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Mathematical knowledge plays a system-forming role in education. The aim of this paper is to find out the didactic features of mathematical knowledge formation. The study involved 50 professors, 300 students. The criteria of mathematical knowledge acquisition include reliability, autonomy, and stability. The main study objectives are as follows. First, to clarify and scientifically substantiate special functions of mathematical knowledge in the process of engineering activity (bijection, formalization, relativity, and robustness). Second, to determine the formation of academic and applied mathematical knowledge. Technical knowledge can be applied using the special functions of mathematical knowledge. For example, in the form of mathematical models, or in a stable mathematical component in the professional outlook. The academic mathematical knowledge is focused on the systematic integrity of purpose, content, forms, methods and means of education. Along with transdisciplinary holistic worldview formation. Applied mathematical knowledge is focused on integration of teaching, educating, developing functions of education to acquire common cultural and professional competences. In modern conditions of rapid computerization, the opportunities to use mathematical analysis and modeling are increasing. However, basic mathematical knowledge is needed to assess the reliability of computer solutions. [ABSTRACT FROM AUTHOR]

Dopaminergic neurons (DAs) of the rodent substantia nigra pars compacta (SNc) display varied electrophysiological properties in vitro. Despite this, projection patterns and functional inputs from DAs to other structures are conserved, so in vivo delivery of consistent, well-timed dopamine modulation to downstream circuits must be coordinated. Here we show robust coordination by linear parameter controllers, discovered through powerful mathematical analyses of data and models, and from which consistent control of DA subthreshold oscillations (STOs) and spontaneous firing emerges. These units of control represent coordinated intracellular variables, sufficient to regulate complex cellular properties with radical simplicity. Using an evolutionary algorithm and dimensionality reduction, we discovered metaparameters, which when regressed against STO features, revealed a 2-dimensional control plane for the neuron’s 22-dimensional parameter space that fully maps the natural range of DA subthreshold electrophysiology. This plane provided a basis for spiking currents to reproduce a large range of the naturally occurring spontaneous firing characteristics of SNc DAs. From it we easily produced a unique population of models, derived using unbiased parameter search, that show good generalization to channel blockade and compensatory intracellular mechanisms. From this population of models, we then discovered low-dimensional controllers for regulating spontaneous firing properties, and gain insight into how currents active in different voltage regimes interact to produce the emergent activity of SNc DAs. Our methods therefore reveal simple regulators of neuronal function lurking in the complexity of combined ion channel dynamics. [ABSTRACT FROM AUTHOR]

We consider the so-called Encrypted Davies-Meyer (EDM) construction, which turns a permutation P on {0,1}n into a function from {0,1}n to {0,1}n defined as P(P(x)⊕x). A similar construction using two independent permutations, namely P′(P(x)⊕x), was previously analyzed by Cogliati and Seurin (Advances in cryptology—CRYPTO 2016 (Proceedings, Part I). LNCS, vol 9814, pp. 121-149, 2016) who showed that when P and P′ are secret and random, then any black-box adversary needs at least roughly 22n/3 queries to distinguish the construction from a uniformly random function from {0,1}n to {0,1}n. In this paper, we focus on the single-permutation variant of the construction. Our main result is that the PRF-security of the single-permutation EDM construction is also (at least) roughly 22n/3, in the sense that any black-box adversary needs at least this number of queries to distinguish the construction from a uniformly random function. This yields the first PRP-to-PRF conversion method which uses a single permutation, does not shrink the original domain nor range of the permutation, and provides security beyond the birthday bound. [ABSTRACT FROM AUTHOR]

Sufficient conditions for the compactness in generalized Morrey spaces of the composition of a convolution operator and the operator of multiplication by an essentially bounded function are obtained. Very weak conditions on the function are also obtained under which the commutator of the operator of multiplication by such a function and a convolution operator is compact. The compactness of convolution operators in domains of cone type is investigated. [ABSTRACT FROM AUTHOR]

