Assumption Of Symmetry Research Articles

Linear Programming with Unitary-Equivariant Constraints

Unitary equivariance is a natural symmetry that occurs in many contexts in physics and mathematics. Optimization problems with such symmetry can often be formulated as semidefinite programs for a dp+q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d^{p+q}$$\\end{document}-dimensional matrix variable that commutes with U⊗p⊗U¯⊗q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U^{\\otimes p} \\otimes {\\bar{U}}^{\\otimes q}$$\\end{document}, for all U∈U(d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U \\in \ extrm{U}(d)$$\\end{document}. Solving such problems naively can be prohibitively expensive even if p+q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p+q$$\\end{document} is small but the local dimension d is large. We show that, under additional symmetry assumptions, this problem reduces to a linear program that can be solved in time that does not scale in d, and we provide a general framework to execute this reduction under different types of symmetries. The key ingredient of our method is a compact parametrization of the solution space by linear combinations of walled Brauer algebra diagrams. This parametrization requires the idempotents of a Gelfand–Tsetlin basis, which we obtain by adapting a general method inspired by the Okounkov–Vershik approach. To illustrate potential applications of our framework, we use several examples from quantum information: deciding the principal eigenvalue of a quantum state, quantum majority vote, asymmetric cloning and transformation of a black-box unitary. We also outline a possible route for extending our method to general unitary-equivariant semidefinite programs.

Hydrodynamic 3D Simulation of Roche Lobe Overflow in High-mass X-Ray Binaries

Abstract While binary merger events have been an active area of study in both simulations and observational work, the formation channels by which a high-mass star extends from Roche lobe overflow (RLO) in a decaying orbit of a black-hole (BH) companion to a binary black-hole (BBH) system merits further investigation. Variable length-scales must be employed to accurately represent the dynamical fluid transfer and morphological development of the primary star as it conforms to a diminishing Roche lobe under the runaway influence of the proximal BH. We have simulated and evolved binary mass flow under these conditions to better identify the key transitional processes from RLO to BBHs. We demonstrate a new methodology to model RLO systems to unprecedented resolution simultaneously across the envelope, donor wind, tidal stream, and accretion disk regimes without reliance upon previously universal symmetry, mass flux, and angular momentum flux assumptions. We have applied this method to the semidetached high-mass X-ray binary M33 X-7 in order to provide a direct comparison to recent observations of an RLO candidate system at two overflow states of overfilling factors f = 1.01 and f = 1.1. We found extreme overflow (f = 1.1) to be entirely conservative in both mass and angular momentum transport, forming a conical L1 tidal stream of density and deflected angle comparable to existing predictions. This case lies within the unstable mass transfer (MT) regime as recently proposed of M33 X-7. The f = 1.01 case differed in stream geometry, accretion disk size, and efficiency, demonstrating nonconservative stable MT through a ballistic uniform-width stream. The nonconservative and stable nature of the f = 1.01 case MT also suggests that existing assumptions of semidetached binaries undergoing RLO may mischaracterize their role and distribution as progenitors of BBHs and common envelopes.

Learning fluid physics from highly turbulent data using sparse physics-informed discovery of empirical relations (SPIDER)

We show how a complete mathematical model of a physical process can be learned directly from data via a hybrid approach combining three simple and general ingredients: physical assumptions of smoothness, locality and symmetry, a weak formulation of differential equations and sparse regression. To illustrate this, we extract a complete system of governing equations of fluid dynamics – the Navier–Stokes equation, the continuity equation and the boundary conditions – as well as the pressure-Poisson and energy equations, from numerical data describing a highly turbulent channel flow in three dimensions. Whether they represent known or unknown physics, relations discovered using this approach take the familiar form of partial differential equations, which are easily interpretable and readily provide information about the relative importance of different physical effects. The proposed approach offers insight into the quality of the data, serving as a useful diagnostic tool. It is also remarkably robust, yielding accurate results for very high noise levels, and should thus be well suited for analysis of experimental data.

