New upper and lower bounds for the numerical radii of Hilbert space operators are given. An application to involution operators is also provided.

We show that Lorenz flows have neither limit shadowing property nor average shadowing property nor the asymptotic average shadowing property where the reparametrizations related to these concepts relies on the set of increasing homeomorphisms with bounded variation.

In this paper, we investigate 2-SG-semisimple rings which are a particular kind of quasi-Frobenius rings over which all modules are periodic of period~2. Namely, we show that local 2-SG-semisimple rings are the same as the known Artinian valuation rings. Also, a relation between Dedekind domains and 2-SG-semisimple rings is established.

An automorphic loop (or A-loop) is a loop whose inner mappings are automorphisms. An open problem was: Does every finite automorphic loop have the elementwise Lagrange property? We give a positive answer to this problem.

We explore a multiplicative analogue of the Reynolds operator that first arose in CR geometry. Comparisons with the classical Reynolds operator are made in connection with the problem of computing algebra generators in invariant rings over $\mathbb{C}$.

According to a formula by Gordon and Litherland \cite{GordonLitherland}, the signature $\sigma (K)$ of a knot $K$ can be computed as $\sigma (K) = \sigma (G) - \mu$, where $G$ is the Goeritz matrix of a projection $D$ of $K$ and $\mu$ is a ``correction term,'' read off from the projection $D$. In this article, we consider the family of two bridge knots $K_{p/q}$ and compute the signature of the Goeritz matrices of their ``standard projections,'' $D_{p/q}$, by explicitly diagonalizing the Goertiz matrix over the rationals. We give applications to signature computations and concordance questions.

We establish a formula for the essential spectral radius of an endomorphism $T$ of Lipschitz algebras under a condition which is equivalent to the quasicompactness of the endomorphism $T$. We also conclude a necessary and sufficient condition for an endomorphism of these algebras to be Riesz. Finally, we get a relation for the spectrum and the set of eigenvalues of a quasicompact and Riesz endomorphism of these algebras.

In this paper, we obtain Gr\"{o}bner-Shirshov (non-commutative) bases for the $n$-extended affine Weyl group $\widetilde{W}$ of type $A_1$, elliptic Weyl groups of types $A_{1}^{(1,1)}$, $A_{1}^{(1,1)^{*}}$ and the $2$-extended affine Weyl group of type $A_{2}^{(1,1)}$ with a generator system of a $2$-toroidal sense. It gives a new algorithm for getting normal forms of elements of these groups and hence a new algorithm for solving the word problem in these groups.

This paper determines the structure of an associative ring $R$ when either all of its additive subgroups, all of its subrings, all of its (right) ideals, or all of its Lie ideals have finite additive index in $R$.

Let $g_1(\omega),g_2(\omega),\ldots, g_n(\omega)$ be independent and normally distributed random variables with mean zero and variance one. We show that, for large values of $n$, the expected number of times the random hyperbolic polynomial $y=g_1(\omega)\cosh x+ g_2(\omega)\cosh 2x+\cdots +g_n(\omega)\cosh nx$ crosses the line $y=L$, where $L$ is a real number, is $\frac{1}{\pi}\log n +O(1)$ if $L=o(\sqrt{n})$ or ${L}/{\sqrt{n}} =O(1)$, but decreases steadily as $O(L)$ increases in magnitude and ultimately becomes negligible when $n^{-1}\log {L}/{\sqrt{n}}\to\infty$.

We establish equivalences between compacta that admit a sequence of retractions that converge uniformly to the identity map and compacta that are inverse limits on subcompacta with retractions for bonding maps. We give partial answers to questions of Charatonik and Prajs, and of Krasinkiewicz. Our results are related to and use results from another paper of the authors \cite{mp}.

Assume that $f:(0,\infty)\rightarrow (0,\infty ) $ is bounded from above on a set of positive Lebesgue measures or on a set of the second category with the Baire property and satisfies the functional equation $f(x+1)=xf(x)$ for $x>0$ and $f(1)=1$. We prove that, if there is a positive sequence $(p_{n})$, $\lim_{n\rightarrow \infty }p_{n}=\infty$, such that for every $n\in \mathbb{N}$, the function $x\mapsto \log ( x^{p_{n}})$ is Jensen convex in the interval $(1,\infty ) $; or there are two positive sequences $(p_{n})$ and $(q_{n})$, $\lim_{n\rightarrow \infty }p_{n}=\infty$, $% \lim_{n\rightarrow \infty }q_{n}=0$ such that, for every $n\in \mathbb{N}$, the function $x\mapsto [f(x^{p_{n}})] ^{q_{n}}$ is Jensen convex in the interval $(1,\infty)$, then $f$ is the Euler gamma function.

