数学前沿

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.

We establish precise upper and lower estimates for harmonic functions vanishing on the slit of a $C^1$ slit domain, with no assumption that the slit lies in a hyperplane. The classical $\sqrt{d}$ growth near the edge, $d$ being the distance to the slit, persists in this generality, up to an explicit factor $$\exp\Big( \pm C \int_\rho^r \omega(s)\, \frac{ds}{s} \Big)$$ determined by the $C^1$-modulus of continuity $\omega$ of the slit, where $\rho < r$ are the two scales being compared. The upper estimates allow a right-hand side and non-zero boundary data. The correction factors remain bounded above and below by positive constants as $\rho\to0$ precisely when $\omega$ satisfies the Dini condition. Moduli of continuity beyond the Dini regime, as is the case for the logarithmic moduli arising at singular sets in relevant free boundary problems, were not covered by the previous $C^{1,\alpha}$ theory. Previously, the $\sqrt{d}$ growth was known for slits contained in a hyperplane, which additionally have a $C^{1,\alpha}$ edge (De Silva, Savin). For Lipschitz slits, there are boundary Harnack principles, but no growth rate is identified precisely. The main technical ingredient is a change of coordinates flattening a Lipschitz slit domain onto the model half-hyperplane slit, with quantitative estimates up to second order. The construction is geometric and does not use the equation, so we expect it to be useful for other boundary regularity problems.

In this paper, we investigate the behavior of the family of graph-based splitting algorithms specialized to the problem of finding a point in the intersection of linear subspaces. The algorithms in this family, which encompasses several classical methods such as the Douglas-Rachford algorithm, are defined by a connected graph and a subgraph. Our main result establishes that when the graph and subgraph coincide, the optimal relaxation parameter is exactly $1$, thereby extending known results for the Douglas-Rachford algorithm to a much broader class of methods. Our analysis hinges on some properties of iso-averaged linear operators, which are defined as the average of an isometry and the identity, and are characterized by a specific symmetry of the norm of their relaxation.

We study the linear stability of spherically symmetric, asymptotically flat steady states of the Einstein-Euler system describing a liquid star surrounded by vacuum. The equation of state has positive surface density, positive causal sound speed, and an asymptotically linear high-density limit. We construct the equilibrium family and derive the linearized equation with a Robin boundary condition. The linearized operator has discrete real spectrum, and its negative Morse index equals the number of growing modes. We prove that the linearized operator around steady states with large central redshift admits a growing mode, while such growing modes do not exist for steady states with small central redshift. A parameter derivative of the equilibrium family provides a regular interior zero solution of the linearized equation, whose boundary defect is a nonzero multiple of the parameter derivative of the total mass. This gives a turning point principle: the kernel is one-dimensional precisely at mass critical points, and the sign of the crossing eigenvalue is determined by the bending of the mass-radius curve.

We show that holomorphic maps from a neighborhood of a compact set in a Stein space to an algebraically elliptic smooth projective variety can be approximated by meromorphic maps, provided that the map extends continuously to the whole space and the compact set is convex with respect to a submonoid of the Picard group. The proof uses algebraic sprays and their polynomial lifts to affine cones, which we call conic lifts. Rational sprays, which generalize algebraic sprays, extend the result to maps avoiding the exceptional set of a rational spray. We also show that, for smooth projective varieties, the existence of an algebraic spray which is dominating at some point is a birational invariant.

We study the truncated Jacobi-symbol determinants $$\{c,d\}_n=\det\!\left[\left(\frac{j^2+cjk+dk^2}{n}\right)\right]_{2\le j,k\le n-2}$$ proposed by Zhi-Wei Sun in \cite{ZWSun} and prove Conjectures 5.1(i), 5.2, 5.3, 5.4, 5.5, 5.6(i), 5.7, 5.8, a case of 5.6(ii) of \cite{ZWSun}, Conjecture 4.8(i) of \cite{Sun2019} and some strengthened forms. All results follow from a single unified approach: for a prime $p$, we diagonalize the nonzero-residue matrix indexed by $\mathbb{F}_p^\times$ and reduce the vanishing of determinants to the supersingular reduction of certain CM elliptic curves. The same framework extends naturally to further families of parameters, suggesting a general mechanism behind identities of this type.

