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To investigate concurrent climate action and poverty eradication, we present combined income growth (GDP/capita) and domestic income inequality (measured as Gini coefficients) pathways that pursue (absolute and relative) poverty eradication reflecting the three narratives of Sustainable Development Pathway. The GDP/capita pathways are modifications of the Shared Socioeconomic Pathway SSP1 scenario, including one post-growth future for high-income countries and higher growth for all currently lower-income countries. Current inequality levels, together with the total national income from the GDP pathways, determine the inequality reductions required to eradicate poverty in individual countries; they are based on a methodology that specifies the relationship between poverty, inequality, and growth. Our pathways show rapid and sustained reductions in within-country inequality (Gini), even with high economic growth. The speed of redistribution is limited to the highest historically observed changes in inequality. We identify which countries face the greatest difficulties in meeting their poverty eradication targets and estimate the level of international transfers needed to fill the gap for those countries. Our findings reconfirm the importance of reducing within-country inequality in eradicating global poverty.

We describe a tabu search algorithm for the vehicle routing problem with split deliveries. At each iteration, a neighbor solution is obtained by removing a customer from a set of routes where it is currently visited and inserting it either into a new route or into an existing route that has enough residual capacity. The algorithm also considers the possibility of inserting a customer into a route without removing it from another route. The insertion of a customer into a route is done by means of the cheapest insertion method. Computational experiments are reported for a set of benchmark problems, and the results are compared with those obtained by the algorithm proposed by Dror and Trudeau.

We show that the Möbius function μ(n) is strongly asymptotically orthogonal to any polynomial nilsequence (F(g(n)Γ)) n∈ℕ . Here, G is a simply-connected nilpotent Lie group with a discrete and cocompact subgroup Γ (so G/Γ is a nilmanifold), g : ℤ → G is a polynomial sequence, and F : G/Γ → ℝ is a Lipschitz function. More precisely, we show that $ \\large{|}{\\normal\\frac{1}{N}} \\large{\\sum\\nolimits^{N}_{n=1}} \\mu(n)F(g(n)\\Gamma)\\large{|}$ ≪ F,G,Γ,A log −A N for all A > 0. In particular, this implies the Möbius and Nilsequence conjecture MN(s) from our earlier paper for every positive integer s. This is one of two major ingredients in our programme to establish a large number of cases of the generalised Hardy-Littlewood conjecture, which predicts how often a collection ψ1,..., ψt : ℤ d → ℤ of linear forms all take prime values. The proof is a relatively quick application of the results in our recent companion paper. We give some applications of our main theorem. We show, for example, that the Möbius function is uncorrelated with any bracket polynomial such as $n\\sqrt{3}\\lfloor n \\sqrt{2} \\rfloor$ . We also obtain a result about the distribution of nilsequences (a n xΓ) n∈ℕ as n ranges only over the primes.

One goal of statistical privacy research is to construct a data release mechanism that protects individual privacy while preserving information content. An example is a random mechanism that takes an input database X and outputs a random database Z according to a distribution Q n (⋅|X). Differential privacy is a particular privacy requirement developed by computer scientists in which Q n (⋅|X) is required to be insensitive to changes in one data point in X. This makes it difficult to infer from Z whether a given individual is in the original database X. We consider differential privacy from a statistical perspective. We consider several data-release mechanisms that satisfy the differential privacy requirement. We show that it is useful to compare these schemes by computing the rate of convergence of distributions and densities constructed from the released data. We study a general privacy method, called the exponential mechanism, introduced by McSherry and Talwar (2007). We show that the accuracy of this method is intimately linked to the rate at which the probability that the empirical distribution concentrates in a small ball around the true distribution.

In the pickup and delivery problem with time windows vehicle routes must be designed to satisfy a set of transportation requests, each involving a pickup and delivery location, under capacity, time window, and precedence constraints. This paper introduces a new branch-and-cut-and-price algorithm in which lower bounds are computed by solving through column generation the linear programming relaxation of a set partitioning formulation. Two pricing subproblems are considered in the column generation algorithm: an elementary and nonelementary shortest path problem. Valid inequalities are added dynamically to strengthen the relaxations. Some of the previously proposed inequalities for the pickup and delivery problem with time windows are also shown to be implied by the set partitioning formulation. Computational experiments indicate that the proposed algorithm outperforms a recent branch-and-cut algorithm.

Background Distance functions are fundamental for evaluating the differences between gene expression profiles. Such a function would output a low value if the profiles are strongly correlated—either negatively or positively—and vice versa. One popular distance function is the absolute correlation distance, d a = 1 - | ρ | , where ρ is similarity measure, such as Pearson or Spearman correlation. However, the absolute correlation distance fails to fulfill the triangle inequality, which would have guaranteed better performance at vector quantization, allowed fast data localization, as well as accelerated data clustering. Results In this work, we propose d r = 1 - | ρ | as an alternative. We prove that d r satisfies the triangle inequality when ρ represents Pearson correlation, Spearman correlation, or Cosine similarity. We show d r to be better than d s = 1 - ρ 2 , another variant of d a that satisfies the triangle inequality, both analytically as well as experimentally. We empirically compared d r with d a in gene clustering and sample clustering experiment by real-world biological data. The two distances performed similarly in both gene clustering and sample clustering in hierarchical clustering and PAM (partitioning around medoids) clustering. However, d r demonstrated more robust clustering. According to the bootstrap experiment, d r generated more robust sample pair partition more frequently ( P -value < 0.05 ). The statistics on the time a class “dissolved” also support the advantage of d r in robustness. Conclusion d r , as a variant of absolute correlation distance, satisfies the triangle inequality and is capable for more robust clustering.

Kinematics remains one of the cornerstones of robotics, and over the decade, Robotica has been one of the venues in which groundbreaking work in kinematics has always been welcome. A number of works in the kinematics community have addressed metrics for rigid-body motions in multiple different venues. An essential feature of any distance metric is the triangle inequality. Here, relationships between the triangle inequality for kinematic metrics and so-called trace inequalities are established. In particular, we show that the Golden-Thompson inequality (a particular trace inequality from the field of statistical mechanics) which holds for Hermitian matrices remarkably also holds for restricted classes of real skew-symmetric matrices. We then show that this is related to the triangle inequality for $SO(3)$ and $SO(4)$ metrics.

We show that if A ⊂ {1,...,N} contains no nontrivial three-term arithmetic progressions then ∥A∥ = O(N/log 1−o(1) N).

This paper investigates the perimetric contraction principle for quadrilaterals, a four-point extension of the Banach contraction principle, within the framework of semimetric spaces using triangle functions introduced by M. Bessenyei and Zs. Páles. We provide new insights into the fixed point theorem for perimetric contractions on quadrilaterals, demonstrating its applicability beyond metric spaces to include ultrametric spaces and distance spaces with power triangle functions.

Quantum correlations are subject to certain distribution rules represented by so-called monogamy relations. Derivation of monogamy relations for multipartite systems is a non-trivial problem, as the multipartite correlations reveal their behaviors in a way different from bipartite systems. We here show that simple geometric properties of probabilistic spaces, in conjunction with no-signaling principle, lead to genuine monogamy relations for a large class of Bell type inequalities for many qubits. The term of ‘genuine’ implies that only one out of N Bell inequalities exhibits a quantum violation. We also generalize our method to qudits. Using the similar geometric approach with a quasi-distance employed, we derive Svetlichny–Zohren–Gill type Bell inequalities for d -dimensional tripartite systems, and show their monogamous nature.