Papers

In this paper, by using the arithmetic properties of character sums over finite fields, we investigate some cyclotomic matrices involving Kloosterman sums over finite fields. For example, let $$K_q(u)=\sum_{x\in\mathbb{F}_q\setminus\{0\}}e^{\frac{2\pi i}{p}{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}\left(x+\frac{u}{x}\right)}$$ be the Kloosterman sum over $\mathbb{F}_q$, where $q=p^f$ is an odd prime power. We prove that matrix $[K_q(s_i+s_j)]_{1\le i,j\le (q-1)/2}$ is singular whenever $q\ge 11$, where $s_1,s_2,\cdots,s_{(q-1)/2}$ are exactly all non-zero squares over $\mathbb{F}_q$.

The Blaschke–Lebesgue problem asks for convex bodies of minimum volume among all convex bodies of prescribed constant width. In the plane, the minimizer is the Reuleaux triangle, whereas the corresponding three-dimensional problem remains open and is also known as Meissner's conjecture. In this paper, we establish the lower bound $(4\pi/33)d^3 \simeq 0.380799\,d^3$ for the volume of any three-dimensional convex body of constant width d. This improves upon Chakerian's lower bound, approximately $0.364916\,d^3$, although it remains below the volume of the conjectured minimizers, Meissner's tetrahedra, whose volume is approximately $0.419860\,d^3$. The proof is based on a support-function formulation, spectral estimates via spherical harmonics, and Bochner's formula. We also show that the resulting lower bound can be interpreted as a Lagrange dual bound for the associated concave quadratic minimization problem. This dual viewpoint suggests possible routes toward sharper lower bounds.

In this article we study periodic orbits of an electron attracted by a proton subject to Lorentz, electric, and Euler forces where each of them is allowed to depend periodically on time. This setup is motivated by the elliptic restricted three-body-problem where the Lorentz force corresponds to Coriolis force, the Coulomb force is replaced by the gravitational force, and the electric force of an external source is a combination of centrifugal forces and gravitational forces of other bodies. This is a singular version of a Euler-Hamilton system as discussed in [FW26b]. The singularity is due to collisions of the electron with the proton, respectively of two masses. Due to the possibility of collisions this problem has to be regularized. We show how periodic collisional solutions of this problem can be detected variationally in a non-local Lagrangian setup as well as in a non-local Hamiltonian setup.

Consider the stochastic partial differential equation, \begin{align*} \partial_t u^{\varepsilon}(t\,,x) = \frac{1}{2} \partial^2_x u^{\varepsilon}(t\,,x) + b(t\,,u^{\varepsilon}(t\,,x)) + \sqrt{\varepsilon}\sigma(t\,,u^{\varepsilon}(t\,,x)) \dot{W}(t\,,x), \end{align*} where $(t\,,x)\in(0\,,\infty)\times\mathbb{R}$, and $\dot{W}$ denotes space-time white noise. Foondun, Khoshnevisan, and Nualart \cite{FKN24} showed that this stochastic partial differential equation is well-posed under the assumptions that the initial condition $u(0)$ is bounded and measurable, while $b$ and $\sigma$ are locally Lipschitz continuous functions with at most linear growth. A Freidlin-Wentzell large deviation principle for the stochastic partial differential equation is established by a weak convergence approach in this paper.

Massera and Schaffer [Ann. Math. (2), 1958] derived a breakthrough upper bound for the Clarkson angle between two nonzero vectors in a normed linear space, which was later improved by Maligranda [Am. Math. Mon., 2006]. Pecaric and Rajic [Math. Inequal. Appl., 2007] extended Maligranda's inequality to finitely many nonzero vectors. We derive a non-Archimedean version of Massera-Schaffer-Maligranda-Pecaric-Rajic inequality.

