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If we consider previously introduced extensions of Stanley's chromatic symmetric function $X_{G}(x_1, x_2, \ldots)$ for a graph $G$ to elements in the algebra $\textsf{QSym}$ of quasisymmetric functions and in the algebra $\textsf{NCSym}$ of symmetric functions in noncommuting variables, this motivates our introduction of a lifting of $X_{G}$ to the dual of $\textsf{QSym}$, i.e., the algebra $\textsf{NSym}$ of noncommutative symmetric functions, as opposed to $\textsf{NCSym}$. For an unlabelled directed graph $D$, our extension of chromatic symmetric functions provides an element $\text{{X}}_{D}$ in $\textsf{NSym}$, in contrast to the analogue $Y_{G} \in \textsf{NCSym}$ of $X_{G}$ due to Gebhard and Sagan. Letting $G$ denote the undirected graph underlying $D$, our construction is such that the commutative image of $\text{{X}}_{D}$ is $ X_{G}$. This projection property is achieved by lifting Stanley's power sum expansion for chromatic symmetric functions, with the use of the $\Psi$-basis of $\textsf{NSym}$, so that the orderings of the entries of the indexing compositions are determined by the directed edges of $D$. We then construct generating sets for $\textsf{NSym}$ consisting of expressions of the form $\text{{X}}_{D}$, building on the work of Cho and van Willigenburg on chromatic generating sets for $\textsf{Sym}$., Comment: Submitted for publication

The pair production of Higgs bosons at the LHC can give information about the triple Higgs boson coupling. We perform an analytic one-loop calculation of the amplitudes for a pair of Higgs bosons in association with three partons, retaining the exact dependence on the quark mass circulating in the loop. These amplitudes constitute the real radiation corrections in the calculation of Higgs boson pair production at next-to-leading order in the strong coupling. The results of an analytic generalised-unitarity computation are simplified via analytic reconstruction in spinor variables. Compact ans\"atze for kinematic pole residues are iteratively fitted via $p$-adic evaluations near said poles and subtracted until no pole remains. A new ansatz construction is introduced to minimally parametrise coefficients of amplitudes with multiple massive external legs. The simplified expressions are faster to evaluate than automatic codes and can lead to more stable results near singular regions., Comment: 44 pages, 4 figures, numerical evaluation of LaTeX formulae available via ancillary files

While the current frontier in fixed-order precision for collider observables is N$^3$LO, important steps are necessary to consolidate NNLO cross-section predictions with improved stability and efficiency. Slicing methods have been successfully applied to obtain NNLO and N$^3$LO predictions, but have shown poor performance in the presence of fiducial cuts due to large kinematical power corrections. In this paper we implement Projection-to-Born-improved $q_T$ (P2B $q_T$) and jettiness (P2B $\tau_0$) subtractions for a large class of color singlet processes in MCFM. This method allows for the efficient evaluation of \emph{fiducial} power corrections in any non-local subtraction scheme using a Projection-to-Born subtraction. We demonstrate the significant numerical improvements of this method based on fiducial Drell-Yan and Higgs cross-sections. Moreover, with fiducial power corrections removed via this method, the leading-logarithmic power corrections that have only been calculated without fiducial cuts can be included, further improving the calculations. For di-photon production with photon isolation, we devise a novel method in combination with P2B-improved subtractions, which we name $P2B_\gamma$ $\tau_0$, and $P2B_\gamma$ $q_T$ for the two subtraction schemes, respectively. This method allows the inclusion of both fiducial power corrections due to kinematic cuts on the photons and a set of isolation power corrections in the fragmentation channel where a quark may enter the isolation cone. We find significant improvements in the convergence of NNLO di-photon cross-sections with photon isolation cuts, demonstrating that it is possible to achieve a stable and efficient calculation of di-photon cross-sections using slicing methods., Comment: 24 pages, 15 figures

Hou, Krattenthaler, and Sun have introduced two $q$-analogues of a remarkable series for $\pi^2$ due to Guillera, and these $q$-identities were, respectively, proved with the use of a $q$-analogue of a Wilf-Zeilberger pair provided by Guillera and with the use of ${}_{3}\phi_{2}$-transforms. We prove a $q$-analogue of Guillera's formula for $\pi^2$ that is inequivalent to previously known $q$-analogues of the same formula due to Guillera, including the Hou-Krattenthaler-Sun $q$-identities and a subsequent $q$-identity due to Wei. In contrast to previously known $q$-analogues of Guillera's formula, our new $q$-analogue involves another free parameter apart from the $q$-parameter. Our derivation of this new result relies on the $q$-analogue of Zeilberger's algorithm.

