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In this note we offer some criticism on the paper “On an identity for zeros of Bessel functions” by Baricz et al. [3] . The paper gives identities of type Stieltjes–Calogero for the sums of reciprocals of differences of fourth powers of zeros of Bessel functions. Although interesting in principle, by containing one too many sums of similar complexity the identities fail to convey the true spirit of the work of Stieltjes and Calogero. We rectify this by providing what we think is the correct type of identity for the above-said sums, in the general setup of entire functions of order

In a recent paper referred to in the title, the author used the concept of quasi self-adjointness to obtain conservation laws for a system of the ( 2 + 1 ) -dimensional Bogoyavlensky–Konopelchenko equation. Apart from the adjoint system of equations, all the results on the quasi self-adjointness and the subsequent conservation laws obtained are inaccurate. In this comment, we clarify these inaccuracies and also generate conservation laws for a potential form of the underlying equation through Noether theorem.

We give a comprehensive account of Chern's Theorem that S^6 admits no omega-compatible almost complex structures. No claim to originality is being made, as the paper is mostly an expanded version of material already in the literature. This article extends the talks that both authors gave in Marburg during the conference "(Non)existence of complex structures on S^6" in April 2017., Misprints corrected

These notes aim to provide a deeper insight on the specifics of two articles dealing with chemotaxis models with nonlinear production. More precisely, we are referring to the papers “Boundedness of solutions to a quasilinear parabolic–parabolic chemotaxis model with nonlinear signal production” by Tao et al. (2019) [2] and “Boundedness for a fully parabolic Keller–Segel model with sublinear segregation and superlinear aggregation” by Frassu and Viglialoro (2021) [1] . These works, independently published in these last years, present results leaving open room for further improvement. Indeed, in the first a gap in the proof of the main claim appears, whereas the cornerstone assumption in the second is not sharp. In these pages we give a more complete picture to the relative underlying comprehension.

The translation into English of Aleksandr Yakovlevich Khinchin's (1894–1959) 1926 paper entitled ‘Ideas of intuitionism and the struggle for a subject matter in contemporary mathematics’ is made available for the first time. Here, Khinchin presented the famous foundational debate between L.E.J. Brouwer and David Hilbert of the 1920s in terms of a search for a mathematics with content. His main aim seems to have been to make intuitionism ideologically acceptable to his audience at the Communist Academy by means of the claim that insofar as Brouwer's intuitionism had a clear ‘subject matter’ and Hilbert's new program was a concession to intuitionism, the alleged victory of intuitionism not only implied the defeat of ‘empty’ formalism, but also showed the compatibility and affinity of Marxism with the newest developments in modern mathematics. This introduction provides a tentative exploration of the issue of what was tactical (or due to ideological pressure) and what was real scientific interest (or due to ignorance) (or what was both) in Khinchin's 1926 paper in the form of a detailed commentary, especially, on the tactical side of his presentation of the positions of Brouwer and Hilbert.

In 1903, Epstein published his proof of meromorphic continuation and a functional equation for Dirichlet series associated with quadratic forms, now called Epstein zeta-functions. However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show. In this article we present and analyze Hurwitz's notes and compare his reasoning with Epstein's paper in detail.

Two limit cycles for the continuous model of the rockscissorspaper (RSP) game is constructed.The Hopf bifurcation method is used for the construction of limit cycles.The results give a partial answer to an open question posed by Neumann and Schuster. In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rockscissorspaper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game.

This paper is an attempt to present the methodologically correct approach of the anisotropy-based theory for discrete-time systems with multiplicative noise. A formulation of sufficient conditions of boundedness of the anisotropic norm is the main goal of this paper. As it is shown, this problem can be reduced to the convex optimization problem. A simple numerical example to illustrate the solution procedure is given.

We would like to present the ideas of Makoto Matsumoto and Andras Rapcsak in the projective theory of Finsler spaces. Finally, we give some remarks using of Matsumoto–Rapcsak Theorem.

In Fritz John's famous paper (1979), he discovered that for the wave equation $\Box u=|u|^p$, where $1, Comment: 11 pages, 3 figures

Assume that A and G are finite groups of coprime orders such that A acts on G via automorphisms. Let p be a prime. The following coprime action version of a well-known theorem of Ito about the structure of a minimal non-p-nilpotent groups is proved: if every maximal A-invariant subgroup of G is p-nilpotent, then G is p-soluble. If, moreover, G is not p-nilpotent, then G must be soluble. Some earlier results about coprime action are consequences of this theorem.

