Frobenius Theorem Research Articles

A note on Frobenius–Wielandt groups

Abstract Let G be a finite group. G is called a Frobenius–Wielandt group if there exists H < G {H<G} such that U = 〈 H ∩ H g ∣ g ∈ G - H 〉 {U=\langle H\cap H^{g}\mid g\in G-H\rangle} is a proper subgroup of H. The Wielandt theorem [H. Wielandt, Über die Existenz von Normalteilern in endlichen Gruppen, Math. Nachr. 18 1958, 274–280; Mathematische Werke Vol. 1, 769–775] on the structure of G generalizes the celebrated Frobenius theorem. From a permutation group point of view, considering the action of G on the coset space G / H {G/H} , it states in particular that the subgroup D = D G ⁢ ( H ) {D=D_{G}(H)} generated by all derangements (fixed-point-free elements) is a proper subgroup of G. Let W = U G {W=U^{G}} , the normal closure of U in G. Then W is the subgroup generated by all elements fixing at least two points. We present the proof of the Wielandt theorem in a new way (Theorem 1.6, Corollary 1.7, Theorem 1.8) such that the unique component whose proof is not elementary or by the Frobenius theorem is the equality W ∩ H = U {W\cap H=U} . This presentation shows what can be achieved by elementary arguments and how Frobenius groups are involved in one case of Frobenius–Wielandt groups. To be more precise, Theorem 1.6 shows that there are two possible cases for a Frobenius–Wielandt group G with H < G {H<G} : (a) W = D {W=D} and G = H ⁢ W {G=HW} , or (b) W < D {W<D} and H ⁢ W < G {HW<G} . In the latter case, G / W {G/W} is a Frobenius group with a Frobenius complement H ⁢ W / W {HW/W} and Frobenius kernel D / W {D/W} .

An analytical solution for bending of axisymmetric circular/annular plates resting on a variable elastic foundation

In this paper, an analytical method is presented in order to determine the static bending response of an axisymmetric thin circular/annular plate with different boundary conditions resting on a spatially inhomogeneous Winkler foundation. To this end, infinite power series expansion of the deflection function is exploited to transform the governing differential equation into a new solvable system of recurrence relations. Singular points of the governing equation are effectively treated by applying the Frobenius theorem in the solution, which in turn permits the use of more-general analytical functions to describe the variation of the foundation modulus along the radius of the plate. Moreover, no special limitation is imposed on the transverse loading function as applied to the system. On employing the proposed method, the deflection response is obtained through an illustrative example for various boundary conditions along the plate edges, considering free, clamped, hinged, and elastically restrained boundaries. In addition, analytical results are validated and compared with those obtained using a finite element analysis, where an excellent agreement is found. Finally, the extension of the method to solve the more-general case of a variable two-parameter (Pasternak) foundation is indicated.

New criteria for mean square exponential stability of stochastic systems with variable and distributed delays

In this study, the authors are interested in the mean square exponential stability of stochastic systems with variable and distributed delays. Different from the traditional methods, based on the well-known Perron–Frobenius theorem and Ito formula, a proof by contradiction to explore some new criteria for the mean square exponential stability of stochastic delay systems is introduced. In particular, the proposed novel stability criteria reduce the traditional restrictions imposed on variable delays and also provide an optimal upper bound for delays. Two examples are given as applications to verify the effectiveness of the obtained results.

The Perron--Frobenius Theorem for Multihomogeneous Mappings

The Perron--Frobenius theory for nonnegative matrices has been generalized to order-preserving homogeneous mappings on a cone and more recently to nonnegative multilinear forms. We unify both approaches by introducing the concept of order-preserving multihomogeneous mappings, their associated nonlinear spectral problems, and spectral radii. We show several Perron--Frobenius type results for these mappings addressing existence, uniqueness, and maximality of nonnegative and positive eigenpairs. We prove a Collatz--Wielandt principle and other characterizations of the spectral radius and analyze the convergence of iterates of these mappings toward their unique positive eigenvectors. On top of providing a new extension of the nonlinear Perron--Frobenius theory to the multidimensional case, our contribution poses the basis for several improvements and a deeper understanding of the current spectral theory for nonnegative tensors. In fact, in recent years, important results have been obtained by recasting certain spectral equations for multilinear forms in terms of homogeneous maps; however, as our approach is more adapted to such problems, these results can be further refined and improved by employing our new multihomogeneous setting.

