Papers

We construct an infinite family of Neumaier graphs of coherent rank five, answering the existence question at the smallest possible coherent rank beyond the strongly regular case. For every prime power $q\geqslant7$ with $q\equiv3\pmod4$, set $n=q+1$. Each graph in our construction has precisely five distinct eigenvalues, Neumaier parameters \[ \left( n(n-1)(n-3), \frac{n^2(n-3)}2, \frac{n(n^2-n-8)}4; \frac{(n-2)^2}{2}, (n-1)(n-3) \right), \] and its adjacency matrix lies in the Bose–Mesner algebra of the four-class association scheme of Holzmann, Kharaghani, and Suda. Paley Hadamard matrices and Desarguesian mutually orthogonal Latin squares yield an infinite family whose smallest member has parameters $(280,160,96;18,35)$.

Homogeneous random graphs, also known as Erd{\H os}-Rényi graphs, are a subset of the family of dense random graphs, specified by a graphon. We analyze several goodness-of-fit tests for these models that are based on subgraph counts. To obtain the limiting null distribution of the test statistics we use a decomposition of graph functionals, which reveals a cancellation effect. Motivated by a quasirandomness result we obtain a test that is consistent against all alternatives, and we use two popular parametric subfamilies to evaluate the other tests. The theoretical results refer to the limit $n\to \infty$ of the size $n$ of the graph, the behavior for finite $n$ is illustrated by simulations.

We consider the invariant subspace problem for operators induced by Lipschitz self-maps on Lipschitz-free spaces. Besides the usual free linearizations $\widehat f$ of basepoint-preserving Lipschitz self-maps, we consider a wider class of operators $T_{f,e}$ which are naturally defined for arbitrary Lipschitz self-maps. We show, among other results, that every $\widehat f$ admits a non-trivial invariant subspace whenever the underlying metric space contains a compact ball, or has at least two connected components one of which has non-empty interior. We also obtain corresponding positive results for $T_{f,e}$ under isolated-point, compact-ball and disconnectedness assumptions, and discuss some consequences for linear dynamics.

We study initialization bias in spherical $K$-means for weakly informative directional mixtures. We model the observations by a $K$-component von Mises-Fisher mixture with a small concentration parameter $\kappa$, corresponding to a high-dispersion regime in which the data provide limited information about the underlying directions. Our analysis begins with the limiting case $\kappa=0$ (corresponding to a uniform distribution over the sphere), where one population spherical $K$-means update is governed entirely by the Voronoi tessellation induced by the initialized templates. For uniformly random initializations in fixed dimension $d$, the updated templates become asymptotically aligned with their initial values as $K\to\infty$: the average squared geodesic error scales as $O(K^{-2/(d-1)})$, while the worst-case error is $O((\log K/K)^{2/(d-1)})$. We then show that, in the weak-signal regime of small positive $\kappa$, the population update remains an $O(\kappa)$ perturbation of this limiting map. Thus, in the weak-signal regime, spherical $K$-means can preserve initialization-induced structure despite the presence of a genuine but highly dispersed directional signal.

This paper studies the category $\mathcal{C}$ of grading-restricted $C_1$-cofinite generalized modules for the vertex operator algebra associated to the $W$-algebra $W(2,2)$. We first show that $\mathcal{C}$ is the same as the category of finite length modules whose simple composition factors are the irreducible highest weight $W(2,2)$–modules $L[r]$ of highest weight $(\frac{1-r}{2}, 0)$ for $r \in \mathbb{Z}_{> 0}$, and hence $\mathcal{C}$ carries a braided tensor category structure. Then we prove the fusion rules for the simple objects $L[r]$ are governed by the $\mathfrak{sl}_2$ Clebsch–Gordan rule. In particular, we prove \[ L[r]\boxtimes L[s]\cong \bigoplus_{i=0}^{\min\{r,s\}-1} L[r+s-1-2i]. \] Using the fusion rules and a recent result of Etingof–Penneys, we establish the rigidity of $\mathcal{C}$. We also show the semisimple subcategory generated by the simple objects $L[r]$ is tensor equivalent to the category Rep $\mathfrak{sl}_2$ of finite dimensional $\mathfrak{sl}_2$-modules.