Examples are given of continuous operators of functions analytic on the unit disk having the monomials as eigenvectors which fail spectral synthesis (that is, which have closed invariant subspaces which are not the closed linear span of collections of eigenvectors). Examples include the diagonal operator having as eigenvalues an enumeration of $\IL\equiv \{m+in: m,n\in\mathbb{Z}\}$ and diagonal operators having as eigenvalues enumerations of $\{n^{1/p}e^{2\pi ij/3p}: 0\leq j \lt p\}$ where $p$ is an integer at least~2.

We introduce some new tree-like Banach spaces, belonging to the class of separable Banach spaces not containing $\ell_1$ with non-separable dual, each one of which satisfies the following: $(1)$~the space has the fixed point property and $(2)$~the space does not satisfy the Opial condition. In addition, one of these spaces contains subspaces isomorphic to $c_0$, whose Banach-Mazur distance from $c_0$ becomes arbitrarily large.

In this work, we present a modification of a well-established measure of dependence appropriate for the analysis of stopping times for adversarial processes on cryptographic primitives. We apply this measure to construct generic criteria for the ideal behavior of fixed functions in both the random oracle and ideal permutation setting. More significantly, we provide a nontrivial extension of the notion of hash function indifferentiability, transporting the theory from the status of providing security arguments for protocols utilizing ideal primitives into the more realistic setting of protocol assurance with fixed functions. The methodology this measure introduces to indifferentiability analysis connects the security of a hash function with an indifferentiable mode to the security of the underlying compression function in a quantitative way; thus, we prove that dependence results on cryptographic primitives provide a direct means of determining the practical resistance or vulnerability of protocols employing such primitives.

In this paper we study the spectrum in $\R$ and $\R^2$ of nonlinear elliptic equations with positive parameters in their nonlinear part. In order to investigate the spectrum in these specific cases, we introduce the monotone method which is an extension of the upper and lower solution methods. Using the Picard iterative process we prove some existence theorems for nonlinear elliptic boundary value problems. We work with both positive and negative solutions.

We consider a strongly coupled reaction-diffusion system describing three interacting species in a simple food chain structure. Based on the Leray-Schauder degree theory, the existence of non-constant positive steady states is investigated. The results indicate that, when the intrinsic growth rate of the middle species is small, cross-diffusions of the predators versus the preys are helpful to create global coexistence (stationary patterns).

Let $\mathcal{R}$ be a commutative ring with identity and $I(X,\mathcal{R})$ the incidence algebra of a locally finite pre-ordered set $X$. In this note, we characterize the derivations of $I(X,\mathcal{R})$ and prove that every Jordan derivation of $I(X,\mathcal{R})$ is a derivation, provided that $\mathcal{R}$ is $2$-torsion free.

In this paper, we make the nonlinear double integral equation of Hammerstein type the background of the research. Computation for the fixed point index of operators such as $A=K_{1}F_{1}K_ {2}F_{2}$ is given. As applications of the main results, we investigate the existence of positive solutions to the nonlinear double integral equation of Hammerstein type and the boundary value problem for the system of elliptic partial differential equations.

Let $R$ and $S$ be rings and ${}_SC_R$ a faithfully semidualizing bimodule. A left $S$-module $M$ is called \textit{Ding $C$-projective} if there exists an exact sequence of $C$-projective left $S$-modules \[ X=\cdots \rightarrow C\otimes_{R} P_{1}\rightarrow C\otimes_{R} P_{0}\rightarrow C\otimes_{R} P^{0}\rightarrow C\otimes_{R} P^{1}\rightarrow \cdots \] such that $M\cong\mbox{Coker\,}(C\otimes_{R} P_{1}\rightarrow C\otimes_{R} P_{0})$ and the complexes $\mbox{Hom}_S(C\otimes_{R}P, X)$ and $\mbox{Hom}_S(X,C\otimes_{R}F)$ are exact for any projective left $R$-module $P$ and any flat left $R$-module $F$. The properties of Ding $C$-projective modules and dimensions are given. Among others, the Foxby equivalences between some subclasses of the Auslander class and the Bass class are also investigated.