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At least 55 records · Page 3

A flexible class of priors for orthonormal matrices with basis function-specific structure

Statistical modeling of high-dimensional matrix-valued data motivates the use of a low-rank representation that simultaneously summarizes key characteristics of the data and enables dimension reduction. Low-rank representations commonly factor the original data into the product of orthonormal basis functions and weights, where each basis function represents an independent feature of the data. However, the basis functions in these factorizations are typically computed using algorithmic methods that cannot quantify uncertainty or account for basis function correlation structure a priori. While there exist Bayesian methods that allow for a common correlation structure across basis functions, empirical examples motivate the need for basis function-specific dependence structure. We propose a prior distribution for orthonormal matrices that can explicitly model basis function-specific structure. The prior is used within a general probabilistic model for singular value decomposition to conduct posterior inference on the basis functions while accounting for measurement error and fixed effects. We discuss how the prior specification can be used for various scenarios and demonstrate favorable model properties through synthetic data examples. Finally, we apply our method to two-meter air temperature data from the Pacific Northwest, enhancing our understanding of the Earth system’s internal variability.

97 MATHEMATICS AND COMPUTING↗

A light weight regularization for wave function parameter gradients in quantum Monte Carlo

The parameter derivative of the expectation value of the energy, ∂E/∂p, is a key ingredient in variational Monte Carlo (VMC) wave function optimization methods. In some cases, a naïve estimate of this derivative suffers from an infinite variance, which inhibits the efficiency of optimization methods that rely on a stable estimate of the derivative. In this work, we derive a simple regularization of the naïve estimator, which is trivial to implement in existing VMC codes, has finite variance, and a negligible bias, which can be extrapolated to zero bias with no extra cost. We use this estimator to construct an unbiased, finite variance estimation of ∂E/∂p for a multi-Slater–Jastrow trial wave function on the LiH molecule and in the optimization of a multi-Slater–Jastrow trial wave function on the CuO molecule. This regularized estimator is a simple and efficient estimator of ∂E/∂p for VMC optimization techniques.

36 MATERIALS SCIENCE↗

SineKAN: Kolmogorov-Arnold Networks using sinusoidal activation functions

Recent work has established an alternative to traditional multi-layer perceptron neural networks in the form of Kolmogorov-Arnold Networks (KAN). The general KAN framework uses learnable activation functions on the edges of the computational graph followed by summation on nodes. The learnable edge activation functions in the original implementation are basis spline functions (B-Spline). Here, we present a model in which learnable grids of B-Spline activation functions are replaced by grids of re-weighted sine functions (SineKAN). We evaluate numerical performance of our model on a benchmark vision task. We show that our model can perform better than or comparable to B-Spline KAN models and an alternative KAN implementation based on periodic cosine and sine functions representing a Fourier Series. Further, we show that SineKAN has numerical accuracy that could scale comparably to dense neural networks (DNNs). Compared to the two baseline KAN models, SineKAN achieves a substantial speed increase at all hidden layer sizes, batch sizes, and depths. Current advantage of DNNs due to hardware and software optimizations are discussed along with theoretical scaling. Additionally, properties of SineKAN compared to other KAN implementations and current limitations are also discussed.

Reinhardt, Eric↗

Amplitudes, supersymmetric black hole scattering at $\mathcal{O}\left({G}^5\right)$, and loop integration

We compute the potential-graviton contribution to the scattering amplitude, the radial action, and the scattering angle of two extremal black holes in $\mathcal{N}$ = 8 supergravity at the fifth post-Minkowskian order and to next-to-leading order in a large mass expansion (first self-force order). Properties of classical unitarity cuts allow us to focus on the integration-by-parts reduction of planar integrals, while nonplanar integrals at this order are obtained from the planar ones by straightforward manipulations. We present the solution to the differential equations for all master integrals necessary to evaluate the classical scattering amplitudes of massive scalar particles at this order in all gravitational theories, in particular in $\mathcal{N}$ = 8 supergravity, and in general relativity. Despite the appearance of higher-weight generalized polylogarithms and elliptic functions in the solution to the differential equation for master integrals, the final supergravity answer is remarkably simple and contains only (harmonic) polylogarithmic functions up to weight 2. The systematic analysis of elliptic integrals discussed here, as well as the particular organization of boundary integrals in $\mathcal{N}$ = 8 observables are independent of supersymmetry and may have wider applications, including to aspects of collider physics.

