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In this investigation, using Opoola differential operator (Dm(µ, β, t)f(z)), a new integral operator: I m,σ t,β,µ(f1, ..., fn)(z) : An → A is defined in the unit disk, U = {z ∈ C : |z| < 1}; and we investigated the Univalence conditions of this generalized operator. Finally, a number of corollaries and remarks which show the extension of our results are presented. [ABSTRACT FROM AUTHOR]

For a given rational number r, a classical theorem of Niven asserts that if $ \cos (r\pi) $ cos ⁡ (rπ) is rational, then $ \cos (r\pi)\in \{0, \pm 1, \pm 1/2\}. $ cos ⁡ (rπ) ∈ { 0 , ± 1 , ± 1 / 2 }. In this note, we extend Niven's theorem to quadratic irrationalities and present an elementary proof of that. [ABSTRACT FROM AUTHOR]

We study the 2‐parity conjecture for Jacobians of hyperelliptic curves over number fields. Under some mild assumptions on their reduction, we prove the conjecture over quadratic extensions of the base field. The proof proceeds via a generalisation of a formula of Kramer and Tunnell relating local invariants of the curve, which may be of independent interest. A new feature of this generalisation is the appearance of terms which govern whether or not the Cassels–Tate pairing on the Jacobian is alternating, which first appeared in work of Poonen–Stoll. We establish the local formula in many instances and show that in remaining cases, it follows from standard global conjectures. [ABSTRACT FROM AUTHOR]

We introduce a new method of constructing complete sequences of key polynomials for simple extensions of tame fields. In our approach the key polynomials are taken to be the minimal polynomials over the base field of suitably constructed elements in its algebraic closure, with the extensions generated by them forming an increasing chain. In the case of algebraic extensions, we generalize the results to countably generated infinite tame extensions over henselian but not necessarily tame fields. In the case of transcendental extensions, we demonstrate the central role that is played by the implicit constant fields, which reveals the tight connection with the algebraic case. [ABSTRACT FROM AUTHOR]

Slice analysis is a generalization of the theory of holomorphic functions of one complex variable to quaternions. Among the new phenomena which appear in this context, there is the fact that the convergence domain of f(q)=Σn∈N(q-p)∗nan, given by a σ-ball Σ(p,r), is not open in H unless p∈R. This motivates us to investigate, in this article, what is a natural topology for slice regular functions. It turns out that the natural topology is the so-called slice topology, which is different from the Euclidean topology and nicely adapts to the slice structure of quaternions. We extend the function theory of slice regular functions to any domains in the slice topology. Many fundamental results in the classical slice analysis for axially symmetric domains fail in our general setting. We can even construct a counterexample to show that a slice regular function in a domain cannot be extended to an axially symmetric domain. In order to provide positive results we need to consider so-called path-slice functions instead of slice functions. Along this line, we can establish an extension theorem and a representation formula in a slice-domain. [ABSTRACT FROM AUTHOR]

In this paper, we extend and improve a common fixed point theorem of G. Jungck. We utilize the notions of weakly commuting and compatible mappings in probabilistic Banach spaces to prove some common fixed point theorems for improved type Jungck contractions. In addition, we present some examples which support our theorems. [ABSTRACT FROM AUTHOR]

The study was conducted in Kerala state in 2021-2022, with the objective of identifying the constraints related to effective role performance as perceived by the Field Extension Functionaries (FEFs) of the Dairy Development Department (DDD). Through non-proportionate stratified random sampling, a sample of 120 FEFs was chosen, of whom 60 were Dairy Extension Officers (DEOs) and 60 were Dairy Farm Instructors (DFIs). The study found that inadequate infrastructure facilities and lack of transport facilities were the major physical constraints perceived by the respondents. Among organisational constraints, inadequate staff strength and heavy administrative work were cited as the major constraints. The most significant technological constraints perceived were inadequate knowledge about dairy innovations among the farmers and lack of technical support for implementing schemes. Inadequate budget allocation for programme execution and lack of monitoring and evaluation of schemes were the major managerial constraints. Under communication constraints, weak research-extension-farmer linkage, and lack of feedback from farmers were perceived as the major ones. [ABSTRACT FROM AUTHOR]