The Upper Extremity Lymphatic System Is Not Symmetrical in Individuals: An Anatomic Study Utilizing ICG Lymphography and SPECT/CT Lymphoscintigraphy.

We evaluated whether superficial lymphatic anatomy and functional lymph node drainage are symmetric between the right and left upper extremities of healthy female volunteers, and if handedness is associated with symmetry of superficial lymphatic anatomy. Symmetry of lymphatic anatomy has been assumed historically. This assumption of individual anatomic symmetry is being utilized clinically and in research without validation. 36 normal female volunteers underwent bilateral indocyanine green (ICG) lymphography and lymphoscintigraphy of the upper extremities. Eight collecting vessel pathways of each upper extremity were mapped on ICG lymphography. 13 lymph node groups were visualized on lymphoscintigraphy. Symmetry of lymphatic anatomy and functional drainage were established by comparing the right and left extremities of each participant. Hand dominance was assessed by hand grip strength on a hand dynamometer. Among the 36 participants, 10 (28%) showed symmetry of all eight upper extremity lymphatic pathways with ICG. However, only 1 (3%) participant demonstrated complete symmetry amongst the 13 lymph node groups. Total symmetry of lymphatic channels was observed on ICG in seven (39%) participants with hand dominance and three (17%) participants without hand dominance (X2 = 2.215, P = 0.137). Lymphatic anatomy and functional drainage of the upper extremities are not consistently symmetric. Functional nodal drainage as demonstrated by lymphoscintigraphy shows less symmetry than anatomic studies of lymphatic channels using ICG. Symmetric lymphatic anatomy does not appear to correlate with hand dominance. These findings challenge the prevailing assumption of left-right lymphatic symmetry.

A Statistical Methodology for Evaluating Asymmetry after Normalization with Application to Genomic Data

This study evaluates the symmetry of data distributions after normalization, focusing on various statistical tests, including a few explored test named Rp. We apply normalization techniques, such as variance stabilizing transformations, to ribonucleic acid sequencing data with varying sample sizes to assess their effectiveness in achieving symmetric data distributions. Our findings reveal that while normalization generally induces symmetry, some samples retain asymmetric distributions, challenging the conventional assumption of post-normalization symmetry. The Rp test, in particular, shows superior performance when there are variations in sample size and data distribution, making it a preferred tool for assessing symmetry when applied to genomic data. This finding underscores the importance of validating symmetry assumptions during data normalization, especially in genomic data, as overlooked asymmetries can lead to potential inaccuracies in downstream analyses. We analyze postmortem lateral temporal lobe samples to explore normal aging and Alzheimer’s disease, highlighting the critical role of symmetry testing in the accurate interpretation of genomic data.

On Augmented Dimensional Analysis

We present an innovative approach to dimensional analysis, referred to as augmented dimensional analysis and based on a representation theorem for complete quantity functions with a scaling-covariant scalar representation. This new theorem, grounded in a purely algebraic theory of quantity spaces, allows the classical π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\pi $\\end{document} theorem to be restated in an explicit and precise form and its prerequisites to be clarified and relaxed. Augmented dimensional analysis, in contrast to classical dimensional analysis, is guaranteed to take into account all relations among the quantities involved. Several examples are given to show that the information thus gained, together with symmetry assumptions, can lead to new or stronger results. We also explore the connection between dimensional analysis and matroid theory, elucidating combinatorial aspects of dimensional analysis. It is emphasized that dimensional analysis rests on a principle of covariance.

How Do Substorms Influence Hemispheric Asymmetries in Equivalent Currents?

AbstractIonospheric dynamics exhibits a distinct hemispheric asymmetry, influenced primarily by the Interplanetary Magnetic Field (IMF) By component, dipole tilt, or a combination of both. Previous studies have indicated a reduction in these asymmetries during substorms. In this study, we conduct a superposed epoch analysis using ground magnetometer data from the northern hemisphere to examine the impact of substorms on ionospheric current asymmetry. This analysis uses the assumption of mirror symmetry between the northern and southern hemispheres when IMF By and dipole tilt are reversed. We observe a significant reduction in nightside equivalent current asymmetry indicating the IMF By and dipole tilt have minimal influence on the substorm current. On the other hand, we find that substorms exert minimal or negligible effects on dayside currents. This difference in response between nightside and dayside currents emphasizes the need to incorporate nightside dynamics into existing climatological models, which presently rely mainly on upstream parameters due to a lack of robust parameters effectively representing them. Our findings provide important insights for future modeling efforts, highlighting the distinct interactions between substorms and ionospheric currents across different hemispheric regions.