We document the evolution of Karl Marx's take on infinitesimals. We contrast his initial favorable stance with later criticisms, and examine the differing perspectives of Marx and Engels on the subject. Marx's favorable assessment was based on his study of Sauri's textbook. Later, influenced by Boucharlat's textbook, Marx reversed his position to an unfavorable stance, describing belief in infinitesimals as a `chimera'. Marxist scholar Guglielmo Carchedi claims that “Marx differentiates with the eyes of the social scientist, of the dialectician” but fails to note dialectician Engels' endorsement of infinitesimals. Struik linked Marx to Abraham Robinson, but missed the fact that the link passes via ... Fermat. Namely, there may be an affinity, as per Struik, between Marx's comments on the calculus and Robinson's nonstandard analysis, but the kernel of such an affinity resides in the techniques already found in the context of Fermat's adequality. To adapt Carchedi's metaphor, we could say that Marx may have differentiated with the eyes of adaequo of Pierre de Fermat. The first editor who worked on some of Marx's mathematical manuscripts in the mid-1920s was Emil J. Gumbel, though he is not mentioned in either the 1933 or the 1968 Soviet edition of Marx's mathematical manuscripts.

Preconditioning is a fundamental technique for accelerating classical linear system solvers, and understanding when its benefits persist in quantum linear system (QLS) solvers is important for assessing the practical resource requirements of quantum linear algebra. In QLS algorithms, however, the potential advantage of preconditioning may be offset by the normalization overhead incurred by composing separate block-encodings of the system matrix and the preconditioner, as observed in recent work. This limitation leaves open whether additional algebraic structure can make preconditioning effective in quantum access models. Motivated by this question, we show that Pauli-structured representations of both the system matrix and the preconditioner allow the preconditioned operator to be accessed through regrouped Pauli expansions. In this setting, algebraic regrouping of Pauli products can reduce the Pauli coefficient weight of the preconditioned operator, thereby altering the normalization parameters relevant to quantum algorithms. We derive explicit size and coefficient-weight bounds for the regrouped Pauli representations, and we trace their consequences for both direct block-encoding constructions and randomized Pauli linear system solvers. These results identify when Pauli-structured preconditioning can reduce the effective complexity parameters of QLS algorithms, rather than merely improving the classical condition number. Numerical experiments on a finite-dimensional synthetic benchmark show reductions in norm-aware direct block-encoding diagnostics and in the randomized QLS per-sample depth proxy.

Mirror descent provides a geometric framework for learning in games, but its last-iterate behavior can fail in weakly stable regimes, where the dynamics may exhibit rotational or recurrent transients. Predictive mirror methods mitigate this issue by modifying the feedback entering the mirror update, yet standard predictive variants are typically introduced algorithmically and analyzed one at a time. This letter gives a variational route to predictive feedback by constructing a stochastic mirror differential game with an auxiliary memory state. Its stage cost couples two Fenchel terms: a strategic term evaluated at a predicted profile and a corrective term driven by realized feedback. The resulting equilibrium feedback induces two-channel predictive mirror dynamics in general mirror geometry. Under local mirror regularity, a quantitative local Bregman growth condition, and bounded Brownian diffusion, we establish finite-horizon local terminal-time bounds in expectation and with high probability, together with an exit-probability estimate for the localization neighborhood. The result provides a unified variational construction of the induced predictive-memory mirror flow together with a local stochastic certificate for last-iterate performance near stable equilibria.

We investigate spaces of prime congruences on tropical algebras and study their geometry, inspired by classical scheme theory. Our main strategy is to use tropical algebras associated to ordered monoids, which play the role of the monomial structure of these algebras. Using the space of prime congruences as local models, we introduce a tropical toric scheme which contains the usual tropical toric variety as a subspace. We show how the separatedness and properness of these schemes are captured by scheme-theoretic points. As an application of our framework, we obtain a necessary and sufficient condition for a prime congruence to be finitely generated.

We use de Sitter space to construct a concrete model for the measured wall structure of finite-dimensional hyperbolic space in hyperbolic model I^n, which will induce a medianization M(I^n). We get an upper bound for the Hausdorff distance between I^n and M(I^n). Especially the Hausdorff distance for I^n case is calculated.

