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We present new designs for quantum random access memory. More precisely, for each function, $f : \mathbb{F}_2^n \rightarrow \mathbb{F}_2^d$, we construct oracles, $\mathcal{O}_f$, with the property \begin{equation} \mathcal{O}_f \left| x \right\rangle_n \left| 0 \right\rangle_d = \left| x \right\rangle_n \left| f(x) \right\rangle_d. \end{equation} Our methods are based on the Walsh-Hadamard Transform of $f$, viewed as an integer valued function. In general, the complexity of our method scales with the sparsity of the Walsh-Hadamard Transform and not the sparsity of $f$, yielding more favorable constructions in cases such as binary optimization problems and function with low-degree Walsh-Hadamard Transforms. Furthermore, our design comes with a tuneable amount of ancillas that can trade depth for size. In the ancilla-free design, these oracles can be $\epsilon$-approximated so that the Clifford + $T$ depth is $O \left( \left( n + \log_2 \left( \tfrac{d}{\epsilon} \right) \right) \mathcal{W}_f \right)$, where $\mathcal{W}_f$ is the number of nonzero components in the Walsh-Hadamard Transform. The depth of the shallowest version is $O \left( n + \log_2 \left( \tfrac{d}{\epsilon} \right) \right)$, using $n + d \mathcal{W}_f$ qubit. The connectivity of these circuits is also only logarithmic in $\mathcal{W}_f$. As an application, we show that for boolean functions with low approximate degrees (as in the case of read-once formulas) the complexities of the corresponding QRAM oracles scale only as $2^{\widetilde{O} \left( \sqrt{n} \log_2 \left( n \right) \right)}$., Comment: 18 pages, 1 figures. Comments are welcome!

We study a Grover-type method for Quadratic Binary Optimization problems. In the unconstrained (QUBO) case, for an $n$-dimensional problem with $m$ nonzero terms, we construct a marker oracle for such problems with a tuneable parameter, $\Lambda \in \left[ 1, m \right] \cap \mathbb{Z}$. At $d \in \mathbb{Z}_+$ precision, the oracle uses $O \left( n + \Lambda d \right)$ qubits, has total depth $O \left( \tfrac{m}{\Lambda} \log_2 \left( n \right) + \log_2 \left( d \right) \right)$, and non-Clifford depth of $O \left( \tfrac{m}{\Lambda} \right)$. Moreover, each qubit required to be connected to at most $O \left( \log_2 \left( \Lambda + d \right) \right)$ other qubits. In the case of a maximal graph cuts, as $d = 2 \log_2 \left( n \right)$ always suffices, the depth of the marker oracle can be made as shallow as $O \left( \log_2 \left( n \right) \right)$. For all values of $\Lambda$, the non-Clifford gate count of these oracles is strictly lower (by a factor of $\sim 2$) than previous constructions. We then introduce a novel \emph{Fixed-point Grover Adaptive Search for QUBO Problems}, using our oracle design and a hybrid Fixed-point Grover Search of Li et al. This method has better performance guarantees than previous Grover Adaptive Search methods. Finally, we give a heuristic argument that, with high probability and in $O \left( \tfrac{\log_2 \left( n \right)}{\sqrt{\epsilon}} \right)$ time, this adaptive method finds a configuration that is among the best $\epsilon 2^n$ ones., Comment: 15 pages; Results substantially improved. Submitted. Comments are welcome!

High-end components that conduct complicated tasks automatically are a part of modern industrial systems. However, in order for these parts to function at the desired level, they need to be maintained by qualified experts. Solutions based on Augmented Reality (AR) have been established with the goal of raising production rates and quality while lowering maintenance costs. With the introduction of two unique interaction interfaces based on wearable targets and human face orientation, we are proposing hands-free advanced interactive solutions in this study with the goal of reducing the bias towards certain users. Using traditional devices in real time, a comparison investigation using alternative interaction interfaces is conducted. The suggested solutions are supported by various AI powered methods such as novel gravity-map based motion adjustment that is made possible by predictive deep models that reduce the bias of traditional hand- or finger-based interaction interfaces

Augmented Reality (AR) solutions are providing tools that could improve applications in the medical and industrial fields. Augmentation can provide additional information in training, visualization, and work scenarios, to increase efficiency, reliability, and safety, while improving communication with other devices and systems on the network. Unfortunately, tasks in these fields often require both hands to execute, reducing the variety of input methods suitable to control AR applications. People with certain physical disabilities, where they are not able to use their hands, are also negatively impacted when using these devices. The goal of this work is to provide novel hand-free interfacing methods, using AR technology, in association with AI support approaches to produce an improved Human-Computer interaction solution.

