Papers

We provide a complete characterization of a finite group in which every non-supersolvable maximal subgroup is a TI-subgroup, which generalizes some known related results.

We study hyperelliptic curves on $K3$ surfaces. We show that any number of general points on a $K3$ surface of Picard rank one can be connected by a hyperelliptic curve. As a by-product we recover the Beauville–Voisin decomposition of the small diagonal, together with some new bounds on its torsion order in the integral Chow group. On the arithmetic side, given a $K3$ surface $X$ of geometric Picard rank one defined over a number field $k$, we show that there exists a finite extension $k\subset k'$ such that quadratic points are dense in $X_{k'}$, extending a theorem of Corvaja–Zucconi to $K3$ surfaces of arbitrary degree.

In this paper, we introduce a new class of unbounded Reinhardt domains in $\mathbb C^n$ ($n\geq 2$),which we call horn domains. We study the dimensions of the Bergman spaces,the holomorphic sectional curvature of the Bergman metric, and the holomorphic equivalence problem for such horn Reinhardt domains.

A tensor product is known for commutative Rota–Baxter algebras through a universal construction, but its underlying algebra need not be the ordinary tensor product $A\otimes B$ and can be substantially larger. We show that, after passing to augmented Rota–Baxter algebras, the ordinary tensor product itself carries a natural Rota–Baxter operator which makes the category monoidal with the usual associator and unit constraints. The resulting structure is intrinsically nonsymmetric: over a field of characteristic zero, this operator and its flip-conjugated counterpart are the only possibilities among linear combinations of five natural terms, while over any field the full category admits no braiding. Its comonoid objects are precisely Rota–Baxter cocycle bialgebras. Moreover, the free Rota–Baxter adjunction yields a bimonad and free cocycle bialgebras on ordinary bialgebras, with compatible Hopf structures for connected filtered inputs, in both the associative and commutative settings.

We study the inverse problem for the fractional conductivity equation with uniformly elliptic conductivities $\gamma\in L^\infty(\mathbb R^n)$. For every $n\geq1$ and $0<s<1$, we prove that the local exterior Dirichlet-to-Neumann map on any nonempty open subset of the exterior of a bounded open domain determines $\gamma$ uniquely almost everywhere in $\mathbb R^n$. No continuity, Sobolev regularity, or prescribed behavior at infinity is required. The proof works directly with the conductivity bilinear form and avoids the fractional Liouville reduction. It combines exterior determination at Lebesgue points, equality of weighted solutions, and a unique continuation principle for the fractional Laplacian in the weighted class \[ L_s=\left\{ g\in L^1_{\mathrm{loc}}(\mathbb R^n): \int_{\mathbb R^n} \frac{|g(x)|}{1+|x|^{n+2s}}\,dx<\infty \right\}. \] Specifically, if $g\in L_s$ and both $g$ and $(-\Delta)^s g$ vanish in a nonempty open set, with the latter equality understood distributionally, then $g=0$ almost everywhere. This result requires neither local boundedness nor Sobolev regularity of $g$. Its proof uses the Caffarelli–Silvestre extension and boundary unique continuation for the associated degenerate elliptic equation.

In this paper, we study planar compressible cavity flows fast-decaying vorticity and establish a Schauder estimate in large scale. As an application, we obtain an a priori asymptotic growth rate estimate of the cavity, that $s^{\frac{1}{2}-\delta}\lesssim h(s)\lesssim s^{\frac{1}{2}+\delta}$. This is by far the most accurate asymptotic estimate for compressible cavity flows and is probably the first paper that takes vorticity into account in the asymptotic analysis.

