Papers

We give an elementary proof that every positive rational number $a/b$ with $b$ squarefree is a finite sum of distinct unit fractions $1/n$, where each $n$ is a product of two distinct primes (Erdős Problem #306). After a reduction to small targets, we take a single complete bipartite graph between the primes in $(y^2,2y^2]$, together with $2$ and the primes of $b$, and a tuned initial segment of the primes in $(y^8,y^9]$, and show that some subgraph has reciprocal sum congruent to $a/b$ modulo $1$; the small total mass then forces equality. Writing the number of such subgraphs as a finite Fourier sum, we sort the frequencies into three cases using a table indexed by the two sides of the graph. The small integer frequencies give a positive main term, and all other frequencies are negligible by a divisor-counting argument and a no-wrap-around form of the Chinese remainder theorem. The only inputs about primes are Chebyshev-type bounds. The circle-method framework comes from Tang's Lean development, which gave the first proof; our construction removes its anchor-synchronisation step. The proof has been formalised in Lean 4, apart from a cited inequality of Ramanujan. This work is a human-AI collaboration: AI tools contributed substantially to the construction, the experiments and the writing.

We study the additive and multiplicative arithmetic of algebraic monogenic semidomains of the form $S_\alpha$. We first relate their structure to the conjugates of $\alpha$, characterize when $S_\alpha$ is a ring, and determine when its additive monoid is free. We then investigate units and prove a Dirichlet-type theorem showing that $S_\alpha^\times$ is the product of a finite cyclic group of roots of unity and a finitely generated free abelian group; in particular, the unit group is free abelian when $\alpha$ is positive. We also give a criterion that produces non-atomic algebraic monogenic semidomains. Turning to finiteness properties, we prove that $\mathbb{Z}[\alpha]$ is a finite factorization domain for every algebraic number $\alpha$, establish criteria ensuring that $S_\alpha$ is a finite factorization semidomain, and construct infinitely many positive cubic generators yielding finite factorization semidomains that are not rings. Finally, we give a membership criterion for rational elements of $\mathbb{Z}[\beta]$ for arbitrary algebraic $\beta$, and prove that, for algebraic integers $\alpha$, unique factorization ascends from $\mathbb{Z}[\alpha]$ to $\mathbb{Z}[\alpha/n]$ for every $n\in\mathbb{N}$.

Let $I\subseteq S$ be a monomial ideal with minimal generator degrees between $e$ and $d$, where $2\leq e\leq d$. We prove that if $t_a(S/I)<ea$, then $$ t_{a+b}(S/I)\leq t_a(S/I)+ \left\lceil\frac{(2d-1)b}{2}\right\rceil $$ for $a\geq2$, $b\geq0$, and $a+b\leq\operatorname{pd}_S(S/I)$. Consequently, $$ t_{a+b}(S/I)\leq ae+db-1-\left\lfloor\frac b2\right\rfloor. $$ In particular, if $t_2(S/I)<2e$, then $$ t_i(S/I)\leq \left\lceil\frac{(2d-1)i}{2}\right\rceil-2(d-e) $$ for $2\leq i\leq\operatorname{pd}_S(S/I)$. We also obtain a regularity bound and give a quadratic family in which our estimate for the last shift is smaller than every bound obtained from ordinary subadditivity. The constant $2d-1$ is sharp for $d=2$.

An $\mathfrak{N}$-semigroup is an archimedean, idempotent-free, cancellative and commutative semigroup. For example, the additive semigroup $\mathbb{R}_{>0}$ of strictly positive real numbers is a $\mathfrak{N}$-semigroup. Hewitt-Zuckerman showed that every $\mathfrak{N}$-semigroup admits a semigroup homomorphism to $\mathbb{R}_{>0}$. We show that every uniquely divisible $\mathfrak{N}$-semigroup $S$ admits enough semigroup homomorphisms to $\mathbb{R}_{>0}$ to detect divisibility in $S$. %They were studied by Tamura and Kobayashi. The main ingredient is a version of the Eidelheit and Kakutani hyperplane separation theorem for vector spaces over $\mathbb{Q}$. A counterexample due to Ravsky explains why this result does not generalise to the larger class of $\mathfrak{N}$-semigroups.

