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...

Let$mathbb{C}^n$ be the space of $n$ complex variables. Let$Omega_{n,p_2,ldots,p_n}$ be a complete Reinhardt on$mathbb{C}^n$. The Minkowski functional on complete Reinhardt$Omega_{n,p_2,ldots,p_n}$ is denoted by $rho(z)$. The concept ofspirallike mapping of type $beta$ and order $alpha$ is defined.So, the concept of the strong and almost spirallike mappings o...

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).

we characterize strictly diagonal type of embeddings offinitary symmetric groups in terms of cardinality and the characteristic. namely, we prove thefollowing.let $kappa$ be an infinite cardinal. if$g=underset{i=1}{stackrel{infty}bigcup} g_i,$ where $ g_i=fsym(kappa n_i)$,$(h=underset{i=1}{stackrel{infty}bigcup}h_i, $ where $ h_i=alt(kappa n_i) ), $ is a group of strictly diagonal type and$xi=(...

We study the boundedness of commutators bi-parameter singular integrals between mixed spaces $$ [b,T]: L^{p_1}L^{p_2} \to L^{q_1}L^{q_2} in off-diagonal situation $q_i,p_i\in(1,\infty)$ where we also allow $q_i\not= p_i.$ Boundedness is fully characterized for several arrangements integrability exponents with some open problems presented.

Abstract In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, $$\text {id}_\tau : {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {B}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$$ id τ : B</mml...

We analyze the quasi-two-body charmed $B$ decays $B^{+,0}_{(s)} \to D_{(s)}^* P_2 D_{(s)} P_1 P_2$ with $P_{1,2}$ as a pion or kaon. The intermediate processes $B_{(s)} $ are calculated factorization-assisted topological-amplitude approach and resonant effects Breit-Wigner formalism. Taking all p-wave resonance states \bar D_{(s)}^*$ into consideration, we present related branching fractions, c...

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...