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In a previous paper, we introduced the notion of magnetic vector fields. More precisely, we consider a vector field ? as a map from a Riemannian manifold into its tangent bundle endowed with the usual almost K?hlerian structure and we find necessary and sufficient conditions for ? to be a magnetic map with respect to ? itself and the K?hler 2-form. In this paper we give new examples of magnetic vector fields.

In the last paper, we introduced a new curvaturlike tensor field in an almost contact Riemannian manifold and we showed some geometrical properties of this tensor field in a Kenmotsu and a Sasakian manifold. In this paper, we define another new curvaturelike tensor field, named (CHR)3-curvature tensor in an almost contact Riemannian manifold which is called a contact holomorphic Riemannian curvature tensor of the second type. Then, using this tensor, we mainly research (CHR)3-curvature tensor in a Sasakian manifold. Then we define the notion of the flatness of a (CHR)3-curvature tensor and we show that a Sasakian manifold with a flat (CHR)3-curvature tensor is flat. Next, we introduce the notion of (CHR)3-?-Einstein in an almost contact Riemannian manifold. In particular, we show that Sasakian (CHR)3- ?-Einstein manifold is ?-Einstain. Moreover, we define the notion of (CHR)3- space form and consider this in a Sasakian manifold. Finally, we consider a conformal transformation of an almost contact Riemannian manifold and we get new invariant tensor fields (not the conformal curvature tensor) under this transformation. Finally, we prove that a conformally (CHR)3-flat Sasakian manifold does not exist.

We study geometric and algebraic approaches to classification problems of differential equations. We consider the so-called Lie problem: provide the point classification of ODEs y?? = F(x, y). In the first part of the paper we consider the case of smooth right-hand side F. The symmetry group for such equations has infinite dimension, so classical constructions from the theory of differential invariants do not work. Nevertheless, we compute the algebra of differential invariants and obtain a criterion for the local equivalence of two ODEs y?? = F(x, y). In the second part of the paper we develop a new approach to the study of subgroups in the Cremona group. Namely, we consider class of differential equations y?? = F(x, y) with rational right hand sides and its symmetry group. This group is a subgroup in the Cremona group of birational automorphisms of C2, which makes it possible to apply for their study methods of differential invariants and geometric theory of differential equations. Also, using algebraic methods in the theory of differential equations we obtain a global classification for such equations instead of local classifications for such problems provided by Lie, Tresse and others.

In this paper we extend some results from our earlier papers on wave-front sets, concerning wave-front sets of Fourier-Lebesgue and modulation space types, to a broader class of spaces of ultradistributions, and relate these wave-front sets with the usual wave-front sets of ultradistributions. Furthermore, we use Gabor frames for the description of discrete wave-front sets, and prove that these wave-front sets coincide with corresponding continuous ones., Comment: 28 pages

Reinhold Baer asked the relationship between certain properties in a nonempty set P with a partial operation (called an ?add" by Baer [1]). The first paper in our sequence [Paper I] answered his question for a special type of an add called a pregroup by Stallings [12]. This paper [Paper II] answers an analogous question for a wider class of adds.

We introduce and systematically analyze the class of convoluted C-groups unifying the well known classes of fractionally integrated groups and C-regularized groups. We relate convoluted C-groups to analytic convoluted C-semigroups and present illustrative examples of differential operators which generate exponentially bounded convoluted groups. 1. Introduction and preliminaries The theory of convoluted C-semigroups is an attractive field of research of many authors and becomes inevitable in the analysis of various kinds of ill-posed abstract Cauchy problems in a Banach space setting. Local convoluted C-semigroups were introduced and studied in the papers of Cior˘ anescu and Lumer (11)-(13) as a unification concept for local integrated semi- groups and local C-semigroups ((1), (48)). The class of exponentially bounded con- voluted semigroups was introduced independently by Keyantuo, Muller and Vieten (22) and the author (27) while global convoluted semigroups which are not neces- sarily exponentially bounded have been recently analyzed in (27) and (29). The important researches of Baumer, Lumer and Neubrander ((7)-(8), (40)) are related to the use of asymptotic Laplace transform techniques in the theory of convoluted semigroups. Of importance is also to stress that K-regularized resolvent families, analyzed by Lizama and his collaborators ((37)-(39)), allow one to consider in a unified treatment the notions of convoluted semigroups and cosine functions as well as to enquire into the abstract Volterra equations of convolution type. On the other hand, global integrated groups (cf. (2), (4)-(6), (15)-(16), (20)- (21), (28) and (42)) were introduced and investigated by El-Mennaoui in his doc- toral dissertation (16). We especially refer the reader to the paper (21) where Keyantuo briefly considered an abstract Laplacian in L p (R n )-type spaces as well as the relations between exponentially bounded integrated cosine functions and global integrated groups. Further study of global α-times integrated groups with