Black Holes↗

The Palomar Transient Factory Core-collapse Supernova Host-galaxy Sample. I. Host-galaxy Distribution Functions and Environment Dependence of Core-collapse Supernovae

Several thousand core-collapse supernovae (CCSNe) of different flavors have been discovered so far. However, identifying their progenitors has remained an outstanding open question in astrophysics. Studies of SN host galaxies have proven to be powerful in providing constraints on the progenitor populations. In this paper, we present all CCSNe detected between 2009 and 2017 by the Palomar Transient Factory. This sample includes 888 SNe of 12 distinct classes out to redshift z ≈ 1. We present the photometric properties of their host galaxies from the far-ultraviolet to the mid-infrared and model the host-galaxy spectral energy distributions to derive physical properties. The galaxy mass function of Type Ic, Ib, IIb, II, and IIn SNe ranges from 105 to 1011.5 M o˙, probing the entire mass range of star-forming galaxies down to the least-massive star-forming galaxies known. Moreover, the galaxy mass distributions are consistent with models of star-formation-weighted mass functions. Regular CCSNe are hence direct tracers of star formation. Small but notable differences exist between some of the SN classes. Type Ib/c SNe prefer galaxies with slightly higher masses (i.e., higher metallicities) and star formation rates than Type IIb and II SNe. These differences are less pronounced than previously thought. H-poor superluminous supernovae (SLSNe) and SNe Ic-BL are scarce in galaxies above 1010 M o˙. Their progenitors require environments with metallicities of < 0.4 and < 1 solar, respectively. In addition, the hosts of H-poor SLSNe are dominated by a younger stellar population than all other classes of CCSNe. Our findings corroborate the notion that low metallicity and young age play an important role in the formation of SLSN progenitors.

