Diagonal Space Research Articles

Adaptive Optics Feedback Control

This paper concentrates on the control aspects of Adaptive Optics (AO) systems and includes a prior exposure to linear control systems from the “classical” point of view. The AO control problem is presented and the well-established optimized modal gain integral control approach is discussed. The design of a controller from a modern control point of view is addressed by means of a linear quadratic Gaussian control methodology. The proposed approach emphasizes the ability of the adaptive optics loop to reject the atmospheric aberration. We derive a diagonal state space system which clearly separates the dynamics of the plant (deformable mirror & wavefront sensor) from the disturbance dynamics (atmospheric model). This representation facilitates the numerical resolution of the problem. A frequency analysis is carried out to check performance and robustness specifications of the multiple-input multiple-output feedback system. The effectiveness of the approach is demonstrated through numerical experiments.

The Number of Intervals in the $m$-Tamari Lattices

An $m$-ballot path of size $n$ is a path on the square grid consisting of north and east steps, starting at $(0,0)$, ending at $(mn,n)$, and never going below the line $\{x=my\}$. The set of these paths can be equipped with a lattice structure, called the $m$-Tamari lattice and denoted by $\mathcal{T}_n^{(m)}$, which generalizes the usual Tamari lattice $\mathcal{T}_n$ obtained when $m=1$. We prove that the number of intervals in this lattice is $$ \frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}. $$ This formula was recently conjectured by Bergeron in connection with the study of diagonal coinvariant spaces. The case $m=1$ was proved a few years ago by Chapoton. Our proof is based on a recursive description of intervals, which translates into a functional equation satisfied by the associated generating function. The solution of this equation is an algebraic series, obtained by a guess-and-check approach. Finding a bijective proof remains an open problem.

Some Designs and Normalized Diversity Product Upper Bounds for Lattice-Based Diagonal and Full-Rate Space–Time Block Codes

In this paper, we first present two tight upper bounds for the normalized diversity products (or product distances) of 2ntn2 diagonal space-time block codes from quadratic extensions on \BBQ (i) and \BBQ ( \mmbzeta6), where i =radic{-1} and \mmbzeta6=exp( i 2pi/6). Two such codes are shown to reach the tight upper bounds and therefore have the maximal normalized diversity products. We present two new diagonal space-time block codes from higher order algebraic extensions on \BBQ (i) and \BBQ ( \mmbzeta6) for three and four transmit antennas. We also present a nontight upper bound for normalized diversity products of 2ntn2 diagonal space-time block codes with QAM information symbols, i.e., in \BBZ [i ], from general 2ntn2 complex-valued generating matrices. We then present an nntnn-diagonal space-time code design method directly from 2n real integers based on extended complex lattices (of generating matrix size nntn2n) that are shown to have better normalized diversity products than the optimal diagonal cyclotomic codes do. We finally use the optimal 2ntn2 diagonal space-time codes from the optimal quadratic extensions to construct two 2ntn2 full-rate space-time block codes and find that both of them have better normalized diversity products than the Golden code does.

Block Diagonal Differential Space–Time–Frequency Codes for Four Transmit Antennas

For Multiple Input Multiple Output-Orthogonal Frequency Division Multiplexing (MIMO-OFDM) system with four transmit antennas, this paper presents a novel block diagonal differential space---time---frequency (DSTF) coded scheme with full diversity and rate one. It overcomes the rate loss of the traditional differential orthogonal space---time---frequency coded scheme for four transmit antennas. The coding and decoding method are described in detail. After analyzing the system error performance, this paper provides two measures to improve the system error performance. One measure is to design good codes which achieve large coding gain. Thus, linear constellation decimation (LCD) codes are imported. The optimal decimation factors are given in the Table 1 under different design parameters. Another measure is to select a good subcarrier grouping method. A grouping method is proposed that all groups have the same subcarrier spacing. By the derived rule of evaluating subcarrier grouping, the optimal subcarrier spacing is obtained by computer search under different channels. Simulation results are presented to illustrate the performance of block diagonal DSTF codes and the proposed subcarrier grouping method.

Optimal Normalized Diversity Product of $2\,\times\,2$ Lattice-Based Diagonal Space–Time Codes From QAM Signal Constellations

In this correspondence, we prove that the optimal normalized diversity product of lattice-based diagonal space-time block codes with Gaussian integer (or QAM) signal constellations, i.e., , and any generating matrices of complex entries (not necessarily algebraic extensions of as commonly used) is . This result implies that lattice-based diagonal space-time block codes with Gaussian integer signal constellations and generating matrices of entries from quadratic algebraic extensions of have already reached the optimal normalized diversity product.

