Abstract:
We apply Tatuzawa's version of Siegel's theorem to derive two lower bounds on the size of the principal genus of positive definite binary quadratic forms.

Abstract:
Based on the feature extraction of principal component,a novel infrared target detection algorithm was proposed which using subspace quadratic synthetic discriminant function (SSQSDF).Firstly,the kernel principal component analysis was extended to mixture probabilistic model,and the latter get the principal component vectors of target samples.Then,training samples and samples to be detected were projected on principal component vectors obtained previously to acquire their low-dimension feature components,and the obtained components are used as the sample parameters for the SSQSDF.The detected samples which had a higher SSQSDF filtering output than given threshold were considered as the detected targets.The proposed algorithm can evidently restrain clutter noise,improve target detection precision.Experimental results under complex scenery demonstrate that the proposed algorithm is feasibility and effectiveness.

Abstract:
We observe structure in the sequences of quotients and remainders of the Euclidean algorithm with two families of inputs. Analyzing the remainders, we obtain new algorithms for computing modular inverses and representating prime numbers by the binary quadratic form $x^2 + 3xy + y^2$. The Euclidean algorithm is commenced with inputs from one of the families, and the first remainder less than a predetermined size produces the modular inverse or representation.

Abstract:
In this paper, we propose a new construction of quadratic bent functions in polynomial forms. Right Euclid algorithm in skew-polynomial rings over finite fields of characteristic 2 is applied in the proof.

Abstract:
Using the circle method, we show that for a fixed positive definite integral quadratic form $A$, the expected asymptotic formula for the number of representations of a positive definite integral quadratic form $B$ by $A$ holds true, providing that the dimension of $A$ is large enough in terms of the dimension of $B$ and the maximum ratio of the successive minima of $B$, and providing that $B$ is sufficiently large in terms of $A$.

Abstract:
In a paper by J. Deutsch [1], a quaternionic proof of the universalityof seven quaternary quadratic forms was given. The proof relieson a construction very similar to that of Hurwitz quaternions, and itsassociated division algorithm. Of course, these results are evident, ifone uses the Conway-Schneeberger Fifteen Theorem [2], as the authoralso mentioned, however it is interesting to give a direct proof for somespecific quadratic forms based on simple argument. It is the purposeof this short note to prove five of the seven quadratic forms mentionedand proven by Deutsch, using the universality of the classical quadraticform associated to the celebrated Lagrange’s Theorem of Four Squaresand Euler’s trick.

Abstract:
We investigate the non-diagonal normal forms of a quadratic form on R^n, in particular for n=3. For this case it is shown that the set of normal forms is the closure of a 5-dimensional submanifold in the 6-dimensional Grassmannian of 2-dimensional subspaces of \R^5.

Abstract:
The generic quadratic form of even dimension n with trivial discriminant over an arbitrary field of characteristic different from 2 containing a square root of -1 can be written in the Witt ring as a sum of 2-fold Pfister forms using n-2 terms and not less. The number of 2-fold Pfister forms needed to express a quadratic form of dimension 6 with trivial discriminant is determined in various cases.

Abstract:
We show a relationship between Chern-Simons 1- and 3-forms and harmonic forms on a principal bundle. Doing so requires one to consider an adiabatic limit. For the 3-form case, assume that G is simple and the corresponding Chern-Weil 4-form is exact. Then, the Chern-Simons 3-form on the princpal bundle G-bundle, minus a canonical term from the base, is harmonic in the adiabatic limit.