A power sum expansion for the Kromatic symmetric function

The works of Brunella and Santos have singled out three special singular holomorphic foliations on projective surfaces having invariant rational nodal curves of positive self-intersection. These foliations can be described as quotients of foliations on some rational surfaces under cyclic groups of transformations of orders three, four, and six, respectively. Through an unexpected connection with the reduced Chazy IV, V and VI equations, we give explicit models for these foliations as degree-two foliations on the projective plane (in particular, we recover Pereira's model of Brunella's foliation). We describe the full groups of birational automorphisms of these quotient foliations, and, through this, produce symmetries for the reduced Chazy IV and V equations. We give another model for Brunella's very special foliation, one with only non-degenerate singularities, for which its characterizing involution is a quartic de Jonquières one, and for which its order-three symmetries are linear. Lastly, our analysis of the action of monomial transformations on linear foliations poses naturally the question of determining planar models for their quotients under the action of the standard quadratic Cremona involution; we give explicit formulas for these as well.

We consider gaps that arise in the possible numbers of non-vanishing components for certain linear systems that arise when studying rational sphere maps. When the source dimension is [Formula: see text], we find an explicit formula for an integer [Formula: see text] such that every number exceeding [Formula: see text] is the value for the minimum embedding dimension of some polynomial (hence rational) sphere map. We conjecture, but do not prove, that [Formula: see text] is sharp. This number is asymptotic to [Formula: see text]; this asymptotic value improves what has been suggested to be quadratic in [Formula: see text]. Our methods are combinatorial.

This paper addresses the approximation of the local volatility function in the Cheyette interest rate model. Its main contribution is an explicit analytical formula for approximating local volatility, derived by extending the classical Dupire framework to interest rate models. In particular, an implicit Dupire-like expression for local volatility is first derived for options written on the short rate. This expression is then approximated using a combination of perturbation methods and probabilistic techniques, resulting in a formula expressed in terms of time and strike derivatives of the Bachelier implied variance. The final formula naturally extends to multi-factor Cheyette models and provides a practical tool for model calibration.

In this paper we count the number of $k$ -potent elements over $\mathbb{H}_{\mathbb{Z}_{p}}$ ,where $\mathbb{H}_{\mathbb{Z}_{p}}$ is the quaternion algebra over $\mathbb{Z}_{p}$ , and we present a descriptive formula for thegeneral case. For $k\in \{3,4,5\}$ , we give an explicit formula forthese values. Moreover, as an application of these results, we count thenumber of solutions of the equation $x^{k}=1$ over $\mathbb{H}_{\mathbb{Z}_{p}}$. For this purpose, we will use computer as a toolto check and understand the behavior of these elements in all cases that will be studied.

In a 3-dimensional generalized trans-Sasakian manifold, explicit formula for Ricci operator, Ricci tensor and curvature tensor are obtained. In particular, expressions for Ricci tensor are obtained in a 3-dimensional generalized trans-Sasakian manifold in cases of the manifold being quasi-Einstein or generalized quasi-Einstein.

Abstract This paper is devoted to the study of the T-tensor associated with a spherically symmetric Finsler metric F = uϕ ( r , s ) on R n . We derive the first explicit formula for the T-tensor in the spherically symmetric case. Furthermore, we characterize all spherically symmetric Finsler metrics satisfying the so-called T-condition, that is, the vanishing of the T-tensor. In addition, we obtain a formula for the mean Cartan tensor and demonstrate that all spherically symmetric Finsler metrics of dimension n ≥3, with a nonzero mean Cartan tensor, are quasi-C-reducible.

We study the resurgent structure of Walcher's real topological string on general Calabi-Yau manifolds. We find trans-series solutions to the corresponding holomorphic anomaly equations, at all orders in the string coupling constant, by extending the operator formalism of the closed topological string, and we obtain explicit formulae for multi-instanton amplitudes. We find that the integer invariants counting disks appear as Stokes constants in the resurgent structure, and we provide experimental evidence for our results in the case of the real topological string on local $\mathbb{P}^2$.

We study the edge-Wiener index of the rank-n skeleton networks associated with the level-4 Sierpinski triangle. Recursive relations are derived for the count of each pattern, and an explicit formula is obtained for the edge-Wiener index of the level-4 Sierpinski skeleton networks. Our result shows that the finite-pattern method remains effective for exact edge-distance computations on higher-order self-similar networks.

