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
A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D_{1} and D_{2} such that A^{-T} = D_{1}AD_{2}, where A^{-T} denotes the transpose of the inverse of A. Denote by J = diag(±1) a diagonal (signature) matrix, each of whose diagonal entries is +1 or −1. A nonsingular real matrix Q is called J-orthogonal if Q^{T} JQ = J. Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation of a J-orthogonal matrix. An investigation into the sign patterns of the J-orthogonal matrices is initiated. It is observed that the sign patterns of the G-matrices are exactly the column permutations of the sign patterns of the J-orthogonal matrices. Some interesting constructions of certain J-orthogonal matrices are exhibited. It is shown that every symmetric staircase sign pattern matrix allows a J-orthogonal matrix. Sign potentially J-orthogonal conditions are also considered. Some examples and open questions are provided., Frank J. Hall, Miroslav Rozložník., and Obsahuje seznam literatury
We derive two identities for multiple basic hyper-geometric series associated with the unitary U(n+1) group. In order to get the two identities, we first present two known q-exponential operator identities which were established in our earlier paper. From the two identities and combining them with the two U(n + 1) q-Chu-Vandermonde summations established by Milne, we arrive at our results. Using the identities obtained in this paper, we give two interesting identities involving binomial coefficients. In addition, we also derive two nontrivial summation equations from the two multiple extensions., Jian-Ping Fang., and Obsahuje seznam literatury
In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point x in a metric measure space (X,d,μ) is called a generalized Lebesgue point of a measurable function f if the medians of f over the balls B(x,r) converge to f(x) when r converges to 0. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function. We show that a function f\in M{s,p}(X),0<s<1,0<p<1, where X is a doubling metric measure space, has generalized Lebesgue points outside a set of Hh-Hausdorff measure zero for a suitable gauge function h., Nijjwal Karak., and Obsahuje bibliografii
We study classifying problems of real hypersurfaces in a complex two-plane Grassmannian G_{2} (\mathbb{C}^{m+2}). In relation to the generalized Tanaka-Webster connection, we consider that the generalized Tanaka-Webster derivative of the normal Jacobi operator coincides with the covariant derivative. In this case, we prove complete classifications for real hypersurfaces in G_{2} (\mathbb{C}^{m+2})satisfying such conditions., Eunmi Pak, Juan de Dios Pérez, Young Jin Suh., and Obsahuje seznam literatury
A classification of dihedral folding tessellations of the sphere whose prototiles are a kite and an equilateral or isosceles triangle was obtained in recent four papers by Avelino and Santos (2012, 2013, 2014 and 2015). In this paper we extend this classification, presenting all dihedral folding tessellations of the sphere by kites and scalene triangles in which the shorter side of the kite is equal to the longest side of the triangle. Within two possible cases of adjacency, only one will be addressed. The combinatorial structure of each tiling is also analysed., Catarina P. Avelino, Altino F. Santos., and Obsahuje bibliografii
We study some geometric properties associated with the t-geometric means A ♯_{t} B:= A^{1/2}(A^{-1/2}BA^{-1/2})^{t} A^{1/2}of two n × n positive definite matrices A and B. Some geodesical convexity results with respect to the Riemannian structure of the n × n positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding m pairs of positive definite matrices is posted., Trung Hoa Dinh, Sima Ahsani, Tin-Yau Tam., and Obsahuje seznam literatury
We consider a convexity notion for complex spaces X with respect to a holomorphic line bundle L over X. This definition has been introduced by Grauert and, when L is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if H0(X,L) separates each point of X, then X can be realized as a Riemann domain over the complex projective space Pn, where n is the complex dimension of X and L is the pull-back of O(1)., Viorel Vâjâitu., and Obsahuje seznam literatury
As a generalization of anti-invariant Riemannian submersions and Lagrangian Riemannian submersions, we introduce the notions of h-anti-invariant submersions and h-Lagrangian submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We obtain characterizations and investigate some properties: the integrability of distributions, the geometry of foliations, and the harmonicity of such maps. We also find a condition for such maps to be totally geodesic and give some examples of such maps. Finally, we obtain some types of decomposition theorems., Kwang-Soon Park., and Seznam literatury