Papers

Small Island Developing States (SIDS) face simultaneous frequency nadir and rate-of-change-of-frequency (RoCoF) violations under high renewable penetration. This paper proposes a Composite Adaptive Higher-Order Control Barrier Function (CA-HOCBF) for BESS-coupled virtual synchronous generators that jointly enforces IEEE~1547 frequency ($\pm0.8$\,Hz) and ENTSO-E RoCoF ($\pm1$\,Hz/s) limits as hard safety constraints under parametric uncertainty. The method unifies three tools: (i)~a dual barrier architecture with a zeroing CBF for frequency and an algebraic RoCoF constraint with a robust inertia-floor fallback; (ii)~a composite energy function coupling logarithmic safety barriers with quadratic parameter and disturbance error terms, which drives safety-weighted adaptation; and (iii)~a disturbance observer integrated into the barrier dynamics. We prove that the closed-form QP is always feasible under a BESS capacity condition and, under explicitly stated assumptions, that the composite energy remains finite along trajectories, rendering the joint safe set forward invariant without requiring parameter convergence. Simulation on a 10\,MW Caribbean microgrid ($H=2$\,s, 70% renewables) shows the CA-HOCBF eliminates all frequency and RoCoF violations under a compound disturbance, achieving steady-state error of 0.001\,Hz and RoCoF of 0.50\,Hz/s, where fixed-gain VSGs suffer a 0.62\,Hz offset with 2.25\,Hz/s RoCoF violations.

We consider the FPUT $\alpha$-model with $N$ particles and initial data with small total energy $E$, independent of $N$. In particular, we prove that if the energy is initially given to Fourier modes of wave number $\pi k/N\leq \theta_*$, then it does not flow to modes of wave number larger than $\theta_*$ up to long times of order $E^{-(1-\epsilon/2)}$, $\theta_*$ and $\epsilon$ being small numbers. The proof relies on the closeness of the Toda lattice to the FPUT system. Unlike previous approaches, we do not make use of the action-angle variables of the Toda system, but we exploit the Flaschka integrals of the latter. Actually, we find a suitable recombination of the Flaschka integrals whose quadratic part is used to control the packet, the higher order part fulfilling very good estimates.

Intelligent reflecting surface (IRS) has emerged as a promising wireless technology for enhancing legitimate communications. In this paper, we investigate an IRS-assisted multiuser MISO downlink network in which an attacker reconfigures the IRS to degrade legitimate data transmission. Specifically, we consider a black-box attack setting in which the attacker has access to neither channel information nor symbol-level observations and can observe only the BER reported by each user. The objective of the attacker is to identify an IRS configuration that maximizes the minimum BER among all users under a limited budget of non-repeated tests. To address this BER-only black-box attack problem, we propose PRIS, an online search method that learns effective phase transitions from previous trials and progressively narrows the search from large-block modifications to single-element refinement. Specifically, an annealed user-balancing reward and prioritized replay are incorporated to exploit noisy BER feedback. Numerical results demonstrate that PRIS achieves a higher minimum BER than existing benchmarks.

We extend the explicit octonionic approach to groups of type $\mathrm{E}_6$ to the twisted groups ${}^2\mathrm{SE}_{6,K}(F)$ associated with a quadratic Galois extension $K/F$. These groups act on the Albert space over $K$, preserving the Dickson–Freudenthal determinant and a Hermitean form. When the field norm $K^{\times}\to F^{\times}$ is surjective, we give explicit generators and coördinate proofs of the three-orbit theorem on white points and of the stabilisers of representative white vectors. Under this hypothesis we also prove simplicity of the central quotient. Over finite fields we recover the classical orbit lengths and group orders from the stabiliser orders and the total number of white points. Finally, without assuming norm surjectivity, we show that the two isotropic white-point orbits remain unchanged, while the non-isotropic orbits are parametrised by $F^{\times}/\mathrm{N}_{K/F}(K^{\times})$.

Let $t \ge 1$ be an integer, and let $G$ be a $t$-tough $n$-vertex graph with degree sequence $d_1, d_2, \ldots, d_n$ in non-decreasing order. In 1995, Hoàng conjectured that if $G$ is Hamiltonian and, for every integer $i$ satisfying $t\le i<n/2$, $d_i\le i$, and $d_{n-i+t}<n-i$, one has $d_j + d_{n-j+t} \ge n$ for all $j$ with $i < j < \frac{n}{2}$, then $G$ is pancyclic or bipartite. In this paper, we disprove the conjecture for $t = 1$ and confirm it for all $t \ge 7$.