We introduce the notions of (τ1, τ2)-interval valued Q1 neutrosophic subbisemirings (IVQ1NSBSs), level sets of a (τ1, τ2)-IVQ1NSBS, and (τ1, τ2)-interval valued Q1 neutrosophic normal subbisemirings ((τ1, τ2)-IVQ1NNSBS) of a bisemiring. Let cZ1 be a (τ1, τ2)-IVQ1NSBS of a bisemiring M and bV be the strongest(τ1, τ2)-interval valued Q1 neutrosophic relation of M. To illustrate cZ1 is a (τ1, τ2)-IVQ1NSBS of M if and only if bV is a (τ1, τ2)-IVQ1NSBS of M ⋇ M. We show that homomorphic image of (τ1, τ2)-IVQ1NSBS is again a (τ1, τ2)-IVQ1NSBS. To determine homomorphic pre-image of (τ1, τ2)-IVQ1NSBS is also a (τ1, τ2)- IVQ1NSBS. Examples are given to strengthen our results. [ABSTRACT FROM AUTHOR]

For a finite group G , a G -module M and a field F , an element u ∈ H d (G , M) is negligible over F if for each field extension L / F and every group homomorphism Gal (L sep / L) → G , u belongs to the kernel of the induced homomorphism H d (G , M) → H d (L , M). We determine the group of negligible elements in H 2 (G , M) for every abelian group M with trivial G -action. [ABSTRACT FROM AUTHOR]

Given a number field extension K/k with an intermediate field K^+ fixed by a central element of \operatorname {Gal}(K/k) of prime order p, there exists an algebraic torus over k whose rational points are elements of K^\times sent to k^\times by the norm map N_{K/K^+}. The goal is to compute the Tamagawa number such a torus explicitly via Ono's formula that expresses it as a ratio of cohomological invariants. A fairly complete and detailed description of the cohomology of the character lattice of such a torus is given when K/k is Galois. Partial results including the numerator of Ono's formula are given when the extension is not Galois, or more generally when the torus is defined by an étale algebra. We also present tools developed in SageMath for this purpose, allowing us to build and compute the cohomology and explore the local-global principles for such an algebraic torus. Particular attention is given to the case when [K:K^+]=2 and K is a CM-field. This case corresponds to maximal tori in \mathrm {GSp}_{2n}, and most examples will be in that setting. This is motivated by the application to abelian varieties over finite fields and the Hasse principle for bilinear forms. [ABSTRACT FROM AUTHOR]

Given a field k and a finite group G, the Beckmann–Black problem asks whether every Galois field extension F/k with group G is the specialization at some t_0 \in k of some Galois field extension E/k(T) with group G and E \cap \overline {k} = k. We show that the answer is positive for arbitrary k and G, if one waives the requirement that E/k(T) is normal. In fact, our result holds if \operatorname {Gal}(F/k) is any given subgroup H of G and, in the special case H=G, we provide a similar conclusion even if F/k is not normal. We next derive that, given a division ring H and an automorphism \sigma of H of finite order, all finite groups occur as automorphism groups over the skew field of fractions H(T, \sigma) of the twisted polynomial ring H[T, \sigma ]. [ABSTRACT FROM AUTHOR]

A purely inseparable field extension K of a field k of characteristic p 6= 0 is said to be !0-generated over k if K=k is not finitely generated, but L=k is finitely generated for each proper intermediate field L. In 1986, Deveney solved the question posed by R. Gilmer and W. Heinzer, which consists in knowing if the lattice of intermediate fields of an !0-generated field extension K=k is necessarily linearly ordered under inclusion, by constructing an example of an !0-generated field extension where [kp-n \ K: k] = p2n for all positive integer n. This example has proved to be extremely useful in the construction of other examples of !0-generated field extensions (of any finite irrationality degree). In this paper, we characterize the extensions of finite irrationality degree which are !0-generated. In particular, in the case of unbounded irrationality degree, any modular extension of unbounded exponent contains a proper subfield of unbounded exponent over the ground field. Finally, we give a generalization, illustrated by an example, of the !0-generated to include modular purely inseparable extensions of unbounded irrationality degree. [ABSTRACT FROM AUTHOR]

