Papers

In this paper, we study the Schrödinger–Bopp–Podolsky system \begin{equation*} \begin{cases} -\Delta u + V(x)u + \phi u = f(x,u), & \text{in } \mathbb{R}^3, -\Delta \phi + a^2 \Delta^2 \phi = 4\pi u^2, & \text{in } \mathbb{R}^3. \end{cases} \end{equation*} We consider the case where the potential \(V\) is indefinite so that the Schrödinger operator \(-\Delta + V\) has a finite-dimensional negative space. Under suitable assumptions on the potential \(V\) and nonlinearity $f(x,u)$, we prove the existence of nontrivial solutions via a local linking argument and Morse theory. Moreover, these solutions are shown to decay exponentially at infinity. Additionally, a ground state solution is obtained by minimization techniques. Finally, if \(f(x,u)\) is odd with respect to \(u\), we obtain an unbounded sequence of solutions using the symmetric mountain pass theorem.

Persistent aerial surveillance using multi-unmanned aerial systems (UASs) requires decentralized coordination, continuous team reconfiguration, and provable mission correctness despite limited onboard energy and communication constraints. This paper develops a distributed framework for continuous aerial surveillance under bounded Linear Temporal Logic (LTL) mission specifications. The proposed approach partitions the UAS team into stationary anchors and mobile workers operating under cyclic replacement modes, and constructs a deep neural network (DNN)-inspired communication topology that enables fully decentralized coordination through local interactions. A hierarchical bounded LTL specification formally captures mode-to-mode reference consistency, cyclic team rotation, finite-time reachability, trajectory tracking, and prescribed surveillance coverage. By proving the finite-time convergence of the worker-agent coordination dynamics, the paper guarantees the finite-time satisfaction of the mission specification. To maximize sensing effectiveness, an information-theoretic optimization framework synthesizes the reference configuration of newly deployed worker agents by minimizing the Kullback–Leibler divergence between the surveillance-node distribution and the induced coverage density. The resulting reference configuration uniquely determines a deterministic, mode-dependent communication topology, eliminating online communication-graph optimization while preserving the formal mission guarantees. Finally, a decentralized quadrotor controller realizes the distributed references using only local communication. Numerical simulations demonstrate cyclic team reconfiguration, decentralized communication-topology synthesis, finite-time formation convergence, and certified persistent surveillance coverage.

We propose an integrated sensing and communications (ISAC) framework that supports chirp signal transmission in CP-OFDM-based multiple access communication systems, enabling efficient coexistence of communication and sensing capabilities. Our framework employs the discrete Fourier transform phase rotated and permuted frequency division multiple access (DFT-p-FDMA) waveform to transmit chirp signals using a portion of the frequency resources, while ensuring interference-free concurrent CP-OFDM data transmissions on other bands. We analyze the effective channel behavior under the DFT-p-FDMA waveform, characterizing how delays and Doppler shifts impact radar target echoes. We also show how processing multiple received symbols improves Doppler resolution in practical scenarios. Our framework allows flexible adjustment of range-Doppler resolution through optimized time-frequency resource allocation, offering a versatile solution for ISAC applications. Simulation results validate the framework's performance in delay and Doppler estimation, highlighting its potential to support ISAC in next-generation wireless networks.

We refine the Matomäki-Radziwiłł method for short averages of multiplicative functions. For the Liouville function and multiplicative functions supported on smooth numbers, we prove decay bounds that are essentially optimal with the Matomäki-Radziwiłł method. The key new ingredient is a sharper treatment of the sieve error, achieved by introducing a more widely separated final prime range when restricting to integers with typical factorizations. We additionally give a weaker but still improved bound for arbitrary 1-bounded multiplicative functions and discuss some limitations of the method. As an application, we give an improved bound for the averaged Chowla conjecture of Matomäki-Radziwiłł-Tao that seems essentially best possible via their method.

The $\mu$-basis is an algebraic tool originating from the theory of moving curves and moving surfaces, and it is widely used in the study of rational curves and surfaces. In this paper, we give the explicit formulas for the $\mu$-basis of planar quartic rational parametric curves based on redefined vector polynomials, and several illustrative examples are provided. Meanwhile, we also discuss the corresponding cases for quadratic and cubic curves.

