This article presents a unified approach to the abstract notions of partial convolution and involution in $L^p$-function spaces over semi-direct product of locally compact groups. Let $H$ and $K$ be locally compact groups and $tau:Hto Aut(K)$ be a continuous homomorphism.  Let $G_tau=Hltimes_tau K$ be the semi-direct product of $H$ and $K$ with respect to $tau$. We define left and right $tau$-c...

this article presents a unified approach to the abstract notions of partial convolution and involution in $l^p$-function spaces over semi-direct product of locally compact groups. let $h$ and $k$ be locally compact groups and $tau:hto aut(k)$ be a continuous homomorphism.  let $g_tau=hltimes_tau k$ be the semi-direct product of $h$ and $k$ with respect to $tau$. we define left and right $tau$-c...

We compute the Nielsen–Borsuk–Ulam number for any selfmap of $$n-$$ torus, $$\mathbb {T}^n$$ , as well free involution $$\tau $$ in with $$n \leqslant 3$$ . Finally, we conclude that tori, {T}^1$$ {T}^2$$ and {T}^3$$ are Wecken spaces theory. Such a is lower bound minimal pair points such $$f(x)=f(\tau (x))$$ given homotopy class maps.

A characterization of the Kadomtsev-Petviashvili hierarchy type C (CKP) in terms KP tau-function is given. Namely, we prove that CKP can be identified with restriction odd times flows on locus turning points second flow. The notion clarified and connected tau function. Algebraic-geometrical solutions particular elliptic are discussed detail. new identity for theta-functions curves holomorphic i...

For an Abelian surface A with a symplectic action by finite group G, one can define the partition function for G-invariant Hilbert schemes $$\begin{aligned} Z_{A, G}(q) = \sum _{d=0}^{\infty } e(\text {Hilb}^{d}(A)^{G})q^{d}. \end{aligned}$$We prove reciprocal \(Z_{A,G}^{-1}\) is modular form of weight \(\frac{1}{2}e(A/G)\) congruence subgroup \(\Gamma _{0}(|G|)\) and give explicit expressions ...

We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda (C-Toda). It can be regarded as certain subhierarchy the 2D lattice obtained by imposing constraint $\bar {\cal L}={\cal L}^{\dag}$ on two Lax operators (in symmetric gauge). prove existence tau-function C-Toda and show that it is square root tau-function. In this some other res...

An involution or anti-involution is a self-inverse linear mapping. In this paper, we will present two real quaternion matrices, one corresponding to a real quaternion involution and one corresponding to a real quaternion anti-involution. Moreover, properties and geometrical meanings of these matrices will be given as reflections in R^3.

It is known that every Fano–Mukai fourfold X of genus 10 acted upon by an involution $$\tau $$ which comes from the center Weyl group simple algebraic type $$\textrm{G}_2$$ , see Prokhorov and Zaidenberg (Eur J Math 4(3):1197–1263 2018, Eur 8:561–572, 2022). This uniquely defined up to conjugation in $${\text {{Aut}}}(X)$$ . In this note we describe set fixed points surface scroll swept out -in...

Let $ (M, g) be a Riemannian manifold with boundary, where g is non-trapping metric. SM the space of spherical tangent to M bundle, and v^g geodesic vector field on $. We study scattering maps C_{v^g}: \partial^+_1SM \to \partial^-_1SM $, generated by $-flow, dynamics billiard B_{v^g, \tau}: \tau denotes an involution, mimicking elastic reflection from boundary \partial getting variety holograp...