Data stored in synthetic DNA is retrieved by shotgun sequencing, which returns short subsequences rather than the stored word itself. A natural abstraction of this readout is the $k$-deck of a word: the vector recording how often each word of length $k$ occurs as a subsequence. Two stored words are distinguishable from their readouts exactly when their $k$-decks differ, so the number $D_{q,k}(n)$ of distinct $k$-decks of words of length $n$ over an alphabet of size $q$ measures what a length-$k$ readout retains. We analyse the degrees of freedom remaining in a $k$-deck once all shorter decks are fixed. Within each class of words having prescribed letter multiplicities, the length-$k$ entries are confined to an affine subspace whose dimension is exactly the number of Lyndon words with the same multiplicities, which we give in closed form as a Möbius sum. Writing $L_q(j)$ for the number of Lyndon words of length $j$ over an alphabet of size $q$, we deduce the improved upper bound \[ D_{q,k}(n)=O\!\left(n^{E_q(k)}\right),\qquad E_q(k)=\sum_{j=1}^{k}j\,L_q(j)-1 . \] In the case of a binary alphabet this bound satisfies $D_{2,k}(n)=O\!\left(n^{4\cdot 2^{k-1}}\right)$. We then prove matching lower bounds in the first two nontrivial cases: $D_{q,2}(n)=\Theta\!\left(n^{q^2-1}\right)$ for every alphabet size $q$, and $D_{2,3}(n)=\Theta(n^{9})$ for the binary alphabet. The latter confirms, for $q=2$ and $k=3$, our conjecture that the upper bound has the correct degree for every $q$ and $k$.

We give a short operator-theoretic proof of the sharp Hoeffding inequality for additive functionals of a Markov chain on a general state space with an $L^2(\pi)$ spectral gap, recovering the bound of Fan et al. (2021). We further show that the same argument yields the Hoeffding inequality of Neeman et al. (2024) for Markov-dependent random matrices. The proof rests on two key observations. First, the transition operator factors as $P=Q_\lambda^{1/2} K Q_\lambda^{1/2}$ through the León-Perron operator $Q_\lambda$, with $\|K\|=1$. This bounds the moment generating function by a product of one-step operator norms. Second, each one-step operator is a rank-one perturbation of a multiplication operator, so bounding its norm reduces to verifying a scalar resolvent condition. Scalar convexity and the classical Hoeffding lemma then give the desired estimate. This bypasses the asymptotic cumulant generating function, essential-spectrum analysis, and extremal two-state comparison used in existing proofs. In the matrix setting, the multi-matrix Golden-Thompson inequality first reduces the moment generating function to an operator problem on a lifted Hilbert space. The same two steps then apply: the projection onto constants becomes finite-rank, and a Schur-complement argument yields the same scalar estimate. The matrix Hoeffding inequality of Neeman et al. (2024) follows. For complex Hermitian summands, working directly on the complex Hilbert space improves the bound by a factor of $2$.

This paper investigates the wave propagation for a non-conservative compressible two-fluid model with unequal pressures in R3. This model is novel and fundamentally distinct from the classical single-phase compressible Navier-Stokes equations proposed by Liu-Wang [28]. This work fills a significant gap, as prior results largely apply only to conservative or partially conservative systems with specific Green's function cancellations. The non-conservative nature and the coexistence of two distinct-speed Huygens waves demand dedicated nonlinear coupling analysis and add further complexity. Our study reveals fundamentally different physical behaviors of this non-conservative two-phase model under equal and unequal interfacial pressures. For equal pressure, the model admits a stationary diffusion wave comprising a single Huygens wave, analogous to classical Navier-Stokes equations. In sharp contrast, under unequal pressure, the stationary diffusion wave consists of two Huygens waves propagating at different speeds–a novel phenomenon absent in the classical Navier-Stokes context.

