Integral Quadratic Form Research Articles

Cost Cumulant Control: State-Feedback, Finite-Horizon Paradigm with Application to Seismic Protection

The paper is concerned with the study of the optimal control which minimizes a finite linear combination of the first k cost cumulants of a finite-horizon integral quadratic form associated with a linear stochastic system, when the controller measures the states. The solution is investigated by adapting dynamic programming techniques to the nontraditional forms evidenced by cumulant representations. The performance of this k-cost cumulant (kCC) controller is compared to that of the best control paradigms published for the American Society of Civil Engineers first-generation structure benchmark for seismically excited buildings; the simulation results indicate that the newly developed control paradigm makes better use of the available control limits and achieves uniform improvement in the officially defined performance statistics for floor vibrations and accelerations.

The universal quaternary quadratic form with the maximal discriminant

A positive definite integral quadratic form over rational integers is said to be universal, if it represents all positive integers. The universal quaternary quadratic form is determined with the maximal discriminant, which is 1073/4.

On Representations of Integers by Indefinite Ternary Quadratic Forms

Let f be an indefinite ternary integral quadratic form and let q be a nonzero integer such that −qdet(f) is not a square. Let N(T, f, q) denote the number of integral solutions of the equation f(x)=q where x lies in the ball of radius T centered at the origin. We are interested in the asymptotic behavior of N(T, f, q) as T→∞. We deduce from the results of our joint paper with Z. Rudnick that N(T, f, q)∼cEHL(T, f, q) as T→∞, where EHL(T, f, q) is the Hardy–Littlewood expectation (the product of local densities) and 0⩽c⩽2. We give examples of f and q such that c takes the values 0, 1, 2.

The design of an active suspension force controller using genetic algorithms with maximum stroke constraints

A novel control scheme is proposed for force control in an active vehicle suspension design using genetic algorithms with a half-car model. The new approach incorporates the constraints of maximum suspension strokes in the objective function to evaluate the compactness of the suspension working space, as opposed to the traditional integral quadratic form of suspension displacement. Genetic algorithms are employed in the design to obtain a more effective search for optimum control parameters. Force cancellation, virtual damper, skyhook damper and road-following concepts are employed in the scheme to achieve a satisfactory performance. Computer simulations are performed to verify the proposed control scheme.

Group schemes and local densities

The subject matter of this paper is an old one with a rich history, beginning with the work of Gauss and Eisenstein, maturing at the hands of Smith and Minkowski, and culminating in the fundamental results of Siegel. More precisely, if L is a lattice over Z (for simplicity), equipped with an integral quadratic form Q, the celebrated Smith-Minkowski-Siegel mass formula expresses the total mass of (L,Q), which is a weighted class number of the genus of (L,Q), as a product of local factors. These local factors are known as the local densities of (L,Q). Subsequent work of Kneser, Tamagawa and Weil resulted in an elegant formulation of the subject in terms of Tamagawa measures. In particular, the local density at a non-archimedean place p can be expressed as the integral of a certain volume form ωld over AutZp(L,Q), which is an open compact subgroup of AutQp(L,Q). The question that remains is whether one can find an explicit formula for the local density. Through the work of Pall (for p 6= 2) and Watson (for p = 2), such an explicit formula for the local density is in fact known for an arbitrary lattice over Zp (see [P] and [Wa]). The formula is obviously structured, though [CS] seems to be the first to comment on this. Unfortunately, the known proof (as given in [P] and [K]) does not explain this structure and involves complicated recursions. On the other hand, Conway and Sloane [CS, §13] have given a heuristic explanation of the formula. In this paper, we will give a simple and conceptual proof of the local density formula, for p 6= 2. The view point taken here is similar to that of our earlier work [GHY], and the proof is based on the observation that there exists a smooth affine group scheme G over Zp with generic fiber AutQp(L,Q), which satisfies G(Zp) = AutZp(L,Q). This follows from general results of smoothening [BLR], as we explain in Section 3. For the purpose of obtaining an explicit formula, it is necessary to have an explicit construction of G. The main contribution of this paper is to give such an explicit construction of G (in Section 5), and to determine its special fiber (in Section 6). Finally, by comparing ωld and the canonical volume form ωcan of G, we obtain the explicit formula for the local density in Section 7. The smooth group schemes constructed in this paper should also be of independent interest.

