This paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``Advanced School and Conference on homological and geometrical methods in representation theory'' at the International Centre for Theoretical Physics (ICTP), Miramare - Trieste, Italy, 11-29 January 2010. We show how to associate to a triple of posit...

in this work, byemploying the leggett-williams fixed point theorem, we study theexistence of at least three positive solutions of boundary valueproblems for system of third-order ordinary differential equationswith $(p_1,p_2,ldots,p_n)$-laplacianbegin{eqnarray*}left { begin{array}{ll} (phi_{p_i}(u_i''(t)))'  +  a_i(t) f_i(t,u_1(t), u_2(t), ldots, u_n(t)) =0 hspace{1cm} 0  leq t leq 1, alpha_i u...

Let $q$ be a large positive integer and let $(a,q)=1$. We prove that there exist primes $p_1,p_2\le q$ number $n\le with at most two prime factors such $p_1p_2n\equiv a\pmod {q}$. This improves upon result of Shparlinski (2018).

In two dimensional digital geometry, lattice points are 4-connected (resp. 8-connected) if their Euclidean distance is at most one \(\sqrt{2}\)). A set \(S \subset \mathbb {Z}^2\) for all pair of \(p_1, p_2\) in S there a path connecting \(p_1\) to \(p_2\) such that every edge consists points. The original definition convexity which states {Z}^d\) convex \(\mathrm {conv}(S) \cap {Z}^d= S\), whe...

In this paper, we classify the solution of following mixed-order conformally invariant system with coupled nonlinearity in $ \mathbb{R}^4 $: \begin{equation} \left\{ \begin{split} & -\Delta u(x) = u^{p_1}(x) e^{q_1v(x)}, \quad x\in \mathbb{R}^4, \\ (-\Delta)^2 v(x) u^{p_2}(x) e^{q_2v(x)}, \end{split} \right. \end{equation}\;\;\;\;\;\;(1) where 0\leq p_1 < 1 $, p_2 >0 q_1 > 0 q_2 \g...

We investigate the impact of neutron-star matter equation state on $f$- and $p_1$-mode oscillations neutron stars obtained within Cowling approximation linearized general relativity. The oscillation frequencies, their damping times are calculated using representative sets Skyrme Hartree-Fock relativistic mean-field models, all which reproduce nuclear systematics support $2M_\odot$ stars. Our st...

We show that if $$N \geq {\rm exp}({\rm exp} (k^{O(1)})))$$ , then any k-colouring of the primes are less than N contains a monochromatic solution to $$p_1 - p_2 = p_3 -1$$ .

this paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``advanced school and conference on homological and geometrical methods in representation theory'' at the international centre for theoretical physics (ictp), miramare - trieste, italy, 11-29 january 2010. we show how to associate to a triple of positive integer...