A class of Möbius invariant function spaces

We introduce a class of Möbius invariant spaces of analytic functions in the unit disk, characterize these spaces in terms of Carleson type measures, and obtain a necessary and sufficient condition for a lacunary series to be in such a space. Special cases of this class include the Bloch space, the diagonal Besov spaces, BMOA, and the so-called $Q_p$ spaces that have attracted much attention lately.

ON THE "CERTAIN" EXACT SEQUENCE OF WHITEHEAD

This paper is devoted to the classification problem of homotopy types of a new class of simply connected CW-complexes called diagonal spaces which contains all spaces X such that H*(X, ℤ) is free. We show that two simply connected diagonal CW-complexes X and Y are homotopic if and only if their Whitehead exact sequences are isomorphic.

Bergman type projections onL p spaces

The paper is a survey on the action of Bergman type projections on various Lp spaces. The focus is on three types of holomorphic function spaces: weighted Bergman spaces, the Bloch space, and diagonal Besov spaces.

Sequential Computability of a Function: Diagonal Space and Limiting Recursion

We consider the real sequences in I=[0,1) and real functions on I. A computability notion with respect to the uniformity {Un}, where Un(x)=[k2n,k+12n) if x∈[k2n,k+12n) will be called D-computability. An R-computable sequence from I will be shown to be approximated by a recursive sequence of rational numbers with a limiting recursive modulus of convergence with respect to {Un}. Using this result, we relate two extended notions of sequential computability of a function or a function sequence, one formulated in terms of limiting recursion and one in terms of {Un}.

A Diagonalisation Algorithm and Its Application in Ambiguity Search

A diagonalisation algorithm of the least squares normal equation is proposed in this paper. Theequivalent observation equations related to the diagonalised normal equations are also derived in detail. For the equivalent observation equations and their normal equations, the related equivalent ambiguity search criteria are outlined. Theoretical application of the proposed algorithm in ambiguity search is briefly sum-marised. Using this algorithm, the ambiguity search turns out to be a search in a diagonal space so that the search can be done very quickly. Numerical examples to illustrate the diagonalisation process of the normal equation and observation equation are also given.

Besov spaces and Bergman projections on the ball

A class of radial differential operators are investigated yielding a natural classification of diagonal Besov spaces on the unit ball of C N . Precise conditions are given for the boundedness of Bergman projections from certain L p spaces onto Besov spaces. Right inverses for these projections are also provided. Applications to complex interpolation are presented. To cite this article: H.T. Kaptanoğlu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 729–732.

Rank-one perturbations of diagonal operators

We study rank-one perturbations of diagonal Hilbert space operators mainly from the standpoint of invariant subspace problem. In addition to proving some general properties of these operators, we identify the normal operators and contractions in this class. We show that two well known results about the eigenvalues of rank-one perturbations and one-codimension compressions of self-adjoint compact operators are equivalent. Sufficient conditions are given for existence of nontrivial invariant subspaces for this class of operators.

Cesaro Summability with Respect to Two-Parameter Walsh Systems

The one- and two-parameter Walsh system will be considered in the Paley as well as in the Kaczmarz rearrangement. We show that in the two-dimensional case the restricted maximal operator of the Walsh–Kaczmarz (C, 1)-means is bounded from the diagonal Hardy space H p to L p for every . To this end we consider the maximal operator T of a sequence of summations and show that the p-quasi-locality of T implies the same statement for its two-dimensional version T α. Moreover, we prove that the assumption is essential. Applying known results on interpolation we get the boundedness of T α as mapping from some Hardy–Lorentz spaces to Lorentz spaces. Furthermore, by standard arguments it will be shown that the usual two-parameter maximal operators of the (C, 1)-means are bounded from L p spaces to L p if . As a consequence, the a.e. convergence of the (C, 1)-means will be obtained for functions such that their hybrid maximal function is integrable. Of course, our theorems from the two-dimensional case can be extended to higher dimension in a simple way.

On the statistical interpretation of density functions: Atomic shell approximation, convex sets, discrete quantum chemical molecular representations, diagonal vector spaces and related problems

Following a statistical interpretation analysis, as a discrete probability distribution, of the atomic shell approximation linear coefficient set, a new description of quantum object sets is given, using a discrete representation, based on diagonal vector spaces. Previous definitions, involving essentially tagged sets and vector semispaces, are used.

