1.
Full Text
The E = E[N, v] functional and the linear response function: a conceptual DFT viewpoint

Theoretical Chemistry Accounts, ISSN 1432-881X, 9/2016, Volume 135, Issue 9, pp. 1 - 8

The energy (E) versus number of electrons (N) and external potential (v) functional E = E[N, v], which has proved to be of fundamental importance in conceptual...

Theoretical and Computational Chemistry | Linear response function | Chemistry | Physical Chemistry | Concavity of the E [ v ] functional | Atomic/Molecular Structure and Spectra | Inorganic Chemistry | Conceptual DFT | Organic Chemistry | Concavity of the E[v] functional | ENERGY | CHEMISTRY, PHYSICAL | EXACT EXCHANGE | MOLECULES | DENSITY | AROMATICITY | KERNEL | CHEMICAL-REACTIVITY | ELECTROPHILICITY INDEX | ATOMS | ELECTRONIC-STRUCTURE | Ionization | Analysis

Theoretical and Computational Chemistry | Linear response function | Chemistry | Physical Chemistry | Concavity of the E [ v ] functional | Atomic/Molecular Structure and Spectra | Inorganic Chemistry | Conceptual DFT | Organic Chemistry | Concavity of the E[v] functional | ENERGY | CHEMISTRY, PHYSICAL | EXACT EXCHANGE | MOLECULES | DENSITY | AROMATICITY | KERNEL | CHEMICAL-REACTIVITY | ELECTROPHILICITY INDEX | ATOMS | ELECTRONIC-STRUCTURE | Ionization | Analysis

Journal Article

Wuli Huaxue Xuebao/ Acta Physico - Chimica Sinica, ISSN 1000-6818, 03/2018, Volume 34, Issue 6, pp. 699 - 707

In view of its use as reactivity theory, Conceptual Density Functional Theory (DFT), introduced by Parr et al., has mainly concentrated up to now on the E = E...

Analogy with thermodynamics | Response kernels | Conceptual DFT | Stability-concavity/convexity | LINEAR-RESPONSE | SOFTNESS | CHEMISTRY, PHYSICAL | CHEMICAL-REACTIVITY INDEXES | AROMATICITY | CHEMISTRY | SYSTEMS | ATOMS | LOCAL HARDNESS | ELECTRONIC MATTER | FUKUI FUNCTION

Analogy with thermodynamics | Response kernels | Conceptual DFT | Stability-concavity/convexity | LINEAR-RESPONSE | SOFTNESS | CHEMISTRY, PHYSICAL | CHEMICAL-REACTIVITY INDEXES | AROMATICITY | CHEMISTRY | SYSTEMS | ATOMS | LOCAL HARDNESS | ELECTRONIC MATTER | FUKUI FUNCTION

Journal Article

Journal of Fixed Point Theory and Applications, ISSN 1661-7738, 9/2018, Volume 20, Issue 3, pp. 1 - 11

We study the stability of functional equations in quasi-Banach spaces where the quasi-norm is not assumed to be a p-norm. To overcome the modulus of concavity...

Mathematical Methods in Physics | Primary 39B82 | 47H10 | functional equation | Analysis | Secondary 39B52 | Mathematics, general | Mathematics | Quasi-Banach | stability | MATHEMATICS | MATHEMATICS, APPLIED | Resveratrol

Mathematical Methods in Physics | Primary 39B82 | 47H10 | functional equation | Analysis | Secondary 39B52 | Mathematics, general | Mathematics | Quasi-Banach | stability | MATHEMATICS | MATHEMATICS, APPLIED | Resveratrol

Journal Article

European Journal of Operational Research, ISSN 0377-2217, 2011, Volume 210, Issue 2, pp. 326 - 335

Utility function properties as monotonicity and concavity play a fundamental role in reflecting a decision-maker’s preference structure. These properties are...

