数学前沿

We obtain bounds on the Kullback--Leibler divergence to equilibrium for mass-action chemical reaction networks (CRNs) with equilibrium. The associated trajectory-dependent bounds are characterized in terms of the singular values of the stoichiometric matrix, convexity parameters, and time-integrated activities via deformed-exponential-type functions. We further extend these bounds within a generalized gradient flow framework. We highlight the biological relevance of this framework: the resulting bounds apply to quasi-steady-state regimes, where long transients and plateau-like behavior are common and functionally important. We illustrate the framework using a catalytic CRN exhibiting plateaus, where the bounds provide an a posteriori characterization of slow relaxation induced by local convexity and provide a bound-based approach to quantifying relaxation in CRNs.

We investigate the semilinear wave equation with potential on weighted graphs. We establish sufficient conditions for the nonexistence of global-in-time solutions. Both nonnegative and sign-changing solutions are considered. In particular, the proof for sign-changing solutions relies on a novel technique for this type of result.

Erd\H{o}s asked about the least possible diameter of an $n$-point set in Euclidean space when any two pairwise distances differ by at least $1$. We consider the uniform-in-dimension variant of his question, in which the dimension $d$ may grow with $n$ and the error term is independent of $d$, and answer it in the negative by an explicit construction. For every prime power $q$, we construct a set $\mathcal X_n$ of $n=q+1$ points in $\mathbb R^{q^2+q}=\mathbb R^{n^2-n}$ such that the $\binom{n}{2}$ pairwise distances are mutually at least $1$ apart, yet $$\operatorname{diam}(\mathcal X_n)\le \Bigl(1-\frac{1}{\pi^2}+o(1)\Bigr)n^2.$$ The construction combines a Singer cyclic difference set with a weighted product of $N=\binom n2$ regular $m$-gons, where $m=q^2+q+1$. The weights are chosen so that the resulting distances increase while the gaps between consecutive distances decrease. After normalizing the smallest gap to $1$, all distinct pairwise distances are separated by at least $1$, while the diameter has asymptotic constant $1-1/\pi^2<1$.

We derive basic differential geometric formulae for surfaces in hyperbolic space represented as envelopes of horospheres. The dual notion of parallel hypersurfaces is also studied. The representation is applied to prove existence and regularity theorems for Weingarten surfaces in H^3, which satisfy (1-a)K = a(2-H), for an a < 0, and have a specified boundary curve at infinity. These surfaces are shown to be closely connected to conformal mappings of domains in S^2 into the unit disk and provide Riemannian interpretations for some conformal invariants associated to such mappings. This paper was originally written in 1984, before I learned to use TeX, and was typed by one of the secretaries in the Princeton Math Department. It was more or less, my first original work after my dissertation. For some reason, I was not able to get this paper published in a timely manner. The results and perspective in this paper have proved to be useful to a variety of people, some of whom asked me to render the article into TeX and post it to the arXiv. I had been seriously thinking about doing this, when Martin Bridgeman sent me a transcription of my original article into TeX. I am extremely grateful to him for the effort he has put into this project. The paper is now formatted in a more or less modern AMS-article style, but for lots of additional punctuation, a few corrections and some minor stylistic changes, the content has been largely reproduced as it originally was. Remarks about the 'state-of-the-art' in hyperbolic geometry are obviously way out of date, as there has been enormous progress in many aspects of this still rich subject.

We construct a comparison morphism $\operatorname{HH}^{\mathrm{Rog}}\to\operatorname{HH}^{\mathrm{Ols}}$ between Rognes' logarithmic Hochschild homology and Olsson's logarithmic Hochschild homology which generalizes the comparison between Gabber's log cotangent complex $\mathbb{L}^{\mathrm{Gab}}$ and Olsson's log cotangent complex $\mathbb{L}^{\mathrm{Ols}}.$ In particular, Olsson's comparison is recovered as the first associated graded piece of our comparison morphism $\operatorname{HH}^{\mathrm{Rog}}\to\operatorname{HH}^{\mathrm{Ols}}.$ In the development of this comparison, we also prove a descent result for the cotangent complex generalizing a result of Bhatt, and construct a notion of HKR completed Hochschild homology for a general morphism of derived stacks which is functorial in the morphism and generalizes existing constructions of Hochschild homology when the base is affine.