We construct nonminimal and irreducible solutions to the Ginzburg-Landau equations on closed manifolds of arbitrary dimension with trivial first real cohomology. Our method uses bifurcation theory where the "bifurcation points" are characterized by the eigenvalues of a Laplace-type operator. To our knowledge these are the first such examples on nontrivial line bundles., Comment: 15 pages, no figures, accepted version, comments are welcome! To appear in the Proceedings of the AMS. arXiv admin note: text overlap with arXiv:2103.05613

Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, we study the noncommutative Bloch transform of Fuchsian groups that we call the hyperbolic Bloch transform. First, we prove that the hyperbolic Bloch transform is injective and "asymptotically unitary" already in the simplest case, that is when the Hilbert space is the regular representation of the Fuchsian group, $\Gamma$. Second, when $\Gamma \subset \mathrm{PSU} (1, 1)$ acts isometrically on the hyperbolic plane, $\mathbb{H}$, and the Hilbert space is $L^2 \left( \mathbb{H} \right)$, then we define a modified, geometric Bloch transform, that sends wave functions to sections of stable, flat bundles over $\Sigma = \mathbb{H} / \Gamma$ and transforms the hyperbolic Laplacian into the covariant Laplacian., Comment: 20 pages, no figures. Comments are welcome!

We produce finite energy BPS monopoles with prescribed arbitrary symmetry breaking from a new class of solutions to Nahm's equation., Comment: 30 pages, no figures. Comments are welcome!

We study a canonical class of perturbations of Dirac operators that are defined in any dimension and on any Hermitian Clifford module bundle. These operators generalize the 2-dimensional Jackiw-Rossi operator, which describes electronic excitations on topological superconductors. We also describe the low energy spectrum of these operators on complete surfaces, under mild hypotheses., Comment: 33 pages, no figures. Typo fixes and citations added since the last version. Submitted. Comments are still welcome!

We study stationary solutions to the Keller--Segel equation on curved planes. We prove the necessity of the mass being $8 \pi$ and a sharp decay bound. Notably, our results do not require the solutions to have a finite second moment, and thus are novel already in the flat case. Furthermore, we provide a correspondence between stationary solutions to the static Keller--Segel equation on curved planes and positively curved Riemannian metrics on the sphere. We use this duality to show the nonexistence of solutions in certain situations. In particular, we show the existence of metrics, arbitrarily close to the flat one on the plane, that do not support stationary solutions to the static Keller--Segel equation (with any mass). Finally, as a complementary result, we prove a curved version of the logarithmic Hardy--Littlewood--Sobolev inequality and use it to show that the Keller--Segel free energy is bounded from below exactly when the mass is $8 \pi$, even in the curved case., Comment: 15 pages, no figures. Accepted by Proceedings of the Royal Society of Edinburgh Section A

We prove the existence of novel, nonminimal and irreducible solutions to the (self-dual) Ginzburg-Landau equations on closed surfaces. To our knowledge these are the first such examples on nontrivial line bundles, that is, with nonzero total magnetic flux. Our method works with the 2-dimensional, critically coupled Ginzburg-Landau theory and uses the topology of the moduli space. The method is nonconstructive, but works for all values of the remaining coupling constant. We also prove the instability of these solutions., Comment: 20 pages, no figures. Submitted version. Comments are welcome!

We study solutions to Nahm's equations with continuous symmetries and, under certain (mild) hypotheses, we classify the corresponding Ans\"atze. Using our classification, we construct novel Nahm data, and prescribe methods for generating further solutions. Finally, we use these results to construct new BPS monopoles with spherical symmetry., Comment: 37 pages, 1 table, 2 figures. Theorem 4.1 improved

This article investigates the asymptotics of $\rm{G}_2$-monopoles. First, we prove that when the underlying $\rm{G}_2$-manifold is nonparabolic (i.e. admits a positive Green's function), finite intermediate energy monopoles with bounded curvature have finite mass. The second main result restricts to the case when the underlying $\rm{G}_2$-manifold is asymptotically conical. In this situation, we deduce sharp decay estimates and that the connection converges, along the end, to a pseudo-Hermitian--Yang--Mills connection over the asymptotic cone. Finally, our last result exhibits a Fredholm setup describing the moduli space of finite intermediate energy monopoles on an asymptotically conical $\rm{G}_2$-manifold., Comment: 83 pages. v4: Corrected minor mistakes/typos on the Bochner--Weitzenb\"ock formulas of Lemmas 5.2, 5.3, 5.5 and 9.1. In particular, this led to a restatement of both Corollary 5.4 and Proposition 6.3, and their proofs were rewritten. Final version to appear in the Memoirs of the American Mathematical Society