The asymmetric effect of fiscal deficits on macroeconomic stability in Nigeria: evidence from nonlinear autoregressive distributed lag (NARDL)

PurposeThe paper examines the asymmetric effects of fiscal deficits on selected macroeconomic variables in Nigeria, which include economic growth, exchange rates and inflation. The existing works of literature are premised on symmetry assumptions with dichotomous findings. In such situations, they suggest using a nonlinear approach as an alternative to checkmate the findings premised on linearity. This is critical, considering the perpetual fiscal deficit trends of Nigeria, which are considered a major economic problem in the country.Design/methodology/approachThe study employs nonlinear autoregressive distributed lag (NARDL) estimator using secondary data collected from the statistical bulletin of the Central Bank of Nigeria (CBN).FindingsThe results show that in the short run, both positive and negative shocks to the fiscal deficit have no effect on Nigeria's economic growth. The same is found on the negative shocks in the long run. However, positive shocks to the fiscal deficit have a long-run positive impact on economic growth. It is further revealed that, in the short run, positive shocks as well as negative shocks to fiscal deficits are positively related to the inflation rate. More so, long-run estimates show that positive shocks to the fiscal deficit have negative impacts on inflation, while negative shocks to the fiscal deficit have positive impacts on inflation.Originality/valueThis study introduces novelties to the understanding of the relationship between fiscal deficits and macroeconomic stability in Nigeria. It accounts for asymmetric and nonlinear features that are more aligned with the socioeconomic realities of real-world phenomena. This study also offers more insightful policy perspectives to enhance the fiscal profile of the country.

The asymmetric effects of exchange rate volatility on UK–Pakistan trade flows: New evidence from nonlinear ARDL approach

Using the nonlinear autoregressive distributed lag (NARDL) method, the current study explores the effects of exchange rate variations on commodities trade between Pakistan and the United Kingdom. The current study contends that the limiting assumption of symmetry between the exchange rate and the trade flows may have hampered the empirical results, despite the fact that earlier studies primarily used the symmetric cointegration approach. From 1980 to 2020, the study examines the symmetric and asymmetries effects of exchange rate variations on 35 sectors in Pakistan that import from the UK and 35 industries that export to the UK. The findings suggest that there is evidence of a considerable influence of asymmetric exchange rate variations on trade flows in both the short and long term in over half of the importing and exporting industries that participate in trade with the UK. By highlighting the significance of taking into account asymmetry in the analysis and exposing the sector-specific effects of exchange rate changes on trade flows, the study adds to the body of current work.

The R-mAtrIx Net

We provide a novel neural network architecture that can: i) output R-matrix for a given quantum integrable spin chain, ii) search for an integrable Hamiltonian and the corresponding R-matrix under assumptions of certain symmetries or other restrictions, iii) explore the space of Hamiltonians around already learned models and reconstruct the family of integrable spin chains which they belong to. The neural network training is done by minimizing loss functions encoding Yang–Baxter equation, regularity and other model-specific restrictions such as hermiticity. Holomorphy is implemented via the choice of activation functions. We demonstrate the work of our neural network on the spin chains of difference form with two-dimensional local space. In particular, we reconstruct the R-matrices for all 14 classes. We also demonstrate its utility as an Explorer, scanning a certain subspace of Hamiltonians and identifying integrable classes after clusterisation. The last strategy can be used in future to carve out the map of integrable spin chains with higher dimensional local space and in more general settings where no analytical methods are available.

Evaluation of the bilateral symmetry assumption in manual wheelchair propulsion: a systematic review of literature in daily-life and sports contexts.