We introduce the $t$-chromatic symmetric function $Z_G(\mathbf{x};t)$, a variant of Stanley's chromatic symmetric function in which the variable $t$ records the number of distinct colors used in a proper coloring. Although $Z_G(\mathbf{x};t)$ is equivalent to the chromatic symmetric function as a graph invariant, its principal specializations retain substantially more information than those of the chromatic symmetric function. This construction is motivated by the work of Bajo--Beck--Vindas-Mel\'endez on the $q$-chromatic polynomial and the conjecture of Loehr--Warrington that the $n$th principal specialization of the chromatic symmetric function distinguishes trees on at most $n$ vertices, both of which stem from Stanley's longstanding question of whether the chromatic symmetric function distinguishes trees. We determine the classes of caterpillars resulting from the second principal specialization of $Z_G(\mathbf{x};t)$, showing that the second specialization already captures some of the structure of trees. We then prove that the third principal specialization distinguishes and explicitly reconstructs caterpillars with three spine vertices as well as elongated bistars. Finally, we conjecture, based on computational data, that the fourth principal specialization distinguishes all caterpillars.

Sports analytics has expanded rapidly in professional sports and mainstream culture, yet defining its academic core remains a challenge. In this paper, we provide an empirical map of the academic sports analytics landscape based on a systematic content analysis of approximately 700 papers published in two leading dedicated journals, the Journal of Quantitative Analysis in Sports (JQAS) and the Journal of Sports Analytics (JSA), between the years 2005 and 2024. We classify each paper by sport discipline, mathematical method, and core research area, identify recurring themes, and examine macro trends. We find game tactics, player evaluation, team ranking, result forecasting, and tournament design to be the most studied research areas in sports analytics. We highlight selected papers in each area, illustrating the wide range of mathematical methods used and sports disciplines analyzed. Our work offers a comprehensive overview of the questions, methods, and disciplines that characterize the field, serving as a foundational reference for practitioners and researchers in sports analytics.

Liquid crystal elastomers (LCEs) are active materials capable of large, reversible thermally induced actuation, offering significant potential for applications such as artificial muscles and soft robotics. However, their broader deployment is hindered by the lack of a unified design-to-fabrication framework, the strong coupling among geometry, director fields, and manufacturing constraints, and the difficulty of reliably translating complex designs into physically realizable architectures. Here, we present an autonomous and monolithic integration of co-optimization and 4D printing that enables the end-to-end design-to-realization of versatile programmable LCE architectures with superior precision, complexity, and versatility. The proposed framework establishes a unified design strategy that co-optimizes the freeform multilayer geometry and fully continuous layer-specific director distributions, and subsequently translates the optimized fields into fabrication-ready DIW toolpaths through an automated manufacturing-aware pipeline. To bridge computation and realization, we develop an optimization-centered 4D printing approach with automated, fabrication-aware toolpath generation tailored for direct ink writing of LCEs. With the proposed design-to-fabrication paradigm, we experimentally realize architectures that exhibit complex, non-intuitive thermo-mechanical behaviors. Experiments achieve highly challenging prescribed functionalities beyond the reach of intuition-based approaches and show excellent agreement with numerical simulations across all samples. This work establishes a generalizable inverse design and physical realization of freeform programmable LCE systems, enabling function-oriented and optimization-driven intelligent LCE applications for next-generation soft robotic and responsive material engineering.

We establish two general quantitative lower bounds on the gap in the classical BK inequality for increasing events. The first is an analogue of Talagrand's quantitative FKG inequality (1996) and depends on the influences. The second is specific to the BK inequality and involves both the influences and the probabilities of symmetric disjoint pivots.