We show that every faithful, transitive, and generically $(n+2)$-transitive action of a connected group $G$ on an irreducible variety $X$ of dimension $n > 0$, all defined over an algebraically closed field $F$, is isomorphic to the natural action of the projective linear group $PGL_{n+1}(F)$ on the projective space $\mathbb{P}^n(F)$. More precisely, we establish the Borovik-Cherlin conjecture for permutation groups $(G,X)$ definable in models of $ACF$.

We prove the existence and uniqueness of a local-in-time solution to the stochastic Cahn-Hilliard equation in space dimensions 1, 2, 3, 4 and 5. Our method relies on the Da Prato-Debussche trick for $d=4$ and paracontrolled distributions for $d=5.$

Distribution dependence generally prevents McKean-Vlasov state solution maps from satisfying a cocycle identity, while the relevant Sobolev embedding is noncompact on unbounded domains. For a class of stochastic reaction-diffusion equations with a dissipative polynomial reaction, a one-sided monotone state-law coupling and finite-dimensional additive noise, we combine the deterministic law semiflow with the pathwise state evolution to construct a continuous random dynamical system on the product of the state space and its quadratic Wasserstein law space. An abstract product-space criterion, mixed-energy estimates, local regularity and uniform far-field estimates yield a unique pullback random attractor without global law contraction. Its law projection is the global attractor of the law semiflow, and its state fibers need not be singletons. Under an additional strict contraction condition, the law attractor reduces to the unique invariant law and the corresponding state fiber to a self-consistent random equilibrium. We also establish zero-noise upper semicontinuity and, in the contractive regime, quantitative convergence of the invariant laws and random equilibria with bounds linear in the noise amplitude.

We compute the $\log p$-part of the asymptotic expansion of the holomorphic torsion of the $p$-th power of a positive line bundle in terms of characteristic classes. This is done for orbifold and equivariant torsions. In the equivariant case the exponents in the expansion are less than or equal to the dimension of the fixed point set. We deduce a corresponding equivariant arithmetic Hilbert-Samuel theorem in Arakelov geometry.

The perihelion of comet 1P/Halley in 2061 is an excellent chance to revisit this object of outstanding scientific and cultural relevance. During its 1986 approach to the Sun, it was targeted by several flyby missions. Due to its retrograde, highly-inclined orbit, the relative velocities during the encounter were large, limiting the scientific return due to the short time spent inside the coma. A rendezvous trajectory would overcome this limitation, but the design is challenging due to the limitations of current propulsion technology. Given the lead times of spacecraft development and the long duration of the interplanetary transfer to the comet, it is imperative to start mission planning as soon as possible. We present a novel rendezvous strategy, combining unpowered Jupiter and Saturn gravity-assists with deep-space low-thrust arcs. It minimizes launch energy and propellant budget, constraining the arrival to occur before the onset of high activity. The double flyby strategy reduces the otherwise prohibitive cost of the plane change maneuver. Closed expressions for the optimal flyby geometry, together with an explicit low-thrust transcription technique, reduce the number of design parameter to three, improving computational efficiency. Crucially, this is the first rendezvous mission concept achievable with well-proven technology (standard radioisotope thermoelectric generators and a Hall-effect thruster) and compatible with several existing launchers. We describe the trajectory optimization strategy and perform a comprehensive exploration of the design space. Finally, we present two promising proof-of-concept trajectories in detail.

Let $0 < \alpha \leq 1/2$. We say that a finite point set $P$ in $\mathbb{R}^d$ is $\alpha$-split by a hyperplane $h$ if each of the closed half-spaces determined by $h$, contains at least $\alpha |P|$ of the points of $P$. We further say $P$ is $\alpha$-split by a $k$-dimensional flat $\tau$ if $P$ is $\alpha$-split by any hyperplane through $\tau$. In the standard notation (which coincides with Tukey depth for $k= 0$), the $k$-flat $\tau$ has depth $\alpha$ with respect to $P$. We establish interesting Helly-type theorems for splitting families of finite point sets in $\mathbb{R}^d$. Unlike the classical sufficient Helly-type criteria for transversals to families of compact convex sets, which exist only for point and hyperplanes, our results extend to splitting families of point sets by collections of $k$-flats of arbitrary dimensionality $ 0 \leq k \leq d-1$.

