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For a fixed integer $r \geq 1$, a distance-$r$ dominating set (D$r$DS) of a graph $G = (V, E)$ is a vertex subset $D \subseteq V$ such that every vertex in $V$ is within distance $r$ from some member of $D$. Given two D$r$DSs $D_s, D_t$ of $G$, the Distance-$r$ Dominating Set Reconfiguration (D$r$DSR) problem asks if there is a sequence of D$r$DSs that transforms $D_s$ into $D_t$ (or vice versa) such that each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. The problem for $r = 1$ has been well-studied in the literature. We consider D$r$DSR for $r \geq 2$ under two well-known reconfiguration rules: Token Jumping ($\mathsf{TJ}$, which involves replacing a member of the current D$r$DS by a non-member) and Token Sliding ($\mathsf{TS}$, which involves replacing a member of the current D$r$DS by an adjacent non-member). It is known that under any of $\mathsf{TS}$ and $\mathsf{TJ}$, the problem on split graphs is $\mathtt{PSPACE}$-complete for $r = 1$. We show that for $r \geq 2$, the problem is in $\mathtt{P}$, resulting in an interesting complexity dichotomy. Along the way, we prove some non-trivial bounds on the length of a shortest reconfiguration sequence on split graphs when $r = 2$ which may be of independent interest. Additionally, we design a linear-time algorithm under $\mathsf{TJ}$ on trees. On the negative side, we show that D$r$DSR for $r \geq 1$ on planar graphs of maximum degree three and bounded bandwidth is $\mathtt{PSPACE}$-complete, improving the degree bound of previously known results. We also show that the known $\mathtt{PSPACE}$-completeness results under $\mathsf{TS}$ and $\mathsf{TJ}$ for $r = 1$ on bipartite graphs and chordal graphs can be extended for $r \geq 2$.

We develop a sparse spectral method for a class of fractional differential equations, posed on $\mathbb{R}$, in one dimension. These equations can include sqrt-Laplacian, Hilbert, derivative and identity terms. The numerical method utilizes a basis consisting of weighted Chebyshev polynomials of the second kind in conjunction with their Hilbert transforms. The former functions are supported on $[-1,1]$ whereas the latter have global support. The global approximation space can contain different affine transformations of the basis, mapping $[-1,1]$ to other intervals. Remarkably, not only are the induced linear systems sparse, but the operator decouples across the different affine transformations. Hence, the solve reduces to solving $K$ independent sparse linear systems of size $\mathcal{O}(n)\times \mathcal{O}(n)$, with $\mathcal{O}(n)$ nonzero entries, where $K$ is the number of different intervals and $n$ is the highest polynomial degree contained in the sum space. This results in an $\mathcal{O}(n)$ complexity solve. Applications to fractional heat and wave equations are considered.

We study the dependent type theory CaTT, introduced by Finster and Mimram, which presents the theory of weak $\omega$-categories, following the idea that type theories can be considered as presentations of generalized algebraic theories. Our main contribution is a formal proof that the models of this type theory correspond precisely to weak $\omega$-categories, as defined by Maltsiniotis, by generalizing a definition proposed by Grothendieck for weak $\omega$-groupoids: Those are defined as suitable presheaves over a cat-coherator, which is a category encoding structure expected to be found in an $\omega$-category. This comparison is established by proving the initiality conjecture for the type theory CaTT, in a way which suggests the possible generalization to a nerve theorem for a certain class of dependent type theories

We give an alternative derivation of $(N,N)$-isogenies between fastKummer surfaces which complements existing works based on the theory oftheta functions. We use this framework to produce explicit formulae for thecase of $N = 3$, and show that the resulting algorithms are more efficient thanall prior $(3, 3)$-isogeny algorithms.

This paper proposes a novel technique for the approximation of strong solutions $u \in C(\overline{\Omega}) \cap W^{2,n}_\mathrm{loc}(\Omega)$ to uniformly elliptic linear PDE of second order in nondivergence form with continuous leading coefficient in nonsmooth domains by finite element methods. These solutions satisfy the Alexandrov-Bakelman-Pucci (ABP) maximum principle, which provides an a~posteriori error control for $C^1$ conforming approximations. By minimizing this residual, we obtain an approximation to the solution $u$ in the $L^\infty$ norm. Although discontinuous functions do not satisfy the ABP maximum principle, this approach extends to nonconforming FEM as well thanks to well-established enrichment operators. Convergence of the proposed FEM is established for uniform mesh-refinements. The built-in a~posteriori error control (even for inexact solve) can be utilized in adaptive computations for the approximation of singular solutions, which performs superiorly in the numerical benchmarks in comparison to the uniform mesh-refining algorithm.

Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the spatial domain. However, obtaining these solutions are often prohibitively costly, limiting the feasibility of exploring parameters in PDEs. In this paper, we propose an efficient emulator that simultaneously predicts the solutions over the spatial domain, with theoretical justification of its uncertainty quantification. The novelty of the proposed method lies in the incorporation of the mesh node coordinates into the statistical model. In particular, the proposed method segments the mesh nodes into multiple clusters via a Dirichlet process prior and fits Gaussian process models with the same hyperparameters in each of them. Most importantly, by revealing the underlying clustering structures, the proposed method can provide valuable insights into qualitative features of the resulting dynamics that can be used to guide further investigations. Real examples are demonstrated to show that our proposed method has smaller prediction errors than its main competitors, with competitive computation time, and identifies interesting clusters of mesh nodes that possess physical significance, such as satisfying boundary conditions. An R package for the proposed methodology is provided in an open repository.

We explicitly construct the first nontrivial extractors for degree $d \ge 2$ polynomial sources over $\mathbb{F}_2^n$. Our extractor requires min-entropy $k\geq n - \tilde{\Omega}(\sqrt{\log n})$. Previously, no constructions were known, even for min-entropy $k\geq n-1$. A key ingredient in our construction is an input reduction lemma, which allows us to assume that any polynomial source with min-entropy $k$ can be generated by $O(k)$ uniformly random bits. We also provide strong formal evidence that polynomial sources are unusually challenging to extract from, by showing that even our most powerful general purpose extractors cannot handle polynomial sources with min-entropy below $k\geq n-o(n)$. In more detail, we show that sumset extractors cannot even disperse from degree $2$ polynomial sources with min-entropy $k\geq n-O(n/\log\log n)$. In fact, this impossibility result even holds for a more specialized family of sources that we introduce, called polynomial non-oblivious bit-fixing (NOBF) sources. Polynomial NOBF sources are a natural new family of algebraic sources that lie at the intersection of polynomial and variety sources, and thus our impossibility result applies to both of these classical settings. This is especially surprising, since we do have variety extractors that slightly beat this barrier - implying that sumset extractors are not a panacea in the world of seedless extraction.

In this paper we develop a classical algorithm of complexity $O(K \, 2^n)$ to simulate parametrized quantum circuits (PQCs) of $n$ qubits, where $K$ is the total number of one-qubit and two-qubit control gates. The algorithm is developed by finding $2$-sparse unitary matrices of order $2^n$ explicitly corresponding to any single-qubit and two-qubit control gates in an $n$-qubit system. Finally, we determine analytical expression of Hamiltonians for any such gate and consequently a local Hamiltonian decomposition of any PQC is obtained. All results are validated with numerical simulations.

In this paper we study the Cayley graph $\mathrm{Cay}(S_n,T)$ of the symmetric group $S_n$ generated by a set of transpositions $T$. We show that for $n\geq 5$ the Cayley graph is normal. As a corollary, we show that its automorphism group is a direct product of $S_n$ and the automorphism group of the transposition graph associated to $T$. This provides an affirmative answer to a conjecture raised by Ganesan in arXiv:1703.08109, showing that $\mathrm{Cay}(S_n,T)$ is normal if and only if the transposition graph is not $C_4$ or $K_n$.

A novel H3N3-2$_\sigma$ interpolation approximation for the Caputo fractional derivative of order $\alpha\in(1,2)$ is derived in this paper, which improves the popular L2C formula with (3-$\alpha$)-order accuracy. By an interpolation technique, the second-order accuracy of the truncation error is skillfully estimated. Based on this formula, a finite difference scheme with second-order accuracy both in time and in space is constructed for the initial-boundary value problem of the time fractional hyperbolic equation. It is well known that the coefficients' properties of discrete fractional derivatives are fundamental to the numerical stability of time fractional differential models. We prove the related properties of the coefficients of the H3N3-2$_\sigma$ approximate formula. With these properties, the numerical stability and convergence of the difference scheme are derived immediately by the energy method in the sense of $H^1$-norm. Considering the weak regularity of the solution to the problem at the starting time, a finite difference scheme on the graded meshes based on H3N3-2$_\sigma$ formula is also presented. The numerical simulations are performed to show the effectiveness of the derived finite difference schemes, in which the fast algorithms are employed to speed up the numerical computation.