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This paper describes the design of the normalising tactic ring_exp for the Lean prover. This tactic improves on existing tactics by extending commutative rings with a binary exponent operator. An inductive family of types represents the normal form, enforcing various invariants. The design can also be extended with more operators.

In this paper, we first generalize to any hyper-Kahler manifold X with $$b_3(X)\not =0$$ results proved by O’Grady for hyper-Kahler manifolds of generalized Kummer type. In the second part, we restrict to hyper-Kahler manifolds of generalized Kummer type and prove, using results of Markman, that their Kuga–Satake correspondence is algebraic.

In a recently published theorem on the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings, Tang et al. (J. Inequal. Appl. 2015:305, 2015) studied a uniformly convex and 2-uniformly smooth real Banach space with the Opial property and best smoothness constant κ satisfying the condition $0 0 < κ < 1 2 , as a real Banach space more general than Hilbert spaces. A well-known example of a uniformly convex and 2-uniformly smooth real Banach space with the Opial property is $E=l_{p}$ E = l p , $2\leq p 2 ≤ p < ∞ . It is shown in this paper that, if κ is the best smoothness constant of E and satisfies the condition $0 0 < κ ≤ 1 2 , then E is necessarily $l_{2}$ l 2 , a real Hilbert space. Furthermore, some important remarks concerning the proof of this theorem are presented.

In this short article, we mainly give some examples to prove that the main results of Section 4 of the paper [W. S. Liao and J. Wu, Ann. Func. Anal. 6 (2015), no. 3, 191–202] are not true. Then we give the corrected matrix inequalities.

Giving the proper citations, it is shown that, except of Lemma 2.4 and Theorem 2.6, almost all the results of the paper in the title have been previously published for ordered hypersemigroups in Eur. J. Pure Appl. Math. and they are not new. There are also two results obtained from ordered semigroups just putting a ``$\circ$" instead of ``$\cdot$" (that isn't a correct way to work), without reference to ordered semigroups on which the results on ordered hypersemigroups are based. One of them can be obtained as corollary to a theorem in Eur. J. Pure Appl. Math. as well, and it is not new.

In this note we offer some criticism on the paper “On an identity for zeros of Bessel functions” by Baricz et al. [3] . The paper gives identities of type Stieltjes–Calogero for the sums of reciprocals of differences of fourth powers of zeros of Bessel functions. Although interesting in principle, by containing one too many sums of similar complexity the identities fail to convey the true spirit of the work of Stieltjes and Calogero. We rectify this by providing what we think is the correct type of identity for the above-said sums, in the general setup of entire functions of order

The paper alluded to in the title contains the following striking result: Let $I$ be the unit interval and $\Delta$ the Cantor set. If $X$ is a quasi Banach space containing no copy of $c_0$ which is isomorphic to a closed subspace of a space with a basis and $C(I, X)$ is linearly homeomorphic to $C(\Delta, X)$, then $X$ is locally convex, i.e., a Banach space. It is shown that Kalton result is sharp by exhibiting non locally convex quasi Banach spaces X with a basis for which $C(I, X)$ and $C(\Delta, X)$ are isomorphic. Our examples are rather specific and actually in all cases X is isomorphic to $C(\Delta, X)$ if $K$ is a metric compactum of finite covering dimension., Comment: 4 pages

This paper is a corrigendum to the article ``On the ideal theorem for number fields''. The main result of this paper proves to be untrue and is replaced by an estimate of a weighted sum with an improved error term.

The purpose of this short note is to present a correction of the proof of the main result given in the paper “Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings,” Demonstr. Math. 53 (2020), 38–43.

In this paper we correct an inaccuracy that appears in the proof of Theorem 1. in Czerwik's article "Contraction mappings in $b$-metric spaces.", Acta Math. Inform. Univ. Ostraviensis, 1:5--11, 1993., 3 pages

We revisit the problem of the predominance of the 'weakest' species in the context of Lotka-Volterra and May-Leonard implementations of a spatial stochastic rock-paper-scissors model in which one of the species has its predation probability reduced by $0 < \mathcal{P}_w < 1$. We show that,despite the different population dynamics and spatial patterns, these two implementations lead to qualitatively similar results for the late time values of the relative abundances of the three species (as a function of $\mathcal{P}_w$), as long as the simulation lattices are sufficiently large for coexistence to prevail --- the 'weakest' species generally having an advantage over the others (specially over its predator). However, for smaller simulation lattices, we find that the relatively large oscillations at the initial stages of simulations with random initial conditions may result in a significant dependence of the probability of species survival on the lattice size and total simulation time., 7 pages and 6 figures

This short and concise note provides a detailed exposition of the approach and results established by (Lin et al, 2015, pp.271-279). We show that the obtained results are not particularly surprising and new. Namely, using an old result due to K. Deimling it is indicated that tvs-cone metric spaces over solid cones are actually cone metric spaces over normal solid cones. Hence, there are only cone metric spaces over normal solid cones or over normal non-solid cones. One question still unanswered is whether an ordered topological vector space with a non-normal non-solid cone exists.

