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1. On the stationarity for nonlinear optimization problems with polyhedral constraints

   https://doi.org/10.1007/s10107-023-01979-9  

   di Serafino, Daniela ; Hager, William W. ; Toraldo, Gerardo ; Viola, Marco ( May 2023 , Mathematical Programming)

   For polyhedral constrained optimization problems and a feasible point$$\textbf{x}$$$x$, it is shown that the projection of the negative gradient on the tangent cone, denoted$$\nabla _\varOmega f(\textbf{x})$$${\nabla }_{\Omega }f\left(x\right)$, has an orthogonal decomposition of the form$$\varvec{\beta }(\textbf{x}) + \varvec{\varphi }(\textbf{x})$$$\beta \left(x\right)+\phi \left(x\right)$. At a stationary point,$$\nabla _\varOmega f(\textbf{x}) = \textbf{0}$$${\nabla }_{\Omega }f\left(x\right)=0$so$$\Vert \nabla _\varOmega f(\textbf{x})\Vert $$$\Vert {\nabla }_{\Omega }f\left(x\right)\Vert$reflects the distance to a stationary point. Away from a stationary point,$$\Vert \varvec{\beta }(\textbf{x})\Vert $$$\Vert \beta \left(x\right)\Vert$and$$\Vert \varvec{\varphi }(\textbf{x})\Vert $$$\Vert \phi \left(x\right)\Vert$measure different aspects of optimality since$$\varvec{\beta }(\textbf{x})$$$\beta \left(x\right)$only vanishes when the KKT multipliers at$$\textbf{x}$$$x$have the correct sign, while$$\varvec{\varphi }(\textbf{x})$$$\phi \left(x\right)$only vanishes when$$\textbf{x}$$$x$is a stationary point in the active manifold. As an application of the theory, an active set algorithm is developed for convex quadratic programs which adapts the flow of the algorithm based on a comparison between$$\Vert \varvec{\beta }(\textbf{x})\Vert $$$\Vert \beta \left(x\right)\Vert$and$$\Vert \varvec{\varphi }(\textbf{x})\Vert $$$\Vert \phi \left(x\right)\Vert$.

    

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2. Generalized Parking Function Polytopes

   https://doi.org/10.1007/s00026-023-00671-1  

   Hanada, Mitsuki ; Lentfer, John ; Vindas-Meléndez, Andrés_R ( November 2023 , Annals of Combinatorics)

   A classical parking function of lengthnis a list of positive integers$$(a_1, a_2, \ldots , a_n)$$$(a_1,a_2,\dots ,a_n)$whose nondecreasing rearrangement$$b_1 \le b_2 \le \cdots \le b_n$$$b_1\le b_2\le \cdots \le b_n$satisfies$$b_i \le i$$$b_i\le i$. The convex hull of all parking functions of lengthnis ann-dimensional polytope in$${\mathbb {R}}^n$$$R^n$, which we refer to as the classical parking function polytope. Its geometric properties have been explored in Amanbayeva and Wang (Enumer Combin Appl 2(2):Paper No. S2R10, 10, 2022) in response to a question posed by Stanley (Amer Math Mon 127(6):563–571, 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of$${\textbf{x}}$$$x$-parking functions for$${\textbf{x}}=(a,b,\dots ,b)$$$x=(a,b,\cdots ,b)$, which we refer to as$${\textbf{x}}$$$x$-parking function polytopes. We explore connections between these$${\textbf{x}}$$$x$-parking function polytopes, the Pitman–Stanley polytope, and the partial permutahedra of Heuer and Striker (SIAM J Discrete Math 36(4):2863–2888, 2022). In particular, we establish a closed-form expression for the volume of$${\textbf{x}}$$$x$-parking function polytopes. This allows us to answer a conjecture of Behrend et al. (2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary.