This paper studies the Fourier spectral method (FSM) for the Schrödinger equation with singular potentials $V \in H^{s}$, where $s > \max\{d/2-2,-1\}$ and $d$ denotes the spatial dimension. This setting includes a broad class of singular potentials, such as the 3D Coulomb potential and the 1D Dirac-delta potential. First, we combine the Feshbach-Schur map with a refined perturbation argument to derive sharp convergence orders for FSM, yielding order $2s+2$ for eigenvalues and order $s+1$ for eigenfunctions in the $H^1$ norm. More importantly, the $H^1$ error with respect to the projected eigenfunction converges with a higher order $s+1+b$, where $b=\min\{s+2-d/2-\varepsilon,\; s+1,\; 2\}>0$ for arbitrarily small $\varepsilon>0$, revealing a super-convergence phenomenon. Second, in the presence of potentials with isolated point singularities, we develop an asymptotic-recovery (AR) technique to post-process the FSM solutions. The resulting method, dubbed AR-FSM, fully exploits the super-convergence property and achieves convergence orders $2s+2+2b$ for eigenvalues and $s+1+b$ for eigenfunctions in the $H^1$ norm, while the AR post-processing requires only a computational cost that is linear in the number of FSM degrees of freedom. The analysis introduces a rigorous definition of point singularities and develops a foundational framework for their study. It further establishes an asymptotic expansion of eigenfunctions consisting of a regular component in $H^{s+4}$ together with $d+1$ asymptotic functions associated with each singular point. Numerical experiments confirm the sharpness of these theoretical bounds.

We show that any entire, capillary minimal graph in a half-space must be linear in low-dimensions or, more generally, when some tangent cone at infinity does not split off a vertical line. We also show that the regular set of any entire, capillary-minimizing hypersurface must be connected, and we discuss connections with the one-phase Bernoulli problem.

We propose FlipItRight, a framework for stable planar pose-targeted throw-flip with a high-DoF manipulator. The task is decomposed into an object-level planner, which generates candidate release states satisfying the desired landing pose, and a robot-level planner, which evaluates executability and constructs a feasible swing motion. Treating the release state as an explicit intermediate representation enables principled candidate filtering, adaptive selection of release and pre-swing configurations, and structured near-release motion design – in particular, approximately constant end-effector velocities during the final swing phase to improve robustness to release-timing uncertainty. We validate on a real platform across objects of varying shape, size, and mass, achieving a 90% success rate across 120 trials. Ablation studies confirm that each design choice contributes to throwing performance, and the framework requires no prior data or learned model, enabling direct deployment on new objects and targets without environment-specific calibration or data collection.

Aubin–Talenti bubbles describe the decaying positive solutions of the zero-frequency critical Lane–Emden equation in Euclidean space. By appealing to bifurcation methods, Dancer constructed in his seminar paper \cite{DancerSolution} positive-frequency solutions to the Lane–Emden equation which decay in the noncompact directions and are periodic in one direction. Alternatively, we give in this paper an energy-based variational construction of such Dancer-type solutions via the semivirial-vanishing geometry developed in author's recent work for studying focusing NLS on waveguide manifolds. The main new ingredient is a strict sub-bubbling estimate below the Euclidean Sobolev threshold. Unlike the usual Brezis–Nirenberg mechanism, no lower-order focusing perturbation is available in our model. Instead, the energy drop is produced by the bounded periodic direction: truncating a Euclidean bubble to one period removes a leading-order part of the gradient tail, while the nonlinear tail is of lower order. This restores compactness of minimizing sequences and yields normalized ground states for every prescribed mass, thereby answering an open question from \cite{Luo_energy_crit}.

