In this paper, we derive new combinatorial formulas for symmetric Macdonald polynomials $P_λ(X;q,t)$ and non-symmetric Macdonald polynomials $E_γ(X;q,t)$, in terms of several new statistics and the major index, for a partition $λ$ and a weak composition $γ$. Compared to previous formulas, these new formulas contain the fewest terms and lead to explicit $(q,t)$-formulas for the coefficients in the monomial expansion of $P_λ(X;q,t)$. In particular, the combinatorial formula for $E_γ(X;q,t)$ extends the one for $E_λ(X;q,t)$ indexed by a partition $λ$, due to Corteel, Mandelshtam and Williams (2022). Three existing formulas for $P_λ(X;q,t)$ established by Corteel, Mandelshtam and Williams (2022), by Corteel, Haglund, Mandelshtam, Mason and Williams (2022), and by Mandelshtam (2025) are recovered. Our proof relies on two new statistics on super fillings, employing the superization formulas of Haglund--Haiman--Loehr (2005) and Ayyer--Mandelshtam--Martin (2023), together with our recent approach to modified Macdonald polynomials.
We conjecture new elementary formulas for computing the greatest common divisor (GCD) of two integers, alongside an elementary formula for extracting the prime factors of semiprimes. These formulas are of fixed-length and require only the basic arithmetic operations of: addition, subtraction, multiplication, division with remainder, and exponentiation. Our GCD formulas result from simplifying a formula of Mazzanti and are derived using Kronecker substitution techniques from our earlier research. By applying these GCD formulas together with our recent discovery of an arithmetic expression for $\sqrt{n}$, we are able to derive explicit elementary formulas for the prime factors of a semiprime $n=p q$.
Solid-state devices such as multistep staircase avalanche photodiodes (APDs) are analogues to the photomultiplier tubes and are considered as a cascade-amplifier. The major source of internal noise in these APDs is due to the randomness in their stepwise impact ionization. Recent literature on staircase APDs by research groups such as Campbell and co-workers have reported the theoretical estimates of total excess noise factors using Capasso's excess noise factor formula. This formula is based on Friis' total noise factor formula for cascade networks. This article proves that Capasso's formula for a staircase APD, erroneously considers the power gains in Friis' total noise factor formula as the gains.
In this paper, we establish the Itô-Wentzell-Lions formulae for flows of both full and conditional measures on general semimartingales. This generalizes the existing works on flows of measures on Itô processes. The key technical components involve an appropriate approximation of random fields by cylindrical functions and localization techniques. Moreover, we present the specific formulae in two special cases, including Itô-Wentzell-Lions formulae for time-space-measure-dependent functions and for functions driven by Poisson random measures.
The single exponential (SE) and double exponential (DE) formulas are widely recognized as efficient quadrature formulas for evaluating integrals with endpoint singularity. For integrals exhibiting algebraic singularity, explicit error bounds in a computable form have been provided, enabling computations with guaranteed accuracy. Such explicit error bounds have also been provided for integrals exhibiting logarithmic singularity. However, these error bounds have two points to be discussed. The first point is on overestimation of divergence speed of logarithmic singularity. The second point is on the case where there exist both logarithmic and algebraic singularity. To address these issues, this study provides new error bounds for integrals with logarithmic and algebraic singularity. Although existing and new error bounds described above pertain to integrals over the finite interval, the SE and DE formulas are also applicable to integrals over the semi-infinite interval. On the basis of the new results, this study provides new error bounds for integrals over the semi-infinite interval with logarithmic and algebraic singularity at the origin.
What is the probability that a random walk in the free group ends in a proper power? Or in a primitive element? We present a formula that computes the exponential decay rate of the probability that a random walk on a regular tree ends in a given subset, in terms of the exponential decay rate of the analogous probability of the non-backtracking random walk. This generalizes the well-known cogrowth formula of Grigorchuk, Cohen and Northshield. We also extend the formula to arbitrary subsets of the biregular tree.
A word $w_1w_2\cdots w_n$ is said to be up-down if $w_1 < w_2 >w_3 \cdots$. Carlitz and Scoville found the generating function for the number of up-down words over an alphabet of size $k$. Using properties of the Chebyshev polynomials we derive a closed-form formula for these numbers.
This brief note revisits the well-established $T_c$ formulas for proximity-coupled thin superconductor-normal conductor (SN) and superconductor-superconductor (SS) bilayers, highlighting their relationships and clarifying their ranges of applicability. Since these formulas are often misapplied in practical contexts, this note provides guidance for their correct use in experimental and applied settings. For SN bilayers, McMillan's formula is recommended for its broad applicability, with its SS counterpart offering similar reliability.
In this paper, we study multiplication formula of $F$-polynomial of representations of Hernandez and Leclerc's quivers with potentials. Since the truncated $q$-characters of some real simple modules over a quantum affine group $U_q(\widehat{\mathfrak{g}})$ can be expressed in terms of such $F$-polynomials, one can describe the product of two simple modules over $U_q(\widehat{\mathfrak{g}})$ using this multiplication formula.