In this article, we investigate the stochastic Camassa–Holm equation driven by additive noise, which models nonlinear shallow-water waves under random external forcing. We establish that, almost surely, any finite-time breakdown occurs through wave breaking: the wave amplitude and the positive part of the slope remain bounded, whereas the minimum slope tends to $-\infty$. Under an additional moment assumption, we derive an initial-slope condition ensuring wave breaking with positive probability. We also formulate the equation as a stochastic Hamiltonian PDE, establish its stochastic multi-symplectic conservation law, and derive the evolution law for the averaged $H^1$ energy. Combining the Fourier–Galerkin method, symplectic Runge–Kutta integration, and exact solution of the linear stochastic subsystem, we propose a fully discrete method which preserves an integrated discrete stochastic multi-symplectic conservation law and the linear growth of the discrete averaged \(H^1\) energy. Under suitable regularity and resolution assumptions, we present that the fully discrete dynamics retain the slope-steepening mechanism responsible for wave breaking, providing a discrete counterpart of the continuous behavior.

The design of optimal causal linear feedback schemes for additive white Gaussian noise (AWGN) channels with noisy output feedback has remained an open problem for over 60 years. Prior work has focused on restricted policy classes, especially passive (uncoded) noisy output feedback, where only the transmitter performs feedback coding. However, passive noisy output feedback fundamentally lacks the degrees of freedom required to attain the information-theoretic performance limit in general. In this paper, we consider the active (coded) noisy output feedback setting, where both the transmitter and the receiver perform feedback coding. We then develop a constructive KKT-optimal active linear feedback design that asymptotically attains the Elias-Butman SNR converse bound, thereby establishing MSE/SNR optimality over the entire class of causal linear schemes. Furthermore, we prove that the optimal passive feedback solution is recovered as a special case of the active design. This passive solution admits a Geometric Toeplitz (GT) structure with a Chance-Love (CL)-style one-shot polynomial characterization, and can be computed with O(log T) complexity. Thus, our results provide an affirmative answer to the long-standing optimality question for noisy output feedback under causal linear feedback coding, and our numerical results support the theoretical findings.

In this paper, we develop a mixed quantization technique for graph vector bundles that combines semiclassical analysis and quantum cat maps. We apply it to study the Kesten–McKay law and quantum ergodicity.

Complementary waveform libraries yield impulse-like aggregate delay responses at zero Doppler, but pulse-dependent Doppler phases degrade their sidelobe cancellation. This paper develops a linear-algebraic formulation for coordinating a (D)-ary transmission schedule and slow-time receive weights for paraunitary MIMO waveform libraries. Using a modal decomposition of the CPI ambiguity matrix, prescribed-order Doppler-nulling conditions for the (D-1) nonzero modes are expressed as homogeneous linear constraints on the receive weights for a fixed schedule. This representation gives the SNR-optimal fixed-schedule weights by subspace projection and reduces the remaining design to a discrete schedule search. Degrees-of-freedom and modal-growth analyses relate the design to the CPI length, null order, and waveform dimension. Numerical examples compare the resulting designs with established binary constructions and demonstrate Doppler suppression across the full ambiguity matrix of a four-waveform configuration.

We analyse a discontinuous Galerkin method for two-dimensional scalar conservation laws on curved domains. Small intersections of a Cartesian grid with the domain are combined by rectangular cell merging. The local space is the restriction of $\mathbb Q_k$ to each physical merged cell. An entropy-conservative flux is supplemented by face viscosity and, for odd degrees, a skew derivative-jump correction on unmerged interior squares. We prove exact semidiscrete $L^2$ dissipation and an optimal error estimate for every fixed $k\ge1$. For smooth solutions with homogeneous trace on the entire boundary, the semi-discrete method satisfies an $O(h^{k+1})$ error bound. The constants are independent of the original cut-cell area fractions. The proof combines a physical-cell $L^2$ projection and cancellation of regular-face projection errors. A separate semidiscrete result covers linear transport with upwind inflow data. Computations for Burgers' equation on a disk and linear advection on a smooth star exhibit orders close to $k+1$ for $k=1,2,3$.