Ultramodular functions | Multiattribute utility theory | Functional ANOVA | Multi-criteria analysis | GLOBAL SENSITIVITY-ANALYSIS | CHOICE | SUPERMODULARITY | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | INDEPENDENCE | DIMENSIONAL MODEL REPRESENTATIONS | MATHEMATICAL-MODELS | Multiattribute utility theory Functional ANOVA Multi-criteria analysis Ultramodular functions | Additives | Utilities | Decision making | Integrals | Risk | Operational research | Derivatives | Concavity | Analysis of variance | Utility theory

Ultramodular functions | Multiattribute utility theory | Functional ANOVA | Multi-criteria analysis | GLOBAL SENSITIVITY-ANALYSIS | CHOICE | SUPERMODULARITY | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | INDEPENDENCE | DIMENSIONAL MODEL REPRESENTATIONS | MATHEMATICAL-MODELS | Multiattribute utility theory Functional ANOVA Multi-criteria analysis Ultramodular functions | Additives | Utilities | Decision making | Integrals | Risk | Operational research | Derivatives | Concavity | Analysis of variance | Utility theory

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 01/2013, Volume 264, Issue 2, pp. 570 - 604

In this paper we define an addition operation on the class of quasi-concave functions. While the new operation is similar to the well-known sup-convolution, it...

Log-concavity | Mixed volumes | Brunn–Minkowski | Alexandrov–Fenchel | Mixed integrals | Quasi-concavity | Rescaling | Alexandrov-Fenchel | Brunn-Minkowski | MATHEMATICS

Log-concavity | Mixed volumes | Brunn–Minkowski | Alexandrov–Fenchel | Mixed integrals | Quasi-concavity | Rescaling | Alexandrov-Fenchel | Brunn-Minkowski | MATHEMATICS

Journal Article

Results in Mathematics, ISSN 1422-6383, 9/2017, Volume 72, Issue 1, pp. 703 - 714

In this paper, some Turán-type inequalities for Mittag–Leffler functions are considered. The method is based on proving monotonicity for special ratio of...

33E12 | Lazarević-type inequalities | Wilker-type inequalities | Turán-type inequalities | Mathematics, general | Mathematics | 26D07 | Mittag–Leffler functions | MATHEMATICS | MATHEMATICS, APPLIED | Mittag-Leffler functions | LOG-CONCAVITY | Lazarevic-type inequalities | CONVEXITY | TURAN TYPE | Turan-type inequalities | HYPERGEOMETRIC-FUNCTIONS | Resveratrol

33E12 | Lazarević-type inequalities | Wilker-type inequalities | Turán-type inequalities | Mathematics, general | Mathematics | 26D07 | Mittag–Leffler functions | MATHEMATICS | MATHEMATICS, APPLIED | Mittag-Leffler functions | LOG-CONCAVITY | Lazarevic-type inequalities | CONVEXITY | TURAN TYPE | Turan-type inequalities | HYPERGEOMETRIC-FUNCTIONS | Resveratrol

Journal Article

Boundary Value Problems, ISSN 1687-2762, 12/2011, Volume 2011, Issue 1, pp. 1 - 10

In this paper, we consider the following viscoelastic equations with initial condition and zero Dirichlet boundary condition. Using the concavity method, we...

Ordinary Differential Equations | Analysis | Approximations and Expansions | Difference and Functional Equations | positive initial energy | Mathematics, general | blow up | Mathematics | Partial Differential Equations | viscoelastic equations | EXISTENCE | MATHEMATICS, APPLIED | NONLINEAR-WAVE EQUATIONS | UNIFORM DECAY | EVOLUTION-EQUATIONS | GLOBAL NONEXISTENCE | SOURCE TERMS | MATHEMATICS | RATES | MEMORY | ASYMPTOTIC STABILITY | Viscoelasticity | Engineering mathematics | Boundary value problems | Research | Mathematical research | Series, Dirichlet | Energy use | Initial conditions | Mathematical analysis | Dirichlet problem | Boundary conditions | Concavity

Ordinary Differential Equations | Analysis | Approximations and Expansions | Difference and Functional Equations | positive initial energy | Mathematics, general | blow up | Mathematics | Partial Differential Equations | viscoelastic equations | EXISTENCE | MATHEMATICS, APPLIED | NONLINEAR-WAVE EQUATIONS | UNIFORM DECAY | EVOLUTION-EQUATIONS | GLOBAL NONEXISTENCE | SOURCE TERMS | MATHEMATICS | RATES | MEMORY | ASYMPTOTIC STABILITY | Viscoelasticity | Engineering mathematics | Boundary value problems | Research | Mathematical research | Series, Dirichlet | Energy use | Initial conditions | Mathematical analysis | Dirichlet problem | Boundary conditions | Concavity

