Papers

For every even integer $m\geq 2$, we construct a closed, orientable, smooth, aspherical $(2m+1)$-manifold whose fundamental group has nontrivial center and for which infinitely many tautological classes in the $\mathbb{F}_2$-cohomology of the classifying space of its group of homotopically trivial diffeomorphisms are nonzero and not nilpotent. Specializing the calculation to cohomological degree $0$ gives examples of such manifolds that do not bound compact smooth manifolds. These provide counterexamples to a conjecture of Hebestreit–Land–Lück–Randal-Williams.

All recent advances \cite{DRS}, \cite{S-Rita1}, \cite{S-Rita2}, \cite{Millet}, \cite{MaS}, \cite{MaS2} on the boundedness and decay for wave equations on subextremal Kerr spacetimes $\KK(a,m)$, with non-small angular rotation, rely in an essential way on Whiting's groundbreaking discovery of a powerful integro-differential transformation with the help of which he ruled out exponentially growing modes in \cite{W} (see also \cite{Yacov}, \cite{AMPW} and \cite{Rita} for further developments). We give a purely physical space, generalized version, of the transformation which takes general solutions of {spin-$\sk$} wave equations in $\KK=\KK(a,m)$ to solutions of a secondary wave equation on a new metric manifold $\KKt=\KKt(a,m)$ with two stationary, asymptotically flat, ends. Moreover, for a certain subdomain $\DDt\subset\DDt_-$, which can be identified as the exterior of a black hole region, the new metric $\gt$ of $\KKt(a, m)$ is Lorentzian and the time translation $\T $ is timelike. Using the geometric properties of the two ends one can then deduce, by classical asymptotic Fourier analysis techniques, the desired bound for the flux of $\psi$ at the future event horizon. This estimate played a crucial role in our companion paper \cite{He-K1}.

We study the generic nullity of generalized commutator operators \[L_{\mathbf{A}}(X)=s_{k+1}(A_1,\cdots,A_k,X)\] on matrix algebras, where $s_{k+1}$ denotes the standard polynomial. Dixon and Pressman conjectured an explicit formula for the generic nullity of $L_{\mathbf{A}}$, and Brassil and Reichstein proved the conjecture when $k$ is even. In this paper, we settle the remaining case where $k$ is odd. Our proof first treats the boundary cases $k=2n-3$ in dimensions $n$ and $n+1$ using degree decompositions and graph-theoretic interpretations of alternating trace forms, and then establishes a dimension-extension argument from $n$ to $n+2$. Consequently, together with the result of Brassil and Reichstein, we obtain a complete proof of the Dixon-Pressman generic nullity conjecture over any field of characteristic zero.

We consider a Type-II variational principle driven by geometric rough path with split boundary conditions naturally suited to adjoint systems. From this rough variational principle we derive rough Hamilton's equations, establish their pathwise conservation laws and associated Hamilton–Jacobi equation. We then specialise the framework to rough adjoint systems, obtaining pathwise conservation and quasi-conservation laws that underpin adjoint sensitivity analysis with respect to initial conditions and parameters. On the discrete side, we construct a rough Galerkin discretisation of the rough Type-II variational principle and show that it generates a symplectic flow with discrete analogues of the continuous conservation laws. We establish it's equivalence to a class of Rough Symplectic Partitioned Runge–Kutta (RSPRK) methods and analyse its convergence and naturality properties. Lastly, we perform numerical experiments to validate the predicted convergence rates and demonstrate that RSPRK methods preserve the adjoint conservation laws to machine precision, yielding more accurate and stable gradients in optimisation problems than non-symplectic alternatives.

This paper is concerned with the Cauchy problem for the one-dimensional bipolar Euler-Poisson equations with space-dependent damping, which arise in the hydrodynamic model of semiconductors. Notably, the two pressure functions are allowed to be different, and the doping profile is nonzero. By means of the time-weighted energy method with carefully chosen weights, we prove that if the initial perturbation around a constant steady-state is small enough, the Cauchy problem admits a unique global smooth solution, which converges to the constant steady-state at algebraic decay rates. We show that the decay rates coincide with those established for the constant-coefficients damping case, thereby overcoming the influence of spatially dependent damping on the decay rates of the solutions. A key feature is that no smallness of the space-dependent damping coefficients is required.

