Papers

Let $p$ be a prime. We provide examples which show that étale endomorphisms of affine planes over an algebraically closed field $k$ of characteristic $p$ can have fibers of arbitrary finite cardinal. Let $(l,m)\in\mathbb N\times\mathbb N^{\ast}$. We provide examples of such étale endomorphisms whose images have complements of cardinality $l$ and whose geometric degrees are $pm$. Several conjectures are disproved, and in particular we provide an analog over $k$ of Kulikov's counterexample to the `Generalized Jacobian Conjecture' of Bass for surfaces. Surjective (resp.\ surjective and non-surjective) counterexamples to Adjamagbo's Jacobian Conjecture over each $k$ are included in dimension $2$ (resp.\ in all dimensions at least $3$); for instance, if $p=2$, then we show for each $m$ there exist surjective étale endomorphisms of the affine spaces over $k$ of dimension at least $3$ of geometric degree $m$. If $e:X\rightarrow X$ is an endomorphism of a variety over an algebraically closed field $K$, then we show that there exists $n\in\mathbb N$ such that $\Imm(e^n)=\Imm(e^{n+1})$ provided either (i) $e$ is quasi-finite or (ii) $\dim(X)\le 2$ and $X\setminus\Imm(e)$ is finite. We provide examples of endomorphisms of affine planes of dimensions at least $3$ over $K$ whose images have complements of cardinality $l$ and for all $n\in\mathbb N$ we have $\Imm(e^n)\neq\Imm(e^{n+1})$. We prove that all affine moduli schemes of étale endomorphisms of affine spaces with Jacobian matrices of determinant 1 over $K$ are connected and classify all those that are smooth. We prove that the Jacobian Conjecture over $\mathbb C$ and Adjamagbo's analog of it over $k$ hold for étale endomorphisms of affine spaces that are composites $g\circ f$, where $f$ is quasi-finite and a locally closed embedding in codimension $1$ outside a specific finite subset and $g$ is a projection that omits one coordinate.

It is shown that a bounded linear map on a complex separable Hilbert space with non-trivial invariant probability measures doesn't have to possess eigenvalues. This resolves a question indicated by Flytzanis in 1995 and concretely asked by Grivaux–López-Martínez from 2023 resp. Grivaux–Matheron–Menet from 2021. Still, as a positive result, we prove that every bounded linear operator on a separable complex Banach space for which a non-fixing invariant probability measure exists, has to have some approximate point spectrum on the unit circle minus $1$.

We describe the topology of the set of tridiagonal symmetric real matrices with fixed non-simple spectrum. The tridiagonal isospectral set is not a submanifold of the corresponding orthogonal conjugacy class, but a collection of submanifolds which intersect cleanly. We show that the tridiagonal isospectral set is an aspherical space and that its Betti numbers are combinatorial numbers. This generalizes to arbitrary spectrum constructions and results known for simple spectrum. We also extend these results to split real semisimple Lie algebras.

We show that the generalized curvatures and more specifically the normal cycle of a compact subset of $\mathbb{R}^d$ are continuous under weak notions of convergence, assuming an a priori mass bound. In particular, provided its mass remains bounded, the normal cycle behaves continuously when the subset is perturbed by a homeomorphism that is $C^0$-close to the identity. Our approach relies on a combination of persistent homology with the geometric measure theory framework classically used in the study of curvatures of singular sets. As an application, we prove that every compact definable set in an o-minimal structure admits a normal cycle by showing that any such set is a limit, in the above sense, of a family of smooth sets whose normal cycles have uniformly bounded mass. We also show that WDC sets are limits of nested smooth sets with uniformly bounded normal cycle masses.

We define a class of infinite dimensional linear port-Hamiltonian delay systems, give a characterization for such systems, and provide an easily verifiable sufficient condition to belong to this class, thereby extending the study by Breiten, Hinsen and Unger in [IEEE Trans. Autom. Control vol. 69, no. 12 (2024), pp. 8924-8930, doi: 10.1109/TAC.2024.3464332]. The results rely on the operator semigroup theoretic treatment of linear delay PDEs.

