Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits

We consider a family of explicit Runge–Kutta pairs of orders six and five without any additional property (reduced truncation errors, Hamiltonian preservation, symplecticness, etc.). This family offers five parameters that someone chooses freely. Then, we train them in order for the presented method to furnish the best results on a couple of Kepler orbits, a certain interval and tolerance. Consequently, we observe an efficient performance on a wide range of orbital problems (i.e., Kepler for a variety of eccentricities, perturbed Kepler with various disturbances, Arenstorf and Pleiades). About 1.8 digits of accuracy is gained on average over conventional pairs, which is truly remarkable for methods coming from the same family and order.

Explainer: Taxing rezoning windfalls (betterment)

In its 2021 budget, the Victorian government announced a new tax on windfall land value gains from rezoning (also known as a betterment tax)This note explains the economic principles behind such a tax, the benefits of applying such a tax, implementation issues that need to be considered, and lessons from the operation of similar taxes elsewhere.Property is, conceptually, a finite bundle of rights. Rezoning grants additional property rights to owners of an existing set of property rights. Those new rights could instead be sold at a market price. A tax on the value gain from rezoning at anything less than 100% is equivalent to selling the new property rights from the community to the current property owner at a discount. Just like selling other property rights from the public to the private sector does not add to market prices in property markets, nor does selling rezoning rights.A tax on rezoning windfalls is uncommon not because it is a bad tax but because it is a good tax.

On Fractional Fragility Rates of Graph Classes

We consider, for every positive integer $a$, probability distributions on subsets of vertices of a graph with the property that every vertex belongs to the random set sampled from this distribution with probability at most $1/a$. Among other results, we prove that for every positive integer $a$ and every planar graph $G$, there exists such a probability distribution with the additional property that for any set $X$ in the support of the distribution, the graph $G-X$ has component-size at most $(\Delta(G)-1)^{a+O(\sqrt{a})}$, or treedepth at most $O(a^3\log_2(a))$. We also provide nearly-matching lower bounds.

Weak approximate unitary designs and applications to quantum encryption

Unitary t-designs are the bread and butter of quantum information theory and beyond. An important issue in practice is that of efficiently constructing good approximations of such unitary t-designs. Building on results by Aubrun (Comm. Math. Phys. 2009), we prove that sampling dtpoly(t,log⁡d,1/ϵ) unitaries from an exact t-design provides with positive probability an ϵ-approximate t-design, if the error is measured in one-to-one norm. As an application, we give a randomized construction of a quantum encryption scheme that has roughly the same key size and security as the quantum one-time pad, but possesses the additional property of being non-malleable against adversaries without quantum side information.

Extremal Square-Free Words

A word is square-free if it does not contain nonempty factors of the form $XX$. In 1906 Thue proved that there exist arbitrarily long square-free words over a $3$-letter alphabet. We consider a new type of square-free words with additional property. A square-free word is called extremal if it cannot be extended to a new square-free word by inserting a single letter at any position. We prove that there exist infinitely many square-free extremal words over a $3$-letter alphabet. Some parts of our construction relies on computer verifications. It is not known if there exist any extremal square-free words over a $4$-letter alphabet.

A Private Quantum Bit String Commitment

We propose an entanglement-based quantum bit string commitment protocol whose composability is proven in the random oracle model. This protocol has the additional property of preserving the privacy of the committed message. Even though this property is not resilient against man-in-the-middle attacks, this threat can be circumvented by considering that the parties communicate through an authenticated channel. The protocol remains secure and private (but not composable) if we realize the random oracles as physical unclonable functions (PUFs) in the so-called bad PUF model.