Let G be a commutative affine algebraic group over a field F , and let H : Fields F → AbGrps be a functor. A (homomorphic) H -invariant of G is a natural transformation Tors (− , G) → H , where Tors (− , G) is the functor Fields F → AbGrps taking a field extension L / F to the group of isomorphism classes of G L -torsors over Spec (L). The goal of this paper is to compute the group Inv hom 1 (G , H) of H -invariants of G when G is a group of multiplicative type, and H is the functor taking a field extension L / F to L × ⊗ Z Q / Z. [ABSTRACT FROM AUTHOR]

The relation between soft element-wise field and soft int-field has been established and then some properties of soft int-field are studied. We define the notions of soft algebraic element and soft purely inseparable element of a soft int-field extension. Some characterizations of soft algebraic and soft purely inseparable int-field extensions are given. Lastly, we define soft separable algebraic int-field extension and study some of its properties. [ABSTRACT FROM AUTHOR]

In this work we construct sequences of locally recoverable AG codes arising from a tower of function fields and give bound for the parameters of the obtained codes. In a particular case of a tower over F q 2 for any odd q , defined by Garcia and Stichtenoth in [3] , we show that the bound is sharp for the first code in the sequence, and we include a detailed analysis for the following codes in the sequence based on the distribution of rational places that split completely in the considered function field extension. [ABSTRACT FROM AUTHOR]

Hypersoft sets have gained more importance as a generalization of soft sets and have been investigated for possible extensions in many fields of mathematics. The main objective of this paper is to introduce Fuzzy Hypersoft Topology and study some of its properties such as neighbourhood of fuzzy hypersoft set, interior hypersoft set and closure fuzzy hypersoft set. Fuzzy hypersoft topology is then extended to Intuitionistic Hypersoft topology, Neutrosophic Hypersoft topology and its basic properties are discussed. [ABSTRACT FROM AUTHOR]

We use the theta lifts between Mp 2 and P D × to study the distinction problems for the pair ( Mp 2 (E) , SL 2 (F)) , where E is a quadratic field extension over a nonarchimedean local field F of characteristic zero and D is a quaternion algebra. With a similar strategy, we give a conjectural formula for the multiplicity of distinction problem related to the pair ( Mp 2 n (E) , Sp 2 n (F)). [ABSTRACT FROM AUTHOR]

We show that each real-representable matroid is a minor of a complex-representable excluded minor for real-representability. More generally, for an infinite field F 1 and a field extension F 2 , if F 1 -representability is not equivalent to F 2 -representability, then each F 1 -representable matroid is a minor of a F 2 -representable excluded minor for F 1 -representability. [ABSTRACT FROM AUTHOR]

Every Lie algebra over a field E gives rise to new Lie algebras over any subfield F ⊆ E by restricting the scalar multiplication. This paper studies the structure of these underlying Lie algebra in relation to the structure of the original Lie algebra, in particular the question how much of the original Lie algebra can be recovered from its underlying Lie algebra over subfields F. By introducing the conjugate of a Lie algebra we show that in some specific cases the Lie algebra is completely determined by its underlying Lie algebra. Furthermore we construct examples showing that these assumptions are necessary. As an application, we give for every positive n an example of a real 2-step nilpotent Lie algebra which has exactly n different bi-invariant complex structures. This answers an open question by Di Scala, Lauret and Vezzoni motivated by their work on quasi-Kähler Chern-flat manifolds in differential geometry. The methods we develop work for general Lie algebras and for general Galois extensions F ⊆ E , in contrast to the original question which only considered nilpotent Lie algebras of nilpotency class 2 and the field extension R ⊆ C. We demonstrate this increased generality by characterizing the complex Lie algebras of dimension ≤4 which are defined over R and over Q. [ABSTRACT FROM AUTHOR]