We study the number of prototypes needed to represent Boolean functions by nearest-neighbour classification. There are two distinct settings: the prototypes may be arbitrary points of Euclidean space, or they may themselves be required to lie in the Boolean cube. For unrestricted prototypes, we strengthen a known lower bound for almost all Boolean functions. The bound applies simultaneously to nearest-neighbour voting rules with any number of voting neighbours, and substantially narrows the gap with the known general upper bound. We obtain a VC-dimension bound for classes with a bounded number of prototypes, and show that it is sharp in order in dimensions at least four. We then study Boolean prototypes, beginning with symmetric threshold functions. A connection with covering designs expresses the minimum number of prototypes at every threshold level exactly in terms of a covering number, and leads to further exact results for related monotone functions, including disjunctive extensions and a characterisation of when a representation with a single negative prototype is possible. For a uniformly random Boolean function, the Boolean nearest-neighbour complexity, as a proportion of the cube, is asymptotically close either to one half or to one, with explicit limiting probabilities. In particular, almost every Boolean function requires at least approximately half as many prototypes as there are points in the cube, and one half is the largest proportion for which such a lower bound holds. Finally, we consider arbitrary symmetric Boolean functions. Their Boolean nearest-neighbour complexity is closely approximated by a weighted vertex-cover problem on paths. As a consequence, a uniformly random symmetric function typically requires prototypes amounting to $11/20$ of the cube. This is much larger than the upper bounds known when the prototypes are allowed to lie anywhere in Euclidean space.

Let $n\geq 1$ and let $\ell$ be an odd prime. Let $G=\mathrm{PGSp}_{2n}(\mathbb{F}_\ell)$ or $\mathrm{GSp}_{2n}(\mathbb{F}_\ell)$, and fix a faithful transitive permutation representation $\pi:G\longrightarrow S_d$. We study degree-$d$ number fields whose Galois closures have Galois group $G$ and whose associated permutation representation is $\pi$. For $\sigma\in S_d$, write $\operatorname{ind}(\sigma)$ for its permutation index, namely $\operatorname{ind}(\sigma) = d-\#\{\text{orbits of $\sigma$ on $\{1,\ldots,d\}$}\}$. If $\tau$ denotes a symplectic transvection, or its image in the projective symplectic group, we prove that the number of such fields with absolute discriminant at most $X$ is bounded below by a constant multiple of $X^{1/(2n\,\operatorname{ind}(\pi(\tau)))}$. For the natural vector and projective actions, the exponents we obtain are asymptotically $1/(4n)$ of those predicted by the weak form of Malle's conjecture as $\ell\to\infty$. The fields are constructed from the mod-$\ell$ Galois representations attached to the Jacobians of a one-parameter family of hyperelliptic curves. The proof combines large symplectic monodromy for this family with a squarefree sieve.

We propose a fast deterministic particle method for the spatially homogeneous Landau equation. The method combines a particle representation of the solution with a Fourier approximation of the nonlinear collision flux. Nonuniform fast Fourier transforms are used to reconstruct the density from the particles and to evaluate the flux and density at the particle locations, while the convolutional structure of the flux enables its efficient computation by FFTs. For a fixed transform tolerance, the cost per time step is $\mathcal{O}(N+M^d\log M)$, where $N$ is the number of particles and $M$ the number of Fourier modes per velocity dimension. We establish a consistency estimate for the reconstructed velocity field on regions where the reference density is bounded away from zero. The estimate separates the spectral truncation error from the particle-density reconstruction error, showing how the latter can dominate for sufficiently smooth densities. Two-dimensional numerical experiments with Maxwellian interactions illustrate the accuracy and efficiency of the method, its sensitivity to particle and spectral resolution, and the effects of filtering. Comparisons with a direct blob implementation demonstrate improved accuracy at a strongly reduced computational cost.

In a spiked covariance model, let $\mathcal{H}$ and $\mathcal{B}$ be selected sample and population eigenvector matrices. Their alignment matrix $\Xi=\mathcal{H}^\top\mathcal{B}$ is a latent oracle diagnostic of direction-specific correspondence beyond subspace overlap. We study its uniform pathwise recovery in a nested high-dimensional, low-sample-size experiment with fixed sample size and growing dimension. At every fixed model, an observable spectral contrast consistently estimates the row Gram matrix $\Xi\Xi^\top$. For two or more spikes, however, no measurable estimator sequence achieves uniformly reliable almost-sure path recovery of $\Xi$ over a family having common spike strengths and a common signal subspace but different ordered eigenframes. For every $0<\epsilon<1/\sqrt{2}$, a finite orbit forces each estimator to have some orbit point at which the approximate path-recovery probability is at most a variational bound $u(\epsilon)$, where $u(\epsilon)$ tends to zero as $\epsilon$ tends to zero. The worst-model exact-recovery probability over a countable orbit is zero. Thus the Gram summary is recoverable model by model, while the full alignment matrix is not uniformly recoverable along the nested path.