We prove that every nonnegative source \(f\in L^1(\Omega)\) in the Monge–Ampère equation \[ \det D^2u=f\quad\text{in }\Omega, \] on a bounded uniformly convex \(C^{1,1}\) domain \(\Omega\subset\mathbb R^n\), \(n\geq2\), is uniquely determined by generalized boundary measurements of Alexandrov solutions. The measurement is the extremal boundary slope, which extends the outward normal derivative and may equal \(+\infty\). For each \(\omega\in\mathbb S^{n-1}\), it suffices to know these slopes for the cylindrical Dirichlet data \[ \frac{t_k^\omega}{2}|P_\omega x|^2,\qquad t_k^\omega\to\infty, \] along any sequence of amplitudes, where \(P_\omega\) projects onto \(\omega^\perp\). The boundary value and extremal slope determine the interior subgradient image up to a Lebesgue null set. An exact change-of-variables formula and a large-amplitude limit then recover the Euclidean X-ray transform of \(f\), yielding uniqueness without a positive lower bound on the source. For nonnegative \(L^\infty\) sources, the extremal slopes are bounded and coincide with the interior normal derivatives, giving a Dirichlet-to-interior-Neumann formulation. The argument also extends to \[ \vartheta(\nabla u)\det D^2u=f, \] where the known weight \(\vartheta\) is locally integrable, Borel measurable, and bounded below by a positive constant. Conditional on existence of the corresponding Alexandrov solutions, the same cylindrical boundary-slope data determine every nonnegative \(L^1\) source.

Let $\varphi\colon A\to B$ be a Jordan homomorphism of associative algebras over a field $\mathbb{F}$ of characteristic different from $2$, with defect ideals $V_{\varphi}$ and $W_{\varphi}$. If $A=\operatorname{Alg}([[A,A],[A,A]])$, then $V_{\varphi}W_{\varphi}=W_{\varphi}V_{\varphi}=0$ and $V_{\varphi}\cap W_{\varphi}\subseteq\operatorname{Ann}(\operatorname{Alg}(\varphi(A)))$. Moreover, the induced map modulo $V_{\varphi}\cap W_{\varphi}$ is standard. The annihilator bound is sharp, even in finite dimension. As an application, when $A$ is unital and $A=K(A)$, every Jordan homomorphism defined on $A$ is splittable and standard. In this setting, the corresponding result of Brešar and Zelmanov holds without the surjectivity hypothesis.

We prove that for every $(X,\mathcal{B},\mu,T)$, an aperiodic, probability preserving system and $\mathcal{P}$ a finite measurable partition of $X$, there exists an Alpern Multitower whose base is independent of $\mathcal{P}$. This removes the ergodicity assumption in the original proof of the independent Alpern Lemma of \cite{MR3373823}. As a consequence, we remove ergodicity assumption from recent results on limit theorems in deterministic systems \cite{Kosloff_Volny_2022,kosloffVolnyStableCLT,MR4955410}.

We maximize the weight of edges covered by selected vertices, together with nonnegative individual rewards, under an arbitrary matroid constraint. The continuous coverage objective is quadratic and need not be concave. Our method searches projections onto a small set of directions and solves a concave subproblem for each projection center. A coverage-specific inequality converts the subproblem bounds into a multiplicative guarantee, while matroid rounding restores an integral solution. Let $A$ be the weighted adjacency matrix and $D$ the diagonal matrix of singleton values, restricted to positive-value elements after objective-preserving preprocessing. Set $\varrho=\lambda_{\max}(D^{-1/2}AD^{-1/2})$. For an integer budget $q\ge0$, let $\tau_q$ be the smallest nonnegative threshold for which $A+\tau_qD$ has at most $q$ negative eigenvalues on the span of differences of matroid bases. We obtain the factor $1-\varrho\tau_q/[2(\varrho+\tau_q)]-\varepsilon$ in time $(2+q/\varepsilon)^{O(q+1)}N^{O(1)}$, where $N$ is the binary input length and a zero-over-zero loss is zero. The algorithm returns a rational upper bound on the integral optimum. A separate exact-algebraic construction produces polynomial-bit low-rank rational corrections with positive threshold slack; its proof isolates eigenvalues, keeps eigenspace bases unnormalized, and approximates scalar factors only at the final step. The numerical proposal prototype is not used to justify this complexity statement. Supplied certificates also yield robustness to locally bounded edge additions. These are threshold-dependent coverage guarantees, not a new general spectral-search principle or a universal approximation improvement.