Let \(\mathcal L\) be the Heisenberg-Virasoro type Lie superalgebra whose even part is the twisted Heisenberg-Virasoro algebra and whose odd part is an abelian ideal. We classify \(\mathbb Z_2\)-graded \(\mathcal L\)-modules that are free of super-rank \(1|1\) over \(U(\mathfrak h)=\mathbb C[L_0,F_0]\). Using polynomial shift operators, we express the defining relations as functional equations for the matrix entries of the generators. Assuming the rank one classification for the even subalgebra, we show that, up to parity reversal, every module with nontrivial odd action belongs to a triangular family \(\Omega^+(\lambda,\beta,d)\), whereas every module with trivial odd action is a direct sum of two rank-one modules for the even subalgebra. We determine the graded isomorphism classes: within the triangular family, the coupling polynomial \(d\) is invariant up to multiplication by a non-zero scalar. We also describe canonical proper submodules and show that every module in the classification is reducible. In particular, we provide a new realization of the Heisenberg-Virasoro type Lie superalgebra using $2\times2$ matrices with entries associated with polynomial shift-differential operators.

Let $F$ be a totally real field and $K/F$ a totally real quadratic extension. For a Hilbert cusp newform $f$ over $F$ with base change $f_K$ to $K,$ we determine the exact level of $f_K$ in terms of the level of $f,$ using only local representation theory: for each place $\mathfrak p$ of $F$ we compute how the conductor of the local component of $f$ changes under quadratic base change. The tame case (odd residue characteristic) follows from existing base change theory, but the case $p=2$ requires a finer $\varepsilon$-factor analysis that does not appear in the literature. We give a complete treatment of the dyadic imprimitive representations and, more delicately, the exceptional supercuspidal representations, for which no closed-form base change formula was previously known; in particular we determine, in terms of local root numbers, the base change of every exceptional supercuspidal representation of $GL(2,\mathbb Q_2)$ with trivial central character and conductor $3.$ This yields an explicit formula for the level of $f_K,$ specialized here to $F=\mathbb Q,$ along with a partial converse.

We give a finite-temperature test for the stability of low-energy spectral subspaces. The test uses a local determinant defect for the alpha-z Renyi kernel of two Gibbs states and bounds the tracial L2 distance between their low-energy projections. The main tool is a spectral-flattening principle for bounded positive injective operators in a semifinite von Neumann algebra. It reduces the local product defect to an exact two-projection formula. The thermal result applies to lower-bounded affiliated Hamiltonians with trace-class Gibbs operators and does not require the Hamiltonians to be close in operator norm. We give an infinite-level example with unbounded Hamiltonians. For matrices, the result gives a sum of principal-angle remainders. For qubits, the defect and the thermal gaps determine the ground-state overlap exactly. The same estimate gives a uniform error bound for bounded observables in an effective low-energy model. We also prove the determinant inequalities, their equality cases, and z-monotonicity in matrix, finite, and semifinite settings.

We prove the Bleiler-Litherland conjecture: every lens space obtained by a nontrivial Dehn surgery on a hyperbolic knot in $S^3$ has order at least 18. The key ingredient is a spectral obstruction: if a hyperbolic knot $K$ of genus $g$ admits a lens space surgery with slope $\pm(4g-2)$, then $\Delta_K(t)$ has a real root outside the unit circle. The proof combines Floer-theoretic restrictions on lens space surgeries, Gabai's degeneracy-slope bound and Gabai-Oertel's persistence theorem for essential laminations, Ni's fixed-point theorem for monodromy, and a mod-2 orientability criterion, together with the relation between homological monodromy and the Alexander polynomial. As a further application of this spectral obstruction, we obtain characterizing-slope results for torus knots.