We prove existence and uniqueness of a càdlàg solution to a planar backward stochastic variational inequality on a general complete filtered probability space, driven by a square integrable martingale, which may have jumps. The multivalued term is the exterior normal cone operator of a bounded uniformly convex planar domain, and the reflection direction is generated by a time-dependent non-symmetric rotation field. The non-symmetry creates a first-order tangential boundary term that destroys the standard monotonicity and quadratic contraction estimates, used in the symmetric oblique-reflection theory. We overcome this obstruction by constructing an explicit symmetric two-point kernel with state dependent coefficients, whose boundary derivative cancels the leading tangential contribution. A weighted martingale-exponential estimate then controls the second order defect and the jump terms. Under suitable geometric and quantitative compatibility constraints, we obtain the unique strong càdlàg solution. We also formulate associated control problems for the rotation angle and prove existence of an optimal control in a compact class of bounded-rate angle paths.

We isolate and analyze the geometric PDE governing the evolution of the vorticity direction in the 3D incompressible (unforced) Navier-Stokes equations (NSE), restricted to the case of a critical spatial point singularity where the vorticity magnitude concentrates as $O(|x|^{-2})$, inhabiting the critical Lorentz space $L^{3/2, \infty}$. The PDE consists of the Harmonic Map Heat Flow (HMHF) into the sphere supplemented with the fluid transport, cross-diffusion and tangential strain. The question is whether the NSE mechanics can propagate logarithmic decay of the local mean oscillations of the direction – the condition $\xi \in \mathrm{bmo}_{1/|\log r|}$ – which (in this setting) was shown in the companion paper to prevent finite time blow-up. A key observation is that the $O(|x|^{-2})$ concentration, factored out of the viscous cross-diffusion, generates an outward radial drift $4\nu\, x/|x|^2$ at the core. Under a suitable condition we prove a barrier lemma for the total drift-diffusion operator and show that the HMHF nonlinearity is harmless for the quantity $\frac12|\xi - e|^2$ which is a subsolution on any hemisphere. This yields a transfer theorem: the $\mathrm{bmo}_{1/|\log r|}$ regularity of the direction at the core, uniformly up to the singular time, is controlled by a logarithmic modulus in time of the direction at the inner scale, together with the physical strain. Crucially, the strain enters the direction equation only through its tangential component which vanishes precisely when the direction is an eigenvector of the strain tensor. Since this includes the eigenvector carrying the maximal stretching, the result is in contrast to the classical geometric regularity theory which is built on depleting the full vortex-stretching term and thus confined to configurations of weak stretching.

We study state tomography when each measurement acts on at most $k$ fresh copies and no quantum memory is retained between blocks. We prove a lower bound matching the upper bound in [arXiv:2510.07788]. Thus the copy complexity of estimating an arbitrary $d$-dimensional state to trace distance $\epsilon$ is, up to absolute constant factors, $\max\{d^3/(\sqrt{k}\epsilon^2),d^2/\epsilon^2\}$ for every $k$ and all sufficiently small $\epsilon$. This removes the earlier restriction that $k$ be small as a function of the accuracy. The lower bound applies to arbitrary measurements within each block and adaptive choices between blocks. The lower bound already applies in a small neighborhood of any state whose smallest eigenvalue is of order $1/d$, even when the center is known. The main ingredient is a uniform Fisher information bound for one measurement block that depends only on the smallest eigenvalue of the state. The proof avoids the perturbative expansion responsible for the restriction in [arXiv:2402.16353]. Fano's inequality for metric balls and a log-Sobolev comparison between mutual and Fisher information then reduce the adaptive protocol to this block bound [arXiv:1607.00550, arXiv:1902.08582].

Let $(M,g)$ be a closed Riemannian manifold of positive Ricci curvature. Let $f$ be a proper isoparametric function on $M$, and $\Gamma$ the unique level set of $f$ which is a minimal hypersurface. For any positive integer $k$, and $\lambda >0$ large enough, we construct a solution of the Allen-Cahn equation $ \Delta u + \lambda (u-u^3 ) =0$ which is constant along the level sets of $f$ and has exactly $k$ nodal components. We prove that the nodal components approach the minimal isoparametric hypersurface $\Gamma$ as $\lambda \rightarrow \infty$ and the energy is uniformly bounded for all such solutions.