79 ASTRONOMY AND ASTROPHYSICS↗

Fundamental Studies of Bifunctional Catalysts for Tandem Reactions

During the past few years of our funding from DOE Basic Energy Sciences, we have focused on the catalytic conversion of biomass-derived 5-hydroxymethyl furfural (HMF) as a platform molecule for sustainable synthesis of value-added chemicals. The combination of the diene and aldehyde functionalities in HMF enables catalytic production of acetalized HMF derivatives with diol or epoxy reactants to allow reversible synthesis of norcantharimide derivatives upon Diels-Alder reaction with maleimides. Whereas the electron-withdrawing nature of aldehyde group inhibits the reactivity of diene group in HMF to undergo Diels-Alder reactions with dienophiles, by converting the aldehyde group in HMF to an electron-donating hydroxyl group, HMF can append maleimide-based chemicals through Diels-Alder coupling. Acetalization of HMF not only produces an electron-rich diene for Diels-Alder reaction, but it also alters the reversibility of acetal formation and thereby allows for the controlled release of chemicals by hydrolysis. Therefore, we synthesized various norcantharimide derivatives from biomass-derived HMF by acetalization over an acid catalyst, Amberlyst-15, followed by Diels-Alder reaction with maleimide. The norcantharimides release the starting materials by retro Diels-Alder reaction that is triggered by acetal hydrolysis under acidic (≤pH 3) conditions. In subsequent work, we explored the synthesis and properties of functional polyurethanes and polyesters with tunable properties from biomass-derived (HMF)-Acetone-HMF (HAH) monomers. HAH can be selectively hydrogenated over Cu and Ru catalysts to produce partially-hydrogenated HAH (PHAH) and fully-hydrogenated HAH (FHAH). HAH contains functional groups, including hydroxyl, furan, enone, and ketone functionalities, that can be exploited to further tune the polymer properties. The π-electron conjugation between the enone and furan groups in HAH monomer stabilizes the furan group to prevent the uncontrolled degradation of furans during reactions, such as etherification. Therefore, HAH is a renewable monomer that can be used to synthesize polymers, having high molecular weight, symmetric functionalities, an inexpensive production price, and being derived from renewable resources without competition with food resources. We demonstrated the synthesis of functional polyurethanes and a renewable polyester from HAH-derived monomers, and 4,4’-methylenebis(phenyl isocyanate) (MDI). We showed that it is essential to be able to selective hydrogenate C=C double bonds in HAH to be able to achieve effective Diels Alder reactions of the furan ring. Therefore, we studied the hydrogenation at temperatures from 313 – 393 K of HAH over Pd, Ru, and Cu based catalysts. HAH was selectively hydrogenated to produce partially-hydrogenated monomers (PHAH) over Cu and Ru catalysts and to fully-hydrogenated HAH monomers (FHAH) over the Ru catalyst. Pd based catalysts yielded a mixture of partially and fully hydrogenated monomers. Reaction kinetics models were employed to quantify the kinetic behavior for hydrogenation over Ru, Cu, and Pd catalysts. A 5-step pathway exhibited over Pd and Ru catalysts consists of both series and parallel reaction steps, where HAH is both converted to fully hydrogenated products sequentially via series reactions of partially hydrogenated intermediates, as well as converted directly in parallel reactions to form the fully hydrogenated products. In contrast, a 3-step pathway over the Cu catalyst consists only of the consecutive reaction steps, where the final product was formed via series reactions of intermediate products. Additionally, reaction over the Cu catalyst did not hydrogenate the furan rings of the HAH molecule and yielded a different final product than those hydrogenation over Pd and Ru catalysts. Using the results from our detailed reaction kinetics studies, we developed universal conditions for the maximum production of the various hydrogenated products for both batch and plug flow systems. In subsequent work, we employed reaction schemes to optimize the yields of products from selective hydrogenations of HAH in isopropanol solvent and in the presence of liquid water. Reaction schemes consisting of 7, 9, and 11 steps were examined to describe the rates of formation of the observed products and reaction intermediates for hydrogenation of HAH over Ru and Pd catalysts, and a 3-step scheme was studied over Cu catalysts. Rate constants and activation energies were calculated using these reaction schemes, and we then apply these reaction schemes to explore the effects of water addition on the hydrogenation pathways. The effects of water addition to isopropanol (IPA) solvents on the hydrogenation of HAH were markedly different over Pd, Ru, and Cu catalysts. Over the Pd catalyst, the addition of water to IPA increased hydrogenation rates and promoted hydrogenation of furan rings. The addition of water to IPA yielded significant carbon losses over the Ru catalyst, and slowed hydrogenation steps over Cu, while significantly inhibiting hydrogenation of the ketone group. This behavior opened routes toward increased production rates of PHAH=O (a partially hydrogenated form of HAH containing a C=O bond), a product in which the diene groups of the furan rings were not hydrogenated. Addition of water also allowed increased feed concentrations of HAH that were previously not possible in pure IPA solvents.

09 BIOMASS FUELS↗

Surrogate construction via weight parameterization of residual neural networks

Surrogate model development is a critical step for uncertainty quantification or other sample-intensive tasks for complex computational models. Here, in this work, we develop a multi-output surrogate form using a class of neural networks (NNs) that employ shortcut connections, namely Residual NNs (ResNets). ResNets are known to regularize the surrogate learning problem and improve the efficiency and accuracy of the resulting surrogate. Inspired by the continuous, Neural ODE analogy, we augment ResNets with weight parameterization strategy with respect to ResNet depth. Weight-parameterized ResNets regularize the NN surrogate learning problem and allow better generalization with a drastically reduced number of learnable parameters. We demonstrate that weight-parameterized ResNets are more accurate and efficient than conventional feed-forward multi-layer perceptron networks. We also compare various options for parameterization of the weights as functions of ResNet depth. We demonstrate the results on both synthetic examples and a large scale earth system model of interest.

97 MATHEMATICS AND COMPUTING↗

Bounded-Confidence Models of Multidimensional Opinions with Topic-Weighted Discordance

