Integral Weight Research Articles

Petersson norms of Eisenstein series and Kohnen–Zagier’s formula

The regularized Petersson norms of Eisenstein series of integral and half-integral weight are computed. We use these results to establish Kohnen–Zagier’s formula for Eisenstein series.

An Uncertainty Analysis Method for Artillery Dynamics with Hybrid Stochastic and Interval Parameters

This paper proposes a non-intrusive uncertainty analysis method for artillery dynamics involving hybrid uncertainty using polynomial chaos expansion (PCE). The uncertainty parameters with sufficient information are regarded as stochastic variables, whereas the interval variables are used to treat the uncertainty parameters with limited stochastic knowledge. In this method, the PCE model is constructed through the Galerkin projection method, in which the sparse grid strategy is used to generate the integral points and the corresponding integral weights. Through the sampling in PCE, the original dynamic systems with hybrid stochastic and interval parameters can be transformed into deterministic dynamic systems, without changing their expressions. The yielded PCE model is utilized as a computationally efficient, surrogate model, and the supremum and infimum of the dynamic responses over all time iteration steps can be easily approximated through Monte Carlo simulation and percentile difference. A numerical example and an artillery exterior ballistic dynamics model are used to illustrate the feasibility and efficiency of this approach. The numerical results indicate that the dynamic response bounds obtained by the PCE approach almost match the results of the direct Monte Carlo simulation, but the computational efficiency of the PCE approach is much higher than direct Monte Carlo simulation. Moreover, the proposed method also exhibits fine precision even in high-dimensional uncertainty analysis problems.

MEMS Sensors for Diagnostics and Treatment in the Fight Against COVID-19 and Other Pandemics

As the world is going through an existential global health crisis, i.e., the outbreak of novel coronavirus-caused respiratory disease (Covid-19), the healthcare systems of all the countries require readily available, low cost and highly precise equipment for the rapid diagnostics, monitoring, and treatment of the disease. The performance and precision of this equipment are solely dependent on the sensors being used. The advancement in research and development of micro-electro-mechanical systems (MEMS) based sensors during recent years, has resulted in the improvement of the conventional equipment being used in biomedical and health care applications. Microfluidics (Lab-on-a-chip) and MEMS sensors are now being used extensively for quick and accurate detection, progression monitoring, and treatment of various diseases including Covid-19. The ongoing miniaturization and design improvements have resulted in more precise sensors and actuators for healthcare applications, even for micro and nanoscale measurements in drug delivery and other invasive applications. This article aims at reviewing the MEMS sensors being used or which can be used in the important equipment for the detection and treatment of Covid-19 or other pandemics. An insight into various designs and working principles of the research-based and commercially available MEMS sensors is presented. The study highlights the role and importance of MEMS sensors in the improvement of equipment with conventional sensors. MEMS sensors outperform the conventional sensors due to their small size ( 1 μm-1mm), negligible weight, prompt response, precise measurements, portability, and ease of integration with electronic circuitry.

Algorithms for Weighted Matching Generalizations II: f-factors and the Special Case of Shortest Paths

For an undirected graph or multigraph $G=(V,E)$ and a function $f:V\to \mathbb{Z_+}$, an $f$-factor is a subgraph whose degree function is $f$. For integral edge weights of maximum magnitude $W$ our algorithm finds a maximum weight $f$-factor in time $\tilde{O}(Wf(V)^{\omega})$, where $f(V)=\sum_{v\in V} f(v)$ and $\omega$ is the exponent of matrix multiplication. The algorithm is randomized and has two versions. For worst-case time the algorithm is correct with high probability. For expected time the algorithm is Las Vegas. The algorithm is based on a detailed analysis of the structure of the optimum blossoms. A special case gives a representation for single-source shortest-paths in conservative undirected graphs, generalizing the standard shortest-path tree to a “tree of cycles”. The representation can be constructed by a randomized algorithm with the same time bound as above, or deterministically by an algorithm for maximum weight matching, achieving time $O(n(m + n \log n))$ or $O(\sqrt{n }\ m \log (nW))$.