In this paper we mainly survey results obtained in (MM3). For example, we give an elementary proof of two versions of Koebe 1/ 4t heorem for analytic functions (see Theorem 1.2 and Theorem 1.4 below). We also show a version of the Koebe theorem for quasiregular harmonic functions. As an application, we show that holomorphic functions (more generally quasiregular harmonic functions) and their modulus have similar behavior in a certain sense. 1. Two versions of Koebe 1/4 theorem for analytic functions This paper can be considered as the review of some results presented in (MM3), but it also contains new results and proofs. We will use the following notation. If r> 0a nda is a complex number B(a; r )= {z ∈ C : |z − a

We consider a class of functional equations with one operational symbol which is assumed to be a quasigroup. Equations are quadratic, level and have four variables each. Therefore, they are of the form x1x2 · x3x4 = x5x6 · x7x8 with xi ∈{ x,y,u,v} (1 i 8) with each of the variables occurring exactly twice in the equation. There are 105 such equations. They separate into 19 equivalence classes defining 19 quasigroup varieties. The paper (partially) generalizes the results of some recent papers of Forg-Rob and Krapez, and Polonijo. 1. Quasigroups One way to define a quasigroup is that it is an algebra (S; ·, \ ,/ ) with three binary operations - multiplication (·), left (\ )a nd right (/) division, satisfying the axioms: x\xy = yx (x\y )= y xy/y = x (x/y)y = x. Very often we say that the operation · is a quasigroup assuming the underlying base set S and the division operations. As usual, whenever unambiguous, the terms like x · y and f (x) are shortened to xy and fx respectively. We review a few basic facts on quasigroups. More can be found in standard references: Belousov (2), Pflugfelder (20), Chein, Pflugfelder and Smith (6). A loop is a quasigroup with unit (e), which is a value of constant terms (x\x and y/y) from the additional axiom: (u) x\x = y/y.

In the paper Free biassociative groupoids, the variety of biasso- ciative groupoids (i.e., groupoids satisfying the condition: every subgroupoid generated by at most two elements is a subsemigroup) is considered and free objects are constructed using a chain of partial biassociative groupoids that satisfy certain properties. The obtained free objects in this variety are not canonical. By a canonical groupoid in a variety V of groupoids we mean a free groupoid (R, ∗ )i nV with a free basis B such that the carrier R is a subset of the absolutely free groupoid (TB, ·) with the free basis B and (tu ∈ R ⇒ t,u ∈ R & t ∗u = tu). In the present paper, a canonical description of free objects in the variety of biassociative groupoids is obtained. 1. Preliminaries Let G =( G, ·) be a groupoid and a, b ∈ G. We denote bya, bthe subgroupoid of G generated by a, b and byathe subgroupoid generated by a. Clearly, � a �⊆ � a, band if b ∈� a� , thena, b� = � a� ; specially, � a, a� = � a� . The subgroupoids � a, bandb, aare equal. Let a1 ,a 2 ,...,a n be a finite sequence of elements in a groupoid G. We denote by a1a2 ··· an the product of the sequence a1 ,a 2 ,...,a n in G defined as follows: i) if n = 3, then a1a2a3 def = a1(a2a3 )a nd ii) if n 3, then a1a2 ··· an def = a1(a2 ··· an). We call a1a2 ··· an the main product of the sequence a1 ,a 2 ,...,a n .I fn =1a nd n = 2, then a1 and a1a2 will also be called the main products of the sequences a1 and a1 ,a 2 respectively. If c = a1a2 ··· an, then we say that c is presented as a main product of the sequence a1 ,a 2 ,...,a n. Let G be a groupoid and A ⊆ G.I fQ is the subgroupoid of G generated by A, i.e., Q = � A� , then Q = {Ak : k 0}, where A0 = A, Ak+1 = Ak ∪ AkAk. If x ∈ Q, then a hierarchy of x in Q is the nonnegative integer χQ(x), defined by χQ(x) = min{k ∈ N0 : x ∈ Ak}, where N0 is the set of nonnegative integers.