People’s opinions on a wide range of topics often evolve over time through their interactions with others. Models of opinion dynamics primarily focus on one-dimensional opinions, which represent opinions on one topic. However, opinions on various topics are rarely isolated; instead, they can be interdependent and correlated. In a bounded-confidence model (BCM) of opinion dynamics, agents are receptive to each other only if their opinions are sufficiently similar. Here, we extend classical agent-based BCMs—namely, the Hegselmann–Krause BCM, which has synchronous interactions, and the Deffuant–Weisbuch BCM, which has asynchronous interactions—to a multidimensional setting, in which the opinions are multidimensional vectors representing opinions of different topics and opinions on different topics are interdependent. To measure opinion differences between agents, we introduce topic-weighted discordance functions that account for opinion differences in all topics. We define regions of receptiveness for our models, and we use them to characterize the steady-state opinion clusters and provide an analytical approach to compute these regions. In addition, we numerically simulate our models on various networks with initial opinions drawn from a variety of distributions. When initial opinions are correlated across different topics, our topic-weighted BCMs yield significantly different results in both transient and steady states compared to baseline models, where the dynamics of each opinion topic are independent.

Mathematics and Computing↗

An improved stochastic weighted particle method for boundary driven flows

Here, the stochastic weighted particle method (SWPM) is a generalization of the Direct Simulation Monte Carlo (DSMC) method where particle weights are variable and dynamic. SWPM is backed by a strong theoretical foundation but has not been critically evaluated for problems of practical interest. A thorough assessment of SWPM for boundary-driven flows reveals significant numerical artifacts near the boundary, notably a diverging heat flux. To correct the boundary heat flux, two modifications to SWPM are proposed: separated grouping and a spatially-dependent weight transfer function. To gauge the relative efficiency of SWPM in comparison to DSMC, a high-Mach-number wheel flow which forms a strong density gradient is also simulated.

97 MATHEMATICS AND COMPUTING↗

Isotope Production Education and Research via a Systematic Study of Photo-nuclear Reaction Yields and Excitation Functions

A new collaboration between the New Mexico Institute of Mining and Technology, Idaho State University, and Idaho National Laboratory builds the domestic workforce in isotope production and related science and technologies by conducting research and education programs related to photo-nuclear reactions. Workforce development was identified by the Department of Energy’s Office of Science in the Reaching a New Energy Sciences Workforce (RENEW) program that seeks to engage underrepresented university students in isotope production activities. This collaboration includes coursework development, student training, and isotope production research utilizing photo-nuclear reactions at electron linear accelerators. Bremsstrahlung-weighted excitation functions and cross-sections will be measured and the use of nanomaterials to exploit kinematic recoil reactions will be investigated. Students will be hosted at the Idaho National Laboratory where mentors will provide opportunities to explore careers in nuclear science and technology. This presentation regards the ongoing progress of the research and education objectives of the collaboration that was performed under the Office of Science Isotopes Program under award number D000-22-2765.

07 ISOTOPE AND RADIATION SOURCES↗

Data from: "Warming of alpine tundra enhances belowground production and shifts community towards resource acquisition traits"

This archive contains data used to draw conclusions in “Warming of alpine tundra enhances belowground production and shifts community towards resource acquisition traits”, by Yang et al. 2020. Data were collected on Niwot Ridge, in an alpine meadow within the Alpine Treeline Warming Experiment (ATWE) field sites in Colorado, USA. Samples were also processed in the U.S. Geological Survey Forest and Rangeland Ecosystem Science Center, in Boise, Idaho. File formats in this archive include comma-separated values (.csv), portable document format (.pdf), Microsoft Excel (.xlsx), and two types of geospatial files: keyhole markup language (.kml), and ESRI shapefiles (.shp). Leaf scans are .jpg images, and root scans are .tiff/.tif images.The .csv files can be opened using R, Microsoft Excel, or any simple text-editing software such as TextEdit and Notepad. Microsoft Excel files can be opened using Microsoft Excel, and .pdf files can be opened with Adobe Acrobat Reader, Preview, or other compatible programs. Scanned images can be opened using any photo and/or picture viewing software.The .kml file can be opened using Google Earth and Google Maps, and the shapefiles can be opened by any programs compatible with shapefiles, such as the ArcGIS Desktop suite, and QGIS.------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Measures of belowground net primary productivity (BNPP) are required to understand whether aboveground net primary production (ANPP) changes reflect changes in allocation or are indicative of a whole plant NPP response. Plant functional traits provide a key way to scale from the individual plant to the community level, and provide insight into drivers of NPP responses to environmental change. We used infrared heaters to warm an alpine plant community at Niwot Ridge, Colorado, and applied supplemental water to compensate for soil water loss induced by warming. We measured ANPP, BNPP, and leaf and root functional traits across treatments after 5 years of continuous warming. Community-level ANPP and total NPP (ANPP + BNPP) did not respond to heating or watering, but BNPP increased in response to heating. Heating decreased community-level leaf dry matter content and increased total root length, indicating a shift in strategy from resource conservation to acquisition in response to warming.

