Triangular Truncation Research Articles

Noncommutative Nest Hardy Spaces and Martingales

Abstract Let $(\mathcal{M},\tau )$ be a finite von Neumann algebra equipped with a normalized faithful trace and let $\mathbb{A}$ be a nest of order type $\mathbb{N}$. The nest algebra is defined by $H_{\infty }^{r}(\mathbb{A})=\{x\in \mathcal{M}:ex=exe,e\in \mathbb{A}\}$, and the related noncommutative Hardy space $H_{p}^{r}(\mathbb{A})$ is the completion of $H_{\infty }^{r}(\mathbb{A})$ in $L_{p}(\mathcal{M})$. In the present paper, we make a connection between $H_{p}^{r}(\mathbb{A})$ and certain noncommutative martingale Hardy space via showing that triangular truncation is a concrete martingale transform. We include several related results for noncommutative nest Hardy spaces in the spirit of recent development of noncommutative martingale theory.

Cut norm discontinuity of triangular truncation of graphons

The space of Lp graphons, symmetric measurable functions w:[0,1]2→R with finite p-norm, features heavily in the study of sparse graph limit theory. We show that the triangular cut operator Mχ—the operator that sets all values of a graphon below the main diagonal of the unit square to 0—acting on this space is not continuous with respect to the cut norm. This is achieved by showing that as n→∞, the norm of the triangular truncation operator Tn on symmetric matrices equipped with the cut norm grows to infinity as well. Due to the density of symmetric matrices in the space of Lp graphons, the norm growth of Tn generalizes to the unboundedness of Mχ. We also show that the norm of Tn grows to infinity on symmetric matrices equipped with the operator norm.

Boundedness of the Hilbert transform in Lorentz spaces and applications to operator ideals

In this paper, it is investigated the optimal range of the classical Hilbert transform in Lorentz spaces of functions and its non-commutative counterparts including a triangular truncation operator in Schatten-Lorentz ideals. Some applications of obtained results to operator Lipschitz functions and commutator estimates in Schatten-Lorentz ideals of compact operators are presented.

The boundedness of the Hilbert transformation from one rearrangement invariant Banach space into another and applications

In this paper, we study the boundedness of the Hilbert transformation in Lorentz function spaces, thereby complementing classical results of Boyd. We also characterize the optimal range of a triangular truncation operator in Schatten-Lorentz ideals. These results further entail sharp commutator estimates and applications to operator Lipschitz functions in Schatten-Lorentz ideals.

The Surface Expression of Deep Columnar Flows

AbstractStudies of the Earth's secular variation have blossomed in recent decades, due to the availability of brand‐new satellite data. In particular, flow inversions at the core surface that incorporate the effects of rapid rotation a priori have been able to provide new insight into flows with time scales as short as a few years. These models have been mostly developed in a cylindrical representation, while core inversion methods are based on a spherical surface representation of the flow. This technical brief links band‐limited expressions for the flow coefficients in a cylindrical representation and those in a spherical surface representation and demonstrates that these also remain band limited, with equal degrees of freedom. Correct regularity conditions on the stream function ensure a regular representation of these flows on the spherical surface. A triangular truncation rule for the cylindrical representation that is applicable for “isotropic resolution” is also presented. The equations we derive will be of use in future data‐assimilation work employing this cylindrical flow representation and may prove helpful for conventional core‐inversion strategies.

The Spectral Density of Hankel Operators with Piecewise Continuous Symbols

In 1966, H. Widom proved an asymptotic formula for the distribution of eigenvalues of the Ntimes N truncated Hilbert matrix for large values of N. In this paper, we extend this formula to Hankel matrices with symbols in the class of piece-wise continuous functions on the unit circle. Furthermore, we show that the distribution of the eigenvalues is independent of the choice of truncation (e.g. square or triangular truncation).

The optimal range of the Calderòn operator and its applications

We identify the optimal range of the Calderòn operator and that of the classical Hilbert transform in the class of symmetric quasi-Banach spaces. Further consequences of our approach concern the optimal range of the triangular truncation operator, operator Lipschitz functions and commutator estimates in ideals of compact operators.

