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It is proved that, for n = 8, 9, 10, the natural algebra of automorphic forms of the group O+2,n(ℤ) acting on the n-dimensional symmetric domain of type IV is free, and the weights of generators are found. This extends results obtained in the author's previous paper for n ≤ 7. On the other hand, as proved in a recent joint paper of the author and O. V. Shvartsman, similar algebras of automorphic forms cannot be free for n > 10. [ABSTRACT FROM AUTHOR]

This paper presents a reliable approach for solving linear systems of ordinary and fractional differential equations. First, the FDEs or ODEs of a system with initial conditions to be solved are transformed to Volterra integral equations. Then Taylor expansion for the unknown function and integration method are employed to reduce the resulting integral equations to a new system of linear equations for the unknown and its derivatives. The fractional derivatives are considered in the Riemann-Liouville sense. Some numerical illustrations are given to demonstrate the effectiveness of the proposed method in this paper. [ABSTRACT FROM AUTHOR]

In their 2010 (Erkenntnis 73:393-412) paper, Dizadji-Bahmani, Frigg, and Hartmann (henceforth 'DFH') argue that the generalized version of the Nagel-Schaffner model that they have developed (henceforth 'the GNS') is the right one for intertheoretic reduction, i.e. the kind of reduction that involves theories with largely overlapping domains of application. Drawing on the GNS, DFH (Synthese 179:321-338, 2011) presented a Bayesian analysis of the confirmatory relation between the reducing theory and the reduced theory and argued that, post-reduction, evidence confirming the reducing theory also confirms the reduced theory and evidence confirming the reduced theory also confirms the reducing theory, which meets the expectations one has about theories with largely overlapping domains. In this paper, I argue that the Bayesian analysis presented by DFH (Synthese 179:321-338, 2011) faces difficulties. In particular, I argue that the conditional probabilities that DFH introduce to model the bridge law entail consequences that run against the GNS. However, I also argue that, given slight modifications of the analysis that are in agreement with the GNS, one is able to account for these difficulties and, moreover, one is able to more rigorously analyse the confirmatory relation between the reducing and the reduced theory. [ABSTRACT FROM AUTHOR]

The power graph $${\fancyscript{P}}(G)$$ of a group $$G$$ is the graph which has a vertex set of the group elements and two elements are adjacent if one is a power of the other. Chakrabarty, Ghosh, and Sen proved the main properties of the undirected power graph of a finite group. The aim of this paper is to generalize some results of their work and presenting some counterexamples for one of the problems raised by these authors. It is also proved that the power graph of a $$p$$ -group is $$2$$ -connected if and only if the group is a cyclic or generalized quaternion group and if $$G$$ is a nilpotent group which is not of prime power order then the power graph $${\fancyscript{P}}(G)$$ is $$2$$ -connected. We also prove that the number of edges of the power graph of the simple groups is less than or equal to the number of edges in the power graph of the cyclic group of the same order. This partially answers to a question in an earlier paper. Finally, we give a complete classification of groups in which the power graph is a union of complete graphs sharing a common vertex. [ABSTRACT FROM AUTHOR]

Since 1909 when Brouwer proved the first fixed-point theorem named after him, the fixed-point results in various settings play an important role in the optimization theory and applications. This technique has proven to be indispensable for the proofs of multiple results related to the existence of solutions to numerous problems in the areas of optimization and approximation theory, differential equations, variational inequalities, complementary problems, equilibrium theory, game theory, mathematical economics, etc. It is also worthwhile to mention that the majority of problems of finding solutions (zero-points) of functions (operators) can be easily reduced to that of discovering of fixed points of properly modified mappings. Not only theoretical but also practical (algorithmic) developments are based on the fixed-point theory. For instance, the well-known simplicial (triangulation) algorithms help one to find the desired fixed points in a constructive way. That approach allows one to investigate the solvability of complicated problems arising in theory and applications. In this paper, making use of the triangulation technique, we extend some antipodal and fixed-point theorems to the case of nonconvex, more exactly, star-shaped sets. Also, similar extensions are made for set-valued mappings defined over star-shaped sets. [ABSTRACT FROM AUTHOR]