Selecting vertices covers their incident edges, but a matroid can prevent selecting the combinations that cover the most weight. We study whether pair-selection information improves rounding from a semidefinite relaxation of this problem. A feasible point $(x,Y)$ records individual and paired selection values. Its relaxation objective is $U$, and the matrix $Y-xx^\top$ has rank $d$. We compare the feasible marginal vectors obtained by fixing one element as selected or excluded, then round only the best vector. For $d\ge1$, the resulting basis has value at least $(3/4+1/(4d))U$. The procedure is deterministic and polynomial-time for a supplied exact rational point and an independence oracle. Weighting the covariance spectrum by the objective gives a stronger version of the bound. To show the rank factor is sharp, we construct a rank-$2m$ rational linear matroid and $m$ disjoint four-cycles with integral optimum $3m+1$, relaxation optimum $4m$, and an optimal covariance matrix of rank $m$. Thus the relaxation has limiting reciprocal integrality gap $4/3$, despite its conditional constraints. One valid pair-exclusion inequality makes it exact on this family, as an explicit semidefinite certificate shows. The result concerns rounding and the strength of this relaxation; it does not supply a low-rank optimal point or a fixed universal approximation improvement.

Let $G$ be a thrackle, where a thrackle is a graph drawing in which each edge meets every other edge exactly once, either at a vertex or at a crossing. It is conjectured that, for all thrackles, the number of edges is less than or equal to the number of vertices. Fulek and Pach created a program to determine whether or not a graph can be a thrackle. Through a reconstruction and improvement of their algorithm, we computationally prove a key dumbbell to be non-thrackleable. Using this exclusion, we improve Xu's 8-face half-edge bound from five to four. Finally, using this stronger restriction with Xu's framework, together with separate arguments for graphs containing triangles and disconnected graphs, we improve the known upper bound toward this conjecture to $m\le1.385(n-1)$ for $n\ge4$.

We study the sharp constant in the differential inequality \[ I(t)I''(t)\geq c\,[I'(t)]^2 \] for Fisher information along the probability-normalized heat flow on $\mathbb R^2$. The unrestricted log-convexity value $c=1$ is false in two dimensions, whereas the known square-root convexity theorem gives the universal value $c=1/2$. We construct an exact, machine-verifiable sum-of-squares certificate proving the stronger universal bound \[ c\geq \frac{231691}{250000}=0.926764. \] Together with an independent exact Fourier/Gaussian-envelope construction, this gives \[ 0.926764\leq c_*<0.94176665012062, \] where $c_*$ is the sharp constant for positive-time Gaussian convolutions of arbitrary probability measures on $\mathbb R^2$. We formulate the proof in explicit two-dimensional index notation. The lower bound follows from an exact rational polynomial identity involving a strictly positive denominator, a weighted total derivative, and eleven positive rational Gram matrices. All $491$ recorded leading principal minors are strictly positive and the complete polynomial identity has zero residual. No symmetry, log-concavity, finite-variance, or finite-entropy assumption is imposed on the initial measure. The sharp value remains open.

We show that matroid Stanley-Reisner rings can have essential deformation obstructions of arbitrarily high order over every field, even at corank three. For every $m\ge3$, we construct a connected cosimple matroid with $m$ fine-homogeneous first-order deformation classes whose proper subcollections integrate jointly and whose pairwise primary cup products vanish. These classes define a joint flat family whose $(m-1)$-jet has no lift to order $m$, even without preserving the grading. For the duals of rank-three truncations of $U_{2,m}\oplus U_{2,n}$, $m,n\ge3$, every nonzero homogeneous class modulo the primary cup image is realized as a higher-order obstruction, and this quotient has an explicit module presentation with $m+n-2$ minimal homogeneous generators. The construction is based on a computation of the full multigraded second cotangent cohomology $T^2$ and its primary cup image in terms of flats and parallel classes of the dual matroid. This computation also proves that $T^2$ vanishes precisely when every connected component has corank at most two, equivalently when the matroid has no $U_{1,4}$-minor, settling the conjecture of Constantinescu, Klein, Nguyen, Singh, and Venturello. The same criterion characterizes formal smoothness of the affine deformation functor.

