Frustration Index Research Articles

Structural and Magnetic Properties of the B-Site-Ordered Double Perovskites Ln2NiTiO6 (Ln = La, Pr, and Nd)

Abstract Structural and magnetic properties of B-site-ordered double perovskites Ln2NiTiO6 (Ln = La, Pr, and Nd) were investigated. The charge formulas were revealed to be Ln23+Ni2+Ti4+O6, which gave a geometrically frustrated face-centered cubic lattice consisting of magnetic Ni2+ (S = 1) cations. In spite of the geometrical spin frustration of Ni2+ spins, the samples show an antiferromagnetic-like magnetic transition at temperatures TN close to their absolute values of the Weiss temperature θ, giving small frustration indices f = |θ|/TN ∼ 1. The small f values are considered to be due to some amount of anti-site disorder between Ni2+ and Ti4+ originating in the small valence difference. The suppression of magnetic ordering in the geometrically frustrated system is strongly affected by the B-site cation ordering in the double perovskites.

Ba9RE2(SiO4)6 (RE = Ho-Yb): A Family of Rare-Earth-Based Honeycomb-Lattice Magnets.

Rare-earth (RE)-based honeycomb-lattice materials with strong spin-orbit coupled Jeff = 1/2 moments have attracted great interest as a platform to realize the Kitaev quantum spin liquid (QSL) state. Herein, we report the discovery of a family of RE-based honeycomb-lattice magnets Ba9RE2(SiO4)6 (RE = Ho-Yb), which crystallize into the rhombohedral structure with the space group R3̅. In these serial compounds, magnetic RE3+ ions are arranged on a perfect honeycomb lattice within the ab-plane and stacked in the "ABCABC"-type fashion along the c-axis. All synthesized Ba9RE2(SiO4)6 (RE = Ho-Yb) polycrystals exhibit the dominant antiferromagnetic interaction and absence of magnetic order down to 2 K. In combination with the magnetization and electron spin resonance results, magnetic behaviors are discussed for the compounds with different RE ions. Moreover, the as-grown Ba9Yb2(SiO4)6 single crystals show large magnetic frustration with frustration index f = θCW/TN > 8 and no long-range magnetic ordering down to 0.15 K, being a possible QSL candidate material. These series of compounds are attractive for exploring the exotic magnetic phases of Kitaev materials with 4f electrons.

Crystal Growth and Magnetism of Transition Metal Pyrochlore Fluorides.

Geometric magnetic frustration arises when the geometry of a structure prevents the simultaneous fulfillment of nearest-neighbor antiferromagnetic interactions and is commonly observed in lattices that exhibit a triangular topology, such as those found in the pyrochlore structure. Via a mild hydrothermal route, we have synthesized seven quaternary β-pyrochlore-related fluorides AxM2+xM3+(2-x)F6 (A = Cs and Rb; M2+ = Co2+, Ni2+, and Zn2+; and M3+ = V3+ and Fe3+). Crystal structures and compositions were determined using a combination of single-crystal X-ray diffraction and energy-dispersive spectroscopy. After adjusting the reaction conditions, phase-pure products of AxM2+xM3+(2-x)F6 were obtained. The magnetic susceptibility and isothermal magnetization data for all seven compounds were collected to interpret the magnetic behavior, which ranged from paramagnetic to antiferromagnetic with and without a ferromagnetic component. We found that the magnetic behavior of the AxM2+xV3+(2-x)F6 pyrochlore structures strongly depends on the presence or absence of unpaired electrons on the M2+ position. The titled pyrochlore compounds, with the exception of the Zn-analogue, can be considered frustrated materials, with frustration indices in the range of 6-13.

Analyzing and visualizing polarization and balance with signed networks: the U.S. Congress case study

Abstract Signed networks and balance theory provide a natural setting for real-world scenarios that show polarization dynamics, positive/negative relationships and political partisanship. For example, they have been proven effective in studying the increasing polarization of the votes in the two chambers of the U.S. Congress from World War II on Andris, Lee, Hamilton, Martino, Gunning & Selden (2015, PLoS ONE, 10, 1–14) and Aref & Neal (2020, Sci. Rep., 10, 1–10). To provide further insights into this particular case study, we propose the application of a pipeline to analyze and visualize a signed graphs configuration based on the exploitation of the corresponding Laplacian matrix spectral properties. The overall methodology is comparable with others based on the frustration index, but it has at least two main advantages: first, it requires a much lower computational cost and second, it allows for a quantitative and visual assessment of how arbitrarily small subgraphs (even single nodes) contribute to the overall balance (or unbalance) of the network. The proposed pipeline allows the exploration of polarization dynamics shown by the U.S. Congress from 1945 to 2020 at different resolution scales. In fact, we are able to spot and point out the influence of some (groups of) congressmen in the overall balance, as well as to observe and explore polarizations evolution of both chambers across the years.