Let $R$ be a ring with identity. The generalized von Neumann inverse graph of $R$, denoted by $\Gamma_{Reg}(R)$, is defined as the graph whose vertex set is $Reg(R)$, where two distinct vertices $a,b \in R$ are adjacent if and only if $aba=a$ or $bab=b$. In this work, we consider the reduced graph $\Gamma'_{Reg}(R)$ obtained by restricting the vertex set to $Reg(R)\setminus{0_R}$, so that $\Gamma_{Reg}(R) \cong K_1 + \Gamma'_{Reg}(R)$, allowing the analysis to focus on its nontrivial structure. We investigate the structure of $\Gamma'_{Reg}(R)$ for finite commutative von Neumann regular rings and establish several results describing its graph-theoretic properties in relation to the algebraic structure of $R$. In particular, we derive conditions that characterize connectivity, acyclicity, and planarity, and examine structural features such as vertex degrees, girth, and the existence of pendant vertices, along with their algebraic implications. We also identify circumstances under which $\Gamma'_{Reg}(R)$ exhibits specific graph classes, including paths, cycles, and wheels, as well as the presence of certain induced subgraphs. Furthermore, an explicit algorithm is provided to construct $\Gamma'_{Reg}(R)$, and connections with the inclusion ideal graph of $R$ are discussed, offering additional insight into the interplay between ring-theoretic properties and graph structures.

In this work, we present a comprehensive rounding error analysis for RCLUPPr, a recently developed randomized CholeskyQR-type algorithm introduced in \cite{RCLUPP}. This method performs LU factorization with partial pivoting (LUPP faztorization) directly on a full-rank tall-skinny matrix $X \in \mathbb{R}^{m \times n}$. Unlike RCLUPP in \cite{RCLUPP}, which applies matrix sketching prior to LUPP factorization, RCLUPPr deploys LUPP as an initial preconditioning step to significantly mitigate the propagation of error significantly. Our rigorous analysis demonstrates that RCLUPPr exhibits superior robustness when applied to the ill-conditioned matrices compared to other CholeskyQR-type algorithms in the high precision. Furthermore, we develop some practical strategies in the implementation to accelerate RCLUPPr. Extensive numerical experiments on the real-world problems validate our theoretical findings, showcasing the robustness and the efficiency of RCLUPPr across the single, double and the mixed-precision frameworks.

We prove, within Bishop's constructive framework, stronger forms of two theorems in the author's 1988 paper about the locatedness of certain Lebesgue measurable complemented sets in R^{N}. In doing so, we expand and improve the proofs of several preliminary results on Lebesgue measurability and convexity that are need en route to our main theorems.

The recent breakthrough work of Chatterjee, Ghosh, Gurjar, Raj and Thierauf [CGGRT26] gives the first deterministic NC algorithm for the bipartite matching problem. They show how to detect as well as find perfect matchings in bipartite graphs in NC. In this note we present an arguably simpler-to-state variation of the NC detection criterion of [CGGRT26], with improved parameters.

In this paper, we analyze the effect of perturbation power in delaying the first error position, i.e., the first information bit incorrectly decoded by the successive cancellation (SC) decoding. It is conducted over the finite-length perturbation-enhanced SC (PE-SC) decoding paradigm. We show that the FEP delaying probability exhibits a non-monotonic dependence on the perturbation power \(\sigma_{p}^{2}\). Based on this property, an efficient perturbation power selection algorithm that maximizes the delay probability is proposed to enhance the perturbation efficiency. It results in a more efficient perturbation power selection in finite-length PE-SC decoding.

We refine Radhakrishnan's entropy proof of the Brégman-Minc bound by introducing a terminal-set framework in which selected vertices are revealed last. This gives new degree-sensitive upper bounds for the number of perfect matchings in bipartite graphs and an explicit formula for single-vertex terminal sets. The bounds recover the standard Brégman equality family and improve the estimate for certain nonuniform degree sequences. We also obtain a $C_4$-free refinement complementary to the edge-count bound of Araujo, Balogh and Wang.

