Showing papers on "Section (fiber bundle) published in 2020"

Variation of canonical height and equidistribution

Abstract: Let $\pi : E\to B$ be an elliptic surface defined over a number field $K$, where $B$ is a smooth projective curve, and let $P: B \to E$ be a section defined over $K$ with canonical height $\hat{h}_E(P) ot=0$. In this article, we show that the function $t \mapsto \hat{h}_{E_t}(P_t)$ on $B(\overline{K})$ is the height induced from an adelically metrized line bundle with non-negative curvature on $B$. Applying theorems of Thuillier and Yuan, we obtain the equidistribution of points $t \in B(\overline{K})$ where $P_t$ is torsion, and we give an explicit description of the limiting distribution on $B(\mathbb{C})$. Finally, combined with results of Masser and Zannier, we show there is a positive lower bound on the height $\hat{h}_{A_t}(P_t)$, after excluding finitely many points $t \in B$, for any "non-special" section $P$ of a family of abelian varieties $A \to B$ that split as a product of elliptic curves.

Energy and scale dependence of q ̂ and the “JET puzzle”

Abstract: We present an attempt to probe the underlying structure of the quark-gluon plasma (QGP) at high resolution, based on the extracted jet transport coefficient $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}$. We argue that the exchanged momentum $k$ between the hard parton and the medium varies over a range of scales, and for $k\ensuremath{\ge}1\phantom{\rule{0.28em}{0ex}}\mathrm{GeV},\phantom{\rule{0.28em}{0ex}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}$ can be expressed in terms of a parton distribution function (PDF). Because the mass of a QGP constituent is unknown, we define a scaling variable ${x}_{N}$ to represent the ratio of the parton momentum to the momentum of a self-contained section of the plasma which has a mass of 1 GeV. This scaling variable is used to parametrize the QGP-PDF. Calculations based on this reconstructed $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}$ are compared to data sensitive to the hardcore of jets, i.e., the single hadron suppression in terms of the nuclear modification factor ${R}_{AA}$ and the azimuthal anisotropy parameter ${v}_{2}$ as a function of transverse momentum ${p}_{\mathrm{T}}$, centrality, and energy of the collision. It is demonstrated that the scale evolution of the QGP-PDF is responsible for the reduction in the normalization of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}$ between fits to the Relativistic Heavy-Ion Collider and the Large Hadron Collider data; a puzzle, first discovered by the JET Collaboration.

Localized Donaldson-Thomas theory of surfaces

Abstract: Let $S$ be a projective simply connected complex surface and $\mathcal{L}$ be a line bundle on $S$. We study the moduli space of stable compactly supported 2-dimensional sheaves on the total spaces of $\mathcal{L}$. The moduli space admits a $\mathbb{C}^*$-action induced by scaling the fibers of $\mathcal{L}$. We identify certain components of the fixed locus of the moduli space with the moduli space of torsion free sheaves and the nested Hilbert schemes on $S$. We define the localized Donaldson-Thomas invariants of $\mathcal{L}$ by virtual localization in the case that $\mathcal{L}$ twisted by the anti-canonical bundle of $S$ admits a nonzero global section. When $p_g(S)>0$, in combination with Mochizuki's formulas, we are able to express the localized DT invariants in terms of the invariants of the nested Hilbert schemes defined by the authors in [GSY17a], the Seiberg-Witten invariants of $S$, and the integrals over the products of Hilbert schemes of points on $S$. When $\mathcal{L}$ is the canonical bundle of $S$, the Vafa-Witten invariants defined recently by Tanaka-Thomas, can be extracted from these localized DT invariants. VW invariants are expected to have modular properties as predicted by S-duality.