We consider an equation with left and right fractional derivatives and with the boundary condition y(0) = lim x→0+ y(x )=0 ,y(b) = lim x→by(x )=0 in the space L 1 (0,b) and in the subspace of tempered distributions. The asymptotic behavior of solutions in the end points 0 and b have been specially analyzed by using Karamata's regularly varying functions. In the last years differential equations of fractional orders have been used in many branches of mechanics and physics. Many results have been published with concrete problems solved in classical spaces of functions and in the spaces of gen- eralized functions. We cite only some of them, recently published or with a new approach: (2)-(4), (7), (8), (13), (15), (17), (19), (20), (22), (23) and with Karamata's regularly varying functions: (11), (24). In this paper we treat such an equation with the boundary condition y(0) = y(b) = 0 in the space L 1 (0 ,b ) and in a subspace of tempered distributions constructed for this problem. We specially discussed as- ymptotic behavior of solutions in the end points 0 and b using Karamata's regularly varying functions and quasi-asymptotics in the space of tempered distributions. As far as we are aware the equation treated in this paper has been solved only in (1) and (18) in some very special cases.

This paper is dedicated to the memory of Tatjana Ostrogorski and also to our co-author Aleksandras Baltrunas who died during the preparation of this paper. Both were infinite dimensional mathematicians and both unfortunately died too young. Abstract. A distribution function F on the nonnegative halfline is called subexponential if limx→∞(1�F ∗n (x))/(1�F (x)) = n for all n 2. We obtain new sufficient conditions for subexponential distributions and related classes of distribution functions. Our results are formulated in terms of the hazard rate. We also analyze the rate of convergence in the definition and discuss the asymptotic behaviour of the remainder term Rn(x )=1 �F ∗n (x)�n(1�F (x)). We use the results in studying subordinated distributions and we conclude the paper with some multivariate extensions of our results.

We partially describe minimal, first-order structures which have a strong form of the strict order property. An infinite first-order structure is minimal if its each definable (possibly with parameters) subset is either finite or co-finite. It is strongly minimal if the mini- mality is preserved in elementarily equivalent structures. While strongly minimal structures were investigated more closely in a number of papers beginning with (4) and (1), there are a very few results on minimal but not strongly minimal structures. For some examples see (2) and (3). In this paper we shall consider minimal, ordered structures. A first-order struc- ture M0 = (M0;:::) is ordered if there is a binary relation < on M0, which is definable possibly with parameters from M0, irreflexive, antisymmetric, transitive and has arbitrarily large finite chains. We usually distinguish (one) such relation by absorbing the involved parameters into the language and assuming that < is an interpretation of a relation symbol from the language, in which case we write

We consider metrics on the set of elliptic curves in short Weierstrass form over a finite field of characteristic greater than three. The metrics have been first found by Mishra and Gupta (2008). Vetro (2011) constructs other metrics which are independent on the choice of a generator of the multiplicative group of the underlying finite field, whereas the metrics found by Mishra and Gupta, are dependent on the choice of a generator of the multiplicative group of the underlying finite field. Hakuta (2015, 2018) constructs metrics on the set of non-supersingular elliptic curves in shortWeierstrass form over a finite field of characteristic two and three, respectively. The aim of this paper is to point out that the metric found by Mishra and Gupta is in fact not a metric. We also construct new metrics which are slightly modified versions of the metric found by Mishra and Gupta.

