Papers

In the proper variant of the classical hat-guessing game on a graph, an adversary properly colors the vertices from a palette of $q$ colors. Each vertex sees only its neighbors' colors and all vertices simultaneously guess their own color. The players win if at least one guess is correct. We study this game on the book graph $B_{k,n}=K_k\vee\overline{K_n}$, with $k$ mutually adjacent spine vertices and $n$ independent pages. We first give a coverability characterization valid for every fixed spine size. Write $\operatorname{HGP}(G)$ for the proper hat-guessing number of $G$. For a finite configuration $P$, let $\operatorname{supp}(P)$ denote the set of colors appearing in its tuples. Let $C_k$ be the minimum of $|P|+|\operatorname{supp}(P)|$ over all non-coverable finite configurations $P$ of proper $k$-tuples. We prove $\sup_{n\geq 1}\operatorname{HGP}(B_{k,n})=C_k$ and that $\operatorname{HGP}(B_{k,n})=C_k$ for all sufficiently large $n$. Thus, the asymptotic problem for every fixed $k$ reduces to a finite extremal invariant. In particular, coverability of two-spine configurations is equivalent to pseudoforestness, and we determine the associated extremal problem exactly: $C_2=11$, with precisely two types of extremal obstruction. Consequently, $\operatorname{HGP}(B_{2,n})\leq 11$ for every $n$, with equality for all sufficiently large $n$; we give an explicit probabilistic estimate with a stabilization threshold of at most $4\times 10^8$. We also resolve the first previously open finite cases. An explicit seven-color construction with affine symmetry proves $\operatorname{HGP}(B_{2,3})=7$. A counting-rigidity argument establishes the linear upper bound $\operatorname{HGP}(B_{2,n})\leq n+3$ for all $n\geq 4$, which together with monotonicity yields $\operatorname{HGP}(B_{2,4})=7$. Finally, a general box obstruction gives explicit uniform bounds on $C_k$.

We propose a randomized iterative method for the ordinary least-squares estimation problem in large-scale linear statistical models, namely the Sequential Preconditioned Conjugate Gradient Method (SPCG). SPCG constructs a sequence of sketched least-squares subproblems with increasing sketch sizes, applies PCG as the inner solver, and warm-starts each subproblem from the previous solution. A final refinement stage is then performed on the full-scale problem. Since most iterations are carried out on smaller subproblems, the overall computational cost is significantly reduced. We establish the convergence theory, prove that SPCG attains OLS prediction accuracy, and derive per-subproblem iteration bounds and complexity estimates. Numerical experiments show that SPCG reaches the target prediction accuracy with fewer iterations and less CPU time than full-data PCG and Iterative Double Sketching (IDS).

Motivated by recent advances in derivative Bohr inequalities and their refinements, we investigate the Bohr phenomenon for holomorphic mappings in complex Banach spaces associated with Schwarz functions. We establish sharp Bohr-type inequalities involving higher-order Fréchet derivatives for both $|F(z)|$ and its refined counterpart $|F(z)|^2$. The corresponding Bohr radii are determined and shown to be best possible. Our results provide affirmative answers to Questions 1.1 and 1.2 and extend several classical and recent Bohr inequalities from the unit disk to the setting of holomorphic mappings on Banach spaces.

Fuel-dominant powered descent can be written as a convex program, but the usual full-state epigraph formulation still carries many state variables, fuel epigraph variables, and dynamics equalities. In addition, the pure-fuel objective provides no strong-convexity curvature. This paper combines three structural reductions. First, a dimensionally consistent low-weight energy term makes the control solution unique. Second, a terminal-state sensitivity recursion eliminates every intermediate state exactly; invertible row normalization turns the 30-node baseline with 300 primal variables and 174 equality multipliers into a problem with 90 control variables and six terminal multipliers. Third, the shared radial structure of the fuel norm and thrust ball gives an exact closed-form proximal operator consisting of group shrinkage followed by magnitude clipping. The condensed problem is solved with a fixed-budget extrapolated proportional–integral projected-gradient iteration implemented in fixed-size C17 arrays. In a Mars powered-descent case with energy weight 0.02 and relative reference-solution tolerance $10^{-3}$, the iteration count decreases from 2744 for the pure-fuel full-state epigraph baseline to 93, while the fuel metric increases by only 0.033%. The mean end-to-end solve time is \SI{68.2}{\micro\second}, and P99 is \SI{128.1}{\micro\second}, on an Intel i7-10875H. Because the thrust set in this test case is already a convex ball, the contribution is fast solution of the convex core rather than a new lossless-convexification theorem.