We investigate whether the geometric parameters of a three-dimensional domain can be recovered from the Dirichlet spectrum of the Laplacian. As a controlled benchmark, we consider rectangular boxes, about which the eigenvalues are explicitly known and the Weyl coefficients can be computed in closed form. Exploiting the short-time asymptotics of the heat trace, we extract the leading Weyl coefficients from finite spectral data and show how they encode volume, surface area, and the third spectral Weyl term. These coefficients uniquely determine the side lengths of the box via an explicit cubic reconstruction formula. Numerical experiments based on several thousand eigenvalues demonstrate that the method is stable, accurate, and robust with respect to spectral truncation. The box setting thus provides a stringent validation of the proposed inverse spectral methodology and serves as a foundation for its extension to smooth curved domains, such as triaxial ellipsoids, where explicit spectral formulas are no longer available.

This research focuses on the Heisenberg Lie group. The aim is to determine the coadjoint orbits and their parametrizations. The method used in this research involves constructing the parametrization of coadjoint orbit for Heisenberg Lie group corresponding to the Heisenberg Lie algebra of dimension 2n+1. Furthermore, the obtained results are specialized to the cases of n=1, 2, and 3 which correspond to the Heisenberg Lie algebras of dimensions 3, 5, and 7. The main results are the explicit formulas of coadjoint orbits for the Heisenberg Lie group H_1, H_2, and H_3 which are expressed by the equations (〖Ad〗^* H_1 ) l_(α,β,γ)={l_(α^',β^',γ^' ):α^',β^',γ^'∈R}, (〖Ad〗^* H_2 ) l_(α,β,γ)={l_(α^',β^',γ^' ):α^',β^'∈R^2,γ^'∈R}, and (〖Ad〗^* H_3 ) l_(α,β,γ)={l_(α^',β^',γ^' ):α^',β^'∈R^3,γ^'∈R}. In addition, their associated parametrizations are given by the explicit formulas ψ(γZ^*,u)=∑_(i=1)^n▒(u_i X_i^*+u_(n+i) Y_i^* ) +γZ^* for n=1, 2, and 3. As a further study, various types of Lie groups can be explored to determine coadjoint orbits and their parametrization. Two Lie groups that are interesting to investigate further regarding their coadjoint orbits and parametrization are the diamond and Jacobi groups.

ABSTRACT This paper generalizes explicit pricing formulas for double barrier first-touch digitals and develops an approximating method for American strangle option prices. By employing a double barrier to emulate the optimal upper/lower exercise boundaries of American strangles, our approach aims to identify the optimal double barrier that maximizes the value of the portfolio of first-touch digitals and knock-out options, whose individual payoffs depend on the first-hitting time of the underlying asset price. The optimized portfolio price provides a viable approximation even with only a few barrier steps. Numerical experiments confirm the effectiveness of our approximation strategy.

Abstract The purpose of this paper is to give moment formulas with the aid of Milovanović [16]. Other aims are to establish new integral formulas in order to define new Apostol-type splines in terms of the Apostol-type polynomials. By the aid of these integral formulas, we derive a novel class of moment-type expressions arising from integrals of these polynomials. By applying generating function techniques and moment computations, we derive explicit representations and approximation formulas for Apostol- Bernoulli, Euler, and Frobenius spline polynomials. Closed-form expansions are established using Goldman’s formula and symbolic moment identities. The connection between cardinal B-splines $$\phi _n(x)$$ ϕ n ( x ) and uniform B-splines $$N_{0,n-1}(x) $$ N 0 , n - 1 ( x ) is given. We compute integrals using beta-type representations and provide recurrence relations for numerical implementation. Furthermore, we develop a comparative numerical table that confirms the validity of the approximation.

In this paper, we investigate the determinants involving some trigonometric functions. We establish a connection between these determinants and the special values of Dirichlet L-functions, thereby extending Guo's results (Determinants of trigonometric functions and class numbers. Linear Algebra Appl. 2022;653:33–43) to arbitrary positive integers n. In addition, we also prove a conjecture raised by Zhi-Wei Sun. Our main tool is the spectral decomposition of some linear operators. By the same method we obtain an explicit formula for the determinants of sine matrices. This formula is expressed as a product of Gauss sums attached to Dirichlet characters.

