Papers

This article proves non-trivial estimates for a bilinear sum involving the Fourier coefficients of a Hecke-holomorphic or Hecke-Maass cusp form for $\mathrm{SL}(2,\mathbb{Z})$. As corollaries, we draw interesting results related to non-trivial bounds of different shifted convolution sums and summatory functions.

In this paper, we study symbolic defect functions of edge ideals through finite antichains of exponent vectors. Let $G$ be a finite simple graph and let $I(G)$ be its edge ideal. For each symbolic degree $s$, we define the symbolic exponent region $\mathcal{P}_s(G)$, the ordinary exponent region $\mathcal{O}_s(G)$, and the symbolic defect antichain $\mathcal{D}_s(G)=\min\big(\mathcal{P}_s(G)\setminus \mathcal{O}_s(G)\big)$, where the minimum is taken with respect to the componentwise partial order. We prove that $\mathcal{D}_s(G)$ gives a finite obstruction set controlling the minimal monomial generators of the quotient $I(G)^{(s)}/I(G)^s$. Our main result is a blow-up transfer formula. If $G^{\mathbf n}$ is the graph obtained from $G$ by replacing each vertex $v_i$ by an independent set of size $n_i$, then for every $s\geq 1$, \[ \operatorname{sdefect}(I(G^{\mathbf n}),s) = \sum_{\mathbf a\in \mathcal D_s(G)} \prod_{i=1}^{r} \binom{a_i+n_i-1}{n_i-1}. \] We further refine this formula to a multigraded symbolic defect series, which records the full multidegree distribution of the minimal generators of $I(G^{\mathbf n})^{(s)}/I(G^{\mathbf n})^s$. As applications, we classify the defect antichains of complete graphs in terms of integer partitions and derive explicit symbolic defect formulas for complete multipartite graphs, complete split graphs, and blow-ups of odd cycles. We also study symbolic defect antichains under graph joins and obtain polynomiality and rational generating-function consequences in the blow-up parameters. The results provide a unified antichain-based framework for symbolic defects of edge ideals and convert several previously case-by-case computations into consequences of a single transfer principle.

The classical Bohr inequality states that if $f(z)=\sum_{n=0}^{\infty} a_n z^n$ is analytic and $|f(z)|<1$ in the unit disk $\mathbb{D}$, then $\sum_{n=0}^{\infty} |a_n| r^n \le 1$ for $|z|=r \le 1/3$, where $1/3$ is sharp. Extending this to harmonic mappings $f=h+\overline{g}$ is central in geometric function theory due to the co-analytic part $g$. This paper establishes sharp Bohr-type inequalities for two classes of sense-preserving close-to-convex harmonic mappings. Let $\mathcal{H}_0$ be the class of harmonic mappings $f=h+\overline{g}$ in $\mathbb{D}$ normalized by $h(0)=g(0)=h'(0)-1=g'(0)=0$. We introduce: \[ \mathcal{P}_{\mathcal{H}_0}(M) := \{ f \in \mathcal{H}_0 : \text{Re}(zh''(z)) > -M + |zg''(z)|, \; z \in \mathbb{D}, \; M > 0 \} \] \[ \mathcal{W}_{\mathcal{H}_0}(\alpha,\beta) := \{ f \in \mathcal{H}_0 : \text{Re}(h'(z) + \alpha zh''(z) - \beta) > |g'(z) + \alpha zg''(z)|, \; z \in \mathbb{D} \} \] where $\alpha \ge 0$, $\beta < 1$. We prove generalized Bohr inequalities by replacing the basis $\{r^n\}$ with non-negative continuous functions $\{\varphi_n(r)\}$. The results are proved using sharp coefficient bounds and growth theorems, providing new insights into the Bohr phenomenon for harmonic mappings and subclasses defined by differential inequalities.