We study linear programming (LP) bounds for sphere packings that fiber over translates of a fixed lattice of lower rank, as well as their dual formulations. In doing so, we place the work of Conway and Sloane on fibered packings in the context of linear programming bounds for packing problems on $\mathbb{R}^k \times A$, where $A$ is a compact abelian group. For these programs we prove that the primal and the dual both attain their optima, so that every instance has an optimal pair satisfying complementary slackness. We study cases in which the LP bounds considered here achieve the best known sphere packing densities in dimensions $\le 9$ with prescribed translational symmetry, recovering analogues of Propositions 2, 3, 5, and 8 of Conway and Sloane, and we show that the Barnes-Wall lattice achieves the optimal sphere packing density for any $16$-dimensional packing that fibers over translates of $E_8$. In dimension $6$, we show that the natural LP bound fibering over translates of $D_4$ is in fact equivalent to the LP bound in dimension $2$ for ordinary sphere packing, while the LP bound for $4$-dimensional packings that fiber over $A_2$ translates is implied by the LP bound in dimension $2$ but has additional rigid structure that may make it more tractable. Finally, using the discrete reduction framework of Li, we show that the linear programming bound for packings fibering over translates of $A_3$ is strictly above $1$, so the linear programming bound alone cannot prove Conjecture 4.1 of Cohn and Rajagopal.

This work presents two different types of proximal gradient methods, with line search and without line search, for solving unconstrained set-valued optimization problems under the lower set-less ordering relation induced by a solid cone that is convex, pointed, and closed. The objective mapping of the problem involves finitely many functions, with each one being the sum of a continuously differentiable function and a convex function that is proper and closed. We present an approach to characterize weakly minimal points of the problem with the help of weakly efficient points of a family of vector optimization problems. Thereafter, we establish a stationarity condition along with its connection with weakly minimal points of the problem under study. Based on the stationary condition, the concept of a descent direction at a non-stationary point is discussed. In view of the line search-based method, we formulate an Armijo-type line search condition and establish the existence of such a step-size. For the proposed methods, global convergence is established under mild assumptions. The convergence analysis of the proximal gradient method with line search provides a theoretical advancement over the convergence results previously established for the steepest descent method in set-valued optimization problems. In addition, we analyze the computational complexity of the proposed methods and show that both methods achieve a convergence rate of $\mathcal{O}(1/\sqrt{k})$. Numerical results are reported to test the performance of the methods in practice.

Arakawa and Moreau constructed explicit singular vectors in a family of negative-level universal affine vertex algebras of types $D$ and $E$ and conjectured that the ideals generated by these vectors are maximal. Previous work established the $n=0$ cases for $D_4$, $E_6$, $E_7$, and $E_8$, as well as the level $-2$ case for $D_\ell$ with $\ell\geq 5$. We prove all the remaining cases: the level $-1$ case for $D_\ell$ with $\ell\geq 5$, and the negative-level cases with $n>0$ for $D_4$, $E_6$, $E_7$, and $E_8$. Together with the previously known results, this completes Arakawa–Moreau Conjecture 1. The proof determines the images of the prescribed singular vectors under minimal Drinfeld–Sokolov reduction and establishes simplicity of the reduced quotients by combining a Ramond–Zhu algebra argument, a Casimir-gap argument, and Li's spectral flow. Exactness and a nonvanishing theorem for the reduction functor then lift simplicity to the corresponding affine quotients. We also formulate a general maximality principle based on minimal reduction, give an alternative reduction-theoretic proof of the known level $-2$ result for $D_\ell$, and obtain a rank-reduction proof of the maximal-ideal theorem for the collapsing family $V^{2-2r}(D_{2r})$. Consequently, every candidate quotient appearing in Arakawa–Moreau Conjecture 1 is the corresponding simple affine vertex algebra.

In this paper, we study the Nijenhuis modules of Nijenhuis algebras. The concepts of free, projective, injective, and flat Nijenhuis modules are introduced, and a construction of free Nijenhuis modules is given. The ring of Nijenhuis operators is introduced, and the relationship between Nijenhuis modules and modules over this ring is established by proving an isomorphism of the corresponding categories. Finally, it is proved that the category of Nijenhuis modules has enough projective, injective, and flat objects.