In 1903, Epstein published his proof of meromorphic continuation and a functional equation for Dirichlet series associated with quadratic forms, now called Epstein zeta-functions. However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show. In this article we present and analyze Hurwitz's notes and compare his reasoning with Epstein's paper in detail.

A description of two alternative coordinate systems and their use in graphing conic sections. This alternative graph paper helps students explore the idea of eccentricity using the definitions of the conic sections. Includes multiple examples of the uses of these alternative graphing sections, along with focus - directrix definitions of conic sections to be used with the new coordinate systems.

We are studying topological properties of the Julia set of the map F ( z , p ) = ( ( 2 z p + 1 − 1 ) 2 , ( p − 1 p + 1 ) 2 ) F(z, p)=\left (\left (\frac {2z}{p+1}-1\right )^2, \left (\frac {p-1}{p+1}\right )^2\right ) of the complex projective plane P C 2 \mathbb {P}\mathbb {C}^2 to itself. We show a relation between this rational function and an uncountable family of “paper folding” plane filling curves.

We study a class of algebras we regard as generalized Rock-Paper-Scissors games. We determine when such algebras can exist, show that these algebras generate the varieties generated by hypertournament algebras, count these algebras, study their automorphisms, and determine their congruence lattices. We produce a family of finite simple algebras., Submitted to the special Topical Collection of Algebra Universalis honoring Ralph Freese, Bill Lampe, and JB Nation

In (2012), Leonetti and Siepe [10] considered solutions to boundary value problems of some anisotropic elliptic equations of the type { ∑ i = 1 n D i ( a i ( x , D u ( x ) ) ) = 0 , x ∈ Ω , u ( x ) = θ ( x ) , x ∈ ∂ Ω . Under some suitable conditions, they obtained an integrability result, which shows that higher integrability of the boundary datum θ forces solutions u to have higher integrability as well. In the present paper, we consider K ψ , θ ( p i ) -obstacle problems of the nonhomogeneous anisotropic elliptic equations ∑ i = 1 n D i ( a i ( x , D u ( x ) ) ) = ∑ i = 1 n D i f i ( x ) under some controllable growth and monotonicity conditions. We obtain an integrability result, which can be regarded as a generalization of the result due to Leonetti and Siepe.

In several recent papers, attempts have been made to apply Wardowski's method of $F$-contractions in order to obtain fixed point results for single and multivalued mappings in $b$-metric spaces. In this article, it is shown that in most cases the conditions imposed on respective mappings are too strong and that the results can be obtained directly, i.e., without using most of the properties of auxiliary function $F$.

Ahmad et al. (2015) consider a Transmuted Kumaraswamy distribution and study certain properties of their distribution. In the title of their paper they mention characterization of this distribution, but no characterization are presented in their paper. In the present short note, we establish certain characterizations of the Transmuted Kumaraswamy distribution in three directions.

We would like to present the ideas of Makoto Matsumoto and Andras Rapcsak in the projective theory of Finsler spaces. Finally, we give some remarks using of Matsumoto–Rapcsak Theorem.

In [2003], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively...

of the paper: "G. Picavet and M. Picavet-L'Hermitte, Modules with finitely many submodules, Int. Electron. J. Algebra, 19 (2016), 119-131.":We characterize ring extensions $R \subset S$ having FCP (FIP), where $S$ is the idealization of some $R$-module. As a by-product we exhibit characterizations of the modules that have finitely many submodules. Our tools are minimal ring morphisms, while Artinian conditions on rings are ubiquitous. $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

In this short note, it is shown that there is a gap in the proof of Theorem 11 in the paper of Meyer and Neutsch (J. of Algebra, 1993). We prove, nevertheless, that the statement of the theorem is true and fix the proof by using a certain extremal property of idempotents which has an independent interest., 6 pages, submitted 2017

The primary role played by analogy in Henri Poincare’s work, and in particular in his “analysis situs” papers, is emphasized. Poincare’s “sixth example” (showing that Betti numbers do not suffice to classify 3-manifolds) and his construction of the homology sphere are discussed in detail.