In this paper, a unique and more distinctive encryption algorithm is proposed. This is based on the complexity of highly nonlinear S box in Flesnelet domain. The nonlinear pattern is transformed further to enhance the confusion in the dummy data using Fresnelet technique. The security level of the encrypted image boosts using the algebra of Galois field in Fresnelet domain. At first level, the Fresnelet transform is used to propagate the given information with desired wavelength at specified distance. It decomposes given secret data into four complex subbands. These complex sub-bands are separated into two components of real subband data and imaginary subband data. At second level, the net subband data, produced at the first level, is deteriorated to non-linear diffused pattern using the unique S-box defined on the Galois field . In the diffusion process, the permuted image is substituted via dynamic algebraic S-box substitution. We prove through various analysis techniques that the proposed scheme enhances the cipher security level, extensively. [ABSTRACT FROM AUTHOR]

In this paper we isolate a property for forcing notions, the *-Prikry condition, that is similar to the Prikry con- dition but that is topological: A forcing P satisfies it iff for every p ∈ P and for every open dense D ⊆ P, there are n ∈ ω and q ≤∗ p such that for any r ≤ q with l(r) = l(q) + n, r ∈ D, for some length notion l. This is implicit in many proofs in literature. We prove this for the tree Prikry forcing and the long extender Prikry forcing. [ABSTRACT FROM AUTHOR]

Finding dense spanning trees (DST) in unweighted graphs is a variation of the well studied minimum spanning tree problem (MST). We utilize established mathematical properties of extremal structures with the minimum sum of distances between vertices to formulate some general conditions on the sum of vertex degrees. We analyze the performance of various combinations of these degree sum conditions in finding dense spanning subtrees and apply our approach to practical examples. After briefly describing our algorithm we also show how it can be used on variations of DST, motivated by variations of MST. Our work provide some insights on the role of various degree sums in forming dense spanning trees and hopefully lay the foundation for finding fast algorithms or heuristics for related problems. [ABSTRACT FROM AUTHOR]

We present new criteria for asymptotic stability of two classes of nonlinear neutral delay differential equations. By using two auxiliary functions on a contraction condition, we extend the results in [12]. Also we give two examples that illustrate our results. [ABSTRACT FROM AUTHOR]

In the paper, the authors find the logarithmically complete monotonicity of the Catalan-Qi function related to the Catalan numbers. [ABSTRACT FROM AUTHOR]

Background: Several studies have shown that total depressive symptom scores in the general population approximate an exponential pattern, except for the lower end of the distribution. The Center for Epidemiologic Studies Depression Scale (CES-D) consists of 20 items, each of which may take on four scores: “rarely,” “some,” “occasionally,” and “most of the time.” Recently, we reported that the item responses for 16 negative affect items commonly exhibit exponential patterns, except for the level of “rarely,” leading us to hypothesize that the item responses at the level of “rarely” may be related to the non-exponential pattern typical of the lower end of the distribution. To verify this hypothesis, we investigated how the item responses contribute to the distribution of the sum of the item scores. Methods: Data collected from 21,040 subjects who had completed the CES-D questionnaire as part of a Japanese national survey were analyzed. To assess the item responses of negative affect items, we used a parameter r, which denotes the ratio of “rarely” to “some” in each item response. The distributions of the sum of negative affect items in various combinations were analyzed using log-normal scales and curve fitting. Results: The sum of the item scores approximated an exponential pattern regardless of the combination of items, whereas, at the lower end of the distributions, there was a clear divergence between the actual data and the predicted exponential pattern. At the lower end of the distributions, the sum of the item scores with high values of r exhibited higher scores compared to those predicted from the exponential pattern, whereas the sum of the item scores with low values of r exhibited lower scores compared to those predicted. Conclusions: The distributional pattern of the sum of the item scores could be predicted from the item responses of such items. [ABSTRACT FROM AUTHOR]

Given a suitably regular nonnegative function ω on (0, 1], let Bω denote the space of all holomorphic functions f on the unit ball Bn of Cn that satisfy |∇f(z)| ≤ C ω(1 - |z|)/1 - |z|, z ∈ Bn, with some fixed C = Cf > 0. We obtain a new characterization of Bω functions in terms of their moduli. [ABSTRACT FROM AUTHOR]

With the arrival of the nineteenth century, a process of change guided the treatment of three basic elements in the development of mathematics: rigour, the arithmetization and the clarification of the concept of function, categorised as the most important tool in the development of the mathematical analysis. In this paper we will show how several prominent mathematicians contributed greatly to the development of these basic elements that allowed the solid underpinning of mathematics and the consideration of mathematics as an axiomatic way of thinking in which anyone can deduce valid conclusions from certain types of premises. This nineteenth century stage shares, possibly with the Heroic Age of Ancient Greece, the most revolutionary period in all history of mathematics. [ABSTRACT FROM AUTHOR]

