Directed Partial Orders on <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">i</Emphasis>) with 1 &gt; 0

Let F be a non-archimedean o-field and C be the field of generalized complex numbers over F. In this paper, we describe all the directed partial orders on C with 1 > 0 by using admissible semigroups of F ⁺.     

On <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">h</Emphasis></Subscript>-supplemented subgroups of finite groups

Suppose that F is a formation of finite groups. We introduce the concept of F _h -supplemented subgroups and investigate the structure of finite groups on assuming that some maximal subgroups of Sylow subgroups, maximal subgroups, minimal subgroups, and 2-maximal subgroup are F _h -supplemented, respectively. Some available results are generalized.     

The <Emphasis Type="Italic">p</Emphasis>-Adic order of the <Emphasis Type="Italic">k</Emphasis>-Fibonacci and <Emphasis Type="Italic">k</Emphasis>-Lucas numbers

Let (F _k,n ) _n and (L _k,n )n be the k-Fibonacci and k-Lucas sequence, respectively, which satisfies the same recursive relation a _n+1 = ka _n + a _n?1 with initial values F _k,0 = 0, F _k,1 = 1, L _k,0 = 2 and L _k,1 = k. In this paper, we characterize the p-adic orders ν _p (F _k,n ) and ν _p (L _k,n ) for all primes p and all positive integers k.     

Ideals of subseries convergence and <Emphasis Type="Italic">F</Emphasis>-spaces

Let X be an F-space and \({\boldsymbol x=(x_n)}\) be a sequence of vectors in X. Ideals \({\mathcal{C}(\boldsymbol x)}\) of subseries convergence are considered. In particular, we show that a characterization of the class of Banach spaces not containing c ₀ obtained by using the ideals \({\mathcal{C}(\boldsymbol x)}\) breaks down in every Fréchet space not isomorphic to a Banach space. On the other hand, the result can be extended to some F-spaces via the definition of a new class of F-spaces satisfying a stronger version of the condition (O) of Orlicz. A theorem discriminating between the finite and infinite dimensional case is obtained about the family \({\mathcal{C}(X)}\) of all ideals associated with the F-space X.     

Abelianization of the <Emphasis Type="Italic">F</Emphasis>-divided fundamental group scheme

Let (X , x ₀) be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for (X , x ₀) produces a homomorphism from the abelianization of the F-divided fundamental group scheme of X to the F-divided fundamental group of the Albanese variety of X. We prove that this homomorphism is surjective with finite kernel. The kernel is also described.     

Asymptotics of the Codimensions <Emphasis Type="Italic">c</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> in the Algebra <Emphasis Type="Italic">F</Emphasis><Superscript>(7)</Superscript>

The paper studies the additive structure of the algebra F⁽⁷⁾, i.e., a relatively free associative countably generated algebra with the identity [x₁,..., x₇] = 0 over an infinite field of characteristic ≠ 2, 3. First, the space of proper multilinear polynomials in this algebra is investigated. As an application, estimates for the codimensions c_n = dimF_n⁽⁷⁾ are obtained, where F_n⁽⁷⁾ stands for the subspace of multilinear polynomials of degree n in the algebra F⁽⁷⁾.     

On Products of <Emphasis Type="Italic">F</Emphasis>-Compact Spaces

An F-compactum or a Fedorchuk compactum is a Hausdorff compact space that admits decomposition into a special well-ordered inverse system with fully closed neighboring projections. We prove that the square of Aleksandroff’s “double arrow” space is not an F-compactum of countable spectral height. Using this, we demonstrate the impossibility of representing the Helly space as the inverse limit of a countable system of resolutions with metrizable fibers. This gives a negative answer to a question posed by Watson in 1992.     

Fixed point theorems for single- and set-valued <Emphasis Type="Italic">F</Emphasis>-contractions in <Emphasis Type="Italic">b</Emphasis>-metric spaces

Very recently, Miculescu and Mihail in (J. Fixed Point Theory Appl 19:2153–2163, 2017) gave a sufficient condition for Cauchyness on sequences in b-metric spaces. In this paper, we give a weaker sufficient condition. Also, to show the new sufficient condition is reasonably weak, we give an example. Using the new sufficient condition, we prove fixed point theorems for set-valued F-contractions in complete b-metric spaces. We also prove a fixed point theorem for single-valued F-contractions in complete b-metric spaces.     

The probability that <Emphasis Type="Italic">r</Emphasis> elements of a rank <Emphasis Type="Italic">n</Emphasis> free group generate a rank <Emphasis Type="Italic">r</Emphasis> subgroup

Granted the three integers n ≥ 2, r, and R, consider all ordered tuples of r elements of length at most R in the free group F _n . Calculate the number of those tuples that generate in F _n a rank r subgroup and divide it by the number of all tuples under study. As R → ∞, the limit of the ratio is known to exist and equal 1 (see [1]). We give a simple proof of this result.     

Strictly singular operators in pairs of <Emphasis Type="Italic">L</Emphasis> <Subscript> <Emphasis Type="Italic">p</Emphasis> </Subscript> space

Let E and F be Banach spaces. A linear operator from E to F is said to be strictly singular if, for any subspace Q ? E, the restriction of A to Q is not an isomorphism. A compactness criterion for any strictly singular operator from L_p to L_q is found. There exists a strictly singular but not superstrictly singular operator on L_p, provided that p ≠ 2.     