The formula for the sums of powers of positive integers, given by Faulhaber in 1631, is proven by using trigonometric identities and some properties of the Bernoulli polynomials. Using trigonometric functions identities and generating functions for some well-known special numbers and polynomials, many novel formulas and relations including alternating sums of powers of positive integers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Fubini numbers, the Stirling numbers, the tangent numbers are also given. Moreover, by applying the Riemann integral and p-adic integrals involving the fermionic p-adic integral and the Volkenborn integral, some new identities and combinatorial sums related to the aforementioned numbers and polynomials are derived. Furthermore, we serve up some revealing and historical remarks and observations on the results of this paper.

We continue the study of single and multiple q-Eulerian integrals in the spirit of Exton, Driver, Johnston, Pandey, Saran and Erd?lyi. The method of proof is often the q-beta integral method with the correct q-power together with the q-binomial theorem. By the Totov method we can prove summation theorems as special cases of multiple q-Eulerian integrals. The Srivastava ? notation for q-hypergeometric functions is used to enable the shortest possible form of the long formulas. The various q-Eulerian integrals are in fact meromorphic continuations of the various multiple q-functions, suitable for numerical computations. In the end of the paper a generalization of the q-binomial theorem is used to find q-analogues of a multiple integral formulas for q-Kamp? de F?riet functions.

A strong triangle blocking arrangement is a geometric arrangement of some line segments in a triangle with certain intersection properties. It turns out that they are closely related to blocking sets. Our aim in this paper is to prove a classification theorem for strong triangle blocking arrangements. As an application, we obtain a new proof of the result of Ackerman, Buchin, Knauer, Pinchasi and Rote which says that $n$ points in general position cannot be blocked by $n-1$ points, unless $n = 2,4$. We also conjecture an extremal variant of the blocking points problem., Comment: 22 pages, 16 figures

Let R be a commutative ring and M a Noetherian R-module. The zero-divisor graph of M, denoted by ?(M), is an undirected simple graph whose vertices are the elements of ZR(M)\AnnR(M) and two distinct vertices a and b are adjacent if and only if abM = 0. In this paper, we study diameter and girth of ?(M). We show that the zero-divisor graph of M has a universal vertex in ZR(M)\r(AnnR(M)) if and only if R = ?Z2?R? and M = Z2?M?, where M? is an R?-module. Moreover, we show that if ?(M) is a complete graph, then one of the following statements is true: (i) AssR(M) = {m1,m2}, where m1,m2 are maximal ideals of R. (ii) AssR(M) = {p}, where p 2 ? AnnR(M). (iii) AssR(M) = {p}, where p 3 ? AnnR(M).

As a generalization of well-known result due to Turan [24] for polynomials having all their zeros in |z| ? 1, Malik [17] proved that, if P(z) is a polynomial of degree n, having all its zeros in |z| ? 1, then for any ? > 0, n{?2?0|P(ei?)|?d?}1/? ? {?2?0|1+ei?|?d?}1/? max |z|=1 |P?(z)|. We generalize the above inequality to polar derivatives, which as special cases include several known results in this area. Besides the paper contains some more results that generalize and sharpen several results known in this direction.

In $n$-dimensional hyperbolic space $\mathbf{H}^n$ $(n\ge2)$ there are $3$-types of spheres (balls): the sphere, horosphere and hypersphere. If $n=2,3$ we know an universal upper bound of the ball packing densities, where each ball volume is related to the volume of the corresponding Dirichlet-Voronoi (D-V) cell. E.g. in $\mathbf{H}^3$ the densest horoball packing is derived from the $\{3,3,6\}$ Coxeter tiling consisting of ideal regular simplices $T_{reg}^\infty$ with dihedral angles $\frac{\pi}{3}$. The density of this packing is $\delta_3^\infty\approx 0.85328$ and this provides a very rough upper bound for the ball packing densities as well. However, there are no "essential" results regarding the "classical" ball packings with congruent balls, and for ball coverings either. The goal of this paper to find the extremal ball arrangements in $\mathbf{H}^3$ with "classical balls". We consider only periodic congruent ball arrangements (for simplicity) related to the generalized, so-called {\it complete Coxeter orthoschemes} and their extended groups. In Theorems 1.1-1.2 we formulate also conjectures for the densest ball packing with density $0.77147\dots$ and the loosest ball covering with density $1.36893\dots$, respectively. Both are related with the extended Coxeter group $(5, 3, 5)$ and the so-called hyperbolic football manifold (look at Fig.~3). These facts can have important relations with fullerens in crystallography., Comment: 23 pages, 3 figures