Condorcet domains are subsets of permutations that ensure pairwise majority voting yields acyclic outcomes, and they form an active area of research at the intersection of social choice theory and combinatorics. In this paper, we extend the theory of Condorcet domains to the broader setting of arbitrary finite Coxeter groups. The core contribution of our approach is the introduction of Condorcet root posets, defined on the chosen root systems. Notably, we establish a natural bijection between closed Condorcet domains and Condorcet root posets, which facilitates the study of Condorcet domains. Using this correspondence, we extend the median graph representation of closed Condorcet domains to arbitrary finite Coxeter groups, demonstrating that these domains can be characterized by the skeletons of their associated Condorcet root posets. These results are novel even in type $A$. Furthermore, these posets give a unified language that efficiently captures a wide range of desirable properties of Condorcet domains, such as being maximal, connected, peak-pit, and of tiling type. Using this framework, we strengthen and generalize several classical results: we establish that a maximal Condorcet domain is connected if and only if it is peak-pit; we prove that the tiling-type property is equivalent to the combination of being maximal and connected, and having maximal width; and we show that strictly positive voting profiles on connected Condorcet domains yield outcomes with only simple ties.

We study generalized \(BMO\)-type spaces associated with a normalized family of local quasi-Banach function spaces \(\mathbb X=\{X_Q\}_{Q\subset\mathbb R^n}\). For such a family we consider two oscillation seminorms: the mean-based seminorm \(BMO_{\mathbb X}\) and the best-constant seminorm \(BMO_{\mathbb X}^{*}\). The main purpose of the paper is to separate the two mechanisms that govern their comparison with classical \(BMO\). First, we introduce a lower median-testing functional \(\Lambda_{\mathbb X}\), which measures the nondegeneracy of the local norms on subsets occupying a fixed positive proportion of a cube. Using the John–Strömberg median oscillation characterization of \(BMO\), we prove that the condition \(\Lambda_{\mathbb X}(\lambda)>0\) for some \(0<\lambda<1/2\) implies the embedding \[ BMO_{\mathbb X}^{*}\cap L^1_{\mathrm{loc}}(\mathbb R^n) \hookrightarrow BMO . \] Second, we introduce a sparse testing seminorm \(T_{\mathbb X}\), which measures the compatibility of the local norms with sparse sums of characteristic functions. Using a sparse domination principle for \(BMO\) oscillation, we prove that \(T_X(\eta_0)<\infty\), where \(\eta_0\) is the sparsity parameter arising from the local sparse domination formula, implies \[ BMO\hookrightarrow BMO_{\mathbb X} . \] We also provide a sufficient small-set criterion for this sparse testing condition in terms of an upper testing functional \(\Psi_{\mathbb X}\).

We prove that the optimal pebbling number of the $n$-dimensional hypercube is \[ o(Q_n)\,=\,\Theta\!\left(\left(\frac43\right)^n\right). \]

The theory of submodular flows, introduced by Edmonds and Giles, is a cornerstone of combinatorial optimization, unifying network flows, matroid intersections and directed cut coverings. In this paper, we establish a measurable-space version of this framework, addressing the structural existence and duality questions raised as part of Problem~10.6 by Lovász in Submodular setfunctions on sigma-algebras, version 2. We develop a theory of submodular flows on standard Borel spaces and establish the measurable analogues of the existence and optimality theorems. Furthermore, we introduce a measurable notion of the residual graph and characterize extreme flows by combining a base-polytope intersection condition with an acyclicity condition for the measurable residual graph, generalizing the discrete geometric intuition to the infinite-dimensional setting. Finally, we apply the theory to constrained supply-demand problems on measurable bipartite graphs and to fractional measurable orientations.

We consider a generalization of the spectral gap of reversible Markov generators to nonreversible processes, following the recent work arXiv:2310.10876 on nonreversible finite-state Markov chains. Extending Chatterjee's observations, we find that this spectral quantity that we call the singular-value gap characterizes the convergence of empirical averages, providing upper and lower bounds for finite-time variance uniformly over $L^2$-functions. A key observation is that when the singular-value gap is positive, the generator is invertible on the $L^2$-orthogonal complement of constant functions. In particular, the Poisson equation $-Lf = g$ can be solved, which enables our proof and connects our results to asymptotic variance and associated central limit theorems. We also compare the singular-value gap with the spectral gap of the reversibilized process, the mixing time in total-variation distance, and the Cheeger constant. Several examples are provided throughout the text. Among other potential applications of the singular-value gap, these examples illustrate that a positive singular-value gap can help with variance reduction for observable classes in MCMC sampling, uncover slow-mixing mechanisms, and certify convergence of empirical averages for diffusion operators with complicated spectrum.

