Showing papers by "Massachusetts Institute of Technology published in 1972"

Dissipative dynamical systems part I: General theory

Abstract: The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper.

The Capital Asset Pricing Model: Some Empirical Tests

Abstract: Considerable attention has recently been given to general equilibrium models of the pricing of capital assets Of these, perhaps the best known is the mean-variance formulation originally developed by Sharpe (1964) and Treynor (1961), and extended and clarified by Lintner (1965a; 1965b), Mossin (1966), Fama (1968a; 1968b), and Long (1972) In addition Treynor (1965), Sharpe (1966), and Jensen (1968; 1969) have developed portfolio evaluation models which are either based on this asset pricing model or bear a close relation to it In the development of the asset pricing model it is assumed that (1) all investors are single period risk-averse utility of terminal wealth maximizers and can choose among portfolios solely on the basis of mean and variance, (2) there are no taxes or transactions costs, (3) all investors have homogeneous views regarding the parameters of the joint probability distribution of all security returns, and (4) all investors can borrow and lend at a given riskless rate of interest The main result of the model is a statement of the relation between the expected risk premiums on individual assets and their "systematic risk" Our main purpose is to present some additional tests of this asset pricing model which avoid some of the problems of earlier studies and which, we believe, provide additional insights into the nature of the structure of security returns The evidence presented in Section II indicates the expected excess return on an asset is not strictly proportional to its B, and we believe that this evidence, coupled with that given in Section IV, is sufficiently strong to warrant rejection of the traditional form of the model given by (1) We then show in Section III how the cross-sectional tests are subject to measurement error bias, provide a solution to this problem through grouping procedures, and show how cross-sectional methods are relevant to testing the expanded two-factor form of the model We show in Section IV that the mean of the beta factor has had a positive trend over the period 1931-65 and was on the order of 10 to 13% per month in the two sample intervals we examined in the period 1948-65 This seems to have been significantly different from the average risk-free rate and indeed is roughly the same size as the average market return of 13 and 12% per month over the two sample intervals in this period This evidence seems to be sufficiently strong enough to warrant rejection of the traditional form of the model given by (1) In addition, the standard deviation of the beta factor over these two sample intervals was 20 and 22% per month, as compared with the standard deviation of the market factor of 36 and 38% per month Thus the beta factor seems to be an important determinant of security returns

Understanding Natural Language

Abstract: This paper describes a computer system for understanding English. The system answers questions, executes commands, and accepts information in an interactive English dialog. It is based on the belief that in modeling language understanding, we must deal in an integrated way with all of the aspects of language—syntax, semantics, and inference. The system contains a parser, a recognition grammar of English, programs for semantic analysis, and a general problem solving system. We assume that a computer cannot deal reasonably with language unless it can understand the subject it is discussing. Therefore, the program is given a detailed model of a particular domain. In addition, the system has a simple model of its own mentality. It can remember and discuss its plans and actions as well as carrying them out. It enters into a dialog with a person, responding to English sentences with actions and English replies, asking for clarification when its heuristic programs cannot understand a sentence through the use of syntactic, semantic, contextual, and physical knowledge. Knowledge in the system is represented in the form of procedures, rather than tables of rules or lists of patterns. By developing special procedural representations for syntax, semantics, and inference, we gain flexibility and power. Since each piece of knowledge can be a procedure, it can call directly on any other piece of knowledge in the system.

The Finite Group Velocity of Quantum Spin Systems

Abstract: It is shown that if Ф is a finite range interaction of a quantum spin system, τ t Ф the associated group of time translations, τ x the group of space translations, and A, B local observables, then $$ \mathop {{\text{lim}}}\limits_{\mathop {\left| t \right| \to \infty }\limits_{\left| x \right| > v\left| t \right|} } \left\| {\left[ {\tau _t^\varphi \tau \left( A \right),B} \right]} \right\|e^{\mu \left( v \right)t} = 0$$ (1) whenever v is sufficiently large (ν > V Ф ,) where μ(ν) > 0. The physical content of the statement is that information can propagate in the system only with a finite group velocity.