This systematic review aimed to 1) verify bilateral symmetry assumption in manual wheelchair (MWC) propulsion in daily-life and sports, and its relationship with injury risk and sports performance; 2) evaluate methods for assessing bilateral symmetry. Scopus, Web-Of-Science, PubMed, and EBSCO databases were searched for articles published before January 2024 investigating bilateral symmetry in MWC users and/or healthy participants during MWC propulsion. Two independent reviewers screened, extracted data, and assessed methodological quality of retrieved papers. Twenty-five studies were included. In daily ground-level propulsion, minimal asymmetries were observed in kinematic, kinetic, and temporal parameters when averaging ≥3 push cycles. In the sports context, diverse findings emerged, ranging from up to 27% side-to-side differences in propulsion kinetics and kinematics during sprinting, to descriptions of both symmetrical and asymmetrical upper extremity motions. Limited evidence exists regarding the role of asymmetry in MWC propulsion as a risk factor for injury and pain, as well as the association between sprinting performance and symmetry. In conclusion, bilateral symmetry assumption in MWC propulsion is valid only under specific conditions (i.e., slow/moderate speed, averaging ≥3 push cycles, smooth level ground). The wheeling environment and inter-individual variability impact symmetry research outcome and require consideration in future studies.

Employing Computed Tomography to Assess Clavicle Symmetry in Healthy Adults in the Indian Population of Dakshina Karnataka

Abstract Background The S-shaped collar bone lies horizontally at the root of the neck and is one of the most common bones to fracture. The symmetry of the clavicle bone differs between males and females, so the present study was conducted to assess the symmetry of the clavicle in healthy adults of the Indian population in the Dakshina Karnataka region. Methods The data for this cross-sectional study were retrospectively collected from 138 adult patients who all underwent chest computed tomography (CT) for clinical reasons. Patients younger than 18 years and those with bony fractures, trauma, or other deformities were excluded. Axial chest CT images were used to trace the anatomy of the clavicle. The curved planar reconstruction technique was used to trace the maximum length of the bilateral clavicle. Results There were 138 patients of both sexes. Patients ranged in age from 20 to 80 years. The chi-square test was used to investigate whether sex and clavicle symmetry were related. In women, bilateral symmetry and asymmetry were 42.5 and 41.2%, respectively, while in men, they were 57.5 and 58.8%, respectively; hence, there was no significant correlation (p-value = 0.877) between clavicle symmetry and sex. A linear-by-linear association test was performed to ascertain the relationship between the variance in clavicle symmetry and age. It was noted that 51 individuals had asymmetry and 87 had symmetry. The patients were further divided into three age groups. There were 34, 67, and 37 individuals in the 20 to 40, 41 to 60, and 61 to 80 years age groups, respectively. Clavicle symmetry was 27.6% in the 20 to 40 and 61 to 80 age groups and 44.8% in the 41 to 60 age group. The percentages of asymmetry were 19.6, 54.9, and 25.5% in the 20 to 40, 41 to 60, and 61 to 80 years age groups, respectively, with no correlation between age and clavicle length symmetry (p-value = 0.643). Conclusion Regardless of age or sex, the population under study had symmetric clavicles. As a result, the assumption of symmetry may be accepted in some clinical scenarios, such as simple fractures, but caution should be exercised when designing fixatives and surgical cases.

Asymmetry Considerations in Constructing Control Charts: When Symmetry Is Not the Norm

Control charts (esp. X¯-chart) are proven and useful tools to preserve process alignment with its design mean. The control charts’ limits are designed symmetrically around the process’s mean. The assumption of symmetry is justifiable when assuming that the measurements are infinite. Typically, these assumptions are warranted since the measuring resolution is significantly (by orders of magnitude) lower than the deviation of the controlled process. However, when the measuring device has a resolution of the same order of magnitude as the standard deviation of the controlled process, the symmetrical nature is no longer justified. In low-resolution measurement process control, symmetry is not the norm and both these control limits should be built asymmetrically. To help remedy this issue, this article explores the asymmetrical nature of the low-resolution measurement and suggests a new (asymmetric) control limit based on false-alarm-required probabilities. This represents a novel approach to the problem

Does high-latitude ionospheric electrodynamics exhibit hemispheric mirror symmetry?