We give a sufficient condition for systems with symmetries to have periodic solutions with equal periods. We show that the main result can be applied both to Hamiltonian and to non-Hamiltonian systems. We apply the main results to produce planar centers with equal period functions. [ABSTRACT FROM AUTHOR]

A signed graph (or sigraph for short) is an ordered pair S = (Su,σ), where Su is a graph, G = (V,E), called the underlying graph of S and σ : E → {+,−} is a function from the edge set E of Su into the set {+,−}. For a sigraph S its •-line sigraph, L•(S) is the sigraph in which the edges of S are represented as vertices, two of these vertices are defined adjacent whenever the corresponding edges in S have a vertex in common, any such L-edge ee′ has the sign given by the product of the signs of the edges incident with the vertex in e ∩ e′. In this paper we establish a structural characterization of •-line sigraphs, extending a well known characterization of line graphs due to Harary. Further we study several standard properties of •-line sigraphs, such as the balanced •-line sigraphs, sign-compatible •-line sigraphs and C-sign-compatible •-line sigraphs. [ABSTRACT FROM AUTHOR]

This is the second of two papers, in which we study the problem of prescribing Webster scalar curvature on the CR sphere as a given function f . Using the Webster scalar curvature flow, we prove an existence result under suitable assumptions on the Morse indices of f . [ABSTRACT FROM AUTHOR]

In this paper, we deduce new conditions for the existence of ground state solutions for the $p$-Laplacian equation $$\begin{equation*} \left \{ \begin{array}{@{}ll} -\mathrm {div}(\nabla u^{p-2}\nabla u)+V(x)u^{p-2}u=f(x, u), \quad x\in {\mathbb {R}}^{N},\\[5pt] u\in W^{1, p}({\mathbb {R}}^{N}), \end{array} \right . \end{equation*}$$ which weaken the Ambrosetti–Rabinowitz type condition and the monotonicity condition for the function $t\mapsto f(x, t)/t^{p-1}$. In particular, both $tf(x, t)$ and $tf(x, t)-pF(x, t)$ are allowed to be sign-changing in our assumptions. [ABSTRACT FROM AUTHOR]

Hill numbers (or the “effective number of species”) are increasingly used to characterize species diversity of an assemblage. This work extends Hill numbers to incorporate species pairwise functional distances calculated from species traits. We derive a parametric class of functional Hill numbers, which quantify “the effective number of equally abundant and (functionally) equally distinct species” in an assemblage. We also propose a class of mean functional diversity (per species), which quantifies the effective sum of functional distances between a fixed species to all other species. The product of the functional Hill number and the mean functional diversity thus quantifies the (total) functional diversity, i.e., the effective total distance between species of the assemblage. The three measures (functional Hill numbers, mean functional diversity and total functional diversity) quantify different aspects of species trait space, and all are based on species abundance and species pairwise functional distances. When all species are equally distinct, our functional Hill numbers reduce to ordinary Hill numbers. When species abundances are not considered or species are equally abundant, our total functional diversity reduces to the sum of all pairwise distances between species of an assemblage. The functional Hill numbers and the mean functional diversity both satisfy a replication principle, implying the total functional diversity satisfies a quadratic replication principle. When there are multiple assemblages defined by the investigator, each of the three measures of the pooled assemblage (gamma) can be multiplicatively decomposed into alpha and beta components, and the two components are independent. The resulting beta component measures pure functional differentiation among assemblages and can be further transformed to obtain several classes of normalized functional similarity (or differentiation) measures, including N-assemblage functional generalizations of the classic Jaccard, Sørensen, Horn and Morisita-Horn similarity indices. The proposed measures are applied to artificial and real data for illustration. [ABSTRACT FROM AUTHOR]

In this note we prove compactness for the Cahn–Hilliard functional without assuming coercivity of the multi-well potential. [ABSTRACT FROM AUTHOR]

Using a generalized Dunkl translation, we obtain an analog of Theorem 5.2 in Younis' paper for the Dunkl transform for functions satisfying the (δ, γ)-Dunkl Lipschitz condition in the space L² (ℝ, |x|2α+1dx). [ABSTRACT FROM AUTHOR]

Abstract: We present some curious formulae for infinite products of twisted -functions. [Copyright &y& Elsevier]