Berezansky et al. [2010] proposed an open problem: How are the dynamic behaviors of the well-known Nicholson’s blowflies model with a delayed linear harvesting. In this paper, we mainly study the existence of Hopf bifurcation of Nicholson’s blowflies model with a delayed linear harvesting. To that end, the stability and Hopf bifurcation of a general functional differential equation with two dependent delays are investigated. We show that if the difference between the two dependent delays is constant, by using one of the delays as a bifurcation parameter, sufficient conditions of stability and Hopf bifurcation are obtained. In addition, in order to determine the direction of Hopf bifurcation and the stability of bifurcated periodic solutions, explicit formulas are given by using the normal form theory. The main results can be applied to guarantee the existence of Hopf bifurcation in Nicholson’s blowflies model with a delayed linear harvesting. Our research partially answers the open problem proposed by Berezansky et al. [2010].

Abstract Studies of density matrices for random quantum states lead naturally to the fixed trace Laguerre ensemble in random matrix theory. Previous studies have uncovered explicit rational function formulas for moments of purity statistic (trace of the squared density matrix), and also a third order linear differential equation satisfied by the eigenvalue density. We further probe the origin of these results from the viewpoint of integrability, which is taken here to mean wider classes of recursions and differential equations, and give extensions. Prominent in our study are first order linear matrix differential equations. One application given is to the derivation of the third order scalar equation for the density. Another is to obtain the explicit rational function formula for the variance of the purity statistic in the β generalised fixed trace Laguerre ensemble. In the original case (β = 2), the purity cumulants are expressed in terms of the large argument expansion of a particular σ-Painlevé IV transcendent. In a different but related direction, the exact computation of the two-point correlation for the fixed determinant circular unitary ensemble SU(N ) is given in the Appendix.Dedicated to the memory of Santosh Kumar and his work on exact results in RMT and their applications 1

Unified explicit K-factor formula and stiffness zoning for restrained columns

We study monoids in Z ( N ) that are invariant under the action of the infinite symmetric group Sym. Our main result establishes a local–global principle characterizing equivariant finite generation for arbitrary Sym-invariant monoids, extending earlier results that required additional assumptions. We further analyze local–global phenomena for other fundamental properties, including positivity, normality, seminormality, and simplicity. In addition, we obtain structural results for symmetric monoids, including characterizations of positivity and non-positivity, a description of their groups of units, and explicit formulas for the ranks of local symmetric monoids and stabilizing Sym-invariant chains.

We study non-Hermitian random matrices belonging to the symmetry classes of the complex and symplectic Ginibre ensemble, and present a unifying and systematic framework for analysing mixed spectral moments involving both holomorphic and anti-holomorphic parts. For weight functions that induce a recurrence relation of the associated planar orthogonal polynomials, we derive explicit formulas for the spectral moments in terms of their orthogonal norms. This includes exactly solvable models such as the elliptic Ginibre ensemble and non-Hermitian Wishart matrices. In particular, we show that the holomorphic spectral moments of complex non-Hermitian random matrices coincide with those of their Hermitian limit up to a multiplicative constant, determined by the non-Hermiticity parameter. Moreover, we show that the spectral moments of the symplectic non-Hermitian ensemble admit a decomposition into two parts: one corresponding to the complex ensemble and the other constituting an explicit correction term. This structure closely parallels that found in the Hermitian setting, which naturally arises as the Hermitian limit of our results. Within this general framework, we perform a large-N asymptotic analysis of the spectral moments for the elliptic Ginibre and non-Hermitian Wishart ensemble, revealing the mixed moments of the elliptic and non-Hermitian Marchenko–Pastur laws. Furthermore, for the elliptic Ginibre ensemble, we employ a recently developed differential operator method for the associated correlation kernel, to derive an alternative explicit formula for the spectral moments and obtain their genus-type large-N expansion.

We study the metaplectic representation of the Jacobi group (the semi-direct product of the Heisenberg group by SU(1, 1)) by using the complex Weyl correspondence. In particular, we give explicit formulas for the complexWeyl symbols of the metaplectic representation operators and we prove that the complex Weyl correspondence is a Stratonovich–Weyl correspondence for the metaplectic representation.