In this paper, we prove a stochastic Fubini theorem by solving a special backward stochastic differential equation (BSDE, for short) which is different from the existing techniques. As an application, we obtain the well-posedness of a class of BSDEs with the Itô integral in drift term under a subtle Lipschitz condition. [ABSTRACT FROM AUTHOR]

In a very influential paper Rota stresses the relevance of mathematical beauty to mathematical research, and claims that a piece of mathematics is beautiful when it is enlightening. He stops short, however, of explaining what he means by 'enlightening'. This paper proposes an alternative approach, according to which a mathematical demonstration or theorem is beautiful when it provides understanding. Mathematical beauty thus considered can have a role in mathematical discovery because it can guide the mathematician in selecting which hypothesis to consider and which to disregard. Thus aesthetic factors can have an epistemic role qua aesthetic factors in mathematical research. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the congruence lattice of a symmetric extended De Morgan algebra $$L$$ . We show that the congruence lattice of the algebra $$L$$ is a pseudocomplemented lattice, and that such a congruence lattice is a Stone lattice if and only if the lattice of the compact congruences on $$L$$ forms a complete Boolean lattice. In particular, we prove that the congruence lattice of $$L$$ is a Boolean lattice if and only if, it is a relative Stone lattice, which is the case, if and only if $$L$$ is finite. [ABSTRACT FROM AUTHOR]

According to a widespread but implicit thesis in Bayesian confirmation theory, two confirmation measures are considered equivalent if they are ordinally equivalent—call this the "ordinal equivalence thesis" (OET). I argue that adopting OET has significant costs. First, adopting OET renders one incapable of determining whether a piece of evidence substantially favors one hypothesis over another. Second, OET must be rejected if merely ordinal conclusions are to be drawn from the expected value of a confirmation measure. Furthermore, several arguments and applications of confirmation measures given in the literature already rely on a rejection of OET. I also contrast OET with stronger equivalence theses and show that they do not have the same costs as OET. On the other hand, adopting a thesis stronger than OET has costs of its own, since a rejection of OET ostensibly implies that people's epistemic states have a very fine-grained quantitative structure. However, I suggest that the normative upshot of the paper in fact has a conditional form, and that other Bayesian norms can also fruitfully be construed as having a similar conditional form. [ABSTRACT FROM AUTHOR]

The term 'continuous' in real analysis wasn't given an adequate formal definition until 1817. However, important theorems about continuity were proven long before that. How was this possible? In this paper, I introduce and refine a proposed answer to this question, derived from the work of Frank Jackson, David Lewis and other proponents of the 'Canberra plan'. In brief, the proposal is that before 1817 the meaning of the term 'continuous' was determined by a number of 'platitudes' which had some special epistemic status. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the family of quotients of internally quasicontinuous functions. Moreover, we study cardinal invariants related to quotients in the case of internally quasicontinuous functions and the complement of this family. [ABSTRACT FROM AUTHOR]

The equivariant CR minimal immersions from the round 3-sphere S3 into the complex projective space CPn have been classified by the third author explicitly (Li in J Lond Math Soc 68:223-240, 2003). In this paper, by employing the equivariant condition which implies that the induced metric is left-invariant and that all geometric properties of S3=SU(2) endowed with a left-invariant metric can be expressed in terms of the structure constants of the Lie algebra su(2), we establish an extended classification theorem for equivariant CR minimal immersions from the 3-sphere S3 into CPn without the assumption of constant sectional curvatures. [ABSTRACT FROM AUTHOR]

Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows studying properties of mathematical objects such as algebraic varieties from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. In this paper we prove a tropical Nullstellensatz, and moreover, we show an effective formulation of this theorem. Nullstellensatz is a natural step in building algebraic theory of tropical polynomials and its effective version is relevant for computational aspects of this field. On our way we establish a simple formulation of min-plus and tropical linear dualities. We also observe a close connection between tropical and min-plus polynomial systems. [ABSTRACT FROM AUTHOR]

This paper is concerned with multiplicity results for semilinear elliptic equations of the type $$-\Delta u+a(x)u=\left| u\right| ^{p-2}u+f\left( x,u\right) $$ in $$\Omega $$ , $$u=0$$ on $$\partial \Omega $$ . We obtain at least two nontrivial solutions for this type of equations with the case that 0 is not a local minimizer of the corresponding functional under suitable hypotheses. The method we used here is based on a linking structure relevant to the Nehari manifold. [ABSTRACT FROM AUTHOR]