    

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3. Eulerian Walks in Temporal Graphs

   https://doi.org/10.1007/s00453-022-01021-y  

   Marino, Andrea ; Silva, Ana ( August 2022 , Algorithmica)

   An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graphGat least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. But what if Euler had to take a bus? In a temporal graph$$\varvec{(G,\lambda )}$$$(G,\lambda )$, with$$\varvec{\lambda : E(G)}\varvec{\rightarrow } \varvec{2}^{\varvec{[\tau ]}}$$$\lambda :E\left(G\right)\to 2^{\left[\tau \right]}$, an edge$$\varvec{e}\varvec{\in } \varvec{E(G)}$$$e\in E\left(G\right)$is available only at the times specified by$$\varvec{\lambda (e)}\varvec{\subseteq } \varvec{[\tau ]}$$$\lambda \left(e\right)\subseteq \left[\tau \right]$, in the same way the connections of the public transportation network of a city or of sightseeing tours are available only at scheduled times. In this paper, we deal with temporal walks, local trails, and trails, respectively referring to edge traversal with no constraints, constrained to not repeating the same edge in a single timestamp, and constrained to never repeating the same edge throughout the entire traversal. We show that, if the edges are always available, then deciding whether$$\varvec{(G,\lambda )}$$$(G,\lambda )$has a temporal walk or trail is polynomial, while deciding whether it has a local trail is$$\varvec{\texttt {NP}}$$$\mathrm{NP}$-complete even if$$\varvec{\tau = 2}$$$\tau =2$. In contrast, in the general case, solving any of these problems is$$\varvec{\texttt {NP}}$$$\mathrm{NP}$-complete, even under very strict hypotheses. We finally give$$\varvec{\texttt {XP}}$$$\mathrm{XP}$algorithms parametrized by$$\varvec{\tau }$$$\tau$for walks, and by$$\varvec{\tau +tw(G)}$$$\tau +tw\left(G\right)$for trails and local trails, where$$\varvec{tw(G)}$$$tw\left(G\right)$refers to the treewidth of$$\varvec{G}$$$G$.

    

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4. The Brown measure of the free multiplicative Brownian motion

   https://doi.org/10.1007/s00440-022-01142-z  

   Driver, Bruce K. ; Hall, Brian ; Kemp, Todd ( October 2022 , Probability Theory and Related Fields)

   The free multiplicative Brownian motion$$b_{t}$$$b_t$is the large-Nlimit of the Brownian motion on$$\mathsf {GL}(N;\mathbb {C}),$$$\mathrm{GL}(N;C),$in the sense of$$*$$$\ast$-distributions. The natural candidate for the large-Nlimit of the empirical distribution of eigenvalues is thus the Brown measure of$$b_{t}$$$b_t$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region$$\Sigma _{t}$$${\Sigma }_t$that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density$$W_{t}$$$W_t$on$$\overline{\Sigma }_{t},$$${\overline{\Sigma }}_t,$which is strictly positive and real analytic on$$\Sigma _{t}$$${\Sigma }_t$. This density has a simple form in polar coordinates:$$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$$\begin{array}{c}{W}_t(r,\theta )=\frac{1}{r^2}{w}_t\left(\theta \right),\end{array}$where$$w_{t}$$$w_t$is an analytic function determined by the geometry of the region$$\Sigma _{t}$$${\Sigma }_t$. We show also that the spectral measure of free unitary Brownian motion$$u_{t}$$$u_t$is a “shadow” of the Brown measure of$$b_{t}$$$b_t$, precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.

    

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5. The $$\pi f_0(500)$$ decay of the $$a_1(1260)$$

   https://doi.org/10.1140/epjc/s10052-021-09574-y  

   Molina, R. ; Döring, M. ; Liang, W_H ; Oset, E. ( September 2021 , The European Physical Journal C)

   We evaluate the$$a_1(1260) \rightarrow \pi \sigma (f_0(500))$$$a_1\left(1260\right)\to \pi \sigma \left(f_0\left(500\right)\right)$decay width from the perspective that the$$a_1(1260)$$$a_1\left(1260\right)$resonance is dynamically generated from the pseudoscalar–vector interaction and the$$\sigma $$$\sigma$arises from the pseudoscalar–pseudoscalar interaction. A triangle mechanism with$$a_1(1260) \rightarrow \rho \pi $$$a_1\left(1260\right)\to \rho \pi$followed by$$\rho \rightarrow \pi \pi $$$\rho \to \pi \pi$and a fusion of two pions within the loop to produce the$$\sigma $$$\sigma$provides the mechanism for this decay under these assumptions for the nature of the two resonances. We obtain widths of the order of 13–22 MeV. Present experimental results differ substantially from each other, suggesting that extra efforts should be devoted to the precise extraction of this important partial decay width, which should provide valuable information on the nature of the axial vector and scalar meson resonances and help clarify the role of the$$\pi \sigma $$$\pi \sigma$channel in recent lattice QCD calculations of the$$a_1$$$a_1$.

    

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