For a set $X$ and a variety $\mathcal{V}$ of bands, let $B_{\mathcal{V}}(X)$ be the relatively free band in $\mathcal{V}$ on $X$. For an arbitrary band variety $\mathcal{V}$ and an arbitrary set $X$, we determine the group of automorphisms of $\mathrm{End}(B_{\mathcal{V}}(X))$, the monoid of endomorphisms of $B_{\mathcal{V}}(X)$.

This paper is concerned with the critical nonlinear Gross–Pitaevskii equation, which describes the attractive Bose–Einstein condensate under a magnetic trap. We derive a sharp threshold between the global existence and the blowing-up of the system. Furthermore, we answer the question: how small are the initial data, such that the system has global solutions for the nonlinear critical power $p=1+(4/N)$?

Incomplete Bessel functions of general order are defined. Their recurrence relations, integral representations and asymptotics are derived.

We prove the existence and uniqueness of maximizing measures for various classes of continuous integrands on metrizable (non-compact) spaces and close subsets of Borel probability measures. We apply these results to various dynamical contexts, especially to hyperbolic mappings of the form $f_\lambda(z)=\lambda\mathrm{e}^z$, $\lambda\ne0$, and associated canonical maps $F_\lambda$ of an infinite cylinder. It is then shown that, for all hyperbolic maps $F_\lambda$, all dynamically maximizing measures have compact supports and, for all $0^+$-potentials $\phi$, the set of (weak) limit points of equilibrium states of potentials $t\phi$, $t\nearrow+\infty$, is non-empty and consists of dynamically maximizing measures.

Under certain assumptions on $g(x)$, we obtain an asymptotic formula for computing integrals of the form

$$ F(x,\alpha)=\int_{-\infty}^\infty g(t)^\alpha\exp\biggl(-\bigg|\int_x^tg(\xi)\,\mathrm{d}\xi\bigg|\biggr)\,\mathrm{d}t,\quad\alpha\in\mathbb{R}, $$

as $|x|\to\infty$. We use this formula to study the properties (as $|x|\to\infty$) of the solutions of the correctly solvable equations in $L_p(\mathbb{R})$, $p\in[1,\infty]$,

\begin{equation} -y''(x)+q(x)y(x)=f(x),\quad x\in\mathbb{R}, \tag{1} \end{equation}

where $0\le q\in L_1^{\mathrm{loc}}(\mathbb{R})$, and $f\in L_p(\mathbb{R})$. (Equation (1) is called correctly solvable in a given space $L_p(\mathbb{R})$ if for any function $f\in L_p(\mathbb{R})$ it has a unique solution $y\in L_p(\mathbb{R})$ and if the following inequality holds with an absolute constraint $c_p\in(0,\infty)$: $\|y\|_{L_p(\mathbb{R})}\leq c(p)\|f\|_{L_p(\mathbb{R})}$, $\forall f\in L_p(\mathbb{R})$.)

Let $R$ be a commutative ring with unity and let $G$ be a group. The group ring $RG$ has a natural involution that maps each element $g\in G$ to its inverse. We denote by $RG^-$ the set of skew symmetric elements under this involution. We study necessary and sufficient conditions for $RG^-$ to be commutative.

The aim of this paper is to investigate and study the possible spectral pair $(\mu_{M,D},\varLambda(M,S))$ associated with the iterated function systems $\{\phi_{d}(x)= M^{-1}(x+d)\}_{d\in D}$ and $\{\psi_{s}(x)=M^{\ast}x+s\}_{s\in S}$ in $\mathbb{R}^n$. For a large class of self-affine measures $\mu_{M,D}$, we obtain an easy check condition for $\varLambda(M,S)$ not to be a spectrum, and answer a question of whether we have such a spectral pair $(\mu_{M,D},\varLambda(M,S))$ in the Eiffel Tower or three-dimensional Sierpinski gasket. Further generalization of the given condition as well as some elementary properties of compatible pairs and spectral pairs are discussed. Finally, we give several interesting examples to illustrate the spectral pair conditions considered here.

This paper presents new existence results for the singular boundary-value problem

\begin{gather*} -u''+p(t)u=f(t,u),\quad t\in(0,1),\\ u(0)=0=u(1). \end{gather*}

In particular, our nonlinearity $f$ may be singular at $t=0,1$ and $u=0$.