Scale-Invariant Formulation of Momentum Diffusion for High-Resolution Atmospheric Circulation Models

A new version of the dynamic Smagorinsky model is presented that applies for nonisotropic momentum diffusion in high-resolution atmospheric circulation models. While the horizontal mixing length is computed in accordance with scale invariance in the mesoscale regime of the horizontal energy cascade, the associated dynamic vertical mixing length (DVML) is based on a recently developed scale invariance criterion and represents an application of the scaling laws of stratified macroturbulence. The DVML is validated in high-resolution simulations with the Kühlungsborn mechanistic general circulation model, using triangular spectral truncation at wavenumber 330 and a vertical level spacing of about 200 m in the upper troposphere. For a proper choice of the test filter, the model simulates a realistic horizontal kinetic energy spectrum in the troposphere along with a realistic intensity of the Lorenz energy cycle. This result is obtained without any hyperdiffusion, and it depends only little on whether the vertical mixing length is prescribed or set to the DVML. The globally averaged Smagorinsky parameter is about c S ≅ 0.53. The latitude–height cross sections show that c S maximizes in regions of strong mesoscale kinetic energy.

Spherical Harmonic Spectral Estimation on Arbitrary Grids

AbstractThis study explores the use of nonuniform fast spherical Fourier transforms on meteorological data that are arbitrarily distributed on the sphere. The applicability of this methodology in the atmospheric sciences is demonstrated by estimating spectral coefficients for nontrivial subsets of reanalysis data on a uniformly spaced latitude–longitude grid, a global cloud resolving model on an icosahedral mesh with 3-km horizontal grid spacing, and for temperature anomalies from arbitrarily distributed weather stations over the United States. A spectral correction technique is developed that can be used in conjunction with the inverse transform to yield data interpolated onto a uniformly spaced grid, with optional triangular truncation, at reduced computational cost compared to other variance conserving interpolation methods, such as kriging or natural spline interpolation. The spectral correction yields information that can be used to deduce gridded observational biases not directly available from other methods.

Periodic Triangle-Truncated DSPSL-Based Antenna With Backfire to Endfire Beam-Scanning Capacity

A backfire to endfire beam-scanning antenna based on double-side parallel-strip lines (DSPSL) is presented in this communication. The antenna consists of periodic offset DSPSL structures with triangular truncation. A series of uniform DSPSL and offset DSPSL equations were established to analyze its operating modes, transmission mode, and first higher order. The working band range in the periodic DSPSL structure is shown to assist the design of the proposed antenna. The main beam of the antenna scans from backfire to endfire as operating frequency increases owing to its periodic structure. Truncating the periodic offset DSPSL improved the current distribution and consequently broadened the impendence, enhancing the radiation patterns. Experimental results showed that the main lobe scans electronically and continuously from the backfire (-z-direction) to endfire in the yz plane toward the absolute end-firing direction (+z-direction) with two open stopbands when operating frequency increases from 1.2 to 5.9 GHz. The peak gain of the antenna is about 9.7 dBi, and the gain in the whole working band exceeds 4 dBi.

Random reordering in SOR-type methods

When iteratively solving linear systems $$By=b$$By=b with Hermitian positive semi-definite B, and in particular when solving least-squares problems for $$Ax=b$$Ax=b by reformulating them as $$AA^*y=b$$AAźy=b, it is often observed that SOR type methods (Gauβ-Seidel, Kaczmarz) perform suboptimally for the given equation ordering, and that random reordering improves the situation on average. This paper is an attempt to provide some additional theoretical support for this phenomenon. We show error bounds for two randomized versions, called shuffled and preshuffled SOR, that improve asymptotically upon the best known bounds for SOR with cyclic ordering. Our results are based on studying the behavior of the triangular truncation of Hermitian matrices with respect to their permutations.