Hartree-Fock Calculations with Skyrme's Interaction. I. Spherical Nuclei

Abstract: Hartree-Fock calculations for spherical nuclei using Skyrme's density-dependent effective nucleon-nucleon interaction are discussed systematically. Skyrme's interaction is described and the general formula for the mean energy of a spherical nucleus derived. Hartree-Fock equations are obtained by varying the mean energy with respect to the single-particle wave functions of occupied states. Relations between the parameters of the Skyrme force and various general properties of nuclear matter and finite nuclei are analyzed. Calculations have been made for closed-shell nuclei using two rather different sets of parameters, both of which give good binding energies and radii for $^{16}\mathrm{O}$ and $^{208}\mathrm{Pb}$. Both interactions give good binding energies and charge radii for all closed-shell nuclei. Calculated electron scattering angular distributions agree qualitatively with experiment, and for one interaction there is good quantitative agreement. The single-particle energies calculated with the two interactions are somewhat different owing to a different nonlocality of the Hartree-Fock potentials, but both interactions give the correct order and density of single-particle levels near the Fermi level. They differ most strongly in their predictions for the energies of $1s$ single-particle states.

Brain Serotonin Content: Physiological Regulation by Plasma Neutral Amino Acids

Abstract: When plasma tryptophan is elevated by the injection of tryptophan or insulin, or by the consumption of carbohydrates, brain tryptophan and serotonin also rise; however, when even larger elevations of plasma tryptophan are produced by the ingestion of protein-containing diets, brain tryptophan and serotonin do not change. The main determinant of brain tryptophan and serotonin concentrations does not appear to be plasma tryptophan alone, but the ratio of this amino acid to other plasma neutral amino acids (that is, tyrosine, phenylalanine, leucine, isoleucine, and valine) that compete with it for uptake into the brain.

Dissipative dynamical systems Part II: Linear systems with quadratic supply rates

Abstract: This paper presents the theory of dissipative systems in the context of finite dimensional stationary linear systems with quadratic supply rates. A necessary and sufficient frequency domain condition for dissipativeness is derived. This is followed by the evaluation of the available storage and the required supply and of a time-domain criterion for dissipativeness involving certain matrix inequalities. The quadratic storage functions and the dissipation functions are then examined. The discussion then turns to reciprocal systems and it is shown that external reciprocity and dissipativeness imply the existence of a state space realization which is also internally reciprocal and dissipative. The paper proceeds with an examination of reversible systems and of relaxation systems. In particular, it is shown how a unique internal storage function may be defined for relaxation systems. These results are applied to the synthesis of electrical networks and the theory of linear viscoelastic materials.

Atomic coherent states in quantum optics

Abstract: The central problem of Quantum Optics (laser theory, super-radiance, resonant propagation, etc.) is the description of the interaction between N atoms and an electromagnetic field confined in a cavity of finite volume. A suitable model Hamiltonian for this problem is the following one (ℏ = 1) $$\begin{array}{*{20}{l}} {{\rm{H = }}\sum\limits_{\rm{k}} {{{\rm{\omega }}_{\rm{k}}}} {\rm{a}}_{\rm{k}}^{\rm{ + }}{{\rm{a}}_{\rm{k}}}{\rm{ + }}\frac{{{{\rm{\omega }}_{\rm{o}}}}}{{\rm{2}}}\sum\limits_{{\rm{i = 1}}}^{\rm{N}} {{{\rm{S}}_{{\rm{3}}\left( {\rm{i}} \right)}}} }\\ {{\rm{ + }}\sum\limits_{{\rm{k,i}}} {{{\rm{g}}_{\rm{k}}}\left( {{{\rm{a}}_{\rm{k}}}{\rm{S}}_{\rm{i}}^{\rm{ + }}{{\rm{e}}^{{\rm{i \bullet }}{{{\rm{}}}_{\rm{i}}}}}{\rm{ + a}}_{\rm{k}}^{\rm{ + }}{\rm{S}}_{\rm{i}}^{\rm{ - }}{{\rm{e}}^{{\rm{ - i \bullet }}{{{\rm{}}}_{\rm{i}}}}}} \right)} {\rm{,}}} \end{array}$$ where ak, a k + are Bose operators describing the k-th field mode and S i ± ,S3i are Pauli operators describing the atom located at position x i as a two-level system.