Sums of squares of integral linear forms

AbstractIn this paper we prove that every positive definite n-ary integral quadratic form with 12 < n < 13 (respectively 14 ≦ n ≤ 20) that can be represented by a sum of squares of integral linear forms is represented by a sum of 2 · 3n + n + 6 (respectively 3 · 4n + n + 3) squares. We also prove that every positive definite 7-ary integral quadratic form that can be represented by a sum of squares is represented by a sum of 25 squares.

Strong genus of nilpotent groups of Hirsch length six

We give a characterization in terms of finitely many arithmetical invariants for two finitely generated, torsion-free, nilpotent groups of nilpotency class two and Hirsch length six to have isomorphic localizations at every finite set of primes. This result is obtained through adequate reduction of an arbitrary binary integral quadratic form. We derive some consequences and, in particular, we characterize completely those groups N in the above class of groups, for which the Mislin genus coincides with the strong genus (i.e. those groups N for which, whenever a nilpotent group M satisfies Mp ≅ Np for every prime p, then MP ≅ NP for every finite set of primes P).

Edge reduction for weakly non-negative quadratic forms

Let \( q\colon \Bbb Z^n\to \Bbb Z \) be an integral quadratic form of the shape¶¶\( q(x)=\sum\limits\limits ^n_{i=1}q_ix^2_i+\sum\limits\limits _{i\le j}q_{ij}x_ix_j \)¶¶with \( q_i\le 1 \), for every \( 1\le i\le n \). Several procedures have been introduced to study these forms. In this paper we consider the edge reduction procedure introduced in [7]. ¶A form \( q\colon \Bbb Z^n\to \Bbb Z \) is weakly non-negative if \( q(x)\ge 0 \) for every vector x with non-negative coordinates. Let \( q^\prime \colon \Bbb Z^{n+1}\to \Bbb Z \) be obtained from q by edge reduction, then q is weakly non-negative if and only if so is \( q^\prime \). We propose an algorithm to decide if q is weakly non-negative.

Representations of Positive Definite Senary Integral Quadratic Forms by a Sum of Squares

As a generalization of the famous four square theorem of Lagrange, it was proved that every positive definite integral quadratic form ofnvariables is represented by the sun ofn+3 squares for 1⩽n⩽5. In this paper, we prove that every positive definite integral quadratic form of six variables that can be represented by a sum of squares is represented by the sum of ten squares, no less.

An inequality for an integral quadratic form

This paper is motivated by the consideration of the stability of a taut string fixed at x = 0 and x = 1 under the weight of a symmetric distributed load. It is intuitively clear that the potential energy of the string will be decreased if portions of the load distribution are moved from the left half of the string to the other half at symmetrical positions. We shall verify this observation by deriving an integral inequality for the energy functional of the string. The mode of proof shows that similar conclusions also hold for symmetrically loaded one-dimensional mechanical systems such as the supported beam.

Additively Indecomposable Positive Integral Quadratic Forms

Call a positive semidefinite integral quadratic form on a lattice additively indecomposable if it cannot be written as the sum of two such nonzero forms. Criteria for additive indecomposability are proved and quick proofs are given for classical results like the finiteness of the number of equivalence classes of such forms in any dimension and the full classification up to dimension 8 (1-dimensional lattice generated by a norm 1 vector, E 6, E 7, E 8). The same finiteness result is proved for invariant forms on lattices with a finite group acting on them.