We construct a simple expansion of the theory of nonabelian free groups in which the group admits a global type with bounded left-translation orbit but is not definably amenable. This extends the construction of Chernikov, Hrushovski, Kruckman, Krupiński, Moconja, Pillay, and Ramsey, and gives a counterexample to Petrykowski's conjecture.

In smooth convex optimization, the gradient norm is a directly observable measure of stationarity. Accelerating a first-order method that minimizes the gradient norm is known to be more delicate than accelerating the minimization of function values. Optimal accelerated methods such as OGM-G (Kim & Fessler, 2021) are known to exist for any prescribed finite horizon, but their coefficients depend explicitly on the length of that horizon, i.e. the number of iterations. We ask what kind of acceleration is feasible when the stopping horizon is unknown to the method, i.e. for horizon-independent methods. Diakonikolas & Wang (2022) conjectured that an $\Omega(N^{-1})$ lower bound on the squared gradient norm holds at every horizon $N$ for any nonadaptive, horizon-independent linear-span first-order method. We disprove this pointwise conjecture by exhibiting a method that achieves near-$N^{-2}$ last-iterate guarantees on a density-one set of horizons. We show instead that an $\Omega(N^{-1})$ lower bound must hold for infinitely many horizons. More precisely, if $\mathcal G_N(\mathcal A)$ denotes the bound on the squared gradient norm after $N$ iterations for a method $\mathcal A$, we prove that $\limsup_{N\to\infty}N \mathcal G_N(\mathcal A) \ge 1/2$ for any method $\mathcal A$. This bound is sharp: the constant $1/2$ is exactly attained by the horizon-independent gradient-descent schedule of Rotaru et al. (2026). In addition, we show that the above two extreme behaviors cannot be achieved by the same method: any method $\mathcal A$ with an $o(N^{-1})$ guarantee on a subsequence of iterates must satisfy $\limsup_N N \mathcal G_N(\mathcal A)=\infty$. In contrast, best-so-far output admits a uniform $O(N^{-2})$ guarantee.

A tubular group has a graph of groups decomposition with $\mathbb{Z}^2$ vertex groups and $\mathbb{Z}$ edge groups. This paper gives a classification of one vertex one loop tubular groups $G_{(k,\ell),(m,n)} = \langle a,b,t: [a,b]=1, t a^m b^n t^{-1} = a^k b^\ell \rangle$ up to commensurability and quasi-isometry. We show that tubular groups where the images of the edge maps have nonzero “intersection number" are all commensurable and so quasi-isometric. When the intersection number is zero, there are two quasi-isometry classes and infinitely many commensurability classes. The nonzero and zero intersection number tubular groups are not quasi-isometric.

Sliced optimal transport replaces a transport problem in Rd by one-dimensional problems along scalar slices, and has been used both to lower bound the Wasserstein distance W1 (max-sliced distances) and, more recently, to upper bound it by lifting a one-dimensional optimal plan back to the ambient supports (min-sliced plans). We show that for 1-Lipschitz slicers these two directions are two sides of a single sandwich inequality and we characterize when it collapses to equality. If the slicer class contains an optimal Kantorovich-Rubinstein potential, then the same slicer simultaneously maximizes the max-sliced lower objective and minimizes the min-sliced upper objective, and both equal W1. We define the upper objective through a tie-broken transport problem, which removes the dependence on the one-dimensional ordering and yields the identities without any uniqueness assumption on the slice-optimal coupling. The result requires the slicer to be nonlinear in general, consistent with the known failure of exact linear max-sliced/W1 equivalence in dimension greater than or equal to 2, and identifies Kantorovich-Rubinstein potentials as optimal generalized slicers.