The discard ZX-calculus, a diagrammatic language for mixed-state quantum mechanics, is used to give a nonperturbative, categorical proof of the Bloch-Nordsieck cancellation of infrared divergences in QED. Soft photons are treated as an open quantum system: the resolved charged particles and hard photons form the system, while photons below a detector resolution form the environment. The reduced hard channel is a completely positive trace-preserving (CPTP) map, and the soft-photon theorem replaces the full S-matrix by a controlled displacement operator whose Feynman-Vernon influence functional satisfies the equal-history normalization ${\cal F}[J,J]=1 $. In the ZX-calculus, this normalization is a single diagrammatic identity: the doubled displacement diagram collapses to the bare wire under the unitarity, cyclicity, and discard rules. The proof therefore serves as a categorical consistency check on the open-system treatment of soft QED given in a companion paper; it confirms that the physical derivation is logically complete and free of hidden assumptions about the infrared limit. For off-diagonal hard-state elements, the same diagram yields the coherent-state overlap, giving a first-principles account of soft-cloud decoherence. The soft-shell coarse graining is then constructed as a CPTP Schur channel whose infinitesimal limit produces the exact Lindblad generator with jump operators determined by the eikonal emission amplitudes. Finally, a local CPTP-certification pipeline is developed for non-Markovian process tensors, enabling constant-time verification of trace preservation in open quantum simulations. The framework bridges categorical quantum semantics, non-equilibrium field theory, and practical open-system compilation.

We study positivity properties of finite flat quotients of normal projective varieties. We show that the numerical groups and positive cones of these quotient varieties are related to those of the original varieties.

In this note, we give some generalizations of Gödel's second incompleteness theorem and study their surroundings. We revisit it from two perspectives. One perspective is the relationship between the definable complexity of a theory and unprovability of its soundness. We clarify the relationship between this perspective and induction axioms. We also determine the logical strength of Craig's trick, which is important for studying the definability of a theory, from the point of view of reverse mathematics. The other perspective is semantic incompleteness. The second incompleteness theorem may be seen as the unprovability of the existence of a model. It is known that `model' is replaced with `$\omega$-model' or `$\beta_n$-model'. We give a new and unified proof of the $\omega$-model and $\beta_n$-model versions of the incompleteness theorem.

A classical theorem of Frucht states that every finite group occurs as the automorphism group of a finite graph. We prove an embedded analogue for regular graphs of arbitrary degree. In particular, we show that for every $d\geq 3$ and every finite group $G$, there exists a $d$-regular graph $\Gamma$ with a strong embedding $\beta$ such that $\mathrm{Aut}(\Gamma) \cong \mathrm{Aut}(\beta(\Gamma)) \cong G.$ Further, we prove that for every such $d$ and $G$ there exists a sequence of $d$-regular graphs with corresponding strong embeddings whose genera form an unbounded sequence and whose automorphism groups are isomorphic to $G$. Along the way, we identify an oversight in Sabidussi's classical construction of regular graphs with prescribed automorphism group. We give an alternative construction that corrects this issue and strengthens Sabidussi's result by producing an automorphism group-invariant proper $d$-edge-colouring.

A simple undirected graph $M$ is called a discrete $d$-pseudomanifold if, for every vertex $v$, the induced subgraph $N_M(v)$ on the neighbors of $v$ is a discrete $(d-1)$-pseudomanifold, where a discrete $1$-pseudomanifold is defined to be an $n$-cycle with $n\geq 4$. These objects arise naturally as graph-theoretic analogues of simplicial pseudomanifolds and provide a purely combinatorial framework for studying manifold-like structures through local neighborhood conditions. Understanding discrete pseudomanifolds with a small number of vertices is therefore a fundamental problem in combinatorial topology and extremal graph theory. In this article, we first prove that every discrete $d$-pseudomanifold has at least $2(d+1)$ vertices. We then provide a complete classification of discrete $d$-pseudomanifolds with at most $2d+6$ vertices by determining all possible combinatorial types of such pseudomanifolds. Further, we establish an equivalence between discrete $d$-pseudomanifolds and edge graphs of flag normal $d$-pseudomanifolds. As a consequence, we derive a purely combinatorial characterization of flag normal $d$-pseudomanifolds with at most $2d+6$ vertices and prove that each such complex is a simplicial $d$-sphere. Finally, we show that this sphere characterization is optimal within the class of flag normal $d$-pseudomanifolds by constructing examples on $2d+7$ vertices that are not spheres. Specifically, we prove that, for $d\geq 3$, every flag normal $d$-pseudomanifold with at most $2d+7$ vertices is either a simplicial $d$-sphere or a flag triangulation of the $(d-2)$-fold suspension of $\mathbb{RP}^{2}$.