Journal Article

Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, 11/2018, Volume 51, Issue 48, p. 483001

Given two density matrices rho and sigma, there are a number of different expressions that reduce to the alpha-Renyi relative entropy of rho with respect to...

data processing inequality | entropy | matrix analysis | convexity | CAPACITY | CONCAVITY | PHYSICS, MULTIDISCIPLINARY | MATRIX TRACE | INFORMATION | STRONG SUBADDITIVITY | PHYSICS, MATHEMATICAL | RELATIVE ENTROPY | ENTROPY INEQUALITIES | TRACE FUNCTIONS | THEOREMS

data processing inequality | entropy | matrix analysis | convexity | CAPACITY | CONCAVITY | PHYSICS, MULTIDISCIPLINARY | MATRIX TRACE | INFORMATION | STRONG SUBADDITIVITY | PHYSICS, MATHEMATICAL | RELATIVE ENTROPY | ENTROPY INEQUALITIES | TRACE FUNCTIONS | THEOREMS

Journal Article

American Journal of Agricultural Economics, ISSN 0002-9092, 11/1999, Volume 81, Issue 4, pp. 865 - 880

Markov Chain Monte Carlo (MCMC) methods are used to estimate a seemingly unrelated regression (SUR) system of input demand functions for U.S. agriculture. Our...

Economic models | Maximum likelihood estimation | Efficiency metrics | Estimation theory | Eigenvalues | Mathematical independent variables | Agriculture | Concavity | Estimators | Cost functions | Markov Chain Monte Carlo | Technical efficiency | Input demands | Bayes | SUBSTITUTION | OUTPUT | technical efficiency | ELASTICITIES | PANEL-DATA | COST FUNCTION | AGRICULTURAL ECONOMICS & POLICY | UNCERTAINTY | ECONOMICS | FLEXIBLE FUNCTIONAL FORMS | concavity | CANADIAN AGRICULTURE | input demands | TECHNOLOGY | Measurement | Evaluation | Monte Carlo method | Usage | Agricultural industry | Markov processes | Management | Industrial efficiency

Economic models | Maximum likelihood estimation | Efficiency metrics | Estimation theory | Eigenvalues | Mathematical independent variables | Agriculture | Concavity | Estimators | Cost functions | Markov Chain Monte Carlo | Technical efficiency | Input demands | Bayes | SUBSTITUTION | OUTPUT | technical efficiency | ELASTICITIES | PANEL-DATA | COST FUNCTION | AGRICULTURAL ECONOMICS & POLICY | UNCERTAINTY | ECONOMICS | FLEXIBLE FUNCTIONAL FORMS | concavity | CANADIAN AGRICULTURE | input demands | TECHNOLOGY | Measurement | Evaluation | Monte Carlo method | Usage | Agricultural industry | Markov processes | Management | Industrial efficiency

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 02/2019, Volume 179, pp. 1 - 14

In this paper, we prove the concavity of -entropy power of probability densities solving the -heat equation on closed Riemannian manifold with nonnegative...

Nash inequality | [formula omitted]-heat equation | Logarithmic Sobolev inequality | Concavity | Entropy power inequality | p-heat equation | MATHEMATICS | MATHEMATICS, APPLIED | PROOF | FORMULAS | Thermodynamics | Nonlinear equations | Riemann manifold | Vector space | Entropy | Curvature

Nash inequality | [formula omitted]-heat equation | Logarithmic Sobolev inequality | Concavity | Entropy power inequality | p-heat equation | MATHEMATICS | MATHEMATICS, APPLIED | PROOF | FORMULAS | Thermodynamics | Nonlinear equations | Riemann manifold | Vector space | Entropy | Curvature

Journal Article

Econometric Reviews, ISSN 0747-4938, 02/2015, Volume 34, Issue 1-2, pp. 198 - 227

In this paper we build on work by Gallant and Golub ( 1984 ), Diewert and Wales ( 1987 ), and Barnett ( 2002 ) and provide a comparison among three different...