Quantum many-body scars are rare nonthermal states embedded within otherwise thermalizing spectra, making them promising candidates for storing and manipulating quantum information. However, a general framework that can both identify such states without prior knowledge of their microscopic scarring mechanism and determine when they enable a quantum error correcting code has remained elusive. Here, we develop an end-to-end framework connecting nonthermal-state discovery to certifiable quantum error correction using a many-body spectral localizer. By searching for states jointly localized in energy and a diagnostic observable in systems with a many-body grading symmetry, the spectral localizer identifies candidate scars within dense many-body spectra in tandem with a topological index whose shifts determine their graded subspace. For suitable pairs of same-graded scars, the grading renders grading-odd errors exactly detectable, while their localization can suppress the logical action of sufficiently local grading-even error products, yielding an approximate Knill–Laflamme condition and an operator-algebra quantum error-correction structure. We demonstrate the scar-identification framework in one- and two-dimensional density-difference-dependent bosonic models and in the PXP model, and explicitly construct the resulting approximate quantum code in the one-dimensional bosonic system. Our results establish a systematic route from identifying nonthermal many-body states to exploiting them for protecting quantum information, and position pseudospectral methods as a broadly applicable tool for discovering useful behaviors in interacting quantum systems.

We study front-type solutions of nonlinear dispersive-dissipative PDEs modeling the propagation of undular bores in a channel in two dimensions. The system extends the Korteweg-de Vries-Burgers (KdVB) equation by incorporating weak transverse motion, and admits the KdVB fronts as one-dimensional solutions. We investigate their stability under general two-dimensional perturbations. We prove that a one-dimensional front is a global asymptotic attractor in the weakly two-dimensional setting when the channel is sufficiently narrow in the transverse direction and the relative dispersion parameter lies in a range for stability to one-dimensional perturbations. Particularly, the front is unique up to spatial translations. The proof extends the energy method for temporally-modulated perturbed solutions, developed previously in the one-dimensional setting, to accommodate the transverse dynamics.

Decomposing a matrix into sparse and low-rank components is central to robust principal component analysis and has broad applications in machine learning, signal processing, and computer vision. Classical formulations seek to recover the underlying sparse and low-rank structure. We instead introduce sparse-plus-low-rank matrix embedding (SLME), whose goal is to construct a computationally efficient surrogate for a large dense matrix, without requiring its components to be interpretable. Given $A\in\mathbb{R}^{m\times n}$, SLME approximates $A \approx S+HW$, where $S$ is sparse, $H\in\mathbb{R}^{m\times r}$, $W\in\mathbb{R}^{r\times n}$, and $r\ll \min\{m,n\}$. The resulting matrix-vector product can be evaluated as $Sx+H(Wx)$ in $\mathrm{nnz}(S)+r(m+n)$ operations, rather than the $mn$ operations required by $Ax$. Our primary motivation arises from optimization problems in cancer radiotherapy treatment planning, where a large dense dose-influence matrix is a major computational bottleneck. We formulate SLME as a bi-objective nonconvex optimization problem that balances approximation error against the computational cost of the downstream tasks. We then develop R3-Trust, an efficient trust-region algorithm that approximates the Pareto frontier in a single parameter-free run. Each point on the resulting frontier provides a sparse-plus-low-rank representation with a different balance between accuracy and downstream computational cost. Experiments on clinical radiotherapy matrices show that decompositions obtained from existing recovery-oriented formulations can be suboptimal for the embedding objective. Conversely, experiments on synthetic instances demonstrate that SLME and R3-Trust can also be applied to sparse-plus-low-rank recovery, where they compare favorably with state-of-the-art recovery methods in both reconstruction accuracy and computational time.

We study the Diophantine problem of representing a Pell or Pell–Lucas number as the base-$b$ concatenation of three other Pell or Pell–Lucas numbers. Under the condition that the index of the middle block does not exceed that of the leading block, we obtain an explicit upper bound for the sequence indices in terms of $b$. For $2\leq b\leq15$, we determine exactly $51$ solutions up to the identification $Q_0=Q_1=2$, while there are $74$ solutions when distinct index choices are counted separately.