Single-Layer Planar Wideband Rat-Race Coupler Using a Shorted Parallel-Coupled Multi-Line Section

Abstract: In this brief, a novel single-layer planar wideband rat-race coupler using a shorted parallel-coupled multi-line section is demonstrated. The proposed coupler consists of three transmission lines and a shorted parallel-coupled six-line. The performances of wider bandwidths and compact size are achieved just by employing a shorted parallel-coupled multi-line microstrip structure, which has a high coupling factor. Explicit design equations of the proposed coupler were derived using the equivalent approach and the even/odd-mode analysis. For demonstration, a 3-dB broadband rat-race coupler operating at 2.2 GHz was designed. Under the criterion of $|{S}_{11}| dB and $|{S}_{33}| dB, the measured relative bandwidths are 68.84% and 56.67%, respectively. Within the above passband, the isolation ( $|{S}_{31}|$ ) is higher than 20.61 dB. The size is 0.034 ${\lambda }_{\mathrm{ g}}^{2}$ . Good agreement is obtained between the experimental and simulated results.

Global dynamics of a general Lotka-Volterra competition-diffusion system in heterogeneous environments

Abstract: Previously in [ 14 ], we considered a diffusive logistic equation with two parameters, \begin{document}$ r(x) $\end{document} – intrinsic growth rate and \begin{document}$ K(x) $\end{document} – carrying capacity. We investigated and compared two special cases of the way in which \begin{document}$ r(x) $\end{document} and \begin{document}$ K(x) $\end{document} are related for both the logistic equations and the corresponding Lotka-Volterra competition-diffusion systems. In this paper, we continue to study the Lotka-Volterra competition-diffusion system with general intrinsic growth rates and carrying capacities for two competing species in heterogeneous environments. We establish the main result that determines the global dynamics of the system under a general criterion. Furthermore, when the ratios of the intrinsic growth rate to the carrying capacity for each species are proportional — such ratios can also be interpreted as the competition coefficients — this criterion reduces to what we obtained in [ 18 ]. We also study the detailed dynamics in terms of dispersal rates for such general case. On the other hand, when the two ratios are not proportional, our results in [ 14 ] show that the criterion in [ 18 ] cannot be fully recovered as counterexamples exist. This indicates the importance and subtleties of the roles of heterogeneous competition coefficients in the dynamics of the Lotka-Volterra competition-diffusion systems. Our results apply to competition-diffusion-advection systems as well. (See Corollary 5.1 in the last section.)

Spherical Harmonics and Linear Representations of Lie Groups

Abstract: This chapter and the next focus on topics that are somewhat different from the more geometric and algebraic topics discussed in the previous chapters. Indeed, the focus of this chapter is on the types of functions that can be defined on a manifold, the sphere Sn in particular, and this involves some analysis. A main theme of this chapter is to generalize Fourier analysis on the circle to higher dimensional spheres. One of our goals is to understand the structure of the space L2(Sn) of real-valued square integrable functions on the sphere Sn, and its complex analog \(L^2_{\mathbb {C}}(S^n)\). Both are Hilbert spaces if we equip them with the inner product $$\displaystyle \langle f, g\rangle _{S^n} = \int _{S^n} f(t)g(t) \, dt = \int _{S^n}fg\,\mathrm {Vol}_{S^n}, $$ and in the complex case with the Hermitian inner product $$\displaystyle \langle f, g\rangle _{S^n} = \int _{S^n} f(t)\overline {g(t)} \, dt = \int _{S^n}f\overline {g}\,\mathrm {Vol}_{S^n}. $$ This means that if we define the L2-norm associated with the above inner product as \(\left \|f\right \| = \sqrt {\langle f, f\rangle }\), then L2(Sn) and \(L^2_{\mathbb {C}}(S^n)\) are complete normed vector spaces (see Section 7.1 for a review of Hilbert spaces). It turns out that each of L2(Sn) and \(L^2_{\mathbb {C}}(S^n)\) contains a countable family of very nice finite-dimensional subspaces \(\mathcal {H}_k(S^n)\) (and \(\mathcal {H}^{\mathbb {C}}_k(S^n)\)), where \(\mathcal {H}_k(S^n)\) is the space of (real) spherical harmonics on Sn, that is, the restrictions of the harmonic homogeneous polynomials of degree k (in n + 1 real variables) to Sn (and similarly for \(\mathcal {H}^{\mathbb {C}}_k(S^n)\)); these polynomials satisfy the Laplace equation $$\displaystyle \Delta P = 0, $$ where the operator Δ is the (Euclidean) Laplacian, $$\displaystyle \Delta = \frac {\partial ^2}{\partial x_1^2} + \cdots + \frac {\partial ^2}{\partial x_{n+1}^2}. $$

Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations

Abstract: We study the topology and geometry of compact complex manifolds associated to Anosov representations of surface groups and other hyperbolic groups in a complex semisimple Lie group $G$. These manifolds are obtained as quotients of the domains of discontinuity in generalized flag varieties $G/P$ constructed by Kapovich-Leeb-Porti (arXiv:1306.3837), and in some cases by Guichard-Wienhard (arXiv:1108.0733). For $G$-Fuchsian representations and their Anosov deformations, where $G$ is simple, we compute the homology of the domains of discontinuity and of the quotient manifolds. For $G$-Fuchsian and $G$-quasi-Fuchsian representations in simple $G$ of rank at least two, we show that the quotient manifolds are not Kahler. We also describe the Picard groups of these quotient manifolds, compute the cohomology of line bundles on them, and show that for $G$ of sufficiently large rank these manifolds admit nonconstant meromorphic functions. In a final section, we apply our topological results to several explicit families of domains and derive closed formulas for topological invariants in some cases. We also show that the quotient manifold for a $G$-Fuchsian representation in $\mathrm{PSL}_3(\mathbb{C})$ is a fiber bundle over a surface, and we conjecture that this holds for all simple $G$.

A Toeplitz-type operator on Hardy spaces in the unit ball

Abstract: We study a Toeplitz type operator $Q_\mu$ between the holomorphic Hardy spaces $H^p$ and $H^q$ of the unit ball. Here the generating symbol $\mu$ is assumed to a positive Borel measure. This kind of operator is related to many classical mappings acting on Hardy spaces, such as composition operators, the Volterra type integration operators and Carleson embeddings. We completely characterize the boundedness and compactness of $Q_\mu:H^p\to H^q$ for the full range $1

Sterile neutrino search with KATRIN - modeling and design-criteria of a novel detector system

Abstract: A fundamental phenomenon in particle physics is the absence of massive objects in our universe: Dark Matter. A promising candidate that could explain these observations are sterile neutrinos with a mass of several $\mathrm{keV}/c^2$. While it is presumed that sterile neutrinos do not interact via the weak force, they, due to their mass, still partake in neutrino oscillation. Consequently, it is experimentally possible to investigate their imprint in beta-decay experiments, such as the Karlsruhe tritium neutrino experiment (KATRIN). A dedicated search for sterile neutrinos however ensues a steep increase in the electron rate and thus requires the development of a new detector system, the TRISTAN detector. In addition, as the imprint of sterile neutrinos is presumably $<10^{-7}$, systematic uncertainties have to be understood and modeled with high precision. In this thesis systematics prevalent at the detector and spectrometer section of KATRIN will be discussed and their impact to a sterile neutrino sensitivity illuminated. The derived model is compared with data of the current KATRIN detector and with characterization measurements of the first TRISTAN prototype detectors, seven pixel silicon drift detectors. It is shown that the final TRISTAN detector requires a sophisticated redesign of the KATRIN detector section. Moreover, the combined impact of the back-scattering and electron charge-sharing systematic lead to an optimal detector magnetic field of $B_\mathrm{det}=0.7\dots0.8\,\mathrm{T}$, which translates to a pixel radius of $r_\mathrm{px}=1.5\dots1.6\,\mathrm{mm}$. The sensitivity analysis discusses individual effects as well as the combined impact of systematic uncertainties. It is demonstrated that the individual effects can be largely mitigated by shifting the tritium \bd energy spectrum above the \bd endpoint. In contrast, their combined impact to the sensitivity leads to an overall degradation and only mixing amplitudes of $\sin^2\theta_4<3\cdot10^{-6}$ would be reachable, even in an optimized case with very low and homogeneous detection deadlayer $z_\mathrm{dl}=20\pm1\,\mathrm{nm}$. Assessing sterile neutrino mixing amplitudes of $\sin^2\theta_4<10^{-7}$ thus requires disentangling of systematic effects. In a future measurement this could be for example achieved by vetoing detector events with large signal rise-times and small inter-event times.