In this note we show that the presented proof of [1, th. 12] , being based on a false statement appearing in this proof, is not viable for all of the proclaimed values of the involved parameters. We also determine necessary and sufficient conditions for those parameters, which provides that the considered statement, and therefore [1, th. 12] itself, still remains valid.

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $\Theta$ which give rise to self-adjoint Laplacians $-\Delta_{\Theta, \Omega}$ in $L^2(\Omega; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $\Omega\subset\bbR^n$, $n\in\bbN$, $n\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $\Omega$, following an approach introduced by Filonov for this type of problems., Comment: 23 pages, added Remark 5.4

In this paper we construct a family of circle-like continua, each admitting a finest monotone map onto S 1 such that there exists a subset of point inverses which is homeomorphic to the Cantor set cross an interval. We then show how to realize some members of this family as the boundaries ∂U of bounded irreducible local Siegel disks U . These boundaries are geometrically rigid in the following sense: there exist arbitrarily small periodic homeomorphisms of the sphere, conformal on U , which keep U invariant. The embedding portion of this paper follows a flexible construction of Herman. These results provide a partial answer to a question of Rogers and a complete answer to a question of Brechner, Guay, and Mayer.

In Vukicevic and Stevanovic (2013), Theorem 2 is erroneous when m = 3 k + 1 for some integer k . The correct one is given now. That is, P I ( G ) ≥ m 2 − 3 m + 2 when m = 3 k + 1 for some integer k , where G is a connected bicyclic graph with m edges.

This article surveys R. Parikh's work on feasibility, bounded arithmetic and the complexity of proofs. We discuss in depth two of Parikh's papers on these subjects and some of the subsequent progress in the areas of feasible arithmetic and lengths of proofs.

For a rational f : C → C with a conformal measure μ we show that if there is a subset of the Julia set J(f) of positive μ -measure whose points are not eventual preimages of critical or parabolic points and have limit sets not contained in the union of the limit sets of recurrent critical points, then μ is non-atomic, μ(J(f))=1 , ω(x)=J(f) for μ -a.e. point x∈J(f) and f is conservative, ergodic and exact. The proof uses a version of the Lebesgue Density Theorem valid for Borel measures and conformal balls.

We point out errors and oversights in a paper by Giordano, Laforgia, and Pecaric [3] on inequalities involving Bessel functions.

In this paper we generalize two results of Khan and Majumdar on the weak sequential convergence of functions in the Lebesgue-Bochner space L X 1 (Ω) and on a Fatou-like convergence theorem.

It is pointed out that the proof of the basic result of Rahmanov's paper has a serious gap. It is documented by original sources that a statement he relied on in the proof contains a misprint, and it is shown by a counterexample that this statement (with the misprint) is, in fact, false. A somewhat weaker statement is proved true.

In this short paper, we extend the result of Shinbrot (1974) to an incompressible fluid with shear dependent viscosity. It is shown that a weak solution to the equations of non-Newtonian fluids lying in a space L q ( 0 , T ; L p ) satisfies an energy equality, where 2 r r − 1 ≤ p ≤ 2 r r − 2 and 1 p + 1 q ≤ r − 1 r , if r > 2 ; p ≥ 2 r r − 1 and r − 1 p + 1 q = r − 1 2 , if 2 ( n + 1 ) n + 2 r ≤ 2 . In particular, our result implies that the weak solution must satisfy the energy equality when r ≥ 3 n + 2 n + 2 , which is consistent with the known fact.

In this short paper, we consider the formation of singularities for the solutions of the Euler–Poisson equations with damping with attractive forces in R n ( n ≥ 1 ) and the repulsive forces in R . Under the appropriate assumptions on initial data, we can obtain the divergence of the velocity will blowup in the finite time.

In this short paper, the Cauchy problem for the Le Roux system is studied and the global weak solutions are obtained by using a new technique from the div–curl lemma in the compensated compactness theorem, where the regular B V estimate is assumed for one characteristic field. This work extends the previous work, (LeVeque and Temple, 1985), which provided the global solutions when both the characteristic fields are B V bounded by using the Godunov’s method.