We consider the sectional curvature function on a 4-dimensional manifold admitting a metric g of neutral signature, (+,+,?,?) together with a review of the situation for the other two signatures. The main results of the paper are: first, that if the sectional curvature function is not a constant function at any m ? M (actually a slightly weaker assumption is made), the conformal class of g is always uniquely determined and in almost all cases g is uniquely determined on M, second, a study of the special cases when this latter uniqueness does not hold, third, the construction of the possible metrics in this latter case, fourth, some remarks on sectional curvature preserving vector fields and finally the complete solution when (M, g) is Ricci flat.

Formal analysis and computer recognition of 2D color images is important branch of modern computer geometry. However, existing algorithms, although they are highly developed, are not quite satisfactory and seem to be much worse than (unknown) algorithms, which our brain uses to analyze eye information. Almost all existing algorithms omit colors and deal with grayscale transformations only. But in many cases color information is important. In this paper fundamentally new method of coding and analyzing color digital images is suggested. The main point of this method is that a full-color digital image is represented, without dropping colors, by special 2D surface in 3D space, after which it is analyzed by methods of differential geometry, rather than traditional gradient-based or Hessian-based methods (like in SIFT, GLOH, SURF, Canny operator, and many other algorithms).

This paper deals with the number of subgroups of a given exponent in a finite abelian group. Explicit formulas are obtained in the case of rank two and rank three abelian groups. An asymptotic formula is also presented., Comment: 15 pages, revised

Let Nn+p q(c) be an (n+p)-dimensional connected indefinite space form of index q(1 ? q ? p) and of constant curvature c. Denote by ? : M ? Nn+p q (c) the n-dimensional spacelike submanifold in Nn+p q (c), ? : M ? Nn+p q(c) is called a Willmore spacelike submanifold in Nn+p q(c) if it is a critical submanifold to the Willmore functional W(?) = ?q M ?n dv =?M (S-nH2)n/2 dv, where S and H denote the norm square of the second fundamental form and the mean curvature of M and ?2 = S ? nH2. If q = p, in [14], we proved some integral inequalities of Simons? type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in a Lorentzian space form Nn+p q(c). In this paper, we continue to study this topic and prove some integral inequalities of Simons? type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in an indefinite space form Nn+p q(c) (1 ? q ? p).

A Grobner basis for the ideal determining mod 2 cohomology of Grassmannian G3,n, obtained in a previous paper, is used, along with the method of obstruction theory, to establish some new immersion results for these manifolds.

Let $V_n$ be the number of equivalence classes of invertible maps from $\{0,1\}^n$ to $\{0,1\}^n$, under action of permutation of variables on domain and range. So far, the values $V_n$ have been known for $n\le 6$. This paper describes the procedure by which the values of $V_n$ are calculated for $n\le 30$., 9 pages, 1 figure

In this paper we obtain a partial answer to a conjecture on the solvabilty of linear difference equations in quasianalytic Carleman classes., 6 pages

In this paper we study the semiring variety V generated by any finite number of finite fields F1,..., Fk and two-element distributive lattice B2, i.e., V = HSP{B2, F1,..., Fk}. It is proved that V is hereditarily finitely based, and that, up to isomorphism, the two-element distributive lattice B2 and all subfields of F1,..., Fk are the only subdirectly irreducible semirings in V.

