We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation x(t)=L(t)x(t)+f(t,x(t)), t R where {L(t) in R}$ is a family of linear operators from a Banach space E into itself and f R E to E. By L(E) we denote the space of linear operators from E into itself. Furthermore, for a<b and d>0, we let C([-d,0],E) be the Banach space of continuous functions from [-d,0] into E and f^d [a,b] C([-d,0],E) E. Let L: [a,b] to L(E) be a strongly measurable and Bochner integrable operator on [a,b] and for t in [a,b] define tau_tx(s)=x(t+s) for each s in[-d,0]. We prove that, under certain conditions, the differential equation with delay x(t)=L(t)x(t)+f^d(t,tau_tx) if t in [a,b], Q has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too., Adel Mahmoud Gomaa., and Seznam literatury
We consider inequalities between sums of monomials that hold for all p-Newton sequences. This continues recent work in which inequalities between sums of two, two-term monomials were combinatorially characterized (via the indices involved). Our focus is on the case of sums of three, two-term monomials, but this is very much more complicated. We develop and use a theory of exponential polynomial inequalities to give a sufficient condition for general monomial sum inequalities, and use the sufficient condition in two ways. The sufficient condition is necessary in the case of sums of two monomials but is not known if it is for sums of more. A complete description of the desired inequalities is given for Newton sequences of less than 5 terms., Charles R. Johnson, Carlos Marijuán, Miriam Pisonero, Michael Yeh., and Obsahuje seznam literatury
We study (non-abelian) extensions of a given hom-Lie algebra and provide a geometrical interpretation of extensions, in particular, we characterize an extension of a hom-Lie algebra g by another hom-Lie algebra h and discuss the case where h has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, Maurer-Cartan formula, curvature and the Bianchi identity for the possible extensions in differential geometry. Moreover, we find a cohomological obstruction to the existence of extensions of hom-Lie algebras, i.e., we show that in order to have an extendible hom-Lie algebra, there should exist a trivial member of the third cohomology., Abdoreza R. Armakan, Mohammed Reza Farhangdoost., and Seznam literatury
A non-regular primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. Let S be a nontrivial finite regular linear space and G ≤ Aut(S). Suppose that G is extremely primitive on points and let rank(G) be the rank of G on points. We prove that rank(G) ≥ 4 with few exceptions. Moreover, we show that Soc(G) is neither a sporadic group nor an alternating group, and G = PSL(2, q) with q + 1 a Fermat prime if Soc(G) is a finite classical simple group., Haiyan Guan, Shenglin Zhou., and Obsahuje seznam literatury
A symmetric positive semi-definite matrix A is called completely positive if there exists a matrix B with nonnegative entries such that A = BB^{\top }. If B is such a matrix with a minimal number p of columns, then p is called the cp-rank of A. In this paper we develop a finite and exact algorithm to factorize any matrix A of cp-rank 3. Failure of this algorithm implies that A does not have cp-rank 3. Our motivation stems from the question if there exist three nonnegative polynomials of degree at most four that vanish at the boundary of an interval and are orthonormal with respect to a certain inner product., Jan Brandts, Michal Křížek., and Obsahuje seznam literatury
The structure of the group (\mathbb{Z}/n\mathbb{Z})* and Fermat’s little theorem are the basis for some of the best-known primality testing algorithms. Many related concepts arise: Euler’s totient function and Carmichael’s lambda function, Fermat pseudoprimes, Carmichael and cyclic numbers, Lehmer’s totient problem, Giuga’s conjecture, etc. In this paper, we present and study analogues to some of the previous concepts arising when we consider the underlying group G_{n}:=\left \{ a+bi\in \mathbb{Z}\left [ i\right ]:a^{2}+b^{2}\equiv 1\left ( mod n \right ) \right \}. In particular, we characterize Gaussian Carmichael numbers via a Korselt’s criterion and present their relation with Gaussian cyclic numbers. Finally, we present the relation between Gaussian Carmichael number and 1-Williams numbers for numbers n ≡ 3 (mod 4). There are also no known composite numbers less than 1018 in this family that are both pseudoprime to base 1 + 2i and 2-pseudoprime., José María Grau, Antonio M. Oller-Marcén, Manuel Rodríguez, Daniel Sadornil., and Obsahuje seznam literatury
We introduce the concept of modified vertical Weil functors on the category \mathcal{F}\mathcal{M}_m of fibred manifolds with m-dimensional bases and their fibred maps with embeddings as base maps. Then we describe all fiber product preserving bundle functors on \mathcal{F}\mathcal{M}_m in terms of modified vertical Weil functors. The construction of modified vertical Weil functors is an (almost direct) generalization of the usual vertical Weil functor. Namely, in the construction of the usual vertical Weil functors, we replace the usual Weil functors T^{a} corresponding to Weil algebras A by the so called modified Weil functors T^{a} corresponding to Weil algebra bundle functors A on the category \mathcal{F}\mathcal{M}_m of m-dimensional manifolds and their embeddings., Włodzimierz M. Mikulski., and Obsahuje seznam literatury
We study the relations between finitistic dimensions and restricted injective dimensions. Let R be a ring and T a left R-module with A = EndRT. If RT is selforthogonal, then we show that rid \left ( T_{A} \right )\leqslant findim(A_{A})\leqslant findim \left ( _{R}T \right )+rid\left ( T_{A} \right ). Moreover, if R is a left noetherian ring and T is a finitely generated left R-module with finite injective dimension, then rid \left ( T_{A} \right )\leqslant findim(A_{A})\leqslant fin.inj.dim \left ( _{R}R \right )+rid\left ( T_{A} \right ). Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension., Dejun Wu., and Obsahuje seznam literatury
We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak σ-additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a σ-additive term-we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures., Dariusz Idczak., and Obsahuje bibliografii
By introducing polynomials in matrix entries, six determinants are evaluated which may be considered extensions of Vandermonde-like determinants related to the classical root systems., Wenchang Chu., and Obsahuje bibliografii