In this short paper, we investigate a definition of Parisian ruin introduced in Czarna (2016), namely Parisian ruin with an ultimate bankruptcy level. We improve the results originally obtained and, moreover, we compute more general Parisian fluctuation identities.

In this short paper, we focus on the dependence on initial data of stochastic Camassa–Holm equation. The results show that the nonlinear noise will have an impact on the property of solutions with respect to initial data, but the linear noise will have no effect.

A graph is edge-primitive if its automorphism group acts primitively on the edge set. In this short paper, we prove that a finite 2-arc-transitive edge-primitive graph has almost simple automorphism group if it is neither a cycle nor a complete bipartite graph. We also present two examples of such graphs, which are 3-arc-transitive and have faithful vertex-stabilizers.

Consider k-binomial equivalence: two finite words are equivalent if they share the same subwords of length at most k with the same multiplicities. With this relation, the k-binomial complexity of an infinite word x maps the integer n to the number of pairwise non-equivalent factors of length n occurring in x. In this paper based on the notion of template introduced by Currie et al., we show that, for all k ≥ 2 , the k-binomial complexity of the Tribonacci word coincides with its usual factor complexity p ( n ) = 2 n + 1 . A similar result was already known for Sturmian words, but the proof relies on completely different techniques that seemingly could not be applied for Tribonacci.

Let H be a fixed forbidden graph and let f be a function of n. Denote by RT(n,H,f(n)) the maximum number of edges a graph G on n vertices can have without containing H as a subgraph and also without having at least f(n) independent vertices. The problem of estimating RT(n,H,f(n)) is one of the central questions of so-called Ramsey–Turán theory.In their recent paper (Discrete Math. 229 (2001) 293–340), Simonovits and Sós gave an excellent survey of this theory and mentioned some old and new interesting open questions. In this short paper we obtain some new bounds for Ramsey–Turán-type problems. These results give partial answers for some of the questions.

This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first algorithm allows one to transform the considered differential systems to the normal form of averaging. Here, we restricted the unperturbed term of the normal form of averaging to be identically zero. The second algorithm is used to derive the computational formulae of the averaged functions at any order. The third algorithm is based on the first two algorithms and determines the exact expressions of the averaged functions for the considered differential systems. The proposed approach is implemented in Maple and its effectiveness is shown by several examples. Moreover, we report some incorrect results in published papers on the averaging method.

The paper discusses the existence of a linear manifold in a vicinity of a steady state for stabilization of the Hamiltonian systems arising in optimal control problems for economic growth models. It is shown that such stable manifold exists for almost all possible values of model parameters guaranteeing the existence of a steady state. Research is based on the qualitative analysis of the Hamiltonian dynamics, which plays a key role for investigating the asymptotic behavior of optimal trajectories. A procedure is proposed for stabilization of the Hamiltonian system, whose trajectories converge to equilibrium and approximate the optimal solution with the quadratic accuracy at a vicinity of the steady state. Basing on properties of the Hamiltonian matrices, the classification of steady states is provided and the sensitivity analysis for identification of their character is implemented with respect to model parameters. The proposed approach is applied to the model dealing with dynamic optimization of the resource productivity. © 2018 18–01–00221a The paper is supported by Russin Foundation for Basic Research (Project No. 18–01–00221a).

In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages

The aim of this paper is twofold. The first is to study the asymptotics of a parabolically scaled, continuous and space-time stationary in time version of the well-known Funaki-Spohn model in Statistical Physics. After a change of unknowns requiring the existence of a space-time stationary eternal solution of a stochastically perturbed heat equation, the problem transforms to the qualitative homogenization of a uniformly elliptic, space-time stationary, divergence form, nonlinear partial differential equation, the study of which is the second aim of the paper. An important step is the construction of correctors with the appropriate behavior at infinity.

Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives. First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, … Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincare compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincare disc for the separable quadratic polynomial differential systems.