Given graphs $X,Y$ on $n$ vertices, Defant and Kravitz defined the friends and strangers graph, denoted $\mathrm{FS}(X,Y)$. The vertices of $\mathrm{FS}(X,Y)$ are bijections $\sigma:V(X)\to V(Y)$, and two bijections $\sigma$ and $\sigma'$ are adjacent in $\mathrm{FS}(X,Y)$ if they differ by a transposition $(ab)$ such that $a$ and $b$ are adjacent in $X$ and $\sigma(a)$ and $\sigma(b)$ are adjacent in $Y$. Our main result resolves a conjecture of Bangachev that if $X,Y$ are connected graphs and $2\min\{\delta(X),\delta(Y)\}+3\max\{\delta(X),\delta(Y)\}\ge 3n$, then $\mathrm{FS}(X,Y)$ is connected. The main step in the proof is showing that there exists an absolute constant $C$ such that for any adjacent vertices $a,b\in V(X)$ and any vertex $\sigma\in V(\mathrm{FS}(X,Y))$, $a$ and $b$ can swap positions in at most $C$ moves. Furthermore, we prove that if $X,Y$ are connected graphs satisfying the above minimum degree condition, then the diameter of $\mathrm{FS}(X,Y)$ is $O(n^2/\min\{\delta(X),\delta(Y)\})$, and that this is sharp as a bound on the maximum diameter of $\mathrm{FS}(X,Y)$ in this regime. This work improves the diameter results of Akella and Li, and resolves many of their questions. Moreover, we show that even for disconnected graphs $X$ and $Y$, the same minimum degree condition implies that the diameter of any connected component of $\mathrm{FS}(X,Y)$ is $O(n^2)$.

Recently, Matomäki, Pandey, Pliego, Teräväinen and Wang improved the Baker–Harman–Pintz estimate of trilinear sums over short intervals of length $x^{\theta}$, which naturally reduces to estimating mean values of products of three Dirichlet polynomials. Based on recent works of Guth and Maynard and Chen, Gupta and Li on new large value estimates for Dirichlet polynomials, they improved the Baker–Harman–Pintz estimate for $\theta > \frac{3}{5}$. In this paper, we use their method to improve the Baker–Harman–Pintz estimate for $\frac{1}{2} < \theta < \frac{4}{7}$ and get new results for all $\frac{1}{2} < \theta < 1$. Our result yields improvements on many problems, including the work of Li on primes in short intervals of length $x^{0.52}$.

We define multidimensional delta functions as generalized functions by pulling back the Dirac delta function, as well as products of such. We prove various formulas including a local version of the coarea formula and we use delta functions to compute marginal probability distributions on spheres.

Developing resilient crops requires models that accurately capture how plant growth dynamics emerge from the continuous interplay between genetic and environmental factors. While high-throughput phenotyping platforms generate data for dynamic growth models, current modelling approaches show limitations: process-based crop growth models (CGMs) capture environmental dynamics but are difficult to scale across many genotypes because they require calibration of many parameters, while quantitative genetic models allow modelling many genotypes but often summarise time-varying environments using static covariates. To address this gap, we propose a data-driven framework to identify reduced systems of ordinary differential equations (ODEs) describing crop growth dynamics as functions of environmental variables and a controllable number of genotype-specific coefficients (GSCs). Using the Sparse Identification of Nonlinear Dynamics (SINDy) method, we infer model structure directly from data via sparse regression on a physiologically informed library of candidate functions. In this framework, GSCs parameterise cultivar-specific differences in response to environmental conditions. The resulting models are linear in these GSCs, making them compatible with mixed-model and genomic prediction approaches. We evaluate the approach on an in silico dataset generated with WOFOST for five potato cultivars across twelve growing seasons. Our identified ODEs generalise to held-out years and achieve $R^2 > 0.9$ for end-of-season yield prediction across held-out years when compared to full-scale WOFOST simulations. These results provide a proof of concept for SINDy-based identification of parsimonious dynamic crop models from longitudinal crop-growth data.