This paper investigates the theory of fractional Sobolev mappings between Carnot groups, with a particular focus on the Korevaar-Schoen energy functional and interpolation properties. We establish that weak contact equations hold for fractional Sobolev mappings on Carnot groups, generalizing classical results from the Euclidean setting to the sub-Riemannian framework. Our main results include: (1) the proof that fractional Sobolev mappings satisfying weak contact equations belong to the horizontal fractional Sobolev space $W_H^{s-\lambda,q}(\Omega;G_2)$ for any sufficiently small $\lambda>0$; (2) interpolation theorems for fractional Sobolev mappings between Carnot groups, demonstrating that the limiting behavior as $s \to 1$ recovers the first-order Sobolev space; (3) a rigorous treatment of the Korevaar-Schoen energy on Carnot groups, including a characterization of its minimizers via horizontal differential forms.

We refine the density-increment argument for $k$-configurations developed by Beker. The point is not to introduce a new inverse theorem, but to retain information that is lost when the graph-counting output is compressed into a single uniform alternative. For the transitive $K_k$ appearing in the $k$-configuration problem we keep a fixed moment throughout the graph recursion and obtain three genuinely different outputs: a row-degree anomaly, a non-centred grid anomaly, and a centred grid anomaly. The centred case is treated directly through positivity of the associated Gram moments, avoiding the usual unbalancing step. Each output is then routed through the appropriate local Kelley–Meka mechanism with its own density gain, rank cost, and width cost. A weighted iteration budget prevents the most expensive local branches from being charged the maximal number of times. For a finite abelian group $G$ of odd order and $A\subseteq G$ of density $\alpha$, write $L=\log(2/\alpha)$ and $\Lambda=L+\log(2k)$. We obtain a lower bound of the form \[ \Pp_{x_1,\ldots,x_k\in G}\!\left(\frac{x_i+x_j}{2}\in A\ \text{for all }1\le i\le j\le k\right) \ge \exp\!\bigl(-C\Phi(k,L,\Lambda)\bigr), \] where \[ \Phi(k,L,\Lambda)=k^{36}L^6\Lambda^6+k^{34}L^8\Lambda^5+k^{32}L^{14}\Lambda+k^{30}L^{16}. \] Consequently, in the Erdős–Moser application one may replace the exponent $68$ in Beker's $k$-configuration route by $36$; in particular the same reduction yields the sum-free lower bound with every exponent $c<1/36$.

Confined cell migration is governed by nuclear and cellular deformation. In this work, we formulate a two-dimensional mathematical framework in which the cell and nuclear membranes are represented as evolving closed surfaces governed by force-balance laws incorporating surface tension, bending elasticity, and external mechanical interactions. Nuclear mechanics depends on several factors, including chromatin organization and compaction, which are regulated by epigenetic modifications. Molecular regulation at the nuclear periphery is described by a surface reaction-diffusion system posed on the evolving nuclear surface. We apply this coupled framework to an experimentally characterized lung cancer cell model in which the balance between the enzymes SETDB1 and SUV39H1 regulates the peripheral deposition of H3K9me3. Using experimentally measured mechanical parameters, we represent control (Ctrl) and SETDB1 knockout (SETDB1 KO) conditions, which display distinct H3K9me3 distributions, especially at the nuclear periphery. Numerical solutions obtained using the evolving surface finite element method qualitatively reproduce the changes in nuclear mechanics as well as part of the experimentally observed increase in the entry time in SETDB1 KO cells. The model therefore reproduces the direction of the experimental observations, while the residual difference between simulated and measured entry times indicates that the cellular response is multifactorial.

