order in which eligible MISs are removed and is the same for corresponding pairs in the structural class. The correction term can therefore be represented using only the MISs in the weighted CNDC, while all relations from the original T-mesh are retained. Dimensional stability is then equivalent to constancy of the dimension of the remaining relation space. In particular, an empty weighted CNDC is sufficient for stability. We also derive an upper bound for the remaining correction term and compare it with Mourrain's upper bound based on all MISs.

Deligeorgaki, Han and Solus introduced colored multiset Eulerian polynomials, derived a generating function identity which generalizes MacMahon's identity, proved their self-interlacing under suitable parameter conditions, and identified these polynomials as the h^*-polynomials of direct products of dilated lattice simplices. In this paper, we introduce an ascent analogue of the colored multiset Eulerian polynomial, derive an explicit generating function identity for this polynomial, show that it is equal to the h^*-polynomial of a family of half-open lattice polytopes, and verify that this ascent polynomial also satisfies self-interlacing under the same parameter conditions. By establishing recurrence relations, we prove that both polynomials are real-rooted for all positive integer parameters. The obtained identities are further applied to interpret combinatorially the h^*-polynomials of Pitman–Stanley polytopes, composition polytopes and a family of reflexive lattice polytopes defined from preorders.

In 1943, Erdős considered the minimum number $f(n)$ of terms between two fractions in the Farey sequence of order $n$ whose numerators and denominators are oppositely ordered. Determining the constant $c$ in $f(n)=(c+o(1))n$ is known as Erdős Problem 1005. Recently, Cipollini solved this asymptotic problem by proving that $f(n)=(1/4+o(1))n$. Following his framework, we give an analytic proof of an exact formula for $f(n)$ for all sufficiently large $n$. Combining this with a finite computer verification, we further determine $f(n)$ for every integer $n\geq 4$.

Facial reduction (FR) is a preprocessing technique used to restore Slater's condition in semidefinite programming (SDP), but each step generally requires solving an auxiliary SDP. Factor-width-\(k\) partial FR restricts the class of exposing matrices. Increasing \(k\) enlarges this class, but, for \(k\geq3\), the auxiliary problem remains an SDP involving positive semidefinite blocks of order \(k\). For SDP relaxations of binary programs, we replace this auxiliary SDP by an LP-based face-identification method. We construct a polyhedron in subset-indexed moment variables that has polynomial size for fixed \(k\). The resulting face contains every feasible binary lift and is contained in every face exposed by a factor-width-\(k\) exposing matrix from the same auxiliary system. Thus, linear programming identifies a face no larger than those obtained from the factor-width-\(k\) auxiliary SDP while preserving all feasible binary points. To iterate the construction, we represent each identified face by linear equations in the original matrix variable, thereby retaining the moment-matrix indexing and the factor-width comparison at every iteration.

Let $\varphi_{d}(|{\,\cdot\,}|)$ be the radial extremal function in the problem for the sharp Nikolskii constant $\mathcal C_d^{-1}=\inf \|f\|_{1}$ over functions $f\in\mathit{PW}\,_{1}^{1}(\mathbb R^{d})$, $f(0)=1$. For every odd dimension, we construct an entire function $\Phi$ of exponential type $1/2$ such that $\varphi_d(z)=\Phi(z)\Phi(-z)$, and $\Phi$ satisfies a quadratic functional equation and a second-order linear differential equation with polynomial coefficients. Thus, the problem of finding the extremal function is reduced to a spectral problem with finitely many parameters. This result extends a recent one-dimensional result, but uses a different method. For example, in dimension $d=3$ it leads to a seven-diagonal spectral scheme that allows us to compute $\mathcal C_3$ to high accuracy. Even dimensions remain open within this approach.