We obtain asymptotic formulas for the three sums \[ \sum_{n\le x}\frac{\psi(\psi(n))}{\psi(n)},\qquad \sum_{n\le x}\frac{\psi(n)}{\psi(\psi(n))},\qquad \sum_{n\le x}\log\frac{\psi(\psi(n))}{\psi(n)}, \] where \(\psi\) denotes Dedekind's arithmetic function. We also determine the normal order of the successive-iterate ratio; more precisely, \[ \frac{\psi(\psi(n))}{\psi(n)} \sim \frac{6e^\gamma}{\pi^2}\log_3 n \] for almost all positive integers \(n\).

We study the iteration of (T(n)=\psi(n)-n), where (\psi) is the Dedekind psi function. We show that, for almost all integers, successive iterates satisfy strong asymptotic growth relations governed by (T(n)/n). As an application, we prove that for every fixed (\ell\ge1), the set of integers satisfying (T^{(\ell)}(n)=n) has asymptotic density zero, and hence so does the set of periodic points of any fixed period.

This thesis investigates Euclidean-space and $\ell_p$-norm representations of different forms of data, with a focus on efficient algorithms for computing these representations in settings where the amount of data is very large. The applications of such representations are also discussed: they may be used to summarize the data in a more compact form for downstream tasks while preserving its most salient properties; and further, they may also be used to extract insights about the data that may not be apparent in its original form. This thesis presents novel algorithms for computing Euclidean representations in the case where the data comes with linear structure, as well as in the case where the data comes only with metric, or distance structure. In addition, we give algorithms for computing $\ell_p$-norm representations in linear structured cases. The analyses of these algorithms use tools from geometry, probability, and optimization, and these tools are used to illuminate other algorithms for similar problems.

For a graph $G$ of order $n$, its positive, negative and non-positive inertia is the number of positive, negative and non-positive eigenvalues of its adjacency matrix $A(G)$, respectively. Mohar asked whether every graph with $k$ non-positive eigenvalues has order $O(k^2)$ as $k\to \infty$. Using NEPS, we construct a sequence of non-singular graphs with negative inertia $k$ and order $\Omega(k^{\frac{7}{3}})$ as $k\to \infty$, thus resolving Mohar's problem. Our result also strongly refutes a recent conjecture of Akbari, Elphick, Kumar, Pragada, and Tang involving positive and negative inertia.

Fixed shipment costs encourage large orders, while inventory and customer backlog costs favor frequent replenishment. In a serial system, the stages must also coordinate their lot sizes and material availability. We study the classical echelon stock $(R,nQ)$ policy: the downstream stage requests a fixed lot, the upstream stage orders an integer multiple of that lot, and transfers contain complete lots. For unit Poisson demand, deterministic lead times, setup and holding costs at both stages, and full backlogging, we prove that the optimized cost in this class is at most $5/3\approx1.667$ of the optimal admissible system cost, uniformly over all nonnegative cost rates, demand rates, and lead times. The guarantee holds even when the selected policy pays a setup charge for every lot while the optimal benchmark pays only once per aggregate shipment. To obtain the guarantee, we use a common inventory calculation that allows independently chosen replenishment frequencies while respecting upstream material availability. We compare its optimized value, an upper bound on the best integer-ratio policy cost, with a novel lower bound on the optimal system cost that combines allocated single-stage costs with the minimum inventory cost when setup costs are zero. We also show that no uniform factor below $(1+\sqrt{3})/2\approx1.366$ is possible for the classical class. Existing numerical experiments place best-found classical policies within $22.7\%$ of an evaluated lower bound across three broad grids. The results quantify both the reliability and the limits of a simple operating rule and explain how shipment consolidation, replenishment timing, and flexibility affect its value.

We analyze an inverse problem for determining the piston speed and the associated flow field from a prescribed leading shock and the initial data in a shock tube. The gas flow is described by the isentropic Euler equations (i.e., the $p$-system), while the trajectory of the leading shock is prescribed as a given Lipschitz curve. Under an Oleĭnik-type entropy condition on the leading shock, we develop a modified wavefront tracking scheme to construct the flow field behind the shock. This construction enables us to determine the corresponding piston speed and the associated flow field.