The notion of extremal subsets was introduced by Assaf, Dranowski, and González as a natural generalization of Demazure crystals. In this paper, we show that the crystal basis of a finite-dimensional irreducible module over a quantum group of finite type has a unique maximal proper extremal subset, whose complement is the set of elements that generate the whole crystal basis as an extremal subset. We conjecture that this complement consists of all nonzero elements obtained by successively applying the lowering Kashiwara operators to a distinguished element, which is constructed from the lowest element of the crystal basis. We prove this conjecture in type A.

A dominating set $D$ of a graph $G$ is a fair dominating set if every two vertices outside $D$ have the same number of neighbors in $D$, and the fair domination number $\mathrm{fd}(G)$ is the minimum cardinality of such a set. Caro, Hansberg and Henning, who introduced this parameter, proved that $\mathrm{fd}(G)<17n/19$ for every maximal outerplanar graph $G$ of order $n\geq3$, and asked whether this bound is asymptotically best possible. We show that it is not the case by proving $\mathrm{fd}(G)\leq(7n-3)/8<7n/8$ for every maximal outerplanar graph $G$ of order $n\geq3$, and we exhibit an infinite family of maximal outerplanar graphs with $\mathrm{fd}(G)/n\rightarrow7/9$, so that the best asymptotic constant lies between $7/9$ and $7/8$.

This paper studies exact controllability of linear mean-field stochastic differential equations with backward-structure. We establish Hautus criteria and uncover a sharp dichotomy between the zero initial time and positive initial times. At time zero, exact controllability is characterized by a Hautus condition for the mean dynamics, with additional control directions generated by the centered stochastic dynamics. At any positive initial time, exact controllability further requires a stochastic Hautus condition for the centered system. Unlike the classical eigenvector-based condition, this stochastic criterion is formulated in terms of positive-semidefinite eigenmatrices of an associated Lyapunov-type operator. The dichotomy arises from the triviality of the initial sigma-field. We also clarify the controllability relations among the corresponding ODE, SDE, and mean-field SDE systems.

A right sub-gamma bound is described by a quadratic proxy $v$ and a pole $c$, and the pair is not unique: enlarging either preserves it. Fixing $v$ at the variance $V$ removes the ambiguity at quadratic order but is a convention, not a consequence. For centered infinitely divisible laws with finite nonzero variance we determine the entire boundary of the feasible set, the map $v\mapsto c_*(v)$. A Beta$(1,2)$ multiplier applied to the normalised Kolmogorov canonical measure turns feasibility into a one-dimensional comparison and yields an exact variational formula: the frontier is convex, nonincreasing, and its feasible set is convex. The abscissa of convergence of the moment generating function imposes a $v$-insensitive floor on $c_*$, so a law whose variance-exact pole sits on it has a flat frontier, whereas the third-cumulant obstruction exists only at $v=V$. Whenever that pole lies strictly above both floors the frontier drops strictly as soon as $v>V$; if the control is also purely local with $\kappa_3^2/(9V^2)>\kappa_4/(12V)$, the drop has a square-root profile, and in the remaining boundary case a cube-root profile, with derivative $-\infty$ at $V$ either way. We then characterise when the optimised frontier reproduces the exact Chernoff deviation: equality holds at a level exactly when some pair on the frontier is tangent to the cumulant generating function at a Legendre maximiser for that level. At the levels tabulated in a worked example $v=V$ costs four to thirty percent; for a centered exponential a gap persists at every level, reaching thirteen percent.