Statistical Exchange-Correlation in the Self-Consistent Field

Abstract: Publisher Summary This chapter discusses statistical exchange-correlation in the self-consistent field. There are two sides to a self-consistent field calculation: the determination of the potential and the solution of Schrodinger's equation for the one-electron problem. The solution of Schrodinger's equation has fortunately advanced far enough through the application of the electronic digital computer so that it can be regarded for most purposes as being a standardized technique. The main point of the method of Gaspar, Kohn, and Sham is that they derive the approximate exchange correction from an approximate Hamiltonian for the system by varying the spin-orbitals to minimize the average value of this Hamiltonian for the ground state. The approximate Hamiltonian has many important and valuable features that are described in the chapter.

Atomic Coherent States in Quantum Optics

Abstract: For the description of an assembly of two-level atoms, atomic coherent states can be defined which have properties analogous to those of the field coherent states. The analogy is not fortuitous, but is shown to be related to the group contraction of exponential operators based on the angular momentum algebra to exponential operators based on the harmonic-oscillator algebra. The derivation of the properties of the atomic coherent states is made easier by the use of a powerful disentangling theorem for exponential angular momentum operators. A complete labeling of the atomic states is developed and many of their properties are studied. In particular it is shown that the atomic coherent states are the quantum analogs of classical dipoles, and that they can be produced by classical fields.

Charge States and Charge-Changing Cross Sections of Fast Heavy Ions Penetrating Through Gaseous and Solid Media

Abstract: This review surveys the experimental and theoretical situation concerning charge states and charge-changing cross sections of heavy ions up to and including uranium, which penetrate through gaseous and solid targets with velocities primarily in the range ${v}_{0}lvlZ{v}_{0}$ (${v}_{0}=\frac{{e}^{2}}{\ensuremath{\hbar}}=2.188\ifmmode\times\else\texttimes\fi{}{10}^{8}$ cm/sec). Particular emphasis is given to ions with atomic numbers in the range $16\ensuremath{\le}Z\ensuremath{\le}92$. The published literature is covered through August 1971. General physical and mathematical relations are outlined which describe the composition of charge states in a heavy-ion beam which passes through matter. Recent experimental techniques and methods of data analysis are summarized. Extensive experimental results on heavy-ion equilibrium charge state distributions, average equilibrium charge states, and cross sections for capture and loss of one or more electrons in single encounters with target atoms are presented and critically examined. The data extend to ions as heavy as uranium and energies up to \ensuremath{\sim}400 MeV. Systematic trends are emphasized and generalizations are discussed which allow interpolations and to some extent extrapolations of the data to be made to ranges which have not been investigated experimentally. Attention is given to the cross sections for multiple-electron loss which are relatively large but which are poorly understood. We deal with effects of residual ion excitation on charge-changing collisions in the light of recent experimental results. It is shown that the average equilibrium charge of heavy ions can be approximated by utilizing both theoretical concepts which originate from the work of Bohr and Lamb, and semiempirical relations which are based on observed regularities of the data. Recent interpretations of phenomena associated with density effects, i.e., with the increase of projectile ionization which is observed for increasing target densities, are scrutinized and refinements of the theory by Bohr and Lindhard are explored.

A mechanism for the strain-induced nucleation of martensitic transformations*

Abstract: The previous work of Professor W. G. Burgers and Dr. A. J. Bogers is used to develop a mechanism of strain-induced martensitic nucleation, involving two intersecting shear systems. We distinguish between strain-induced nucleation and stress-assisted nucleation, the latter involving the same sites and embryos as does the regular spontaneous transformation. The strain-induced nucleation, on the other hand, depends on the creation of new sites and embryos by plastic deformation; this phenomenon may also contribute in a major way to autocatalytic nucleation during the course of martensitic transformation. For the case of strain-induced nucleation, it is possible to focus on specific intersecting-shear systems when the austenitic stacking-fault energy is low and e (h.c.p.) martensite can form as a part of the shear displacements. It then becomes feasible to extend the intersecting-shear mechanism from this special case to alloys of higher stacking-fault energy, where e is no longer stable relative to the austenite. It should be noted, however, that these events are very early stages in the formation of martensitic plates and relate primarily to the genesis of embryos; the actual growth start-ups which determine the operational (measured) nucleation rates may be controlled by subsequent processes.