In this paper, we consider the problem of updating the directed minimum spanning tree (DMST), when the given sample tree is subject to the weight changes, edge deletions and edge insertions. We present an implementation for updating the tree to a DMST using the weighted matroid intersection algorithm. Our algorithm focuses on maintaining a dynamic auxiliary graph, which plays a central role in the matroid intersection algorithm, and governs the iterations from the given tree to a DMST. Each iteration is guaranteed to yield an improved solution. We also provide an implementation of this algorithm and some experimental analysis.

We study diagonal cycles on moduli spaces of shtukas for groups of the form $H\times H$, where $H$ is a split almost simple reductive group, allowing arbitrary modification types. We relate their self-intersection numbers, with insertions of determinant line bundles, to higher derivatives of adjoint $L$-functions. This gives a general Gross–Zagier-type identity over function fields for groups of arbitrary type, and suggests a parallel conjectural picture for arithmetic intersections on Shimura varieties.

We prove the local analytic Bertini theorem, confirming a conjecture of Boucksom in full generality.

We consider a harmonically forced linear impact oscillator, where impact events are instantaneous with energy loss. We study the dynamics at the grazing bifurcation of the non-impacting periodic solution in the limit that the damping coefficient of the oscillator is zero. Through numerical computations we show that a recurring sequence of bifurcations exists between points of resonance. Specifically, resonance creates a stable periodic solution that subsequently loses stability in a secondary grazing bifurcation, then regains stability in a saddle-node bifurcation, then transitions to a chaotic attractor through a period-doubling cascade. The dynamics persist under mild parameter variation, so apply to weakly-damped impact oscillators near grazing.

The quantum electrodynamics for massive fermions carrying electric and magnetic charges, the dyons, is proposed based on the 1960's seminal works by Cabibbo, Ferrari, and Salam, with the gauge group being $U(1)\times U(1)$, which is associated to a vector field (photon) and a pseudo-vector field (metaphoton), wherein the Dirac quantization is set aside. At the tree level the spectrum consistency of the model is analyzed, and all continuous and discrete symmetries are established. The quantum analysis is performed by using the Becchi-Rouet-Stora (BRS) algebraic renormalization method, which is independent on any regularization scheme. Moreover, thanks to the presence of massless gauge fields, the Lowenstein-Zimmermann (LZ) subtraction scheme is required within the framework of the Bogoliubov-Parasiuk-Hepp-Zimmermann-Lowenstein (BPHZL) renormalization procedure. Finally, it is verified that the proposed model, the dyon quantum electrodynamics (dQED), is free from any anomaly and is multiplicative renormalizable at all orders in perturbation theory, proving, therefore, its quantum consistency.

Let Gamma be a Coxeter matrix and q be a generic parameter, let Hq (Gamma) be the Iwahori-Hecke algebra. We propose a concept H-representation of type [Gamma;q], which are representations of certain quiver satisfying several sets of relations. We show type [Gamma;q] H-representation and Hq (Gamma)-representation almost determine each other. By using this we construct all nonintersecting type Hq (Gamma)-representations . H-representation is compatible with the Kazhdan-Lusztig theory . Some relations between H-representation and W-graph are proposed, which produce an equation system for a W-graph to giving a W-representation of Hq (Gamma) . With the intention of constructing general type Hq (Gamma)-representations we introduce the concept H^2 -representation , which is another quiver representation hidden in a H-representation. A program to construct general type H-representations from a chosen [Gamma;q]-realizable H^2 -representation is developed ,by using it we prove that if Gamma satisfies m_(i,j) >= 5 for any i not equal to j, then any H^2 -representation is [ Gamma;q]-realizable.