The computation of the numerical index of classical Banach spaces is one of the original problems in the theory. In this paper, we compute the numerical index of two-dimensional real \(\ell_p\)-spaces for all \(p\ge1\). More precisely, we prove that \[ n(\ell_p^2)=v(J), \qquad J= \begin{pmatrix} 0&1 -1&0 \end{pmatrix}, \] confirming the conjectured formula in the two-dimensional real case.

I was taught that $e^x = x$ has no solution, and taught to leave it at that. But in mathematics "no solution" has usually meant "not on this line yet": $x^2 = -1$ waited for the complex plane, and $e^x = x$ turns out to be waiting there too. Over $\mathbb{C}$ the exponential has a fixed point $\varrho = 0.318\ldots + 1.337\ldots\,i$, the unique solution of $\exp(z) = z$ in the strip $0 < \operatorname{Im} z < \pi$ (equivalently $-W_{-1}(-1)$), and it carries more structure than its one-line definition lets on. At $\varrho$ the rectangular and log-polar coordinates of a complex number coincide, forcing the identities $\operatorname{Re}\varrho = \log|\varrho|$ and $\arg\varrho = \operatorname{Im}\varrho$. As a dynamical point $\varrho$ is repelling for $\exp$ and attracting for $\log$, linearizable for both by one Koenigs coordinate, and the base of a transpose identity $w^\varrho = \varrho^{\log w}$. It generates an aperiodic log-polar lattice and sits a hair off a clean relation with $\pi$, namely $\operatorname{Re}(\varrho)\,\pi = 0.99944\ldots$. Passing to the octonions, the fixed points of $\exp_{\mathbb{O}}$ fill concentric six-spheres, the innermost $\operatorname{Re}(\varrho) + \operatorname{Im}(\varrho)\,S^6$, whose triples obey an exact identity $I_4^2 + \tfrac14 I_5^2 = \operatorname{Gram}$ carrying one invariant, a twist angle, absent from ordinary spherical trigonometry. Throughout, what is proved is kept apart from what is only computed.

Fix a multivariate polynomial $\mathfrak{p}$ in $n$ non-commuting variables of arbitrary degree, and consider $n$ independent $N\times N$ complex Ginibre matrices $X_1^N,\cdots,X_n^N$. We prove that the empirical spectral distribution of $P^N=\mathfrak{p}(X_1^N,\cdots,X_n^N)$ converges as $N$ tends to infinity to the so-called Brown measure of $\mathfrak{p}$ evaluated at free circular variables. For polynomials of degree at most 2, the convergence was proven by Cook, Guionnet, and Husson \cite{cook2022spectrum}, and we prove that the convergence in fact holds for polynomials $\mathfrak{p}$ of any degree. The main step in the proof is a least singular value lower bound for $P^N-z$ for almost all complex shifts $z$, and we prove this via a least singular value lower bound for a wide class of tensorized Ginibre matrices of finite type with a deterministic shift, which is of independent interest. We further show that the Brown measure convergence holds beyond Gaussians: the same convergence holds when the entry law has mean 0, variance 1, bounded density on $\mathbb{C}$ and finite moments of all orders.

The Bayesian paradigm offers principled tools for sequential decision-making under uncertainty, but its reliance on a probabilistic model for all parameters can hinder the incorporation of complex structural constraints. We introduce a minimalist Bayesian framework that places a prior only on the location of the optimum, while eliminating nuisance parameters through profile likelihood. This yields a generalized posterior that naturally accommodates structural constraints. As a direct instantiation, we develop MINimalist Thompson Sampling (MINTS). For multi-armed bandits with mean constraints, we establish near-optimal non-asymptotic regret guarantees and sharp almost-sure asymptotic regret characterizations. In particular, MINTS attains the classical Lai–Robbins constant in the unstructured setting and automatically adapts to unimodal structure, achieving the sharp constant determined only by the immediate neighbors of the optimal arm.