A NOTE ON RÉNYI'S ENTROPY RATE FOR TIME-INHOMOGENEOUS MARKOV CHAINS

AbstractIn this note, we use the Perron–Frobenius theorem to obtain the Rényi's entropy rate for a time-inhomogeneous Markov chain whose transition matrices converge to a primitive matrix. As direct corollaries, we also obtain the Rényi's entropy rate for asymptotic circular Markov chain and the Rényi's divergence rate between two time-inhomogeneous Markov chains.

The steady-state lifting bifurcation problem associated with the valency on networks

In this paper we consider homogeneous coupled cell networks with asymmetric inputs — networks where each cell receives exactly one input of each edge type. The coupled cell systems associated with a network are the dynamical systems that respect the network structure. There are subspaces, determined solely by the network structure, that are flow-invariant under any such coupled cell system — the synchrony subspaces. For a homogeneous network with asymmetric inputs, one of the eigenvalues of the Jacobian matrix of any coupled cell system at an equilibrium in the full-synchrony subspace corresponds to the valency of the network. In this work, we study the codimension-one steady-state bifurcations of coupled cell systems with a bifurcation condition associated with the valency. We start by giving an adaptation of the Perron–Frobenius Theorem for the eigenspace associated with the valency showing that the dimension of that eigenspace equals the number of the network source components. A network source component is a strongly connected component of the network whose cells receive inputs only from cells in the component. Each synchrony subspace determines a smaller network called quotient network. The lifting bifurcation problem addresses the issue of understanding when the bifurcation branches of a network can be lifted from one of its quotient networks. We consider the lifting bifurcation problem when the bifurcation condition is associated with the valency. We give sufficient conditions on the number of source components for the answer to the lifting bifurcation problem to be positive and prove that those conditions are necessary and sufficient for a class of networks.

Perron–Frobenius theorem for hypermatrices in the max algebra

Recently, there have been many intriguing new developments in the study of hypermatrices and their associated eigenvalue problems. In particular, results coming from the matrix setting when studying the max algebra have shown especially attractive combinatorial features. We now extend this max algebra setting into the realm of hypermatrices. Considering that the max algebra has shown particular significance in optimization problems for the matrix setting, we look to examine and extend these results in the higher order conditions. Furthermore, we establish some algebraic properties for hypermatrices and then proceed to extend the Perron–Frobenius Theorem for this setting and prove the existence of a unique eigenvalue. We continue by stating a result from Nussbaum, that the Min–Max theorem holds, and provide a proof for completeness. For strongly increasing hypermatrices, an iterative algorithm which converges to our unique eigenvalue is given. Finally, we conclude with an analysis of our results in the hypergraph setting.

On the characteristic polynomials of multiparameter pencils

In this paper we study the characteristic polynomial of multiparameter pencil z1A1+z2A2+⋯+zsAs. The main theorem states that a unitary representation of a finitely generated group contains a one-dimensional representation if and only if the characteristic polynomial of its generators contains a linear factor. It follows that a two or three dimensional unitary representation of a finitely generated group is irreducible if and only if the characteristic polynomial of the pencil of its generators is irreducible. The result is of kin to the Dedekind and Frobenius theorem on finite group determinant.

An optimal transportation approach to the decay of correlations for non-uniformly expanding maps

We consider the transfer operators of non-uniformly expanding maps for potentials of various regularity, and show that a specific property of potentials (‘flatness’) implies a Ruelle–Perron–Frobenius theorem and a decay of the transfer operator of the same speed as that entailed by the constant potential. The method relies neither on Markov partitions nor on inducing, but on functional analysis and duality, through the simplest principles of optimal transportation. As an application, we notably show that for any map of the circle which is expanding outside an arbitrarily flat neutral point, the set of Hölder potentials exhibiting a spectral gap is dense in the uniform topology. The method applies in a variety of situations, including Pomeau–Manneville maps with regular enough potentials, or uniformly expanding maps of low regularity with their natural potential; we also recover in a united fashion variants of several previous results.