Strongly enforced Dirichlet boundary conditions require highly refined near-wall meshes to resolve steep velocity and thermal gradients. This introduces high computational costs, especially for practical flow simulations. Weakly enforced boundary conditions alleviate this burden by acting as a variationally consistent near-wall model. By allowing a controlled slip at the solid wall, weak enforcement recovers accurate flow quantities on coarse boundary-layer meshes across both incompressible and compressible regimes. Furthermore, weak boundary conditions serve as the fundamental enabling technology for immersogeometric analysis. Because the weak operator evaluates boundary integrals independently of the background mesh, high-fidelity flow analysis can be performed directly on complex geometries without fitting a mesh to the surface. This capability has facilitated direct geometry-to-analysis workflows for boundary-representation CAD models, raw point clouds, photogrammetric reconstructions, and segmented medical images. This review examines the unified mathematical development of the weak boundary condition framework from scalar advection-diffusion equations to the full Navier-Stokes equations, illustrating its versatility and robustness across incompressible and compressible flows, whether in traditional boundary-fitted, sliding-interface, or advanced immersogeometric applications.

Let $L^2_m(\mathbb{R}^2)$ be the space of square-integrable functions on $\mathbb{R}^2$ with $m$-fold rotational symmetry. Let $\overline{\omega}(r,\theta) = r^{-\alpha}f(\theta)$ be a homogeneous steady state of the two-dimensional incompressible Euler equations with $(m\cdot l)$-fold rotational symmetry. If $f(\theta)$ is a constant function we say that $\overline{\omega}$ is a radial power-law vortex. We prove that the incompressible Euler equations in vorticity form, linearized around any homogeneous steady state $\overline{\omega}$ that is not a radial power-law vortex, are ill-posed on $L^2_m(\mathbb{R}^2)$ for any $m\geq 2$, any $l\geq 1$ and $\alpha\in (0,1)$.

We study several $k$-fold generalizations of the Lovász theta function associated with the maximum $k$-colorable induced subgraph problem. The first is the Narasimhan–Manber parameter $\vartheta_k$. We prove that, for graphs whose adjacency matrix belongs to a homogeneous partially coherent algebra, this parameter is recovered by the theta number of the Cartesian product with the complete graph on $k$ vertices. This class includes distance-regular and $1$-walk-regular graphs, and thus our result generalizes a theorem by Sinjorgo and Sotirov (2022) for graphs that are vertex- and edge-transitive. We introduce a new parameter $\varphi_k$ obtained from orthonormal representations of graphs and show the inequality $\varphi_k \leq \vartheta_k$. For both parameters, we study the smallest $k$ for which the parameter is equal to the number of vertices; these saturation parameters yield lower bounds on the chromatic number. We determine which vertex-weighted versions of these parameters are gauges, and discuss a natural definition for the $k$-fold theta body of a graph. We conclude with open questions comparing $\vartheta_k$, $\varphi_k$, $\vartheta(G\square K_k)$, and related convexifications.

We propose a symmetry-preserving Boson-informed neural network (BINN) for the variational Monte Carlo computation of interacting-boson ground states in continuous space. The framework explicitly separates the two dominant physical scales of the system. At the microscopic interaction scale, a cell Jastrow factor absorbs the leading near-contact stiffness, while at the macroscopic scale, a potential-aware envelope encodes the external confinement, leaving the neural network to capture a smoother many-body residual. Across both two- and three-dimensional tests, the BINN approaches the Hartree and Gross-Pitaevskii limits with an empirical $\mathcal{O}(N^{-1})$ finite-size trend in both energy and density. The energies agree with diffusion Monte Carlo to within $0.06\%$ for the particle number tested up to $350$ in the three-dimensional shrinking-range case. Applications are made under various trapping scenarios, justifying robust performance of BINN.

