Twistor constructions for higher-spin extensions of (self-dual) Yang-Mills

We present the inverse Penrose transform (the map from spacetime to twistor space) for self-dual Yang-Mills (SDYM) and its higher-spin extensions on a flat background. The twistor action for the higher-spin extension of SDYM (HS-SDYM) is of $$ \mathcal{BF} $$ BF -type. By considering a deformation away from the self-dual sector of HS-SDYM, we discover a new action that describes a higher-spin extension of Yang-Mills theory (HS-YM). The twistor action for HS-YM is a straightforward generalization of the Yang-Mills one.

Irregular perverse sheaves

We introduce irregular constructible sheaves, which are ${\mathbb {C}}$-constructible with coefficients in a finite version of the Novikov ring $\Lambda$ and special gradings. We show that the bounded derived category of cohomologically irregular constructible complexes is equivalent to the bounded derived category of holonomic ${\mathcal {D}}$-modules by a modification of D’Agnolo and Kashiwara's irregular Riemann–Hilbert correspondence. The bounded derived category of cohomologically irregular constructible complexes is equipped with the irregular perverse $t$-structure, which is a straightforward generalization of usual perverse $t$-structure, and we prove that its heart is equivalent to the abelian category of holonomic ${\mathcal {D}}$-modules. We also develop the algebraic version of the theory.

Finsler spacetime geometry in physics

Finsler geometry naturally appears in the description of various physical systems. In this review, I divide the emergence of Finsler geometry in physics into three categories: dual description of dispersion relations, most general geometric clock and geometry being compatible with the relevant Ehlers–Pirani–Schild axioms. As Finsler geometry is a straightforward generalization of Riemannian geometry there are many attempts to use it as generalized geometry of spacetime in physics. However, this generalization is subtle due to the existence of non-trivial null directions. I review how a pseudo-Finsler spacetime geometry can be defined such that it provides a precise notion of causal curves, observers and their measurements as well as a gravitational field equation determining the Finslerian spacetime geometry dynamically. The construction of such Finsler spacetimes lays the foundation for comparing their predictions with observations, in astrophysics as well as in laboratory experiments.

Recognizable languages of arrows and cospans

In this article, we generalize Courcelle's recognizable graph languages and results on monadic second-order logic to more general structures.First, we give a category-theoretical characterization of recognizability. A recognizable subset of arrows in a category is defined via a functor into the category of relations on finite sets. This can be seen as a straightforward generalization of finite automata. We show that our notion corresponds to recognizable graph languages if we apply the theory to the category of cospans of graphs.In the second part of the paper, we introduce a simple logic that allows to quantify over the subobjects of a categorical object. Again, we show that, for the category of graphs, this logic is equally expressive as monadic second-order graph logic (msogl). Furthermore, we show that in the more general setting of hereditary pushout categories, a class of categories closely related to adhesive categories, we can recover Courcelle's result that everymsogl-expressible property is recognizable. This is done by giving an inductive translation of formulas of our logic into automaton functors.

The GKLS Master Equation in High Energy Physics

Utilizing the GKLS master equation we show that the decay property of a particle can be straightforwardly incorporated. In standard particle physics the decay is often described by an efficient non-hermitian Hamiltonian, in accord with the seminal Wigner-Weisskopf approximation. We show that by enlarging the Hilbert space and defining specific GKLS operators we have attained a formalism with a hermitian Hamiltonian and probability conserving states. This proves that the Wigner-Weisskopf approximation is Markovian and completely positive. In addition, this formalism allows a straightforward generalization to many-particle decays. Last, but not least, some impacts of the GKLS master equation onto systems at high energies are reported, such as for neutral meson, neutrino and hyperon systems.