A Bilevel Integrated Model With Data-Driven Layer Ensemble for Multi-Modality Image Fusion

Image fusion plays a critical role in a variety of vision and learning applications. Current fusion approaches are designed to characterize source images, focusing on a certain type of fusion task while limited in a wide scenario. Moreover, other fusion strategies (i.e., weighted averaging, choose-max) cannot undertake the challenging fusion tasks, which furthermore leads to undesirable artifacts facilely emerged in their fused results. In this paper, we propose a generic image fusion method with a bilevel optimization paradigm, targeting on multi-modality image fusion tasks. Corresponding alternation optimization is conducted on certain components decoupled from source images. Via adaptive integration weight maps, we are able to get the flexible fusion strategy across multi-modality images. We successfully applied it to three types of image fusion tasks, including infrared and visible, computed tomography and magnetic resonance imaging, and magnetic resonance imaging and single-photon emission computed tomography image fusion. Results highlight the performance and versatility of our approach from both quantitative and qualitative aspects.

Adaptive Cooperative Control With Guaranteed Convergence in Time-Varying Networks of Nonlinear Dynamical Systems.

In this paper, we investigate the adaptive cooperative control problem with guaranteed convergence for a class of nonlinear multiagent systems with unknown control directions and time-varying topologies. A key lemma is first derived which involves dynamically changing interaction topologies, and then a new kind of distributed control algorithms with Nussbaum-type functions are proposed based on this lemma. It is proven that if the topologies are time varying with integral weight uniform upper bound and reciprocity, then convergence is guaranteed with the proposed algorithms for nonlinear multiagent systems with nonidentical unknown control directions. An important feature of this paper is that, under time-varying topologies, the designed algorithms can deal with nonidentical unknown control directions by using classical Nussbaum-type functions. Moreover, with the proposed algorithms, we extend the adaptive cooperative control results to the case of δ -connected graphs. In particular, the adaptive leaderless consensus of high-order nonlinear agents with nonidentical unknown control directions and a directed graph having a spanning tree is also tackled as a special case. Finally, theoretical results are illustrated by a group of Genesio-Tesi systems with distributed control algorithms under time-varying topologies and some special network topologies.

On coefficients of Poincar\xe9 series and single-valued periods of modular forms

We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level varGamma _0(N) and integral weight kge 2 coincides with the field generated by the single-valued periods of a certain motive attached to varGamma _0(N). This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres–Broadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann–Ono’s construction of harmonic lifts of Poincaré series.

On modular signs of half integral weights

In this paper, we consider the first negative eigenvalue of eigenforms of half-integral weight k+1∕2 and obtain an almost type bound.

Fusion rules for ℤ2-orbifolds of affine and parafermion vertex operator algebras

This paper is about the orbifold theory of affine and parafermion vertex operator algebras. It is known that the parafermion vertex operator algebra K(sl2,k) associated to the integrable highest weight modules for the affine Kac—Moody algebra $$A_1^{(1)}$$ is the building block of the general parafermion vertex operator K( $$K(\mathfrak{g},k)$$ ,k) for any finite-dimensional simple Lie algebra $$\mathfrak{g}$$ and any positive integer k. We first classify the irreducible modules of ℤ2-orbifold of the simple affine vertex operator algebra of type $$A_1^{(1)}$$ and determine their fusion rules. Then we study the representations of the ℤ2-orbifold of the parafermion vertex operator algebra K(sl2, k). The quantum dimensions, and more technically, fusion rules for the ℤ2-orbifold of the parafermion vertex operator algebra K(sl2, k) are completely determined.

Long-Term Correlation between Water Deficit and Quality Markers in HydroSOStainable Almonds