This is the fourth part of a four-article series containing a Mizar [3], [2], [1] formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker's classical proof using F [X]/ as the desired field extension E [6], [4], [5]. In the first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ F [X]/ < p > as sets, so F is not a subfield of F [X]/, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ: F → F [X]/ and show that the translated polynomial ϕ (p) has a root over F [X]/. Because F is not a subfield of F [X]/ we construct in the second part the field (E \ ϕF)∪F for a given monomorphism ϕ: F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that "one can identify F with its image ϕF in F [X]/ and therefore consider F as a subfield of F [X]/". Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker's construction can be formalized for fields F with F ∩ F [X] = ∅. Surprisingly, as we show in the third part, this condition is not automatically true for arbitrary fields F : With the exception of ℤ2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar's representations of ℤn, ℚ and ℝ we have ℤn ∩ ℤn[X] = ∅, ℚ ∩ ℚ[X] = ∅ and ℝ ∩ ℝ[X] = ∅, respectively. In this fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/ with the canonical monomorphism ϕ: F → F [X]/. Together with the first part this gives – for fields F with F ∩ F [X] = ∅ – a field extension E of F in which p ∈ F [X]\F has a root. [ABSTRACT FROM AUTHOR]

This is the third part of a four-article series containing a Mizar [3], [1], [2] formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker's classical proof using F [X]/ as the desired field extension E [6], [4], [5]. In the first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ F [X]/ < p > as sets, so F is not a subfield of F [X]/, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ: F → F [X]/ and show that the translated polynomial ϕ (p) has a root over F [X]/. Because F is not a subfield of F [X]/ we construct in the second part the field (E \ ϕF)∪F for a given monomorphism ϕ: F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that "one can identify F with its image ϕF in F [X]/ and therefore consider F as a subfield of F [X]/". Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker's construction can be formalized for fields F with F ∩ F [X] = ∅. Surprisingly, as we show in this third part, this condition is not automatically true for arbitrary fields F : With the exception of ℤ2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar's representations of ℤn, ℚ and ℝ we have ℤn ∩ ℤn[X] = ∅, ℚ ∩ ℚ[X] = ∅ and ℝ ∩ ℝ[X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/ with the canonical monomorphism ϕ: F → F [X]/. Together with the first part this gives – for fields F with F ∩ F [X] = ∅ – a field extension E of F in which p ∈ F [X]\F has a root. [ABSTRACT FROM AUTHOR]

We present a formalisation of the theory of finite fields, from basic axioms to their classification, both existence and uniqueness, in HOL4 using the notion of subfields. The tools developed are applied to the characterisation of subfields of finite fields, and to the cyclotomic factorisation of polynomials of the form , with coefficients over a finite fields. [ABSTRACT FROM AUTHOR]

We consider a polymer chain confined by a harmonic potential in θ solvents using the Zimm and Rouse model to elucidate the chain relaxation behavior in weak and strong fields, respectively. We investigate a case in which the center of the field is tuned to match the center of mass of the polymer at the instant when the field is switched on. The closed-form expressions are obtained for these models. When the field strength is weak enough so that the chain conformation is close to ideal Gaussian, the Zimm model predicts that the chain molecule would fluctuate within the confined space induced by the applied field. Moreover, the molecular rotation relaxes faster than the translational motion of the center of mass of the polymer molecule. However, under a strong field, the polymer molecule contracts continuously from a random coil to a collapsed conformation after the field is switched on. The Rouse model makes predictions that the center of mass of the confined polymer molecule would achieve its equilibrium state first. After the relaxation of the center of mass, the polymer molecule reaches the equilibrium chain conformation, followed by the molecular rotation. Furthermore, the Rouse model also predicts that in the presence of a strong field, the Rouse time is predominated by the field strength only. [ABSTRACT FROM AUTHOR]