Miniaturized Single-Ended-to-Balanced Arbitrary Four-Section Coupled-Line Coupler With Inherent Impedance Matching

Abstract: In this brief, a single-ended-to-balanced arbitrary four-section coupled-line coupler with inherent impedance matching is proposed. The proposed coupler consists of four coupled-line sections which achieves a compact circuit layout. Besides, the even- and odd-mode equivalent circuits combining with the standard and mixed-mode ${S}$ -parameters are adopted to derive the analytical equations. For demonstration, two prototypes operating at 2 GHz with different power division ratios ( ${k}=1$ and 3) and different input/output impedances of 50/ $50~{\Omega }$ and 50/ $100~{\Omega }$ , have been fabricated and measured. The sizes of two prototypes are 0.21 ${\lambda }_{\mathrm{ g}}^{2}$ and 0.14 ${\lambda }_{\mathrm{ g}}^{2}$ . A good agreement between the simulated and measured results is observed, which demonstrates the validity of the proposed design method and the circuit model.

The vanishing cohomology of non-isolated hypersurface singularities

Abstract: We employ the perverse vanishing cycles to show that each reduced cohomology group of the Milnor fiber, except the top two, can be computed from the restriction of the vanishing cycle complex to only singular strata with a certain lower bound in dimension. Guided by geometric results, we alternately use the nearby and vanishing cycle functors to derive information about the Milnor fiber cohomology via iterated slicing by generic hyperplanes. These lead to the description of the reduced cohomology groups, except the top two, in terms of the vanishing cohomology of the nearby section. We use it to compute explicitly the lowest (possibly nontrivial) vanishing cohomology group of the Milnor fiber.

Low-Loss Broadband Silicon TM-Pass Polarizer Based on Periodically Structured Waveguides

Abstract: A compact and high performance integrated silicon TM-pass polarizer is proposed and experimentally demonstrated. The device is formed by periodically structuring a waveguide with a slot section within each period, and is implemented on a silicon-on-insulator platform. The fabricated device has low insertion loss for the TM mode (average 0.7dB) and high extinction for the TE mode (average 41.25dB), over a 1.5 $\mu \text{m}$ to 1.6 $\mu \text{m}$ wavelength range. Numerical simulations indicate that a large fraction of the blocked TE mode radiates out of the periodic structure, resulting in weak back reflections. The compact footprint of the device (21 $\mu \text{m}\,\,\times $ 0.5 $\mu \text{m}$ ) makes it suitable for dense integration in photonic integrated circuits.

A Reeb Flow on the Three-Sphere Without a Disk-Like Global Surface of Section

Abstract: We show that there are Reeb flows on the standard, tight three-sphere that do not admit global surfaces of section with one boundary component. In particular, the Reeb flows that we construct do not admit disk-like global surfaces of section. These Reeb flows are constructed using integrable systems, and a connected sum construction that extends the integrable system.

Problem-Complexity Adaptive Model Selection for Stochastic Linear Bandits

Abstract: We consider the problem of model selection for two popular stochastic linear bandit settings, and propose algorithms that adapts to the unknown problem complexity. In the first setting, we consider the $K$ armed mixture bandits, where the mean reward of arm $i \in [K]$, is $\mu_i+ \langle \alpha_{i,t},\theta^* \rangle $, with $\alpha_{i,t} \in \mathbb{R}^d$ being the known context vector and $\mu_i \in [-1,1]$ and $\theta^*$ are unknown parameters. We define $\|\theta^*\|$ as the problem complexity and consider a sequence of nested hypothesis classes, each positing a different upper bound on $\|\theta^*\|$. Exploiting this, we propose Adaptive Linear Bandit (ALB), a novel phase based algorithm that adapts to the true problem complexity, $\|\theta^*\|$. We show that ALB achieves regret scaling of $O(\|\theta^*\|\sqrt{T})$, where $\|\theta^*\|$ is apriori unknown. As a corollary, when $\theta^*=0$, ALB recovers the minimax regret for the simple bandit algorithm without such knowledge of $\theta^*$. ALB is the first algorithm that uses parameter norm as model section criteria for linear bandits. Prior state of art algorithms \cite{osom} achieve a regret of $O(L\sqrt{T})$, where $L$ is the upper bound on $\|\theta^*\|$, fed as an input to the problem. In the second setting, we consider the standard linear bandit problem (with possibly an infinite number of arms) where the sparsity of $\theta^*$, denoted by $d^* \leq d$, is unknown to the algorithm. Defining $d^*$ as the problem complexity, we show that ALB achieves $O(d^*\sqrt{T})$ regret, matching that of an oracle who knew the true sparsity level. This methodology is then extended to the case of finitely many arms and similar results are proven. This is the first algorithm that achieves such model selection guarantees. We further verify our results via synthetic and real-data experiments.