Maximization of Minimum Rate for Wireless Powered Communication Networks in Interference Channel

We investigate wireless powered communication networks in an interference channel. In this system, due to asymmetric time allocation of the downlink and the uplink among multiple cells, cross-link interference may occur, which significantly affects overall performance. Considering this interference issue, we study a minimum rate maximization problem to overcome a severe imbalance on a rate distribution among users. The minimum rate maximization problem is non-convex; thus we propose an algorithm that updates the time allocation and the users’ transmit power based on the Lagrangian duality method and the Perron–Frobenius theorem, respectively. Simulation results verify that the proposed methods outperform conventional schemes.

Asymptotic and non-asymptotic analysis for a hidden Markovian process with a quantum hidden system

We focus on a data sequence produced by repetitive quantum measurement on an internal hidden quantum system, and call it a hidden Markovian process. Using a quantum version of the Perron–Frobenius theorem, we derive novel upper and lower bounds for the cumulant generating function of the sample mean of the data. Using these bounds, we derive the central limit theorem and large and moderate deviations for the tail probability. Then, we give the asymptotic variance by using the second derivative of the cumulant generating function. We also derive another expression for the asymptotic variance by considering the quantum version of the fundamental matrix. Further, we explain how to extend our results to a general probabilistic system.

Stochastic fixed points and nonlinear Perron–Frobenius theorem

We provide conditions for the existence of measurable solutions to the equation ξ ( T ω ) = f ( ω , ξ ( ω ) ) \xi (T\omega )=f(\omega ,\xi (\omega )) , where T : Ω → Ω T:\Omega \rightarrow \Omega is an automorphism of the probability space Ω \Omega and f ( ω , ⋅ ) f(\omega ,\cdot ) is a strictly nonexpansive mapping. We use results of this kind to establish a stochastic nonlinear analogue of the Perron–Frobenius theorem on eigenvalues and eigenvectors of a positive matrix. We consider a random mapping D ( ω ) D(\omega ) of a random closed cone K ( ω ) K(\omega ) in a finite-dimensional linear space into the cone K ( T ω ) K(T\omega ) . Under the assumptions of monotonicity and homogeneity of D ( ω ) D(\omega ) , we prove the existence of scalar and vector measurable functions α ( ω ) > 0 \alpha (\omega )>0 and x ( ω ) ∈ K ( ω ) x(\omega )\in K(\omega ) satisfying the equation α ( ω ) x ( T ω ) = D ( ω ) x ( ω ) \alpha (\omega )x(T\omega )=D(\omega )x(\omega ) almost surely.

Optimization Algorithms for Multiaccess Green Communications in Internet of Things

The exponential increase of the intelligent connected devices and the dramatic growth of the wireless data traffic have motivated the development of the green wireless networks as well as the Internet of Things (IoT). In this paper, we study the minimization problem of the total power to satisfy the required rate constraints in IoT, where the users simultaneously communicate through multiple independent channels. This problem is complicated due to the nonlinear data rate function based on the Shannon capacity formula. To this end, we first transfer the initial problem in power domain to an equivalent problem in rate domain instead of direct approximation for the high data rate. Then, we approximate it to a convex problem with the spectral radius constraints by the use of the Neumann expansion and nonlinear Perron–Frobenius theorem. By doing so, we achieve the close upper bound for this total power minimization problem. Moreover, we obtain the lower bound by making use of the convex relaxation technique, and finally get the global optimal solution by leveraging the branch-and-bound method. Simulation results verify that our proposed algorithms have a good approximation to the global optimal value for the power and rate allocations.

An extension of a theorem of Frobenius and Stickelberger to modules of projective dimension one over a factorial domain

Let R be a Cohen–Macaulay ring. A quasi-Gorenstein R-module is an R-module such that the grade of the module and the projective dimension of the module are equal and the canonical module of the module is isomorphic to the module itself. After discussing properties of finitely generated quasi-Gorenstein modules, it is shown that this definition allows for a characterization of diagonal matrices of maximal rank over a Cohen–Macaulay factorial domain R extending a theorem of Frobenius and Stickelberger to modules of projective dimension 1 over a commutative factorial Cohen–Macaulay domain.