Let T_0={L} and T_{r+1}={L} union {N(a,b):a,b in T_r}. On the nonleaf terms E_r, require each event N(a,b) to occur after its nonleaf children, and let mu_r be the uniform measure on the linear extensions of this poset. We study the restriction rho_43 that deletes the new level-4 events while preserving the relative order of the level-3 events. We prove that the pushforward of mu_4 under rho_43 is not mu_3. Two explicit orders on the 25 level-3 events have different numbers of level-4 extensions. If b_i is the number of T_2 terms, including the leaf, seen in a prefix of length i, the number of newly released events is (i+1)^2-b_i^2. The release profile of a depth-priority order dominates that of a level order pointwise and is strictly larger for 3<=i<=15. An explicit injection between admissible interleavings therefore gives a strict analytic fiber inequality. Two algorithmically independent exact computations reproduce both 1557-digit fiber counts; their reduced ratio is 614690215260160000/479048686862260621, approximately 1.2831476885707443. The analogous restrictions through level 3 are consistent, so 4-to-3 is the first failure in this grammar. The result is specific to this grammar, uniform measures, and restriction map.

In this project, I shall establish a large number of new Fejér-type inequalities which reduce all results of the following Subsections 1.1-1.3. A number of applications for the extended incomplete Beta function and the extended special means are given.

We study repetition avoidance in a word ${\bf w}$ and its curling-number transform $C({\bf w})$. For alphabets of sizes $2$, $3$, and $4$, we use Thue-Morse-based morphic constructions and exhaustive finite searches. A ternary word for which both ${\bf w}$ and $C({\bf w})$ are overlap-free has length at most $84$, whereas over four letters an infinite example exists. Hence $4$ is the smallest alphabet size admitting simultaneous infinite overlap-freeness. The infinite constructions are verified in Walnut; the finite maxima are obtained by exhaustive breadth-first search and checked independently.

We study continuous-time multi-asset portfolio choice and consumption under smooth pointwise constraints, including state-dependent feasible sets. The method separates dynamic information acquisition from local constrained recovery. A pointwise-feasible neural actor generates reference rollouts; after training, its realized latent outputs are frozen and first- and second-order adjoints are harvested from a fixed-latent open-loop backpropagation-through-time graph. Feedback therefore generates the reference trajectory without restricting the adjoint formulation to Markov controls. Conditional on the harvested adjoints, deployment solves a local generalized Pontryagin-Hamiltonian problem: quadratic-affine portfolio blocks are recovered exactly by a quadratic program, while a log barrier approximates more general regular KKT branches. We establish local chart representations, an OL-BPTT-to-adjoint correspondence retaining orthogonal martingale residuals, and an end-to-end bound from reference value loss and numerical errors to recovered-policy and local QP-gap errors. Analytical constant- and predictable-opportunity benchmarks validate the adjoints. Common-input experiments show that recovery reduces residual PMP/KKT error left by finite-budget direct policy optimization, including under a state-dependent consumption cap and with up to 100 risky assets. Scalability concerns the constrained action block rather than dimension-free state-space complexity.

A small n-gon is a planar n-gon whose diameter is at most one. For odd n, Reinhardt proved that the regular polygon is optimal. For even n, the maximizer is nonregular, and only a few low orders were known exactly. We prove that for every even n >= 8 the maximum-area small n-gon is unique up to Euclidean isometry and reflection. Foster and Szabo's description of the diameter graph reduces a maximizer to an (n-1)-cycle of unit distances together with one pendant diameter. We determine the compatible boundary order, interpret the cycle as the centers of a Reuleaux (n-1)-gon, and eliminate the pendant vertex by a one-variable area calculation. The remaining first-order equations have conserved translation and rotation quantities. In the resulting radial variables they become the critical-point equations of an explicit function on a convex domain. We prove that this function is strictly concave by factoring the relevant principal minors of its local Hessian. Compactness gives existence, while strict concavity gives uniqueness. The proof is analytic. The symbolic scripts supplied with the paper check algebraic identities but are not used as part of the proof. Combined with the classical odd-order and low-order results, this determines the maximal area for every n >= 3.