Let $\K$ be an arbitrary field. Extending the structural frameworks established in \cite{JZ2025-2}, this paper introduces novel techniques utilizing mirror reduction orders, prime words, and the generalized lifting methodology. We establish a definitive necessary and sufficient criterion for the affineness of Noetherian Hopf algebras, thereby providing a significant advancement toward resolving the long-standing Wu–Zhang question \cite{WZ2003}. Specifically, we prove that a left or right Noetherian Hopf algebra over $\K$ is affine if and only if its Hopf coradical is affine. This characterization fundamentally reduces the verification to the first filtration step, yielding a criterion that is structurally transparent and highly operational. To establish necessity, we provide an essential intrinsic result demonstrating that the Hopf coradical of an affine Hopf algebra inherits the property of being affine. Furthermore, as direct applications of this equivalence, we prove that a left or right Noetherian Hopf algebra $H$ is affine provided that its coradical $H_{(0)}$ forms a subalgebra (the dual Chevalley property), its coradical $H_{(0)}$ is cocommutative, or its Hopf coradical $H_{[0]}$ is commutative.

Büyükaşik et al. [Publ. Math. Debrecen, 2024] introduced a family of generalizations of Euler's totient function $\varphi(n)$, by setting $\varphi_k(n) = \sum_{a} a^k$ for $a \in [1, n]$ such that $(a, n) = 1$, with $\varphi_0(n) = \varphi(n)$. Letting $\mathcal{D}_{s} = \{ k \geq s : \forall n \geq 1 \ \varphi_s(n) \mid \varphi_k(n) \}$, Büyükaşik et al. proved that $\mathcal{D}_{s}$ is finite for each $s \geq 0$, and conjectured that $\mathcal{D}_{1} = \{ 1, 3, 15 \}$ and provided computations to support this conjecture. We succeed in proving this conjecture, using an argument based on our extensive interactions with GPT-5.5 Pro.

Consider a random geometric graph $G_M(n;r)$ whose vertex set consists of $n$ points chosen independently and uniformly from a Riemannian manifold $M$, with edges joining pairs of vertices whose distance in the metric $d_M$ is at most $r$. Let $\Delta$ denote the expected average degree of the graph. As is the case for Erdős-Rényi graphs, there is a critical value $\Delta_c$, depending only on the dimension of $M$, such that if $\Delta > \Delta_c$ then $G_M(n;r)$ has a giant component. We show that whenever $\Delta > \Delta_c$, the giant component of $G_M(n;r)$, equipped with the graph distance, converges to the underlying manifold $M$ in the Gromov-Hausdorff distance after rescaling by an appropriate deterministic factor. Our result holds for $\Delta$ depending on $n$ as well, provided $\Delta = o(n)$ and $\Delta \geq \Delta_c + \varepsilon$ for any fixed $\varepsilon > 0$. As a consequence, we show that for any pair of non-isometric compact Riemannian manifolds $M_1$ and $M_2$, there is a polynomial-time algorithm that distinguishes random geometric graphs on $M_1$ and $M_2$ throughout this regime of $\Delta.$ In the thermodynamic regime – i.e.\ when $\Delta$ is constant – our results appear to be new even in the classical cases where $M$ is a sphere or a torus. Our proof makes use of techniques from first-passage percolation which allow us to understand the long-range behavior of the graph distance on small, approximately Euclidean patches of $M$, together with global arguments that glue these local estimates into a global description.

We propose the Alternating Learning-Based Collocation (ALBC) method for solving inverse elliptic problems. Our approach employs sinusoidal shallow networks as adaptive basis generators. By alternately updating the state variable and the unknown parameter, we decompose the original nonconvex joint optimization problem into a sequence of tractable linear subproblems. This strategy effectively overcomes the fixed-basis limitations of classical collocation methods while avoiding the slow convergence typically encountered in deep learning approaches. Theoretically, we establish stability estimates and prove the convergence of the proposed algorithm. Numerical experiments on five benchmark problems demonstrate the efficacy of ALBC, which consistently outperforms the standard collocation method in accuracy. Furthermore, it achieves performance comparable to or better than that of physics-informed neural networks at a substantially lower computational cost. Finally, the method remains robust under noise levels of up to twenty percent.