Ferrimagnetic Ordering and Spin-Glass State in Diluted GdFeO3-Type Perovskites (Lu0.5Mn0.5)(Mn1-xTix)O3 with x = 0.25, 0.50, and 0.75.

ABO3 perovskite materials with small cations at the A site, especially those with ordered cation arrangements, have attracted a great deal of interest because they show unusual physical properties and deviations from the general characteristics of perovskites. In this work, perovskite solid solutions (Lu0.5Mn0.5)(Mn1-xTix)O3 with x = 0.25, 0.50, and 0.75 were synthesized by means of a high-pressure, high-temperature method at approximately 6 GPa and approximately 1550 K. All the samples crystallize in the GdFeO3-type perovskite structure (space group Pnma) and have random distributions of the small Lu3+ and Mn2+ cations at the A site and Mn4+/3+/2+ and Ti4+ cations at the B site, as determined by Rietveld analysis of high-quality synchrotron X-ray powder diffraction data. Lattice parameters are a = 5.4431 Å, b = 7.4358 Å, c = 5.1872 Å (for x = 0.25); a = 5.4872 Å, b = 7.4863 Å, c = 5.2027 Å (for x = 0.50); and a = 5.4772 Å, b = 7.6027 Å, c = 5.2340 Å (for x = 0.75). Despite a significant dilution of the A and B sublattices by non-magnetic Ti4+ cations, the x = 0.25 and 0.50 samples show long-range ferrimagnetic order below TC = 89 K and 36 K, respectively. Mn cations at both A and B sublattices are involved in the long-range magnetic order. The x = 0.75 sample shows a spin-glass transition at TSG = 6 K and a large frustration index of approximately 22. A temperature-independent dielectric constant was observed for x = 0.50 (approximately 32 between 5 and 150 K) and for x = 0.75 (approximately 50 between 5 and 250 K).

Molybdenum chloride double perovskites: dimensionality control of optical and magnetic properties.

Halide double perovskites are a promising class of semiconducting materials for applications in solar cells and other optoelectronic devices. Recently, there has been a surge of interest in these materials to study phenomena beyond optoelectronics, especially magnetism. Here, we report three new Mo3+ (4d3) based chloride double perovskites: a 3-D rock-salt ordered Cs2NaMoCl6, a 1-D chain (MA)2AgMoCl6 and a Dion-Jacobson type 2-D layered (1,4-BDA)2AgMoCl8 (MA = methylammonium; 1,4-BDA = 1,4-butanediammonium). Their structures and dimensionalities can be tuned by means of the A-cation. The measured bandgaps are relatively narrow (2.0-2.1 eV) which show a blueshift on reducing the dimensionality. At low temperatures, we observe antiferromagnetic coupling between the nearest-neighbour Mo3+ ions in all these systems. Cs2NaMoCl6 shows stronger coupling with a frustration index f of 5 which we attribute to the geometrically frustrating fcc lattice of Mo3+ ions. This work expands the scope of halide double perovskites beyond main group metals and beyond optoelectronics, and we hope that it will lead to future developments in magnetic halide perovskites.

Синтез, структурные и магнитные свойства NaZnFe-=SUB=-2-=/SUB=-(VO-=SUB=-4-=/SUB=-)-=SUB=-3-=/SUB=-

