Brazil et al. [‘Maximal subgroups of infinite symmetric groups’, Proc. Lond. Math. Soc. (3) 68(1) (1994), 77–111] provided a new family of maximal subgroups of the symmetric group $G(X)$ defined on an infinite set X. It is easy to see that, in this case, $G(X)$ contains subsemigroups that are not groups, but nothing is known about nongroup maximal subsemigroups of $G(X)$. We provide infinitely many examples of such semigroups.

Let V be an infinite-dimensional vector space over a field F and let $I(V)$ be the inverse semigroup of all injective partial linear transformations on V. Given $\alpha \in I(V)$, we denote the domain and the range of $\alpha $ by ${\mathop {\textrm {dom}}}\,\alpha $ and ${\mathop {\textrm {im}}}\,\alpha $, and we call the cardinals $g(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {dom}}}\,\alpha $ and $d(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {im}}}\,\alpha $ the ‘gap’ and the ‘defect’ of $\alpha $. We study the semigroup $A(V)$ of all injective partial linear transformations with equal gap and defect and characterise Green’s relations and ideals in $A(V)$. This is analogous to work by Sanwong and Sullivan [‘Injective transformations with equal gap and defect’, Bull. Aust. Math. Soc. 79 (2009), 327–336] on a similarly defined semigroup for the set case, but we show that these semigroups are never isomorphic.

We introduce the concept of a $\{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\}$-Rota–Baxter operator, as a twisted version of a Rota–Baxter operator of weight zero. We show how to obtain a certain $\{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\}$-Rota–Baxter operator from a solution of the associative (Bi)Hom-Yang–Baxter equation, and, in a compatible way, a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.

Spot prices in energy markets exhibit special features, such as price spikes, mean reversion, stochastic volatility, inverse leverage effect, and dependencies between the commodities. In this paper a multivariate stochastic volatility model is introduced which captures these features. The second-order structure and stationarity of the model are analyzed in detail. A simulation method for Monte Carlo generation of price paths is introduced and a numerical example is presented.

In Benth and Vos (2013) we introduced a multivariate spot price model with stochastic volatility for energy markets which captures characteristic features, such as price spikes, mean reversion, stochastic volatility, and inverse leverage effect as well as dependencies between commodities. In this paper we derive the forward price dynamics based on our multivariate spot price model, providing a very flexible structure for the forward curves, including contango, backwardation, and hump shape. Moreover, a Fourier transform-based method to price options on the forward is described.

Let ${{M}_{n}}$ be the algebra of all $n\,\times \,n$ matrices over $\mathbb{C}$ . We say that $A,B\in {{M}_{n}}$ quasi-commute if there exists a nonzero $\xi \,\in \,\mathbb{C}$ such that $AB\,=\,\xi BA$ . In the paper we classify bijective not necessarily linear maps $\Phi :{{M}_{n}}\to {{M}_{n}}$ which preserve quasi-commutativity in both directions.

Hua’s fundamental theorem of the geometry of hermitian matrices characterizes bijective maps on the space of all $n\times n$ hermitianmatrices preserving adjacency in both directions. The problem of possible improvements has been open for a while. There are three natural problems here. Do we need the bijectivity assumption? Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only? Can we obtain such a characterization formaps acting between the spaces of hermitian matrices of different sizes? We answer all three questions for the complex hermitian matrices, thus obtaining the optimal structural result for adjacency preserving maps on hermitian matrices over the complex field.

It is proved that every adjacency preserving continuous map on the vector space of real matrices of fixed size, is either a bijective affine tranformation of the form $A\,\mapsto \,PAQ\,+\,R$ , possibly followed by the transposition if the matrices are of square size, or its range is contained in a linear subspace consisting of matrices of rank at most one translated by some matrix $R$ . The result extends previously known theorems where the map was assumed to be also injective.

Let K be the field of real or complex numbers. Let $(X \cong {\bf K}^{2n}, \omega)$ be a symplectic vector space and take $0 < k < n,N = ({2n \atop 2k})$. Let $L_1,\ldots,L_N \subset X$ be $2k$-dimensional linear subspaces which are in a sufficiently general position. It is shown that if $F : X \longrightarrow X$ is a linear automorphism which preserves the form $\omega^k$ on all subspaces $L_1,\ldots,L_N$, then $F$ is an $\epsilon_k$-symplectomorphism (that is, $F^*\omega = \epsilon_k\omega$, where $\epsilon_k^k = 1$). In particular, if ${\bf K} = \R$ and $k$ is odd then $F$ must be a symplectomorphism. The unitary version of this theorem is proved as well. It is also observed that the set ${\cal A}_{l,2r}$ of all $l$-dimensional linear subspaces on which the form $\omega$ has rank $\le 2r$ is linear in the Grassmannian $G(l,2n)$, that is, there is a linear subspace $L$ such that ${\cal A}_{l,2r} = L \cap G(l, 2n)$. In particular, the set ${\cal A}_{l,2r}$ can be computed effectively. Finally, the notion of symplectic volume is introduced and it is proved that it is another strong invariant.

