Let$G$denote a totally disconnected locally compact metric abelian group with translation invariant metric$d$and character group$Γ$_{$G$}. The Lipschitz spaces are defined by $$Lip\left( {\alpha ;p} \right) = \left\{ {f \in L^p \left( G \right):\left\| {\tau _a f - f} \right\|_p = O\left( {d\left( {a,0} \right)^\alpha } \right),a \to 0} \right\},$$ where$τ$_{$a$}$f$:$x$→$f$($x-a$) and α∈(0,1). For a suitable choice of metric it is shown that Lip (α;$p$)⊂$L$^{$r$}($Γ$_{$G$}), where α>1/$p$+1/$r$−1≧0 and 1≦$p$≦2. In the case$G$is compact the corresponding result holds for α>1/$r$−1/2 and$p$>2. In addition for$G$non-discrete the above result is shown to be sharp, in the sense that the range of values of α cannot be extended. The results include classical theorems of S. N. Bernstein, O. Szász and E. C. Titchmarsh.

We supply a detailed proof of the result by P.S. Green and T. H$\ddot{\text{u}}$bsch that all complete intersection Calabi--Yau 3-folds in product of projective spaces are connected through projective conifold transitions (known as the standard web). We also introduce a subclass of small transitions which we call \emph{primitive} small transitions and study such subclass. More precisely, given a small projective resolution $\pi : \widehat{X} \rightarrow X$ of a Calabi--Yau 3-fold $X$, we show that if the natural closed immersion $\Def(\widehat{X}) \hookrightarrow \Def(X)$ is an isomorphism then $X$ has only ODPs as singularities.

We provide an explicit desingularization and study the resulting fiber geometry of elliptically fibered four-folds defined by Weierstrass models admitting a split ilde {A} _4 singularity over a divisor of the discriminant locus. Such varieties are used to geometrically engineer SU (5) grand unified theories in F-theory. The desingularization is given by a small resolution of singularities. The ilde {A} _4 fiber naturally appears after resolving the singularities in codimension-one in the base. The remaining higher codimension singularities are then beautifully described by a four-dimensional affine binomial variety which leads to six different small resolutions of the elliptically fibered four-fold. These six small resolutions define distinct four-folds connected to each other by a network of flop transitions forming a dihedral group. The location of these exotic fibers in the base is mapped to conifold points of the three-folds that defines the type IIB

We study the initial value problem of the thermal-diffusive combustion systemu 1, t= u 1, x, x u 1 u 2 2, u 2, t= du 2, xx+ u 1 u 2 2, x R 1, for non-negative spatially decaying initial data of arbitrary size and for any positive constantd. We show that if the initial data decay to zero sufficiently fast at infinity, then the solution (u 1, u 2) converges to a self-similar solution of the reduced systemu 1, t= u 1, xx u 1 u 2 2, u 2, t= du 2, xx, in the large time limit. In particular, u 1 decays to zero like O (t 1/2 ), where> 0 is an anomalous exponent depending on the initial data, andu 2 decays to zero with normal rate O (t 1/2). The idea of the proof is to combine the a priori estimates for the decay of global solutions with the renormalization group method for establishing the self-similarity of the solutions in the large time limit.

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