Papers

We prove the Tate conjecture in every codimension on every power of an abelian variety of dimension at most four over a finite field. For geometrically simple fourfolds, we use Broe's theorem and the classification of Frobenius relations. For dihedral pairs of abelian surfaces, we construct algebraic classes using families of abelian threefolds, the Gross–Schoen height formula, and polarization contractions. We also prove the Hodge conjecture for every power of a complex CM abelian fourfold, using Markman's theorem on Weil classes and Milne's criterion. The finite field results imply standard conjecture~D and the expected pole orders of the zeta function on every power.

The Variational Quantum Eigensolver (VQE) requires a large number of measurements to evaluate molecular Hamiltonians. Expressing a molecular Hamiltonian as a linear combination of Pauli strings creates a measurement bottleneck: non-commuting Pauli strings cannot be measured simultaneously. Consequently, mutually commuting Pauli strings must be grouped and measured together to minimise the number of quantum-state preparations. This task maps to the minimum clique cover problem on a commutativity graph, an NP-hard problem typically addressed using classical heuristics. Although an Ising-model formulation has recently been explored on fully connected CMOS Ising machines and demonstrated on physical quantum annealers only at small scale, the embedding cost that governs its behaviour on hardware with sparse connectivity, where each logical variable must be represented by a chain of physical qubits, has not been characterised. In this work, we formulate the Pauli-grouping problem as a standard QUBO colouring model and study its scalability on a D-Wave quantum annealer. Across a series of molecular systems and for both qubit-wise and full commutativity, we quantify the growth in the number of logical variables and the QUBO interaction density, characterise the physical-qubit and chain-length overhead required for embedding on the Zephyr topology, and compare the annealer's time-to-solution and solution quality with those of heuristic classical baselines. This analysis identifies the threshold of molecular complexity beyond which hardware connectivity prevents viable embedding, and shows that a second, practical limit is reached earlier. The threshold therefore measures how far current annealers are from the regime in which pre-optimising a VQE measurement scheme on hardware would be worth considering.

Let $R_N(m)$ count unordered partitions of $m$ into reduced positive fractions whose numerators and denominators are at most $N$, excluding integer parts. Uniformly for $\rho$ in compact positive intervals, $\log R_N(\lfloor\rho N\rfloor)=\sqrt{2\rho}\,N^{3/2}-\kappa(\rho)N^{3/2}/\log N+o(N^{3/2}/\log N)$. The positive, continuously differentiable function $\kappa$ is an explicit sum of lattice-distance integrals, with $\kappa(1)\approx 0.00264713$. For a uniform partition of $n$, all but $o_{\mathbb{P}}(n/\log n)$ prime-denominator blocks in $(n/2,n]$ have total 2 or 3, according to whether $p/n$ lies below or above an explicit threshold near 0.7522. For fixed $N$, we give the Ehrhart numerator and determine how its poles control quasipolynomial coefficients. Its residue distribution is asymmetric for $N\ge 4$, but agrees with an independent model in every moment below order $\lceil N/2\rceil$. We identify the first discrepancy and prove a Gaussian limit. We also obtain joint denominator and size laws under two sampling rules.

Let $M^n$ be a closed minimal hypersurface of the unit sphere with constant squared norm $S$ of the second fundamental form. For $n\geq4$, we prove that $S>n$ implies $S>85n/59$, improving the bound $10n/7$ of Suh and Yang. No constancy assumption is imposed on the cubic trace $f_3$. The proof combines a symmetrized Hessian estimate, an integral identity for a mixed moment, and a weighted estimate for the repeated-index components of the covariant derivative of the second fundamental form. An exact polynomial argument treats the entire excluded interval. We also prove an explicit positive lower bound for the average of the normalized quadratic defect $P/S^2$, where $P=f_4-f_3^2/S-S^2/n$. This second estimate holds in every dimension $n\geq2$ and gives a quantitative obstruction to concentration near the locus of two principal curvatures.

Many stochastic systems in operations and economics exhibit feedback between their long-run state distribution and the transition law governing their dynamics. In this paper, we develop a computational framework for stationary equilibria in such measure-dependent Markov systems when this feedback operates through a finite-dimensional aggregate. We show that the original stationary-equilibrium problem can be reduced to a finite-dimensional self-consistency equation, separating steady-state analysis of the underlying Markov system from equilibrium computation. We use properties of the resulting self-consistency map to guide the choice among fixed-point iteration, relaxed fixed-point iteration, and minimization of the fixed-point residual. The last approach requires derivatives of the self-consistency map, which are typically unavailable in closed form. We therefore develop finite-time infinitesimal perturbation analysis estimators for these derivatives, with error bounds that separate Monte Carlo error from finite-time bias. We illustrate the framework through a strategic $G/G/c$ queue and an opinion-dynamics model, showing how different structural properties lead naturally to different equilibrium-computation methods.