Rejoinder: On nearly assumption-free tests of nominal confidence interval coverage for causal parameters estimated by machine learning

Abstract: For many causal effect parameters of interest, doubly robust machine learning (DRML) estimators $\hat{\psi}_{1}$ are the state-of-the-art, incorporating the good prediction performance of machine learning; the decreased bias of doubly robust estimators; and the analytic tractability and bias reduction of sample splitting with cross-fitting. Nonetheless, even in the absence of confounding by unmeasured factors, the nominal $(1-\alpha)$ Wald confidence interval $\hat{\psi}_{1}\pm z_{\alpha/2}\widehat{\mathsf{s.e.}}[\hat{\psi}_{1}]$ may still undercover even in large samples, because the bias of $\hat{\psi}_{1}$ may be of the same or even larger order than its standard error of order $n^{-1/2}$. In this paper, we introduce essentially assumption-free tests that (i) can falsify the null hypothesis that the bias of $\hat{\psi}_{1}$ is of smaller order than its standard error, (ii) can provide a upper confidence bound on the true coverage of the Wald interval, and (iii) are valid under the null under no smoothness/sparsity assumptions on the nuisance parameters. The tests, which we refer to as Assumption Free Empirical Coverage Tests (AFECTs), are based on a U-statistic that estimates part of the bias of $\hat{\psi}_{1}$. Our claims need to be tempered in several important ways. First no test, including ours, of the null hypothesis that the ratio of the bias to its standard error is smaller than some threshold $\delta$ can be consistent [without additional assumptions (e.g., smoothness or sparsity) that may be incorrect]. Second, the above claims only apply to certain parameters in a particular class. For most of the others, our results are unavoidably less sharp. In particular, for these parameters, we cannot directly test whether the nominal Wald interval $\hat{\psi}_{1}\pm z_{\alpha/2}\widehat{\mathsf{s.e.}}[\hat{\psi}_{1}]$ undercovers. However, we can often test the validity of the smoothness and/or sparsity assumptions used by an analyst to justify a claim that the reported Wald interval’s actual coverage is no less than nominal. Third, in the main text, with the exception of the simulation study in Section 1, we assume we are in the semisupervised data setting (wherein there is a much larger dataset with information only on the covariates), allowing us to regard the covariance matrix of the covariates as known. In the simulation in Section 1, we consider the setting in which estimation of the covariance matrix is required. In the simulation, we used a data adaptive estimator which performs very well in our simulations, but the estimator’s theoretical sampling behavior remains unknown.

On the group of self-homotopy equivalences of an elliptic space

Abstract: Let $X$ be a simply connected rational elliptic space of formal dimension $n$ and let $\E(X)$ denote the group of homotopy classes of self-equivalences of $X$. If $X^{[k]}$ denotes the $k^{\text{th}}$ Postikov section of $X$ and $X^{k}$ denotes its $k^{\text{th}}$ skeleton, then making use of the models of Sullivan and Quillen we prove that $\E(X)\cong\E(X^{[n]})$ and if $n>m=max\big\{k \,| \,\pi_{k}(X) eq 0\big\}$ and $\E(X)$ is finite, then $\E(X)\cong\E(X^{m+1})$. Moreover, in case when $X$ is 2-connected, we show that if $\pi_{n}(X) eq0$, then the group $\E(X)$ is infinite.