Let R be a commutative ring with identity and A(R) be the set of ideals with nonzero annihilator. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)* = A(R)\{0} and two distinct vertices I and J are adjacent if and only if IJ = 0. In this paper, we study the domination number of AG(R) and some connections between the domination numbers of annihilating-ideal graphs and zero-divisor graphs are given.

An antichain is here regarded as a hypergraph that satisfies the following property: neither of every two different edges is a subset of the other one. The paper is devoted to the enumeration of antichains given on an n-set and having one or more of the following properties: being labeled or unlabeled; being ordered or unordered; being a cover (or a proper cover); and finally, being a T0-, T1- or T2-hypergraph. The problem of enumeration of these classes comprises, in fact, different modifications of Dedekind?s problem. Here a theorem is proved, with the help of which a greater part of these classes can be enumerated. The use of the formula from the theorem is illustrated by enumeration of labeled antichains, labeled T0-antichains, ordered unlabeled antichains, and ordered unlabeled T0-antichains. Also a list of classes that can be enumerated in a similar way is given. Finally, we perform some concrete counting, and give a table of digraphs that we used in the counting process.

Let X be a Banach space. Let K be a nonempty subset of X. Let T : K ? K be an I-asymptotically quasi-nonexpansive type mapping and I : K ? K be an asymptotically quasi-nonexpansive type mappings in the Banach space. Our aim is to establish the necessary and sufficient conditions for the convergence of the Ishikawa iterative sequences with errors of an I-asymptotically quasi-nonexpansive type mappping in Banach spaces to a common fixed point of T and I. Also, we study the convergence of the Ishikawa iterative sequences to common fixed point for nonself I-asymptotically quasinonexpansive type mapping in Banach spaces. The results presented in this paper extend and generalize some recent work of Chang and Zhou [1], Wang [19], Yao and Wang [20] and many others.

Let G be a finite group and m, n two positive coprime integers. We prove that the set of conjugacy class sizes of primary and biprimary elements of G is {1, m, n} if and only if G is quasi-Frobenius with abelian kernel and complement. Throughout this paper all groups considered are finite and G always denotes a group. A primary element is an element of prime power order and a biprimary element is an element whose order is divisible by precisely two distinct primes. We will denote by x G the conjugacy class containing x, and |x G | the conjugacy class size of x G. A positive integer a is a Hall number of group G if a is a divisor of |G| and (a, |G|/a) = 1. We say that G is quasi-Frobenius if G/Z(G) is Frobenius. The inverse image in G of the kernel and a complement of G/Z(G) are called the kernel and a complement of G. The other notation and terminology are standard, as in (2).

A systematic exposition of Sheffer polynomial sequences via determinantal form is given. A general linear interpolation problem related to Sheffer sequences is considered. It generalizes many known cases of linear interpolation. Numerical examples and conclusions close the paper.

We describe an algorithm for the numerical solution of an integral equation of the form ? 1/? ?1?1 [(y ? x) ln |y ? x| ? h(x, y)] u(y) dy/?1?y2 = f(x), ?1 < x < 1, which is based on a collocation-quadrature method and which has the same convergence rate as this method, but only O(n log n) complexity. This integral equation turns out to be an ill-posed problem in (the best possible choice of) a pair of non-periodic Sobolev-like spaces. The present paper presents the technique, how to overcome this peculiarity in the investigation of the fast algorithm.

We present some open problems pertaining to the approximation theory involved in the solution of the Nonlinear Schr?dinger (NLS) equation. For this important equation, any Initial Value Problem (IVP) can be theoretically solved by the Inverse Scattering Transform (IST) technique whose main steps involve the solution of Volterra equations with structured kernels on unbounded domains, the solution of Fredholm integral equations and the identification of coefficients and parameters of monomial-exponential sums. The aim of the paper is twofold: propose a method for solving the above mentioned problems under particular hypothesis; arise interest in the issues illustrated to achieve an effective method for solving the problem under more general assumptions.