In this paper, we study the harmonicity and existence of algebraic generalized Ricci solitons. Firstly, we characterize harmonic torsion of algebraic generalized Ricci solitons on arbitrary metric Lie algebras by an identity involving the Killing form. In particular, positive semidefiniteness of the Killing form implies harmonicity without a unimodularity assumption. Then, we construct generalized nilsolitons with nonzero torsion on indecomposable three-step nilpotent Lie algebras admitting no classical nilsoliton, in dimension seven and in dimensions $6m+d$ for $m>d\geq1$. Furthermore, we provide a spectral obstruction for nilpotent Lie algebras with abelian derived algebra and an obstruction based on the action of derivations on the quotient by the center. Combining these obstructions with explicit constructions, we prove that the filiform Lie algebra $\mathfrak m_2(n)$, $n\geq5$, admits a generalized nilsoliton if and only if $5\leq n\leq8$.

We report on a Lean 4 formalization of Sullivan's No Wandering Domains theorem: every Fatou component of a rational self-map of the Riemann sphere of degree at least two is eventually periodic. The project formalizes the relevant background in normal families, the Montel-Carathéodory theorem, Julia and Fatou sets, local Sobolev regularity and Wirtinger derivatives, the Cauchy and Beurling transforms, the equivalence of analytic and geometric quasiconformality, and the measurable Riemann mapping theorem. This paper describes the translation of this mathematics into Lean, the proof architecture, the reusable components, and the autoformalization workflow.

In this paper, we investigate the qualitative properties, long-term dynamic behavior, and stability of a longitudinal mathematical model describing the progression of Type 2 Diabetes Mellitus (T2DM). We significantly extend and refine the mathematical analysis of the model recently proposed by De Gaetano et al. (Journal of Theoretical Biology, 2024). Specifically, we analyze both the autonomous system with a fixed age parameter and the non-autonomous system incorporating explicit time-dependent aging dynamics under generalized conditions, providing rigorous proofs for global existence, uniqueness, positivity, asymptotic bounds, and stability properties.

The problem of the Nile has been stated by R. Fisher in 1936. This problem concerns the fair division of an agricultural land and is the eponymous problem of a general class of problems on measures and similar regions. In 1938, W. Feller has shown that the general problem has no solution. However, we will show that under a natural hypothesis, the original problem of the Nile is solvable with connected plots.

We establish Malliavin nondegeneracy for every weighted integral against a singular Hermite kernel, with all negative moments of the Malliavin derivative norm finite, uniformly over admissible compact families of bounded deterministic weights. In particular, all finite-dimensional distributions of Hermite processes of every fixed finite order have Schwartz densities, as do vectors of non-overlapping increments and, more generally, finite families of Wiener integrals with respect to a Hermite process. For such vectors, linear independence of the weights is both necessary and sufficient for absolute continuity; under this condition, all inverse Malliavin determinant moments exist and the joint density belongs to the Schwartz space, with bounds that are again uniform over compact families. The proof rests on a nonvanishing condition on compact weight families that is preserved under directional differentiation. An analytic-tail property of fractional transforms of the directions verifies this condition at every chaos level. Uniform Malliavin estimates then convert lower-order gradient bounds into bounded joint densities of arbitrarily many first directional derivatives, and Bessel's inequality closes the induction. The argument works directly with fixed non-Gaussian laws and singular kernels, without a Gaussian-limit assumption and without a separate local-nondeterminism transfer.

We study fifteen properties of binary relations that have established uses in modal logic, order and preference theory, and relation algebra. The organising question is pragmatic: once some properties of a relation are known, which further properties follow, which combinations force degeneracy, and which properties remain independent? We give a proved Horn basis of elementary and compound entailments, explicit countermodels for non-entailments, and a relation-algebraic translation of all fifteen properties. The modal discussion includes the usual Scott–Lemmon correspondences and the logic of transitive dense frames studied by Ghilardi and Mints. Exhaustive finite enumeration and targeted model search are used for discovery, while the catalogue is certified in Lean. A kernel-checked coverage calculation ranges over all 2^15 = 32,768 antecedent sets and all fifteen possible consequents. Its 51-rule Horn manifest is coupled definitionally to Lean proofs of semantic soundness, and its 49 witness rows are coupled definitionally to Lean-certified finite or infinite relations. Consequently, over non-empty domains, the Horn closure is complete for positive entailment among the fifteen selected properties, and the consistency classification is complete for their positive combinations.