Triangular truncation and its extremal matrices

SummaryThe triangular truncation operator is a linear transformation that maps a given matrix to its strictly lower triangular part. The operator norm (with respect to the matrix spectral norm) of the triangular truncation is known to have logarithmic dependence on the dimension, and such dependence is usually illustrated by a specific Toeplitz matrix. However, the precise value of this operator norm as well as on which matrices can it be attained is still unclear. In this article, we describe a simple way of constructing matrices whose strictly lower triangular part has logarithmically larger spectral norm. The construction also leads to a sharp estimate that is very close to the actual operator norm of the triangular truncation. This research is directly motivated by our studies on the convergence theory of the Kaczmarz type method (or equivalently, the Gauß‐Seidel type method), the corresponding application of which is also included. Copyright © 2016 John Wiley & Sons, Ltd.

Simulation of the Indian Summer Monsoon in the Superparameterized Climate Forecast System Version 2: Preliminary Results

Abstract An analysis of a 5-yr (from 1 January 2009 to 31 December 2013) free run of the superparameterized (SP) Climate Forecast System (CFS) version 2 (CFSv2) (SP-CFS), implemented for the first time at a spectral triangular truncation at wavenumber 62 (T62) atmospheric horizontal resolution, is presented. The SP-CFS simulations are evaluated against observations and traditional convection parameterized CFSv2 simulations at T62 resolution as well as at some higher resolutions. The metrics for evaluating the model performance are chosen in order to mainly address the improvement in systematic biases observed in the CFSv2 documented in earlier studies. While the primary focus of this work is on evaluating the improvement of the simulation of the Indian summer monsoon (ISM) by the SP-CFS model, some results are also presented within the context of the global climate. The SP-CFS significantly reduces the dry bias of precipitation over the Indian subcontinent and better captures the monsoon intraseasonal oscillation (MISO) modes. SP-CFS also improves the northward and eastward propagation of high- and low-frequency modes of ISM. Compared to CFSv2, the SP-CFS model simulates improved convectively coupled equatorial waves; better temperature structures both spatially and vertically, leading to a significantly improved relative distribution of variance for the synoptic disturbances and low-frequency tropical intraseasonal oscillations (ISOs). This analysis of the development of SP-CFS is particularly important as it shows promise for improving the cloud process representation through an SP framework and is able to improve the mean as well as intraseasonal characteristics of CFSv2 within the context of the ISM.

River discharge assessment under a changing climate in the Chao Phraya River, Thailand by using MRI-AGCM3.2S

In recent years, General Circulation Model (GCM) has been widely used in climate change impact studies. It is an effective tool for well understanding of a change in climate behavior in a long term. Nowadays, more than twenty GCMs have been developed in many research institutes around the world. We selected the latest version of GCM developed by Meteorological Research Institute (MRI), Japan Meteorology Agency, MRI-AGCM3.2S. The model has a horizontal resolution of triangular truncation 959 (TL959), and the transform grid uses 1920 x 960 grid cells, corresponding to approximately a 20-km grid interval with 64 vertical layers (top at 0.01 hPa).We used a regional distributed hydrologic model based on the concept of the variable infiltration capacity to generate runoff intensity and a kinematic wave model including effects of dam operation and inundation to simulate river discharge. The C.2 gauging station at Nakhon Sawan was selected to monitor changes in the river discharge. Input data of the distributed hydrological model, GCM precipitation and evapotranspiration, were corrected to remove biases using the quantile-quantile bias-correction method for precipitation and the different factor bias- correction method for evapotranspiration. The results of the experiment in projecting discharge of the Chao Phraya River under the near future climate (2015-2043) and the future climate (2075-2103) by using the bias- corrected GCM data set showed that 1) the mean annual discharge tends to increase in both near future and future projection periods, 2) During a dry season the tendency of low flow in the near future period tends to decrease. However, the flood frequency analysis using Generalized Extreme Value distribution (GEV) indicates that flood risk in the future will have more severities and damages to the country as the result of the analysis shows that, in the near future the magnitude of 80-year return period flood is greater than the devastating 2011 Thai flood. .