We study braid representations obtained from character-twisted Burau inputs by the Katz-Long-Moody (KLM) construction, which corresponds to multiplicative middle convolution for KZ-type equations. For $n\ge2$, let $q=e^{2\pi ia}$, $\tau=e^{2\pi ic}$, where $a,c>0$ and $nc+(n+1)a<1$. Known form and signature results give a positive-definite invariant form for $\lambda=e^{2\pi il}$, $0<l<1-nc-(n+1)a$. For fixed input, unitary realizations are unique up to simultaneous unitary conjugation. We address two questions. First, we determine spectra and invariant summands of the $M=n(n+1)$-dimensional output. The generator spectrum recovers $(q,\tau)$. The output decomposes into two trivial and two reduced Burau summands, and an $n(n-1)$-dimensional summand isomorphic to Long's reduced construction, generically irreducible for $n=3,4$. If $a$ or $c$ is irrational, every generator is non-Clifford in every finite-dimensional Weyl-Heisenberg frame; if $1,a,c$ are linearly independent over $\mathbb{Q}$, every generator is entangling for every nontrivial bipartition. The full braid image is not projectively dense. For general positive-definite unitary inputs, irreducibility of the full braid output at a nonexceptional parameter forces the free-group input operators to commute. Second, we construct one orthonormal basis giving each generator one block of size $2(n+1)$, $n-2$ blocks of size two, and identity elsewhere. Each generator is a product of $2n$ rank-one phase reflections and, in a fixed binary encoding, an $O(M)$-length word in one-qubit unitaries and CNOTs, without auxiliary qubits and including direction preparation and its inverse. We give finite-alphabet approximation, conditioning and error bounds, and a matched-resource comparison with a Quantum Shannon reference compiler. The gate bound is exponential in the encoded qubit count and excludes classical preprocessing.

We develop an intermediate-derivative theory for Banach-valued evolution classes with Morrey control in time. Let $E_0\hookrightarrow E$ be Banach spaces, $m\ge2$, $1<p<\infty$, and $0\le\lambda<1$. If $u\in\mathcal M^{p,\lambda}(E_0)$ and $u^{(m)}\in\mathcal M^{p,\lambda}(E)$, then every intermediate derivative satisfies \[ u^{(j)}\in\mathcal M^{p,\lambda}\bigl((E_0,E)_{j/m,\infty}\bigr),\qquad 0<j<m, \] with no UMD, Fourier type, or operator-valued multiplier assumption. The proof is based on an exact rational-frequency splitting of $D^j$, the Peetre $K$-functional, and the Hardy–Littlewood maximal operator on Morrey spaces. On the line the estimate has the scale-sharp multiplicative form. On finite intervals we construct a graph-Morrey extension and obtain the same embedding. The weak fine index is optimal for arbitrary Banach couples. Combining the derivative theorem with the exact Morrey trace theorem yields an exact onto trace formula for every $u^{(j)}$, together with mixed Hölder/interpolation embeddings and a trace-powered $p$-to-$q$ Morrey gain. If $E_0\hookc E$, these embeddings become compact both in the same Morrey norm after an arbitrarily small loss of interpolation smoothness and in time-Hölder spaces above the derivative-trace threshold. For positive selfadjoint Hilbert scales we further improve the weak interpolation target to the fractional domain $D(B^{1-j/m})$ and show that this exponent is sharp.

How much entropy must a physical classifier produce to achieve a prescribed accuracy? Rate–distortion theory specifies the minimum information required, but does that information threshold suffice to determine the physical cost? We show that it does not, even for a binary task and a two-state memory. For a uniform binary target observed through a finite symmetric experiment, replacing the classification-error constraint with its necessary mutual-information threshold strictly lowers the infimum of entropy production under a common operation time, integrated mobility budget, and sufficiently large finite transition-rate cap. The separation holds whenever the target error lies strictly between the Bayes error of the observations and chance. Two results establish this physical gap. First, ordering observations by posterior confidence gives exact transport–risk and transport–information frontiers. Observations with the same Bayes error can have different cost frontiers. Second, we construct bounded-rate protocols that realize prescribed encoders with write probabilities below one, starting from exact reset, with an explicit excess cost above the transport bound. An achievable information-constrained cost then falls below a lower bound valid for every task-feasible protocol. Examples with repeated noisy observations illustrate the separation. The results identify a limitation of information-only benchmarks for physical classification: task accuracy and kinetic constraints must be retained explicitly. The cost analyzed is total entropy production during memory writing, excluding data acquisition, controller operation, and subsequent reset.