Finite model finders cannot witness an Austin law: an identity whose finite models are all trivial but which has a nontrivial infinite model. We introduce rank-decreasing sparse trace-tree magmas, finitely presented total operations on a countably infinite constructor-tree carrier. The default product pairs its arguments; finitely many positive Horn clauses define exceptions. Our main procedure derives clauses from symbolic evaluation traces. For every model found, it proves functionality of the exceptional relation by descent on constructor size, proves the identity by exhaustive symbolic case analysis, and emits a self-contained Lean 4 certificate. A least simultaneous fixed point gives an implementation-independent semantics, so bounded search may miss models but cannot invalidate certified results. On ETP's 96 order-five Austin candidates, we discover and Lean-verify infinite countermodels for 28 identities with no prior public classification in our audit. They form 14 duality classes and establish 28 new Austin classifications. Four ALPS-known cases bring the total to 32 certified candidates. On Canonical-4187, the deduplicated union of Order5-130 and the 4,141-row ALPS pool, a fresh trace run produces 636 certificates, all accepted by Judge v3. At equal resource limits, Vampire 5.0.1, E 3.5.1, and complete Twee 2.6.1 jointly prove implications in 94 canonical classes. Only Twee returns trusted counter-satisfiable outcomes, for 18 classes; independent finite-side certificates force 16 to be infinite. None of these ATPs emits an explicit model or Lean certificate, and none decides the 28 new classifications. To the best of our audit, this is the first automated system to synthesize this trace-tree model family, generate well-founded inversion proofs, and emit self-contained Lean 4 certificates.

We construct bounded Bell observables for a free scalar field in complementary spacetime wedges using measurable involutions of the spectral variable of a local canonical quadrature. These involutions yield wedge-local Hermitian unitary operators that anticommute with a chosen bounded observable. For a cosine observable, we characterize the admissible translations and reflections and derive exact correlations at finite modular bandwidth. The affine-reflection construction yields a violation of the Bell-CHSH inequality of approximately 2.06.

We consider the family of operators of stack sorting (s_m)_{m >= 1} acting on words according to the following rule: s_m(w) is the output produced by the usual West algorithm of stack sorting on the input word w, but allowing for at most m repetitions of the same letter stacked in succession. Our first main theorem gives an explicit formula for the action of the operator s_m on binary words w = a^p b a^q (with distinct letters a > b): s_m(a^p b a^q) = a^max(p-m,0) b a^min(p,m)+q, d_m(a^k b) = ceil(k/m). From the above formulas it follows that for every m < m' we have d_m(w) >= d_{m'}(w) for binary words w = a^k b, and moreover, the ratio d_m(w)/d_{m'}(w) is always possible to be chosen exactly m'/m, so the ratios d_m/d_{m'} are unbounded for this family on the specified class – this is an explicit form of the expected separation-speedup phenomenon. Additionally, we provide explicit constructions of words of length 4 for which s_m and s_{m'} do not commute, and words of length 7 for which the speed d_m is not monotonic in m. Further, we provide a full recursive characterization – generalizing the classical 231-avoidance result of Knuth and West – of words which can be sorted using s_m only once, and then we employ it to show that there cannot exist an m-independent classical pattern avoidance condition for one-pass sortable words. Finally, we give a complete proof of the inequality d_1(w) >= d_m(w) >= d_infinity(w) for all m for two-letter words, together with a structural result establishing it for a single pass in general; nevertheless, we show that the family of operators (s_m) does not, after all, lie entirely between s_1 and s_infinity, exhibiting for every n >= 7 an explicit word w_n with d_1(w_n) = n-4 < n-3 = d_m(w_n) for every m >= 2, including m = infinity, so that the inequality above is false in general.

We (the author and the AI systems that did the heavy lifting) show that the quantum query complexity of minimizing a convex function over a convex subset of $\mathbb{R}^n$ with evaluation and membership queries is $\tilde\Omega(n)$, nearly matching the best known upper bound. In particular, we show this even for quadratic minimization, which is equivalent to inverting an $n \times n$ real matrix using matrix-vector queries. We also show linear or nearly linear lower bounds on the quantum query complexity of computing the trace, the sign of the determinant, and the magnitude of the determinant of a real matrix in the matrix-vector query model. We use a novel quantum lower bound technique, the determinantal witness method, based on identifying a witness whose Fourier transform vanishes on low-rank matrices and that correlates well with the function being computed.