In this paper, we study the unramified Iwasawa module over the cyclotomic $\mathbb{Z}_2$-extension of the real quadratic field $\mathbb{Q}(\sqrt{p_1p_2p_3p_4})$, where $p_1, p_2, p_3$, and $p_4$ are distinct odd prime numbers. We give a criterion for the finiteness of an unramified Iwasawa module of a number field. The criterion is based on the nonexistence of a $2$-group with a certain prescribed quotient. Using this criterion, we construct an infinite family of such real quadratic fields with unramified Iwasawa module of type $\mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ and $2$-class group of type $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$. This gives the first example of an infinite family of real quadratic fields whose ideal class group has $2$-rank three and whose cyclotomic $\mathbb{Z}_2$-extension is totally ramified and satisfies Greenberg's conjecture.

The Time-Evolving Natural Gradient (TENG) method evolves neural approximations of time-dependent partial differential equations by projecting time-discrete target states onto the neural-network manifold through local natural-gradient iterations. For stiff diffusion-dominated problems, however, explicit Runge–Kutta methods face severe stability restrictions. High-frequency diffusive components can drive stage targets beyond the locally reachable region of the neural manifold, worsening the conditioning of local least-squares projections and destabilizing parameter updates. We propose TASE–TENG, which applies the highly stable explicit operator $T_p(hW_0)$ to each Runge–Kutta stage increment before projection, where $W_0$ approximates the dominant stiff diffusion operator. To construct this surrogate in a mesh-free setting, we develop a learnable operator whose local interaction coefficients are generated by a shared geometry-dependent neural rule. A structured factorization enforces discrete self-adjointness, dissipativity, and constant preservation. The operator is trained offline and remains fixed during online integration, where it is used only within the TASE stabilization operator. Numerical experiments on stiff diffusion and reaction–diffusion problems demonstrate improved stability and accuracy at the same time-step size, with reduced errors from high-frequency diffusive modes. The learned rule also allows the stabilization operator to be reconstructed at different spatial point resolutions without retraining.

We prove the local version of the Pólya–Szegö conjecture for the torsional rigidity of polygons: for every \(n\geq5\), the regular $n$-gon is a strict local maximizer of torsional rigidity among convex $n$-gons of prescribed area. Our proof is entirely analytic. It builds on a locally stable proportional triangular covering inspired by Solynin and Zalgaller and an associated weighted Voronoi-type partition, together with a quantitative asymptotic analysis of the loss produced by truncating the overlapping triangles to the partition cells. The result is then obtained by establishing two key ingredients: the optimality of isosceles triangles for mixed torsional rigidity at fixed area and vertex angle, and a strict concavity property of the mixed torsional rigidity of isosceles triangles.

We establish the asymptotic-preserving property of the exponential Euler method for stochastic reaction-diffusion-advection equations in $\mathbb R^2$ under fast Hamiltonian advection with multiple critical points. In the fast-advection limit, the dynamics reduces to a stochastic partial differential equation on a noncompact metric graph. A key ingredient is the strong convergence analysis of the limiting exponential Euler scheme, which is complicated by the graph's noncompactness, its multiple edges and vertices, and the nonuniform ellipticity and vertex degeneracy of the graph operator. Using a weighted $L^2$ framework and analytic-semigroup smoothing estimates, we prove a temporal convergence order arbitrarily close to $1/2$ for weighted $L^2$ initial data and exactly $1/2$ under a half-order fractional-domain condition. Combining these estimates with the fast-advection limits yields the asymptotic-preserving property in the multiple-critical-point setting. Numerical experiments illustrate the asymptotic behavior and confirm the predicted temporal convergence rate.

We consider a nonlinear model motivated by polychromatic computed tomography (CT). Here, $W$ distinct $d$-dimensional signals $x^*_1, \ldots, x^*_W \in \mathbb{R}^d$ must be recovered from $n$ measurements $(a_i, y_i)_{i = 1}^n$ that obey the nonlinear model $\mathbb{E}[y_i|a_i] = h(\langle a_i, x^*_1 \rangle, \ldots, \langle a_i, x^*_W \rangle)$, where $h: \mathbb{R}^W \to \mathbb{R}$ is a known nonlinearity that models a certain type of exponential attenuation law. Even when there is no noise in the measurements, the sample size $n$ (for any measurement ensemble $\{a_i\}_{i = 1}^n$) must exceed the number of unknowns $Wd$ to guarantee that the underlying signals are identifiable. We construct a measurement ensemble that ensures perfect signal recovery almost surely provided $n \geq Wd + W - 1$, thereby isolating the injectivity threshold up to an additive factor $W - 1$. We also study computational complexity of signal recovery in this model with a general measurement ensemble $\{a_i\}$. We show that if $W \geq 2$, then there is a measurement ensemble for which deciding whether there exist signals consistent with the measurements is NP-hard. In particular, this implies that polychromatic, multimaterial CT reconstruction is NP-hard in general. This finding stands in sharp contrast to the single-material setting, for which a polynomial-time algorithm can provably perform signal recovery for any measurement ensemble.