Numerical experiments reported by Quan and Wu [SIAM J Numer Anal 61 (2023) 2106-2132] show that the observed temporal convergence rates of nonuniform L2 methods for subdiffusion models are not consistent with the theoretically predicted order $3-\alpha$. This discrepancy suggests that a more refined analysis is needed and motivates the development of a nonuniform BDF3-L2 method for the subdiffusion equation. To account for the initial solution singularity, we employ the multiscale solution decomposition to decompose the original solution and approximate a smoother unknown variable that satisfies the subdiffusion model with a smoother source term. The resulting formulation, however, involves restrictive high-order boundary conditions on the source term and initial data. To overcome this difficulty, we introduce a spectral truncation technique that requires only slightly stronger regularity of the data and a controllable truncation error. We establish high-order regularity estimates of the solution to the truncated problem and develop a nonuniform BDF3-L2 method for its numerical approximation, based on which we derive a rigorous error estimate of temporal convergence order $2+\alpha$. Numerical experiments are carried out to substantiate the theoretical findings.

We establish a Serre-type presentation for $\imath$quantum enveloping superalgebras arising from quantum supersymmetric pairs of type A. We first prove, in arbitrary type, that evaluating a quantum Serre polynomial on the coideal generators produces a correction of strictly smaller weight in the natural filtration using the projection technique. We then compute these remainder terms for all type A super Satake diagrams. In particular, we determine the new degree-four relations associated with isotropic odd simple roots of the relevant local Satake diagrams.

We investigate inverse scattering at a fixed energy for semiclassical Schrödinger operators with smooth potentials. For compactly supported potentials satisfying a natural virial condition, we prove that, under the sharp condition, $\liminf_{h\to0}\|S_1(\lambda,h)-S_2(\lambda,h)\|<2$ implies $V_1=V_2$, provided the fixed energy $\lambda$ lies above both potentials. We then consider radial short-range repulsive potentials. Under a monotonicity assumption on the radial force, we show that the semiclassical differential cross section at a fixed energy, up to $o(1)$ as $h\to0$, determines the potential in the classically accessible region. Finally, we obtain an analogous rigidity result for nontrapping compactly supported perturbations of the Euclidean metric under a strict convexity assumption.

Let $K$ be a number field containing a primitive $p$-th root of unity $\zeta_p$. Let $f(x)\in K[x]$ be a monic integral polynomial, and let $f_0$ denote its radical. Let $C/K$ be the superelliptic curve defined by $y^p=f(x)$, and let $J$ be its Jacobian variety. The variety $J$ admits multiplication by $\zeta_p$ over $K$; in particular, the endomorphism induced by $\Pi:=1-\zeta_p$ gives an isogeny of $J$ over $K$. Let $\operatorname{Sel}_\Pi(J)$ denote the Selmer group associated to $\Pi$. Under suitable hypotheses, we obtain bounds for $\operatorname{Sel}_\Pi(J)$ in terms of the $p$-torsion subgroup of the class group of $L:=K[x]/(f_0)$. Several examples illustrating the results are also discussed.

Let $A$ be a complex abelian surface. We prove that $A$ admits a dominant rational map of degree three to $\PP^2$ if and only if it admits a polarization of type $(1,1)$, $(1,2)$, $(1,3)$, or $(1,6)$. Consequently, the degree of irrationality of $A$ is $3$ precisely in these four cases and is $4$ otherwise.