Resolving a prescribed multidimensional control window by a single continuous, serial scan is a task distinct from driving between fixed endpoints. We formulate this task as metric coverage at resolution $\eps$. Classical covering estimates, applied to a compact regular $d$-dimensional Riemannian window, give a required length of order at least $\eps^{1-d}$; local grid traversals attain the same exponent. When a physical friction tensor uniformly dominates the coverage metric, this geometric cost implies a necessary duration of order at least $\eps^{-2(d-1)}$ at fixed quadratic excess-work budget. Fixed-duration divergence refers to the slow-driving quadratic action and is not a claim about exact work outside that regime. A detailed-balance three-state process and an overdamped harmonic trap provide microscopic friction examples. For the trap, independently integrated Gaussian moment dynamics approach the quadratic prediction as the prescribed raster is slowed, while refinement at a controlled work budget increases scan duration. A quasi-uniform operational resolution floor cuts off the geometric divergence. Landau-response and van der Waals proxy metrics serve separately as prescribed diagnostic geometries: independent quadrature tests their coordinate- and protocol-dependent critical prefactors without asserting universal critical dissipation. Morton ordering supplies an ordering/connector benchmark with larger finite-resolution amplitudes. The resulting resource statements apply to exhaustive serial coverage, with metric, interpolation and driving assumptions specified explicitly.

We study $H$-packings in plane triangulations for the three-vertex graphs $H\in\{P_3,K_3,P_2\cup P_1\}$. For a graph $H$, let $\lambda_H(G)$ denote the maximum size of an $H$-packing in $G$, with the convention that for $H=P_2\cup P_1$ the copies are required to be induced. For $P_3$-packings, we prove that every triangulation $G$ on $n$ vertices satisfies $\lambda_{P_3}(G)\ge \left\lfloor \frac n5\right\rfloor$, and show that this lower bound is asymptotically tight. We also study triangle packings in triangulations and provide lower bounds for $\lambda_{K_3}(G)$ in terms of the maximum degree and the degree sequence. We give a face-path characterization of triangle factors in $4$-connected plane triangulations using a hamiltonian cycle and the weak duals of the two associated maximal outerplanar graphs. Finally, for induced packings by $P_2\cup P_1$, we prove that every plane triangulation $T$ on $n$ vertices satisfies $\lambda_{P_2\cup P_1}(T)\ge \left\lfloor \frac n3\right\rfloor-2$, and show that such a packing can be found in polynomial time.

We investigate a variant of Nim called Halve Nim, which in addition to the standard moves of Nim, we allow replacing each pile of coins with half its amount. We determine the P-positions of all two-pile games of Halve Nim. Also, we determine the P-positions of all three-pile games of Halve Nim in which one pile has at most ten coins.

Classical hypothesis testing frameworks break down in contemporary settings in which null hypotheses are increasingly abstract, the same data are used to both generate and test hypotheses, and minimal assumptions about the underlying data are made. In this work, we propose a new framework for conducting valid hypothesis tests in broad contexts. We propose to add and subtract external noise generated from a symmetric shift-family to our data, $X$, to partition it into two pieces, $X^{(1)}$ and $X^{(2)}$. We provide a generic strategy for orthogonalizing $X^{(2)}$ against $X^{(1)}$ under the null hypothesis $H_0$, then show that testing whether the orthogonalization was successful provides a valid test of $H_0$ under mild assumptions. Remarkably, this framework extends naturally to the post-selection inference setting: we simply select a hypothesis on $X^{(1)}$, then perform orthogonalization under the selected null. As our approach neither requires pre-specification of the selection mechanism, nor is restricted to a small class of data-generating distributions, it dramatically expands the settings for which valid post-selection inference can be conducted. We showcase the flexibility of our proposal in several case studies involving challenging pre-specified null hypotheses and post-selection inference scenarios.