This paper presents a generalized version of the classic projective reconstruction theorem which helps to choose or assess depth constraints for projective depth estimation algorithms. The theorem shows that projective reconstruction is possible under a much weaker constraint than requiring all estimated projective depths to be nonzero. This result enables us to present classes of depth constraints under which any reconstruction of cameras and points projecting into given image points is projectively equivalent to the true camera-point configuration. It also completely specifies the possible wrong configurations allowed by other constraints. We demonstrate the application of the theorem by analysing several constraints used in the literature, as well as presenting new constraints with desirable properties. We mention some of the implications of our results on iterative depth estimation algorithms and projective reconstruction via rank minimization. Our theory is verified by running experiments on both synthetic and real data. [ABSTRACT FROM AUTHOR]

In the present paper, we study the extrinsic and intrinsic geometry of submanifolds of an almost contact metric manifold admitting a quarter-symmetric metric connection. We deduce Gauss, Codazzi and Ricci equations corresponding to the quarter-symmetric metric connection and show some applications of these equations. Finally, we give an example verifying the results. [ABSTRACT FROM AUTHOR]

Björner, Lovász and Shor introduced a chip-firing game on a finite graph $$G$$ as follows. We put some chips on each vertex of $$G$$ , we say that a vertex is ready if it has at least as many chips as its degree, in which case we can fire it and the result is that it distributes one chip to each of its neighbors, this may cause other vertices to be ready, and so on. This game continues until no vertex can be fired. In this paper, we study chip-firing games on complete graphs. We obtain a sufficient and necessary condition for chip-firing games on complete graphs to be finite. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a concept of graph convergence for the $$H(\cdot ,\cdot )$$ -co-accretive mapping in Banach spaces and prove an equivalence theorem between graph convergence and resolvent operator convergence for the $$H(\cdot ,\cdot )$$ -co-accretive mapping. Further, we consider a system of generalized variational inclusions involving $$H(\cdot ,\cdot )$$ -co-accretive mapping in real $$q$$ -uniformly smooth Banach spaces. Using resolvent operator technique, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions under some suitable conditions. Further, we discuss the convergence of iterative algorithm using the concept of graph convergence. Our results can be viewed as a refinement and generalization of some known results in the literature. [ABSTRACT FROM AUTHOR]

Let $$G$$ be a graph with degree sequence $$d_1,d_2,\ldots ,d_n$$ . Given a positive integer $$p$$ , denote by $$e_p(G)=\sum _{i=1}^n d_i^p$$ . Caro and Yuster introduced a Turán-type problem for $$e_p(G)$$ : given an integer $$p$$ , how large can $$e_p(G)$$ be if $$G$$ has no subgraph of a particular type. They got some results for the subgraph of particular type to be a clique of order $$r+1$$ and a cycle of even length, respectively. Denote by $$ex_p(n,H)$$ the maximum value of $$e_p(G)$$ taken over all graphs with $$n$$ vertices that do not contain $$H$$ as a subgraph. Clearly, $$ex_1(n,H)=2ex(n,H)$$ , where $$ex(n,H)$$ denotes the classical Turán number. In this paper, we consider $$ex_p(n, C_5)$$ and prove that for any positive integer $$p$$ and sufficiently large $$n$$ , there exists a constant $$c=c(p)$$ such that the following holds: if $$ex_p(n, C_5)=e_p(G)$$ for some $$C_5$$ -free graph $$G$$ of order $$n$$ , then $$G$$ is a complete bipartite graph having one vertex class of size $$cn+o(n)$$ and the other $$(1-c)n+o(n)$$ . [ABSTRACT FROM AUTHOR]