The effect on eigenvalues of connected graphs by adding edges

By the well-known Perron–Frobenius Theorem [3], for a connected graph G, its largest eigenvalue strictly increases when an edge is added. We are interested in how the other eigenvalues of a connected graph change when edges are added. Examples show that all cases are possible: increased, decreased, unchanged. In this paper, we consider the effect on the eigenvalues by suitably adding edges in particular families, say the family of connected graphs with clusters. By using the result, we also consider the effect on the energy by suitably adding edges to the graphs of the above families.

A Frobenius–Nirenberg theorem with parameter

Abstract The Newlander–Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the complex structure in the complex Euclidean space. We will show two results about the Newlander–Nirenberg theorem with parameter. The first extends the Newlander–Nirenberg theorem to a parametric version, and its proof yields a sharp regularity result as Webster’s proof for the Newlander–Nirenberg theorem. The second concerns a version of Nirenberg’s complex Frobenius theorem and its proof yields a result with a mild loss of regularity.

A note on field-valued measures

Abstract We consider the Horn-Tarski condition for the extension of (signed) measures (resp., non-negative measures) in the setup of field-valued assignments. For a finite collection 𝓒 of subsets of Ω, we find that the extension from 𝓒 over the collection exp Ω of all subsets of Ω is implied by, and indeed equivalent to, a certain type of Frobenius theorem (resp. a certain type of Farkas lemma). This links classical notions of linear algebra with a generalized version of Horn-Tarski condition on extensions of measures. We also observe that for a general (infinite) 𝓒 the Horn-Tarski condition guarantees the extension of signed measures (here the standard Zorn lemma applies). However, we find out that the extensions for non-negative ordered-field-valued measures are generally not available.

On some geometric properties of currents and Frobenius theorem

In this note we announce some results, due to appear in [2], [3], on the structure of integral and normal currents, and their relation to Frobenius theorem. In particular we show that an integral current cannot be tangent to a distribution of planes which is nowhere involutive (Theorem 3.6), and that a normal current which is tangent to an involutive distribution of planes can be locally foliated in terms of integral currents (Theorem 4.3). This statement gives a partial answer to a question raised by Frank Morgan in [1].

Aggregation may or may not eliminate reproductive uncertainty

In matrix population models, there cannot be any ‘reproductive uncertainty’ when the life cycle graph contains only one reproductive stage. Otherwise, it is logical to expect that aggregating all the reproductive stages into a single one would exclude the very basis of uncertainty. However, can the aggregation change principally the model characteristics such as the dominant eigenvalue λ1 of the projection matrix, thus signifying the aggregation failure? I demonstrate that it can with the data mined in a case study on the dynamics of a local stage-structured population of Eritrichium caucasicum, a short-lived perennial plant species inhabiting an alpine lichen heath. Frobenius Theorem for nonnegative matrices specifies the upper and lower bounds for λ1 via the row (or column) sums of matrix elements, and the lower bound, when it exceeds the maximal possible λ1 of the original, disaggregated matrix, does explain why the aggregation may fail to eliminate reproductive uncertainty.

A Perron–Frobenius Type of Theorem for Quantum Operations

We define a special class of quantum operations we call Markovian and show that it has the same spectral properties as a corresponding Markov chain. We then consider a convex combination of a quantum operation and a Markovian quantum operation and show that under a norm condition its spectrum has the same properties as in the conclusion of the Perron–Frobenius theorem if its Markovian part does. Moreover, under a compatibility condition of the two operations, we show that its limiting distribution is the same as the corresponding Markov chain. We apply our general results to partially decoherent quantum random walks with decoherence strength $$0 \le p \le 1$$ . We obtain a quantum ergodic theorem for partially decoherent processes. We show that for $$0 < p \le 1$$ , the limiting distribution of a partially decoherent quantum random walk is the same as the limiting distribution for the classical random walk.