Total Roman {3}-domination in Graphs

For a graph G = ( V , E ) with vertex set V = V ( G ) and edge set E = E ( G ) , a Roman { 3 } -dominating function (R { 3 } -DF) is a function f : V ( G ) → { 0 , 1 , 2 , 3 } having the property that ∑ u ∈ N G ( v ) f ( u ) ≥ 3 , if f ( v ) = 0 , and ∑ u ∈ N G ( v ) f ( u ) ≥ 2 , if f ( v ) = 1 for any vertex v ∈ V ( G ) . The weight of a Roman { 3 } -dominating function f is the sum f ( V ) = ∑ v ∈ V ( G ) f ( v ) and the minimum weight of a Roman { 3 } -dominating function on G is the Roman { 3 } -domination number of G, denoted by γ { R 3 } ( G ) . Let G be a graph with no isolated vertices. The total Roman { 3 } -dominating function on G is an R { 3 } -DF f on G with the additional property that every vertex v ∈ V with f ( v ) ≠ 0 has a neighbor w with f ( w ) ≠ 0 . The minimum weight of a total Roman { 3 } -dominating function on G, is called the total Roman { 3 } -domination number denoted by γ t { R 3 } ( G ) . We initiate the study of total Roman { 3 } -domination and show its relationship to other domination parameters. We present an upper bound on the total Roman { 3 } -domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. Finally, we investigate the complexity of total Roman { 3 } -domination for bipartite graphs.

Column-orthogonal strong orthogonal arrays of strength two plus and three minus

Summary Strong orthogonal arrays have better space-filling properties than ordinary orthogonal arrays for computer experiments. We consider column-orthogonal strong orthogonal arrays of strength two plus and three minus, and present methods of constructing such designs. Several situations are examined, including those of four or higher levels and mixed levels. The methods are based on both regular and nonregular designs. The resulting designs inherit the good property of strong orthogonal arrays of strength two plus or three and have the additional property of column orthogonality. This type of design is a better choice for computer experiments.

Spanning Properties of Yao and 𝜃-Graphs in the Presence of Constraints

We present improved upper bounds on the spanning ratio of constrained [Formula: see text]-graphs with at least 6 cones and constrained Yao-graphs with 5 or at least 7 cones. Given a set of points in the plane, a Yao-graph partitions the plane around each vertex into [Formula: see text] disjoint cones, each having aperture [Formula: see text], and adds an edge to the closest vertex in each cone. Constrained Yao-graphs have the additional property that no edge properly intersects any of the given line segment constraints. Constrained [Formula: see text]-graphs are similar to constrained Yao-graphs, but use a different method to determine the closest vertex. We present tight bounds on the spanning ratio of a large family of constrained [Formula: see text]-graphs. We show that constrained [Formula: see text]-graphs with [Formula: see text] ([Formula: see text] and integer) cones have a tight spanning ratio of [Formula: see text], where [Formula: see text] is [Formula: see text]. We also present improved upper bounds on the spanning ratio of the other families of constrained [Formula: see text]-graphs. These bounds match the current upper bounds in the unconstrained setting. We also show that constrained Yao-graphs with an even number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text] and constrained Yao-graphs with an odd number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text]. As is the case with constrained [Formula: see text]-graphs, these bounds match the current upper bounds in the unconstrained setting, which implies that like in the unconstrained setting using more cones can make the spanning ratio worse.

A longitudinal analysis of the effect of public rail infrastructure on proximate residential property transactions

Value capture offers the promise of recouping the additional value that public transportation infrastructure investments confer to local property. The totality of the empirical evidence supports the contention that such investments do indeed add value to proximate residential property. However, little research to estimate just how much any additional property value changes from year to year is evident in the empirical literature. The absence of evidence on the reliability of accessibility premiums is problematic from a policy perspective because transit operators need dedicated sources of funds to compensate for the retrenchment of government sources. Adoption of a multilevel approach expands the temporal scale of analysis to more than a decade and the spatial scale of analysis to an entire heavy rail system, Metro in Washington, District of Columbia (DC), in order to estimate just how much accessibility premiums change year over year. Results advance the state of knowledge on value capture since public rail infrastructure adds property value across an entire heavy rail system, but the additional value is modest and the year-to-year changes are dramatic. Overall, a strategy to capture any additional property value to fund public transportation may technically qualify as a dedicated source of funds, but it is not a reliable source of funds.