Unimodular congruence of the Laplacian matrix of a graph

Let G be a graph with vertices 1, 2, …, n. Associated with G, there is an integral quadratic form, Q( x), on the n-tuple of indeterminates x = (x 1, …, x n) , given by Q( x ) = Σ(x i − x j) 2 , where the sum is taken over all edges ( i, j) of G. In this paper we prove that the quadratic forms Q 1, Q 2 associated with graphs G 1, G 2 are congruent by a unimodular matrix if and only if G 1 and G 2 are cycle isomorphic.

An elementary remark on the distribution of integers representable by a positive-definite integral binary quadratic form

In this note we prove an elementary result concerning the distribution of the positive integers which are represented by a positive-definite integral binary quadratic form Theorem 1. Let f(X, Y) = aX ~ + b X Y + c y2 be a positive-definite integral binary quadratic form of discriminant - A (= b 2 - 4ac = mi , there exist integers x and y such that n < ax 2 q- bxy + cy 2 < n + 2ml/~A1/4n ~m + m i . P r 0 0 f. Replacing the form f by an equivalent form we may suppose that m~ = c. We define integers x and y by =F(4cn~l/2q I!4cn-Ax2)l/2-bx l x L\~) /' Y= 2c -+1.

Rings, quadratic forms, and complete degeneracy for a subclass of highly overmoded rectangular waveguides

In this article eigenvalue degeneracy formulas for a subclass of highly overmoded rectangular waveguides are determined. This subclass consists of those rectangular geometries with side length ratio, a/b, equal to the square root of a rational integer, √σ, σ∈Z, and contains an unusual amount of degeneracy even when the side length ratio is irrational. The complete degeneracy for members of this subclass (σ&amp;lt;10) is determined by interpreting its normalized eigenvalue formula as (a) the norm of an associated ring of algebraic integers and (b) a special case of a positive definite binary integral quadratic form.

Small solutions to a given quadratic form with a variable modulus

For a positive definite integral quadratic form Q( x) in at least 4 variables, we show that there is a constant c = c( Q) so that for any m > 0, there is a non-zero integral vector x = ( x i ) such that Q( x) ≡ 0 mod( m), and max | x i | ≤ c√ m.

Small prime solutions of some additive equations

Essentially sharp bounds for small prime solutionspj,qi of the following two different types of equations are obtained. $$\begin{gathered} a_1 p_1^2 + \ldots + a_5 p_5^2 = b, \hfill \\ c_1 q_1 + c_2 q_2 + c_3 q_3^k = d \hfill \\ \end{gathered}$$ where the two sets of integers,a1,...,a5,b andc1,c2,c3,d satisfy the congruent solubility condition andk≥1 is an integer. These results are comparable with the well-known Meyer theorem on indefinite integral quadratic form and the celebrated Vinogradov three primes theorem.

Spinor Regular Positive Ternary Quadratic Forms

Refining the notion of regularity introduced by Dickson, an integral quadratic form is said to be spinor regular if it represents all integers represented by its spinor genus. Examples of positive definite primitive integral ternary quadratic forms which have this property are presented, and it is proved that there exist only finitely many equivalence classes containing such forms.

One point extensions of linear quivers and quadratic forms

In representation theory of finite-dimensional algebras over a field k, the idea of associating an integral quadratic form qA to a given algebra A, in order to study the relation between the representation type of A and the sign of qA in the positive integral cone, has its origins in a fundamental paper of Gabriel [4]. In the present paper we examine, for any n E M, one-point extensions of the form

The application of the Shimura lifting to the representation of large numbers of ternary quadratic forms

One obtains an asymptotic formula for the number of the representations of numbers, divisible by a large square, by a positive ternary integral quadratic form. One gives an estimate of the remainder, unimprovable with respect to the quadratic part and uniform with respect to the square-free part of the represented number.

Waring's problem for a ternary quadratic form and an arbitrary even power

One obtains asymptotic formulas for the number of solutions of the equation n= f(x,y,z)+w2k, where f is a primitive integral quadratic form. One gives an estimate of the remainder, having a logarithmic reducing factor in the general case and a powerlike one when f(x,y,z)=x2+y2+z2.