A semiring variety is d-semisimple if it is generated by the distributive lattice of order two and a finite number of finite fields. A d-semisimple variety V = HSP{B2, F1,..., Fk} plays the main role in this paper. It will be proved that it is finitely based, and that, up to isomorphism, the two-element distributive lattice B2 and all subfields of F1,..., Fk are the only subdirectly irreducible members in it. [The first author is supported by China Postdoctoral Science Foundation, Grant 2011M501466 and the Natural Science Foundation of Shannxi Province, Grant 2011JQ1017. The second authoris Supported by the Ministry of Education, Science and Technological Development of Serbia,Grant 174018. The third author is Supported by the Ministry of Education, Science and TechnologicalDevelopment of Serbia, Grant 174026]

Contrary to the notion of a set or a tuple, a multiset is an unordered collection of elements which do not need to be different. As multisets are already widely used in combinatorics and computer science, the aim of this paper is to get on track to algebraic multiset theory. We consider generalizations of known results that hold for equivalence and order relations on sets and get several properties that are specific to multisets. Furthermore, we exemplify the novelty that brings this concept by showing that multisets are suitable to represent partial orders. Finally, after introducing the notion of an algebra on multisets, we prove that two algebras on multisets, whose root algebras are isomorphic, are in general not isomorphic.

The aim of this research paper is to establish sufficient conditions for the nonexistence of global solutions for the following nonlinear fractional differential equation D?0|tu + (??)?/2|u|m?1u + a(x)??|u|q?1u = h(x, t)|u|p, (t,x) ? Q, u(0, x) = u0(x), x ? RN where (??)?/2, 0 < ? < 2 is the fractional power of ??, and D?0|t, (0 < ? < 1) denotes the time-derivative of arbitrary ? ? (0; 1) in the sense of Caputo. The results are shown by the use of test function theory and extended to systems of the same type.

First I will explain my motivation to introduce the ?-invariants for Riemannian manifolds. I will also recall the notions of ideal immersions and best ways of living. Then I will present a few of the many applications of ?-invariants in several areas in mathematics. Finally, I will present two optimal inequalities involving ?-invariants for Lagrangian submanifolds obtained very recently in joint papers with F. Dillen, J. Van der Veken and L. Vrancken.

This paper deals with the rectangle packing problem, of filling a big rectangle with smaller rectangles, while the rectangle dimensions are real numbers. A new nonlinear programming formulation is presented and the validity of the formulation is proved. In addition, two cases of the problem are presented, with and without rotation of smaller rectangles by 90?. The mixed integer piecewise linear formulation derived from the model is given, but with a simple form of the objective function.

The study of ill-posed abstract Volterra equations is a recent subject. In this paper, we investigate equations on the line, continue the research of (a, k)-regularized C-resolvent families, subordination principles, abstract semilinear Volterra integrodifferential equations, and provide several illustrative examples.

The problem discussed in this paper was stated by Alexander O. Ivanov and Alexey A. Tuzhilin in 2009. It stands at the intersection of the theories of Gromov minimal fillings and Steiner minimal trees. Thus, it can be considered as one-dimensional stratified version of the Gromov minimal fillings problem. Here we state the problem; discuss various properties of one-dimensional minimal fillings, including a formula calculating their weights in terms of some special metrics characteristics of the metric spaces they join (it was obtained by A.Yu. Eremin after many fruitful discussions with participants of Ivanov-Tuzhilin seminar at Moscow State University); show various examples illustrating how one can apply the developed theory to get nontrivial results; discuss the connection with additive spaces appearing in bioinformatics and classical Steiner minimal trees having many applications, say, in transportation problem, chip design, evolution theory etc. In particular, we generalize the concept of Steiner ratio and get a few of its modifications defined by means of minimal fillings, which could give a new approach to attack the long standing Gilbert-Pollack Conjecture on the Steiner ratio of the Euclidean plane.

The gamma class ??(g) consists of positive and measurable functions that satisfy f(x + yg(x))/f(x) ? exp(?y). In most cases the auxiliary function g is Beurling varying and self-neglecting, i.e., g(x)/x ? 0 and g??0(g). Taking h = log f, we find that h?E??(g, 1), where E??(g, a) is the class of positive and measurable functions that satisfy (f(x + yg(x))? f(x))/a(x) ? ?y. In this paper we discuss local uniform convergence for functions in the classes ??(g) and E??(g, a). From this, we obtain several representation theorems. We also prove some higher order relations for functions in the class ??(g) and related classes. Two applications are given.