A new magnetic compound NaZnFe2(VO4)3 obtained by solid-phase synthesis has been studied using X-ray diffraction, Mössbauer spectroscopy, electron paramagnetic resonance, measurement of the temperature dependence of the dielectric permeability, and magnetometry. The crystalline structure of NaZnFe2(VO4)3 is described by a triclinic spatial group of symmetry P1 with the parameters of an elementary crystalline chain: a = 6.74318 (7) Å, b = 8.1729 (1) Å, c = 9.8421 (1) Å, α = 106.2611 (9)º, β = 104.55 (1)º, γ = 102.337 (1)º, V = 479.88 (1) Å3, Z = 2. Magnetic Fe3+ cations in the cell occupy six positions populated together with diamagnetic Zn2+ cations, which leads to states of magnetic inhomogeneity and local violation of charge neutrality. Data from resonance and magnetic studies of NaZnFe2(VO4)3 confirm the main role of high-spin Fe3+ iron cations in the formation of magnetism with competing exchange magnetic interactions and a high value of the frustration index. It is shown that the magnetic subsystem of a sample with a negative asymptotic Neel temperature undergoes a magnetic transition from the paramagnetic state to the magnetic state of the spin glass when the temperature decreases. T.V. Drokina 1, M.S. Molokeev M.C. 1, 2, D.A. Velikanov 1, O.A. Bayukov 1, A.M. Vorotynov 1, A.L. Freidman 1, G.A. Petrakovskii 1 1 L.V. Kirensky Institute of Physics, Federal Research Center KSC Siberian Branch of Russian Academy of Science, Krasnoyarsk, 660036, Russia 2 Siberian Federal University, Krasnoyarsk, 660074, Russia

Frustration-critical signed graphs

A signed graph $(G,\Sigma)$ is a graph $G$ together with a set $\Sigma \subseteq E(G)$ of negative edges. A circuit is positive if the product of the signs of its edges is positive. A signed graph $(G,\Sigma)$ is balanced if all its circuits are positive. The frustration index $l(G,\Sigma)$ is the minimum cardinality of a set $E \subseteq E(G)$ such that $(G-E,\Sigma-E)$ is balanced, and $(G,\Sigma)$ is $k$-critical if $l(G,\Sigma) = k$ and $l(G-e, \Sigma - e)<k$, for every $e \in E(G)$. We study decomposition and subdivision of critical signed graphs and completely determine the set of $t$-critical signed graphs, for $t \leq 2$. Critical signed graphs are characterized. We then focus on non-decomposable critical signed graphs. In particular, we characterize the set $S^*$ of non-decomposable $k$-critical signed graphs not containing a decomposable $t$-critical signed subgraph for every $t \leq k$. We prove that $S^*$ consists of cyclically 4-edge-connected projective-planar cubic graphs. Furthermore, we construct $k$-critical signed graphs of $S^*$ for every $k \geq 1$.

Total graph of a signed graph

The total graph is built by joining the graph to its line graph by means of the incidences. We introduce a similar construction for signed graphs. Under two similar defnitions of the line signed graph, we defne the corresponding total signed graph and we show that it is stable under switching. We consider balance, the frustration index and frustration number, and the largesteigenvalue. In the regular case we compute the spectrum of the adjacency matrix of the total graph and the spectra of certain compositions, and we determine some with exactly two main eigenvalues.

A simple and effective iterated greedy algorithm for structural balance in signed networks

Recently, problems related to structural balance have received widespread attention in the research fields of signed networks, such as signed network partition, community detection and correlation clustering. Structural balance problems aim to find a partition of a signed network such that the frustration index, the sum of positive edges between subsets and negative edges within subsets, is minimized, which is known to be nondeterministic polynomial-time (NP)-complete and remains an open challenge. Many heuristic and meta-heuristic algorithms are developed to minimize the frustration index. However, most extant algorithms require quite a few parameters and are not efficient enough for large signed networks. In this study, we present a simple and effective iterated greedy (IG) algorithm for solving the structural balance problems with the objective of frustration minimization. In the algorithm, a constructive greedy heuristic is proposed to generate and reconstruct solutions to the problem with high efficiency. A two-stage local search procedure is designed to exploit the search space at different levels of granularity. An adaptive destruction method is developed to enhance the exploitation ability of the algorithm in the early stage and maintain the exploration ability later. Additionally, two acceleration methods are proposed to help the algorithm get a better performance in large signed networks. The experimental results indicate that the proposed IG algorithm outperforms other meta-heuristics in the literature especially in terms of the computational time.

