Remarks on E11 approach

We consider a few topics in $E_{11}$ approach to superstring/M-theory: even subgroups ($Z_2$ orbifolds) of $E_{n}$, n=11,10,9 and their connection to Kac-Moody algebras; $EE_{11}$ subgroup of $E_{11}$ and coincidence of one of its weights with the $l_1$ weight of $E_{11}$, known to contain brane charges; possible form of supersymmetry relation in $E_{11}$; decomposition of $l_1$ w.r.t. the $SO(10,10)$ and its square root at first few levels; particle orbit of $l_1 \ltimes E_{11}$. Possible relevance of coadjoint orbits method is noticed, based on a self-duality form of equations of motion in $E_{11}$.Comment: Two references adde

Effects of nature of cooling surface on radiator performance

This report discusses the effects of roughness, smoothness, and cleanness of cooling surfaces on the performance of aeronautic radiators, as shown by experimental work, with different conditions of surface, on (1) heat transfer from a single brass tube and from a radiator; (2) pressure drop in an air stream in a single brass tube and in a radiator; (3) head resistance of a radiator; and (4) flow of air through a radiator. It is shown that while smooth surfaces are better than rough, the surfaces usually found in commercial radiators do not differ enough to show marked effect on performance, provided the surfaces are kept clean

Head Resistance Due to Radiators

Part 1 deals with the head resistance of a number of common types of radiator cores at different speeds in free air, as measured in the wind tunnel at the bureau of standards. This work was undertaken to determine the characteristics of various types of radiator cores, and in particular to develop the best type of radiator for airplanes. Some 25 specimens of core were tested, including practically all the general types now in use, except the flat plate type. Part 2 gives the results of wind tunnel tests of resistance on a model fuselage with a nose radiator. Part 3 presents the results of preliminary tests of head resistance of a radiator enclosed in a streamlined casing. Special attention is given to the value of wing radiator and of the radiator located in the open, especially when it is provided with a properly designed streamlined casing

Synopsis of aeronautic radiator investigations for years 1917 and 1918

Extensive series of experiments have been conducted at the Bureau of Standards to determine the properties of cooling radiator cores manufactured for airplanes and to develop improvements in design. The analysis of the problem on which this work was based, and consequently the experimental method employed, is different from that commonly used. Instead of attempting to test complete radiators, either full size or in model, uniform sections representing different types of core construction have been tested and an analysis of the results made with a view to determining independently the various factors which influence its performance. This report describes referenced method of analysis in predicting the performance of radiators designed for aeronautic use

Results of Tests on Radiators for Aircraft Engines

Part 1 is to present the results of tests on 56 types of core in a form convenient for use in the study of the performance of and possible improvements in existing designs. Working rules are given by which the data contained in the report may be used, and the most obvious conclusions as to the behavior of cores are summarized. Part 2 presents the results of tests made to determine the pressure necessary to produce water flows up to 50 gallons per minute through an 8-inch square section of radiator core. These data are of special value in evaluating the hydraulic head against which the circulating pump is required to operate

Gradient Representations and Affine Structures in AE(n)

We study the indefinite Kac-Moody algebras AE(n), arising in the reduction of Einstein's theory from (n+1) space-time dimensions to one (time) dimension, and their distinguished maximal regular subalgebras sl(n) and affine A_{n-2}^{(1)}. The interplay between these two subalgebras is used, for n=3, to determine the commutation relations of the `gradient generators' within AE(3). The low level truncation of the geodesic sigma-model over the coset space AE(n)/K(AE(n)) is shown to map to a suitably truncated version of the SL(n)/SO(n) non-linear sigma-model resulting from the reduction Einstein's equations in (n+1) dimensions to (1+1) dimensions. A further truncation to diagonal solutions can be exploited to define a one-to-one correspondence between such solutions, and null geodesic trajectories on the infinite-dimensional coset space H/K(H), where H is the (extended) Heisenberg group, and K(H) its maximal compact subgroup. We clarify the relation between H and the corresponding subgroup of the Geroch group.Comment: 43 page

Finite and infinite-dimensional symmetries of pure N=2 supergravity in D=4

We study the symmetries of pure N=2 supergravity in D=4. As is known, this theory reduced on one Killing vector is characterised by a non-linearly realised symmetry SU(2,1) which is a non-split real form of SL(3,C). We consider the BPS brane solutions of the theory preserving half of the supersymmetry and the action of SU(2,1) on them. Furthermore we provide evidence that the theory exhibits an underlying algebraic structure described by the Lorentzian Kac-Moody group SU(2,1)^{+++}. This evidence arises both from the correspondence between the bosonic space-time fields of N=2 supergravity in D=4 and a one-parameter sigma-model based on the hyperbolic group SU(2,1)^{++}, as well as from the fact that the structure of BPS brane solutions is neatly encoded in SU(2,1)^{+++}. As a nice by-product of our analysis, we obtain a regular embedding of the Kac-Moody algebra su(2,1)^{+++} in e_{11} based on brane physics.Comment: 70 pages, final version published in JHE

Eisenstein series for infinite-dimensional U-duality groups

We consider Eisenstein series appearing as coefficients of curvature corrections in the low-energy expansion of type II string theory four-graviton scattering amplitudes. We define these Eisenstein series over all groups in the E_n series of string duality groups, and in particular for the infinite-dimensional Kac-Moody groups E9, E10 and E11. We show that, remarkably, the so-called constant term of Kac-Moody-Eisenstein series contains only a finite number of terms for particular choices of a parameter appearing in the definition of the series. This resonates with the idea that the constant term of the Eisenstein series encodes perturbative string corrections in BPS-protected sectors allowing only a finite number of corrections. We underpin our findings with an extensive discussion of physical degeneration limits in D<3 space-time dimensions.Comment: 69 pages. v2: Added references and small additions, to be published in JHE

An M-theory solution from null roots in E11

We find a purely gravitational classical solution of M-theory/eleven-dimensional supergravity which corresponds to a solution of the E10 brane sigma-model involving a null root. This solution is not supersymmetric and is regularly embedded into E11.Comment: 10 page

Few smooth d-polytopes with n lattice points

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with at most 12 lattice points. In fact, it is sufficient to bound the singularities and the number of lattice points on edges to prove finiteness.Comment: 20+2 pages; major revision: new author, new structure, new result