We characterize metric measure spaces that satisfy intrinsic parabolic Harnack inequalities for doubly nonlinear equations of slow diffusion type. The considered parabolic Harnack inequalities are intrinsic in the sense that the waiting time depends on the solution. Our main results shows that in a metric measure space with a doubling measure, such an intrinsic parabolic Harnack inequality holds if and only if a Poincar\'e inequality holds. We thus show that estimates in scale and function dependent intrinsic geometries can characterize scale and location invariant functional inequalities.

We investigate a predator-prey model with constant prey harvesting and a generalized Holling type III functional response, focusing on the biological and dynamical consequences of allowing the parameter $b$ to be negative. In this regime, predation efficiency declines at high prey densities, potentially reflecting group defense. This change permits an additional coexistence equilibrium and introduces new bifurcation mechanisms affecting population persistence. We establish the existence of a codimension-three Bogdanov--Takens bifurcation, derive a codimension-three nilpotent-saddle unfolding, and investigate a codimension-two generalized Hopf bifurcation and the emergence of multiple limit cycles. Our analysis show that moderate harvesting weakens the protection conferred by high prey abundance; excessive harvesting leaves too few prey to sustain predator growth; and low harvesting preserves strong group defense, requiring high predator density for population persistence.

In this paper, we introduce the notion of stable imaginary elements in quantum groups and use crystal basis theory to construct explicit one-parameter families of such elements for all finite types. We show that all of Leclerc's examples of imaginary elements are stable imaginary. Using the generalized quantum Schur--Weyl duality, we also construct explicit examples of corresponding imaginary modules over quantum affine algebras of untwisted affine types.

We study martingality and strict local martingality in a broad class of stochastic and path-dependent Volterra volatility models and explain the underlying mechanisms. Our martingality criteria allow rough kernels and rely on non-accessibility of an explosive direction or one-sided growth bounds on the Girsanov drift. For smooth kernels, we derive sufficient conditions for strict local martingality from an explosion criterion allowing multiplicative noise, coupled factors and general initial input curves. Our results recover classical criteria and yield new results for Volterra and path-dependent models.

For a graph $G$, a hypergraph is Berge-$G$-saturated if it contains no Berge copy of $G$, but adding any missing hyperedge creates one. We determine the complete saturation spectrum of three-uniform Berge five-stars for every order. For $n\ge7$, every integer from $\lceil4n/3\rceil-3$ to $2n-5$ occurs; the additional values depend only on $n$ modulo five. The structural ingredient is a sharp bound on the remainder after deleting all complete five-vertex components: for an $n$-vertex, $m$-edge Berge-five-star-saturated hypergraph, this remainder has at most $15(2n-m)/7$ vertices. We characterize all equality cases. We also prove that every three-uniform Berge-$K_4$-saturated hypergraph on $n\ge7$ vertices has at least $n$ edges. Together with known constructions, this determines its saturation number for every order and settles the conjecture of Chen, He, and Xie; the new lower bound covers the previously unresolved orders $9\le n\le95$. Both arguments use the distinct representatives forced by saturation to obtain local incidence constraints. For five-stars these constraints bound the size of the remainder and characterize equality. For $K_4$, they reduce a putative counterexample to a configuration on at most eight vertices; incidence estimates and a sharp bound on the vertices covered by short cycles in a bipartite graph exclude all remaining cases.

Very recently, the papers "Point Convergence of Nesterov's Accelerated Gradient Method: An AI-Assisted Proof" by Jang and Ryu, and "The Iterates of Nesterov's Accelerated Algorithm Converge in the Critical Regimes" by Bot, Fadili, and Nguyen have simultaneously resolved a long-standing open problem concerning Nesterov's accelerated gradient method. These works show that the iterates of the algorithm (known in its composite form as FISTA) indeed converge to an optimal solution. In this work, we extend these results and prove that, in infinite dimensional Hilbert spaces, the iterates of such an algorithm still converge (in the weak sense) even when the proximity operator and the gradient are computed inexactly, with the latter possibly stochastic. Our framework covers both absolute and relative errors in proximal computations. In the stochastic setting, it allows for uniformly bounded variance as well as proximal counterparts of strong growth conditions, under which iterate convergence can be established while retaining accelerated objective-value rates.