We study digit-appending walks in base $b\geq 2$ along integer linear recurrences with minimal characteristic polynomials having a unique simple dominant root $\alpha$ that lies outside the unit circle. We prove that if $\alpha$ and $b$ are multiplicatively independent, then no infinite walk can have uniformly bounded appended digit lengths. In contrast, we show that when the appended digit length can be unbounded, the same independence hypothesis implies the existence of infinite walks.

We present short proofs for the main parts of the recently completed solution of Erdős Problem 506, which asks for the minimum number of distinct circles determined by a finite planar point set that is neither collinear nor concyclic. An inversion argument and a seven-point line lemma show that eight points determine at least $17$ circles, and an explicit configuration attains this bound. A single argument combining Melchior's inequality with Langer's incidence bound establishes the sharp formula for every $n\ge15$. The proofs give concise alternatives to longer arguments in the recent literature; the remaining finite cases are cited separately.

We present an elementary but powerful method for proving that a two-variable equation has no integer solutions. The method produces quadratic identities which every integer solution must satisfy. Once these identities are established, a standard quadratic reciprocity argument finishes the proof. We illustrate the power of the method by resolving $25$ out of $30$ two-variable equations of size up to $34$ for which the problem of integer solubility remained open. We also resolve the remaining $5$ equations by deeper methods, and further determine the integer solubility of all two-variable equations of size up to $35$ and of length up to $12$, with two exceptions. Before this work, there were $211$ equations in this category that resisted the previously known methods, and we proved that $209$ of them have no integer solutions, with all proofs formalized in Lean. In addition, we show that our method can be used to solve certain two-variable equations whose integer solution set is non-empty.

A continuous shape parameter enters both a radial density and the normalization of its transform coefficients. This coupling allows changes in shape to be compensated by changes in scale and exponential tilt. We study the coefficients b_k=e^a E[Y^k ]/[k!(d)_k ] for densities proportional to y^(d-1) exp(-{\alpha}y^q+e̱ta}y), with known integer q>=2, d,{\alpha}>0, and unrestricted real a,e̱ta}. For q=2, the first four coefficients determine all parameters globally, whereas three coefficients leave an analytic curve of solutions. We characterize the exact image of the two shape ratios and prove that two alternating corners decide whether an entire coefficient box is realizable and give its exact shape range. For every fixed q, the first q(q+1) coefficients give global identification and an everywhere full-rank map. A finite-difference criterion yields generic identification from 2q+1 coefficients, including every zero-tilt model against the full tilted family. For q=3, an exact rational certificate constructs two distinct shapes with identical first four coefficients and separated fifth coefficients. The results distinguish exact identification, realizability, and sensitivity for a parameter-dependent coefficient normalization.

A set of $m$ distinct positive integers is called a Diophantine $m$-tuple if the product of any two distinct elements, increased by one, is a perfect square. A Diophantine quadruple $\{a,b,c,d\}$ with $a<b<c<d$ is called regular if $$d=a+b+c+2abc+2\sqrt{(ab+1)(ac+1)(bc+1)}.$$ We prove that every irregular Diophantine quadruple satisfies $c<10^{338}$ and $d<10^{10^{24}}$.