A variety $X$ over a suitable base $Z$ gives rise to a highly structured algebra $C^*(X)$ in motives over $Z$. In turn, a $Z$-point gives rise to an augmentation $C^*(X) \to 1$. This assignment $X(Z) \to Aug(C^*(X))$ factors the so-called “unipotent Kummer map” to torsors under the unipotent fundamental group in realizations. In one direction, this suggests the possibility of “performing” Chabauty-Kim theory motivically without waiting for a motivic t-structure. In a different (largely independent) direction, we may hope to extract arithmetic information for use in bounding sets of integral points from the full rational homotopy type going beyond $\pi_1$. In both directions, the coarse space for motivic augmentations $Aug(C^*(X))$ would benefit from a structure of finite type $\mathbb{Q}$-variety, and similarly, the coarse space $Aug(C^*_{F\phi}(X))$ of augmentations in filtered $\phi$ modules would benefit from a structure of finite type $\mathbb{Q}_p$-variety. We establish two criteria for representability, compute these spaces in several examples, and construct a comparison with Selmer varieties. Finally, we demonstrate how these constructions lead to K-theoretic finiteness criteria in an example.

We prove that every global classical stable solution to the one-phase Bernoulli problem in $\mathbb R^4$ is one-dimensional. In particular, if nonempty, its free boundary consists of one or two parallel hyperplanes. This settles the stability conjecture for the Bernoulli problem in dimension four. As consequences, we obtain universal local Hessian and free-boundary curvature estimates for classical stable solutions in $\mathbb R^4$, as well as axial symmetry of global classical solutions with finite Morse index in $\mathbb R^4$.

A wide range of sparse integer programs admit a block structure in which subproblems interact through a small set of shared variables. Dualizing the linking equalities yields a decomposable Lagrangian relaxation, but generally introduces a duality gap. Recent work shows that this gap can be closed while preserving decomposability by dualizing exponentially large families of redundant nonlinear consistency constraints on the shared variables. We develop a computational framework for exploiting this idea without explicitly constructing the resulting exponentially large relaxation. Our framework combines nonlinear edge encodings of shared-variable consistency with a column-and-row generation (CRG) algorithm that generates local integer solutions by pricing and encoding constraints by separation. With complete encodings and exact separation, the framework recovers the exact relaxation while retaining independent optimization over the blocks. We introduce several encoding families and establish exponential separations among them: the Generalized family can require exponentially fewer constraints than the Vertex, Monomial, or Reflected families, yet can itself require exponentially many constraints on instances for which a single problem-specific encoding suffices. Computational experiments on decomposed stable-set and dominating-set instances show that CRG substantially outperforms a monolithic formulation across a range of tree topologies and coupling strengths. The results also show that richer encoding families need not perform better computationally, highlighting the choice of encoding as a central issue in effective decomposition.

Let $(X,T)$ be a topological dynamical system, and let $\dim(X,T)$ denote Meyerovitch's dynamical dimension. We prove that, for every integer $n\geq1$, if $\dim(X,T)<n/2$, then the set of maps $f\in C(X,[0,1]^n)$ for which the orbit map $$ I_f: X\longrightarrow([0,1]^n)^{\mathbb{Z}}, \qquad I_f(x)=\bigl(f(T^k x)\bigr)_{k\in\mathbb{Z}}, $$ is a topological embedding is a dense $G_\delta$ set. Consequently, $(X,T)$ embeds into $(([0,1]^n)^{\mathbb{Z}},\sigma)$ for some integer $n\geq1$ if and only if $\dim(X,T)<\infty$. The constant $1/2$ and the strict inequality are optimal.