Global warming enhances the rainfall and temperature irregularity, producing a collapse in water resources and generating an urgent need for hydro-sustainable thinking in agriculture. The aim of this study was to evaluate the correlation between the water stress of almond trees and quality parameters of fruits, after 3 years of experiments, with the objective of establishing quality markers necessary in the certification process of hydroSOStainable almonds. The results showed positive correlations among the stress integral (SI) and dry weight, color coordinates (L*, a* and b*), minerals (K, Fe, and Zn), organic acids (citric acid), sugars (sucrose, fructose, and total sugars), antioxidant activity, and fatty acids [linoleic acid, polyunsaturated (PUFA)/monounsaturated (MUFA) ratio, PUFA and SFA, among others]. As well as negative correlations of SI with water activity, weight (almond, kernel, and shell), kernel size, minerals (Ca and Mg), fatty acids (oleic acid, oleic/linoleic ratio, MUFA, and PUFA/SFA ratio), and sensory attributes (size, bitterness, astringency, benzaldehyde, and woody). Finally, this research helped to prove key quality parameters that can be used as makers of hydroSOStainable almonds. In addition, it was demonstrated that controlling water stress in almond trees by using deficit irrigation strategies can lead to appropriate yields, improve the product quality, and consequently, lead to a final added value.

Connectedness of Lakshmibai-Seshadri path crystals for hyperbolic Kac-Moody algebras of rank 2

Let be a hyperbolic Kac-Moody algebra of rank 2. We give a necessary and sufficient condition for the crystal graph of the Lakshmibai-Seshadri path crystal to be connected for an arbitrary integral weight λ.

Equivariant K -theory of semi-infinite flag manifolds and the Pieri–Chevalley formula

We propose a definition of equivariant (with respect to an Iwahori subgroup) $K$-theory of the formal power series model $\mathbf{Q}_{G}$ of semi-infinite flag manifold and prove the Pieri-Chevalley formula, which describes the product, in the $K$-theory of $\mathbf{Q}_{G}$, of the structure sheaf of a semi-infinite Schubert variety with a line bundle (associated to a dominant integral weight) over $\mathbf{Q}_{G}$. In order to achieve this, we provide a number of fundamental results on $\mathbf{Q}_{G}$ and its Schubert subvarieties including the Borel-Weil-Bott theory, whose special case is conjectured in [A. Braverman and M. Finkelberg, Weyl modules and $q$-Whittaker functions, Math. Ann. 359 (2014), 45--59]. One more ingredient of this paper besides the geometric results above is (a combinatorial version of) standard monomial theory for level-zero extremal weight modules over quantum affine algebras, which is described in terms of semi-infinite Lakshmibai-Seshadri paths. In fact, in our Pieri-Chevalley formula, the positivity of structure coefficients is proved by giving an explicit representation-theoretic meaning through semi-infinite Lakshmibai-Seshadri paths.

Path model for an extremal weight module over the quantized hyperbolic Kac-Moody algebra of rank 2

Let be a hyperbolic Kac-Moody algebra of rank 2, and set where are the fundamental weights. Denote by the extremal weight module of extremal weight λ with the extremal weight vector, and by the crystal basis of with the element corresponding to We prove that (i) is connected, (ii) the subset of elements of weight μ in is a finite set for every integral weight μ, and (iii) every extremal element in is contained in the Weyl group orbit of (iv) is irreducible. Finally, we prove that the crystal basis is isomorphic, as a crystal, to the crystal of Lakshmibai-Seshadri paths of shape λ.

On Weights Which Admit Reproducing Kernel of Szegő Type

In this paper we generalize the concept of the Szego kernel by putting the weight of integration in the definition of the inner product in the Szego space. We give some sufficient conditions for the weight in order for the Szego kernel of the correspoding space to exist. We give examples of weights on unit ball for which there is no Szego kernel of the corresponding Szego space. Then using biholomorphisms we prove that there exist such weights for a large class of domains. At the end we show that weighted Szego kernel depends continuously in some sense on weight of integration.

Bases of spaces of harmonic weak Maass forms and Shintani lifts of harmonic weak Maass forms

We construct bases of the space of harmonic weak Maass forms of weight $$\kappa \in \frac{1}{2}{\mathbb {Z}}$$ . Using these bases, we obtain a Shintani lift from a positive integral weight harmonic weak Maass form to a half-integral weight harmonic weak Maass form, which reduces to the classical Shintani lift on the space of cusp forms.

All modular forms of weight 2 can be expressed by Eisenstein series

We show that every elliptic modular form of integral weight greater than 1 can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central mathrm{L}-values present in all previous work. For weights greater than 2, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice.