We compare motivic nearby cycles at infinity with classes computed directly on rational compactifications. Let $f$ be a regular function on a smooth variety over a field of characteristic zero, and let $X$ be a smooth proper compactification. Suppose that a reduced divisor containing the boundary and the support of $\operatorname{div}_X(f)$ has simple normal crossings near the polar locus. We prove that the open-pair class defined by $1/f$ agrees with Raibaut's compactification-independent motivic nearby cycle at infinity. The key step is a local cancellation under blow-ups of transverse zero-pole intersections. For Laurent polynomials nondegenerate at infinity, this comparison yields a direct toric proof of Raibaut's essential-face formula. Equivariant Hodge realization and weighted Ehrhart theory give Newton formulas retaining cohomological degree and the limit-weight filtration, and determine irregular Hodge numbers through the limiting-irregular comparison. The weight-forgetting specialization computes the zero-potential stringy invariant of toric stacky fans with the same radial support, yielding a motivic-Hodge derivation of the corresponding Fano-Landau-Ginzburg mirror identity. In the full-dimensional weak Fano case, the refined formulas identify monodromy eigenvalues and Jordan-block sizes with inertia ages and anticanonical Lefschetz strings, giving a noncanonical comparison over $\mathbb C$.

We study corank-two distributions $D$ of odd rank $2N+1$ whose associated pencil of skew-symmetric forms $\{d\alpha|_{D}(q)\mid \alpha\in\Omega^1(M),\ \alpha|_{D}=0\}$ lies, at every point, in the generic orbit of the natural $\mathrm{GL}\bigl(D(q)\bigr)$-action; equivalently, $D$ has maximal first Kronecker index. In corank two, this class is the analogue of contact and even-contact distributions. We prove that the corresponding differential relation is ample and hence satisfies the multi-parametric $C^0$-close $h$-principle, and we deduce a complete topological characterization for a $(2N+3)$-dimensional manifold to admit such a distribution. The criterion is satisfied by non-parallelizable manifolds as well. To the best of our knowledge, ampleness has been verified so far for two classes of distributions defined by an open $\mathrm{Diff}$-invariant relation: even-contact distributions (McDuff, 1987) and distributions of rank greater than two with maximal small growth vector, in arbitrary ambient dimension (Martínez-Aguinaga, 2026); for $N\ge2$, our class is a proper open subclass of the latter, singled out by a further condition on the pencil. The verification of ampleness reduces, via Kronecker–Weierstrass normal forms of the restrictions of the pencil to hyperplanes, to computing the convex hulls of the connected components of the complement of the resultant hypersurface in the space of pairs of real binary forms of fixed degrees, components separated by a winding number, i.e., by the classical Cauchy index. This is in contrast with the even-contact case of McDuff, where the set to be removed is thin (i.e., of codimension at least two), so that its complement is automatically connected and ampleness is immediate.

Let D_N be the determinant of the generic symmetric N x N matrix, and let differential operators act by ordinary differentiation over a field of odd characteristic p. We determine the apolar ideal of D_N in every size. In addition to the classical quadratic relations, it is generated by one (p-1) x (p-1) determinant of differential variables for each set of 2p-2 indices. These additional generators are minimal modulo the quadratic ideal. An exterior-algebra realization identifies the remaining apolar relations with radicals of invariant pairings for SL_2. Semisimplification gives a Hilbert series in terms of Dyck paths of height at most p-2, and the Steinberg symmetrizer gives the explicit generators of the entire radical. We also compute the ranks of powers of a general linear form. The weak Lefschetz property holds exactly when N <= 2p-2, and the strong Lefschetz property holds exactly when N < p.

In 1918, Gross proved that a regular local inverse of a meromorphic function in the plane can be continued analytically along every ray from its centre except for directions in a set of Lebesgue measure zero. Eremenko asked whether the same conclusion holds for an implicit function defined by an entire relation in two variables. We prove that there exist an entire function $F$ of two variables and a regular implicit germ $\varphi$, defined by $F(z,\varphi(z))=0$, whose analytic continuation fails on every ray from its centre. In fact, along each ray, the modulus of the continuation tends to infinity as the singular point is approached.

