Stable categories of higher preprojective algebras

We introduce (n+1)$(n+1)$-preprojective algebras of algebras of global dimension n$n$. We show that if an algebra is n$n$-representation-finite then its (n+1)$(n+1)$-preprojective algebra is self-injective. In this situation, we show that the stable module category of the (n+1)$(n+1)$-preprojective algebra is (n+1)$(n+1)$-Calabi–Yau, and, more precisely, it is the (n+1)$(n+1)$-Amiot cluster category of the stable n$n$-Auslander algebra of the original algebra. In particular this stable category contains an (n+1)$(n+1)$-cluster tilting object. We show that even if the (n+1)$(n+1)$-preprojective algebra is not self-injective, under certain assumptions (which are always satisfied for n∈{1,2}$n\in \{1,2\}$) the results above still hold for the stable category of Cohen–Macaulay modules.     

The approximation characteristic of diagonal matrix in probabilistic setting

In this paper, the approximation characteristic of a diagonal matrix in probabilistic and average case settings is investigated. And the asymptotic degree of the probabilistic linear (n,δ)$(n,\delta )$-width and p$p$-average linear n$n$-width of diagonal matrix M$M$ are determined.     

Necessary and sufficient condition for uniqueness of solution to the first boundary value problem for the diffusion equation in unbounded domains

This paper introduces a notion of regularity of t=-∞$t=-\infty$ for the diffusion (or heat) equation and establishes a necessary and sufficient condition for the existence of a unique bounded solution to the first boundary value problem for the diffusion equation in a general domain Ω⊂R^N+1$\Omega \subset {\mathbb{R}}^{N+1}$ which extends up to t=-∞$t=-\infty$.     

The Collatz conjecture and De Bruijn graphs

We study variants of the well-known Collatz graph, by considering the action of the 3n+1$3n+1$ function on congruence classes. For moduli equal to powers of 2, these graphs are shown to be isomorphic to binary De Bruijn graphs. Unlike the Collatz graph, these graphs are very structured, and have several interesting properties. We then look at a natural generalization of these finite graphs to the 2-adic integers, and show that the isomorphism between the resulting infinite graphs is exactly the conjugacy map previously studied by Bernstein and Lagarias. Finally, we show that for generalizations of the 3n+1$3n+1$ function, such as the family of an+b$an+b$ functions and Collatz-like functions, we get similar relations with 2-adic and p$p$-adic De Bruijn graphs respectively.     

Cycle-maximal triangle-free graphs

We conjecture that the balanced complete bipartite graph _{K⌊n/2⌋,⌈n/2⌉}$K_{\lfloor n/2\rfloor ,\lceil n/2\rceil }$ contains more cycles than any other n$n$-vertex triangle-free graph, and we make some progress toward proving this. We give equivalent conditions for cycle-maximal triangle-free graphs; show bounds on the numbers of cycles in graphs depending on numbers of vertices and edges, girth, and homomorphisms to small fixed graphs; and use the bounds to show that among regular graphs, the conjecture holds. We also consider graphs that are close to being regular, with the minimum and maximum degrees differing by at most a positive integer k$k$. For k=1$k=1$, we show that any such counterexamples have n≤91$n\le 91$ and are not homomorphic to _C5$C_5$; and for any fixed k$k$ there exists a finite upper bound on the number of vertices in a counterexample. Finally, we describe an algorithm for efficiently computing the matrix permanent (a #P$\#P$-complete problem in general) in a special case used by our bounds.     

Avoiding σ-porous sets in Hilbert spaces

We give a constructive proof that any σ  -porous subset of a Hilbert space has Lebesgue measure zero on typical C¹$C^1$ curves. Further, we discover that this result does not extend to all forms of porosity; we find that even power-p   porous sets may meet many C¹$C^1$ curves in positive measure.     

Nonlinear maps preserving Jordan *-products

Let η$\eta$ be a non-zero scalar. In this paper, we investigate a bijective map ?$?$ between two von Neumann algebras, one of which has no central abelian projections, satisfying ?(AB+ηBA^∗)=?(A)?(B)+η?(B)?(A)^∗$?(AB+\eta B{A}^{\ast })=?\left(A\right)?\left(B\right)+\eta ?\left(B\right)?{\left(A\right)}^{\ast }$ for all A,B$A,B$ in the domain. It is showed that ?$?$ is a linear *-isomorphism if η$\eta$ is not real and ?$?$ is a sum of a linear *-isomorphism and a conjugate linear *-isomorphism if η$\eta$ is real.