Let $M$ be a closed, connected, smooth $n$-dimensional manifold. We prove that $M$ is dominated by the underlying space of the $n$-skeleton of a finite simplicial complex. Furthermore, the total number of simplices in the $n$-skeleton is bounded above by a constant depending only on $n$ and the embolic volume of $M$.

Motivated by the correspondence between ideal cotorsion pairs in Frobenius exact categories and those in their stable categories, we introduce the notion of an ideal $n$-cotorsion pair in an extriangulated category. We study the relationship between ideal $n$-cotorsion pairs in a Frobenius extriangulated category $\mathcal C$ and those in its stable category $\underline{\mathcal C}=\mathcal C/\omega$. Our main result shows that $(\mathcal I,\mathcal J)$ is an ideal $n$-cotorsion pair in $\mathcal C$ if and only if $(\mathcal I/\omega,\mathcal J/\omega)$ is an ideal $n$-cotorsion pair in $\underline{\mathcal C}$. This provides a bridge between higher ideal approximation theory in Frobenius extriangulated categories and its counterpart in their stable categories. Additionally, in Krull–Schmidt exact categories, we establish a bijective correspondence between complete cotorsion pairs and complete ideal cotorsion pairs, answering a question of Fu, Guil Asensio, Herzog and Torrecillas.

Let $(\Omega,\mu)$ be a finite measure space with $M=\mu(\Omega)>0$. We investigate the integral form, stability, and metric geometry associated with a square-root complex. After proving the inequality and determining all equality cases, we analyze its phase stability near the intersection of the two branches of the equality set. In general phase directions, the quadratic term is precisely a Cauchy–Schwarz deficit; along the corresponding degenerate cone, the leading term is of fourth order and is strictly positive. A symmetric two-point example shows that the exponent four is unavoidable in any uniform distance-stability estimate. Finally, on the group of measurable circle-valued functions, we introduce the LY-metric \[ d_\mu(f,g)=\left|M-\int_\Omega f\overline g\,d\mu\right|^{1/2}. \] We prove that this metric is bi-invariant and complete, and that it induces the same topology as the $L^2$ metric. On finite-dimensional tori, we establish the optimality of the exponent $1/2$, derive explicit formulas for the intrinsic distance and geodesics, describe the anisotropic geometry and volume growth of small metric balls, and show that the Hausdorff dimension is $n+1$.

Of concern is a class of non-autonomous evolution equations of second order in Hilbert spaces, with a nonnegative self-adjoint operator $A$, time-varying damping and nonlinear source term. We give an upper decay rate of the energy, valid for all solutions and solely based on the damping coefficient and the geometrical index of the source term. Furthermore, we prove under suitable conditions that for all initial data, except for those in the kernel of $A$, the solutions decay (in the energy norm) at most as fast as this decay rate. The result not only shows the optimality of the decay rate, but also reveals an unusual phenomenon: “slow solutions", i.e. those that decay at exactly this rate, are dominant in amount. Moreover, specialized to the case when the nonlinear source is absent, our result improves relevant existing ones to a large extent.