Let $M^5$ be an oriented real hypersurface of a six-dimensional strict nearly Kähler manifold $(\widetilde M^6,\widetilde g,J)$, and let $\xi=-JN$ and $\eta=g(\,\cdot\,,\xi)$ denote the induced Reeb field and Reeb one-form. We prove the pointwise obstruction \[ (d\eta)_p\neq0\qquad\text{for every }p\in M. \] The proof is local and uses only the hypersurface identity \[ \nabla_X\xi=\phi AX-G(X,N) \] together with the six-dimensional strict nearly Kähler algebra. If $(d\eta)_p=0$, then \[ \phi A+A\phi=2G_N,\qquad G_NX=G(X,N), \] which forces the Hopf condition. On the contact hyperplane $\mathcal D_p=\ker\eta_p$, the endomorphism $\phi A+A\phi$ then commutes with $\phi$, whereas $G_N$ anti-commutes with $\phi$ and satisfies \[ G_N^2=-\alpha I_{\mathcal D_p},\qquad \alpha>0. \] This contradiction proves the obstruction. As a connection-theoretic consequence, for the standard Yano semi-symmetric metric connection associated with $\xi$, \[ \widehat\nabla_XY=\nabla_XY+\eta(Y)X-g(X,Y)\xi, \] one has in dimension five \[ \operatorname{Skew}\operatorname{Ric}^{\widehat\nabla}=-\frac32\,d\eta, \] and hence $\operatorname{Ric}^{\widehat\nabla}$ is nowhere symmetric. The obstruction also rules out any local induced almost contact class whose defining conditions include $d\eta=0$. We illustrate the result on the standard homogeneous strict nearly Kähler spaces.

We study gradient Ricci solitons on partially totally real (PTR) submanifolds of complex space forms. In nonzero ambient holomorphic sectional curvature, the Gauss–soliton equation reconstructs the square $P^2$ of the tangential Kähler operator from the Hessian endomorphism and an extrinsic Ricci operator. Consequently, the PTR tangency spectrum is encoded by intrinsic soliton data together with the second fundamental form. For compact minimal PTR solitons with positive ambient curvature we obtain $\lambda\le c(n-1)/4$, with a strict version under a relative-nullity condition. We also prove compact triviality criteria, an exact trace-free anisotropy identity, a parallel-data rigidity result, and a nonexistence theorem for compact real hypersurfaces. A flat multi-level Gaussian model and a nonflat Segre $CR$-product provide explicit consistency checks and identify the limiting cases of the reconstruction formulas.

A question posed in the literature asks whether the divisoriality of a prime can alternate indefinitely along an ascending chain of overrings of a Prüfer domain or a descending chain of same-quotient-field underrings, and what happens at the limit. We construct such chains with all finite-stage domains Bézout and show, in both directions, that the limit prime can be either divisorial or nondivisorial.

We introduce an observable distance that compares realvalued features through a fixed bounded coordinate. The coordinate retains the distinction between finite feature values while compressing their independent escape to infinity, the source of nonseparability for the classical observable distance on all geometric data sets. The new distance makes the full class separable and geodesic: every pair is joined by a constant-speed path, and on metric measure spaces the induced topology agrees with the concentration topology. From the same coordinate we construct compact screens, whose features take values in one fixed compact interval. The screened class is Polish and geodesic for the Box distance. Organizing its finite-feature quotients by the feature order yields a compact pyramid space. Pyramids generated by single compact screens form a dense subspace, so this pyramid space compactifies the original class after passage to compact screens. Convergence is detected through the Box-Hausdorff behavior of every finite measurement layer. Finally, taking sum-metric products with a common metric measure factor is nonexpansive for the new distance. More precisely, the comparison determined by a prescribed coupling of the original spaces and the diagonal coupling of the common factor is preserved, whereas optimization over all product couplings yields the nonexpansive inequality.