We investigate the operator distance between the Identity and the Copson operator on the cone of nonnegative, nonincreasing sequences within the Lebesgue sequence spaces $\ell^p(\mathbb{N})$. By proving the exact value of this distance, we confirm a conjecture originally formulated by A. Ben Said, S. Boza, and J. Soria in “The norm of Hardy-type oscillation operators in the discrete and continuous settings” (Analysis and Applications, 2025).

In this paper, we investigate higher-order symmetries and non-geometric symmetries within the framework of $k$-cosymplectic Hamiltonian field theory. We introduce the concept of generalized infinitesimal symmetry, which relaxes the standard symmetry condition, and then define two new classes of symmetries: higher-order general infinitesimal Cartan symmetries and good non-geometric symmetries. For each of these symmetry classes, we establish a Noether-type theorem that provides explicit conserved quantities for exact $k$-cosymplectic Hamiltonian systems. These results extend and generalize previous work on standard Cartan symmetries in the $k$-cosymplectic setting. Furthermore, we extend all of these results in a parallel fashion to Hamiltonian systems on $k$-symplectic manifolds, demonstrating the unifying nature of our approach. Concrete examples are provided to illustrate the existence and applicability of the proposed symmetries. Our framework offers a systematic geometric treatment of generalized symmetries and their associated conservation laws in classical field theories.

For two even integers $r_1,r_2$ with $4\leq r_1\leq r_2\leq N/2$, a graph $G$ of even order $N\geq8$ is \emph{2-DCC edge $[r_1,r_2]$-bipancyclic} if for every ordered pair $(e,f)$ of independent edges and every even integer $\ell\in[r_1,r_2]$, there exist two vertex-disjoint cycles $C_1$ and $C_2$ with $V(C_1)\cup V(C_2)=V(G)$, $|C_1|=\ell$, $|C_2|=N-\ell$, $e\in E(C_1)$, and $f\in E(C_2)$. We study this property for graphs defined by Cartesian products. Let $G=H_1\square\cdots\square H_d$, where $d\geq2$ and each factor $H_i$ is either $K_2$ or a bipartite graph of even order at least four that has a Hamiltonian cycle and in which every edge lies on a Hamiltonian cycle. We prove that $G$ is 2-DCC edge $[4,|V(G)|/2]$-bipancyclic if one of the following holds: (i) $d\geq4$; (ii) $d=3$ and at least two factors have order at least four; (iii) $d=3$, exactly one factor has order at least four, and every set of at most two edges of that factor lies on a common Hamiltonian cycle; (iv) $d=2$, both factors have order at least four, and every set of at most two edges of each factor lies on a common Hamiltonian cycle; (v) $G\cong K_2\square K_2\square K_2$. The theorem applies to several families of interconnection networks. In this paper, we give a sharp characterization for undirected toroidal meshes, star-cube networks, and hyperstar networks.

This paper concerns the variational inequality arising from a specific singular control problem: portfolio selection under proportional transaction costs. This variational inequality is a gradient-constrained partial differential equation (PDE). Generally, the literature only guarantees the W{2,\infty} regularity of the solution to such a PDE, and counterexamples exist for higher regularity. In this paper, by exploiting the concavity of the value function, we connect this gradient-constrained problem to an obstacle problem, and finally show that the solution is C^{2,\alpha}. This paper concerns a two-dimensional portfolio selection setup, but our approach can potentially be extended to more general multi-dimensional singular control problems when the value function is concave.