Abstract For a finite separable field extension K / k , all subfields can be obtained by intersecting so-called principal subfields of K / k. In this work we present a way to quickly compute these intersections. If the number of subfields is high, then this leads to faster run times and an improved complexity. [ABSTRACT FROM AUTHOR]

This is the second part of a four-article series containing a Mizar [2], [1] formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker's classical proof using F [X]/ as the desired field extension E [5], [3], [4]. In the first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ [X]/ < p > as sets, so F is not a subfield of F [X]/, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ : F → F [X]/ and show that the translated polynomial ϕ (p) has a root over F [X]/. Because F is not a subfield of F [X]/ we construct in this second part the field (E \ ϕF)∪F for a given monomorphism ϕ : F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that "one can identify F with its image ϕF in F [X]/ and therefore consider F as a subfield of F [X]/". Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker's construction can be formalized for fields F with F ∩ F [X] = ∅. Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields F : With the exception of 𝕑2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar's representations of 𝕑n, 𝕈 and 𝕉 we have 𝕑n ∩ 𝕑n[X] = ∅, 𝕈 ∩ 𝕈 [X] = ∅ and 𝕉 ∩ 𝕉 [X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/ with the canonical monomorphism ϕ : F → F [X]/. Together with the first part this gives - for fields F with F ∩ F [X] = ∅ - a field extension E of F in which p ∈ F [X]\F has a root. [ABSTRACT FROM AUTHOR]

This is the first part of a four-article series containing a Mizar [3], [1], [2] formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker's classical proof using F [X]/ as the desired field extension E [9], [4], [6]. In this first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ [X]/ < p > as sets, so F is not a subfield of F [X]/, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ: F → F [X]/ and show that the translated polynomial ϕ(p) has a root over F [X]/. Because F is not a subfield of F [X]/ we construct in the second part the field (E \ ϕF)∪F for a given monomorphism ϕ : F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that "one can identify F with its image ϕF in F [X]/ and therefore consider F as a subfield of F [X]/". Interestingly, to do so we need to assume that F ∩ E =∅, in particular Kronecker's construction can be formalized for fields F with F \ F [X] =∅. Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields F : With the exception of 𝕑2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠∅. We also prove that for Mizar's representations of 𝕑n, 𝕈 and 𝕉 we have 𝕑n ∩ 𝕑n[X] = ∅, 𝕈 ∩ 𝕈[X] = ∅and 𝕉 ∩ 𝕉[X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F i F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/ with the canonical monomorphism ϕ : F → F [X]/. Together with the first part this gives - for fields F with F ∩ F [X] = ∅ - a field extension E of F in which p ∈ F [X]\F has a root. [ABSTRACT FROM AUTHOR]

The celebrated Primitive Normal Basis Theorem states that for any n ≥ 2 and any finite field F q , there exists an element α ∈ F q n that is simultaneously primitive and normal over F q . In this paper, we prove some variations of this result, completing the proof of a conjecture proposed by Anderson and Mullen (2014). [ABSTRACT FROM AUTHOR]

Abstract In this note, for p > 3 , we calculate the dimensions of Ext SL 2 (Q p) 1 (τ , σ) , for any two irreducible supersingular representations τ and σ of SL 2 (Q p). [ABSTRACT FROM AUTHOR]