Future greening of the Earth may not be as large as previously predicted

Changes in global vegetation growth and its drivers during recent decades have been well studied with satellite data, ecosystem models and field experiments. However, a systematic understanding of how global vegetation will respond to projected changes in climate and atmospheric composition is still lacking. Here, we analyze changes in projected global leaf area index (LAI) from 16 Coupled Model Intercomparison Project phase 5 (CMIP5) and 17 phase 6 (CMIP6) Earth System Models (ESMs) during the 21st century under future scenarios RCP2.6, RCP4.5 and RCP8.5 and future scenarios SSP1–2.6, SSP2–4.5, SSP3–7.0 and SSP5–8.5, respectively. In addition to the widely-used multi-model ensemble mean (MME) method, we employed the Reliability Ensemble Averaging (REA) strategy to integrate the modeled results. The REA integration weights were determined by combining the performance of the ESMs in simulating present-day LAI changes, which was based on a comparison with three long-term remote sensing LAI data sets, with their convergence of future predictions. The results suggest that global LAI will increase under all seven future scenarios with both integration methods. The magnitude of LAI growth is expected to increase with the forcing levels of the scenarios. The ESMs integrated with the REA weights predicted significantly smaller magnitudes and lower uncertainties in global LAI growth by the end of the 21st century than those integrated with the MME method. Both REA and MME results suggest that the growth in atmospheric CO2 concentration is the main positive driver of the projected increment of global LAI, which is partly offset by the negative effects of global warming. Our improved comprehensive prediction based on CMIP5 and CMIP6 ESMs using the REA integration strategy provides a more robust estimation of global vegetation change during the 21st century, and is expected to help better understand the state of the planet in the coming decades.

Comprehensive evaluation model of wind farm site selection based on ideal matter element and grey clustering

China has carried out the relevant research on the evaluation of the wind farm site selection, but the evaluation index are mainly from the economic aspect and comprehensive benefit evaluation research of the wind farm site selection exists defects. In order to comprehensively to consider the explicit and implicit benefits of wind farms, firstly, a comprehensive evaluation index system for the wind farm site selection is established, which mainly includes economic, environmental and social aspects. Secondly, an integrated weight model with subjective and objective weight method is constructed, and the ideal matter element extension evaluation model and the grey clustering evaluation model for the wind farms site selection are proposed. Finally, case studies show that the proposed comprehensive evaluation model can rank the comprehensive benefits of wind farms site selection. Based on the comprehensive evaluation results of wind farm site selection, a comprehensive evaluation model of wind farm site selection is proposed to this paper, which can provide the decision basis of the decision makers. This paper establishes an index evaluation model from three aspects of economy, environment and society, examines the advantages and disadvantages of each access location of wind power system, so as to make a reasonable site selection. In view of the current development of the electricity market, this paper constructs a differential weighting model based on the built evaluation model for wind power system location. The simulation results show that the proposed model can effectively guide the location optimization of wind power system.

LEVEL-ZERO VAN DER KALLEN MODULES AND SPECIALIZATION OF NONSYMMETRIC MACDONALD POLYNOMIALS AT t = ∞

Let λ ∈ P+ be a level-zero dominant integral weight, and w the coset representative of minimal length for a coset in W/Wλ, where Wλ is the stabilizer of λ in a finite Weyl group W. In this paper, we give a module $$ {\mathbbm{K}}_w^{-}\left(\uplambda \right) $$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at t = ∞ of the nonsymmetric Macdonald polynomial Ewλ(q, t) multiplied by a certain explicit finite product of rational functions of q of the form (1 − q−r)−1 for a positive integer r. This module $$ {\mathbbm{K}}_w^{-}\left(\uplambda \right) $$ (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module $$ {V}_w^{-}\left(\uplambda \right) $$ by the sum of the submodules $$ {V}_z^{-}\left(\uplambda \right) $$ for all those coset representatives z of minimal length for cosets in W/Wλ such that z > w in the Bruhat order < on W.

Some remarks on the Fourier coefficients of cusp forms

In this paper, we consider the angular changes of Fourier coefficients of half integral weight cusp forms and sign changes of [Formula: see text]-exponents of generalized modular functions.