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Integrated entangled photon-pair sources are key elements for enabling large-scale quantum photonic solutions, and addresses the challenges of both scaling-up and stability. Here we report the first demonstration of an energy-time entangled…
For any three nonzero vectors $a,b,c$ in $\mathbb R^2$, we obtain a necessary and sufficient condition for the sum of the three pairwise angles between these vectors to equal $2\pi$. As an easy consequence of this, a proof of Euclid's…
For a superelliptic curve $\mathcal X$, defined over $\mathbb Q$, let $\mathfrak p$ denote the corresponding moduli point in the weighted moduli space. We describe a method how to determine a minimal integral model of $\mathcal X$ such…
Let n be a positive integer and t a non-zero integer. We consider the elliptic curve over Q given by E : y 2 = x 3 + tx 2 -- n 2 (t + 3n 2)x + n 6. It is a special case of an elliptic surface studied recently by Bettin, David and Delaunay…
We discuss photon number bounds for a system of non-relativistic particles coupled to the quantized electromagnetic field (non-relativistic QED), below the ionization threshold. Such a bound was assumed in the proof of asymptotic…
We present a systematic framework to quantify the interplay between coherence and wave-particle duality in generic two-path interference systems. Our analysis reveals a closed-form duality ellipse (DE) equality, that rigorously unifies…
We present a relativistic formulation of pion loop corrections to the coupling of photons with nucleons on the light-front. Vertex and wave function renormalization constants are computed to lowest order in the pion field, including their…
We study surfaces in Euclidean space ${\mathbb R}^3$ that are minimal for a log-linear density $\phi(x,y,z)=\alpha x+\beta y+\gamma y$, where $\alpha,\beta,\gamma$ are real numbers not all zero. We prove that if a surface is $\phi$-minimal…
Modular curves like X_0(N) and X_1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL_2(Z), they allow for a more arithmetic description as…
Image inpainting involves filling in damaged or missing regions of an image by utilizing information from the surrounding areas. In this paper, we investigate a highly nonlinear partial differential equation inspired by the modified…
We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equations in two steps. We first obtain a numerical approximation…
Compared to classical optical coherence theory based on Maxwell's electromagnetic theory and Glauber's quantum optical coherence theory based on matrix mechanics formulation of quantum mechanics, quantum optical coherence theory based on…
We get new results (and rederive some know ones) on smooth surfaces in $\mathbb{R}^n$ by unifying several view points into a coherent general view. Namely, we show and use new relations of the evolute (caustic) with the curvature ellipse,…
We present a novel method for deciding whether a given n-dimensional ellipsoid contains another one (possibly with a different center). This method consists in constructing a particular concave function and deciding whether it has any value…
Infinitely many elliptic curves over ${\bf Q}$ have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect in their subgroups of order 2. Let $N_i(X)$ denote the number of elliptic curves over ${\bf Q}$…
Let $P$ be an arbitrary point on an elliptic curve over the complex numbers of the form $y^2=x^3+a_4\,x+a_6$ or of the form $y^2=x^3+a_2\,x^2+a_4\,x$. We provide explicit formulae to compute the points $P/2$, i.e., the points $Q$ such that…
We calculate $p_T$ spectra and elliptic flow for tip-tip and body-body configurations of full overlap uranium-uranium (U+U) collisions using a hydrodynamic model with smooth initial density distribution and compare the results with those…
The modular degree m_E of an elliptic curve E/Q is the minimal degree of any surjective morphism X_0(N) -> E, where N is the conductor of E. We give a necessarily set of criteria for m_E to be odd. Specializing to N prime our results imply…
In this paper we give an affirmative answer to the Euclidean analogue of a question of Bourgain and Brezis concerning the optimal Lorentz estimate for a Div-Curl system: The function $Z=\operatorname*{curl} (-\Delta)^{-1} F$ satisfies…
Let $\Phi\in\mathbb{R}^{m\times n}$ be a sparse Johnson-Lindenstrauss transform [KN14] with $s$ non-zeroes per column. For a subset $T$ of the unit sphere, $\varepsilon\in(0,1/2)$ given, we study settings for $m,s$ required to ensure $$…