One method of generalizing a distribution is to add new parameter to the baseline distribution. In this paper the author discusses this methodology to create a new six parameters generalized Weibull-Kumaraswamy-{Lomax) Distribution. The author utilizes the approach of the T-X{Y} family to add new parameters the baseline distribution Kumaraswamy distribution defined on the unit interval with two shape parameters a & b>0. The author used the Lomax distribution with two parameters {\alpha} & {\lambda}>0 , scale and shape parameters respectively, to link the baseline distribution with the generator distribution. The generator distribution is the Weibull distribution with the two parameters, c & {\gamma}>0 , shape and scale parameters respectively as will be explained in the paper. The result is a new distribution with six parameters. The author derived the PDF, CDF, and quantile functions. The author used a new technique to expand the pdf in a linear series expansion that is simple and does not require special function. This makes the computational statistics for the raw moments easier and simpler. The author discusses the Shannon entropy and the paper contains figures for the PDF, CDF and the quantile functions.

We survey a geometric approach to the Navier-Stokes equations developed by the author over the past fifteen years. The central object is a Riemannian space of fourteen dimensions whose metric is Ricci-flat precisely on the solutions of the Navier-Stokes system, so that the behaviour of a viscous incompressible fluid becomes a property of curvature. The space decomposes into a flat six-dimensional part and two dual quadruples of coordinates, Eulerian and Lagrangian, which exhibits the two classical descriptions of a fluid as two halves of a single geometry. We also recall the associated six-dimensional metric, whose integrability condition is the incompressibility of the fluid; the link with the projective geometry of second-order ordinary differential equations in the sense of E. Cartan; and the use of Cartan's invariants to construct a three-dimensional analogue of the Taylor-Green vortex. An application of the same construction to the equations of rotation of a rigid body is indicated.

A blob grows on the plane (as in films called "The Blob" of 1958 and 1988). Suppose that it grows in all directions at unit rate, and can be stopped only by a certain kind of fence that can be manufactured at rate $\lambda$. What is the critical value $\lambda_C$ for this rate, above which the blob can be surrounded and contained, and below which we are all doomed? This problem, introduced by Bressan in 2007 and independently by Barghi and Winkler in 2013, was in both cases motivated by consideration of strategies in fighting forest fires. In particular, does it pay to place "gambit" barriers which will subsequently be overwhelmed by the fire, in order to slow it down enabling later containment? A non-gambit strategy, which will be described, succeeds when $\lambda > 2$ and is widely believed to be optimal; but up until now no one has succeeded in obtaining a lower bound greater than 1 for $\lambda_C$. We introduce a novel weakened version of the blob which is used to raise the lower bound to 1.5.

Quasispecies theory describes how mutation and selection shape highly variable RNA virus populations, but most models explicitly consider neither RNA-dependent RNA polymerases nor the delays associated with their synthesis and functional activation. A recent minimal model showed that delayed polymerase availability can induce extinction by reorganizing basins of attraction, a mechanism termed {\tau}-tipping. Here, we extend this framework to a delay differential equation model in which master and mutant genomes encode distinct functional polymerases. We characterize its equilibrium and bifurcation structure, identifying master-mutant coexistence, mutant-only persistence (error catastrophe), and complete extinction governed by transcritical and saddle-node bifurcations. Although delays leave the equilibria unchanged, they reorganize their basins of attraction and can redirect populations that would otherwise persist at fixed mutation and replication parameters toward extinction, thereby extending {\tau}-tipping to systems with autonomous mutant replication. We also recover the previously studied complementation model as a limiting case in which defective genomes depend on master-derived polymerase, providing a comprehensive analysis of the bifurcation structure. Comparing the two models shows that the ability of mutant genomes to encode functional polymerases determines whether loss of the master sequence results in mutant replacement or complete population extinction. The occurrence of delay-induced extinction in both settings demonstrates the robustness of {\tau}-tipping and connects intracellular replication kinetics, complementation, and quasispecies extinction. Finally, we translate this mechanism into concrete virological predictions and propose experimental strategies to test the effects of replicase timing, RNA degradation, and functional complementation on viral persistence.