This paper is concerned with a stochastic optimal control problem under inside information. The control process depends on an $\mathcal{F}_{T_0}$-measurable random variable $Y$, representing the static inside information, and is adapted to the enlarged filtration generated by the underlying Brownian motion and the random variable $Y$. Accordingly, the traditional stochastic integral fails to be well-defined in this non-adapted setting; we adopt forward integrals to formulate the stochastic integral terms in the system. By means of the Donsker delta function, the original controlled system is transformed into a $y$-parameterized system. We further establish the existence of solutions to forward stochastic differential equations (SDEs), and prove the uniqueness of solutions via flow transformation techniques. Under a Gaussian assumption on $Y$, we derive both necessary and sufficient optimality conditions for the aforementioned control problem. Subsequently, we formulate the linear-quadratic (LQ) optimal control problem under inside information. Through the $y$-parameterized transformation, the original problem is converted into an LQ control problem with random coefficients. A numerical example for the LQ case is provided at the end to validate our theoretical findings.

We solve the Euclidean-ball case of the higher-dimensional Shepp covering problem. More precisely, we give an exact criterion for full limsup coverage of the $d$-dimensional torus, $d\ge 2$, by independently centered Euclidean balls with an arbitrary decreasing sequence of radii. Let $X_1,X_2,\ldots$ be independent Haar-uniform points, let $r_1\ge r_2\ge\cdots\downarrow 0$, and put $u_n(z)=m(B(0,r_n)\cap B(z,r_n))$ and $H(z)=\sum_{n\ge 1}u_n(z)$. Then every point belongs to infinitely many of the balls $B(X_n,r_n)$ almost surely if and only if $\int_{\mathbb{T}^d}\exp(H) dm=\infty$. In dimension one this condition is equivalent to Shepp's criterion. No regular-variation or comparable-scale assumption is imposed on the radii. The main difficulty is shared noise: after spatial decomposition, the same Poisson input acts on many uncovered cells, so their descendants are not conditionally independent. We overcome this by establishing an extinction bound for monotone population recursions driven by positively associated innovations. Together with Poissonization and spatial localization this proves sufficiency, while a second-moment estimate and the zero-one law prove necessity.

The Johnson-Lindenstrauss (JL) lemma guarantees that a random projection of $n$ points to $m=O(\varepsilon^{-2}\log n)$ dimensions preserves pairwise squared distances within relative error $\varepsilon$ with high probability, and this dimension order is asymptotically optimal. In high dimensions, however, distances concentrate around a baseline while key geometric information lies in much smaller fluctuations. We show that the JL bound can therefore be uninformative about retained geometry: an independent Gaussian replacement map can satisfy it even though the replacement cloud is independent of the original data. We then ask how well any decoder can recover a feature $f(D)$ of a squared distance $D$ from a linear sketch. Under squared-error loss, the optimal decoder is conditional expectation, so recovery defines a linear operator whose singular values quantify feature recovery. For isotropic Gaussian data ($\Sigma=\sigma^2 I_d$), we diagonalize this operator in closed form. For fixed $k$ with $m,d-m\to\infty$, its $k$th singular value satisfies $\ell_k\approx(m/ d)^{k/2}$. This yields three sharp consequences. A rank-$m$ sketch retains at most an $m/d$ fraction of the variance of any feature of one squared distance. If $m\to\infty$ and $m/d\to0$, the expected Kendall correlation is $\frac{2}{\pi}\sqrt{m/d}(1+o(1))$; for fixed $q$, nearest- neighbor agreement tends to $1/q$. Yet one projection can satisfy the JL bound while mean Kendall correlation vanishes when $\log n\ll m\ll d$. After removing scale, Haar-averaged retained covariance-shape information is $(m/d)^2$. Thus JL distance preservation does not quantify the geometry available for comparison or inference.