Diagonal Cauchy spaces

A diagonal condition is defined which internally characterises those Cauchy spaces which have topological completions. The T2 diagonal Cauchy spaces allow both a finest and a coarsest T2 diagonal completion. The former is a completion functor, while the latter preserves uniformisability and has an extension property relative to θ-continuous maps.

Sums, products, and mappings of weakly pseudocompact spaces

Each open subspace of a weakly pseudocompact space is either weakly pseudocompact or locally compact Lindelöf. A topological sum is weakly pseudocompact if and only if 1. (1) each summand is either weakly pseudocompact or locally compact Lindelöf and 2. (2) the sum is either compact or not Lindelöf. If X is realcompact and the lattice of compactifications of X is a b-lattice or if X is a not Čech-complete G δ -diagonal space then X is not weakly pseudocompact. Weak pseudocompactness is neither an inverse invariant of perfect maps nor an invariant of open maps with compact fibers. There is a not pseudocompact space in which each zero set is weakly pseudocompact.

Exact recursion relation for pseudobidimensional arrays of dumbbells

Exact relationships are developed that describe the occupation statistics for pseudobidimensional arrays of dumbbells. It is found that E(k,N), the number of ways of arranging k indistinguishable dumbbells on a 3×N diagonal array of compartments is exactly described by the recursion relation E(k,N) = 4E(k−1,N−1)+E(k,N−1)−2E(k−2,N−2). The 3×N diagonal lattice space provides to the central sites of the lattice their full coordination number of nearest-neighboring compartments; therefore the solution of the present system provides a 1×N diagonal (central sites) ’’window’’ to watch the behavior of a true bidimensional array of dumbbells. For large values of N, the dumbbell freedom per lattice site on any given site of the central diagonal is found to be 1.8477⋅⋅⋅, which is only 3% higher than the exact value (1.7916⋅⋅⋅) found for a ’’locked’’ dumbbell on a bidimensional array.

Radiation patterns and gain of a four-antenna array located at the corners of a square around a central parasitic antenna

An array of four driven antennas, located at the corners of a square, around a central parasitic antenna, is analysed. All antennas are taken to be of one-quarter wavelength and grounded to a perfectly conducting plane. It can be easily proved that, with the antennas shorter than one-quarter wavelength, grounded to an imperfectly conducting earth, the shape of the radiation patterns and the field strength gain remain practically unaffected because the field strength is then reduced uniformly in the same ratio.The procedure followed in this analysis is, first, to calculate the total impedance at the base of each antenna as a function of the antenna diagonal spacing. This spacing is the fundamental variable used in all subsequent calculations.Next, the ratio of the parasitic antenna induction current to the current fed into each of the four driven antennas is determined. An equation is obtained, giving the field strength produced at an arbitrary point in space. This equation gives immediately the horizontal and vertical radiation patterns in terms of the diagonal spacing of the array. Under certain conditions it exhibits pronounced directive characteristics.Finally, a formula giving the root mean square field strength in the horizontal plane is obtained. This formula is transformed later in order to give the root mean square field strength which is produced from the total power fed into the array. The same total power fed into a single antenna of the same design produces a different horizontal field. Comparing the two fields produced by the array and the single antenna respectively, a relation giving the average field strength gain is obtained.It is confirmed that this average gain, under certain favourable conditions, having regard to the diagonal spacing, is increased as much as 10 to 20 per cent. An even greater field strength gain, up to 60 per cent., is obtained in certain directions, with a given diagonal spacing of the array.

Relation between the Reciprocal Impenetrability of Matter and Pauli's Exclusion Principle: a Correction

SOME paradoxes which have occurred to me and have been pointed out to me also by some of my colleagues (especially Dr. Fues, Copenhagen), show that the fundamental statement of my letter (NATURE, Feb. 5) published under the above title, is incorrect. It is not true that the reciprocal impenetrability of the molecules allows only of the Heisenberg-Dirac determinant solutions, and excludes all others. On the contrary, all the symmetrical and antisymmetrical characteristic solutions which existed for absolutely penetrable molecules remain for a (not one-dimensional) gas with molecules having a radius very small compared with the mean distance; only the characteristic values are a little changed, and the characteristic functions undergo a deformation in the immediate neighbourhood of the ‘diagonal spaces.’ If, therefore, the Pauli principle is valid not only for the electrons of an atom but also for the translatory motion of gas molecules, with radius almost zero, then such a remarkable relation between the molecules cannot be so simply explained by wave mechanics as my mistake led me to believe.