We are interested in the following conjecture of Wilson from 1981: every locally nilpotent countably categorical group is nilpotent. Following our previous work on the Lie algebra analogue of the conjecture, we use Lie methods to deduce the first nontrivial cases of the Wilson conjecture for n-Engel groups: countably categorical 3-Engel groups and 4-Engel 5-groups are nilpotent, from which we also conclude that countably categorical 4-Engel groups of odd exponent are nilpotent. This is implemented via an exceptional case of the Lazard correspondence, checked using computer algebra systems. We also study the transfer of nilpotency results between the three categories: groups, Lie algebras, associative algebras. Among other things, we prove that the Wilson conjecture implies the analogous nilpotency statement for associative algebras (modulo the commutative case).

Different field species can begin with finite anisotropic propagation geometries whose principal axes need not coincide. We ask whether local Yukawa interactions on a finite graph can nevertheless generate one common infrared spatial cone, even when the interactions that drive the locking become marginally irrelevant. Retaining the full finite mismatch between positive-definite spatial kinetic matrices, we derive the one-loop nonlinear matrix flow and show that its Thompson diameter is globally nonexpansive. On a connected graph, persistent normalized edge activity upgrades this to a uniform finite-window contraction, so all interacting sectors asymptotically share one relative propagation geometry. An explicit two-channel Pauli-Dirac Yukawa-quartic completion realizes the persistence conditions on an open weak-coupling set of finite-mismatch initial data. In four dimensions the Yukawa couplings vanish in the infrared, yet their accumulated interaction time diverges, leaving a Gaussian endpoint with a common spatial cone. For a controlled two-plateau extremal class, graph distance also fixes the first nonzero short-time order of the diameter decrease. Thus common-cone locking can emerge from renormalization-group dynamics rather than being imposed as a microscopic common structure.

We develop two Brownian stopping methods: one uses hitting probabilities and capacity, the other moments of weighted exit distributions. On a closed $m$-dimensional Riemannian manifold, $m\geq 2$, the first gives low-energy spectral inequalities for equilibrium measures supported on sets of zero volume. Combining scales gives heat observability and null controllability from a full-support measure carried by a dense set of Hausdorff dimension $m-2$, minimal under a bounded $H^1$-trace condition. For every nondecreasing unbounded $\Phi$ with $\Phi(4E)\leq D\Phi(E)$, the measure can be chosen so that the optimal spectral constant is comparable to $\Phi(E)$ and the small-time control cost to $T^{-1/2}\Phi(T^{-1})^{1/2}$. Squaring the Feynman-Kac martingale doubles the potential: for radial $V$, the boundary second moment is log-convex in $\log r$ if $r^2V(r)$ is nondecreasing, wherever the regular solution of $(\tfrac12\Delta+2V)g=0$ stays positive; the factor $2$ is sharp when $V(0)>0$. On small geodesic balls, weighted second moments give an almost-monotone frequency and a weighted three-radius inequality; bounds for its logarithmic derivative give doubling estimates, which, after a ground-state transform, feed the Remez theorem of Logunov-Malinnikova. The normalised first-moment exit law gives an averaged boundary-variance identity for Laplace eigenfunctions. At $r=c\lambda^{-1/2}$, outside a set carrying $O(c^{2/m})$ of the $\varphi^2$-mass, the component of $\{\varphi(x)\varphi>0\}\cap B(x,r)$ through $x$ contains a concentric ball of nearly full radius, fills nearly all of $B(x,r)$, and nearly matches its first Dirichlet eigenvalue. First-moment stopping also controls positive superlevel components. Reflected Brownian motion and local time give boundary-to-collar $L^\infty$ estimates for Steklov eigenfunctions on bounded Lipschitz domains, and an $L^2$ estimate for $C^2$ domains.

The popular CART algorithm for regression trees combines a greedy splitting rule with a stopping rule, but while the splitting rule has been well studied, the statistical role of stopping rules is less well understood. Meanwhile, although regression trees fit using Bayesian methods or via empirical risk minimization (ERM) have been shown to be spatially adaptive to local smoothness and anisotropy, it is unknown whether CART can achieve the same adaptation. We address these gaps by proving that, under spatially heterogeneous and anisotropic smoothness and appropriate structural assumptions on the regression function and covariate distribution, CART with the minimum impurity decrease (MID) stopping rule and a suitable threshold achieves pointwise rates that are minimax up to logarithmic factors. These rates hold simultaneously over all points in the domain. Moreover, we prove that spatial adaptation cannot be achieved under the widely used minimum leaf size stopping rule. Together, these results establish a precise statistical role for the MID stopping rule and provide a theoretical basis for the empirical success of CART.