Conventional pinching antenna (PA)-assisted integrated sensing and communication (ISAC) architectures typically assume static receiver locations or predetermined receive waveguides, thereby underutilizing the inherent spatial degrees of freedom. This paper proposes a novel mode-selectable PA-assisted ISAC framework to maximize the post-combining sensing signal-to-noise ratio while satisfying multi-user quality-of-service constraints by jointly optimizing the waveguide mode selection, transmit beamforming, and transmit/receive PA positions. To tackle the resulting mixed-integer nonconvex optimization problem, we develop a low-complexity block-coordinate descent algorithm that leverages a penalty-based majorization-minimization method to achieve high-quality suboptimal solutions. Numerical results demonstrate that the proposed design significantly outperforms both traditional PA and fixed-antenna benchmarks by synergistically harnessing spatial adaptability and modal reconfigurability. In particular, the mode-selectable design enables the coordinated optimization of transmit/receive operations and sensing-communication resource allocation, thereby maintaining sensing robustness under stringent communication requirements.

This paper focuses on the stability of a closed characteristic and invariant sets on compact star shaped hypersurfaces in ${\bf R}^{2n}$ with finitely many simple closed characteristics. Firstly, it is proved that all closed characteristics are non-hyperbolic, under some index condition weaker than dynamical convexity. Secondly, when $n=3$, it is proved that all closed characteristics are elliptic when their number is exactly $3$, under non-degeneracy and some minor index condition. Lastly, it is proved that all closed characteristics are either degenerate at some iteration or not locally maximal under dynamical convexity.

In these notes, we explain in details the construction of the universal C*-algebra as done by Blackadar, starting with the construction of the free *-algebra. At the end, we explain the extension done by Boava and the author to include relations described using the strong operator topology. As an example, we show that a graph C*-algebra can be defined using infinite sums.

Logic and optimization can, in combination, make valuable contributions to rule-based AI. Logic is the obvious medium for encoding a rule base and drawing inferences from it, while optimization provides a powerful technology for computing inferences. Their combination has taken on new relevance amid a growing concern for transparency in AI. which is important for reproducibility, explainability, trustworthiness, and fairness. Rule-based AI provides a natural solution to transparency that is becoming increasingly practical due to today's highly advanced optimization methods. This article surveys several areas of logic-optimization partnership, including probabilistic logic, Bayesian logic, belief logics and Dempster-Shafer theory, nonmonotonic (default) logic, many-valued logics, and inference of logical formulas from noisy data based on Boolean regression. It shows how to compute projections, the fundamental problem of both logic and optimization, using decision diagrams and logic-based Benders decomposition. It describes the use of postoptimality analysis to explain how conclusions are reached, further enhancing transparency, as well as the role of optimization in answer set programming modulo theories. The paper concludes by suggesting possible future research directions.

Binary Iterative Hard Thresholding (BIHT) is a simple, yet effective, greedy method for recovering a sparse vector from one-bit sign measurements. In its original form, BIHT performs a “gradient-descent” step, followed by hard thresholding. A convergence analysis of this algorithm was left open in the introductory work of Jacques, Laska, Boufounos, and Baraniuk [IEEE Trans. Inform. Theory (2013)] and has remained unresolved for over a decade, with subsequent sharp analyses studying a normalized variant instead, that additionally projects every iterate onto the unit sphere. This paper resolves that gap and characterizes when per-iterate normalization is algorithmically necessary. In the noiseless setting, we prove a universal, sample-optimal convergence theorem for the original BIHT algorithm. Specifically, with $\widetilde O(s/\epsilon)$ measurements, a deterministic, computable finite-time iterate has directional error at most $\epsilon$, simultaneously for every $s$-sparse unit vector. This matches the optimal sample dependence achieved by normalized BIHT in prior work. Under sign corruptions, we prove a sharp separation. If at most a $\tau$ fraction of signs are flipped adversarially, then BIHT, without per-iterate normalization, still reaches the robust error floor at an early iterate with a matching $\widetilde O(s/\epsilon)$ sample complexity rate as its normalized variant. This recovery, however, is not stable. We prove a scalar lower bound showing that any nontrivial corruption pattern, even one that involves only one flipped sign together with one clean sign, forces the iterates to oscillate indefinitely. Consequently, no general last-iterate convergence theorem can hold for BIHT under sign corruptions, while its normalized surrogate provably escapes this instance.