For $r \ge 0$, let $\mathcal P_r=\{\binom{r}{0},\binom{r}{1},\ldots,\binom{r}{\lfloor r/2\rfloor}\}$ be the set of distinct entries in row $r$ of Pascal's triangle. We study the least possible least common multiple of $n$ entries chosen from one row, with the row itself also free: $a(n)=\min_{r\ge0,\ S\subseteq\mathcal P_r,\ |S|=n}\operatorname{lcm}(S)$. Whereas the least common multiple of an entire row is given by a classical identity of Farhi, allowing both the row and the selected coefficients to vary creates a different optimization problem. We first recast the fixed-row problem exactly as a weighted prime-power exclusion problem, which explains how omitting a few coefficients can remove expensive prime-power contributions. Our main asymptotic result is $\log a(n)=2n+O\!\left(n\exp\!\left(-c(\log n)^{3/5}(\log\log n)^{-1/5}\right)\right)$ for some absolute $c>0$, so $a(n)^{1/n}\to e^2$. A two-band refinement further shows that every optimal row satisfies $r_n=2n+O\!\left(n\exp\!\left(-c(\log n)^{3/5}(\log\log n)^{-1/5}\right)\right)$, and that the minimum prefix defect of an optimal support is $o(n)$.

We investigate the free boundary propagation of a non-linear degenerate parabolic problem with both fourth and second order non-linearities, which may be used to model viscous thin-film flow in the regime of weak slippage with an additional contributions from a singular disjoining pressure brought forth by inter-molecular forces. By carefully treating the individual scaling behaviors of the non-linearities, we obtain asymptotically optimal upper and lower bounds on the rate of forward support propagation. These rates show that the fourth-order operator dominates the propagation rate on short time scales, but the long-term rate of propagation is controlled by the second-order operator. We estimate further the time at which this trade-off in propagation rate occurs. Lastly, we examine sufficient conditions to ensure that the free boundary does not immediately advance in a given direction, leading to a detailed picture of how the competition between the second/fourth-order operators influence the free boundary dynamics of solutions.

We prove the uniqueness conjecture of Fomin and Neville for their long mutation cycles in all ranks $n\ge4$. More generally, we show that every nucleus-extension quiver has a unique simple cycle in its labeled mutation graph. These quivers allow an acyclic part of arbitrary size outside the nucleus. Our proof uses three mutation statuses determined by a closed nucleus and the full subquivers obtained by deleting one part of its nucleus decomposition at a time from the ambient quiver. We analyze transitions between these statuses to prove that every mutation leaving the prescribed cycle is performed at an exit.

Multi-user multiple-input multiple-output (MU-MIMO) is a key technology that improves wireless capacity through multiple antennas. In MU-MIMO downlink precoding, vector perturbation (VP) is a representative nonlinear method that achieves high performance. However, its search for the integer perturbation vector reduces to a closest vector problem, whose complexity grows rapidly as the number of users increases. We propose a method that relaxes the discrete structure of the integer perturbation into a continuous mixture of $t$-distributions, enabling efficient search via gradient-based Hamiltonian Monte Carlo (HMC). Complexity analysis and numerical experiments demonstrate the effectiveness of the proposed method. Its search complexity scales as O(N^2) in the number of users N. At a symbol error rate of 10^-3, it performs within 2.4 dB of a hypersphere approximation benchmark, which approximates the performance limit of VP. This paper reframes the VP perturbation search as a probabilistic inference problem, providing a general formulation for handling high-dimensional discrete search in a continuous space.

We study the Bregman proximal point method for the obstacle problem, a fundamental variational inequality arising in contact mechanics, optimal design, and mathematical finance. Each Bregman proximal step regularizes the energy through a Bregman divergence generated by a Legendre function, leading to a semilinear elliptic subproblem. We first establish a well-posedness and strict feasibility theory for these subproblems. We then analyze the convergence of the resulting iteration in the $H^1$-norm. Our convergence analysis is based on a sequential strict local minimality inequality. We show abstractly how the order of minimality determines the convergence rate. Assuming that the initial guess lies above the exact solution and a one-sided Bregman growth condition holds, we establish sequential strict local minimality of order $s \geq 2$ and derive sublinear convergence rates in the $H^1$-norm of the form $O(k^{-\zeta})$, where $\zeta\in(1/2,1]$ depends on the choice of Legendre function. For the Shannon and Tsallis entropies, we show that these rates are sharp in a uniform worst-case sense. Under additional assumptions on the free boundary and the obstacle, we establish sequential strict local minimality of order $s = 1$ with the Shannon and Spence entropies. This yields linear convergence of the form $O(\varrho^k)$ for some $\varrho\in(0,1)$.