Carsten Thomassen and I proved that in any graph G , the number of cycles containing a specified edge as well as all the vertices of odd degree is odd if and only if G is eulerian. Where all vertices have even degree this is a theorem of Shunichi Toida and where all vertices have odd degree it is Andrew Thomason’s extension of Smith’s Theorem. Our proof was not algorithmic. In this paper, I extend Thomason’s algorithm to one which, in a non-eulerian graph, finds a second cycle containing a specified edge and all the odd-degree vertices. Kenneth Berman extended Thomason’s Theorem to spanning trees: he proved that if T is a spanning tree such that the degree of every vertex in G − E ( T ) is odd, then there is an even number of spanning trees of G with the same degree as T at each vertex. In this paper, I give a common generalization of all of these theorems to a parity theorem about “good” trees in general graphs. My proof provides an algorithm for finding a second “good” tree.

In this paper, we study various aspects of the ODE's flow $X$ solution to the equation $\partial_t X(t,x)=b(X(t,x))$, $X(0,x)=x$ in the $d$-dimensional torus $Y_d$, where $b$ is a regular $Z^d$-periodic vector field from $R^d$ in $R^d$. We present an original and complete picture in any dimension of all logical connections between the following seven conditions involving the field $b$: - the everywhere asymptotics of the flow $X$, - the almost-everywhere asymptotics of the flow $X$, - the global rectification of the vector field $b$ in $Y_d$, - the ergodicity of the flow related to an invariant probability measure which is absolutely continuous with respect to Lebesgue's measure, - the unit set condition for Herman's rotation set $C_b$ composed of the means of $b$ related to the invariant probability measures, - the unit set condition for the subset $D_b$ of $C_b$ composed of the means of $b$ related to the invariant probability measures which are absolutely continuous with respect to Lebesgue's measure, - the homogenization of the linear transport equation with oscillating data and the oscillating velocity $b(x/\varepsilon)$ when $b$ is divergence free. The main and surprising result of the paper is that the almost-everywhere asymptotics of the flow $X$ and the unit set condition for $D_b$ are equivalent when $D_b$ is assumed to be non empty, and that the two conditions turn to be equivalent to the homogenization of the transport equation when $b$ is divergence free. In contrast, using an elementary approach based on classical tools of PDE's analysis, we extend the two-dimensional results of Oxtoby and Marchetto to any $d$-dimensional Stepanoff flow: this shows that the ergodicity of the flow may hold without satisfying the everywhere asymptotics of the flow.

The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new invariant $\gamma_G(M)$ for a finite dimensional module $M$ of a finite group $G$ which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for all the signed permutation modules of the symmetric group using tools from representation theory and combinatorics., Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:2012.00341

This paper studies local properties of Jacobson-Witt algebras over fields of prime characteristic, i.e., initiates the study on 2-local derivations of Lie algebras of prime characteristic. Let W n be a simple Jacobson-Witt algebra over a field F of prime characteristic p with | F | ≥ p n . In this paper, it is shown that every 2-local derivation on W n is a derivation.

This paper investigates algebraic objects equipped with an operator, such as operated monoids, operated algebras etc. Various free object functors in these operated contexts are explicitly constructed. For operated algebras whose operator satisfies a set $\Phi$ of relations (usually called operated polynomial identities (aka. OPIs)), Guo defined free objects, called free $\Phi$-algebras, via universal algebra. Free $\Phi$-algebras over algebras are studied in details. A mild sufficient condition is found such that $\Phi$ together with a Gr\"obner-Shirshov basis of an algebra $A$ form a Gr\"obner-Shirshov basis of the free $\Phi$-algebra over algebra $A$ in the sense of Guo et al.. Ample examples for which this condition holds are provided, such as all Rota-Baxter type OPIs, a class of differential type OPIs, averaging OPIs and Reynolds OPI., Comment: Slightly revised version of the published paper in Journal of Algebra