We prove that every connected simple graph of bounded degree satisfying the classical dimension-free Bakry–Émery condition $\mathrm{CD}(0,\infty)$ for the unnormalised Laplacian is volume doubling and supports, at all integer graph scales, a scale-invariant $L^2$-Poincaré inequality with dilation two, with constants depending only on the maximum degree. This settles the polynomial-growth conjecture of Cushing, Liu, and Peyerimhoff in a stronger form. The main novelty is a dimension-free adaptation of the graph-theoretic modified nonlinear heat-flow method introduced by Münch and extended to infinite weighted graphs by Pajot and Russ: point-mass consequences of $\Gamma_2\geq0$ and positive-resolvent smoothing replace any global $\mathrm{CD}(0,n)$ reduction, while diffusive exit-time control and finite-volume localisation yield the Poincaré inequality.

Newman polynomials have coefficients in {0, 1} and constant term 1, whereas Littlewood polynomials have coefficients in {-1, 1}. We study two questions concerning the existence of Newman and Littlewood multiples for certain integer polynomials. For Newman multiples, we revisit a question of Hare and Mossinghoff [6]: whether there exists a real number sigma > 1 such that every P in Z[x] with no nonnegative real root and Mahler measure less than sigma has a Newman multiple. To bound any possible value of sigma from above, we seek polynomials with no nonnegative real root and no Newman multiple, and with Mahler measure as small as possible. Using the updated database of known small-Mahler-measure polynomials up to degree 200 [11], we test, for each listed polynomial p(x), both p(x) and its sign transform p(-x). These two polynomials have the same Mahler measure, but the existence of a Newman multiple is not invariant under the substitution x -> x. After a degree-bounded prefilter formulated as a mixed-integer linear optimization problem, we apply the Hare-Mossinghoff certification procedure to the remaining candidates. This yields 14 irreducible polynomials of Mahler measure less than 1.3, with no nonnegative real root and no Newman multiple. The smallest of their Mahler measures is 1.263095875491..., showing that any such sigma is at most this value. For Littlewood multiples, we return to the three Newman polynomials listed in Table 3 of Drungilas, Jankauskas, Junevicius, Klebonas and Siurys [3], for which the existence of a Littlewood multiple of smallest possible degree was left unresolved. We show that one of them attains this degree, whereas the other two have no Littlewood multiple at either of their first two possible degrees.

We give a finite-dimensional boundary-residence mechanism for Takens-type historic behavior. For a class of continuous self-maps of a simplex, a multiplicative Lyapunov drift toward the boundary, combined with cyclic residence near separated boundary regions, forces every non-fixed interior orbit to have non-convergent Cesàro averages and non-convergent empirical measures. Thus the historic set has full relative Lebesgue measure. The mechanism is independent of hyperbolic specification, wandering domains and positive entropy: in fact, for the maps considered here the topological entropy is carried entirely by the boundary restriction. The non-convergence persists under all positive powers and, more generally, under multiplicatively thick sampling times. The abstract criterion applies to Stein–Ulam and Lotka–Volterra type stochastic operators once the model-specific cyclic coding and logarithmic residence estimates are verified. In explicit cyclic-collar models we identify the full weak-* accumulation set of empirical measures; this accumulation set is a non-trivial circle of probability measures and yields polynomial pointwise emergence of order \(\varepsilon^{-1}\). We also obtain examples with positive-dimensional omega-limit sets by combining the mechanism with a Barański–Misiurewicz type boundary-tracing construction.