The 2026 measles resurgence in Bangladesh motivates a spatial modelling framework that accounts for heterogeneous local transmission, vaccination activity, and interdivisional connectivity. We formulate a multipatch susceptible-vaccinated-infectious-recovered model for the eight administrative divisions of Bangladesh. Division-specific parameters are estimated from reported measles cases \cite{DGHS} and smoothed vaccination data using bounded nonlinear least squares, with uncertainty quantified through 1000 parametric bootstrap replicates. The fitted divisional systems are then connected through compartment-specific movement and cross-patch infection coefficients adjusted for connectivity and the source division's relative reproduction number. The fitted model captures the main division-specific epidemic patterns. Estimated local reproduction numbers range from \(1.1386\) in Khulna (95% CI:\(0.5703\)–\(1.3834\)) to \(4.3491\) in Dhaka (95% CI: \(1.4449\)–\(5.6307\)), while Chattogram also exhibits high transmission potential (\(\mathcal{R}_0^i=4.2275\); 95% CI: \(4.0019\)–\(4.3185\)). Baseline spatial coupling increases the network-wide epidemic peak from \(190.96\) to \(313.20\) infectious individuals and more than doubles cumulative infections from \(1086.42\) to \(2345.35\). Relative to baseline coupling, 50% and 80% restrictions reduce the epidemic peak by \(26.69\%\) and \(34.37\%\), and cumulative infections by \(25.07\%\) and \(41.48\%\), respectively. Conversely, a 50% increase in spatial coupling raises the peak by \(30.14\%\) and cumulative infections by \(23.54\%\). Although local transmission remains dominant in every division, cross-patch infection accounts for \(28.83\%\) of new infections in Rajshahi at the baseline epidemic peak. Source-sink analysis identifies Dhaka as the dominant cross-patch source and Chattogram as a smaller secondary source.

Let $K$ be an imaginary quadratic field with fundamental discriminant $D=-5n$, where $n>0$ is odd, and cyclic ideal class group $\text{Cl}(K)=\langle g\rangle\simeq C_{4M}$. Assume that the ramified prime class above $5$ is the unique involution $g^{2M}$. For the Kronecker eta invariant \[ F(A)=\frac{|\eta(z_A)|^2}{\sqrt{a_A}}, \] where $z_A$ and $a_A$ arise from a positive definite quadratic form representing the ideal class $A$, we show that the logarithms of the mirror ratios \[ R_j=\frac{F(g^{2M-j})}{F(g^j)},\qquad 0\le j<M, \] are related by an explicit discrete cosine transform to the odd class-character Hecke $L(1)$-values. A single level-$5$ generalized-Weber orbit realizes all of these mirrors. Combining Fourier nonvanishing, lower-power Weber class invariants and a ray-class cancellation argument, we prove \[ \mathbb{Q}(R_j)=H^{\langle c\rangle},\qquad R_j^{\sigma^{2M}}=R_j^{-1},\qquad \mathbb{Q}(R_j+R_j^{-1})=H^{\langle c,\sigma^{2M}\rangle}. \] Here $H$ is the Hilbert class field, $c$ denotes complex conjugation, and $\sigma$ corresponds to $g$ under the Artin map. Consequently \[ [\mathbb{Q}(R_j):\mathbb{Q}]=4M,\qquad [\mathbb{Q}(R_j+R_j^{-1}):\mathbb{Q}]=2M, \] and the latter degree becomes $M$ over the real genus field $\mathbb{Q}(\sqrt5)$. Thus the degree pattern $2^r,2^{r-1},2^{r-2}$ in the cyclic $2$-power hierarchy is a specialization of a uniform $C_{4M}$ theorem. We give exact low-class-number evaluations and certified computations at $D=-895,-3295,-8495$, separating respectively four, eight and sixteen non-genus Hecke $L(1)$-values and producing corresponding alternating Rogers-Ramanujan singular values. Classical mixed modular equations then give ordinary Rogers–Ramanujan companions at the same arguments.

In this paper, we establish large deviations of invariant measures for fractional stochastic reaction-diffusion equations with polynomial reaction term driven by unbounded multiplicative noise on R^n. Under a dissipativity condition, we first prove the Freidlin-Wentzell large deviation principle of solution trajectories uniformly over all initial data in L^2(R^n). The key ingredient is a positive-time L^\infty-estimate for differences of controlled solutions which is independent of all initial data and hence yields uniform convergence of controlled stochastic solutions over the entire tphase space. Then using the uniform trajectory estimates, contraction and a second-moment estimate, we further prove the large deviations of invariant measures of the equation with a good quasipotential without introducing weighted spaces or establishing exponential tightness. The non-compactness of Sobolev embeddings in R^n is circumvented by uniform spatial tail-ends estimates. This is the first result on large deviations of invariant measures of the equation with multiplicative noise on unbounded domains.