A complex unit gain graph ($\mathbb{T}$-gain graph), $\Phi = (G, \varphi)$ is a graph where the function $\varphi$ assigns a unit complex number to each orientation of an edge of $G$, and its inverse is assigned to the opposite orientation. A $ \mathbb{T} $-gain graph $\Phi$ is balanced if the product of the edge gains of each cycle (with a fixed orientation) is $1$. Signed graphs are special cases of $\mathbb{T}$-gain graphs. The adjacency matrix of $\Phi$, denoted by $ \mathbf{A}(\Phi)$ is defined canonically. The gain Laplacian for $\Phi$ is defined as $\mathbf{L}(\Phi) = \mathbf{D}(\Phi) - \mathbf{A}(\Phi)$, where $\mathbf{D}(\Phi)$ is the diagonal matrix with diagonal entries are the degrees of the vertices of $G$. The minimum number of vertices (resp., edges) to be deleted from $\Phi$ in order to get a balanced gain graph the frustration number (resp, frustration index). We show that frustration number and frustration index are bounded below by the smallest eigenvalue of $\mathbf{L}(\Phi)$. We provide several lower and upper bounds for extremal eigenvalues of $\mathbf{L}(\Phi)$ in terms of different graph parameters such as the number of edges, vertex degrees, and average $2$-degrees. The signed graphs are particular cases of the $\mathbb{T}$-gain graphs, all the bounds established in paper hold for signed graphs. Most of the bounds established here are new for signed graphs. Finally, we perform comparative analysis for all the obtained bounds in the paper with the state-of-the-art bounds available in the literature for randomly generated Erd\H{o}s-Re\'yni graphs. Some of the major highlights of our paper are the gain-dependent bounds, limit convergence of the bounds to the extremal eigenvalues, and optimal extremal bounds obtained by posing optimization problems to achieve the best possible bounds., Comment: Preliminary version. 32 pages

A generalization of the well–known Fibonacci sequence is the l-Fibonacci sequence ( F m ( l ) ) m whose first l terms are 0 , … , 0 , 1 and each term afterwards is the sum of the preceding l terms. The k-Pell sequence ( P n ( k ) ) n , which is a generalization of the classical Pell sequence, can be defined similarly. In this paper, we find all coincidences between these two families of sequences. That is, we find all the solutions of the Diophantine equation P n ( k ) = F m ( l ) in positive integers n , k , m , l with k , l ≥ 2 . This paper continues and extends prior results which dealt with the above problem for some particular cases of k and l. In particular, it extends the previous work [2] that found all Fibonacci numbers in the Pell sequence.

The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GU m ( q ) × GU n ( q ) on M m × n ( q 2 ) (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form A A ∗ and A ∗ A over a finite field and A A ⊤ and A ⊤ A over arbitrary fields.

It is known that every graph of sufficiently large chromatic number and bounded clique number contains, as an induced subgraph , a subdivision of any fixed forest, and a subdivision of any fixed cycle. Equivalently, every forest is pervasive, and K 3 is pervasive, in the class of all graphs, where we say a graph H is “pervasive” (in some class of graphs) if for all l ≥ 1 , every graph in the class of bounded clique number and sufficiently large chromatic number has an induced subgraph that is a subdivision of H, in which every edge of H is replaced by a path of at least l edges. Which other graphs are pervasive? It was proved by Chalopin, Esperet, Li and Ossona de Mendez that every such graph is a “forest of lanterns”: roughly, every block is a “lantern”, a graph obtained from a tree by adding one extra vertex, and there are rules about how the blocks fit together. It is not known whether every forest of lanterns is pervasive in the class of all graphs; but in another paper two of us prove that all “banana trees” are pervasive, that is, multigraphs obtained from a forest by adding parallel edges, thus generalizing the two results above. This paper contains the first half of the proof, which works for any forest of lanterns, not just for banana trees. Say a class of graphs is “ρ-controlled” if for every graph in the class, its chromatic number is at most some function (determined by the class) of the largest chromatic number of a ρ-ball in the graph. In this paper we prove that for every ρ ≥ 2 , and for every ρ-controlled class, every forest of lanterns is pervasive in this class. These results turn out particularly nicely when applied to string graphs. A “chandelier” is a special lantern, a graph obtained from a tree by adding a vertex adjacent to precisely the leaves of the tree. A “string graph” is the intersection graph of a set of curves in the plane. There are string graphs with clique number two and chromatic number arbitrarily large. We prove that the class of string graphs is 2-controlled, and consequently every forest of lanterns is pervasive in this class; but in fact something stronger is true, that every string graph of sufficiently large chromatic number and bounded clique number contains each fixed chandelier as an induced subgraph (not just as a subdivision); and the same for most forests of chandeliers (there is an extra condition on how the blocks are attached together).

We study the sequence of momenta of the Coxeter transformation Φ G → associated to a bipartite quiver G → (that is, the sequence of traces of the powers of Φ G → ). Among other results, we derive A'Campo like formulas for such momenta in terms of the spectrum of the corresponding underlying graph G of G → (using Chebyshev polynomials), and analyze the maximal number of spikes in a tree graph (that is, eigenvalues with norm greater than 1), to construct general bounds for the Coxeter momenta. We apply the main results of the paper to produce formulas and bounds for the trace of the exponential of Coxeter matrices of tree graphs (called Coxeter-Estrada index in the paper).