A kinetic treatment of glass formation

Abstract: A kinetic treatment of glass formation is presented. This treatment is based on the construction of time-temperature-transformation curves corresponding to some barely detectable degree of crystallinity. From such curves, the minimum cooling rates required to form glasses of various materials are estimated. The most important factors determining the glass-forming abilities of different materials are suggested to be the magnitude of the viscosity at the melting point and the rate of increase in viscosity with falling temperature below the melting point.

Rheological behavior of Sn-15 pct Pb in the crystallization range

Abstract: Rheological behavior of Sn-15 pct Pb alloy in the solidification range has been investigated using a Couette type viscometer. In samples partially solidified before shearing, deformation is localized and primarily intergranular. Samples containing more than about 0.15 fraction solid exhibit an “apparent yield point” which is on the order of 106 dyne per sq cm and increases with increasing fraction solid. When shearing is conducted continuously while the alloy is cooled from above the liquidus to the desired final fraction solid, shear stresses required for flow are reduced by about three orders of magnitude. The solid-liquid mixture now behaves as a fluid slurry. Structural examination shows that shear takes place throughout the cross section of the specimen and that the solid is present as a fine grained particulate suspension. Flow behavior can be described by a viscosity which depends on fraction solid, decreases with increasing shear rate and exhibits hysteresis when shear rate is changed. For shear rates of 200 sec−1, at 0.40 fraction solid, viscosity is about 5 poise which is equivalent to that of heavy machine oil at room temperature. The fact that the slurry is highly fluid at large fractions solid suggests potential applications in new and existing metal casting processes.

Self-Consistent-Field X α Cluster Method for Polyatomic Molecules and Solids

Abstract: This paper describes a practical self-consistent-field (SCF) method of calculating electronic energy levels and eigenfunctions, adapted for polyatomic molecules and solids. The one-electron Schr\"odinger equation is set up for a so-called "muffin-tin" approximation to the true potential, spherically symmetrical within spheres surrounding the various nuclei, constant in the region between the spheres, spherically symmetrical outside a sphere surrounding the molecule. The method of solving this equation is a multiple-scattering method, equivalent to the Korringa-Kohn-Rostoker (KKR) method often used for crystals. Once the eigenfunctions and eigenvalues of this problem are determined, one assumes that the orbitals of lowest eigenvalue are occupied, up to a Fermi level. From the resulting charge densities, one can compute a total energy, using a statistical approximation for the exchange correlation. This approximation has an undetermined factor $\ensuremath{\alpha}$ (whence the name $X\ensuremath{\alpha}$ method). The spin orbitals and occupation numbers are varied to minimize this total energy, resulting in one-electron equations. The value of $\ensuremath{\alpha}$ for an isolated atom is determined by requiring that the total energy, using the statistical approximation, should equal the precise Hartree-Fock energy. This leads to very accurate spin orbitals. In a molecule or crystal, one uses the $\ensuremath{\alpha}'\mathrm{s}$ characteristic of the various atoms within the atomic spheres, and a suitable average in the region between. The computer programs for making these self-consistent calculations, for such radicals and polyatomic molecules as S${\mathrm{O}}_{4}^{\ensuremath{-}2}$, Cl${\mathrm{O}}_{4}^{\ensuremath{-}}$, Mn${\mathrm{O}}_{4}^{\ensuremath{-}}$, and S${\mathrm{F}}_{6}$ have been worked out and calculations made. They are more than 100 times as fast as comparable programs using the LCAO (linear-combination-of-atomic-orbitals) method, and the results appear to be in better agreement with experiment than such LCAO results. For calculating the frequencies of optical transitions, one must make a self-consistent calculation, not for the initial or final state, but for what we call the transition state, in which occupation numbers are halfway between the initial and final states. Then it can be proved that the differences of eigenvalues of the $X\ensuremath{\alpha}$ method are more accurate than Hartree-Fock energy values, in that they take account of the modification or relaxation of the orbitals in going from the initial to the final states. These transition states, for a crystal, involve a localized perturbation at the site of the excited atom. The multiple-scattering method is adapted to the use of such perturbed crystals, as well as to isolated molecules, and to perfect crystals. It results, in such problems as x-ray absorption, in the use of localized orbitals rather than bandlike functions. The method is adapted to the calculation of magnetic problems, by use of a spin-polarized version of the method. The method can also be used for calculations of cohesive energy of crystals, and has been used successfully for several types of metals. Unlike most other SCF methods, long-range correlation is automatically included, so that the energy of a system as a function of internuclear positions automatically reduces to the proper values at infinite internuclear distances.