Basing on the priori assumption principle, the main goal of this paper is to propose the rational expansion method that can be used to handle the exact solution of the nonlinear partial differential equation. It is extension of tanh function method. As an application, three classes of exact solutions to Klein-Gordon-SchrÄodinger equation are obtained. [ABSTRACT FROM AUTHOR]

This paper contains a new elementary proof of the Fundamental Theorem of Calculus for the Lebesgue integral. The hardest part of our proof simply concerns the convergence in L1 of a certain sequence of step functions, and we prove it using only basic elements from Lebesgue integration theory. [ABSTRACT FROM AUTHOR]

We generalize, simultaneously, the ring of real-valued continuous functions and the ring of real-valued measurable functions [ABSTRACT FROM AUTHOR]

A function : ℝ ℝ is said to be a Fibonacci function if f(x + 2) = f(x + 1) + f(x) for all x ∈ ℝ. In 2012, some properties on the Fibonacci functions were presented. In this paper, for any positive integer k, a function f : ℝ ℝ is said to be a Fibonacci function with period k if f(x + 2k) = f(x+k) + f(x) for all x ∈ ℝ we present some properties on the Fibonacci functions with period k. [ABSTRACT FROM AUTHOR]

Let be a simple graph. Graph polynomials are a well-developed area useful for analyzing properties of graphs. We consider domination polynomial, matching polynomial, and edge cover polynomial of G. Graphs which their polynomials have few roots can sometimes give surprising information about the structure of the graph. This paper is primarily a survey of graphs whose domination polynomial, matching polynomial, and edge cover polynomial have few distinct roots. In addition, some new unpublished results and questions are concluded. [ABSTRACT FROM AUTHOR]

We provide a sufficient condition for the existence of a cycle C in a connected graph G which is edge-disjoint from two connected subgraphs H1 and H2 of G such that G -- E(C) is connected. [ABSTRACT FROM AUTHOR]

Given a horizontal monoid M in a duoidal category F, we examine the relationship between bimonoid structures on M and monoidal structures on the category F"M of right M-modules which lift the vertical monoidal structure of F. We obtain our result using a variant of the so-called Tannaka adjunction; that is, an adjunction inducing the equivalence which expresses Tannaka duality. The approach taken utilizes hom-enriched categories rather than categories on which a monoidal category acts ("actegories"). The requirement of enrichment in F itself demands the existence of some internal homs, leading to the consideration of convolution for duoidal categories. Proving that certain hom-functors are monoidal, and so take monoids to monoids, unifies classical convolution in algebra and Day convolution for categories. Hopf bimonoids are defined leading to a lifting of closed structures on F to F"M. We introduce the concept of warping monoidal structures and this permits the construction of new duoidal categories. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider non-linear differential equations which are closely related to the generating functions of Frobenius–Euler polynomials. From our non-linear differential equations, we derive some new identities between the sums of products of Frobenius–Euler polynomials and Frobenius–Euler polynomials of higher order. [Copyright &y& Elsevier]

Abstract: We use some recent results of Fitzgerald (2009) on quadratic functions with two trace terms in characteristic 2 to generalize some results of Lahtonen, McGuire and Ward (2007) on Gold and Kasami–Welch functions. Then we get the Walsh spectrum of under certain conditions introduced in Lahtonen et al. (2007) and finally introduce some new results on Gold and Kasami–Welch functions when the degree of extension is even. [Copyright &y& Elsevier]

Gowers norms have been studied extensively both in the direct sense, starting with a function and understanding the associated norm, and in the inverse sense, starting with the norm and deducing properties of the function. Instead of focusing on the norms themselves, we study associated dual norms and dual functions. Combining this study with a variant of the Szemerédi Regularity Lemma, we give a decomposition theorem for dual functions, linking the dual norms to classical norms and indicating that the dual norm is easier to understand than the norm itself. Using the dual functions, we introduce higher order algebras that are analogs of the classical Fourier algebra, which in turn can be used to further characterize the dual functions. [ABSTRACT FROM AUTHOR]

In this paper we introduce double entire rate sequence spaces defined by a modulus function. We study their different properties and obtain some inclusion relations involving these double entire rate sequence spaces. [ABSTRACT FROM AUTHOR]