13C/12C isotope ratio↗

Deep Eutectic Solvent Extraction of High-Purity Lignin from a Corn Stover Hydrolysate

Here, a lactic acid/chlorine chloride-based deep eutectic solvent (DES) was used for the extraction of high-purity lignin (up to 94.7%) in high yield (up to 75%) from the hydrolysis/fermentation residue corn stover hydrolysate (CSH), which was generated from a pilot-plant-scale biorefinery. A range of extraction conditions were investigated, which involved varying reaction temperature, time, and DES composition. The relationship between lignin yield, purity, and structural characteristics with DES treatment conditions was determined. The extraction of high-purity lignin from hydrolysis/fermentation residues presents a promising approach for enhancing the economic feasibility of a lignocellulose biorefinery. It was also determined that DES extraction can produce lignin with a controlled range of molecular weight and functional group content.

09 BIOMASS FUELS↗

Physics-based adaptivity of a spectral method for the Vlasov–Poisson equations based on the asymmetrically-weighted Hermite expansion in velocity space

We propose a spectral method for the 1D-1V Vlasov–Poisson system where the discretization in velocity space is based on asymmetrically-weighted Hermite functions, dynamically adapted via a scaling α and shifting u of the velocity variable. Specifically, at each time instant an adaptivity criterion selects new values of α and u based on the numerical solution of the discrete Vlasov–Poisson system obtained at that time step. Once the new values of the Hermite parameters α and u are fixed, the Hermite expansion is updated and the discrete system is further evolved for the next time step. The procedure is applied iteratively over the desired temporal interval. The key aspects of the adaptive algorithm are: the map between approximation spaces associated with different values of the Hermite parameters that preserves total mass, momentum and energy; and the adaptivity criterion to update α and u based on physics considerations relating the Hermite parameters to the average velocity and temperature of each plasma species. For the discretization of the spatial coordinate, we rely on Fourier functions and use the implicit midpoint rule for time stepping. The resulting numerical method possesses intrinsically the property of fluid-kinetic coupling, where the low-order terms of the expansion are akin to the fluid moments of a macroscopic description of the plasma, while kinetic physics is retained by adding more spectral terms. Moreover, the scheme features conservation of total mass, momentum and energy associated in the discrete, for periodic boundary conditions. A set of numerical experiments confirms that the adaptive method outperforms the non-adaptive one in terms of accuracy and stability of the numerical solution.

97 MATHEMATICS AND COMPUTING↗

Data Imbalance, Uncertainty Quantification, and Transfer Learning in Data‐Driven Parameterizations: Lessons From the Emulation of Gravity Wave Momentum Transport in WACCM

Abstract Neural networks (NNs) are increasingly used for data‐driven subgrid‐scale parameterizations in weather and climate models. While NNs are powerful tools for learning complex non‐linear relationships from data, there are several challenges in using them for parameterizations. Three of these challenges are (a) data imbalance related to learning rare, often large‐amplitude, samples; (b) uncertainty quantification (UQ) of the predictions to provide an accuracy indicator; and (c) generalization to other climates, for example, those with different radiative forcings. Here, we examine the performance of methods for addressing these challenges using NN‐based emulators of the Whole Atmosphere Community Climate Model (WACCM) physics‐based gravity wave (GW) parameterizations as a test case. WACCM has complex, state‐of‐the‐art parameterizations for orography‐, convection‐, and front‐driven GWs. Convection‐ and orography‐driven GWs have significant data imbalance due to the absence of convection or orography in most grid points. We address data imbalance using resampling and/or weighted loss functions, enabling the successful emulation of parameterizations for all three sources. We demonstrate that three UQ methods (Bayesian NNs, variational auto‐encoders, and dropouts) provide ensemble spreads that correspond to accuracy during testing, offering criteria for identifying when an NN gives inaccurate predictions. Finally, we show that the accuracy of these NNs decreases for a warmer climate (4 × CO 2 ). However, their performance is significantly improved by applying transfer learning, for example, re‐training only one layer using ∼1% new data from the warmer climate. The findings of this study offer insights for developing reliable and generalizable data‐driven parameterizations for various processes, including (but not limited to) GWs.