Equivariant completions of affine spaces

Abstract: We survey recent results on open embeddings of the affine space $\mathbb{C}^n$ into a complete algebraic variety $X$ such that the action of the vector group $\mathbb{G}_a^n$ on $\mathbb{C}^n$ by translations extends to an action of $\mathbb{G}_a^n$ on $X$. The current version of the text includes the introduction and the section on equivariant embeddings into the projective space $\mathbb{P}^n$. Comments and suggestions are very welcome.

Auto-equivalences of the modular tensor categories of type $A$, $B$, $C$ and $G$

Abstract: We compute the monoidal and braided auto-equivalences of the modular tensor categories $\mathcal{C}(\mathfrak{sl}_{r+1},k)$, $\mathcal{C}(\mathfrak{so}_{2r+1},k)$, $\mathcal{C}(\mathfrak{sp}_{2r},k)$, and $\mathcal{C}(\mathfrak{g}_{2},k)$. Along with the expected simple current auto-equivalences, we show the existence of the charge conjugation auto-equivalence of $\mathcal{C}(\mathfrak{sl}_{r+1},k)$, and exceptional auto-equivalences of $\mathcal{C}(\mathfrak{so}_{2r+1},2)$, $\mathcal{C}(\mathfrak{sp}_{2r},r)$, $\mathcal{C}(\mathfrak{g}_{2},4)$. We end the paper with a section discussing potential applications of these computations.

On pseudo-Frobenius elements of submonoids of $\mathbb{N}^d$

Abstract: In this paper we study those submonoids of $$\mathbb {N}^d$$ with a nontrivial pseudo-Frobenius set. In the affine case, we prove that they are the affine semigroups whose associated algebra over a field has maximal projective dimension possible. We prove that these semigroups are a natural generalization of numerical semigroups and, consequently, most of their invariants can be generalized. In the last section we introduce a new family of submonoids of $$\mathbb {N}^d$$ and using its pseudo-Frobenius elements we prove that the elements in the family are direct limits of affine semigroups.

Global hypersurfaces of section in the spatial restricted three-body problem

Abstract: We propose a contact-topological approach to the spatial circular restricted three-body problem, for energies below and slightly above the first critical energy value. We prove the existence of an $S^1$-family of global hypersurfaces of section for the regularized dynamics, which are copies of $(\mathbb{D}^*S^2,\omega)$, where $\omega$ is deformation equivalent to the standard symplectic form. This is the page of an open book in $S^*S^3=S^3\times S^2$, with binding $S^*S^2=\mathbb{R}P^3$, and the situation at the binding reduces to the planar problem. The first return map is Hamiltonian, restricting to the boundary as the time-$1$ map of a positive reparametrization of the Reeb flow in $\mathbb{R}P^3$ giving the planar problem. This construction holds for any choice of mass ratio, and is therefore non-perturbative. We illustrate the technique in the completely integrable case of the rotating Kepler problem, where the return map can be studied explicitly.

Global surfaces of section for dynamically convex Reeb flows on lens spaces

Abstract: We show that a dynamically convex Reeb flow on the standard tight lens space $(L(p, 1),\xi_{\mathrm{std}})$, $p>1,$ admits a $p$-unknotted closed Reeb orbit $P$ which is the binding of a rational open book decomposition with disk-like pages. Each page is a rational global surface of section for the Reeb flow and the Conley-Zehnder index of the $p$-th iterate of $P$ is $3$. We also check dynamical convexity in the Henon-Heiles system for low positive energies. In this case the rational open book decomposition follows from the fact that the sphere-like component of the energy surface admits a $\mathbb{Z}_{3}$-symmetric periodic orbit and the flow descends to a Reeb flow on the standard tight $(L(3,2),\xi_{\mathrm{std}})$.

