Define for a smooth compact hypersurface $M^n$ of ${\mathbf{R}}^{n+1}$ its crumpleness $\kappa \left(M^n\right)$ as the ratio ${diam}_{{\mathbf{R}}^{n+1}}(M^n)/r(M^n)$, where $r\left(M^n\right)$ is the distance from $M^n$ to its central set. (In other words, $r\left(M^n\right)$ is the maximal radius of an open non-selfintersecting tube around $M^n$ in ${\mathbf{R}}^{n+1}.)$ We prove that any $n$-dimensional non-singular compact algebraic hypersurface of degree $d$ is rigidly isotopic to an algebraic hypersurface of degree $d$ and of crumpleness $\le exp\left(c\right(n\left)d^{\alpha \left(n\right)d^{n+1}}\right)$. Here $c\left(n\right)$, $\alpha \left(n\right)$ depend only on $n$, and rigid isotopy...

First we find effective bounds for the number of dominant rational maps $f:X\to Y$ between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type $\{A·K_X^n{\}}^{\{B·K_X^n{\}}^2}$, where $n=\dim X$, $K_X$ is the canonical bundle of $X$ and $A,B$ are some constants, depending only on $n$. Then we show that for any variety $X$ there exist numbers $c\left(X\right)$ and $C\left(X\right)$ with the following properties: For any threefold $Y$ of general type the number of dominant rational maps $f:X\to Y$ is bounded above by $c\left(X\right)$. ...

For $\mathbf{r}=(r_0,...,r_{d-1})\in {ℝ}^d$ define the function $${\tau }_{\mathbf{r}}:{\mathbb{Z}}^d\to {\mathbb{Z}}^d,\mathbf{z}=(z_0,...,z_{d-1})\mapsto (z_1,...,z_{d-1},-⌊\mathbf{rz}⌋),$$ where $\mathbf{rz}$ is the scalar product of the vectors $\mathbf{r}$ and $\mathbf{z}$. If each orbit of ${\tau }_{\mathbf{r}}$ ends up at $\mathbf{0}$, we call ${\tau }_{\mathbf{r}}$ a shift radix system. It is a well-known fact that each orbit of ${\tau }_{\mathbf{r}}$ ends up periodically if the polynomial $t^d+r_{d-1}{t}^{d-1}+\cdots +r_0$ associated to $\mathbf{r}$ is contractive. On the other hand, whenever this polynomial has at least one root outside the unit disc, there exist starting vectors that give rise to unbounded orbits. The present paper deals...

We prove the existence of at least three solutions to the following fractional boundary value problem: ⎧ $-d/dt(1/2_0{D}_t^{-\sigma }\left(u^{\text{'}}\left(t\right)\right)+1/2_t{D}_T^{-\sigma }\left(u^{\text{'}}\left(t\right)\right))-\lambda \beta \left(t\right)f\left(u\left(t\right)\right)-\mu \gamma \left(t\right)g\left(u\left(t\right)\right)=0$, a.e. t ∈ [0, T], ⎨ ⎩ u (0) = u (T) = 0, where ${}_0{D}_t^{-\sigma }$ and ${}_t{D}_T^{-\sigma }$ are the left and right Riemann-Liouville fractional integrals of order 0 ≤ σ < 1 respectively. The approach is based on a recent three critical points theorem of Ricceri [B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal. 74 (2011), 7446-7454].

Let $C$ be a smooth real quartic curve in ${\mathbb{P}}^2$. Suppose that $C$ has at least $3$ real branches $B_1,B_2,B_3$. Let $B=B_1\times B_2\times B_3$ and let $O\in B$. Let ${\tau }_O$ be the map from $B$ into the neutral component Jac$\left(C\right){\left(ℝ\right)}^0$ of the set of real points of the jacobian of $C$, defined by letting ${\tau }_O\left(P\right)$ be the divisor class of the divisor $\sum P_i-O_i$. Then, ${\tau }_O$ is a bijection. We show that this allows an explicit geometric description of the group law on Jac$\left(C\right){\left(ℝ\right)}^0$. It generalizes the classical geometric description of the group law on the neutral component of the set of real...

It is well-known that if r is a rational number from [-1,0), then there is no polynomial f in two complex variables and a fiber $f^{-1}\left(t₀\right)$ such that r is the Łojasiewicz exponent of grad(f) near the fiber $f^{-1}\left(t₀\right)$. We show that this does not remain true if we consider polynomials in real variables. More exactly, we give examples showing that any rational number can be the Łojasiewicz exponent near the fiber of the gradient of some polynomial in real variables. The second main result of the paper is...

Let $X\to B$ an elliptic fibration with general fibre $F$. Let $n_e,n_s,n_a,n_v$ be the minima of the non-zero intersection numbers $(ℒ,F)$ where $ℒ$ runs successively through the following sets: effective divisors on $X$, invertible sheaves spanned by global sections, ample divisors and very ample divisors. Let $m$ be the maximum of the multiplicities of the fibres of $X\to B$. We prove that $n_e=n_s$ if and only if $n_e\ge 2m$ and that $n_a=n_v$ if and only if $n_a\ge 3m$.

We prove that a self-map f: M → M of a compact PL-manifold of dimension ≥ 3 is homotopic to a map with no periodic points of period n iff the Nielsen numbers $N\left(f^k\right)$ for k dividing n all vanish. This generalizes the result from [Je] to dimension 3.

We describe the polynomials P ∈ ℂ[x,y] such that $P(1/vⁿ,A₁vⁿ+A₂v^{2n}+...+A_{m-1}{v}^{n(m-1)}+v^{nm-k}w)\in ℂ[v,w]$. As applications we give new examples of bad field generators and examples of families of polynomials with smooth and irreducible fibers.

In the paper the conditions for the existence of a $2\pi$-periodic solution in $t$ of the system $u_{tt}-u_{rr}-(2/r)u_r=ϵf(t,r,u,u_t,u_r)$, $\left|u(t,0)\right|<+\infty ,u(t,\pi )=0$ are investigated provided that $f$ is sufficiently smooth and $2\pi$-periodic in $t$.