We analyze the onset of superconductivity, in a type II superconductor adjacent to a normal material, via a generalized Ginzburg–Landau energy functional, which models the effects of superconducting electron pairs diffusing into the normal part. We consider a superconductor and a normal material, each filling a half-space, in the presence of a constant magnetic field parallel to their interface. Among other results, we show that if the normal state conductivity of the superconductor is less than or equal to the conductivity of the normal material, then normal states are the only global minimizers down to the second critical field $H_{c_2}$. Hence, we analytically confirm experimental predictions that surface superconductivity may be suppressed by coating a superconductor with a normal metal.

An element $a$ in a ring $R$ with identity is called strongly clean if it is the sum of an idempotent and a unit that commute. And $a\in R$ is called strongly $\pi$-regular if both chains $aR\supseteq a^2R\supseteq\cdots$ and $Ra\supseteq Ra^2\supseteq\cdots$ terminate. A ring $R$ is called strongly clean (respectively, strongly $\pi$-regular) if every element of $R$ is strongly clean (respectively, strongly $\pi$-regular). Strongly $\pi$-regular elements of a ring are all strongly clean. Let $\sigma$ be an endomorphism of $R$. It is proved that for $\varSigma r_ix^i\in R[[x,\sigma]]$, if $r_0$ or $1-r_0$ is strongly $\pi$-regular in $R$, then $\varSigma r_ix^i$ is strongly clean in $R[[x,\sigma]]$. In particular, if $R$ is strongly $\pi$-regular, then $R[[x,\sigma]]$ is strongly clean. It is also proved that if $R$ is a strongly $\pi$-regular ring, then $R[x,\sigma]/(x^n)$ is strongly clean for all $n\ge1$ and that the group ring of a locally finite group over a strongly regular or commutative strongly $\pi$-regular ring is strongly clean.

Based on the identification of four-dimensional Möbius transformations

$$ g(x)=(ax+b)(cx+d)^{-1} $$

by the matrix group $\mathrm{PS}_\triangle L(2,\mathbb{H})$ of quaternionic $2\times2$ matrices with Dieudonné determinant equal to $1$, we give an explicit expression for the classification of $g$ in terms of $a$, $b$, $c$ and $d$.

Let $G$ be the abelian Lie group $\mathbb{R}^n\times\mathbb{R}^k/\mathbb{Z}^k$, acting on the complex space $X=\mathbb{R}^{n+k}\times\ri G$. Let $F$ be a strictly convex function on $\mathbb{R}^{n+k}$. Let $H$ be the Bergman space of holomorphic functions on $X$ which are square-integrable with respect to the weight $e^{-F}$. The $G$-action on $X$ leads to a unitary $G$-representation on the Hilbert space $H$. We study the irreducible representations which occur in $H$ by means of their direct integral. This problem is motivated by geometric quantization, which associates unitary representations with invariant Kähler forms. As an application, we construct a model in the sense that every irreducible $G$-representation occurs exactly once in $H$.

In this paper we give Bessel- and Grüss-type inequalities in an inner product module over a proper $H^*$-algebra or a $C^*$-algebra.

We show that if a locally compact group $G$ is non-discrete or has an infinite amenable subgroup, then the second dual algebra $L^1(G)^{**}$ does not admit an involution extending the natural involution of $L^1(G)$. Thus, for the above classes of groups we answer in the negative a question raised by Duncan and Hosseiniun in 1979. We also find necessary and sufficient conditions for the dual of certain left-introverted subspaces of the space $C_b(G)$ (of bounded continuous functions on $G$) to admit involutions. We show that the involution problem is related to a multiplier problem. Finally, we show that certain non-trivial quotients of $L^1(G)^{**}$ admit involutions.