In this work, we address the stability of $N$-soliton solutions to the completely integrable modified Camassa–Holm (mCH) equation. Recently, Li, Liu, and Zhu (Math. Ann. 392 (2025), 899–932) established the orbital stability of 2-soliton solutions in $H^4(\mathbb{R})$ with respect to the solution $u$ and highlighted the stability of mCH $N$-soliton solutions remains an urgent challenge. Motivated by their work, we systematically investigate the stability of mCH $N$-solitons. We first employ the bi-Hamiltonian structure of mCH to construct a novel hierarchy of explicit conservation laws with well-defined regularity domains. Then by formulating an appropriate Lyapunov functional, we apply the Inverse Scattering Transform to conduct a rigorous spectral analysis on the recursion operators. Finally, we demonstrate that the mCH $N$-solitons are non-isolated constrained minimizers of a variational problem. Our analysis proves that the $N$-soliton solutions of the mCH equation are both dynamically and orbitally stable in $H^{N+1}(\mathbb{R})$. Notably, when reduced to the 2-soliton case, our framework establishes stability in $H^3(\mathbb{R})$, which improves upon the existing regularity threshold.

In this paper, we introduce a new framework for designing stabilization-free virtual element methods (VEMs) based on an finite element interpolation-based strategy, where we can simultaneously eliminate the stabilization terms in the discretizations of diffusion and reaction terms. The core idea is to construct a computable, polynomial-preserving, and norm-equivalent interpolation operator from the virtual element space to a (local) finite element space. Leveraging the properties of this operator, we design two types of stabilization-free schemes. The first scheme requires the interpolation to preserve the polynomial consistency related to the bilinear forms, thereby maintaining both consistency and stability as in the standard VEM. The second scheme relaxes this consistency requirement. While it may not satisfy the standard polynomial consistency, the second scheme retains optimal convergence with simpler construction, fewer degrees of freedom and, more importantly, applicable to more complex problems such as those involving nonlinearities or variable coefficients. We construct concrete interpolation operators for both conforming and nonconforming virtual elements in two and three dimensions. These operators are then employed to realize stabilization-free schemes for conforming and nonconforming VEMs. Numerical experiments confirm the optimal convergence rates of the proposed methods. The presented framework can be extended to design stabilization-free schemes for other polytopal discretization methods, such as the hybrid high-order method and the weak Galerkin method.

Let $G$ be a connected reductive algebraic group defined over $\mathbb{R}$. In the 1980s, Arthur conjectured the existence of certain packets of irreducible admissible representations of $G(\mathbb{R})$ satisfying various remarkable properties. These packets were given a precise definition in the book of Adams, Barbasch, and Vogan in terms of microlocal geometry on a space of Langlands parameters. A longstanding conjecture, originally due to Arthur, is that all Arthur packets consist of $\textit{unitary}$ representations. In this paper, we prove this conjecture in general. The main new idea is a `Jordan decomposition' for Arthur packets: a canonical two-step process for realizing an arbitrary Arthur packet via real parabolic and cohomological induction from a unipotent Arthur packet for a certain Levi subgroup. This process is analogous to the decomposition of an element of a complex algebraic group as a (unique) commuting product of elliptic, hyperbolic, and unipotent parts. Using our Jordan decomposition, we reduce the question of unitarity to the case of unipotent Arthur packets, where the answer is already known (by work of Adams-Arancibia-Mezo, Adams-van Leeuwen-Miller-Vogan, Arthur, Barbasch, Barbasch-Ma-Sun-Zhu, and Davis-Mason-Brown). As an application of the same methods, we also give a proof of Jiang's conjecture for real reductive groups, which gives an upper bound on the wavefront sets of the members of an Arthur packet in terms of the Barbasch-Vogan dual of the Arthur $SL_2(\mathbb{C})$.