C3 | Flexible functional forms | C13 | Translog cost function | Curvature | Monotonicity | C51 | UNITED-STATES | SYSTEM | MARKOV-CHAINS | CONCAVITY | COST-FUNCTIONS | EQUATIONS | STATISTICS & PROBABILITY | DEMAND | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | SOCIAL SCIENCES, MATHEMATICAL METHODS | ECONOMICS | CURVATURE CONDITIONS | TECHNOLOGY | Gallant, A. Ronald | Wales, T.J | Walter, Erwin D | Golub, Stephen S | Barnett, William A | Data points | Constraints | Methodology | Mathematical analysis | Imposition | Bayesian analysis | Regularity | Concavity

C3 | Flexible functional forms | C13 | Translog cost function | Curvature | Monotonicity | C51 | UNITED-STATES | SYSTEM | MARKOV-CHAINS | CONCAVITY | COST-FUNCTIONS | EQUATIONS | STATISTICS & PROBABILITY | DEMAND | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | SOCIAL SCIENCES, MATHEMATICAL METHODS | ECONOMICS | CURVATURE CONDITIONS | TECHNOLOGY | Gallant, A. Ronald | Wales, T.J | Walter, Erwin D | Golub, Stephen S | Barnett, William A | Data points | Constraints | Methodology | Mathematical analysis | Imposition | Bayesian analysis | Regularity | Concavity

Journal Article

Integral Transforms and Special Functions, ISSN 1065-2469, 11/2018, Volume 29, Issue 11, pp. 875 - 892

In this paper we prove some monotonicity, log-convexity and log-concavity properties for the Volterra and incomplete Volterra functions. Moreover, as...

33B15 | log-convex functions | 26A51 | 11M35 | incomplete Volterra function | Turán type inequalities | completely monotonic | Volterra function | functional inequalities | 33D05 | MATHEMATICS | Turan type inequalities | MATHEMATICS, APPLIED | Lower bounds | Convexity | Concavity | Inequalities | Mathematics - Classical Analysis and ODEs

33B15 | log-convex functions | 26A51 | 11M35 | incomplete Volterra function | Turán type inequalities | completely monotonic | Volterra function | functional inequalities | 33D05 | MATHEMATICS | Turan type inequalities | MATHEMATICS, APPLIED | Lower bounds | Convexity | Concavity | Inequalities | Mathematics - Classical Analysis and ODEs

Journal Article

PeerJ, ISSN 2167-8359, 2016, Volume 2016, Issue 1, p. e1572

The family Cetotheriidae has played a major role in recent discussions of baleen whale phylogenetics. Within this group, the enigmatic, monotypic Metopocetus...

Metopocetus | Paroccipital concavity | Baleen whales | Cetotheriidae | Phylogenetics | Late Miocene | Primary dorsal infraorbital foramen | Mysticeti | BASIN NORTHERN BELGIUM | PHYLOGENY | DINOFLAGELLATE CYST | MULTIDISCIPLINARY SCIENCES | MAMMALIA | STRATIGRAPHY | MIDDLE MIOCENE | BALEEN WHALE | EVOLUTION | PLIOCENE | MORPHOLOGY | Discovery and exploration | Phylogeny | Cetacea | Geology | Zoology | Museums | Whales & whaling | Paleontology | Maxilla | Basicranium | Microorganisms | Cysts | Functional morphology | Sinus | Morphology | New species | Stratigraphy

Metopocetus | Paroccipital concavity | Baleen whales | Cetotheriidae | Phylogenetics | Late Miocene | Primary dorsal infraorbital foramen | Mysticeti | BASIN NORTHERN BELGIUM | PHYLOGENY | DINOFLAGELLATE CYST | MULTIDISCIPLINARY SCIENCES | MAMMALIA | STRATIGRAPHY | MIDDLE MIOCENE | BALEEN WHALE | EVOLUTION | PLIOCENE | MORPHOLOGY | Discovery and exploration | Phylogeny | Cetacea | Geology | Zoology | Museums | Whales & whaling | Paleontology | Maxilla | Basicranium | Microorganisms | Cysts | Functional morphology | Sinus | Morphology | New species | Stratigraphy

Journal Article

Journal of Productivity Analysis, ISSN 0895-562X, 4/2016, Volume 45, Issue 2, pp. 121 - 130

We use a Monte Carlo experiment to compare the quadratic and translog functional forms in terms of their ability to approximate known frontiers that possess...

