Narrow Results

The quantum H₄ integrable system is a 4D system with rational potential related to the non-crystallographic root system H₄ with 600-cell symmetry. It is shown that the gauge-rotated H₄ Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H₄, is in algebraic form: it has polynomial coefficients in front of the derivatives. Any eigenfunction is a polynomial multiplied by ground-state function (factorization property). Spectra correspond to one of the anisotropic harmonic oscillators. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector α = (1, 5, 8, 12).

An appropriate generalization of the unitary parasupersymmetry algebra of Beckers–Debergh to arbitrary order is presented in this paper. A special representation for realizing the even arbitrary order unitary parasupersymmetry algebra of Beckers–Debergh is analyzed by one-dimensional shape invariance solvable models, 2D and 3D quantum solvable models obtained from the shape invariance theory as well. In particular, in the special representation, it is shown that the isospectrum Hamiltonians consist of the two partner Hamiltonians of the shape invariance theory.

The quantum H₃ integrable system is a three-dimensional system with rational potential related to the noncrystallographic root system H₃. It is shown that the gauge-rotated H₃ Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H₃, is in algebraic form: it has polynomial coefficients in front of derivatives. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector . One among possible integrals is found (of the second order) as well as its algebraic form. A hidden algebra of the H₃ Hamiltonian is determined. It is an infinite-dimensional, finitely-generated algebra of differential operators possessing finite-dimensional representations characterized by a generalized Gauss decomposition property. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model on the uniform lattice in a space of H₃-invariants "polynomially"-isospectral to the quantum H₃ model is defined.

It is shown that the E₈ trigonometric Olshanetsky–Perelomov Hamiltonian, when written in terms of the fundamental trigonometric invariants, is in algebraic form, i.e. it has polynomial coefficients, and preserves two infinite flags of polynomial spaces marked by the Weyl (co)-vector and E₈ highest root (both in the basis of simple roots) as characteristic vectors. The explicit form of the Hamiltonian in new variables has been obtained both by direct calculation and by means of the orbit function technique. It is shown the triangularity of the Hamiltonian in the bases of orbit functions and of algebraic monomials ordered through Weyl heights. Examples of first eigenfunctions are presented.

In the paper we pose and discuss new Burnside-type problems, where the role of nilpotency is replaced by that of solvability.

We characterize H-solvable semigroups with respect to a pseudovariety H of groups. Applications are given to iterated Mal'cev products with H.

We show that the right ideal of a Novikov algebra generated by the square of a right nilpotent subalgebra is nilpotent. We also prove that a $G$-graded Novikov algebra $N$ over a field $K$ with solvable $0$-component $N_0$ is solvable, where $G$ is a finite additive abelean group and the characteristic of $K$ does not divide the order of the group $G$. We also show that any Novikov algebra $N$ with a finite solvable group of automorphisms $G$ is solvable if the algebra of invariants $N^G$ is solvable.

This paper is devoted to the so-called complete Leibniz algebras. We construct some complete Leibniz algebras with complete radical and prove that the direct sum of complete Leibniz algebras is also complete. It is known that a Lie algebra with a complete ideal is split. We discuss the analogs of this result for the Leibniz algebras and show that it is true for some special classes of Leibniz algebras. Finally, we consider derivations of Leibniz algebras and present some classes of Leibniz algebras which are not complete, since they admit outer derivation.

The purpose of this paper is to deal with a system governed by a system of semilinear nonlocal fractional evolution inclusions with partial Clarke subdifferential and its optimal control. First, we establish an existence theorem of the mild solution for the presented control system by applying the measure of noncompactness, a fixed point theorem of a condensing multivalued map and some properties of partial Clarke subdifferential. Moreover, under some mild conditions, we obtain a result on the existence of an optimal control to the presented control system. Finally, an example is provided to demonstrate the main results. The results presented in this paper improve, extend and develop the corresponding results in the earlier and recent literature.

The work considers the problem of solvability of a fuzzy relation equation with max-min composition. It presents the necessary and sufficient conditions for the existence of solutions, then derives a fast algorithm for finding all minimal solutions. The results are compared with those of previous publications regarding this subject.

In this paper, a new kind of fuzzy relational equations (FREs for short) A ∘^R_*x = b is first introduced, and then the problem of solving solution to the FREs is discussed, where A is an m × n matrix, x and b are an n and an m dimensional column vectors, respectively. More specifically, their solvability and unique solvability are investigated, the corresponding necessary and sufficient conditions are presented, the complete solution set is obtained. It is worth noting the method to construct the complete solution set.

For second-order nonlinear nonautonomous differential equations, new sufficient conditions on the existence of periodic solutions are established.

We find a pair of finite groups, one nilpotent and the other perfect, with the same set of character degrees.