This paper is concerned with Friedrichs extensions for a class of Hamiltonian systems. The non-symmetric problems are usually complicated and have unexpected properties. Here, Friedrichs extensions of a class of singular Hamiltonian systems including non-symmetric cases are characterized by imposing some constraints on each element of domains D ( H ) of the maximal operators H. These characterizations are given independent of principal solutions. It is interesting that by the results in the paper the Friedrichs extension of each of a large class of non-symmetric Hamiltonian systems has similar form to that of a symmetric Hamiltonian system. In addition, Friedrichs extensions of regular Hamiltonian systems are characterized incidently, J -self-adjoint Friedrichs extensions are studied, and a result is given for elements of D ( H ) , which makes the expression of the Friedrichs extension domain simpler. All results for Hamiltonian systems are finally applied to Sturm-Liouville operators with matrix-valued coefficients.

Let M be an Anderson t-motive of dimension n and rank r. Associated are two F q [ T ] -modules H 1 ( M ) , H 1 ( M ) of dimensions h 1 ( M ) , h 1 ( M ) ≤ r - analogs of H 1 ( A , Z ) , H 1 ( A , Z ) for an abelian variety A. There is a theorem (Anderson): h 1 ( M ) = r ⇔ h 1 ( M ) = r ; in this case M is called uniformizable. It is natural to expect that always h 1 ( M ) = h 1 ( M ) . Nevertheless, we explicitly construct a counterexample. Further, we answer a question of D. Goss: is it possible that two Anderson t-motives that differ only by a nilpotent operator N are of different uniformizability type, i.e. one of them is uniformizable and other not? We give an explicit example that this is possible. Finally, explicit formulas for calculation of h 1 ( M ) , h 1 ( M ) obtained in the present paper will be used in future for systematic calculation of h 1 , h 1 of all Anderson t-motives. Moreover, the first step of this calculation (for a class of t-motives) is already made in a forthcoming paper of the authors.

In this paper we study a generalisation of the Igusa-Todorov functions which gives rise to a vast class of algebras satisfying the finitistic dimension conjecture. This class of algebras is called Lat-Igusa-Todorov and includes, among others, the Igusa-Todorov algebras (defined by J. Wei) and the self-injective algebras which in general are not Igusa-Todorov algebras. Finally, some applications of the developed theory are given in order to relate the different homological dimensions which have been discussed through the paper., 18 pages, submitted to a peer-reviewed journal

In this paper we consider existence, nonexistence and multiplicity of solutions for a class of indefinite quasilinear elliptic problems in the upper half-space involving weights in anisotropic Lebesgue spaces. One of our basic tools consists in a Hardy type inequality proved in the present paper that allows us to establish Sobolev embeddings into Lebesgue spaces with weights in anisotropic Lebesgue spaces.

The aim of the paper is demonstration of equality max σ ∈ S n σ ≠ id { 1 , … , n } { ∏ i = 1 n | 〈 x i , x σ ( i ) 〉 | } = max τ ∈ S n τ is a transposition ⁡ { ∏ i = 1 n | 〈 x i , x τ ( i ) 〉 | } for each sequence of vectors x 1 , … , x n in a unitary space. The obtained equality has an important application in optimizing of certain generalization of the Hadamard's inequality proved by Xiaodong Zhang and Shangjun Yang and by the authors of the paper independently. Some illustrating examples are presented in the article. Also a problem connected with the mentioned equality is formulated.

We are concerned with two kinds of singular limits of the Cauchy problem to the two-layer rotating shallow water equations as the Rossby number and the Froude number tend to zero. First we construct the uniform estimates for the strong solutions to the system under the condition that the Froude number is small enough. Different from the previously studied cases, the large operator of this model is not skew-symmetric. One of the key new ideas in this paper is to obtain the uniform estimates using the special structure of the system rather than the antisymmetry of the large operator. After that the convergence of the equations with ill-prepared data to a two-layer incompressible Navier-Stokes system is proved with the help of Strichartz estimates constructed in this paper.