Inspired by Lomonaco-Kauffman paper on quantum knots and knot mosaics we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. We introduce codes for mirror-curves treated as knot or link diagrams placed in rectangular square grid, suitable for software implementations and discuss possible applications of mirror-curves. [Projekat Ministarstva nauke Republike Srbije, br. 174012]

We present some fixed point results for mappings which satisfy Hardy-Rogers rational type, quasicontraction type, weak contraction type and generalized f? type cyclic conditions in 0-complete partial metric spaces. Presented results generalize or improve many existing fixed point theorems in the literature. To demonstrate our results, we give throughout the paper some examples. One of the possible applications of our results to well-posed and limit shadowing property of fixed point problems is also presented.

A graph is a cactus if any two of its cycles have at most one common vertex. In this paper, we determine the first sixteen largest Laplacian spectral radii together with the corresponding graphs among all connected cacti with n vertices and k cycles, where n > 2k + 8.

In the first part of the paper [5], we gave a new definition of real or complex 2-normed algebras and 2-Banach algebras. Here we give two examples which establish that not all 2-normed algebras are normable and a 2-Banach algebra need not be a 2-Banach space. We conclude by deriving a new and interesting spectral radius formula for 1-Banach algebras from the basic properties of 2-Banach algebras and thus vindicating our definitions of 2-normed and 2-Banach algebras given in [5].

Consider the solution of the time-dependent Schr?dinger equation with initial data f. It is shown by Sj?gren and Sj?lin (1989) that there exists f in the Sobolev space Hs(Rn), s=n/2 such that tangential convergence can not be widened to convergence regions. In this paper we show that the corresponding result holds when -?x is replaced by an operator ?(D), with special conditions on ?.

In [3] and [4] A. Torgasev described all finite and infinite connected graphs having 3, 4 or 5 nonzero eigenvalues (not necessarily distinct). In the same papers he has given a general method how to describe all connected graphs with any fixed number of nonzero eigenvalues. In [2] M. Lepovic applying his method described all finite connected graphs which have exactly 6 nonzero eigenvalues. We here describe all finite connected graphs with exactly 7 nonzero eigenvalues.

The distance or D-eigenvalues of a graph G are the eigenvalues of its distance matrix. The distance or D-energy ED(G) of the graph G is the sum of the absolute values of its D-eigenvalues. Two graphs G1 and G2 are said to be D-equienergetic if ED(G1) = ED(G2). Let F1 be the 5-vertex path, F2 the graph obtained by identifying one vertex of a triangle with one end vertex of the 3-vertex path, F3 the graph obtained by identifying a vertex of a triangle with a vertex of another triangle and F4 be the graph obtained by identifying one end vertex of a 4-vertex star with a middle vertex of a 3-vertex path. In this paper we show that if G is r-regular, with diam(G)? 2, and Fi,i = 1,2,3,4, are not induced subgraphs of G, then the k-th iterated line graph Lk(G) has exactly one positive D-eigenvalue. Further, if G is r-regular, of order n, diam(G)?2, and G does not have Fi,i=1,2,3,4, as an induced subgraph, then for k ?1, ED(Lk(G)) depends solely on n and r. This result leads to the construction of non D-cospectral, D-equienergetic graphs having same number of vertices and same number of edges.

Many statistics are based on functions of sample moments. Important examples are the sample variance s2(n), the sample coefficient of variation SV (n), the sample dispersion SD(n) and the non-central t-statistic t(n). The definition of these quantities makes clear that the vector defined by (?ni=1Xi, ?ni=1Xi2)plays an important role. In the paper we obtain conditions under which the vector (X,X2) belongs to a bivariate domain of attraction of a stable law. Applying simple transformations then leads to a full discussion of the asymptotic behaviour of SV(n) and t(n).