In a mechanistic model of the dawn chorus, Kaye showed that heterogeneous activation thresholds and a shared feedback signal determined by the population's active fraction can produce abrupt collective activation and hysteresis. We extend this mechanism to a network of interacting agents. Each node has a continuous activation level, and a nonnegative row-stochastic matrix determines how node activities contribute to individual feedback. We prove that sufficiently weak feedback yields a unique globally attracting equilibrium. For all feedback strengths, the homogeneous dynamics exactly reproduce Kaye's scalar equation; consequently, network topology does not alter the folds or cusp of the homogeneous branch, and no heterogeneous mode becomes unstable before the homogeneous mode. For equitable partitions, the network admits an exact quotient system in which nodes within a block receive the same aggregate input from every block. When blocks are uncoupled, the quotient reduces to independent copies of Kaye's scalar equation. We show that every assignment of stable scalar equilibria to blocks persists under sufficiently weak interblock coupling, producing a combinatorial family of stable quotient equilibria that lift to stable full-network equilibria and remain under small perturbations that break exact equitability. For two symmetrically coupled blocks, branches with unequal block activities terminate at a pair of symmetry-related cusp bifurcations. Near the onset of bistability, we derive scaling laws for the interblock coupling at which these bifurcations occur, the activity difference between the blocks at bifurcation, and the corresponding shift of the external stimulus from the scalar cusp. Numerical continuation confirms the scaling laws for gamma, logistic, and normal threshold distributions.

We present a machine-checked Lean~4 formalization of Dong and Yang's classification of optimal finite-length $(n,4)$ binary block codes for binary symmetric channels. The formalization was developed mainly by feeding the paper's proofs to an AI tool. To establish correctness, the authors verified the main theorem statements in Lean and the accepted axioms. This note discusses the corrections and simplifications made to the AI-generated formalization, and records discrepancies found in the paper during the formalization. The Lean code is available at https://github.com/shhyang/n4code_lean.

While studying for a linear algebra final, the first named author prepared some test questions for herself to see how well she understood the material, and asked the second named author: \emph{If $A$ and $A^T$ have the same eigenvalues and eigenvectors, is $A$ a symmetric matrix?} We show how this excellent question is a great springboard to related questions, in particular when do equal eigenvalues and eigenvectors imply the matrices are, if not equal, at least closely related (such as similar or the transpose/complex conjugate transpose of each other)? The answer depends on how we interpret the question, and provides a great opportunity to talk about creating good questions. In particular, we characterize matrices $A$ for which the transpose or conjugate transpose shares the same eigenvectors (regardless of eigenvalues) and, for each eigenvalue, the same eigenpair (equivalently, the same eigenspace). Thus, a square matrix $A$ is Hermitian if and only if $A^*$ has the same eigenpairs as $A$; moreover, if $A$ is a real matrix with real eigenvalues and $A^T$ has the same eigenvectors as $A$, then $A$ is symmetric.

By utilizing two known hypergeometric summation identities, we extend two \mbox{well-known} Ramanujan series for $1/\pi$ by making each of them a member of an infinite family of series. We also find the corresponding families involving harmonic numbers and odd harmonic numbers. To further illustrate our method we derive families of \mbox{Ramanujan-like} series for $1/\pi$ associated with a hypergeometric series derived by Lavoie, two hypergeometric series from Bailey's book and a recent hypergeometric series found by Campbell. Thus, by leveraging known hypergeometric summation identities, our approach bypasses the more tedious, traditional limits or complex Fourier-Legendre expansions; offering a cleaner, unified template for generating Ramanujan-like series for $1/\pi$.

Borisov and Yeung showed that the Cartwright-Steger surface $X$ is defined over ${\mathbb {Q}}$, hence complex conjugation gives an antiholomorphic involution on $X$. We show that the orbit space $Y$ is a closed oriented smooth four-manifold homeomorphic to ${\mathbb {CP}}^2\# (S^1 \times S^3)$, presenting $X$ as a double branched cover $X\to Y$, with branch locus diffeomorphic to the connected sum of three copies of the real projective plane.

Can a LEGO model built from many kinds of rectangular bricks be rebuilt using only the familiar $2\times4$ brick? Enlarging every dimension by four makes the pieces fit, but they must also connect. We give a four-layer arrangement that joins every finite face-connected union of lattice cubes, using eight upright bricks per source cube, and show that no smaller universal enlargement works. A second question has an unexpected answer: if connectivity is set aside, the minimum integer scale is always $1$, $2$, or $4$, never $3$. The missing value follows from a checkerboard argument applied separately inside each enlarged cell.