In this paper, we study normalized solutions to the following $L^2$-supercritical nonlinear Schrödinger equation with a periodic potential \begin{equation*} \begin{dcases} -\Delta u +V (x)u + \lambda u=\chi_{{\Omega}}(x)f(u)\quad \text{in } \mathbb{R}^N, u>0 \quad \text{in} \ \mathbb{R}^N, \int_{\mathbb{R}^N}\abs{u}^2\, dx =\mu. \end{dcases} \end{equation*} Here $N \geq 1$, $\mu>0$ is prescribed, $\lambda \in \mathbb{R}$ is a Lagrange multiplier, $V\in C(\mathbb{R}^N)$ is $1$-periodic in $x_1,...,x_N$, $f \in C^1(\mathbb{R})$ is a nonlinearity having $L^2$-supercritical growth at infinity, $\Omega\subset \mathbb{R}^N$ is either a (nonempty) bounded open set with smooth boundary $\partial \Omega$ or the whole space \(\mathbb{R}^N\), and $\chi_{{\Omega}}$ is the characteristic function of ${\Omega}$. In both cases, we prove the existence of normalized solutions of mountain pass type when $\mu>0$ is small and $f$ behaves like a power function $|t|^{p-2}t$ with $2+4/N<p<2^*$ at infinity. Moreover, if $\Omega$ is bounded, $1 \leq N \leq 4$ and $f$ has $L^2$-supercritical growth near the origin, then the existence of normalized solutions is obtained for all $\mu>0$. On the other hand, when $\Omega=\mathbb{R}^N$, we prove the existence of solutions corresponding to local minimizers when $\mu>0$ is small. When $\Omega$ is bounded, the results are obtained via a monotonicity trick for mountain pass values and blow-up analysis with the Morse index estimates. In the case where $\Omega = \mathbb{R}^N$, we develop the concentration-compactness argument based on the Morse index for the existence of mountain pass type solutions.

We consider in this paper complex Lie groups $\mathrm{SL}_n(\mathbb{C})$ for $n\geq 2$, $\mathrm{SO}_n(\mathbb{C})$ for $n\geq 5$, and $\mathrm{Sp}_{2n}(\mathbb{C})$ for $n\geq 1$. We show that for any such Lie group, the Lie algebra of polynomial vector fields can be generated by three $\mathbb{C}$-complete polynomial vector fields.

Semidefinite programming plays a key role in optimization, with broad impact across control theory, machine learning, and combinatorial optimization. Although semidefinite programs are polynomially solvable, several commonly used algorithms rest on a linear-algebraic step whose running time grows cubically with the matrix dimension and which requires the matrix itself to be held in memory, at quadratic cost. In this study, we propose replacing it with a variational quantum eigensolver, whose qubit requirement is logarithmic in the matrix dimension, and present the first end-to-end implementation of such an approach within a cutting-plane framework, together with an operator-derived ansatz whose entanglement structure is read directly from the Pauli support of the candidate matrix. Evaluated on the control family of SDPLIB against an identical scheme driven by an exact eigendecomposition, the variational oracle produces valid cuts throughout, closing 32 to 82% of the initial optimality gap against a near-constant 75 to 82% for the exact oracle. Implementing and measuring the method end to end surfaces several effects not visible from theoretical analyses alone: where memory is actually consumed, how the padding required to fit a matrix onto a quantum register can mislead the variational optimizer, and why the candidate matrices prove dense in the Pauli basis, reducing the operator-derived ansatz to full entanglement. We report these findings and discuss their implications for near-term hybrid quantum-classical approaches.

We construct an $\mathfrak{sl}_2$-action by derivations on the Lie algebra $\mathfrak B$ of swap invariant alternil bimoulds with the uri bracket of Kühn and Schneps. This Lie algebra plays the role for formal multiple Eisenstein series which Racinet's double shuffle Lie algebra $\mathfrak{dm}_0$ plays for multiple zeta values. The kernel $\mathfrak m$ of the lowering operator is a Lie subalgebra, $\mathfrak B$ is the direct sum of its iterates under the raising operator, and this gives Rankin-Cohen type operators on $\mathfrak m$. We define a Lie subalgebra $\mathfrak d$ of $\mathfrak B$ and show that it is isomorphic to $\mathfrak{dm}_0$ extended by an additional element in weight one.