Let x be an element of a finite group G, and p a prime factor of the order of G. It is clear that there always exists a unique minimal subnormal subgroup containing x, say $G_x$. We call the conjugacy class of x in $G_x$ the sub-class of x in G, see [G. Qian and Y. Yang, On sub-class sizes of finite groups, J. Aust. Math. Soc. (2020) 402–411]. In this paper, assume that $G(=AB)$ is the product of the subgroups A and B, we investigate the solvability, p-nilpotence and supersolvability of the group G under the condition that the sub-class sizes of prime power order elements in $A\cup B$ are $2^2$ free, $p^2$ free and square free, respectively, so that some known results relevant to conjugacy class sizes are generalized.

Let a finite group $A$ act on a finite group $G$ via automorphism and let $s$ be a positive integer. In this paper, we introduce the probabilistic zeta function of ordered $s$-tuples from $G$ which $A$-generate $G$, denoted by ${\mathrm{P}}^A(G,s)$. When $(|A|,|G|)=1$, it is proved that $G$ is solvable if and only if ${\mathrm{P}}^A(G,s)$ has an Euler product expansion with all factors of the form $1-c/q^s$, where each $q$is a prime power. This is a generalization of Detomi and Lucchini's result in 2003.

Let $a,b$ be fixed positive integers such that $a$ is not a perfect square and $b$ is squarefree, and let $\omega (b)$ denote the number of distinct prime divisors of $b$. Let $(u_1,v_1)$ denote the least solution of Pell equation $u^2-a{v}^2=1$. Further, for any positive integer $n$, let $u_n=\frac{{\alpha }^n+{\bar{\alpha }}^n}{2}$ and $v_n=\frac{{\alpha }^n-{\bar{\alpha }}^n}{2\sqrt{a}}$, where $\alpha =u_1+v_1\sqrt{a}$ and $\bar{\alpha }=u_1-v_1\sqrt{a}$. In this paper, using the basic properties of Pell equations and some known results on binary quartic Diophantine equations, a necessary and sufficient condition for the system of equations $(\ast )x^2-a{y}^2=1$ and $y^2-b{z}^2=v_1^2$ to have positive integer solutions $(x,y,z)$ is obtained. By this result, we prove that if $(\ast )$ has a positive integer solution $(x,y,z)$ for $\omega (b)\le 2$ or $3$ according to $2\nmid b$ or not, then $4u_1^2-\delta =b{g}^2$ and $(x,y,z)=(u_r,v_r,g{v}_{[\frac{r+1}{2}]})$, where $g$ is a positive integer, $\delta =1$ or $2$ and $r=2$ or $3$ according to $2\nmid b$ or not, $[\frac{r+1}{2}]$ is the integer part of $\frac{r+1}{2}$, except for $(a,b,x,y,z)=(24,2134,47525,9701,210)$

In 2011, Beeler and Hoilman introduced the peg solitaire on graphs. The peg solitaire on a connected graph is a one-player combinatorial game starting with exactly one hole in a vertex and pegs in all other vertices and removing all pegs but exactly one by a sequence of jumps; for a path $xyz$, if there are pegs in $x$ and $y$ and exists a hole in $z$, then $x$ can jump over $y$ into $z$, and after that, the peg in $y$ is removed. A problem of interest in the game is to characterize solvable (respectively, freely solvable) graphs, where a graph is solvable (respectively, freely solvable) if for some (respectively, any) vertex $s$, starting with a hole $s$, a terminal state consisting of a single peg can be obtained from the starting state by a sequence of jumps. In this paper, we consider the peg solitaire on graphs with large maximum degree. In particular, we show the necessary and sufficient condition for a graph with large maximum degree to be solvable in terms of the number of pendant vertices adjacent to a vertex of maximum degree. It is a notable point that this paper deals with a question of Beeler and Walvoort whether a non-solvable condition of trees can be extended to other graphs.

It was a great surprise when Hans Lewy in 1957 presented a non-vanishing complex vector field that is not locally solvable. Actually, the vector field is the tangential Cauchy—Riemann operator on the boundary of a strictly pseudocon-vex domain. Hörmander proved in 1960 that almost all linear partial differential equations are not locally solvable. This also has connections with the spectral instability of non-selfadjoint semiclassical operators.

Nirenberg and Treves formulated their well-known conjecture in 1970: that condition (Ψ) is necessary and sufficient for the local solvability of differential equations of principal type. Principal type essentially means simple characteristics, and condition (Ψ) only involves the sign changes of the imaginary part of the highest order terms along the bicharacteristics of the real part.

The Nirenberg-Treves conjecture was finally proved in 2006. We shall present the background, the main ideas of the proof and some open problems.