For a Haar-distributed matrix $U \in U(N)$, Rains's power-map theorem represents the spectrum of $U^m$, $m \in \mathbb{N}$, as a superposition of independent lower-dimensional CUE spectra. We determine how this exact decomposition appears in trace moments and in sharp Gaussian and Poisson limit laws. For averages of mixed trace products, a Charlier--Ewens convolution separates the stable Wick term from its full finite-$N$ correction. A Fredholm factorisation gives sharp total-variation asymptotics for the normalised trace in four regimes, from fixed powers to $m\geq N$. For a fixed trace power $p$, coherent summation of the leading Laguerre correction yields the exact factor $e^{2/p}$. At the mean-spacing scale, we obtain a sharp Poisson limit for the unfolded spectrum of $U^m$ on a fixed interval as $N, m \rightarrow \infty$; its first-order correction, obtained via the second Charlier polynomial, quantifies the residual under-dispersion of local eigenvalue counts.

The main purpose of this paper is to develop the duality theory of the homogeneous Besov spaces associated with the Euclidean flag, using the discrete Littlewood--Paley--Stein analysis. We first establish an exact discrete sampling formula together with bounded analysis and synthesis maps, by which the space is identified with a sequence space of coefficients. The continuous dual is then computed through a coefficient norm. In this computation the geometry of the sampling rectangles plays an essential role: when the spatial exponent is below one, the volume of a dyadic rectangle forces an explicit shift of both smoothness indices. We devote particular attention to the supremum sequence exponent. At that endpoint the closure of the flag test functions is characterized by vanishing scale tails, and the limiting tail is comparable to the distance to this closure; every continuous functional on the full space decomposes into a canonical regular part and a part that annihilates the closure, and the latter may be nonzero. For finite exponents the dual is again a flag Besov space. As applications, finite Calder\'on sums give a quantitative approximation theorem, and the common analysis and synthesis framework yields real and complex interpolation as well as the lifting property. The arguments are based on the exact sampling formula, scalar maximal estimates and the transpose of one bounded coefficient projection.

We associate a compact genus-two real spectral core with a transverse logarithmic-derivative discriminant of the Riemann $\Xi$-function. Directed interval arithmetic certifies its branch geometry, a real $P_{\mathrm{III}}(D_6)$ scalar-normal-form realization, and associated Kummer and Shioda--Inose models. Zero-action resonance reduces to rationality of two real period ratios. Their certified heat motion yields generic local nonresonance and Diophantine separation, while explicit Farey windows impose denominator barriers of $1{,}036{,}003$ and $140{,}769$ throughout $|\kappa|\le3\times10^{-10}$ in the heat time $\kappa$. Fixed-point irrationality and exact Picard rank remain open. The heat tangent differs from the matched fixed-exponent isomonodromic direction, but admits a local moving-exponent curve lift. Separately, a reciprocal-head canopy certificate combined with published finite-RH and rectangular time-barrier verifications bounds the de Bruijn--Newman constant by $\Lambda\le433/2500=0.1732$ without a QRH assumption. These local arithmetic and global analytic results share a phase--transverse bridge; no equivalence between core nonresonance and RH is asserted.

Let $V$ be a nonempty relatively compact open subset of a smooth Riemannian manifold $(M^n,g)$ without boundary. We prove that $C_{\mathrm N}(V,g)\ge C_{\mathrm N}(\mathbb R^n)$ and $\int_V \mathrm{Sc}_g d\mu_g\ge0$ imply $\mathrm{Rm}_g=0$ on $V$, where $C_{\mathrm N}$ denotes the optimal constant in the Nash inequality and $\mathrm{Sc}_g$ denotes the scalar curvature. This improves an earlier rigidity result of Druet, Hebey, and Vaugon [Int. Math. Res. Not. (1999), no. 14, 735--779].