Every norm $\mathcal{V}$ on ${{\text{C}}^{\text{n}}}$ induces two norm numerical ranges on the algebra ${{M}_{n}}\,\text{of}\,\text{all}\,\text{n}\,\times \,\text{n}$ complex matrices, the spatial numerical range

$$W\left( A \right)=\left\{ {{x}^{*}}Ay:x,y\in {{\text{C}}^{n}},{{v}^{D}}\left( x \right)=v\left( y \right)={{x}^{*}}y=1 \right\},$$

where ${{\mathcal{V}}^{D}}$ is the norm dual to $\mathcal{V}$ , and the algebra numerical range

$$V\left( A \right)=\left\{ f\left( A \right):f\in S \right\},$$

where $S$ is the set of states on the normed algebra ${{M}_{n}}$ under the operator norm induced by $\mathcal{V}$ . For a symmetric norm $\mathcal{V}$ , we identify all linear maps on ${{M}_{n}}$ that preserve either one of the two norm numerical ranges or the set of states or vector states. We also identify the numerical radius isometries, i.e., linear maps that preserve the (one) numerical radius induced by either numerical range. In particular, it is shown that if $\mathcal{V}$ is not the ${{\ell }_{1}},{{\ell }_{2}},\,\text{or}\,{{\ell }_{\infty }}$ norms, then the linear maps that preserve either numerical range or either set of states are “inner”, i.e., of the form $A\mapsto {{Q}^{*}}AQ$ , where $Q$ is a product of a diagonal unitary matrix and a permutation matrix and the numerical radius isometries are unimodular scalar multiples of such inner maps. For the ${{\ell }_{1}}$ and the ${{\ell }_{\infty }}$ norms, the results are quite different.

Let $U(n)$ be the group of $n\,\times \,n$ unitary matrices. We show that if $\phi $ is a linear transformation sending $U(n)$ into $U(m)$ , then $m$ is a multiple of $n$ , and $\phi $ has the form

$$A\,\mapsto \,V[(A\,\otimes \,{{I}_{s}})\,\otimes \,({{A}^{t}}\,\otimes \,{{I}_{r}})]W$$

for some $V,\,W\,\in \,U(m)$ . From this result, one easily deduces the characterization of linear operators that map $U(n)$ into itself obtained by Marcus. Further generalization of the main theorem is also discussed.

Let $c\,=\,\left( {{c}_{1}},\ldots ,{{c}_{n}} \right)$ be such that ${{c}_{1}}\,\ge \,\cdots \,\ge \,{{c}_{n}}$ . The $c$ -numerical range of an $n\,\times \,n$ matrix $A$ is defined by

$${{W}_{c}}\left( A \right)\,=\,\left\{ \sum\limits_{j=1}^{n}{{{c}_{j}}\left( A{{x}_{j}},\,{{x}_{j}} \right)\,:\,\left\{ {{x}_{1}},\ldots ,{{x}_{n}} \right\}\,\text{an}\,\text{orthonormal basis for }{{\mathbf{C}}^{n}}} \right\}\,,$$

and the $c$ -numerical radius of $A$ is defined by ${{r}_{c}}\left( A \right)\,=\,\max \left\{ \left| z \right|\,:\,z\,\in \,{{W}_{c}}\left( A \right) \right\}$ . We determine the structure of those linear operators $\phi$ on algebras of block triangular matrices, satisfying

$${{W}_{c}}\left( \phi \left( A \right) \right)={{W}_{c}}\left( A \right)\text{for}\,\,\text{all}\,\,A\,\text{or}\,\,{{r}_{c}}\left( \phi \left( A \right) \right)={{r}_{c}}\left( A \right)\text{for}\,\,\text{all}\,A.$$

It is proved that linear mappings of matrix algebras which preserve idempotents are Jordan homomorphisms. Applying this theorem we get some results concerning local derivations and local automorphisms. As an another application, the complete description of all weakly continuous linear surjective mappings on standard operator algebras which preserve projections is obtained. We also study local ring derivations on commutative semisimple Banach algebras.