Generalizations of Wiener polarity index and terminal Wiener index

In theoretical chemistry, distance-based molecular structure descriptors are used for modeling physical, pharmacologic, biological and other properties of chemical compounds. We introduce a generalized Wiener polarity index $W_k (G)$ as the number of unordered pairs of vertices ${u, v}$ of $G$ such that the shortest distance $d (u, v)$ between $u$ and $v$ is $k$ (this is actually the $k$-th coefficient in the Wiener polynomial). For $k = 3$, we get standard Wiener polarity index. Furthermore, we generalize the terminal Wiener index $TW_k (G)$ as the sum of distances between all pairs of vertices of degree $k$. For $k = 1$, we get standard terminal Wiener index. In this paper we describe a linear time algorithm for computing these indices for trees and partial cubes, and characterize extremal trees maximizing the generalized Wiener polarity index and generalized terminal Wiener index among all trees of given order $n$.Comment: 3pages, 4 figure

Krylov subspace approximations for the exponential Euler method: error estimates and the harmonic Ritz approximant

We study Krylov subspace methods for approximating the matrix-function vector product φ(tA)b where φ(z) = [exp(z) - 1]/z. This product arises in the numerical integration of large stiff systems of differential equations by the Exponential Euler Method, where A is the Jacobian matrix of the system. Recently, this method has found application in the simulation of transport phenomena in porous media within mathematical models of wood drying and groundwater flow. We develop an a posteriori upper bound on the Krylov subspace approximation error and provide a new interpretation of a previously published error estimate. This leads to an alternative Krylov approximation to φ(tA)b, the so-called Harmonic Ritz approximant, which we find does not exhibit oscillatory behaviour of the residual error

Time, Ethics and Experience: Review of David O. Brink\u27s Prospects for Temporal Neutrality

Are temporal locations of harms and benefits important to human existence? Conventional wisdom unambiguously suggests so, albeit interpretations of various dogmatic texts and beliefs. Discussions about pain, grief, and suffering are commonly favored within past temporal settings, unlike those of happiness, comfort, and wellbeing that permeate conversations with future temporal locales. Past pain is preferred to future pain, even when this choice includes more total pain (Callender, 2011). Should these positive and negative qualifiers that constitute conscious existence have privileged temporal locations? This ethical question, like many others surrounding temporality, inherits both theoretical and pragmatic inquiries - becoming indispensable within moral and juridical dispositions. The concept of temporal neutrality, which posits that agents should not attach normative significance to temporal locations of benefits and harms, all else being equal, is central to the present philosophical investigation. In his Prospects for Moral Neutrality chapter of The Oxford Handbook of Philosophy of Time (2011), David O. Brink articulates what exactly temporal neutrality requires and why we ought to care about its precepts. As they are assessed by how they distribute benefits and harms across people’s lives through interpersonal distributive justice, actions and policies can also be assessed by their distribution of benefits and harms across time. This concept of intertemporal distribution is a normative demand of temporal neutrality, and according to some philosophers it makes temporal neutrality an essential part of rationality (Brink, 2011). However, establishing an impartial foundation for temporal neutrality often appears controversial and counterintuitive