Spin freezing in the van der Waals material Mn2Ga2S5

Geometrical frustration in low-dimensional magnetic systems has been an intriguing research aspect, where the suppression of conventional magnetic order may lead to exotic ground states such as spin glass or spin liquid. In this work we report the synthesis and magnetism study of the monocrystalline Mn2Ga2S5, featuring both the van der Waals structure and a bilayered triangular Mn lattice. Magnetic susceptibility reveals a significant antiferromagnetic interaction with a Curie–Weiss temperature θ w ∼ –260 K and a high spin S = 5/2 Mn2+ state. However, no long range magnetic order has been found down to 2 K, and a spin freezing transition is found to occur at around 12 K well below its θ w. This yields a frustration index of f = –θ w/T f ≈ 22, an indication that the system is highly frustrated. The absence of a double-peak structure in magnetic specific heat compared with the TM 2S4 compounds implies that the spin freezing behavior in Mn2Ga2S5 is a result of the competition between exchange interactions and the 2D crystalline structure. Our results suggest that the layered Mn2Ga2S5 would be an excellent candidate for investigating the physics of 2D magnetism and spin disordered state.

Residue-Frustration-Based Prediction of Protein-Protein Interactions Using Machine Learning.

The study of protein-protein interactions (PPIs) is important in understanding the function of proteins. However, it is still a challenge to investigate the transient protein-protein interaction by experiments. Hence, the computational prediction for protein-protein interactions draws growing attention. Statistics-based features have been widely used in the studies of protein structure prediction and protein folding. Due to the scarcity of experimental data of PPI, it is difficult to construct a conventional statistical feature for PPI prediction, and the application of statistics-based features is very limited in this field. In this paper, we explored the application of frustration, a statistical potential, in PPI prediction. By comparing the energetic contribution of the extra stabilization energy from a given residue pair in the native protein with the statistics of the energies, we obtained the residue pair's frustration index. By calculating the number of residue pairs with a high frustration index, the highly frustrated density, a residue-frustration-based feature, was then obtained to describe the tendency of residues to be involved in PPI. Highly frustrated density, as well as structure-based features, were then used to describe protein residues and combined with the long short-term memory (LSTM) neural network to predict PPI residue pairs. Our model correctly predicted 75% dimers when only the top 2‰ residue pairs were selected in each dimer. Our model, which considers the statistics-based features, is significantly different from the models based on the chemical features of residues. We found that frustration can effectively describe the tendency of residue to be involved in PPI. Frustration-based features can replace chemical features to combine with machine learning and realize the better performance of PPI prediction. It reveals the great potential of statistical potential such as frustration in PPI prediction.

The Highly Frustrated 5d2 Double Perovskite Doppelgängers, SrLaMgReO6 and SrLaLiOsO6. A Comparison including Isostructural La2LiReO6.

The synthesis and characterization of the double perovskite SrLaLiOsO6 is presented. It is isostructural (P21/n) and isoelectronic (5d2) with SrLaMgReO6, which has been reported previously. The cell volumes are the same to within 1.4%: i.e., these perovskites are doppelgängers. In a previous study SrLaMgReO6 showed no sign of spin order to 2 K. New data at lower temperatures disclose a maximum in the dc susceptibility near 1.5 K. As the Curie-Weiss (C-W) temperature (Θ) for this material is -161 K, an enormous frustration index, f ≈ 100, is implied (f = |Θ|/Tord). On the other hand, SrLaLiOsO6 does not follow the C-W law over the investigated susceptibility range, 2-300 K. Fitting with an added temperature independent term (TIP) gives μeff = 1.96 μB, Θ = -102 K, and TIP = 1.01 × 10-3 emu/mol. A clear zero-field-cooled (ZFC), field-cooled (FC) divergence in the dc data occurs at ∼10 K, suggesting a much reduced frustration index, f ≈ 10, relative to SrLaMgReO6. The real part of the ac susceptibility data, χ'max, shows a frequency shift that is consistent with a spin glass ground state according to the Mydosh criterion. Heat capacity data for SrLaLiOsO6 show no sign of a λ peak at 10 K and a linear dependence on temperature below 10 K, also supporting a spin glass ground state. A spin frozen ground state for SrLaMgReO6 could not be established from χ' data due to a much weaker signal. Nonetheless, the 10-fold difference in f between these doppelgänger materials is remarkable. It is possible that the enhanced covalency with the oxide ligands for Os6+ relative to Re5+ plays a major role here. Finally, a comparison with isostructural La2LiReO6 (with a much smaller f ≈ 4) is made and a correlation between the frustration level and the sense of the local distortion of the Re(Os)-O octahedron is pointed out.