In combinatorics on words, the well-studied factor complexity function $\rho_{\bf x}$ of a sequence ${\bf x}$ over a finite alphabet counts, for any nonnegative integer $n$, the number of distinct length-$n$ factors of $\mathbf{x}$. In this paper, we introduce the reflection complexity function $r_{\bf x}$ to enumerate the factors occurring in a sequence ${\bf x}$, up to reversing the order of symbols in a word. We introduce and prove general results on $r_{\bf x}$ regarding its growth properties and relationship with other complexity functions. We prove a Morse-Hedlund-type result characterizing eventually periodic sequences in terms of their reflection complexity, and we deduce a characterization of Sturmian sequences. Furthermore, we investigate the reflection complexity of quasi-Sturmian, episturmian, $(s+1)$-dimensional billiard, and complementation-symmetric Rote, and rich sequences. Furthermore, we prove that if ${\bf x}$ is $k$-automatic, then $r_{\bf x}$ is computably $k$-regular, and we use the software $\mathtt{Walnut}$ to evaluate the reflection complexity of automatic sequences, such as the Thue-Morse sequence. We note that there are still many unanswered questions about this measure.

The numerically stable evaluation of scattering matrix elements near the infrared limit of gauge theories is of great importance for the success of collider physics experiments. We present a novel algorithm that utilizes double precision arithmetic and reaches higher precision than a naive quadruple precision implementation at smaller computational cost. The method is based on physics-driven modifications to propagators, vertices and external polarizations., Comment: 5 pages, 1 figure, 2 pages of supplemental material with one additional figure

Schur-Weyl duality concerns the actions of $\text{GL}_{n}(\mathbb{C})$ and $S_{k}$ on tensor powers of the form $V^{\otimes k}$ for an $n$-dimensional vector space $V$. There are rich histories within representation theory, combinatorics, and statistical mechanics involving the study and use of diagram algebras, which arise through the restriction of the action of $\text{GL}_{n}(\mathbb{C})$ to subgroups of $\text{GL}_{n}(\mathbb{C})$. This leads us to consider further variants of Schur-Weyl duality, with the use of variants of the tensor space $V^{\otimes k}$. Instead of taking repeated tensor products of $V$, we make use of a freest commutative bilinear operation in place of $\otimes$, and this is motivated by an associated invariance property given by the action of $S_{k}$. By then taking the centralizer algebra with respect to the action of the group of permutation matrices in $\text{GL}_{n}(\mathbb{C})$, this gives rise to a diagram-like algebra spanned by a new class of combinatorial objects. We construct orbit-type bases for the centralizer algebras introduced in this paper, and we apply these bases to prove a combinatorial formula for the dimensions of our centralizer algebras., Comment: Submitted for publication

Chu and Zhang, in 2014, introduced hypergeometric transforms derived through the application of an Abel-type summation lemma to Dougall's ${}_{5}H_{5}$-series. These transforms were applied by Chu and Zhang to obtain accelerated rates of convergence, yielding rational series related to the work of Ramanujan and Guillera. We apply a variant of an acceleration method due to Wilf using what we refer to as shifted indices for Pochhammer symbols involved in our first-order, inhomogeneous recurrences derived via Zeilberger's algorithm, to build upon Chu and Zhang's accelerations, recovering many of their accelerated series and introducing many inequivalent series for universal constants, including series of Ramanujan type involving linear polynomials as summand factors, as in Ramanujan's series for $\frac{1}{\pi}$., Comment: Submitted for publication

Quantum state comparison, utilizing metrics like fidelity and trace distance, underpins the assessment of quantum networks within quantum information theory. While recent research has expanded theoretical understanding, incorporating error analysis and scalability considerations remains crucial for practical applications. The primary contribution of this letter is to address these gaps by deriving the novel operational trace distance for multi-node networks, establishing a trace distance vs. fidelity benchmark incorporating error bounds, and bridging quantum operations with tensor network analysis. We further explore the application of tensor network tools to quantum networks, offering new analytical avenues. This comprehensive approach provides a robust framework for evaluating quantum network performance under realistic error conditions, facilitating the development of reliable quantum technologies.