Let $N\geq2$, let $F\in GL_N(\mathbb C)$, and let $O_F^+$ be the associated free orthogonal quantum group of Kac type, with tracial Haar state $h$. We prove that, for $1<p\leq r<\infty$, the normalized heat semigroup $P_t=e^{-tL}$ on $O_F^+$ \cite{CFK14,BVY21} is hypercontractive with the optimal time \[ \|P_t:L_p(O_F^+)\to L_r(O_F^+)\|\leq1 \quad\Longleftrightarrow\quad t\geq\frac12\log\frac{r-1}{p-1}. \] In particular, this solves a conjecture of Brannan, Vergnioux and Youn \cite{BVY21}. To prove this hypercontractivity result, we establish the equivalent logarithmic Sobolev inequality \[ \operatorname{Ent}(|x|^2)\leq2\mathsf{E}(x,x), \qquad x\in D(\mathsf{E}), \] with sharp constant $2$, where $\operatorname{Ent}(y)=h(y\log y)-h(y)\log h(y)$ is the relative entropy and $\mathsf E(x,x)=\|L^{1/2}x\|_2^2$ is the Dirichlet form associated with $P_t$. The main ingredients are the cubic-majorant idea from the recent work of Frank–Ivanisvili \cite{FI26} and Xie–Zhang \cite{XZC26,XZ26}, and a centered third-moment estimate coming from the representation theory. We also prove the sharp Beckner inequalities \[ \frac{\|x\|_p^2-\|x\|_2^2}{p-2}\leq\mathsf E(x,x), \qquad 1\leq p\leq3,\quad p\ne2, \] whose limit as $p\to2$ recovers the logarithmic Sobolev inequality above. Moreover, when $N=2$, the results can be sharpened by replacing the heat semigroup with $\mathsf P_t=e^{-t\Lambda}$, where $\Lambda=\sqrt{I+3L}-I$. The key step is to use the graded-twist realization of $SU_{-1}(2)$ to transfer Beckner's sharp $p=3$ inequality on $SU(2)$ \cite{Beckner93}.

Let $F$ be a number field. Given finitely many $F$-valued points on a commutative algebraic group defined over $F$, a question of interest to number theorists is the determination of the group of their linear relations. In this article, we investigate an analogous problem in the $t$-module setting. Let $L$ be a global function field, and $E$ be a $d$-dimensional $t$-module defined over $L$. Given finitely many points on $E$ with entries in $L$, we establish the connection between their $\mathbb{F}_q[t]$-linear relations and polynomial solutions of Frobenius difference equations. Consequently, we deduce an algorithm to compute the module of their $\mathbb{F}_q[t]$-linear relations.

We report a machine-verified resolution of a problem open for over a decade in quantum optimization: the Farhi, Goldstone and Gutmann (FGG) conjecture that depth-$p$ Quantum Approximate Optimization Algorithm (QAOA) on the ring of disagrees attains approximation ratio $(2p+1)/(2p+2)$ exactly. We found the proof using a large language model, Claude Fable 5, and verified its correctness end-to-end by the Lean 4 proof assistant. Our methodology includes several ingredients: building on a substantial Lean library of quantum information, we formalized the QAOA components and the known parts of the problem, and reduced the conjecture to a single open mathematical statement. The model was then handed the library and our agentic toolkit, and tasked with closing that gap by constructing a proof in Lean. The resulting process is a feedback loop between the model's natural-language reasoning and Lean's mechanical verification, which converged to a machine-verified proof. Human verification is required only for the structural scaffolding - that the formal statement faithfully encodes the intended claim - while the proof itself is supplied by the model and certified mechanically by Lean. The proof is nevertheless striking - the model uncovered a hidden dynamical symmetry of the problem and exploited it, borrowing tools and machinery from an adjacent field to turn a hard existence problem into an explicit construction. This work paves the way for resolving open conjectures in quantum information science and beyond.

We prove a scaling limit theorem for a double sequence of probability measures involving additive free convolution $\boxplus$ and additive Boolean convolution $\uplus$. Let $\mu$ be a probability measure on $\mathbb{R}$ with mean zero and variance one, and let $M=M(N)>0$ satisfy $MN^{\alpha+1/2}\to t>0$. We study the weak limits, as $N\to \infty$, of the double arrays $D_{N^\alpha}((\mu^{\boxplus N})^{\uplus M})$. We show that the limit distribution is the Cauchy distribution with scale parameter $t$ if $\alpha>-1/2$, the $t$-fold Boolean convolution power of the standard semicircle law if $\alpha=-1/2$, and the point mass at the origin if $\alpha<-1/2$.