Abstract: A nonbinary Ford sequence is a de Bruijn sequence generated by simple rules that determine the priorities of what symbols are to be tried first, given an initial word of size which is the order of the sequence being generated. This set of rules is generalized by the concept of a preference function of span , which gives the priorities of what symbols to appear after a substring of size is encountered. In this paper, we characterize preference functions that generate full de Bruijn sequences. More significantly, we establish that any preference function that generates a de Bruijn sequence of order also generates de Bruijn sequences of all orders higher than , thus making the Ford sequence no special case. Consequently, we define the preference function complexity of a de Bruijn sequence to be the least possible span of a preference function that generates this de Bruijn sequence. [Copyright &y& Elsevier]

Abstract: The main goal of this work is to introduce the relation between the partial boolean derivatives of an -variable boolean function and their directional boolean derivatives. [Copyright &y& Elsevier]

Abstract: The aim of the paper is to improve known estimates of the Wallis ratio. Moreover, we show that these improvements are valid, because certain functions involving the continuous version of the Wallis ratio are completely monotone. [Copyright &y& Elsevier]

In this paper, we considered the following second order multi-valued iterative equation λ1F(x)+λ2F²(x)=G(x), and the decreasing solution for decreasing G which may be multi-valued at endpoints is given. [ABSTRACT FROM AUTHOR]

We study some relationships between holomorphic motions, continuous motions, and monodromy. We also study extensions of holomorphic motions over Riemann surfaces and characterize the extendability of holomorphic motions over some planar regions in terms of monodromy. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to generalize the functions with bounded Mocanu variation and define the class of functions with bounded Mocanu variation which map the open unit disk onto conic domain. Many interesting properties of these functions are investigated. [Copyright &y& Elsevier]

Abstract: In this paper, we define generalized -Horadam sequence . After that, we study the properties of the generalized -Horadam sequence and prove some of these properties by means of determinant. Also, we obtain a generating function for the generalized -Horadam sequence. [Copyright &y& Elsevier]

Abstract: In this paper, we establish a functional generalization of the diamond- integral reverse Hölder inequality on time scales. Some related inequalities are also considered. [Copyright &y& Elsevier]

Abstract: We consider retarded functional differential equations in the setting of Kurzweil–Henstock integrable functions and we state an averaging result for these equations. Our result generalizes previous ones. [Copyright &y& Elsevier]

Abstract: We generalize Springer representations to the context of groups over a global function field. The global counterpart of the Grothendieck simultaneous resolution is the parabolic Hitchin fibration. We construct an action of the graded double affine Hecke algebra (DAHA) on the direct image complex of the parabolic Hitchin fibration. In particular, we get representations of the degenerate graded DAHA on the cohomology of parabolic Hitchin fibers, providing the first step towards a global Springer theory. [Copyright &y& Elsevier]

Abstract: Let be a family of meromorphic functions defined in a domain such that for each , all zeros of are of multiplicity at least 3, and all zeros of are of multiplicity at least 2 in . If for each , has at most 1 zero in , ignoring multiplicity, then is normal in . [Copyright &y& Elsevier]

It has been shown that every n-person, perfect information game with no chance moves and bounded, lower semicontinuous payoffs has a subgame perfect-equilibrium in pure strategies. Here the same is proved when the payoffs are bounded and upper semicontinuous. [ABSTRACT FROM AUTHOR]

A commuting tuple $ T =(T_1, \ldots, T_n) \in B(H)^n$ $ \sum_{i=1}^n T_i^*T_i = 1_H$-Toeplitz operators, which he defined to be the solutions $ X \in B(H)$ $ \sum_{i=1}^n T_i^*XT_i = X$ $ X \in B(H)$-Toeplitz operator precisely when $ X$ for every isometry $ J$ $ \mathcal{A}_T \subset B(H)$. As a consequence we deduce that a spherical isometry $ T$-Toeplitz operator is the zero operator. Moreover, we show that if $ \sigma_p(T) = \emptyset$ $ \mathcal{A}_T$-Toeplitz operator. [ABSTRACT FROM AUTHOR]

In this paper, an ancient Chinese algorithm is used to estimate the length constant of a long cooling fin, and an approximate solution formulation is obtained. The obtained results show that this method is a simple but promising method without any requirement for advanced calculus. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a functional generalization of diamond- integral Hölder’s inequality on time scales is established. Some related inequalities are also considered. [Copyright &y& Elsevier]