Abstract. Ionospheric electrodynamics is a problem of mechanical stress balance mediated by electromagnetic forces. Joule heating (the total rate of frictional heating of thermospheric gases and ionospheric plasma) and ionospheric Hall and Pedersen conductances comprise three of the most basic descriptors of this problem. More than half a century after identification of their central role in ionospheric electrodynamics, several important questions about these quantities, including the degree to which they exhibit hemispheric symmetry under reversal of the sign of dipole tilt and the sign of the y component of the interplanetary magnetic field (so-called “mirror symmetry”), remain unanswered. While global estimates of these key parameters can be obtained by combining existing empirical models, one often encounters some frustrating sources of uncertainty: the measurements from which such models are derived, usually magnetic field and electric field or ion drift measurements, are typically measured separately and do not necessarily align. The models to be combined moreover often use different input parameters, different assumptions about hemispheric symmetry, and/or different coordinate systems. We eliminate these sources of uncertainty in model predictions of electromagnetic work J⋅E (in general not equal to Joule heating ηJ2) and ionospheric conductances by combining two new empirical models of the high-latitude ionospheric electric potential and ionospheric currents that are derived in a mutually consistent fashion: these models do not assume any form of symmetry between the two hemispheres; are based on Apex magnetic coordinates (denoted Apex), spherical harmonics, and the same model input parameters; and are derived exclusively from convection and magnetic field measurements made by the Swarm and CHAMP satellites. The model source code is open source and publicly available. Comparison of high-latitude distributions of electromagnetic work in each hemisphere as functions of dipole tilt and interplanetary magnetic field clock angle indicates that the typical assumption of mirror symmetry is largely justified. Model predictions of ionospheric Hall and Pedersen conductances exhibit a degree of symmetry, but clearly asymmetric responses to dipole tilt and solar wind driving conditions are also identified. The distinction between electromagnetic work and Joule heating allows us to identify where and under what conditions the assumption that the neutral wind corotates with the Earth is not likely to be physically consistent with predicted Hall and Pedersen conductances.

A virtual wind tunnel for deforming airborne wind energy kites

This paper presents a quasi-steady simulation framework for soft-wing kites with suspended control unit employed for airborne wind energy. The kites are subject to actuation-induced and aero-elastic deformation and are described by a coupled aero-structural model in a virtual wind tunnel setup. Key contributions of the present work are a kinetic dynamic relaxation algorithm and a procedure to define a physically consistent initial state. For symmetric actuation, the kite is pitch-statically stable and the simulations converge to a static equilibrium state. Most soft-wing kites are not roll-statically stable and do not find a static equilibrium without a symmetry assumption, as this introduces non-zero roll- and yaw moments. Another important contribution is the introduction of a steady circular flight state that enables convergence without a symmetry assumption. By neglecting gravity, the kite can fly in a perfectly circular turning motion around the wind vector with a constant radius and constant rotational velocity without requiring active control input. In an idealized wind-aligned tether case, the difference in aerodynamic- and centrifugal force application centers makes it impossible to achieve both a force- and moment equilibrium. This was resolved by including an elevation angle that introduces a radial tether force component, which introduces a centrifugal and aerodynamic force difference. Therefore, an operating point with roll equilibrium can be found where the kite finds a static equilibrium, enabling the first quasi-steady simulations of turning flights. Simulated quantifications of soft-wing kite turning behavior, i.e., turning laws, contribute to better kite- and control design.

Lower-extremity inter-joint coordination variability in active individuals with transtibial amputation and healthy males during gait.