Enhanced Learning Experiences Based on Regulatory Fit Theory Using Affective State Detection

Predicting learners' affective states through the internet has great impact on their learning experiences. Hence, it is important for an intelligent tutoring system (ITS) to consider the learners' affective state in their learning models. This research work focuses on finding learners' frustration levels during learning. Motivating the learners appropriately can enhance their learning experiences. Therefore, the authors also bring in a strategy to respond to learners' affective states in order to motivate them. This work uses Behavioral theory for goal generation, and frustration index is calculated. Based on the frustration level of the learner, motivational messages are displayed to the learners using Regulatory fit theory. The authors evaluated the model using t-test by collecting learners' data from MoodleCloud. The results of the evaluation demonstrate that 80% of the learners' performance significantly increases statistically as an impact of motivational messages provided in response to the learners' frustration.

Characterizing attitudinal network graphs through frustration cloud

Attitudinal network graphs are signed graphs where edges capture an expressed opinion; two vertices connected by an edge can be agreeable (positive) or antagonistic (negative). A signed graph is called balanced if each of its cycles includes an even number of negative edges. Balance is often characterized by the frustration index or by finding a single convergent balanced state of network consensus. In this paper, we propose to expand the measures of consensus from a single balanced state associated with the frustration index to the set of nearest balanced states. We introduce the frustration cloud as a set of all nearest balanced states and use a graph-balancing algorithm to find all nearest balanced states in a deterministic way. Computational concerns are addressed by measuring consensus probabilistically, and we introduce new vertex and edge metrics to quantify status, agreement, and influence. We also introduce a new global measure of controversy for a given signed graph and show that vertex status is a zero-sum game in the signed network. We propose an efficient scalable algorithm for calculating frustration cloud-based measures in social network and survey data of up to 80,000 vertices and half-a-million edges. We also demonstrate the power of the proposed approach to provide discriminant features for community discovery when compared to spectral clustering and to automatically identify dominant vertices and anomalous decisions in the network.

A Geometrically Frustrated Family of MIIMIIIF5(H2O)2 Mixed-Metal Fluorides with Complex Magnetic Interactions.

Inverse weberites are of interest as geometrically frustrated magnetic materials due to their unique cation arrangement. We have synthesized nine isostructural materials that adopt the inverse weberite crystal structure, which consists of cross-linked kagome layers. These materials, having the general formula MIIMIIIF5(H2O)2 (MII = Co, Mn, Ni, Zn; MIII = Ga, Cr, Fe, V), were synthesized using mild hydrothermal conditions, which yielded phase-pure samples after optimization of the reaction conditions. Their crystal structures and optical, thermal, and magnetic behavior were characterized using single-crystal X-ray diffraction, UV-vis spectroscopy, thermogravimetric analysis, and measurement of the magnetic susceptibility and isothermal magnetization data, respectively. Three distinct types of magnetism were observed, including simple paramagnetism, antiferromagnetism, and canted antiferromagnetism; the last type is accompanied by a high frustration index fin the range 4.16-8.09. We demonstrated that the magnetic behavior of inverse weberites depends on the presence or absence of unpaired-electron-containing cations on the two distinct crystallographic sites, which can be employed for the prediction of the magnetic properties of other compounds in this rich and diverse family.

Bounds for the extremal eigenvalues of gain Laplacian matrices

A complex unit gain graph (T-gain graph), Φ=(G,φ) is a graph where the function φ assigns a unit complex number to each orientation of an edge of G, and its inverse is assigned to the opposite orientation. A T-gain graph Φ is balanced if the product of the edge gains of each cycle (with a fixed orientation) is 1. Signed graphs are special cases of T-gain graphs.The adjacency matrix of Φ, denoted by A(Φ) is defined canonically. The gain Laplacian for Φ is defined as L(Φ)=D(Φ)−A(Φ), where D(Φ) is the diagonal matrix with diagonal entries are the degrees of the vertices of G. The minimum number of vertices (resp., edges) to be deleted from Φ in order to get a balanced gain graph is the frustration number (resp, frustration index). We show that frustration number and frustration index are bounded below by the smallest eigenvalue of L(Φ). We provide several lower and upper bounds for extremal eigenvalues of L(Φ) in terms of different graph parameters such as the number of edges, vertex degrees, and average 2-degrees. The signed graphs are particular cases of the T-gain graphs, all the bounds established in this paper hold for signed graphs. Most of the bounds established here are new for signed graphs. Finally, we perform comparative analysis for all the obtained bounds in the paper with the state-of-the-art bounds available in the literature for randomly generated Erdős-Reýni graphs.Some of the major highlights of our paper are the gain-dependent bounds, limit convergence of the bounds to the extremal eigenvalues, and optimal extremal bounds obtained by posing optimization problems to achieve the best possible bounds.