Let $\varOmega$ be a bounded, simply connected domain in $\mathbb{C}$ with $0\in\varOmega$ and $\partial\varOmega$ analytic. Let $S(\varOmega)$ denote the class of functions $F(z)$ which are analytic and univalent in $\varOmega$ with $F(0)=0$ and $F'(0)=1$. Let $\{\varPhi_{n}(z)\}_{n=0}^{\infty}$ be the Faber polynomials associated with $\varOmega$. If $F(z)\in S(\varOmega)$, then $F(z)$ can be expanded in a series of the form

$$ F(z)=\sum_{n=0}^{\infty}A_{n}\varPhi_{n}(z),\quad z\in\varOmega, $$

in terms of the Faber polynomials. Let

$$ E_{r}=\bigg\{(x,y)\in\mathbb{R}^{2}:\frac{x^{2}}{(1+(1/r^{2}))^{2}}+\frac{y^{2}}{(1-(1/r^{2}))^{2}}\lt1\bigg\}, $$

where $r\gt1$.

In this paper, we obtain sharp bounds for certain linear combinations of the Faber coefficients of functions $F(z)$ in $S(E_{r})$ and in certain related classes.

We study surface tension effects for two-dimensional Darcy flow with a free boundary in a corner between two non-parallel walls. The analytic solution is based on two governing expressions constructed in an auxiliary parameter domain, namely a complex velocity and a derivative of the complex potential. These expressions admit a general solution for the problem in a corner geometry for the flow generated by a source/sink at the corner vertex or at infinity. We derive an integral equation in terms of the velocity modulus and angle at the free surface, determined by the dynamic boundary condition. A numerical procedure, used to solve the obtained system of equations, and numerical results concerning the effect of surface tension on the time evolution of the free boundary, are discussed.

The propagation of a solitary wave in a horizontal fluid layer is studied. There is an interfacial free surface above and below this intrusion layer, which is moving at constant speed through a stationary density-stratified fluid system. A weakly nonlinear asymptotic theory is presented, leading to a Korteweg–de Vries equation in which the two fluid interfaces move oppositely. The intrusion layer solitary wave system thus forms a widening bulge that propagates without change of form. These results are confirmed and extended by a fully nonlinear solution, in which a boundary-integral formulation is used to solve the problem numerically. Limiting profiles are approached, for which a corner forms at the crest of the solitary wave, on one or both of the interfaces.

For the Cauchy problem for the nonlinear infiltration equation $$\left\{\begin{array}{@{}l@{\qquad}l} u_{t}=\frac{1}{m}(u^{m})_{xx},&x\in{\mathbb{R}}, t>0,m\geq{}1,\\[3pt] u|_{t=0}=u_{0}(x),&x\in{\mathbb{R}}, \end{array} \right.$$ we use its linear solution $u(x,t,1)$ to approach the nonlinear solution $u(x,t,m)$, and obtain the explicit estimate: $$\int_{0}^{T}\int_{\mathbb{R}}|u(x,t,m)-u(x,t,1)|^{2}\,dx\,dt{} \leq{}(C^{\ast}(m-1))^{2},$$ where $C^{\ast}=O(T^{\gamma})$ and $\gamma=\frac{1+m-\alpha}{2(1+m)}$ for any $0<\alpha<1$.

Sufficient conditions are obtained for the existence of positive periodic solutions of a class of neutral delay differential equations of the form \begin{equation*} \left\{\begin{array}{@{}l@{}} {\normalsize N}^{\prime}{\normalsize (t)=N(t)F[t,N(t),N(t-\tau (t,N(t))),N}^{\prime}{\normalsize (t-\gamma (t)),P(t),P(t-\mu (t))]}\\ {\normalsize P}^{\prime}{\normalsize (t)=-}e(t)P{\normalsize (t)+k(t)N(t)+h(t)N(t-\sigma (t))}\end{array}\right. \end{equation*} by using the theory of topological degree. These results extend substantially the existing relevant existence results in the literature. As a demonstration, applying the obtained analysis results to a real complex neutral Lotka-Volterra population model, the existence criterion for positive periodic solutions is easily obtained and an example is used to give an impression of how restrictive these conditions are. Especially, this method is more suitable to state-dependent delay, and which have further applications in many fields.

In this paper, three iterative procedures (Landweber-Fridman, conjugate gradient and minimal error methods) for obtaining a stable solution to the Cauchy problem in slow viscous flows are presented and compared. A section is devoted to the numerical investigations of these algorithms. There, we use the boundary element method together with efficient stopping criteria for ceasing the iteration process in order to obtain stable solutions.