For a prime p >= 5 let f_I(p) and f_II(p) count the Type I and Type II solutions of the Erdos-Straus equation 4/p = 1/x + 1/y + 1/z, in the sense of Elsholtz and Tao. We do not address the Erdos-Straus conjecture itself; our subject is the number of solutions. We turn a bijection of Bradford into an exact counting identity: f_I(p) and f_II(p) are sums, over the denominators n coprime to p, of the number of divisors of n^2 in a single residue class modulo e_n = 4n - p. A dual identity puts both counts on the same integer and the same modulus, distinguished only by the classes -p and -1. Our main unconditional result holds for every numerator m: the number of Type II solutions of m/p = 1/x + 1/(pY) + 1/(pZ) with Y <= K is at most sum_{n <= K} tau(n) = K log K + O(K), uniformly in p and m; on average over p <= N it is only O_m((log K)^3 log log K). Hence Type II solutions concentrate at x/p -> 1/m, while Type I solutions cannot. The numerator carries arithmetic information in two independent ways. A real character chi modulo e_n that is trivial on the primes dividing n annihilates Type II when chi(-1) = -1, and Type I when chi(-1)chi(m_0) = -1, where m_0 is the squarefree kernel of m; these coincide exactly when m is a square, so for the numerator 4 no criterion modulo squares can separate the two types. Separately, f_I(n) = f_II(n) = 0 for every odd perfect square n whenever 4 divides m, generalising a theorem of Elsholtz and Tao. Finally we prove the sharp local inequality 2B(n) >= A(n) for e_n in {1,3,5}, with a construction showing it fails for every odd e with 7 <= e <= 2*10^4; we conjecture f_I(p) > f_II(p) for all p > 5 with p not congruent to 1 mod 8, verify this for p <= 10^6, and reduce it to a finite window. We also locate exactly the factor log log N in the Elsholtz-Tao upper bound for the sum of f_I(p).

A convex semilattices is a convex algebra endowed with an additional semilattice operation that satisfies a distributive law. We investigate free convex semilattices by studying the homomorphisms into products R{\Omega} and, more specifically, the homomorphisms between two free finitely generated convex semilattices. We characterise the finitely generated subalgebras of a free finitely generated convex semilattice.

We consider a random walk on the integers that at each site is biased to step right until the first time it has stepped left from that site, thereafter stepping left or right from that site with equal chance, introduced by Pinsky. Pinsky calculated the speed of the walk and the probability that such a walk escapes to infinity for some ranges of \(p\) and conjectured these formulas hold over a wider range. To study this walk, we introduce a branching process featuring geometric emigration and answer some questions concerning its life-periods, transience and recurrence, and the mean of its stationary distribution when positive recurrent. We apply these results to confirm Pinsky's predictions. For another application, we calculate the speed and escape probability of a related self-interacting random walk.

The compactness of the tiling space and the continuity of the action of the Euclidean space by translation, as well as the minimality of this action for repetitive tilings, are among the most frequently mentioned foundational facts in the dynamical theory of aperiodic order. Nonetheless, despite their classical status, a comprehensive treatise with complete and detailed proofs, to the best of our knowledge, is not available in the literature: existing arguments are typically sketched, deferred to other sources, or rely on unstated results or hypotheses. The purpose of this manuscript is to give a self-contained and pedagogical treatment of these foundations, providing complete and carefully explained proofs. We hope that this exposition will provide a useful reference and contribute to a clearer and more accessible understanding of the fundamentals of the theory of tiling dynamical systems.

We prove that every nonzero integer polynomial with at least two distinct complex roots has lattice point visibility density one, resolving the generalized form of the Visibility Density Conjecture for nonzero polynomials. This extends the origin-passing case established by Chaubey, Pandey, and Regavim. Visibility is taken along the curves $y=tF(x)$ with rational $t$, with a point visible if no positive lattice point on the same curve has a smaller horizontal coordinate. The proof uses a greatest common divisor cutoff to reduce the problem to finitely many equations of the form $F(b)=qF(a)$, with $0<q<1$; for each equation, the positive integers $a$ admitting a positive integer solution $b<a$ form a set of density zero. This elementary argument removes the proper-power hypothesis of earlier work without requiring estimates uniform in the ratio.