Economics | C63 | Accounting/Auditing | D24 | Microeconomics | Operation Research/Decision Theory | Econometrics | Parameterization | Distance functions | BENEFIT | CONCAVITY | SOCIAL SCIENCES, MATHEMATICAL METHODS | BUSINESS | ECONOMICS | Monte Carlo method | Usage | Production possibility frontier | Analysis | Studies | Productivity measurement | Monte Carlo simulation | Economics and Finance | Humanities and Social Sciences

Economics | C63 | Accounting/Auditing | D24 | Microeconomics | Operation Research/Decision Theory | Econometrics | Parameterization | Distance functions | BENEFIT | CONCAVITY | SOCIAL SCIENCES, MATHEMATICAL METHODS | BUSINESS | ECONOMICS | Monte Carlo method | Usage | Production possibility frontier | Analysis | Studies | Productivity measurement | Monte Carlo simulation | Economics and Finance | Humanities and Social Sciences

Journal Article

Empirical Economics, ISSN 0377-7332, 12/2014, Volume 47, Issue 4, pp. 1411 - 1440

This paper empirically investigates the implications of the choice of dates when imposing local restrictions to maintain convexity in output prices and...

C30 | Flexible functional forms | Elasticities | Statistics for Business/Economics/Mathematical Finance/Insurance | Curvature violations | D24 | Economic Theory | GNP function | Econometrics | Economics / Management Science | UNITED-STATES | CANADA | DEMAND SYSTEMS | CONCAVITY | COST-FUNCTIONS | SINGULAR EQUATION SYSTEMS | MODEL SELECTION | IMPORTS | SOCIAL SCIENCES, MATHEMATICAL METHODS | ECONOMICS | EXPORTS | Prices | Statistical analysis | Macroeconomics | Gross National Product--GNP | Output

C30 | Flexible functional forms | Elasticities | Statistics for Business/Economics/Mathematical Finance/Insurance | Curvature violations | D24 | Economic Theory | GNP function | Econometrics | Economics / Management Science | UNITED-STATES | CANADA | DEMAND SYSTEMS | CONCAVITY | COST-FUNCTIONS | SINGULAR EQUATION SYSTEMS | MODEL SELECTION | IMPORTS | SOCIAL SCIENCES, MATHEMATICAL METHODS | ECONOMICS | EXPORTS | Prices | Statistical analysis | Macroeconomics | Gross National Product--GNP | Output

Journal Article

16.
Full Text
Sectoral interfuel substitution in Canada: An application of NQ flexible functional forms

Energy Journal, ISSN 0195-6574, 2016, Volume 37, Issue 2, pp. 181 - 199

This paper focuses on the aggregate demand for electricity, natural gas, and light fuel oil in Canada as a whole and six of its provinces Quebec, Ontario,...

Flexible functional forms | NQ cost function | NQ expenditure function | Global concavity | ENERGY DEMAND | DEMAND SYSTEMS | ENERGY & FUELS | SHORT-RUN | ELASTICITIES | INTERNATIONAL EVIDENCE | CURVATURE | ECONOMICS | ENVIRONMENTAL STUDIES | TECHNOLOGY | Energy industry | Planning | Microeconomics | Fuel oil | Natural gas | Substitutes | Residential areas | Economic models | Statistical analysis | Electricity | Electricity consumption | Fuel oils | Curvature

Flexible functional forms | NQ cost function | NQ expenditure function | Global concavity | ENERGY DEMAND | DEMAND SYSTEMS | ENERGY & FUELS | SHORT-RUN | ELASTICITIES | INTERNATIONAL EVIDENCE | CURVATURE | ECONOMICS | ENVIRONMENTAL STUDIES | TECHNOLOGY | Energy industry | Planning | Microeconomics | Fuel oil | Natural gas | Substitutes | Residential areas | Economic models | Statistical analysis | Electricity | Electricity consumption | Fuel oils | Curvature

Journal Article

Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 6/2018, Volume 57, Issue 3, pp. 1 - 39

A prominent model for transportation networks is branched transport, which seeks the optimal transportation scheme to move material from a given initial to a...