The existence of eigenvalues for operators acting in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We present conditions that allow us to prove the existence of eigenvalues and characteristic values for operator F(D) ? C(λ): L ²(R ^m ) → L ²(R ^m ), where F(D) is a pseudo-differential operator with a symbol F(iξ) and C(λ): L ²(R ^m ) → L ²(R ^m ) is a linear continuous operator.     

The Center of <Emphasis Type="Italic">Dist</Emphasis> (<Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">m</Emphasis>|<Emphasis Type="Italic">n</Emphasis>)) in Positive Characteristic

The purpose of this paper is to investigate central elements in distribution algebras D i s t(G) of general linear supergroups G = G L(m|n). As an application, we compute explicitly the center of D i s t(G L(1|1)) and its image under Harish-Chandra homomorphism.     

Real elements in groups of type <Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>

Let G be a group. An element x ∈ G is called real if x is conjugate to x ^?1 in G. In this paper we study the structure of real elements in the compact connected Lie group of type F ₄ and algebraic groups of type F ₄ defined over an arbitrary field.     

On <Emphasis Type="Italic">MP</Emphasis>-closed saturated formations of finite groups

A class of groups F is called MP-closed, if it contains every group G = AB such that the F-subgroup A permutes with every subgroup of B and the F-subgroup B permutes with every subgroup of A. We prove that the formation F that contains the class of all supersoluble groups is MP-closed if and only if the formation F(p) is MP-closed for any prime number p, where F is a maximal inner local screen of F. In particular, we prove that the formation of all groups with supersoluble Schmidt subgroups is MP-closed.     

On the sequential order continuity of the <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)-space

As shown in [1], for each compact Hausdorff space K without isolated points, there exists a compact Hausdorff P′-space X but not an F-space such that C(K) is isometrically Riesz isomorphic to a Riesz subspace of C(X). The proof is technical and depends heavily on some representation theorems. In this paper we give a simple and direct proof without any assumptions on isolated points. Some generalizations of these results are mentioned.     

Periods and nonvanishing of central <Emphasis Type="Italic">L</Emphasis>-values for GL(2<Emphasis Type="Italic">n</Emphasis>)

Let π be a cuspidal automorphic representation of PGL(2n) over a number field F, and η the quadratic idèle class character attached to a quadratic extension E/F. Guo and Jacquet conjectured a relation between the nonvanishing of L(1/2, π)L(1/2, π ? η) for π of symplectic type and the nonvanishing of certain GL(n,E) periods. When n = 1, this specializes to a well-known result of Waldspurger. We prove this conjecture, and related global results, under some local hypotheses using a simple relative trace formula.We then apply these global results to obtain local results on distinguished supercuspidal representations, which partially establish a conjecture of Prasad and Takloo-Bighash.     

<Emphasis Type="Italic">X</Emphasis>-decomposable finite groups for <Emphasis Type="Italic">X</Emphasis>={1, <Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">m</Emphasis> + 1, <Emphasis Type="Italic">m</Emphasis> + 2}

A normal subgroup N of a finite group G is called n-decomposable in G if N is the union of n distinct G-conjugacy classes. We study the structure of nonperfect groups in which every proper nontrivial normal subgroup is m-decomposable, m+1-decomposable, or m+2-decomposable for some positive integer m. Furthermore, we give classification for the soluble case.     

Facets of the <Emphasis Type="Italic">r</Emphasis>-Stable (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">k</Emphasis>)-Hypersimplex

Let k, n, and r be positive integers with k < n and \({r \leq \lfloor \frac{n}{k} \rfloor}\). We determine the facets of the r-stable n, k-hypersimplex. As a result, it turns out that the r-stable n, k-hypersimplex has exactly 2n facets for every \({r < \lfloor \frac{n}{k} \rfloor}\). We then utilize the equations of the facets to study when the r-stable hypersimplex is Gorenstein. For every k > 0 we identify an infinite collection of Gorenstein r-stable hypersimplices, consequently expanding the collection of r-stable hypersimplices known to have unimodal Ehrhart \({\delta}\)-vectors.     

Largest <Emphasis Type="Italic">H</Emphasis>-eigenvalue of uniform <Emphasis Type="Italic">s</Emphasis>-hypertrees

The k-uniform s-hypertree G = (V,E) is an s-hypergraph, where 1 ≤ s ≤ k - 1; and there exists a host tree T with vertex set V such that each edge of G induces a connected subtree of T. In this paper, some properties of uniform s-hypertrees are establised, as well as the upper and lower bounds on the largest H-eigenvalue of the adjacency tensor of k-uniform s-hypertrees in terms of the maximal degree Δ. Moreover, we also show that the gap between the maximum and the minimum values of the largest H-eigenvalue of k-uniform s-hypertrees is just Θ(Δ ^s/k ).     

On the Second Parameter of an (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">p</Emphasis>)-Isometry

A bounded linear operator T on a Banach space X is called an (m, p)-isometry if it satisfies the equation \({\sum_{k=0}^{m}(-1)^{k} {m \choose k}\|T^{k}x\|^{p}=0}\) , for all \({x \in X}\) . In this paper we study the structure which underlies the second parameter of (m, p)-isometric operators. We concentrate on determining when an (m, p)-isometry is a (μ, q)-isometry for some pair (μ, q). We also extend the definition of (m, p)-isometry, to include p = ∞ and study basic properties of these (m, ∞)-isometries.