Time-Bounded Best-First Search for Reversible and Non-reversible Search Graphs

Time-Bounded A* is a real-time, single-agent, deterministic search algorithm that expands states of a graph in the same order as A* does, but that unlike A* interleaves search and action execution. Known to outperform state-of-the-art real-time search algorithms based on Korf's Learning Real-Time A* (LRTA*) in some benchmarks, it has not been studied in detail and is sometimes not considered as a ``true'' real-time search algorithm since it fails in non-reversible problems even it the goal is still reachable from the current state. In this paper we propose and study Time-Bounded Best-First Search (TB(BFS)) a straightforward generalization of the time-bounded approach to any best-first search algorithm. Furthermore, we propose Restarting Time-Bounded Weighted A* (TB_R(WA*)), an algorithm that deals more adequately with non-reversible search graphs, eliminating ``backtracking moves'' and incorporating search restarts and heuristic learning. In non-reversible problems we prove that TB(BFS) terminates and we deduce cost bounds for the solutions returned by Time-Bounded Weighted A* (TB(WA*)), an instance of TB(BFS). Furthermore, we prove TB_R(WA*), under reasonable conditions, terminates. We evaluate TB(WA) in both grid pathfinding and the 15-puzzle. In addition, we evaluate TB_R(WA*) on the racetrack problem. We compare our algorithms to LSS-LRTWA*, a variant of LRTA* that can exploit lookahead search and a weighted heuristic. A general observation is that the performance of both TB(WA*) and TB_R(WA*) improves as the weight parameter is increased. In addition, our time-bounded algorithms almost always outperform LSS-LRTWA* by a significant margin.

A note on CVA and wrong way risk

Hull and White approach to Wrong Way Risk in the computation of Credit Value Adjustment (CVA) is considered the most straightforward generalization of the standard Basel approach. The model is financially intuitive and it can be implemented by a slight modification of existing algorithms for CVA calculation. However, path dependency in the key quantities has non-elementary consequences in the calibration of model parameters. We propose a simple and fast approach for computing these quantities via a recursion formula. We show in detail the calibration methodology on market data and CVA computations in two relevant cases: a FX forward and an interest rate swap.

A Class of Bistochastic Positive Optimal Maps in Md(ℂ)

We provide a straightforward generalization of a positive map in [Formula: see text] considered recently by Miller and Olkiewicz [5]. It is proved that these maps are optimal and indecomposable. As a byproduct we provide a class of PPT entangled states in [Formula: see text].

Partitioning of diversity: the "within communities" component

It is routinely understood that the total diversity within a metacommunity (γ-diversity) can be partitioned into one component summarizing the diversity within communities (α-diversity) and a second component representing the contribution of diversity (or differences) between communities (β-diversity). The underlying thought is that merging differentiated communities should raise the total diversity above the average level of diversity within the communities. The crucial point in this partitioning criterion is set by the notion of "diversity within communities" (DWC) and its relation to the total diversity. The common approach to summarizing DWC is in terms of averages. Yet there are many different ways to average diversity, and not all of these averages stay below the total diversity for every measure of diversity, corrupting the partitioning criterion. This raises the question of whether conceptual properties of diversity measures exist, the fulfillment of which implies that all measures of DWC obey the partitioning criterion. It is shown that the straightforward generalization of the plain counting of types (richness) leads to a generic diversity measure that has the desired properties and, together with its effective numbers, fulfills the partitioning criterion for virtually all of the relevant diversity measures in use. It turns out that the classical focus on DWC (α) and its complement (β as derived from α and γ) in the partitioning of total diversity captures only the apportionment perspective of the distribution of trait diversity over communities (which implies monomorphism within communities at the extreme). The other perspective, differentiation, cannot be assessed appropriately unless an additional level of diversity is introduced that accounts for differences between communities (such as the joint "type-community diversity"). Indices of apportionment IA (among which is GST and specially normalized versions of β) and differentiation ID are inferred, and it is demonstrated that conclusions derived from IA depend considerably on the measure of diversity to which it is applied, and that in most cases an assessment of the distribution of diversity over communities requires additional computation of ID.

Hypermap-homology quantum codes

We introduce a new type of sparse CSS quantum error correcting code based on the homology of hypermaps. Sparse quantum error correcting codes are of interest in the building of quantum computers due to their ease of implementation and the possibility of developing fast decoders for them. Codes based on the homology of embeddings of graphs, such as Kitaev's toric code, have been discussed widely in the literature and our class of codes generalize these. We use embedded hypergraphs, which are a generalization of graphs that can have edges connected to more than two vertices. We develop theorems and examples of our hypermap-homology codes, especially in the case that we choose a special type of basis in our homology chain complex. In particular the most straightforward generalization of the m × m toric code to hypermap-homology codes gives us a [(3/2)m2, 2, m] code as compared to the toric code which is a [2m2, 2, m] code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.