Coherent Hamiltonian drifts and systematic errors can pass through quantum error correction differently than stochastic Pauli noise: they generate phase components of the logical channel that are lost under full Pauli twirling. We consider the Lindbladian dynamics of \(n\) physical qubits propagated through an effective logical channel of a fixed encoding, recovery, and decoding scheme. The work is methodological in character: standard ideas of observability and minimal realization from linear systems theory are applied to the channel \[ \omega \mapsto \Phi_t(\omega) = \cD e^{t\cL}\cE(\omega), \] where \(\cE\) encodes the logical state and \(\cD\) is the chosen recovery map with final decoding. The minimal code-visible space \(\cK_{\rm code}\) is defined as the Krylov closure of the decodable logical observables with respect to \(\cL^\dagger\); its observable factorization yields a minimal realization of the logical channel. In this form, the Knill–Laflamme conditions serve as a short-time consistency check, Pauli noise on stabilizer codes reduces to a chain on syndrome-logical classes, and the choice of recovery map acts as a spectral filter on the visible modes. For quantum memory, the same language yields an estimate of the storage time in terms of the dominant visible eigenvalues of the logical channel of the correction cycle. Particular attention is paid to the spectral signatures of coherent noise and to how these signatures depend on the chosen decoder.

The sign problem is the non-positive path-integral weight that makes a quantum system hard to simulate classically. It shows up in problems that look unrelated, from fermion antisymmetry and geometric frustration to real-time evolution, topological angles, and finite density. We study it here for compact continuous variables, modeled as networks of quantum rotors, $\hat H=\sum_i 4E_C(\hat n_i-n_{g,i})^2-\sum_{\langle ij\rangle} E_{ij}\cos(\hat\phi_i-\hat\phi_j-\theta_{ij})$. For these, a single gauge-invariant quantity controls the obstruction, the frustration flux through the independent loops of the interaction graph. We prove a sign rule, the rotor analogue of Marshall's. The charge-basis Hamiltonian is sign-free if and only if every loop flux vanishes modulo $2\pi$. Exact diagonalization and density-matrix renormalization then show that the sign cost is a gauge-invariant function of the flux. For a single loop it vanishes at zero flux, peaks at $\pi$, and is exponentially small in the loop's perimeter. Summed over many loops it is extensive, growing with system size. The same loop flux generates the sign of three problems usually treated apart, namely the Mott transition (through the slave-rotor mapping), compact $U(1)$ lattice gauge theory, and frustrated continuous optimization. A superconducting rotor array realizes both the model and its sign natively.

This article describes Scaffold Write Debug, a three-stage instructional approach used in an introductory real analysis course to support students learning mathematical proof. The approach combines guided proof completion, independent proof writing, and the analysis and correction of faulty proofs. The goal is to help students understand proof structure, develop confidence in writing proofs, and become more careful readers of mathematical arguments. The paper presents classroom examples, instructor observations, and student responses, and discusses how the approach can be incorporated into a proof-based course.

Bilu and Linial conjectured that if $d\ge 2$, then every $d$-regular graph $G$ has an edge signing $\sigma:E(G)\to\{-1,1\}$ such that its signed adjacency matrix $A_\sigma$ satisfies the Ramanujan bound \begin{equation*} \rho(A_\sigma)\le 2\sqrt{d-1} \end{equation*} and they proved that \begin{equation*} \rho(A_\sigma)=O\!\left(\sqrt{d\log^3 d}\right). \end{equation*} Using a method of interlacing polynomials, Marcus, Spielman, and Srivastava confirmed one side of this conjecture that there is a signing $\sigma$ for which \[ \lambda_{\max}(A_\sigma)\le 2\sqrt{d-1}. \] By constructing an auxiliary bipartite graph from a balanced orientation of $G$ and applying a bipartite signing argument, we prove that every finite simple graph $G$ of maximum degree $d\ge 3$ admits an edge signing $\sigma$ such that \[ \rho(A_\sigma) \le 4\sqrt{\left\lceil \frac d2\right\rceil-1} \le 2\sqrt{2(d-1)}. \] Thus we obtain a bound within a factor at most $\sqrt2$ of the conjectured Ramanujan bound.