This paper introduces a discrete-time event-triggered extremum seeking framework for real-time optimization of multivariable nonlinear systems. In contrast to conventional discrete-time extremum seeking implementations that rely on periodic input updates, the proposed scheme updates the control action only when a state-dependent triggering condition is met, enabling aperiodic execution and substantial reduction of actuation and communication effort. The resulting closed-loop architecture reconciles two structurally different paradigms: the periodic excitation required for gradient recovery in extremum seeking and the aperiodic philosophy of event-triggered control. By combining discrete-time averaging arguments with Lyapunov-based analysis, we prove practical convergence of the system trajectories to a neighborhood of the unknown extremum and exponential stability of the corresponding average dynamics despite the loss of uniform sampling. The results show that appropriately designed triggering rules preserve the essential optimization mechanisms of classical extremum seeking while drastically reducing the number of input updates. Numerical simulations demonstrate that comparable optimization performance can be achieved with significantly fewer actuation events, highlighting the suitability of the method for resource-aware digital and networked control implementations.

Recently, Samanta et al. introduced a unifying perspective to common graph-associated matrices through the matrix $B_{\alpha}(G)=\alpha A(G)+(1-\alpha)L(G)$ where $\alpha\in [0,1]$. We resolve an open problem regarding the positive semi-definiteness of $B_{\alpha}$ and similar matrices. We also explore eigenvector properties of $B_{\alpha}$ through edge and vertex removal. As a consequence, we provide a basis for results on the common graph-associated matrices.

We establish optimal convergence rates for Steklov eigenvalues and harmonically extended eigenfunctions toward their weighted Laplace–Beltrami counterparts on a closed manifold perforated by many small geodesic balls. The holes have radii that scale critically with respect to their spacing in the sense that the boundary area of each hole balances with the weighted volume of its Voronoi cell. We then derive a higher-order expansion of the Steklov eigenvalues; in dimensions two and three, we identify two correction scales. The expansion is governed by an indefinite Coulomb-type energy of the discrepancy between the boundary and bulk measures, mediated by the reduced Green function of the limiting operator. The proof relies on sharp estimates for certain auxiliary functions that we introduce in order to quantify the discrepancy measure between the surface measure on the holes and their background density. Our paper serves to bridge an emerging literature in spectral geometry with one on systems of points interacting via Coulomb-type energies.

In this work, we study the homogenization limit for Fokker–Planck equations on the flat torus with rapidly oscillating diffusion coefficients at the time-discrete level. We discretize the equation in time using a minimizing movement scheme and compare two choices of metric: a transport-type metric and a weighted L2 metric. We prove that this choice is crucial: the two schemes lead to different homogenization limits as the scaling parameter tends to zero, and consequently to different evolution equations as the time step is sent to zero. In particular, only the scheme based on the weighted L2 metric recovers the correct effective dynamics.

In this paper, we propose three projection–contraction algorithms for solving variational inclusion problems in the setting of Hilbert spaces, each incorporating a double inertial technique into the projection–contraction framework. The first algorithm \algoO achieves weak convergence under monotonicity and Lipschitz continuity of the single-valued operator, with an adaptive stepsize rule that does not require prior knowledge of the Lipschitz constant. A modified variant \algoT of \algoO attains $R$-linear convergence under the strong monotonicity assumption. The third algorithm \algoDIM obtains strong convergence to the minimum-norm solution without requiring strong monotonicity. We illustrate the proposed methods on an abstract variational inclusion problem and apply them to the split feasibility problem and the elastic net regularization problem, comparing them with existing methods in the literature. The $R$-linear convergence of \algoT is also verified through numerical simulations.

Let \[ \Bbbk[\Bbbk^3\otimes\Bbbk^2\otimes\Bbbk^2]^{U_3\times U_2\times U_2}. \] We construct an ordinary cluster family with Markov principal part. At $\zeta=-1$, the intersection of its initial Laurent ring with the three adjacent Laurent rings equals $\mathscr U$, and \[ \mathscr U=\mathcal M_u[u_\Delta], \] where $\mathcal M_u$ is the middle algebra. Its theta functions are indexed by a cone with a sixteen-element Hilbert basis. Multiplication by $u_\Delta$ pairs the Hilbert generators of mutable degrees $1$ and $-1$ and reduces each triple-weight space to the slice $\ell=\ell_0$. Counting the lattice points in this slice gives a finite sum with nonnegative summands. Determinant reduction extends the formula to all $n\times2\times2$ Kronecker coefficients.

A $(1, 1, 2, k)$-packing coloring of a graph $G$ is a partition of $V(G)$ into two independent sets, a 2-packing, and a $k$-packing. Recently, the question was posed in [A short proof that every claw-free cubic graph is (1, 1, 2, 2)-packing colorable, arXiv:2512.24001v1] as to whether every claw-free cubic graph is $(1, 1, 2, 3)$-packing colorable. We provide an answer in the affirmative in the case that $G$ is a diamond-free, claw-free cubic graph.