This study was aimed to compare the variability of inter-joint coordination in the lower-extremities during gait between active individuals with transtibial amputation (TTAs) and healthy individuals (HIs). Fifteen active male TTAs (age: 40.6±16.24years, height: 1.74±0.09m, and mass: 71.2 ± 8.87kg) and HIs (age: 37.25±13.11years, height: 1.75±0.06m, and mass: 74 ± 8.75kg) without gait disabilities voluntarily participated in the study. Participants walked along a level walkway covered with Vicon motion capture system, and their lower-extremity kinematics data were recorded during gait. The spatiotemporal gait parameters, lower-extremity joint range of motion (ROM), and their coordination and variability were calculated and averaged to report a single value for each parameter based on biomechanical symmetry assumption in the lower limbs of HIs. Additionally, these parameters were separately calculated and reported for the intact limb (IL) and the prosthesis limb (PL) in TTAs individuals. Finally, a comparison was made between the averaged values in HIs and those in the IL and PL of TTAs subjects. The results showed that the IL had a significantly lower stride length than that of the PL and averaged value in HIs, and the IL had a significantly lower knee ROM and greater stance-phase duration than that of HIs. Moreover, TTAs showed different coordination patterns in pelvis-to-hip, hip-to-knee, and hip-to-ankle couplings in some parts of the gait cycle. It concludes that the active TTAs with PLs walked with more flexion of the knee and hip, which may indicate a progressive walking strategy and the differences in coordination patterns suggest active TTA individuals used different neuromuscular control strategies to adapt to their amputation. Researchers can extend this work by investigating variations in these parameters across diverse patient populations, including different amputation etiologies and prosthetic designs. Moreover, Clinicians can use the findings to tailor rehabilitation programs for TTAs, emphasizing joint flexibility and coordination.

Allen–Cahn solutions with triple junction structure at infinity

AbstractWe construct an entire solution to the elliptic system where is a ‘triple‐well’ potential. This solution is a local minimizer of the associated energy in the sense that minimizes the energy on any compact set among competitors agreeing with outside that set. Furthermore, we show that along subsequences, the ‘blowdowns’ of given by approach a minimal triple junction as . Previous results had assumed various levels of symmetry for the potential and had not established local minimality, but here we make no such symmetry assumptions.

The effect of (a)symmetry on the optimal toxin allocation

When mathematical models of biological phenomena deal with an unknown parameter, it is often assumed that such a parameter follows a normal distribution. This introduces a symmetry assumption into the model. The purpose of this paper is to investigate and quantify the effect of asymmetry on model prediction. We introduce an asymmetry into a model of sexual conflict and toxin allocation by replacing a normal distribution by a shifted beta distribution. This way, we can naturally consider a large family of continuously changing distributions. We isolate the effect of skewness on the model prediction and demonstrate that in most cases, increasing skewness causes a slight increase in optimal toxicity allocation. We conclude that overall, the effect of the skewness is much smaller than the effect of the mean. In fact, for the particular model we studied, skewness does not seem to affect qualitative predictions.

Nonparametric measure-transportation-based methods for directional data

Abstract This article proposes various nonparametric tools based on measure transportation for directional data. We use optimal transports to define new notions of distribution and quantile functions on the hypersphere, with meaningful quantile contours and regions and closed-form formulas under the classical assumption of rotational symmetry. The empirical versions of our distribution functions enjoy the expected Glivenko–Cantelli property of traditional distribution functions. They provide fully distribution-free concepts of ranks and signs and define data-driven systems of (curvilinear) parallels and (hyper)meridians. Based on this, we also construct a universally consistent test of uniformity and a class of fully distribution-free and universally consistent tests for directional MANOVA which, in simulations, outperform all their existing competitors. A real-data example involving the analysis of sunspots concludes the article.

New multiplicity results in prescribing Q-curvature on standard spheres

Abstract In this paper, we study the problem of prescribing Q-Curvature on higher dimensional standard spheres. The problem consists in finding the right assumptions on a function K so that it is the Q-Curvature of a metric conformal to the standard one on the sphere. Using some pinching condition, we track the change in topology that occurs when crossing a critical level (or a virtually critical level if it is a critical point at infinity) and then compute a certain Euler-Poincaré index which allows us to prove the existence of many solutions. The locations of the levels sets of these solutions are determined in a very precise manner. These type of multiplicity results are new and are proved without any assumption of symmetry or periodicity on the function K.