We study centered Gamma approximation and a same-kernel Poisson–Gaussian invariance principle on fixed Poisson chaoses. For Gamma approximation, a martingale-core argument extends the carré-du-champ and $d_2$ estimates of Döbler and Peccati (Ann. Probab., 2018) from regular kernels to every fourth-integrable chaos element. In the diffuse regime, characterized by vanishing fourth add-one energy, this yields an exact four-moment criterion under uniform integrability of fourth powers. In the rare-jump regime, ordinary moments do not determine the approximation mechanism: convergence of the full moment sequence may coexist with a nonvanishing fourth add-one energy, and we construct such centered Gamma limits in every fixed chaos order. The invariance principle is independent of the Gamma target. For Poisson and Gaussian multiple integrals with the same kernel, we bound both smooth-test discrepancies and the Wasserstein distance in terms of the variance and the fourth add-one energy. Thus, vanishing fourth add-one energy is an intrinsic Lindeberg condition under which the two chaoses are asymptotically indistinguishable in distribution. Combined with a moment-transfer estimate and the Gaussian fourth-moment theorem, this gives an alternative proof of the qualitative normal fourth-moment theorem on a fixed Poisson chaos. A rainbow example shows that the Lindeberg condition is essential: the Gaussian analogue may be asymptotically normal while the Poisson integral converges to a centered compound-Poisson law. The same comparison also explains the different behavior of even and odd chaos orders for diffuse centered Gamma limits.

We study the problem of finding shortest self-orthogonal embeddings of a given code which have the largest possible minimum distance by employing a constraint-preserving genetic algorithm. We show that the orthogonal group is generated by transvections corresponding to weight-four vectors. Using this result together with the fact that all shortest self-orthogonal embeddings can be obtained through the action of the orthogonal group, we construct a genetic algorithm in which each chromosome is represented by a sequence of weight-four vectors. Applying our algorithm to codes with dimensions $9\le k\le 15$ and lengths $22\le n\le 256$, we obtain 280 inequivalent optimal self-orthogonal codes. In particular, we obtain 34 codes with new parameters and 100 codes which are inequivalent to previously known codes with the same parameters.

We introduce a broad family of groups $G(\Gamma)$ associated with transverse stratifications of moduli spaces. For a wide class of stratifications satisfying a natural local condition (transverse niceness), we construct a homomorphism from the fundamental group of the moduli space to the corresponding group $G(\Gamma)$. This general theorem, which is proved in the companion paper \cite{Manturov2026}, provides a far-reaching generalization of the $G_n^k$-theory. The main goal of the present paper is to discuss numerous examples and situations where the transverse niceness property actually occurs. We describe various moduli spaces and stratifications that are transversely nice. In particular, we study the hierarchy of configuration spaces of points in the plane defined by algebraic curves and establish the transversality property for conic configurations. The general hierarchy problem is formulated in the final section.

This paper develops a unified framework for instance optimal sparse recovery from nonlinear observations. The main ingredient is a signal-dependent restricted approximate invertibility condition (RAIC) of some gradient, which leads to the instance optimality of iterative hard thresholding. Under Gaussian designs, we apply the proposed framework to phaseless, one-bit, and ReLU measurements, which correspond to the problems of sparse phase retrieval, one-bit compressed sensing, and sparse ReLU regression, respectively. For sparse phase retrieval, we propose a variant of thresholded amplitude flow and show its instance optimality under $O(s^3)$ measurements (up to logarithmic factors), where $s$ is the sparsity level. To our best knowledge, this is the first instance optimal efficient algorithm for sparse phase retrieval and complements Gao, Wang and Xu (2016) that achieved this via a computationally intractable program. In one-bit compressed sensing, we establish the instance optimality of normalized binary iterative hard thresholding and strengthen the recent result of Matsumoto and Mazumdar (2024). In sparse ReLU regression, it is shown that a slight variant of the algorithm in Soltanolkotabi (2017) is instance optimal. Moreover, $(\ell_2,\ell_2)$ non-uniform instance optimal guarantees are obtained for these problems. The analysis is built upon a number of high-dimensional concentration bounds, including bounds on restricted eigenvalues and a novel instance-dependent hyperplane tessellation result.