We show that an arithmetic path integral over the $\ell$-torsion of a Jacobian $J[\ell]$ is equal to the trace of the Frobenius action on a representation of the Heisenberg group $H(J[\ell])$, up to an explicitly determined sign. This is an arithmetic analogue of trace--path integral formulae which arise in quantum field theory, where path integrals over a space of sections of a fibration over a circle can be expressed as the trace of the monodromy action on a Hilbert space.
In this article, we develop four types of analytical recurrence formulas for non-trivial zeros of the Riemann zeta function on critical line assuming (RH). Thus, all non-trivial zeros up to the $n$th order must be known in order to generate the $n$th+1 non-trivial zero. All the presented formulas are based on certain closed-form representations of the secondary zeta function family, which are already available in the literature. We also present a formula to generate the non-trivial zeros directly from primes. Thus all primes can be converted into an individual non-trivial zero, and we also give a set of formulas to convert all non-trivial zeros into an individual prime. We also extend the presented results to other Dirichlet-L functions, and in particular, we develop an analytical recurrence formula for non-trivial zeros of the Dirichlet beta function. Throughout this article, we also numerically compute these formulas to high precision for various test cases and review the computed results.
Let $r_{k}(n)$ denote the number of representations of the positive integer $n$ as the sum of $k$ squares. We prove a new summation formula involving $r_{k}(n)$ and the Bessel functions of the first kind, which constitutes an analogue of a result due to the Russian mathematician A. I. Popov.
We prove a general formula for the $p$-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above $p$. The formula is in terms of the cyclotomic derivative of a Rankin-Selberg $p$-adic $L$-function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author, to the context of the work of Yuan-Zhang-Zhang on the archimedean Gross-Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is $+1$ rather than $-1$, by an anticyclotomic version of the Waldspurger formula. When combined with work of Fouquet, the anticyclotomic Gross-Zagier formula implies one divisibility in a $p$-adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.
We present a new family of hook-length formulas for the number of standard increasing tableaux which arise in the study of factorial Grothendieck polynomials. In the case of straight shapes our formulas generalize the classical hook-length formula and Stanley's formula. For skew shapes, our formulas generalize the Naruse hook-length formula and its $q$-analogues, which were studied in previous papers of the series.
We prove an analog of the holomorphic Lefschetz formula for endofunctors of smooth compact dg-categories. We deduce from it a generalization of the Lefschetz formula of V. Lunts that takes the form of a reciprocity law for a pair of commuting endofunctors. As an application, we prove a version of Lefschetz formula proposed by Frenkel and Ngo. Also, we compute explicitly the ingredients of the holomorphic Lefschetz formula for the dg-category of matrix factorizations of an isolated singularity w(x). We apply this formula to get some restrictions on the Betti numbers of a Z/2-equivariant module over k[[x_1,...,x_n]]/(w) in the case when w(-x)=w(x).
In the paper we formulate and derive the family blowup formula of family Seiberg-Witten invariants. The formula has been used in the enumerative application of counting singular curves on algebraic surfaces. We first give a topological derivation of the formula by using family index theorem. Then we define the algebraic (family) Seiberg-Witten invariants for algebraic surfaces and then give an algebraic derivation of the family blowup formula for the algebraic family Seiberg-Witten invariants.
A general way of representing Stochastic Differential Equations (SDEs) on smooth manifold is based on Schwartz morphism. In this manuscript we are interested in SDEs on a smooth manifold $M$ that are driven by p-dimensional Wiener process $W_t \in \mathbb{R}^p$. In terms of Schwartz morphism, such SDEs are represented by Schwartz morphism that morphs the semi-martingale $(t,W_t)\in\mathbb{R}^{p+1}$ into a semi-martingale on the manifold $M$. We show that it is possible to construct such Schwartz morphisms using special maps that we call as diffusion generators. We show that one of the ways of constructing diffusion generator is by using regular Lagrangian. Using this diffusion generator approach, we also give extended Ito formula (also known as generalized Ito formula or Ito-Wentzell's formula) for SDEs on manifold.
Wang and Sun proved a certain summatory formula involving derangements and primitive roots of the unit. We study such a formula but for the particular case of the set of affine derangements in $\overrightarrow{GL}(\mathbb{Z}/2k\mathbb{Z})$ and its subset of involutive affine derangements in particular; in this last case its value is relatively simple and it is related to even unitary divisors of $k$.
We observe that the linearization coefficients for ultraspherical polynomials are the orthogonality weights for Racah polynomials with special parameters. Then it turns out that the linearization sum with such a Racah polynomial as extra factor inserted, can also be evaluated. The corresponding Fourier--Racah expansion is an addition type formula which is dual to the well-known addition formula for ultraspherical polynomials. The limit to the case of Hermite polynomials of this dual addition formula is also considered. Similar results as for ultraspherical polynomials, although only formal, are given by taking the Ruijsenaars--Hallnäs dual product formula for Gegenbauer functions as a starting point and by working with Wilson polynomials.
In this paper, with the help of a result by Matsumoto--Yor (2000), we prove a Girsanov-type formula for a class of anticipative transforms of Brownian motion which possesses exponential functionals as anticipating factors. Our result unifies existing formulas in earlier works. As an application, we also consider the law of Brownian motion perturbed by a positive weight of a fairly wide class, and prove its invariance under an anticipative transformation associated with the perturbation. In the course of our exploration, a disintegration formula for the Wiener measure related to exponential functionals plays a key role.