Peter Lax's 1968 discovery that certain nonlinear wave equations possess operator pairs (Lax pairs) opened the theory of completely integrable systems, yielding exact solutions, infinite conservation hierarchies, and perfectly elastic soliton collisions. When a nonlinear dispersive equation is provably non-integrable (admitting no Lax pair and only finitely many conservation laws), what mathematical structure governs its dynamics? The K(m,m) compacton equations, u_t + (u^m)_x + (u^m)_{xxx} = 0 for m >= 2, provide a testing ground: they support traveling waves with compact support, yet fail every known integrability test. We prove universal algebraic formulas determining the shape, width, and speed of every K(m,m) compacton. We develop a constraint-deficit mechanism explaining how the five independent conservation laws constrain emergent amplitudes and suppress the fundamental Fourier mode of the collision debris, while higher radiation modes remain unconstrained. The mechanism's conclusions rest on explicit hypotheses supported by numerical evidence. A phase-plane analysis reveals the compacton as a singular limit of smooth solitary waves through simultaneous regularity loss and support compactification. These results delineate intermediate territory between integrable order and generic nonlinear dynamics, organized by scaling universality and by the constraint deficit of a finite conservation structure.

We consider the 1-D focusing cubic NLS with a smooth, spatially-dependent potential which decays rapidly as $x \rightarrow \pm \infty$. As is well known, under sufficient regularity conditions, solutions of the cubic NLS with such a spatially-dependent potential conserve both mass and energy. Additionally, in the absence of a potential, the 1-D focusing cubic NLS admits boosted soliton solutions. Furthermore, in the presence of a potential, the interaction of the potential with boosted soliton solutions to the free case is of active interest in the literature, particularly with respect to the dependence of this soliton-potential interaction on the velocity of the incident soliton. We adapt a finite-difference scheme to the setting of a smooth, rapidly-decaying spatially-dependent potential and show that this adaptation preserves the appropriate mass and energy invariants. Additionally, we use this algorithm to model soliton-potential interactions at a variety of incident soliton velocities.

Prescribed-time controllers drive the state to the origin at a time $T$ chosen in advance, with gains that grow without bound as $t$ approaches $T$. This paper shows the growth is a necessity and computes the exact price of avoiding it. The starting point is a simple estimate: any trajectory with $\|\dot z(t)\|\le L(t)\|z(t)\|$ satisfies $\|z(t)\|\ge\|z(0)\|e^{-\int_0^t L}$. From this: a gain profile $L$ supports exact prescribed-time convergence, over the unrestricted class of closed-loop vector fields admitting it, if and only if its integral over $[0,T)$ is infinite; so every design must either use an envelope of infinite integral or violate the linear growth bound at the origin, as finite-time and fixed-time designs do. Under a finite gain budget $\mathcal B=\int_0^T L$ the best worst-case contraction over the class is exactly $e^{-\mathcal B}$, a floor valid for every plant, and a finite-budget trajectory converges to a definite point away from the origin. A comparison lemma then carries the theory beyond linear growth: for envelopes $\|\dot z\|\le L(t)\,\omega(\|z\|)$, first arrival at the origin requires budget at least $\int_0^{\|z_0\|}du/\omega(u)$; arrival is impossible whenever this integral diverges, and when it is finite it is the exact threshold, attained inside the class. The classical settling expressions of finite-time control reappear as hitting-time floors, and the known fixed-time condition is the two-ended convergence of the same integral. For linear time-varying loops the results refine to logarithmic-norm budgets, corollaries of a known two-sided estimate, with the terminal error of gain freezing described exactly.

Furusho, Hirose and Komiyama reformulated the pentagon equation using an involution called the flip, which led to a new characterisation of the graded Grothendieck-Teichmüller group $\mathsf{GRT}_1$. We show that the flip is a reflection in the cyclic structure of the operad of framed Drinfeld-Kohno Lie algebras, with the framings removed. Applying the reflection twice to the pentagon gives the two-cycle relation for every Drinfeld associator. This relation is precisely the condition for cyclic compatibility on the associativity morphism, giving a new proof that $\mathsf{GRT}_1$ acts by cyclic automorphisms of parenthesized ribbon chord diagrams.