Let $${\mathcal {F}}$$ denote the class of all functions univalent in the unit disk $$\Delta \equiv \{\zeta \in {\mathbb {C}}\,:\,\left| \zeta \right| <1\}$$ and convex in the direction of the real axis. The paper deals with the subclass $${\mathcal {F}}^{(n)}$$ of these functions $$f$$ which satisfy the property $$f(\varepsilon z)=\varepsilon f(z)$$ for all $$z\in \Delta $$ , where $$\varepsilon =e^{2\pi i/n}$$ . The functions of this subclass are called $$n$$ -fold symmetric. For $${\mathcal {F}}^{(n)}$$ , where $$n$$ is odd positive integer, the following sets, $$\bigcap _{f\in {\mathcal {F}}^{(n)}} f(\Delta )$$ -the Koebe set and $$\bigcup _{f\in {\mathcal {F}}^{(n)}} f(\Delta )$$ -the covering set, are discussed. As corollaries, we derive the Koebe and the covering constants for $${\mathcal {F}}^{(n)}$$ . [ABSTRACT FROM AUTHOR]

There are many articles in the literature dealing with the first-order and the second-order differential subordination and differential superordination problems for analytic functions in the unit disk, but there are only a few articles dealing with the third-order differential subordination problems. The concept of third-order differential subordination in the unit disk was introduced by Antonino and Miller, and studied recently by Tang and Deniz. Let $$\Omega $$ be a set in the complex plane $$\mathbb {C}$$ , let $$\mathfrak {p}(z)$$ be analytic in the unit disk $$\mathbb {U}=\{z:z\in \mathbb {C}\quad \text {and} \quad |z|<1\}$$ , and let $$\psi : \mathbb {C}^4\times \mathbb {U}\rightarrow \mathbb {C}$$ . In this paper, we investigate the problem of determining properties of functions $$\mathfrak {p}(z)$$ that satisfy the following third-order differential superordination: As applications, we derive some third-order differential superordination results for analytic functions in $$\mathbb {U}$$ , which are associated with a family of generalized Bessel functions. The results are obtained by considering suitable classes of admissible functions. New third-order differential sandwich-type results are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we use a result of N. S. Feldman to show that there are no supercyclic subnormal tuples in infinite dimensions. Also, we investigate some spectral properties of hypercyclic tuples of operators. Besides, we prove that if $$T$$ is a supercyclic $$\ell $$ -tuple of commuting $$n\times n$$ complex matrices, then $$\ell \ge n$$ and also there exists a supercyclic $$n$$ -tuple of commuting diagonal $$n\times n$$ matrices. Furthermore, we see that if $$T=(T_{1},\ldots ,T_{n})$$ is a supercyclic $$n$$ -tuple of commuting $$n\times n$$ complex matrices, then $$T_{j}$$ 's are simultaneously diagonalizable. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the central limit theorem and the law of the iterated logarithm for Galton-Watson branching processes with time-dependent immigration in varying environments. [ABSTRACT FROM AUTHOR]

In this paper, the semilocal convergence for a class of multi-point modified Chebyshev-Halley methods in Banach spaces is studied. Different from the results in reference [11], these methods are more general and the convergence conditions are also relaxed. We derive a system of recurrence relations for these methods and based on this, we prove a convergence theorem to show the existence-uniqueness of the solution. A priori error bounds is also given. The R-order of these methods is proved to be 5+ q with ω−conditioned third-order Fréchet derivative, where ω( μ) is a non-decreasing continuous real function for μ > 0 and satisfies ω(0) ≥ 0, ω( tμ) ≤ t ω( μ) for μ > 0, t ∈ [0,1] and q ∈ [0,1]. Finally, we give some numerical results to show our approach. [ABSTRACT FROM AUTHOR]

Recently Carrie S. Jenkins formulated an epistemology of mathematics, or rather arithmetic, respecting apriorism, empiricism, and realism. Central is an idea of concept grounding. The adequacy of this idea has been questioned e.g. concerning the grounding of the mathematically central concept of set (or class), and of composite concepts. In this paper we present a view of concept formation in mathematics, based on ideas from Carnap, leading to modifications of Jenkins's epistemology that may solve some problematic issues with her ideas. But we also present some further problems with her view, concerning the role of proof for mathematical knowledge. [ABSTRACT FROM AUTHOR]

The article focuses on mathematical tastes working with mathematical area tend to have consensus as to theorem, proof or conjecture and mentions elementary mathematical reasoning based on character. Topics discussed include sum of consecutive odd numbers adding up to square number, association between different mathematical arguments and different works of art and ability to recognize aesthetic aspects of mathematical arguments.