In quantum information theory, the state space of a multipartite quantum system is modeled by a tensor product. In the tensor-product space $\C^{d_1}\otimes\cdots\otimes\C^{d_p}$, a nonzero vector is a product state if it can be written as $\ket{\varphi_1}\otimes\cdots\otimes\ket{\varphi_p}$ with $\ket{\varphi_j}\in\C^{d_j}\setminus\{0\}$. An unextendible product basis (UPB) is a finite family of pairwise orthogonal product states such that no nonzero product state is orthogonal to all of them. UPBs play a key role in investigating quantum entanglement and nonlocal phenomena. Finding a smallest UPB is a natural extremal problem: it asks how few pairwise orthogonal product states suffice to prevent any further product state from being added. The general minimum-size problem for UPBs has been studied for over two decades since the seminal work of Alon and Lovász. Let $f_m(d_1,\ldots,d_p)$ be the minimum cardinality of a UPB in the space, and let \(f_{LB}(d_1,\ldots,d_p)=1+\sum_{j=1}^{p}(d_j-1)\) be the natural lower bound. Alon and Lovász determined exactly when \(f_m\) attains the lower bound \(f_{LB}\), but the obstructed multipartite cases remained open in general. We prove a stabilization theorem: for every non-all-qubit system with $p\ge3$, whenever the natural lower bound $f_{LB}$ is unattainable, the true minimum is exactly $f_{LB}+1$. Precisely, if the number of even local dimensions $d_i$ is positive and even, and at least one local dimension is greater than two, then \(f_m(d_1,\ldots,d_p)=f_{LB}(d_1,\ldots,d_p)+1\). The proof is built on a unified graph-theoretic framework. Our result, together with earlier work, settles the minimum size problem for UPBs in all finite quantum systems.

Thermoacoustic instability in premixed flames results from the coupling between the flame's heat release, as determined by combustion parameters, and surrounding acoustics, as determined by combustor geometry. A primary instability results in flame flattening as intrinsic flame instability modes are stabilised. Secondary thermoacoustic instability results in a parametric flame instability and drastic growth of acoustic amplitudes. Due to their relative expense, numerical simulations of these phenomena remain scarce. In this work, Direct Numerical Simulations (DNS) of thermoacoustically unstable idealised premixed flames in a tube with acoustically closed upstream and open downstream ends are presented. Results herein demonstrate nonlinear saturation of the primary instability as the flame flattens as well as an oscillating flame fingering characteristic of the unsteady Rayleigh-Taylor effect. To reduce computational cost, we perform DNS only on the region surrounding the flame. Acoustics at in- and outflows are described using the Navier-Stokes Characteristic Boundary Condition (NSCBC) method to model their delayed reentry into the domain in a formulation referred to as the Acoustic Delay Characteristic Boundary Condition (ADCBC) method. A new Averaged Proportional and Integral Linear Relaxation (APILR) method is also introduced, which modifies the Classic Linear Relaxation (CLR) method to maintain time-averaged values of inflow velocity and outflow pressure. Here, an integral control term is used to remove non-zero equilibrium time-averaged inflow velocities which impinge control over flame position. Both new methods demonstrate their capability in inert and counterflow flames test cases. These methods enable the numerical simulation of combustion instabilities at significantly reduced computational expense.

The group of polynomial automorphisms of the affine n-space is an interesting large group. A slightly simpler group is its subgroup of tame automorphisms. Natural problems about these groups include the existence of normal subgroups, the classification of finite subgroups, the Tits alternative, and the possible dynamical degrees of their elements. One method to investigate these questions is via some actions on some metric spaces, namely the coset complex and the valuation complex, that we introduce in detail. This paper is a survey focusing on the following three cases: dimension 2, dimension 3, and dimension 4 for tame automomorphism preserving a nondegenerate quadratic form.

We develop two transition principles for lower-bounding value sets generated by structured sequences over prime fields. A reciprocal-affine family with $M$ internal transitions and bounded quotient multiplicity has image size $\gg \min{M,p}^{8/15}$. This recovers the factorial-residue bound and yields the same exponent for arithmetic Pochhammer products, Gaussian $q$-factorials, derangement numbers, and the numbers of ordered subsets. A second theorem treats nonzero sequences whose consecutive ratios evolve under a nondegenerate M"obius transformation: their value sets have size $\gg \min{M,p}^{1/2+\eta}$ for an absolute constant $\eta>0$. As consequences, fixed rows of Pascal's triangle and the initial half-blocks of the Catalan and central binomial sequences exceed the square-root scale. The proofs combine transition quotients with, respectively, Cartesian-product point-line incidence geometry and Bourgain's expansion-based incidence theorem in $\mathrm{SL}_2(\mathbb{F}_p)$.

We determine the maximum rank of $4\times 4\times 4$ and $k\times 4\times 3$ multidimensional arrays over the finite field $\mathbb{F}_2$. These results are made possible by a new method of enumerating non-isomorphic tensors.