Magnetoencephalography: Detection of the Brain's Electrical Activity with a Superconducting Magnetometer

Abstract: Measurements of the brain's magnetic field, called magnetoencephalograms (MEG's), have been taken with a superconducting magnetometer in a heavily shielded room. This magnetometer has been adjusted to a much higher sensitivity than was previously attainable, and as a result MEG's can, for the first time, be taken directly, without noise averaging. MEG's are shown, simultaneously with the electroencephalogram (EEG), of the alpha rhythm of a normal subject and of the slow waves from an abnormal subject. The normal MEG shows the alpha rhythm, as does the EEG, when the subject's eyes are closed; however, this MEG also shows that higher detector sensitivity, by a factor of 3, would be necessary in order to clearly show the smaller brain events when the eyes are open. The abnormal MEG, including a measurenment of the direct-current component, suggests that the MEG may yield some information which is new and different from that provided by the EEG.

Fourier Preprocessing for Hand Print Character Recognition

Abstract: A pattern-recognition method, making use of Fourier transformations to extract features which are significant for a pattern, is described. The ordinary Fourier coefficients are difficult to use as input to categorizers because they contain factors dependent upon size and rotation as well as an arbitrary phase angle. From these Fourier coefficients, however, other more useful features can easily be derived. By using these derived property constants, a distinction can be made between genuine shape constants and constants representing size, location, and orientation. The usefulness of the method has been tested with a computer program that was used to classify 175 samples of handprinted letters, e.g., 7 sets of the 25 letters A to Z. In this test, 98 percent were correctly recognized when a simple nonoptimized decision method was used. The last section contains some considerations of the technical realizability of a fast preprocessing system for reading printed text.

Complementary sets of sequences

Abstract: A set of equally long finite sequences, the elements of which are either + 1 or - 1, is said to be a complementary set of sequences if the sum of autocorrelation functions of the sequences in that set is zero except for a zero-shift term. A complementary set of sequences is said to be a mate of another set if the sum of the cross-correlation functions of the corresponding sequences in these two sets is zero everywhere. Complementary sets of sequences are said to be mutually orthogonal complementary sets if any two of them are mates to each other. In this paper we discuss the properties of such complementary sets of sequences. Algorithms for synthesizing new sets from a given set are given. Recursive formulas for constructing mutually orthogonal complementary sets are presented. It is shown that matrices consisting of mutually orthogonal complementary sets of sequences can be used as operators so as to per form transformations and inverse transformations on a one- or two-dimensional array of real time or spatial functions. The similarity between such new transformations and the Hadamard transformation suggests applications of such new transformations to signal processing and image coding.

The Origin and Structure of Easterly Waves in the Lower Troposphere of North Africa

Abstract: This study investigates the origin and structure of easterly waves that form in the lower troposphere of North Africa and have a periodicity of 3–5 days,. From June to early October these waves propagate across the Atlantic and occasionally reach the eastern Pacific. Although only a few of these disturbances actually intensify after reaching the Atlantic, they account for approximately half of the tropical cyclones that form in the Atlantic. Spectral analysis of five years of upper air data shows that African waves produce a spectral peak of the meridional wind at periods of 3–5 days with a maximum amplitude of 1–2 m sec−1 near 700 mb. These waves normally originate between Khartoum (32E) and Ft. Lamy (I5E) and affect a greater depth of the atmosphere as they propagate westward. Wind statistics at stations flanking the mountains in Ethiopia indicate that airflow over these mountains is not the cause of the easterly waves. This study shows that the African waves are directly related to the mid-tro...