Approximate Bayesian computation (ABC) replaces likelihood evaluation by simulation and comparison of observed and synthetic data. We establish minimax-rate guarantees for nonparametric ABC under random-series priors with simulable finite-dimensional coordinates. The contraction theorem uses local prior mass, bounds on ABC acceptance probabilities, and control of prior mass outside a sieve. In fixed-design orthogonal-series regression with centered $g$-and-$k$ errors, an infinite Gaussian series prior with a compact scale hyperprior yields minimax-rate contraction and a minimax-rate clipped posterior mean. In compound Poisson decompounding, only random sums are observed and the target is the underlying jump density. With an unknown count intensity in a fixed compact subinterval of $(0,\pi/2)$, we prove stability of the zero-count-augmented trigonometric population summaries and use a square-root Gaussian series prior on the space of probability density functions. Over bounded periodic Sobolev classes of smoothness $\alpha>d/2$, a polynomially enlarged synthetic sample yields ABC contraction at rate $n^{-\alpha/(2\alpha+d)}$ and posterior mean squared risk of order $n^{-2\alpha/(2\alpha+d)}$, matching a lower bound for the aggregate-observation model. Rejection-ABC Monte Carlo approximations inherit these rates under sufficient sampling budgets.

Let \[\mathfrak O=\{(x,y)\in[0,1]^2:x+y\le \sqrt2\} \] be the first-quadrant sector of a regular octagon. For quadratic forms \(P(x,y)=ax^2+bxy+cy^2\), we study the supremum norm over \(\mathfrak O\). We obtain a complete five-region formula for the norm, according to whether the norming contact occurs at an endpoint or in the interior of one of the three radial sides. We then prove that the projection of the unit ball onto the \(ac\)-plane is exactly \([-1,1]^2\), compute both endpoints of every vertical section, and thereby parametrize the entire unit sphere. Finally, we characterize the extreme points of the unit ball as four explicit curves, their negatives, and four pairs of isolated points. The resulting description is fully explicit and reduces subsequent convex extremal problems on this polynomial space to four one-parameter families and finitely many isolated polynomials.

This paper addresses the Turán extremal problem on the symmetric three-interval set $\Omega_\lambda = (-1,1) \cup (\lambda-1,\lambda+1) \cup (-\lambda-1,-\lambda+1)$, $\lambda \geq 0$. We establish a discrete approximation principle relating the continuous Turán problem on $\Omega_\lambda$ to the discrete Turán problem on an associated finite discrete set. This enables the continuous problem to be exactly expressed as a limit of finite nonnegative trigonometric polynomial problems, which are equivalent to finite semidefinite programs. Using this framework, we compute the Turán constant for integer and rational parameters $\lambda$. We further analyze the dependence of the Turán constant on $\lambda$ and derive upper bounds for all $\lambda > 5$.

In this paper, we construct new sign-changing solutions for the Hénon problem \begin{equation*} \begin{cases} -\Delta u= |x|^{\alpha} |u|^{p-1}u,\,\,\,\text{in}\,\,\, \Omega, \\[1mm] u=0,\,\,\,\,\,\,\text{on}\,\,\, \partial\Omega, \end{cases} \end{equation*} where $\Omega\subseteq \mathbb{R}^2$ is a bounded smooth domain containing 0, $\alpha\in\mathbb{R}$ is a positive parameter and $p$ approaches $+\infty$.

A graph $G$ is said to be $k$-collapsible if $G$ has minimum degree $k$ and every non-null proper induced subgraph of $G$ has minimum degree less than $k.$ In 2018, Bickle conjectured that the minimum number of vertices of degree $k$ in a $k$-collapsible graph of order $n$ with $k\ge 3$ is ${\rm max}\{\lceil 2n/(2k-1)\rceil,\, k^2-k-2-(k-3)n\}.$ We prove this conjecture.