The article focuses on Jordan loop which is set of form of homeomorphism and mentions that mathematician Otto Toeplitz conjectured that Jordan Loop contains points that are vertices of square. Topics discussed include loop in space which consists of vertices of inscribed parallelogram, rectangles having all possible aspect ratios and nonsqueezing theorem from symplectic topology.

In this paper, we characterize the family of quotients of internally quasicontinuous functions. Moreover, we study cardinal invariants related to quotients in the case of internally quasicontinuous functions and the complement of this family. [ABSTRACT FROM AUTHOR]

We introduce the concept of asymmetry number for finite tournaments, as a natural generalization of that for graphs by Erdős and Rényi (Acta Math Acad Sci Hungar 14:295-315, 1963). We prove an upper bound for the asymmetry number of finite tournaments and discuss its asymptotically best possibility. We also give an alternative proof of a theorem by Jaligot and Khelif (Aequat Math 67:73-79, 2004) which states that the random tournament can be expressed by a Cayley tournament on an arbitrary countable group without involutions. We slightly improve a result by Cameron and Johnson (Math Proc Camb Philos Soc 102:223-231, 1987). [ABSTRACT FROM AUTHOR]

Programming-based activities are becoming more widespread in curricula. Our theoretical and empirical investigation seeks to identify appropriate ways to connect computer programming and algorithmics to mathematical learning. We take the intermediate value theorem as our starting point, as it is covered by the French school curriculum, and because of its links with the bisection algorithm. We build upon the theory of mathematical working spaces, distinguishing between algorithmic and mathematical working spaces. Both working spaces are explored from the semiotic, instrumental, and discursive dimensions that support learning. Our two research questions focus on the suitable algorithmic and mathematical working spaces in which students develop an understanding of the intermediate value theorem, and the bisection algorithm. Our method starts at the reference level, with an epistemological and curricular analysis. Then, a series of tasks is designed for students working in adidacticity, and suitable working spaces are determined a priori. The tasks have been implemented in French classrooms with students aged 16–19. An analysis of their work supports an a posteriori examination of the working spaces. Our findings demonstrate that the students were able to make connections between algorithmics and mathematics in each of the three dimensions, semiotic, instrumental, and discursive, and point out the interplay between these dimensions. [ABSTRACT FROM AUTHOR]

This paper deals with the first-order delayed differential systems {u′+a(t)u=h(t)v+f(t,u(t−τ(t))),v′+b(t)v=g(t,u(t−τ(t))), where a, b, τ, h are continuous ω-periodic functions with ∫0ωa(t)dt=0 and ∫0ωb(t)dt>0; f∈C(R×[0,∞),R) and g∈C(R×[0,∞),[0,∞)) are ω-periodic with respect to t. By means of the fixed point theorem in cones, several new existence theorems on positive periodic solutions are established. Our main results enrich and complement those available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and uniqueness of solutions for a new class of multi-point and multi-strip boundary value problems of multi-term fractional differential equations by using standard fixed point theorems. We demonstrate the application of the obtained results with the aid of examples. Some new results are also deduced by fixing the parameters involved in the problem at hand. [ABSTRACT FROM AUTHOR]

A personal narrative is presented which explores problems faced by the author while deriving a theorem.

The conservativeness argument poses a dilemma to deflationism about truth, according to which a deflationist theory of truth must be conservative but no adequate theory of truth is conservative. The debate on the conservativeness argument has so far been framed in a specific formal setting, where theories of truth are formulated over arithmetical base theories. I will argue that the appropriate formal setting for evaluating the conservativeness argument is provided not by theories of truth over arithmetic but by those over subject matters 'richer' than arithmetic, such as set theory. The move to this new formal setting provides deflationists with better defence and brings a broader perspective to the debate. [ABSTRACT FROM AUTHOR]

A new method for obtaining lower bounds on the number of matrices over a finite field with nonzero permanent is developed. Some earlier results are improved. [ABSTRACT FROM AUTHOR]

We consider the space M of Euclidean similarity classes of framed loops in R3. Framed loop space is shown to be an infinite-dimensional Kähler manifold by identifying it with a complex Grassmannian. We show that the space of isometrically immersed loops studied by Millson and Zombro is realized as the symplectic reduction of M by the action of the based loop group of the circle, giving a smooth version of a result of Hausmann and Knutson on polygon space. The identification with a Grassmannian allows us to describe the geodesics of M explicitly. Using this description, we show that M and its quotient by the reparameterization group are nonnegatively curved. We also show that the planar loop space studied by Younes, Michor, Shah and Mumford in the context of computer vision embeds in M as a totally geodesic, Lagrangian submanifold. The action of the reparameterization group on M is shown to be Hamiltonian, and this is used to characterize the critical points of the weighted total twist functional. [ABSTRACT FROM AUTHOR]