We show a curious connection between an example of dynamical forcing, where an existence of one periodic orbit for a dynamical system forces the existence of other periodic orbits, and the existence of different fibrations of a 3 dimensional manifold. Specifically, we establish a one to one correspondence between fibrations of the Whitehead link complement and the set of `simple orbit pairs' on the torus. The set of orbits is equipped with a complete order coming from a dynamical forcing relation, that corresponds to the Farey tree order of rational numbers. This corresponds to the order of the rational point on a fibered cone of the Whitehead link. For the proof we obtain an explicit description of all monodromies of the different fibrations of the Whitehead link complement.

Distributed optimization becomes particularly challenging when nonconvex objectives are combined with nonlinear equality constraints: feasibility is network coupled, while most existing exponentially convergent methods rely on convexity or affine constraints. This paper introduces a feedback-linearization and singular-perturbation framework for two representative problem classes involving, respectively, distributed local nonlinear equality constraints and nonlinear aggregate equality constraints. The framework separates the regulation of consensus and feasibility residuals from optimization along the feasible manifold. Specifically, an ideal feedback-linearized dynamics is constructed whose output behavior can be explicitly assigned, while its zero-output dynamics coincides with the projected gradient flow of the aggregate objective on the feasible manifold. The ideal feedback-linearizing input, however, is determined by a state-dependent, network-coupled algebraic equation and is therefore not directly implementable in a distributed manner. To overcome this obstruction, we replace the nonlocal algebraic solution with a fast residual-tracking dynamics, yielding a singular-perturbation realization that uses only local and neighboring information. Under a local quadratic-growth condition imposed only on the feasible manifold, we establish local exponential convergence of both the ideal and distributed dynamics. For sufficiently strong time-scale separation, the convergence rate of the distributed realization can be chosen arbitrarily close to that of the ideal dynamics. Explicit Euler discretizations are also proved to preserve local exponential convergence. Under stronger manifold regularity conditions, the region of attraction extends to a tubular neighborhood of the feasible manifold and, for affine local equality constraints, to the whole admissible state space.

We establish a universal block-diagonalization framework for Discrete Exterior Calculus (DEC) operators on symmetric meshes, enabling embarrassingly parallel solvers with provable FLOP reductions. We prove that the two fundamental DEC operators, the discrete exterior derivative $d$ and the Hodge star $\star$, are equivariant under isometric finite group actions on simplicial complexes. The proof exploits the permutation representation induced on cochain spaces by the group action. As a consequence, any operator assembled from $d$ and $\star$ (including the Hodge Laplacian, the codifferential, Maxwell-type operators, and elasticity operators) inherits a block-diagonal structure in a single symmetry-adapted basis, which is computed only once per mesh. Unlike spectral methods restricted to flat Platonic domains, the framework applies natively to curved manifolds and is applicable in principle to computational electromagnetism and geometric fluid simulation on symmetric domains. Numerical experiments on a geodesic sphere ($I_h$ symmetry) and a hexagonal torus ($D_{6h}$ symmetry) yield FLOP-based parallel speedups, relative to a dense direct factorization, of up to $62\times$ and $182\times$, respectively. A further experiment on a body-centred-cubic (BCC) tessellation of the flat 3-torus $T^3$ with $T_d$ symmetry confirms equivariance of the exterior derivative, Hodge star, and Hodge Laplacian at machine precision for form degrees $k=0,1,2$ across three mesh resolutions. The FLOP-based sequential speedup approaches its theoretical asymptote of $\approx 9.07\times$, which a standard Schur-multiplicity reduction deepens by a further factor of order $|G|$. These results show that a single symmetry-adapted basis reduces the linear-solve cost of structure-preserving DEC computations on curved and three-dimensional meshes.

We use the Birman–Craggs–Johnson (BCJ) homomorphism to study the intersection of the handlebody group with the Torelli group and with the Johnson kernel. For genus $\geq 3$, we determine the images of the BCJ homomorphism restricted to these subgroups. We then explicitly compute cup products of pairs of cohomology classes in $H^1(-;\mathbb{F}_2)$ detected by the BCJ homomorphism for genus $\geq 4$. For the handlebody Johnson kernel, our results lift to integral coefficients.