With the increased adoption of robotic agents operating in human environments by scanning and sharing 3D representations (e.g., for fleet learning, cloud-based planning, or collaborative mapping), collected point clouds reveal not just the objects in a scene but also sensitive spatial context, such as room function or information that occupants never consented to disclose. Traditional point cloud encoders offer no principled control over this: either all is preserved, or none. Hence, we introduce RoboShape, an information theory guided compression head following the frozen Sonata encoder. We project voxel-level embeddings using the Donsker-Varadhan formulation of mutual information (MI). Specifically, we maximize the MI between embeddings and object-level understanding while minimizing it for private attributes. RoboShape leads to 87.5% smaller embeddings that retain 98.7% of object classification utility while collapsing sensitive attribute predictions by 39.3% across the three real-world indoor LiDAR datasets. Its privacy-preserving embeddings are cheaper to transmit over the network or to train a model for any downstream tasks. We release the RoboShape codebase to give the robotics community a practical, encoder-agnostic tool for building perception pipelines that are compact, privacy-aware, and deployment-ready.

Derivative-free optimization (DFO) is challenging when the observation channel varies over time and evaluations are noisy. Conventional model-based methods assume stationary observations; under temporal drift, historical data biases gradient estimates and acceptance tests confound latent-objective decrease with temporal variation. We propose TOBYQA (Time-augmented Optimization BY Quadratic Approximation), a regularized model-based DFO framework that jointly incorporates spatial geometry and temporal variations within a single saddle-point interpolation system. TOBYQA augments the classical least-Frobenius-norm quadratic interpolation system with a linear-in-time drift term and a ridge regularization on the residual kernel, which accommodates noise and ensures well-posedness when the constraint block has full column rank, relaxing the geometric poisedness requirements of classical interpolation. We prove that under affine temporal drift, the recovered gradient is algebraically invariant to the drift rate for any noise scale and sample radius. This property leads to a drift-compensated acceptance test that subtracts the estimated temporal component from the observed reduction. Driven by an adaptive cubic regularization scheme with a closed-form step and a geometry-guarded statistical stationarity stopping rule, TOBYQA achieves an expected oracle complexity of $O(\varepsilon^{-2})$. Benchmark evaluations across diverse temporal drift regimes show that, at tolerance $\tau=10^{-3}$, TOBYQA solves 71.0%, 60.8%, and 41.9% of instances at $n=6$, $10$, and $20$, respectively, compared with 25.1%, 18.5%, and 14.9% for the best-performing comparison method. These results demonstrate higher solve rates under temporal drift while retaining comparable performance in static environments.

Wasserstein distributionally robust optimization (DRO) is commonly built around the empirical distribution, with the ambiguity radius selected from a concentration bound. Although this construction provides useful statistical guarantees, it can be conservative and does not fully exploit predictive information about the underlying distribution or the difficulty of a particular decision problem. We develop a more flexible framework in which a predictive model determines the nominal distribution and a separate model estimates a data-dependent radius. The key requirement is not that the ambiguity set be centered at the empirical distribution, but that it contain the unknown data-generating distribution with the desired probability. We establish finite-sample guarantees and asymptotic consistency for arbitrary learned centers, derive tractable reformulations for non-uniform discrete predictive distributions, separate predictive-model and scenario-discretization errors, and prove stability under simultaneous perturbations of the center and radius. We further characterize the oracle conditional-quantile radius as the smallest conditionally valid rule and introduce a split-conformal procedure for finite-sample marginal calibration. Experiments on newsvendor problems, synthetic portfolios, distribution shifts, and real financial data show that learned and calibrated ambiguity sets can improve reliability, but do not automatically yield smaller radii or better decisions. Overall, the proposed framework treats calibration as a practical mechanism for reliable decision making rather than a universal guarantee of improved optimization performance.