This paper proves Conjecture 5.10 of Gajdzica, Visser, and Zakarczemny on restricted partitions of a rectangle. For every fixed positive integer k, the generating function for feasible multisets of bars of lengths at most k tiling a 2 x n rectangle has denominator dividing the product of (1 - x^j) for j = 1,...,k. Its coefficients are eventually quasipolynomial of degree k - 1 and period dividing the least common multiple of 1,...,k. The key observation is that appending a two-bar slab makes feasibility upward closed within each parity class of multiplicities. Dickson's lemma and finite inclusion-exclusion then give the precise denominator.

We study how amphichirality of a knot constrains its braid index and, in the smallest nontrivial case, the combinatorics of its braid words. Using the Dynnikov–Prasolov resolution of the Jones conjecture, we show the braid index of an amphichiral knot is odd. This answers, in the negative, a question of Stoimenow on amphichiral knots of even braid index. We then classify prime amphichiral knots of braid index~$3$. Every amphichiral knot of braid index~$3$ is alternating and admits a minimal $3$-braid representative in a standard form encoded by a word~$c$. Building on the Birman–Menasco classification of closed $3$-braids and the Murasugi normal form, we describe a dihedral action on~$c$ under which the mirror and mirror-reverse operations are realized by a rotation and a reflection. For a standard form whose closure is a prime knot of braid index~$3$, this yields a complete criterion: $\widehat{\beta_c}$ is amphichiral if and only if $c$ is a palindrome or has odd period, together with a determination of the precise symmetry type in terms of these properties and the presence of a non-degenerate flype.

We introduce two new graph classes, the bicoloured-interval graphs and the interval-sandwich graphs, arising from the study of the robust line fitting problem in computer vision. In a bicoloured-interval graph, each vertex is assigned a real interval and one of two colours. Vertices of different colours are adjacent precisely when their intervals intersect, and vertices of the same colour precisely when the centre of one interval lies in the other. Discarding the colouring yields the interval-sandwich graphs, sandwiched between the $50\%$-tolerance graph and the intersection graph of a family of intervals. We prove structural results for both classes, determining which holes, antiholes, trees, and complete bipartite graphs each contains. Consequently, neither class is characterised by finitely many forbidden induced subgraphs. We place both classes strictly between the unit tolerance graphs and the co-comparability graphs in the tolerance hierarchy, separating them from the neighbouring classes. In particular, we construct an infinite family of proper tolerance graphs, each of which is a minimal forbidden induced subgraph for the bicoloured-interval graphs. Finally, we consider the recognition problems for both classes. We show that every graph in either class has a polynomial-size integer representation, and hence that both problems lie in~$\textsf{NP}$.

Let $\Gamma$ be the Cayley graph of Thompson's group $F$ with its standard generators. Belk and Brown constructed finite marked-forest subgraphs of limiting density $7/2$, and Guba showed that deleting certain low-degree vertices gives density greater than $3.5004$. We refine this approach in two directions. First, a deterministic interval-deletion rule followed by root-sensitive triple and pair deletions yields finite induced subgraphs with \[ \operatorname{dens}(\Gamma)>3.50074529. \] The interval rule is evaluated by a nine-state recurrence, while the simultaneous deletion conditions created by split and merge operations are computed exactly by a common-suffix first-passage argument. Second, we determine the joint distribution of the categories of a tree and its two children and reduce optimization over every fixed category-window retention rule to a weighted densest-subgraph problem. An explicit edge-allocation certificate shows that, for every fixed $a\ge0$, the optimal limiting density of a rule depending on the window $[-a,1]$ is exactly $7/2$. Thus arbitrary finite left context together with one right-hand category does not improve the Belk–Brown limit, whereas the root-sensitive whole-segment construction does. All constants are explicit elements of $\mathbb Q(\sqrt3,\sqrt{2\sqrt3-1})$.

We give a self-contained proof, in the bounded variation setting, of the Hambly–Lyons uniqueness theorem, which states that (total) signature identifies the path up to tree-like equivalences. The argument is organized around two key geometric observations. First, tree-like paths have trivial signature because factorization over a loop in a tree $\tau:[0,1]\to T$ is preserved under signature lifts, which follows from an elementary property of planar curves. Second, a path with trivial total signature contains a nontrivial subpath with trivial total signature (the sub-interval lemma). This is proven by applying a winding-number argument to a two-dimensional projection of the signature lift. Collapsing all trivial-signature sub-intervals then defines a compact metric tree $T$ through which the original path factors by virtue of the sub-interval lemma.