Let $T$ be a positive $dd^c$-closed current of bidimension $(q,q)$ on a compact Kähler manifold $X$. For every $c>0$, let $E_c(T)$ be the set of points of $X$ where the Lelong number of $T$ is larger or equal to $c$. We show that $E_c(T)$ is an analytic subset of dimension at most $q$ of $X$. Moreover, the following Siu decomposition holds $$T=\sum_{i\in I} \lambda_i[V_i] +T_0,$$ where $\{V_i\}_{i\in I}$ is a (possibly empty) finite or countable family of compact analytic subsets of dimension $q$ in $X$, $\lambda_i\in\mathbb{R}^+$, and $T_0$ is a positive $dd^c$-closed current such that $E_c(T_0)$ is an analytic subset of dimension at most $q-1$ of $X$ for every $c>0$. The proof relies on the duality between the pseudoeffective cone and the movable cone of a compact Kähler manifold, recently obtained by Tosatti (2026), together with a theorem of Vigny (2009) and the theory of density currents for positive $dd^c$-closed currents developed by Sibony, the first and third authors.

We study direct recovery of visible phase centers and concentric radial interfaces from full-aperture acoustic far-field data under the Born approximation. Two complementary moment structures are extracted from the two-angle Fourier matrix. At fixed positive total Fourier order, the nonnegative-order channels share the same leading phase-center moment, and a generalized least-squares combination yields an exact variance gain over a single Fourier row. The corresponding Hankel rank and shifted pencil recover distinct phase centers. Signed moments reveal off-center cavities but become degenerate when material and cavity centers coincide. Zero-total-order coefficients retain complementary radial Bessel moments. For a piecewise-constant radial average, low-frequency extrapolation produces a second finite exponential sequence whose nodes are the squared interface radii. We establish rank and perturbation results for both reductions. Numerical experiments verify multichannel stabilization, concentric-cavity resolution, and recovery of multiple radial interfaces. A final full-wave Helmholtz experiment, generated without the Born substitution, assesses the Born-derived reconstruction under model mismatch and shows how nonlinear scattering eventually appears as an additional Hankel tail.

An integer palindrome is a self-reciprocal polynomial evaluated at its base, so its roots are symmetric about the unit circle – where the coordinate is angle, in turns of $\tau$. Read this way, the date $\texttt{6/28/26}\to 62826$ secretly contains the primitive cube roots of unity – at angle $\tau/3$ – along with one further pair of roots on the circle.

We give a thorough analysis of the time complexity of computing Reshetikhin–Turaev knot polynomials via tensor contractions on the associated tensor networks, showing that the computation is fixed-parameter tractable with respect to a parameter at most linear in the tree-width of the input knot diagram. When combined with existing approximation algorithms for tree decomposition, this recovers the sub-exponential bound $e^{O(\sqrt{n})}$ for the time complexity of computing any Reshetikhin–Turaev knot polynomial. We accompany this paper with an implementation of such an algorithm in SnapPy, which computes any Reshetikhin–Turaev knot polynomial given its $R$-matrix and ribbon element.

We study a singular mean-field control problem in which a central planner allocates losses generated by bank defaults to maximize terminal system health. Every generated loss must be borne by the surviving banks, without external capital. We prove that a cutoff, or "taxing-the-richest" rule is optimal for every non-decreasing concave terminal payoff, including the survival indicator. We construct the continuous-time optimizer by solving a time-discretized problem and passing to the limit, and establish its uniqueness in law for every non-constant payoff. The optimal flow is characterized by a novel reflected free-boundary problem. We establish well-posedness in a probabilistic solution class and prove positive-time smoothness of both the free boundary and the density of surviving banks. The main difficulty is the dependence of the limiting feedback on a moving support endpoint, which is unstable under Wasserstein convergence. We address this through stochastic-order comparisons, compactness estimates for the cutoff boundaries, and an augmented call-function formulation.