This paper is concerned with the time-asymptotic stability of the generic Riemann solution for the one-dimensional system of heat-conductive ideal gas without viscosity, where the generic Riemann solution consists of a shock, a contact discontinuity, and a rarefaction wave. We prove that, as time tends to infinity, the solution of the non-viscous and heat-conductive ideal gas system converges uniformly to a composite wave composed of rarefaction wave, viscous contact wave, and viscous shock wave with a time-dependent shift. Motivated by the recent work of Kang-Vasseur-Wang [Arch. Ration. Mech. Anal. 249: 42 (2025)], we overcome the difficulties arising from the concurrence of shock and rarefaction waves for the partially dissipative hyperbolic-parabolic system with dissipation acting only on a single variable. More notably, the absence of velocity dissipation gives rise to new and intrinsic difficulties when handling the terms associated with the density and velocity. To resolve this, we exploit the precise structure of the governing equations and the additional properties of shock waves. Furthermore, we utilize the wave structure of the system without viscosity and perform separate space-time estimates for the density and velocity.

Consider the polynomial $f=x^N+y^N-z^N$ where $x,\,y$ and $z$ are positive integers and $N \ge 3$ is an integer. By Fermat's Last Theorem, $f$ is never zero so that its reciprocal, $1/f$, has no singularities. We therefore study the finite sum of the reciprocal: $S(m,N)=\sum_{x=1}^m\sum_{y=1}^m\sum_{z=1}^{m}\frac{1}{f}$. The terms $1/f$ can be positive, negative and their magnitude is less than unity. A key observation is that $S(m,N)$ can be split into two convenient parts: a dominant contribution $D(m,N)$ that has a simple analytical expression and a remainder $R(m,N)$ which is more complicated but negligible compared to $D(m,N)$. Therefore, $S(m,N)$ is almost identical to $D(m,N)$. The analytical expression for $D(m,N)$ is $(2\,m-1)\,H_m^{(N)}$ where $H_m^{(N)}=\sum_{x=1}^m\frac{1}{x^N}$ approaches quickly the Riemann zeta function $\zeta(N)$ as $m$ increases. Therefore, the original sum $S(m,N)$ has a simple expression: it is basically linear in $m$ with slope equal to $2\,\zeta(N)$. Its linear behavior is not an asymptotic result; plots of $S(m,N)$ vs. $m$ for different $N$ show a straight line starting at $m=1$. $S(m,N)$ deviates slightly from a straight line over a small interval $8\le m\le 12$ for the case $N=3$. This slight deviation is due to Fermat near misses where $x^3+y^3-z^3=\pm 1$ (for $z\ne x$ and $z\ne y$); these create a jump in the remainder $R(m,3)$ at $m=9$. We make a numerical and analytical study of the remainder $R(m,N)$. From the numerical analysis, $R(m,N)$ converges for $N\ge 4$ but it was harder to tell whether $N=3$ converged. An analytical study based on a comparison of $R(m,N)$ to its Cauchy principal value integral, shows that $R(m,3)$ likely diverges logarithmically. It also shows that $R(m,N)$ converges for $N\ge 4$ in agreement with the numerical analysis. We discuss in the conclusion some interesting questions for future investigation.

We introduce contact symbol systems, a noncommutative analogue of symbol systems for projective fundamental forms, by replacing the polynomial algebra on a vector space by the graded dual of the universal enveloping algebra of a Heisenberg algebra. For a complex projective submanifold equipped with a contact structure, we define contact fundamental forms and prove that, at a general point, they form a contact symbol system, which gives a contact version of the classical result due to E. Cartan. Conversely, we prove that every contact symbol system can be realized as the contact fundamental forms of a projective variety with a dense open Heisenberg orbit, called the Heisenberg-symmetric variety associated to the contact symbol system. We show that the closure of a projectivized nilpotent orbit in a simple Lie algebra is Heisenberg-symmetric if and only if it is the adjoint variety, namely, the projectivization of the minimal nilpotent orbit. For adjoint varieties of non-symplectic simple Lie algebras, we prove the contact analogue of the Landsberg–Manivel strict prolongation property by using Yamaguchi's prolongation theory.