The existence of weak solutions for a class of $(p,q)$ -Laplacian systems is obtained by using a new linking theorem on the product space $W^{1,p}_{0}(\Omega)\times W^{1,q}_{0}(\Omega)$ . [ABSTRACT FROM AUTHOR]

In this article, using an infinite-dimensional equivariant Conley index, we prove a generalization of the profitable Liapunov center theorem for symmetric potentials. Consider a system $$(*)\; \ddot{q}= -\nabla U(q)$$ , where U( q) is a $$\Gamma $$ -invariant potential and $$\Gamma $$ is a compact Lie group acting linearly on $${\mathbb {R}}^n$$ . If system $$(*)$$ possess a non-degenerate orbit of stationary solutions $$\Gamma (q_0)$$ with trivial isotropy group, such that there exists at least one positive eigenvalue of the Hessian $$\nabla ^2 U(q_0)$$ , then in any neighborhood of $$\Gamma (q_0)$$ there is a non-stationary periodic orbit of solutions of system $$(*)$$ . [ABSTRACT FROM AUTHOR]

We study the spectrum of prime ideals in the tensor-triangulated category of compact equivariant spectra over a finite group. We completely describe this spectrum as a set for all finite groups. We also make significant progress in determining its topology and obtain a complete answer for groups of square-free order. For general finite groups, we describe the topology up to an unresolved indeterminacy, which we reduce to the case of p-groups. We then translate the remaining unresolved question into a new chromatic blue-shift phenomenon for Tate cohomology. Finally, we draw conclusions on the classification of thick tensor ideals. [ABSTRACT FROM AUTHOR]

The article presents information regarding the appearance of Hopf fibration, mathematical field of topology discovered by Heinz Hopf, in other areas of mathematics and physics, wherein it is called as four fiber bundles in spheres. Topics discussed include the Hurwitz-Radon Function which is evaluated at powers of two, the history of Hurwitz-Radon Function and the theorem of Adams about the vector fields on spheres.

A numerical method for solving a system of hypersingular integral equations of the second kind is presented. The theorem on the existence and uniqueness of a solution to such a system is proved. The rate of convergence of an approximate solution to the exact solution is estimated. [ABSTRACT FROM AUTHOR]

A theta graph is the union of three internally disjoint paths that have the same two distinct end vertices. We show that every graph of order $$n\ge 9$$ and size at least $$\lfloor \frac{7n-13}{2}\rfloor $$ contains two disjoint theta graphs. We also show that every 2-edge-connected graph of order $$n\ge 6$$ and size at least $$3n-5$$ contains two disjoint cycles, such that any specified vertex with degree at least three belongs to one of them. The lower bound on size in both is sharp in general. [ABSTRACT FROM AUTHOR]

Let $$D$$ be a simple digraph with vertex set $$V(D)$$ , and let $$f:V(D)\rightarrow \{-1,1\}$$ be a two-valued function. If $$k\ge 1$$ is an integer and $$\sum _{x\in N^-[v]}f(x)\ge k$$ for each $$v\in V(D)$$ , where $$N^-[v]$$ consists of $$v$$ and all vertices of $$D$$ from which arcs go into $$v$$ , then $$f$$ is a signed $$k$$ -dominating function on $$D$$ . A set $$\{f_1,f_2,\ldots ,f_d\}$$ of distinct signed $$k$$ -dominating functions on $$D$$ with the property that $$\sum _{i=1}^df_i(x)\le k$$ for each $$x\in V(D)$$ , is called a signed $$(k,k)$$ -dominating family (of functions) on $$D$$ . The maximum number of functions in a signed $$(k,k)$$ -dominating family on $$D$$ is the signed $$(k,k)$$ -domatic number of $$D$$ . In this article, we mainly present upper bounds on the signed $$(k,k)$$ -domatic number, in particular for regular digraphs. [ABSTRACT FROM AUTHOR]