Starting with the vertices of a simplex in $\R^n$, we build a cubic vector field. If the semiaxes of the simplex's John ellipsoid are distinct, the Jacobian of this field has repeated eigenvalues exactly on the ellipsoid's focal sets. This works in every dimension $n\geq2$. We give examples in $\R^4$, $\R^3$, and $\R^2$: three focal quadrics, an ellipse and a hyperbola, and finally two points. In the plane, these two points are also the zeros of the derivative in the Siebeck–Marden theorem. Complex numbers explain why the planar case has a shorter formula.

Let $M_f$ be the closed hyperbolic $3$–manifold which is the mapping torus of a pseudo-Anosov homeomorphism $f$ of a closed surface $S$ of genus at least two. The inclusion of the fiber subgroup \(\pi_1(S)\hookrightarrow \pi_1(M_f)\) has a Cannon–Thurston boundary map \(\partial\pi_1(S)\cong S^1\to S^2\cong\partial\pi_1(M_f)\), which is a Peano curve. We relate this map to two variational representatives of the fiber homotopy class: the harmonic map of Eells–Sampson and Hartman, and a least-area embedded minimal surface. Equivariant lifts of homotopic maps are a bounded distance apart; hence, for every marked conformal structure on $S$, the lift of the harmonic representative extends continuously to the closed disk with boundary value the Cannon–Thurston map, and a least-area fiber lifts to a properly embedded minimal plane in $\Hh^3$ with the same boundary map. We give area and curvature bounds for a least-area fiber, show that its largest principal curvature is at least one, and study the energy of the harmonic representatives as a function on Teichmüller space, its critical points, which are the zeros of the Hopf differential, and its invariance under the monodromy.

We show that, in $\MG^{-}$ — $\MG$, the Hilbert plane, deprived of the Side-Angle-Side axiom — a hypotenuse-angle criterion for right triangles, together with the existence of a perpendicular from an external point to a line and a ray correspondence principle governing the ordering of angles, suffices to reconstruct SAS in full — and hence, as standard consequences, SSS and SAA as well. This offers a more elementary alternative to Hilbert's own choice of SAS as an axiom, confined throughout to right triangles and provable without any continuity assumption. Several recent papers have proposed related alternatives on comparable grounds; we compare our approach to theirs in the concluding remarks. The result builds on, and is best read alongside, two earlier papers of ours on related reconstructions of SAS \cite{Volpe2026I,Volpe2026II}.

We show that there exists a subadditive bijection $f:(0,\infty)\to(0,\infty)$ such that $\liminf_{x\to 0}f(x)=0$ and $\limsup_{x\to 0}f(x)=1$. This answers a question by Matkowski and Swiatkowski in [Proc. Amer. Math. Soc. 119 (1993), 187–197].

Guo and Igusa introduced the derived and sub-derived delooping levels as refinements of the delooping level and asked how these two invariants compare, whether the derived delooping level always coincides with the opposite big finitistic dimension, and, if not, how large their difference can be. We answer these questions. First, we prove that every Artin algebra $A$ satisfies $\operatorname{ddell} A\leq \operatorname{sub-ddell} A$. This gives a positive answer to the ordinary-degree comparison problem, whereas a single explicit eight-dimensional algebra $C$ satisfies $k\text{-}\operatorname{sub-ddell} C=0<1=k\text{-}\operatorname{ddell} C$ for $k\geq 2$. We then compute, for every $k\geq 1$, all $k$-derived and $k$-sub-derived delooping levels of the monomial family of Barrios–Lanzilotta–Mata. In particular, that family satisfies $k\text{-}\operatorname{ddell} A_{n,s}=\operatorname{Findim}(A_{n,s}^{\mathrm{op}})$ for $k\geq 1$. In ordinary degree, this recovers an equality previously announced by Lam. We also construct a two-vertex, eleven-dimensional monomial algebra $\Lambda$, over an arbitrary field, such that $\operatorname{Findim}(\Lambda^{\mathrm{op}})=1<2=\operatorname{ddell}\Lambda=\operatorname{dell}\Lambda$. The same strict inequality is obtained by Gao, Liu and Xu using different algebras. A ten-dimensional non-monomial variant with a commutative corner algebra is also given. Finally, we construct a family of finite-dimensional monomial algebras $B_r$ satisfying $\operatorname{Findim}(B_r^{\mathrm{op}})=1$ and $\operatorname{ddell} B_r=2r+1$. Thus the difference $\operatorname{ddell} A-\operatorname{Findim}(A^{\mathrm{op}})$ is unbounded even among monomial algebras with fixed opposite big finitistic dimension.