Inverses of acyclic or treediagonal matrices are nicely characterized in the landmarked paper by Klein \cite{KLe82}. But also in \cite{Nab01} a characterization of these inverses is given which is related to the structure of inverses of tridiagonal matrices. Here we use this characterization to determine and construct matrices whose inverses are acyclic $Z$-matrices. Moreover, we establish relations between several classes of matrices, namely the classes of $Z$-matrices, inverse $Z$-matrices, ultrametric matrices, generalized ultrametric matrices, shifted ultrametric matrices, acyclic matrices, inverse acyclic matrices, matrices of tree structure, cyclopes and generalized cyclopes.

It is well-known that the absolute values of the entries of the inverse of a diagonally dominant tridiagonal matrix decay away from the diagonal. Moreover, lower and upper bounds for these absolute values are known for a long time. The graph of a tridiagonal matrix can be viewed as a line. Thus tridiagonal matrices are special acyclic matrices, i.e. matrices whose graph is a tree. In this paper we explore the structure of inverses of acyclic matrices. However, there is no strict decay from the diagonal in the inverses of diagonally dominant acyclic matrices. The decay is somehow hidden. Here we identify this decay which can be described nicely. Moreover we give bounds for the absolute value of the entries of the inverses of acyclic matrices. All of our results generalize results for tridiagonal matrices.

We propose a generalized fine-tuning framework for diffusion models from the perspective of stochastic control. Beyond entropy-regularized formulations, we introduce a general running cost that induces a relative generalized path cost, encompassing both Kullback-Leibler divergence and optimal transport metrics as special cases. This leads to a fully nonlinear Hamilton-Jacobi-Bellman equation. We characterize the value function through a forward-backward stochastic differential equation system and establish existence and uniqueness under standard regularity conditions. The optimal control admits a nonlinear feedback form driven by the backward stochastic differential equation gradient component. Finally, we show that generalized fine-tuning naturally introduces an additional gradient-dependent penalty, providing a unified framework for diffusion fine-tuning under general distributional constraints.

Let $\mathcal{S}$ denote the class of functions $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ that are analytic and univalent in the open unit disk $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$. In this paper, we determine the sharp bounds of the Toeplitz determinants whose entries are the logarithmic coefficients of $f \in \mathcal{S}$. Furthermore, we investigate the corresponding Toeplitz determinants for the logarithmic coefficients of the associated inverse functions. These sharp bounds are established for functions belonging to several well-known subclasses of $\mathcal{S}$, namely, the classes $\mathcal{S}^*(\alpha)$ of starlike functions of order $\alpha$, $\mathcal{C}(\alpha)$ of convex functions of order $\alpha$, $\mathcal{S}^*_{\alpha}$ and $\mathcal{C}_{\alpha}$ of strongly starlike and strongly convex functions of order $\alpha$, and $\mathcal{R}(\alpha)$ of functions with bounded turning. As special cases of our main results, we obtain the exact bounds of these determinants for the classical classes of starlike, convex, and bounded turning functions.

We determine the asymptotic maximum number of unlabelled copies of $P_{2r+1}$ in graphs with prescribed edge density, where $r\ge1$ is fixed and $P_{2r+1}$ denotes the path on $2r+1$ vertices. If an $n$ vertex graph $G$ has edge density $c=2e(G)/n^2$, then the maximum is $\frac12S_r(c)n^{2r+1}+O(n^{2r})$ for $0<c\le c_r$, and $\frac12c^{r+1/2}n^{2r+1}+O(n^{2r})$ for $c_r\le c<1$, where $S_r(c)$ is the value given by the quasi-star construction and $c_r\in(0,1)$ is an explicit algebraic transition point. Thus the quasi-star construction is asymptotically extremal below the transition, while the quasi-clique construction is asymptotically extremal above the transition. This extends the quasi-star versus quasi-clique theorem of Ahlswede and Katona for $P_3$ and the theorem of Nagy for $P_5$ to all paths with an odd number of vertices. The proof reduces the problem to threshold graphons and then to two endpoint families. The three-step endpoint is handled by reducing the required inequality to coefficient nonnegativity in a Bernstein expansion, which is proved by a direct combinatorial argument.