The (Mahler-Popken) complexity $\| n \|$ of a natural number $n$ is the smallest number of ones that can be used via combinations of multiplication and addition to express $n$, with parentheses arranged in such a way so as to form legal nestings. We generalize $\| \cdot \|$ by defining $\| n \|_{m}$ as the smallest number of possibly repeated selections from $\{ 1, 2, \ldots, m \}$ (counting repetitions), for fixed $m \in \mathbb{N}$, that can be used to express $n$ with the same operational and bracket symbols as before. There is a close relationship, as we explore, between $\|\cdot\|_{2}$ and lengths of shortest addition chains for a given natural number. This illustrates how remarkable it is that $(\| n \|_{2} : n \in \mathbb{N} )$ is not currently included in the On-Line Encyclopedia of Integer Sequences and has, apparently, not been studied previously. This, in turn, motivates our exploration of the complexity function $\| \cdot\|_{2}$, in which we prove explicit upper and lower bounds for $\|\cdot\|_{2}$ and describe some problems and further areas of research concerning $\|\cdot\|_{2}$., Comment: Accepted for publication in INTEGERS: The Electronic Journal of Combinatorial Number Theory

Through the application of an evaluation technique based on cyclotomic multiple zeta values recently due to Au, we solve open problems on inverse binomial series that were included in a 2010 analysis textbook by Chen., Comment: a sequel to arXiv:2210.17243v4 and arXiv:2306.04638v3

We introduce and prove evaluations for families of multiple elliptic integrals by solving special types of ordinary and partial differential equations. As an application, we obtain new expressions of Ramanujan-type series of level 4 and associated singular values for the complete elliptic integral $\mathbf K$ with integrals involving $\mathbf K$., Comment: 8 pages

Using an infinite family of generalizations of the Chudnovsky brothers' series recently obtained via the analytic continuation of the Borwein brothers' formula for Ramanujan-type series of level 1, we apply the Gauss-Salamin-Brent iteration for $\pi$ to obtain a new, Ramanujan-type series that yields more digits per term relative to current world record given by an extension of the Chudnovsky algorithm from Bagis and Glasser that produces about 110 digits per term. We explicitly evaluate the required nested radicals over $\mathbb{Q}$ involved in the our summation, which yields about 153 digits per term, and we provide a practical way of implementing our higher-order version of the Chudnovsky algorithm via the the PARI/GP system. An evaluation due to Berndt and Chan for the modular $j$-invariant associated with their order-3315 extension of the Chudnovskys' Ramanujan-type series provides a key to our applications of recursions for the elliptic lambda and elliptic alpha functions., Comment: Submitted for publication

Using the Wolfram NumberTheory package and the Recognize command, together with numerical estimates involving the elliptic lambda and elliptic alpha functions, Bagis and Glasser, in 2013, introduced a conjectural Ramanujan-type series related to the class number $h(-d) = 1$ for a quadratic form with discriminant $d = 163$. This conjectured series is of level one and has positive terms, and recalls the Chudnovsky brothers' alternating series of the same level, given the connection between the Chudnovsky-Chudnovsky formula and the Heegner number $d = 163$ such that $\mathbb{Q}\left( \sqrt{-d} \right)$ has class number one. We prove Bagis and Glasser's conjecture by proving evaluations for $\lambda^{\ast}(163)$ and $\alpha(163)$, which we derive using the Chudnovsky brothers' formula together with the analytic continuation of a formula due to the Borwein brothers for Ramanujan-type series of level one. As a byproduct of our method, we obtain an infinite family of Ramanujan-type series for $\frac{1}{\pi}$ generalizing the Chudnovsky algorithm., Comment: To appear in the Journal of Number Theory

Let $R(q)$ denote the Rogers-Ramanujan continued fraction for $|q| < 1$. By applying the RootApproximant command in the Wolfram language to expressions involving the theta function $f(-q) := (q;q)_{\infty}$ given in modular relations due to Yi, this provides a systematic way of obtaining experimentally discovered evaluations for $R\big(e^{-\pi\sqrt{r}}\big)$, for $r \in \mathbb{Q}_{> 0}$. We succeed in applying this approach to obtain explicit closed forms, in terms of radicals over $\mathbb{Q}$, for the Rogers-Ramanujan continued fraction that have not previously been discovered or proved. We prove our closed forms using the icosahedral equation for $R$ together with closed forms for and modular relations associated with Ramanujan's $G$- and $g$-functions. An especially remarkable closed form that we introduce and prove is for $R\big( e^{-\pi \sqrt{48/5} } \big)$, in view of the computational difficulties surrounding the application of an order-25 modular relation in the evaluation of $G_{48/5}$., Comment: Submitted for publication

This document explores the connections between singular value decomposition (SVD) and various tensor operations in the context of quantum state calculations and tensor networks. The relationships between SVD and quantities such as trace, trace distance, Frobenius norm, and fidelity are examined, highlighting their similarities and differences. The intuition behind these mathematical connections is explained, providing insights into the underlying principles and applications of SVD and tensor operations in quantum physics and computational mathematics, Comment: 8 pages, 1 figure, 1 table