90B10 | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | Mathematics | 49Q10 | 49Q20 | MATHEMATICS | MATHEMATICS, APPLIED | FLAT CHAINS | REGULARITY | DISTANCES | PATHS | NETWORKS | IRRIGATION PATTERNS | MICROPATTERNS

90B10 | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | Mathematics | 49Q10 | 49Q20 | MATHEMATICS | MATHEMATICS, APPLIED | FLAT CHAINS | REGULARITY | DISTANCES | PATHS | NETWORKS | IRRIGATION PATTERNS | MICROPATTERNS

Journal Article

ESAIM - Control, Optimisation and Calculus of Variations, ISSN 1292-8119, 07/2012, Volume 18, Issue 3, pp. 693 - 711

Motivated by a long-standing conjecture of Polya and Szego about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in...

Shape derivatives | Convex bodies | Capacity | Concavity inequalities | Optimization | MATHEMATICS, APPLIED | BRUNN-MINKOWSKI INEQUALITY | AREA | MINIMIZATION | optimization | CALCULUS | POLYA | concavity inequalities | shape derivatives | AUTOMATION & CONTROL SYSTEMS | capacity | Studies | Mathematical problems | Mathematical models | Mathematics | Numerical analysis

Shape derivatives | Convex bodies | Capacity | Concavity inequalities | Optimization | MATHEMATICS, APPLIED | BRUNN-MINKOWSKI INEQUALITY | AREA | MINIMIZATION | optimization | CALCULUS | POLYA | concavity inequalities | shape derivatives | AUTOMATION & CONTROL SYSTEMS | capacity | Studies | Mathematical problems | Mathematical models | Mathematics | Numerical analysis

Journal Article

Advances in Mathematics, ISSN 0001-8708, 2005, Volume 194, Issue 1, pp. 105 - 140

Recently, several inequalities of Brunn–Minkowski type have been proved for well-known functionals in the Calculus of Variations, e.g. the first eigenvalue of...

Convex body | Minkowski problem | Elliptic partial differential equations | Brunn–Minkowski inequality | Variational functionals | Brunn-Minkowski inequality | MATHEMATICS | CAPACITY | CONVEX-BODIES | CONCAVITY | elliptic partial differential equations | ELLIPTIC-EQUATIONS | convex body | variational functionals

Convex body | Minkowski problem | Elliptic partial differential equations | Brunn–Minkowski inequality | Variational functionals | Brunn-Minkowski inequality | MATHEMATICS | CAPACITY | CONVEX-BODIES | CONCAVITY | elliptic partial differential equations | ELLIPTIC-EQUATIONS | convex body | variational functionals

Journal Article

Advances in Difference Equations, ISSN 1687-1847, 12/2019, Volume 2019, Issue 1, pp. 1 - 14

This research demonstrates the log-convexity and log-concavity of the modified Bessel function of the first kind and the related functions. The method of...

Ordinary Differential Equations | Functional Analysis | Analysis | Hypergeometric Function | Difference and Functional Equations | Kibble’s bivariate gamma distribution | Mathematics, general | Modified Bessel Function | Mathematics | Convexity | Partial Differential Equations | Concavity | RATIOS | MATHEMATICS | MATHEMATICS, APPLIED | INEQUALITIES | BOUNDS | Kibble's bivariate gamma distribution

Ordinary Differential Equations | Functional Analysis | Analysis | Hypergeometric Function | Difference and Functional Equations | Kibble’s bivariate gamma distribution | Mathematics, general | Modified Bessel Function | Mathematics | Convexity | Partial Differential Equations | Concavity | RATIOS | MATHEMATICS | MATHEMATICS, APPLIED | INEQUALITIES | BOUNDS | Kibble's bivariate gamma distribution

Journal Article

No results were found for your search.

Cannot display more than 1000 results, please narrow the terms of your search.