Abstract We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in B α ∈ L [ y ] , where L = Q (α) is a simple algebraic extension of the rational numbers. We revisit two approaches for the problem. In the first approach, using resultant computations, we perform a reduction to a polynomial with integer coefficients and we deduce a bound of O ˜ B (N 8) for isolating the real roots of B α , where N is an upper bound on all the quantities (degree and bitsize) of the input polynomials. The bound becomes O ˜ B (N 7) if we use Pan's algorithm for isolating the real roots. In the second approach we isolate the real roots working directly on the polynomial of the input. We compute improved separation bounds for the roots and we prove that they are optimal, under mild assumptions. For isolating the real roots we consider a modified Sturm algorithm, and a modified version of descartes' algorithm. For the former we prove a Boolean complexity bound of O ˜ B (N 12) and for the latter a bound of O ˜ B (N 5). We present aggregate separation bounds and complexity results for isolating the real roots of all polynomials B α k , when α k runs over all the real conjugates of α. We show that we can isolate the real roots of all polynomials in O ˜ B (N 5). Finally, we implemented the algorithms in C as part of the core library of MATHEMATICA and we illustrate their efficiency over various data sets. [ABSTRACT FROM AUTHOR]

Abstract The cyclotomic p -adic multi-zeta values are the p -adic periods of π 1 u n i (G m ∖ μ M , ⋅) , the unipotent fundamental group of the multiplicative group minus the M -th roots of unity. In this paper, we compute the cyclotomic p -adic multi-zeta values at all depths. This paper generalizes the results in [9] and [10]. Since the main result gives quite explicit formulas we expect it to be useful in proving non-vanishing and transcendence results for these p -adic periods and also, through the use of p -adic Hodge theory, in proving non-triviality results for the corresponding p -adic Galois representations. [ABSTRACT FROM AUTHOR]

Let L=F be a Galois extension of fields with Galois group isomorphic to the quaternion group of order 8. We describe all of the Hopf-Galois structures admitted by L=F, and determine which of the Hopf algebras that appear are isomorphic as Hopf algebras. In the case that F has characteristic not equal to 2 we also determine which of these Hopf algebras are isomorphic as F-algebras and explicitly compute their Wedderburn-Artin decompositions. [ABSTRACT FROM AUTHOR]

We characterize pointwise minimal extensions of rings, introduced by Cahen et al. (Rocky Mt J Math 41:1081-1125, 2011), in the special context of domains. We show that pointwise minimal extensions are either integral or integrally closed. In the closed case, they are nothing but minimal extensions. Otherwise, there are four cases: either all minimal sub-extensions are of the same type (ramified, decomposed, or inert) or coexist as only ramified and inert minimal sub-extensions. [ABSTRACT FROM AUTHOR]

Given an indivisible field F , let D be a finite dimensional noncommutative F -central division algebra. It is shown that if GL n (D) is radicable, then D is the ordinary quaternion division algebra and F (− 1) is divisible. Also, it is shown that when F is a field of characteristic zero and m ≥ 5 , then GL m (F) is radicable if and only if for any field extension K / F with [ K : F ] ≤ m , K is divisible. [ABSTRACT FROM AUTHOR]

Slater introduced the point-addition operation on graphs to characterize 4-connected graphs. The Γ-extension operation on binary matroids is a generalization of the point-addition operation. In general, under the Γ-extension operation the properties like graphicness and cographicness of matroids are not preserved. In this paper, we obtain forbidden minor characterizations for binary matroids whose Γ-extension matroids are graphic (respectively, cographic). [ABSTRACT FROM AUTHOR]

Design of bearing layers (granular fill material layers) is important for a house with a soil depressurisation (SD) system for indoor radon mitigation. These layers should not only satisfy the bearing capacity and serviceability criteria but should also provide a sufficient degree of the air permeability for the system. Previous studies have shown that a critical parameter for a SD system is the sub-slab pressure field extension in the bearing layers, but this issue has not been systematically investigated. A series of two-dimensional computational fluid dynamic simulations that investigate the behaviour of the sub-slab pressure field extension developed in a SD system is presented in this paper. The SD system considered in this paper consists of a granular fill material layer and a radon sump. The granular fill materials are ‘T1 Struc’ and ‘T2 Perm’, which are standard materials for building in the Republic of Ireland. Different conditions, which might be encountered in a practical situation, were examined. The results show that the air permeability and thickness of the granular fill materials are the two key factors which affect the sub slab pressure field extension (SPFE) significantly. Furthermore, the air permeability of native soil is found to be a fundamental factor for the SPFE so that it should be well understood when designing a SD system. Therefore, these factors should be considered sufficiently in each practical situation. Finally, a significant improvement of the pressure field extension can be achieved by ensuring air tightness of the SD system. [ABSTRACT FROM AUTHOR]