Impact study of integrated precipitable water estimated from Indian GPS measurements

The Global Positioning System – Integrated Precipitable Water (IPW) data from Indian stations namely Chennai, Guwahati, Kolkata, Mumbai and New Delhi have been assimilated in the National Centre for Medium Range Weather Forecasting’s (NCMRWF) Global Data Assimilation System (GDAS). Gridpoint Statistical Interpolation (GSI) Scheme of GDAS analysis is experimented with the global model T254L64. The analyses and forecasts are carried out at triangular truncation of wave number 254 and with 64 levels in vertical. Global analyses are carried four times (0000 UTC, 0600 UTC, 1200 UTC and 1800 UTC) daily with intermittent time scheme. Model integrations are carried up to 168 hours. The present study examines the impact that integrated precipitable water has over various meteorological parameters. The study reveals that the assimilation of IPW data influences the analyses and corresponding forecasts of the weather model T254L64. This is an attempt of assimilation of IPW data of the aforesaid five Indian stations in the global model and examination of corresponding impact on various meteorological parameters over Indian region. It is seen that for the layers above 750 hPa the zonal and meridional wind components for IPW analyses have less biases. Forecasts from IPW simulations are found to have consistently by lower 850 hPa wind vector root mean square error (RMSE) where as at 250 hPa, improvement in IPW runs are seen only for day-1 and day-4 forecasts. For temperature at 850 hPa, IPW forecasts valid for day-4 & day-5 are better. At 250 hPa, temperature RMSE for IPW runs is lower for day-1 forecasts. Mean error of IPW forecasts at 250 hPa is lower for all the days of forecasts. Also, geo-potential RMSE for the IPW runs at 250 hPa is lower for all the days of forecasts. Forecasts vs analyses study shows positive impact of IPW assimilation on the anomaly and pattern correlations.

Triangular truncation of k-Fibonacci and k-Lucas circulant matrices

We prove a general theorem that gives tight bounds on the spectral norms of triangularly truncated k-Fibonacci and k-Lucas circulant matrices. The bounds are good enough to enable the calculation of the limit‖C‖‖τ(C)‖, as the dimension n approaches infinity, where τ(C) denotes the triangular truncation of C, and C is any n×n circulant matrix built using a sequence (si) satisfyingsi=ksi−1+si−2. In particular, we have that this limit is equal to the golden ratio, if C is built using either the ordinary Fibonacci or Lucas sequence.

Circular Polarization in Defected Hexagonal Shaped Microstrip Antenna

A novel circular polarization (CP) design of a single-feed Hexagonal shaped microstrip antenna for enhanced antenna parameters is presented. The CP design is obtained by placing three isosceles triangular defects at proper locations in the hexagonal radiating patch as well as in the square ground plane. By adjusting one of the side lengths of the triangular defect, slightly longer or shorter than that of the others, two orthogonal near degenerate resonant modes for CP radiation are excited. Moreover, the 3 dB axial ratio bandwidth is surrounded by the 10 dB impedance bandwidth, to ensure that the antenna radiates CP waves in its impedance band. The antenna gain of the proposed antenna has increased by 5.34 dB (7.93 vs. 2.59 dB), as compared to the hexagonal microstrip antenna with triangular truncations in patch but without triangular defects in the ground plane having same substrate material, height and same radiating patch size and shape. Prototype antenna is fabricated for experimental evaluation. Simulated and measured results are in good agreement with each other.

Impact of Resolution and Design on the U.S. Navy Global Ensemble Performance in the Tropics

Abstract The performance of the U.S. Navy global atmospheric ensemble prediction system is examined with a focus on tropical winds and tropical cyclone tracks. Ensembles are run at a triangular truncation of T119, T159, and T239, with 33, 17, and 9 ensemble members, respectively, to evaluate the impact of resolution versus the number of ensemble member tradeoffs on ensemble performance. Results indicate that the T159 and T239 ensemble mean tropical cyclone track errors are significantly smaller than those of the T119 ensemble out to 4 days. For ensemble forecasts of upper- and lower-tropospheric tropical winds, increasing resolution has only a small impact on ensemble mean root-mean-square error for wind speed, but does improve Brier scores for 10-m wind speed at the 5 m s−1 threshold. In addition to the resolution tests, modifications to the ensemble transform initial perturbation methodology and inclusion of stochastic kinetic energy backscatter are also evaluated. Stochastic kinetic energy backscatter substantially increases the ensemble spread and improves Brier scores in the tropics, but for the most part does not significantly reduce ensemble mean tropical cyclone track error.