54 ENVIRONMENTAL SCIENCES↗

Convergence acceleration of Monte Carlo many-body perturbation methods by direct sampling

In the Monte Carlo many-body perturbation (MC-MP) method, the conventional correlation-correction formula, which is a long sum of products of low-dimensional integrals, is first recast into a short sum of high-dimensional integrals over electron-pair and imaginary-time coordinates. These high-dimensional integrals are then evaluated by the Monte Carlo method with random coordinates generated by the Metropolis–Hasting algorithm according to a suitable distribution. The latter algorithm, while advantageous in its ability to sample nearly any distribution, introduces autocorrelation in sampled coordinates, which in turn increases the statistical uncertainty of the integrals and thus the computational cost. It also involves wasteful rejected moves and an initial “burn-in” step as well as displays hysteresis. Here, an algorithm is proposed that directly produces a random sequence of electron-pair coordinates for the same distribution used in the MC-MP method, which is free from autocorrelation, rejected moves, a burn-in step, or hysteresis. Furthermore, this direct-sampling algorithm is shown to accelerate second- (MC-MP2) and third-order Monte Carlo many-body perturbation (MC-MP3) calculations by up to 222% and 38%, respectively.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Para-particle oscillator simulations on a trapped-ion quantum computer

Deformed oscillators allow for a generalization of the standard fermions and bosons, namely, for the description of para-particles. Such particles remain hypothetical and unobserved in nature; yet, they can model physical phenomena, such as topological phases of matter. Here, we report the digital quantum simulation of para-particle oscillators by mapping para-particle states to the state of a qubit register, which allows us to identify the para-particle oscillator Hamiltonian as an XY model and further digitize the system onto a universal set of gates. In both instances, the gate depth grows polynomially with the number of qubits used. To establish the validity of our results, we experimentally simulate the dynamics of para-fermions and para-bosons, demonstrating full control of para-particle oscillators on a quantum computer. Furthermore, we compare the overall performance of the digital simulation of dynamics of the driven para-Fermi oscillator to a recent analog quantum simulation result.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Comparison of heavy-ion transport simulations: Mean-field dynamics in a box

Within the transport model evaluation project (TMEP) of simulations for heavy-ion collisions, the mean-field response is examined here. Specifically, zero-sound propagation is considered for neutron-proton symmetric matter enclosed in a periodic box, at zero temperature and around normal density. The results of several transport codes belonging to two families (BUU-like and QMD-like) are compared among each other and to exact calculations. For BUU-like codes, employing the test particle method, the results depend on the combination of the number of test particles and the spread of the profile functions that weight integration over space. These parameters can be properly adapted to give a good reproduction of the analytical zero-sound features. QMD-like codes, using molecular dynamics methods, are characterized by large damping effects, attributable to the fluctuations inherent in their phase-space representation. Moreover, for a given nuclear effective interaction, they generally lead to slower density oscillations, as compared to BUU-like codes. The latter problem is mitigated in the more recent lattice formulation of some of the QMD codes. Furthermore, the significance of these results for the description of real heavy-ion collisions is discussed.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Aspects of propagator sparsening in lattice QCD

In lattice field theory, field sparsening aims to replace quantum fields, or objects constructed from them, with approximations that preserve the appropriate symmetries and maintain many aspects of the physics that the fields determine. For example, an effective sparsening of a quark propagator provides an efficient map from a quark propagator on a fine lattice geometry to a quark propagator defined on a coarser geometry in order to reduce storage and computational costs of subsequent calculational stages while maintaining long-distance correlations and corresponding low-energy physical information. Previous studies have focused on decimating lattice sites or randomly sampling lattice sites to reduce the size of the propagator and subsequent costs of Wick contractions. Here, we extend the study of sparsening to incorporate covariant averaging of spatial sites and examine the effects on two-point and three-point correlation functions involving various hadrons. We find that sparsening is most effective in reproducing the unsparsened versions of these correlation functions when weighted covariant-averaging is sequentially applied many times.

Lattice QCD↗