What does a vector field know about volume

Abstract: This note provides an affirmative answer to a question of Viterbo concerning the existence of nondiffeomorphic contact forms that share the same Reeb vector field. Starting from an observation by Croke-Kleiner and Abbondandolo that such contact forms define the same total volume, we discuss various related issues for the wider class of geodesible vector fields. In particular, we define an Euler class of a geodesible vector field in the associated basic cohomology and give a topological characterisation of vector fields with vanishing Euler class. We prove the theorems of Gauss-Bonnet and Poincare-Hopf for closed, oriented 2-dimensional orbifolds using global surfaces of section and the volume determined by a geodesible vector field. This volume is computed for Seifert fibred 3-manifolds and for some transversely holomorphic flows.

On the Volume of the John–Löwner Ellipsoid

Abstract: We find an optimal upper bound on the volume of the John ellipsoid of a k-dimensional section of the n-dimensional cube, and an optimal lower bound on the volume of the Lowner ellipsoid of a projection of the n-dimensional cross-polytope onto a k-dimensional subspace, which are respectively $$\bigl (\frac{n}{k}\bigr )^{{k}/{2}}$$ and $$\bigl (\frac{k}{n}\bigr )^{{k}/{2}}$$ of the volume of the unit ball in $$\mathbb {R}^k$$. Also, we describe all possible vectors in $$\mathbb {R}^n,$$ whose coordinates are the squared lengths of a projection of the standard basis in $$\mathbb {R}^n$$ onto a k-dimensional subspace.

Dynamic Response of Alternating-Current-Driven Light-Emitting Diodes Based on Hybrid Halide Perovskites

Abstract: In this work, we report the dynamic response of alternating-current-driven light-emitting diodes based on a range of hybrid halide perovskites. Electroluminescence (EL), which appears only in the positive section of a dc voltage, arises in both cycles under a sinusoidal ac voltage. Appearance of EL emission in negative cycles of the ac voltage is explained in terms of available residual charges injected in the previous positive cycle of the sinusoidal voltage; the relative intensity of EL emission in the negative cycle can be correlated to the ambipolarity of charge carriers in the ${\mathrm{CH}}_{3}{\mathrm{NH}}_{3}{\mathrm{Pb}\mathrm{Br}}_{x}{\mathrm{I}}_{3\text{\ensuremath{-}}x}$ series. At low frequencies of the ac voltage, the dynamic response of EL emission in any cycle is in phase with the applied ac voltage; interestingly, the EL emission starts to lag the sinusoidal voltage at higher frequencies. The phase lag also has a correlation to the ambipolarity of the active perovskite material. Finally, the frequency dependence of EL emission under an ac voltage provides the limiting frequency of device operation (\ensuremath{-}3 dB frequency) and thereby the effective carrier mobility in the active heterostructure.

Sturm's theorem on the zeros of sums of eigenfunctions: Gelfand's strategy implemented

Abstract: In the second section ``Courant-Gelfand theorem'' of his last published paper (Topological properties of eigenoscillations in mathematical physics, Proc. Steklov Institute Math. 273 (2011) 25--34), Arnold recounts Gelfand's strategy to prove that the zeros of any linear combination of the $n$ first eigenfunctions of the Sturm-Liouville problem $$-\, y''(s) + q(x)\, y(x) = \lambda\, y(x) \mbox{ in } ]0,1[\,, \mbox{ with } y(0)=y(1)=0\,,$$ divide the interval into at most $n$ connected components, and concludes that ``the lack of a published formal text with a rigorous proof \dots is still distressing.''\\ Inspired by Quantum mechanics, Gelfand's strategy consists in replacing the ana\-lysis of linear combinations of the $n$ first eigenfunctions by that of their Slater determinant which is the first eigenfunction of the associated $n$-particle operator acting on Fermions.\\ In the present paper, we implement Gelfand's strategy, and give a complete proof of the above assertion. As a matter of fact, refining Gelfand's strategy, we prove a stronger property taking the multiplicity of zeros into account, a result which actually goes back to Sturm (1836).