We obtain explicit non-asymptotic bounds for the eigenvalue transition of multiband prolate matrices. Let $S\subset[-1/2,1/2]$ be a finite union of pairwise disjoint intervals, and let $\lambda_0\ge\cdots\ge\lambda_{N-1}$ be the eigenvalues of $B_{N,S}$. If $\nu$ is the number of boundary points of $S$ on the frequency torus, then, for every $0<\eps<1/2$, $$\#\{k:\eps<\lambda_k<1-\eps\} \le \nu\ceil{\frac{\log(4N)}{\pi^2} \log\!\left(\frac{4}{\eps(1-\eps)}\right)}.$$ Compared with the multiband estimate of Zhu and Wakin~\cite{wakin2017approximating}, this improves the dependence on $\eps$ from order $1/[\eps(1-\eps)]$ to logarithmic dependence. We also prove a second bound in which the main logarithmic term depends on the time--bandwidth product $N|S|$ rather than only on $N$. We also compare $S$ with frequency sets having fewer boundary points. Using a classical large-sieve estimate, we take into account not only the total size of the change in the frequency set, but also how that change is distributed. For sets with many small gaps, this can give a bound growing like $\log N$ for fixed $\eps$, even when the bounds based only on the total size of the gaps grow at least linearly in $N$. We also obtain exponential bounds for individual eigenvalues and use them to estimate the error in approximating sampled multiband signals by leading eigenvectors. A discretization argument gives corresponding results for the continuous problem. Numerical examples compare the bounds and examine how the locations of the frequency intervals affect the eigenvalues.

This paper studies the nonlinear matrix-valued stochastic Riccati equation with jumps (SRE-J) arising from indefinite stochastic linear-quadratic optimal control problems with random coefficients and Poisson jumps. We demonstrate that the presence of jumps may cause the matrix-valued optimal state process $\mathbf{X}$ to become singular, even under the standard condition. Consequently, the classical representation $P=\mathbf{Y}\mathbf{X}^{-1}$ used in the diffusion case is generally unavailable. To overcome this difficulty, we construct the Riccati process directly from the stochastic value flow under the uniform convexity condition on the cost functional, an approach that requires no invertibility condition on the jump coefficient. We then prove that the SRE-J admits a maximal strongly regular solution, which yields a closed-loop representation of the unique open-loop optimal control. Furthermore, we establish verifiable sufficient conditions for uniform convexity and provide examples of indefinite stochastic linear-quadratic problems with jumps.

We present a logical system CFP (Concurrent Fixed Point Logic) supporting the extraction of nondeterministic and concurrent programs that are provably total and correct. CFP is an intuitionistic first-order logic with inductive and coinductive definitions extended by two propositional operators: Restriction (binary), a strengthening of implication, and a unary operator for total concurrency. The source of the extraction is formal CFP proofs, the target is a lambda calculus with constructors and recursion extended by a constructor Amb (for McCarthy's amb) which is interpreted operationally as globally angelic choice and is used to implement nondeterminism and concurrency. The correctness of extracted programs is proven via an intermediate domain-theoretic denotational semantics. We demonstrate the usefulness of our system by extracting a nondeterministic program that translates infinite Gray code into the signed digit representation. A noteworthy feature of CFP is the fact that the proof rules for restriction and concurrency involve variants of the classical law of excluded middle that would not be interpretable computationally without Amb. This is a revised and extended version of the conference paper presented at ESOP 2022 with the same title that contains full proofs of all major results.

We prove a topological generalization of the Figiel--Lindenstrauss--Milman inequality, answering a recent question of Frick, Hosseini, and Vasileuski: every nonempty strongly regular CW complex with $v$ vertices, $f$ maximal cells, and a free involution that permutes its cells has $\mathbb Z/2$-index at most $4\log v\log f$. The classical inequality for centrally symmetric polytopes is the special case of polytope boundaries. As an application, we show that every finite simplicial complex homotopy equivalent to $\mathbb RP^d$ has at least $\exp(\Omega(\sqrt d))$ vertices. The previous best lower bound was $\Omega(d^2)$. We also construct triangulations with $\exp(O(\sqrt d))$ vertices. Using the same method, we prove a low-dimensional Postnikov factorization theorem, which gives similar vertex lower bounds for several classical families, such as complex and quaternionic projective spaces, the $d$-torus, closed symplectic and aspherical manifolds, Grassmannians, classical Lie groups, and spherical space forms. Further applications include results in learning theory, convex geometry, computational topology, and topological bounds for chromatic numbers and Vietoris--Rips complexes.

We show that each of the infinitely many extensions of the basic inquisitive logic has the Craig interpolation property, and deduce from this that each of these extensions also has the Beth definability property. These results stand in contrast to existing characterizations of interpolation within lattices of extensions, in which the distribution of logics with interpolation is sparse.