We study the selection and stable realization of local approximation spaces in Trefftz discontinuous Galerkin discretizations of the Helmholtz equation. A scaled Cauchy-trace inner product places plane waves and Fourier-Bessel functions in a common geometry: Fourier-Bessel modes are orthogonal with explicit weights, while the same weights determine the circulant spectrum of an equispaced plane-wave trace Gram matrix. This separates amplitude scaling from genuine trace-rank loss and yields an exact best-approximation identity for mixed plane-wave–Fourier–Bessel spaces. With complex plane-wave angles, the unresolved modal tail is an exponential sequence, so propagating and evanescent components can be identified by the same ESPRIT/variable-projection procedure. We prove exact recovery and a perturbation estimate for the recovered angles and, under the standard PWDG quasi-optimality bound, transfer these perturbations to the DG error. Trace-Riesz orthonormalization is then separated from a graph–Riesz normalization of the assembled operator. Numerical experiments verify the identities, recover sparse ray fields to roundoff, and resolve a propagating-to-evanescent transition without a prescribed critical angle.

We determine how p-adic slopes of modular forms vary as their associated Galois representations modulo p vary over moduli of Galois representations. As a consequence, we show that the authors' ghost conjecture recently proven by Liu, Truong, Xiao, and Zhao is "stable under direct sums".

We establish a sharp pathwise signal-to-noise criterion for quasi-maximum-likelihood (QML) estimation of a dominant breakpoint in the second-moment structure of a multivariate time series: the QML estimator is consistent whenever the between-regime contrast exceeds the within-regime fluctuation by an explicit factor, and below this threshold global recovery can fail. Two innovations drive the result. First, the framework is pathwise: no stochastic model is imposed on the data. Second, the log-determinant objective carries an additive ridge regularization: it removes the endpoint boundary layers, so no trimming of the candidate set is required, and tuning the ridge weakens the consistency condition. As an application of the main theorem, we establish consistency of QML breakpoint estimation in pervasive factor models whose dimension diverges with the sample size, for completely general error terms – not necessarily independent, idiosyncratic, or even random.

It is known that the Sherman–Morrison (SM) formula is not numerically stable. In recent work, we introduced SMIR, an algorithm that incorporates iterative refinement to enhance the SM backward error. In this paper we take a different route: adapting an algorithm of Govaerts, originally designed for general bordered linear systems, we develop a modified Sherman–Morrison (MSM) method whose built-in self-correction makes it surprisingly resilient. Whereas standard SM requires the solution of two $n\times n$ linear systems, MSM requires three; by contrast, SMIR requires $2+k$ solves, where $k$ is the number of IR steps and can be substantially larger than three, when IR converges slowly. We then derive backward and forward error bounds for both SM and MSM. The SM backward error bound established here is stronger than the one proved in [Hashemi & Nakatsukasa 2026]: it accounts for every rounding error and holds with no conditions on the size of the capacitance. From these bounds we extract growth factors that are cheap to compute a posteriori and can be used to certify the backward and forward stability of SM and MSM on a given problem. In our experiments, MSM consistently produces backward stable solutions, and is therefore observed to be forward stable as well. A formal proof of the stability — or instability — of MSM remains an open problem.

We study a deformation of the central heat trace on $\mathrm U(N)$ obtained by deforming Schur polynomials to Jack polynomials, and prove that it admits an asymptotic expansion to arbitrary order. Its coefficients are governed by elliptic $b$-Hurwitz numbers, which are genus-one counterparts of the $b$-Hurwitz theory of Chapuy and Dołęga \cite{ChapuyDolega22}. We construct the associated elliptic $b$-Hurwitz theory by means of generalized coverings on a torus and identify it with the genus-one closure of the genus-zero simple $b$-Hurwitz theory. At $b=0$ the construction recovers ordinary elliptic Hurwitz theory, while at $b=1$ it gives an automorphism-weighted geometric interpretation of the connected and disconnected twisted elliptic Hurwitz numbers of Hahn–Markwig \cite{HahnMarkwig26}. Our results extend the topological expansion obtained in the classical case in \cite{LemMai25,LM2} and take the form of a coupling between chiral and antichiral elliptic $b$-Hurwitz generating functions.