We study solutions to the Kapustin--Witten equations on ALE and ALF gravitational instantons. On any such space and for any compact structure group, we prove asymptotic estimates for the Higgs field. We then use it to prove a vanishing theorem in the case when the underlying manifold is $\mathrm{R}^4$ or $\mathrm{R}^3 \times \mathbb{S}^1$ and the structure group is $\mathrm{SU} (2)$., Comment: 12 pages, no figures. A mistake in the main theorem is fixed; supersedes published version

We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator; these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to three dimensions, thus we call them Haydys monopoles. We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on $\mathbb{R}^3$ is a K\"ahler manifold containing the ordinary Bogomolny moduli space as a minimal Lagrangian submanifold -- an $A$-brane. Moreover, using a gluing construction we construct an open neighborhood of this submanifold modeled on a neighborhood of the zero section in the tangent bundle to the Bogomolny moduli space. This is analogous to the case of Higgs bundles over a Riemann surface, where the (co)tangent bundle of holomorphic bundles canonically embeds into the Hitchin moduli space. These results contrast immensely with the case of finite energy Kapustin--Witten monopoles for which we have shown a vanishing theorem in [12]., Comment: 28 pages, no figures. Published version (with a few extra typo/reference fixes)

We consider the speed planning problem for a robotic manipulator. In particular, we present an algorithm for finding the time-optimal speed law along an assigned path that satisfies velocity and acceleration constraints and respects the maximum forces and torques allowed by the actuators. The addressed optimization problem is a finite dimensional reformulation of the continuous-time speed optimization problem, obtained by discretizing the speed profile with N points. The proposed algorithm has linear complexity with respect to N and to the number of degrees of freedom. Such complexity is the best possible for this problem. Numerical tests show that the proposed algorithm is significantly faster than algorithms already existing in literature.

The first irreducible solution of the $\mathrm{SU} (2)$ self-duality equations on the Euclidean Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on $\mathbb{R}^4$ were discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on $\mathbb{R}^4$, the case of the Euclidean Schwarzschild manifold has resisted many efforts for the past 40 years. By exploring a correspondence between the planar Abelian vortices and spherically symmetric instantons on ES, we obtain: a complete description of a connected component of the moduli space of unit energy $\mathrm{SU} (2)$ instantons; new examples of instantons with non-integer energy (and non-trivial holonomy at infinity); a complete classification of finite energy, spherically symmetric, $\mathrm{SU} (2)$ instantons. As opposed to the previously known solutions, the generic instanton coming from our construction is not invariant under the full isometry group, in particular not static. Hence disproving a conjecture of Tekin., Comment: 37 pages, no figures. Submitted version

Ginzburg-Landau fields are the solutions of the Ginzburg-Landau equations which depend on two positive parameters, $\alpha$ and $\beta$. We give conditions on $\alpha$ and $\beta$ for the existence of irreducible solutions of these equations. Our results hold for arbitrary compact, oriented, Riemannian 2-manifolds (for example, bounded domains in $\mathbb{R}^2$, spheres, tori, etc.) with de Gennes-Neumann boundary conditions. We also prove that, for each such manifold and all positive $\alpha$ and $\beta$, the Ginzburg-Landau free energy is a Palais-Smale function on the space of gauge equivalence classes, Ginzburg-Landau fields exist for only a finite set of energy values, and the moduli space of Ginzburg-Landau fields is compact., Comment: 16 pages, 1 figure

We analyze 2-dimensional Ginzburg-Landau vortices at critical coupling, and establish asymptotic formulas for the tangent vectors of the vortex moduli space using theorems of Taubes and Bradlow. We then compute the corresponding Berry curvature and holonomy in the large volume limit., Comment: 28 pages, 2 figures, published version

Definition of the partition function of U(1) gauge theory is extended to a class of four-manifolds containing all compact spaces and certain asymptotically locally flat (ALF) ones including the multi-Taub--NUT spaces. The partition function is calculated via zeta-function regularization with special attention to its modular properties. In the compact case, compared with the purely topological result of Witten, we find a non-trivial curvature correction to the modular weights of the partition function. But S-duality can be restored by adding gravitational counter terms to the Lagrangian in the usual way. In the ALF case however we encounter non-trivial difficulties stemming from original non-compact ALF phenomena. Fortunately our careful definition of the partition function makes it possible to circumnavigate them and conclude that the partition function has the same modular properties as in the compact case., Comment: LaTeX; 22 pages, no figures