We assume F is a purely inseparable field extension of the field K. The characteristic of K is p ≠ 0, and we assume F and K are not perfect. For x ∈ F, the exponent of x over K is the smallest

non-negative integer e such that and will be denoted by

e (x); ⨱ will denote . For any subset S of F, e(x; S) will denote the exponent of x over K(S); in case S = {y} we will write e(x; y) for e(x; S).

In this paper, we solved an open problem raised by Cecil and Ryan (2015, Geometry of Hypersurfaces, Springer Monographs in Mathematics, Springer, New York, p. 531) by proving the nonexistence of non-Hopf Ricci-semisymmetric real hypersurfaces in $\mathbb {C}P^{2}$ and $\mathbb {C}H^{2}$.

In this note we use functional analysis arguments to prove the existence of families of analytic solutions for the singular system of complex ordinary differential equations zW′ = h(z, W).

An epimorphism $\phi :\,G\,\to \,H$ of groups, where $G$ has rank $n$, is called coessential if every (ordered) generating $n$-tuple of $H$ can be lifted along $\phi $ to a generating $n$-tuple for $G$. We discuss this property in the context of the category of groups, and establish a criterion for such a group $G$ to have the property that its abelianization epimorphism $G\,\to \,{G}/{[G,G]}\;$, where $[G,\,G]$ is the commutator subgroup, is coessential. We give an example of a family of 2-generator groups whose abelianization epimorphism is not coessential. This family also provides counterexamples to the generalized Andrews–Curtis conjecture.

A Lidstone series provides a (formal) two-point expansion of a given function f(x) in terms of its derivatives of even order at the nodes 0 and 1 and takes the form.

We provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an $n$-dimensional flat torus ${{\mathbb{T}}^{n}}$, and the Fourier transform of squares of the eigenfunctions ${{\left| \varphi \lambda \right|}^{2}}$ of the Laplacian have uniform ${{l}^{n}}$ bounds that do not depend on the eigenvalue $\lambda $. The proof is a generalization of an argument by Jakobson, et al. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on ${{\mathbb{T}}^{n+2}}$. We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an $n$-dimensional sphere ${{S}^{n}}\,\left( \text{ }\!\!\lambda\!\!\text{ } \right)$ of radius $\sqrt{\lambda }$, and we use it in the proof.

Given any homogeneous complex manifold X = G/H, there exists a natural coset map π :G/H → G/K satisfying π (X1) = π (x2) if and only if dx(x1 x2) = 0, where dx denotes the Kobayashi pseudometric on X. Its typical fiber Z : = K/H is a connected complex submanifold of X. Also G/K has a (7-invariant complex structure, provided K satisfies a certain technical assumption (see Theorem 3). If Z is compact as well, then G/K is biholomorphic to a homogeneous bounded domain.

Let X be a topological space. We say that cat X ≤ n if there exists a map ϕ: X → T1(X, …, X) such that jϕ≃Δ: X → Xn+1, where T1(X, …, X) is the “fat wedge”, j is the inclusion and Δ is the diagonal map. This is an example of a right structure system. This right structure system leads to an associated weak structure system, namely weak category in this particular case.

We prove that every topologically stable homeomorphism with global attractor of $\mathbb {R}^n$ is topologically stable on its global attractor. The converse is not true. On the other hand, if a homeomorphism with global attractor of a locally compact metric space is expansive and has the shadowing property, then it is topologically stable. This extends the Walters stability theorem (Walters, On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems, 1978, pp. 231–244).

In this note we prove:

Theorem. Let R be a right primitive ring with pair-wise non-isomorphic faithful irreducible modules Ml, M2,…, Mk. Let Di = EndR Mi. For each i, let be elements of Mi linearly independent over Di. For each i, let be a set of elements of Mi. Then there exists an element r of R such that uij = vijr, for i = 1, 2, …, k and j = 1, 2, …, ni.

We show that an $\text{RA}$ loop has a torsion-free normal complement in the loop of normalized units of its integral loop ring. We also investigate whether an $\text{RA}$ loop can be normal in its unit loop. Over fields, this can never happen.

For certain real quadratic number fields, we prove density results concerning 4-ranks of tame kernels. We also discuss a relationship between 4-ranks of tame kernels and 4-class ranks of narrow ideal class groups. Additionally, we give a product formula for a local Hilbert symbol.

A commutative semigroup G of continuous, selfmappings on (X, d) is called proximately nonexpansive on X if for every x in X and every (β > 0, there is a member g in G such that d(fg(x),fg(y)) ≤ (1 + β) d (x, y) for every f in G and y in X. For a uniformly convex Banach space it is shown that if G is a commutative semigroup of continuous selfmappings on X which is proximately nonexpansive, then a common fixed point exists if there is an x0 in X such that its orbit G(x0) is bounded. Furthermore, the asymptotic center of G(x0) is such a common fixed point.

Let B be a Brownian motion on R, B(0) = 0, and let f (t, x) be continuous. T. Salisbury conjectured that if the total variation of f (t, B(t)), 0 ≤ t ≤ 1, is finite P-a.s., then f does not depend on x. Here we prove that this is true if the expected total variation is finite.

Associated with any closed quantum subgroup $G\,\subset \,U_{N}^{+}$ and any index set $I\,\subset \,\{1,\,.\,.\,.\,,\,N\}$ is a certain homogeneous space ${{X}_{G,I}}\subset S_{\mathbb{C},+}^{N-1},$ called an affine homogeneous space. Using Tannakian duality methods, we discuss the abstract axiomatization of the algebraic manifolds $X\subset S_{\mathbb{C},+}^{N-1}$ that can appear in this way.

We extend results of P. M. Gauthier, W. Hengartner and A. A. Nersesyan on simultaneous approximation and interpolation on Arakelian sets.

In the study of algorithms for the iterative solution of an arbitrary analytic equation f(z) = 0, acceleration procedures are of importance in practice and of considerable interest in the theory of the subject. Let

.

be an iteration formula which has a zero ξ of f(z) as attractive fixed point. An algorithm of this type is said to converge towards a root ξ of f(z) = 0 for all initial approximations z = z0 in a vicinity of ξ, of order k > 0, when

1.

Unique global strong solutions of a Cauchy problem arising in electrophoretic separation are constructed with arbitrary initial data in L1, thus generalizing an earlier global existence result. For small diffusion coefficients, the solutions can be viewed as approximate solutions for the corresponding zero-diffusion Riemann problem.

The main result of this paper is a theorem about three conies in the complex or the real complexified projective plane. Is this theorem new? We have never seen it anywhere before. But since the golden age of projective geometry so much has been published about conies that it is unlikely that no one noticed this result. On the other hand, why does it not appear in the literature? Anyway, it seems interesting to "repeat" this property, because several theorems in connection with straight lines and (or) conies in projective, affine or euclidean planes are in fact special cases of this theorem. We give a few classical examples: the theorems of Pappus-Pascal, Desargues, Pascal (or its converse), the Brocard points, the point of Miquel. Finally, we have never seen in the literature a proof of these theorems using the same short method see the proof of the main theorem).