Given a polynomial ring quotient $R=\mathbb{C}[x_1,\ldots,x_n]/I$ with an action of the symmetric group induced by permuting the variables, a higher Specht basis is a collection of bases for each irreducible $\mathfrak{S}_n$-module in its decomposition that mimics the behavior of the classical Specht polynomial construction in the lowest degrees. Higher Specht bases have now been constructed for the coinvariant ring, the full polynomial ring, the rings $R_{n,k}$ appearing in the $t=0$ Delta conjecture, the hook shape Garsia-Haiman modules, and the two-row Garsia-Procesi modules, and more. We establish a general theory for determining when a higher Specht polynomial is nonzero, and give a proof of a conjectural higher Specht basis for all Garsia-Procesi modules in the cases of three row shapes and hook shapes.

We study numerical invariants of homogeneous ideals generated by three forms, with particular emphasis on the interaction between the initial degree of the syzygy module, the multiplicity of the codimension-two part of the ideal, and the Bourbaki degree. We extend the Bourbaki degree from gradient ideals of plane curves to arbitrary three-generated homogeneous ideals and obtain sharp numerical bounds in both the general and the equigenerated settings. In the equigenerated case, these bounds extend the du Plessis–Wall picture and lead to the study of numerical, structural, dimensional, and integrable gaps. We show that numerical admissibility does not in general imply realizability and obtain further restrictions from the minimal graded free resolution. As an application, we determine the possible Bourbaki degrees of reduced singular quartic plane curves and describe them in terms of their singularity configurations; the possible values are 0,1,2,3,4,6,7,8.

For $1<p<2$, the $p$-Laplace equation and its $p$-Stokes generalization are difficult to solve numerically. Newton's method converges rapidly only close to the solution, with iteration counts that grow under mesh refinement and deteriorate as $p\to1$. The more robust Picard iteration converges only linearly. Rather than globalizing or preconditioning Newton's method, we modify the system to which it is applied. We lift the equation by introducing the flux $|\nabla u|^{p-2}\nabla u$ as an auxiliary (or ”lifting”) variable, apply a nonlinear transformation to the resulting constitutive relation, linearize, and eliminate the auxiliary variable by static condensation. Lifting alone leaves the linearization unchanged; it is the preceding transformation that yields the new method. The elimination is algebraic and pointwise at the quadrature points, so the flux variable is never discretized, no inf-sup condition or indefinite system arises, and the cost per iteration is that of a standard Newton step. Together with a pointwise feasibility bound on the lifting variable that keeps the diffusion tensor uniformly positive definite, this yields an iteration that we prove, in finite dimensions and for the $p$-Laplace equation, to converge globally and locally at a quadratic rate; a one-dimensional model problem explains why the lagged flux variable removes the zig-zag behavior of standard Newton for $p$ close to one. Firedrake-based experiments for $p$-Laplace problems in two and three dimensions and for stationary and time-dependent $p$-Stokes flows show iteration counts largely insensitive to $p$ and to mesh refinement, and up to an order of magnitude fewer iterations than standard Newton for $p$ close to one.

We introduce a new method for obtaining sharp $L^p$ estimates for martingales under predictable analogues of differential subordination. By allowing suitable comparisons between continuous and jump variation adapted to the relevant range of $p$, we retain sharp constants in settings where pathwise differential subordination fails. The key idea is to study together the continuous and jump contributions arising from the appropriate Bellman functions. Although these contributions need not be nonpositive separately, we quantify their defects and show that they compensate at the predictable level. This compensation mechanism is inspired by the author's earlier work. As an application, we substantially improve the explicit dimension-free bounds for the Riesz vectors on the Hamming cube and on $\mathbb{Z}^n$.

In this article, we study Ricci-Yamabe solitons on the Lie group $\mathrm{Sol} \times \mathbb{R}^n$ equipped with a natural left-invariant Riemannian metric, explicitly determining the vector fields that characterize them. We then deduce that, in the case of a Ricci soliton, it is expanding, whereas in the case of a Yamabe soliton, it is shrinking. Finally, we show that if this Lie group is a gradient Ricci-Yamabe soliton, the vector field belongs to $\operatorname{Span}\{\partial_{t_1}, \dots, \partial_{t_n}\}$, and we explicitly provide the Perelman potential.