We present a constructive atlas of finite simple groups with proofs of their orders, simplicity, and structural properties. The eight completed families are cyclic groups of prime order, alternating groups, the classical series $A_r(q)$, $B_r(q)$, $C_r(q)$, $D_r(q)$, and the exceptional series $G_2(q)$ and the small Ree groups $ {}^2G_2(3^{2m+1})$, $m\geq1$. The fifteen sporadic entries are $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$, $M_{24}$, $\mathrm{Co}_1$, $\mathrm{Co}_2$, $\mathrm{Co}_3$, $\mathrm{McL}$, $\mathrm{HS}$, $\mathrm{Suz}$, $J_2$, and $\mathrm{Fi}_{22}$, $\mathrm{Fi}_{23}$, $\mathrm{Fi}_{24}'$. The simple parameter ranges and exceptional cases are stated explicitly. The models arise from codes, lattices, forms, algebras, and finite geometries, including the split octonions and the Conway–Parker algebra. They retain natural actions, stabilizers, central quotients, and comparison maps for subsequent group theory. Several classical comparison isomorphisms relate the models, while involution-class counts distinguish the equal-order orthogonal and symplectic families in odd characteristic and rank at least three. Drawing on classical sources and companion mathematical work, the project develops the construction side of finite simple group theory, not the exhaustiveness proof of the classification. The completed models and stated proofs are formalized in Lean for verification and reuse.

We classify the simultaneous half-bandwidths \(p,q\) of a symmetric oscillatory matrix \(A\) and its inverse. At order \(n\ge2\), the possible pairs are \((1,n-1)\), \((n-1,1)\), and all pairs with \(2\le p,q\le n-1\). Every pair has an integer realization of determinant one. Let \(e_k(A)\) be the least positive integer \(m\) for which all minors of order \(k\) of \(A^m\) are positive. We prove \[ e_k(A)\le \min\{\max\{k,n-p\},\max\{n-k,n-q\}\}\qquad(1\le k<n). \] When \(p+q\ge n\), one realization attains all these bounds simultaneously. The maximum of \(\sum_{k=1}^{n-1}e_k(A)\) is determined for every bandwidth pair. Write \(A=U^T\Delta U\), with \(U\) unit upper triangular and \(\Delta\) positive diagonal. Let \(\ell(U)\) be the least number of positive elementary upper bidiagonal factors of \(U\). At fixed \((p,q)\), its minimum is \(\max\{n-1,p+q-1\}\). The bound \(\sum_{k=1}^{n-1}e_k(A)+\ell(U)\le n(n-1)/2+n-1\) is sharp at each factor count; equality is characterized using \(U\), \(U^2\) and the zero corner minors of \(A\). In particular, \(\sum_{k=1}^{n-1}e_k(A)\le n(n-1)/2\), with equality exactly for basic matrices. In the basic oscillatory class, without a symmetry assumption, a positive integer vector \((f_1,\ldots,f_{n-1})\) is realizable if and only if \(f_k\le\max\{k,n-k\}\) and \(f_k+f_{n-k}\ge n\) for \(1\le k<n\), and \(\lvert f_{k+1}-f_k\rvert\le1\) for \(1\le k<n-1\). Explicit factor orders realize every such vector.