In this note, we establish a distortion theorem for locally biholomorphic Bloch mappings $$f$$ satisfying $$||f||_{0}=1$$ and $$\det f'(0)=\alpha \in (0,1],$$ where $$\Vert f\Vert _{0}=\mathrm {sup}\{(1-|z|^{2})^\frac{n+1}{2n}|\det f'(z)| ^\frac{1}{n}:z\in \mathcal {B}^{n}\}.$$ This result extends the result of Bonk, Minda, and Yanagihara of one complex variable to higher dimensions. Moreover, a lower estimate for the radius of the largest univalent ball in the image of $$f$$ centered at $$f(0)$$ is given. [ABSTRACT FROM AUTHOR]

Let $$\mathcal {U}$$ denote the class of normalized analytic functions $$f$$ in the open unit disk $$\mathbb {D}$$ satisfying The $$\mathcal {U}$$ -radius is obtained for several classes of functions. These include the class of normalized analytic functions $$f$$ satisfying the inequality $${{\mathrm{Re}}}\, f(z)/g(z)>0$$ or $$\left| f(z)/g(z)-1\right| < 1$$ in $$\mathbb {D},$$ where $$g$$ belongs to a certain class of functions, the class of functions $$f$$ satisfying $$|f'(z)-1|<1$$ in $$\mathbb {D},$$ and functions $$f$$ satisfying $${{\mathrm{Re}}}\, f(z)/z>\alpha , 0 \le \alpha < 1,$$ in $$\mathbb {D}.$$ A recent conjecture by Obradović and Ponnusamy concerning the radius of univalence for a product involving univalent functions is also shown to hold true. [ABSTRACT FROM AUTHOR]

A ring $$R$$ is $$CN$$ if and only if for any $$x\in N(R)$$ and $$y\in R$$ , $$((1+x)y)^{n+k}=(1+x)^{n+k}y^{n+k}$$ , where $$n$$ is a fixed positive integer and $$k=0, 1,2$$ ; (2) Let $$R$$ be a $$CN$$ ring and $$n\ge 1$$ . If for any $$x, y\in R\backslash N(R)$$ , $$(xy)^{n+k}=x^{n+k}y^{n+k}$$ , where $$k=0, 1,2$$ , then $$R$$ is commutative; (3) Let $$R$$ be a ring and $$n\ge 1$$ . If for any $$x\in R\backslash N(R)$$ and $$y\in R$$ , $$(xy)^k=x^ky^k$$ , $$k=n, n+1, n+2$$ , then $$R$$ is commutative; (4) $$NLI $$ exchange rings are clean. [ABSTRACT FROM AUTHOR]

We show that m points and n two-dimensional algebraic surfaces in $${\mathbb {R}}^4$$ can have at most $$O(m^{{k}/({2k-1})}n^{({2k-2})/({2k-1})}+m+n)$$ incidences, provided that the algebraic surfaces behave like pseudoflats with k degrees of freedom, and that $$m\le n^{(2k+2)/3k}$$ . As a special case, we obtain a Szemerédi-Trotter type theorem for 2-planes in $${\mathbb {R}}^4$$ , provided $$m\le n$$ and the planes intersect transversely. As a further special case, we obtain a Szemerédi-Trotter type theorem for complex lines in $${\mathbb {C}}^2$$ with no restrictions on m and n (this theorem was originally proved by Tóth using a different method). As a third special case, we obtain a Szemerédi-Trotter type theorem for complex unit circles in $${\mathbb {C}}^2$$ . We obtain our results by combining several tools, including a two-level analogue of the discrete polynomial partitioning theorem and the crossing lemma. [ABSTRACT FROM AUTHOR]

Let G be a finite group. We give a criterion of nilpotency of G based on the existence of elements of certain order in each section of G. [ABSTRACT FROM AUTHOR]

We consider the problem of the optimal recovery of harmonic functions in the ball from inaccurate information on the Radon transform. Presented are the error of the optimal recovery and the set of optimal methods for which this error is attained. [ABSTRACT FROM AUTHOR]

Differential substitutions of arbitrary order for vectorial generalizations of the mKdV equation are considered. The fact that these equations admit only substitutions of the form u = F( v, v) is established. [ABSTRACT FROM AUTHOR]