Let N ≥ 2 be an integer, F a quadratic form in N variables over Q, and Z ⊆ Q N an L-dimensional subspace, 1 ≤ L ≤ N . We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z, F ). This provides an analogue over Q of a wellknown theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bil...

‎Let $X,Y$ be normed spaces with $L(X,Y)$ the space of continuous‎ ‎linear operators from $X$ into $Y$‎. ‎If ${T_{j}}$ is a sequence in $L(X,Y)$,‎ ‎the (bounded) multiplier space for the series $sum T_{j}$ is defined to be‎ [ ‎M^{infty}(sum T_{j})={{x_{j}}in l^{infty}(X):sum_{j=1}^{infty}%‎ ‎T_{j}x_{j}text{ }converges}‎ ‎]‎ ‎and the summing operator $S:M^{infty}(sum T_{j})rightarrow Y$ associat...

‎let $x,y$ be normed spaces with $l(x,y)$ the space of continuous‎ ‎linear operators from $x$ into $y$‎. ‎if ${t_{j}}$ is a sequence in $l(x,y)$,‎ ‎the (bounded) multiplier space for the series $sum t_{j}$ is defined to be‎ [ ‎m^{infty}(sum t_{j})={{x_{j}}in l^{infty}(x):sum_{j=1}^{infty}%‎ ‎t_{j}x_{j}text{ }converges}‎ ‎]‎ ‎and the summing operator $s:m^{infty}(sum t_{j})rightarrow y$ associat...

Stationary Problem I We consider a mapping, A : X 7→ Y , given explicitly by A(u) = Lu+ a(u, u, · ) + F (u, · ), where L : X 7→ X is an isomorphism. The structure of L is induced by a continuous, coercive bilinear form B(· , · ) on X ⊗X: 〈Lu, v〉 = B(u, v), B(u, u) ≥ c‖u‖. Charge Transport in anIncompressible Fluid Medium – p. 31/63 Abstract Stationary Problem I We consider a mapping, A : X 7→ Y...

Let N ≥ 2 be an integer, F a quadratic form in N variables over Q, and Z ⊆ Q N an L-dimensional subspace, 1 ≤ L ≤ N . We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z, F ). This provides an analogue over Q of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bi...

Let X,Y, Z be Banach spaces and let u : X×Y → Z be a bounded bilinear map. Given a locally compact abelian group G , and two functions f ∈ L(G,X) and g ∈ L(G,Y ), we define the u -convolution of f and g as the Z -valued function f ∗u g(t) = ∫ G u(f(t− s), g(s))dμG(s) where dμG stands for the Haar measure on G . We define the concepts of vector-valued approximate identity and summability kernel ...

Let N ≥ 2 be an integer, F a quadratic form in N variables over Q, and Z ⊆ Q N an L-dimensional subspace, 1 ≤ L ≤ N . We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z, F ). This provides an analogue over Q of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bi...

An N=1 supersymmetric generalization of coupled dispersionless (SUSY-CD) integrable system has been proposed by writing its superfield Lax representation. It shown that under a suitable variable transformation, the SUSY-CD is equivalent to sine-Gordon equation. A bilinear form using super Hirota operator. Explicit expressions soliton solutions have obtained method.

The X s,b spaces, as used by Beals, Bourgain, Kenig-Ponce-Vega, Klainerman-Machedon and others, are fundamental tools to study the low-regularity behaviour of non-linear dispersive equations. It is of particular interest to obtain bilinear or multilinear estimates involving these spaces. By Plancherel's theorem and duality, these estimates reduce to estimating a weighted convolution integral in...

The common topic of this thesis is boundedness of integral and supremal operators between function spaces with weights. The results of this work have the form of characterizations of validity of weighted operator inequalities for appropriate cones of functions. The outcome can be divided into three categories according to the particular type of studied operators and function spaces. The first p...