In order to represent the complex dynamics of contemporary energy systems, a novel fractional-order energy supply-demand model with time delay is presented in this investigation. The fractional model, compared to traditional integer-order models, naturally accommodates for memory and genetic effects, and the incorporation of delay component takes into account unavoidable lags in energy production, transmission, and consumption. To ensure the mathematical rigor of the proposed model, we prove the existence and uniqueness of solutions. We additionally examine into the model's stability within the Ulam-Hyers concept and demonstrate that it is resilient to minor uncertainties and perturbations. To handle fractional derivatives with delay systems, we utilize the Grnwald-Letnikov (GL) discretization scheme, which offers a straightforward and effective method for approximating the solutions. The impact of delay parameters and fractional orders on system behavior is investigated numerically, demonstrating how they shape oscillations, convergence rates, and equilibrium states. According to the results, fractional-order modeling with delay, reinforced by the GL discretization scheme, provides a flexible and realistic framework for examining energy dynamics. This framework gives important insights for long-term policy planning, supply management, and demand forecasting. Additionally, the framework lays the groundwork for upcoming additions that incorporate optimization techniques, stochastic effects, and the integration of renewable energy sources, all of which will further the development of effective and sustainable energy systems. We also discuss three sensitivity analysis scenarios including high demand combined with low supply, high renewable and reduced and imports low demand with high imports which the results show stable and efficient results.

We study ideal class maps on projective spaces and Grassmannians over a number field. We prove weak approximation in every fixed ideal class fiber on positive-dimensional linear subspaces and on Grassmannians. We also introduce local defects of morphisms of projective spaces and use them to describe the ideal classes arising from the image of a linear subspace as a finite union of cosets in the class group.

This work is part of a research program to compute the Hochschild homology groups $\text{HH}_*(\mathbb{C}[x_1, ..., x_d]/(x_1, ..., x_d)^3; \mathbb{C})$ via Coxeter cohomology, which utilizes the isomorphism \[ \text{HH}_i(\mathbb{C}[x_1, ..., x_d]/(x_1, ..., x_d)^3; \mathbb{C}) \cong \sum_{0 \leq j \leq i} H_C^j\bigg( S_{i+j}, V^{\otimes(i+j)} \bigg)\] provided by Larsen and Lindenstrauss. Here, $H^*_C$ denotes Coxeter cohomology, $S_{i+j}$ is the symmetric group on $i+j$ letters, and $V$ is the standard representation of $\text{GL}_d(\mathbb{C})$ on $\mathbb{C}^d$. While previous work has focused exclusively on the case $d=2$, we extend some results to all values of $d$. Notably, we show that when the tensor representation is replaced by an irreducible representation, the Euler characteristic of the corresponding Coxeter cohomology is given by a polynomial with bounded degree. Although the problem is motivated by algebra and topology, the solution relies primarily on combinatorial arguments.

Varkonyi and Domokos proved that homogeneous convex bodies with any prescribed numbers $S\ge1$ of stable and $U\ge1$ of unstable static equilibria exist, the case $S=U=1$ being the mono-monostatic Gomboc. We give a complete constructive answer for the simplest nondegenerate shapes: a homogeneous body whose boundary is the unit sphere perturbed radially by a single real spherical harmonic $Y_\ell^m$ of degree $\ell\ge2$ and order $1\le m\le\ell$ has exactly $S=U=m(\ell-m+1)$ stable and unstable equilibria, for every amplitude in the convex range when $m\ge2$, and for small amplitude when $m=1$. The reduction to the critical points of the perturbing harmonic is an identity when $m\ge2$: the symmetry of a single tesseral harmonic pins the centroid at the origin exactly, so the centroid-to-surface distance is a strictly increasing function of the harmonic. We count the critical points of $Y_\ell^m$ from its separable structure and verify the Poincare-Hopf balance in index form. Three consequences follow. (i) A single harmonic populates only the diagonal $S=U$ of the equilibrium-class lattice; no single harmonic of degree $\ge2$ is mono-monostatic. (ii) Any centrally symmetric (even-degree) perturbation has even $S$ and $U$ and cannot be mono-monostatic, a parity obstruction. (iii) Mono-monostaticity is therefore intrinsically multi-harmonic. A predict-then-confirm numerical study on eight single-harmonic bodies matches the formula exactly. For the multi-harmonic regime we give a certified equilibrium counter (interval arithmetic with Krawczyk uniqueness and Hessian classification) that, given the centroid, provably neither under- nor over-counts. It certifies that a member of a known analytic family, at an amplitude of our own choosing inside the strictly convex sub-regime, is mono-monostatic. For that body the centroid is exactly the origin, so its certificate is unconditional.