Rare-earth pyrocoboltates [formula omitted]Co2O7 (R = Ce, Yb): Effective spin-[formula omitted] antiferromagnetic insulators with strong geometrical frustration

The two new rare-earth pyrocoboltates R2Co2O7 (R = Ce, Yb) adopt the layered monoclinic Ca2Nb2O7-type structure with the space group P21(No. 4). Both compounds have been investigated by DC magnetic susceptibility χdc(T), AC magnetic susceptibility χac(T) and isothermal magnetization M(B) measurements. The observed effective magnetic moment of Ce2Co2O7 indicates a contribution from the Co sublattice while that of Yb2Co2O7 is mainly from the localized 4f13 electronic configuration of Yb3+ ion. A magnetic ordering of the antiferromagnetic-type is observed in Ce2Co2O7 at TN≃ 34 K. This ordering is accompanied by the opening of a magnetic spectral gap, Δ=57.7 K (~ 5.0 meV) in the spin-wave spectrum and it survives in an external magnetic field of 7 T. In addition to the transition at TN,χac(T) measurements reveal a dome-like feature centered at ≃ 20 K in high frequency which indicates the presence of possible short-range correlation. In Yb2Co2O7, a magnetic phase transition is observed below TN=2.94 K, which is suppressed below 1.9 K in an external magnetic field of 3 T. In the low-T region where the crystal electric field effects can be neglected, an effective spin-1/2 state is realized in Yb2Co2O7, with a Landé-g factor of 3.76 which is comparable to values observed from electron spin resonance (ESR) measurements on related systems. In addition, M(B) of Yb2Co2O7 shows a metamagnetic-like feature at a critical field of HC≈2.5 T. Signatures from both compounds reveal the presence of strong magnetic frustrations with frustration index f of 9.17 and 15.12 for Ce2Co2O7 and Yb2Co2O7, respectively.

Kinetic Frustration Effects on Dense Two-Dimensional Packings of Convex Particles and Their Structural Characteristics.

The study of hard-particle packings is of fundamental importance in physics, chemistry, cell biology, and discrete geometry. Much of the previous work on hard-particle packings concerns their densest possible arrangements. By contrast, we examine kinetic effects inevitably present in both numerical and experimental packing protocols. Specifically, we determine how changing the compression/shear rate of a two-dimensional packing of noncircular particles causes it to deviate from its densest possible configuration, which is always periodic. The adaptive shrinking cell (ASC) optimization scheme maximizes the packing fraction of a hard-particle packing by first applying random translations and rotations to the particles and then isotropically compressing and shearing the simulation box repeatedly until a possibly jammed state is reached. We use a stochastic implementation of the ASC optimization scheme to mimic different effective time scales by varying the number of particle moves between compressions/shears. We generate dense, effectively jammed, monodisperse, two-dimensional packings of obtuse scalene triangle, rhombus, curved triangle, lens, and "ice cream cone" (a semicircle grafted onto an isosceles triangle) shaped particles, with a wide range of packing fractions and degrees of order. To quantify these kinetic effects, we introduce the kinetic frustration index K, which measures the deviation of a packing from its maximum possible packing fraction. To investigate how kinetics affect short- and long-range ordering in these packings, we compute their spectral densities χ̃V(k) and characterize their contact networks. We find that kinetic effects are most significant when the particles have greater asphericity, less curvature, and less rotational symmetry. This work may be relevant to the design of laboratory packing protocols.

Magnetic-Sparseness and Schr\xf6dinger Operators on Graphs

We study magnetic Schrödinger operators on graphs. We extend the notion of sparseness of graphs by including a magnetic quantity called the frustration index. This notion of magnetic-sparseness turns out to be equivalent to the fact that the form domain is an ell ^{2} space. As a consequence, we get criteria of discreteness for the spectrum and eigenvalue asymptotics.