As a proper generalization of the modules with the properties (E) and (EE) that were introduced by Zöschinger in terms of supplements, we say that a module M has the property (WRE) (respectively, (WREE)) if M has a weak Rad-supplement (respectively, ample weak Rad-supplements) in every extension. In this paper, we prove that if every submodule of a module M has the property (WRE), then M has the property (WREE). We show that a ring R is semilocal if and only if every left R-module has the property (WRE). Also we prove that over a commutative Von Neumann regular ring a module M has the property (WRE) if and only if M is injective. [ABSTRACT FROM AUTHOR]

We generalize Amitsur's construction of central simple algebras over a field F which are split by field extensions possessing a derivation with field of constants F to nonassociative algebras: for every central division algebra D over a field F of characteristic zero there exists an infinite-dimensional unital nonassociative algebra whose right nucleus is D and whose left and middle nucleus are a field extension K of F splitting D , where F is algebraically closed in K . We then give a short direct proof that every p -algebra of degree m , which has a purely inseparable splitting field K of degree m and exponent one, is a differential extension of K and cyclic. We obtain finite-dimensional division algebras over a field F of characteristic p > 0 whose right nucleus is a division p -algebra. [ABSTRACT FROM AUTHOR]

The main problem of constant-dimension subspace coding is to determine the maximal possible size ${\mathbf{A}}_{q}(n,d,k)$ of a set of $k$ -dimensional subspaces in ${\mathbf{F}}_{q}^{n}$ such that the subspace distance satisfies $d(U,V) \geq d$ for any two different subspaces $U$ and $V$ in this set. In this paper, we give a direct construction of constant-dimension subspace codes from two parallel versions of maximum rank-distance codes. The problem about the sizes of our constructed constant-dimension subspace codes is transformed into finding a suitable sufficient condition to restrict number of the roots of $L_{1}(L_{2}(x))-x$ where $L_{1}$ and $L_{2}$ are $q$ -polynomials over the extension field ${\mathbf{F}}_{q^{n}}$. New lower bounds for ${\mathbf{A}}_{q}(4k,2k,2k)$ , ${\mathbf{A}}_{q}(4k+2,2k,2k+1)$ , and ${\mathbf{A}}_{q}(4k+2,2(k-1),2k+1)$ are presented. Many new constant-dimension subspace codes better than previously best known codes with small parameters are constructed. [ABSTRACT FROM AUTHOR]

Let E / F be an extension of fields. We investigate the Steinberg module of G L ( n , E ) restricted to G L ( n , F ) . We give examples for n = 2 and n = 3 where we compute the homology of a subgroup of G L ( n , F ) with coefficients in the Steinberg module of G L ( n , E ) . [ABSTRACT FROM AUTHOR]

In this paper, we prove that some recent Rotfel’d theorem extensions are equivalent. [ABSTRACT FROM AUTHOR]

A Theorem of Hou, Leung and Xiang generalised Kneser’s addition Theorem to field extensions. This theorem was known to be valid only in separable extensions, and it was a conjecture of Hou that it should be valid for all extensions. We give an alternative proof of the theorem that also holds in the non-separable case, thus solving Hou’s conjecture. This result is a consequence of a strengthening of Hou et al.’s theorem that is inspired by an addition theorem of Balandraud and is obtained by combinatorial methods transposed and adapted to the extension field setting. [ABSTRACT FROM AUTHOR]

Iwasawa showed that there are non-cyclotomic Z p -extensions with positive μ -invariant. We show that these μ -invariants can be evaluated explicitly in many situations when p = 2 and p = 3 . [ABSTRACT FROM AUTHOR]