Large-Scale Surface Mass Balance of Ice Sheets from a Comprehensive Atmospheric Model

The surface mass balance for Greenland and Antarctica has been calculated using model data from an AMIP-type experiment for the period 1979–2001 using the ECHAM5 spectral transform model at different triangular truncations. There is a significant reduction in the calculated ablation for the highest model resolution, T319 with an equivalent grid distance of ca 40 km. As a consequence the T319 model has a positive surface mass balance for both ice sheets during the period. For Greenland, the models at lower resolution, T106 and T63, on the other hand, have a much stronger ablation leading to a negative surface mass balance. Calculations have also been undertaken for a climate change experiment using the IPCC scenario A1B, with a T213 resolution (corresponding to a grid distance of some 60 km) and comparing two 30-year periods from the end of the twentieth century and the end of the twenty-first century, respectively. For Greenland there is change of 495 km3/year, going from a positive to a negative surface mass balance corresponding to a sea level rise of 1.4 mm/year. For Antarctica there is an increase in the positive surface mass balance of 285 km3/year corresponding to a sea level fall by 0.8 mm/year. The surface mass balance changes of the two ice sheets lead to a sea level rise of 7 cm at the end of this century compared to end of the twentieth century. Other possible mass losses such as due to changes in the calving of icebergs are not considered. It appears that such changes must increase significantly, and several times more than the surface mass balance changes, if the ice sheets are to make a major contribution to sea level rise this century. The model calculations indicate large inter-annual variations in all relevant parameters making it impossible to identify robust trends from the examined periods at the end of the twentieth century. The calculated inter-annual variations are similar in magnitude to observations. The 30-year trend in SMB at the end of the twenty-first century is significant. The increase in precipitation on the ice sheets follows closely the Clausius-Clapeyron relation and is the main reason for the increase in the surface mass balance of Antarctica. On Greenland precipitation in the form of snow is gradually starting to decrease and cannot compensate for the increase in ablation. Another factor is the proportionally higher temperature increase on Greenland leading to a larger ablation. It follows that a modest increase in temperature will not be sufficient to compensate for the increase in accumulation, but this will change when temperature increases go beyond any critical limit. Calculations show that such a limit for Greenland might well be passed during this century. For Antarctica this will take much longer and probably well into following centuries.

Impact of Resolution Degradation of the Initial Condition on Typhoon Track Forecasts

Abstract The operational numerical weather prediction (NWP) systems at the Japan Meteorological Agency (JMA) indicated that the typhoon track forecasts made by the control member of the ensemble prediction system (EPS) tended to be worse than those made by the high-resolution global NWP. The control forecast of the EPS with horizontal triangular truncation at 319 wavenumbers and 60 vertical levels (T319/L60 resolution) was initialized by eliminating the higher-wavenumber components of the global analysis at T959/L60 resolution. When the data assimilation cycle was performed at the lower T319/L60 resolution, the forecast gave typhoon track forecasts closer to the high-resolution global NWP. Therefore, it stands to reason that the resolution transform of the initial condition must be responsible for the degradation of the typhoon track forecasts at least to considerable extent. To improve the low-resolution forecast, two approaches are tested in this study: 1) applying a smoother spectral truncation for the resolution transform and 2) performing noncycled lower-resolution data assimilation during preprocessing. Results from the single case study of Typhoon Nuri (2008) indicate almost no impact from the former approach, but a significant positive impact when using the latter approach. The results of this study illuminate the importance of considering a model’s resolving capability during data assimilation. Namely, if the initial conditions contain features caused by unresolved scales, degraded forecasts may result.