A dominating set in a graph $G$ is a subset $S$ of its vertices such that each vertex in $G$ is either in $S$ or adjacent to a vertex in $S$. Nordhaus-Gaddum inequalities relate the values of a graph parameter on a graph and its complement. In this setting, Keough and Shane conjecture that any graph $G$ on $n$ vertices satisfies $\partial(G) + \partial(\bar{G}) \leq 2(2^{\lfloor n/2 \rfloor} - 1)(2^{\lceil n/2 \rceil} - 1) + 2$, where $\partial(G)$ is the number of dominating sets in $G$. We partially resolve this conjecture for the bipartite case by proving the stronger bound: for a bipartite graph $G$ with nonempty bipartition $(A,B)$, it holds that $\partial(G) + \partial(\bar{G}) \leq 2(2^{|A|} - 1)(2^{|B|} - 1) + 2$. We also characterize the bipartite graphs for which equality holds.

Curvature effects are expected to smooth flame-front wrinkles and thereby reduce turbulent flame speed. We construct a smooth three-dimensional periodic shear flow for which introducing Markstein curvature diffusivity instead increases the effective flame speed predicted by the level-set G-equation. This gives the first counterexample, within this model, to monotone slowdown under curvature smoothing and contrasts with the rigorous monotonicity result for two-dimensional shear flows. The example reveals a genuinely multidimensional mechanism in which local curvature smoothing can enhance, rather than suppress, large-scale front propagation.

We establish the following Miyaoka-Yau inequality for any $n$-dimensional klt Fano variety $X$, possibly K-unstable, in terms of its delta invariant: \[\left(2(n+1)\widehat{c}_2(X)-n c_1(X)^2\right)\cdot c_1(X)^{n-2} \ge -n \left(1-\min\{1,\delta(X)\}\right)^2 \cdot c_1(X)^n.\] Moreover, if equality holds in this inequality, then there exist a finite quasi-étale cover $\widehat{X} \to X$ and a positive integer $p$ such that $\widehat{X} \cong \mathbb{P}(1,\ldots, 1, p)$. We also discuss the relationship between Donaldson's conjecture and the Miyaoka-Yau inequality.

We present an efficient Fourier spectral method based on the convex splitting framework of Eyre~\cite{eyre1998unconditionally} for the phase field crystal (PFC) equation. The proposed first-order scheme is unconditionally energy stable and conserves mass to machine precision. An energy stability theorem is established using a truncated potential argument, and a semi-analytical neutral stability curve is derived from a dominant-mode energy balance, providing a closed-form characterization of the practical stability boundary. The classical sufficient condition $a \geq 2$ is shown to be conservative: a computational stability map obtained from 40,000 GPU-accelerated PFC simulations reveals that energy-stable solutions persist for values of $a$ significantly below this threshold. Crucially, accuracy analysis demonstrates that smaller values of $a$ within the stable region consistently yield lower $L^2$ errors. This high-fidelity regime is rigorously verified through an extended asymptotic stress test consisting of a long-time three-dimensional simulation on a $256^3$ grid up to $T_f = 10,000$, successfully executing $10^6$ continuous temporal increments deep within the relaxed stability regime, well below the classical convex splitting limit ($a < 2$), while preserving strict monotonic energy dissipation and machine-precision mass conservation. Finally, two-dimensional and three-dimensional simulations at resolutions up to $512^3$ are performed on a single consumer GPU, demonstrating the scalability of the proposed framework for resolving complex phase-field dynamics without requiring HPC infrastructure. The code is made publicly available on GitHub.

We refine the Explicit Hypergeometric Modularity Method (EHMM) and develop a variant that applies to a broader class of hypergeometric data. As an application, we establish the modularity of hypergeometric Galois representations arising from length-$4$ data that are not necessarily defined over $\mathbb{Q}$. We also use this method to give explicit constructions of nine modular forms associated with hypergeometric rigid Calabi-Yau threefolds conjectured to be modular by Rodriguez-Villegas. The modularity of these threefolds was first proved by Long-Tu-Yui-Zudilin using a different approach based on the Faltings-Serre method. Moreover, the explicit nature of the method makes it well suited to the computation of special $L$-values of the associated modular forms.