Space and time complexity for infinite time Turing machines

Abstract: We consider notions of space complexity for Infinite Time Turing Machines (ITTMs) that were introduced by B. Lowe and studied further by J. Winter. We answer several open questions about these notions, among them whether low space complexity implies low time complexity (it does not) and whether one of the equalities P=PSPACE, P$_{+}=$PSPACE$_{+}$ and P$_{++}=$PSPACE$_{++}$ holds for ITTMs (all three are false). We also show various separation results between space complexity classes for ITTMs. This considerably expands our earlier observations on the topic in section 7.2.2 of \cite{Ca2}, which appear here as Lemma $6$ up to Corollary $9$.

Homotopy fibrations with a section after looping

Abstract: We analyze a general family of fibrations which, after looping, have sections. Methods are developed to determine the homotopy type of the fibre and the homotopy classes of the map from the fibre to the base. The methods are driven by applications to two-cones, Poincare Duality complexes, the connected sum operation, and polyhedral products.

Estimation of upper bounds of certain matrix operators on Binomial weighted sequence spaces

Abstract: In this article, we introduce binomial weighted sequence spaces $$b_p^{r,s}(w)$$ $$(1\le p<\infty )$$ , where $$w=(w_n)$$ is a non-negative decreasing sequence of real numbers, and investigate some topological and inclusion properties of the new spaces. We give an upper estimation of $$\left\| A\right\| _{\ell _p(w),b_p^{r,s}(w)}$$ , where A is the Hausdorff matrix operator or Norlund matrix operator. A Hardy type formula is established in the case of Hausdorff matrix operator. In the final section, we give an upper estimation for the transpose of Norlund matrix as an operator from $$\ell _p$$ to $$b_p^{r,s}$$ .

High Gain Metamaterial Loaded Antipodal Tapered Slot Antenna for Millimeter Wave Applications

Abstract: We propose a novel compact antipodal elliptically tapered slot antenna loaded with left-handed metamaterial over 20-40 GHz for millimetre wave applications. The conventional antipodal tapered slot antenna is modified by incorporating a novel transition between the feed and taper section of the antenna to stabilize the radiation pattern in E-plane and to reduce the beam tilting. We employ a semi-dodecagon shaped dielectric lens with unit cells of a novel broadband metamaterial placed on it to improve the gain of the proposed antenna. The proposed antenna is designed on Rogers RT/Duroid 5880 substrate with dielectric constant of 2.2. The antenna is compact with a volume of $ 30\times 49\times$ 0.254 $\mathrm{m}\mathrm{m}^{3}$. The antenna reflection coefficient $(|\mathrm{S}_{11}|\lt -10\mathrm{d}\mathrm{B})$ shows good impedance matching over a wide frequency range of 20-40GHz. We exploit the left-handedness of the proposed broadband metamaterial structure to guide the electromagnetic waves along end-fire which results in gain enhancement up to 13.78 dB at the highest frequency (40GHz). The proposed antenna exhibits high gain, small area and stable radiation pattern as compared to other state-of-the-art designs.

Floer homology in the cotangent bundle of a closed Finsler manifold and noncontractible periodic orbits

Abstract: We show that the existence of noncontractible periodic orbits for compactly supported time-dependent Hamiltonian on the disk cotangent bundle of a Finsler manifold provided that the Hamiltonian is sufficiently large over the zero section. We generalize the BPS capacities and earlier constructions of Weber (2006 Duke Math. J. 133, 527-568) and other authors Biran et al (2003 Duke Math. J. 119, 65-118) to the Finsler setting. We then obtain a number of applications including: (1) generalizing the main theorem of Xue (2017 J. Symplectic Geom. 15, 905-936) to the Lie group setting, (2) preservation of minimal Finsler length of closed geodesics in any given free homotopy class by symplectomorphisms, (3) existence of periodic orbits for Hamiltonian systems separating two Lagrangian submanifolds, (4) existence of periodic orbits for Hamiltonians on noncompact domains, (5) existence of periodic orbits for Lorentzian Hamiltonian in higher dimensional case, (6) partial solution to a conjecture of Kawasaki (2016 Heavy subsets and non-contractible trajectories (arXiv:1606.01964)), (7) results on squeezing/nonsqueezing theorem on torus cotangent bundles, etc.