Theory of monomer-dimer systems

Abstract: We investigate the general monomer-dimer partition function, P(x), which is a polynomial in the monomer activity, x, with coefficients depending on the dimer activities. Our main result is that P(x) has its zeros on the imaginary axis when the dimer activities are nonnegative. Therefore, no monomer-dimer system can have a phase transition as a function of monomer density except, possibly, when the monomer density is minimal (i.e. x = 0). Elaborating on this theme we prove the existence and analyticity of correlation functions (away from x = 0) in the thermodynamic limit. Among other things we obtain bounds on the compressibility and derive a new variable in which to make an expansion of the free energy that converges down to the minimal monomer density. We also relate the monomer-dimer problem to the Heisenberg and Ising models of a magnet and derive Christoffell-Darboux formulas for the monomer-dimer and Ising model partition functions. This casts the Ising model in a new light and provides an alternative proof of the Lee-Yang circle theorem. We also derive joint complex analyticity domains in the monomer and dimer activities. Our considerations are independent of geometry and hence are valid for any dimensionality.

Numerical Simulation of Three-Dimensional Homogeneous Isotropic Turbulence

Abstract: This Letter reports numerical simulations of three-dimensional homogeneous isotropic turbulence at wind-tunnel Reynolds numbers. The results of the simulations are compared with the predictions of the direct-interaction turbulence theory.

Thermal conductivity of Earth materials at high temperatures

Abstract: The total thermal conductivity (lattice plus radiative) of several important earth materials is measured in the temperature range from 500 to 1900 K. A new technique is used in which a CO2 laser generates a low-frequency temperature wave at one face of a small disk-shaped sample, and an infrared detector views the opposite face to detect the phase of the emerging radiation. Phase data at several frequencies yield the simultaneous determination of the thermal diffusivity and the mean extinction coefficient of the material. The lattice, radiative, and total thermal conductivities are then calculated. Results for single-crystal and polycrystalline forsterite-rich olivines and an enstatite indicate that, even in relatively pure large-grained material, the radiative conductivity does not increase rapidly with temperature. The predicted maximum total thermal conductivity at a depth of 400 km in an olivine mantle is 0.020 cal/cm/sec/deg C, which is less than twice the surface value.

Moving visual scenes influence the apparent direction of gravity.

Abstract: When an observer views a wide-angled display rotating around his line of sight, he both feels his body tilted and sees a vertical straight edge tilted opposite to the moving stimulus. Displacement of the perceived vertical increases with stimulus speed to reach a maximum (averaging 15 degrees) at 30 degrees per second.

Some Elementary Examples of Unirational Varieties Which are Not Rational

Abstract: An outstanding problem in the algebraic geometry of varieties of dimension n ^ 3 over an algebraically closed field k has been whether there exist unirational varieties which are not rational. Here V is unirational if it has the equivalent properties: (a) there exists a rational surjective map / : V -> V, or, there exists an embedding k(V) V, or, there exists an isomorphism k(V) = k(Xv ...,1^). For n—\, these are equivalent (Liiroth's theorem). For n = 2 they are equivalent in characteristic 0 (Castelnuovo's theorem) or if the map / in (a) is assumed separable (Zariski's extension of Castelnuovo's theorem). In 1959 ([13]), Serre clarified classical work on this problem for n = 3. It has been generally accepted since then that none of the examples proposed by Fano or Roth had been correctly proved irrational. In the past year, two solutions of this problem have been found: Clemens and Griffiths ([6]) showed that all non-singular cubic hypersurfaces in P are irrational, and Iskovskikh and Manin ([16]) showed that all non-singular quartic hypersurfaces in P are irrational. Some are unirational (Segre ([11])). Both of these solutions are quite deep and it seems worth while to have an elementary example as well, even if our method applies to a very special kind of variety. Ramanujam suggested using torsion in H and this led us to the examples presented here. We construct varieties, of all dimensions n ^ 3 and all characteristics £> # 2, which are unirational and which have 2-torsion in H. With the present state of resolution of singularities, we can show that such a V cannot be rational if the characteristic is 0 or if the characteristic is not 2 and n = 3.