The aim of this paper is to study a new equivalent form of the weak maximum principle for a large class of differential operators on Riemannian manifolds. This new form has been inspired by the work of Berestycki, Hamel and Rossi for trace operators, and allows us to shed new light on it and to introduce a new sufficient bounded Khas’minskii type condition for its validity. We show its effectiveness by applying it to obtain some uniqueness results in a geometric setting.

Let [Formula: see text] be a projective variety of dimension [Formula: see text] over an algebraically closed field of arbitrary characteristic. We prove a Fujiki–Lieberman type theorem on the structure of the automorphism group of [Formula: see text]. Let [Formula: see text] be a group of zero entropy automorphisms of [Formula: see text] and [Formula: see text] the set of elements in [Formula: see text] which are isotopic to the identity. We show that after replacing [Formula: see text] by a suitable finite-index subgroup, [Formula: see text] is a unipotent group of the derived length at most [Formula: see text]. This result was first proved by Dinh et al. for compact Kähler manifolds.

We complete a gap in the proof that exceptional polynomials are complete orthogonal systems in the associated Hilbert spaces.

Perron-Frobenius theorem describe five properties of irreducible nonnegative matrix. Leslie matrix is one of nonnegative matrix. Leslie matrix that used in this research is limited to the irreducible Leslie matrix. In previous research has been proven that irreducible Leslie matrix satisfies three properties in Perron-Frobenius theorem by spectral radius. This research completed the previous research, proving that irreducible Leslie matrix has a unique Perron vector and satisfies Collatz Wielandt formula. Leslie matrix is a primitive matrix. It is used to calculate the number of populations in the future. Growth of population is interpreted by value spectral radius of Leslie matrix.

To prove Theorem 2, we use the following two facts about the class F: 1. F is a differential ring [2]. This means that F is closed under addition, multiplication and differentiation. 2. All functions y ∈ F are entire functions of completely regular growth in the sense of Levin–Pfluger [4], with piecewise-trigonometric indicators, the notions which we recall now. Let f be a holomorphic function in an angular sector S = {re : |θ−θ0| 0}. We say that f has completely regular growth with respect to order ρ > 0 if the following finite limit exists

In (Barbour, 1990) foundations for diffusion approximation via Stein's method are laid. This paper has been cited more than 130 times and is a cornerstone in the area of Stein's method. A semigroup argument is used therein to solve a Stein equation for Gaussian diffusion approximation. We prove that, contrary to the claim in (Barbour, 1990), the semigroup considered therein is not strongly continuous on the Banach space of continuous, real-valued functions on D[0,1] growing slower than a cubic, equipped with an appropriate norm. We also provide a proof of the exact formulation of the solution to the Stein equation of interest, which does not require the aforementioned strong continuity. This shows that the main results of (Barbour, 1990) hold true., Comment: 8 pages

In this paper, we introduce some notions of invariancy for the Ricci tensor on real hypersurfaces in complex two-plane Grassmannians G 2 ( C m + 2 ) , namely, F -invariant and invariant Ricci tensor. Using these notions, we give non-existence theorem and characterization for the special case among the real hypersurfaces of Type ( A ) in G 2 ( C m + 2 ) , respectively. Here the distribution F is defined by F = [ ξ ] ∪ D ⊥ where D ⊥ = Span { ξ 1 , ξ 2 , ξ 3 } and [ ξ ] = Span { ξ } .

In this paper we introduce a notion of a t-filter on residuated lattices which is a generalization of several special types of filters. We provide some basic properties of t-filters and show how particular results about special types of filters (e.g. Extension property, Triple of equivalent characteristics, and Quotient characteristics) are uniformly covered by this simple general framework.

Theorem 3.30 in the published paper [1] is not correct. A valid reasoning concerning this error is given.

This is a continuation of our first paper in [WY16]. There are two purposes of this paper: One is to give a proof of the main result in [WY16] without going through the argument depending on numerical effectiveness. The other one is to provide a proof of our conjecture, mentioned in [TY], where the assumption of negative holomorphic sectional curvature is dropped to quasi-negative. We should note that a solution to our conjecture is also provided by Diverio-Trapani [DT]. Both proofs depend on our argument in [WY16]. But our argument here makes use of the argument given by the second author and Cheng in [CY75].

In the paper (HA14), unfortunately, a computational error occurred in one estimate. Although the wrong estimate does not affect the main results, we want to present the necessary corrections. Essentially, Lemma 5.2 has to be corrected and, since it is used in the proof of Theorem 5.1, the proof of this theorem also requires an adaptation. (i) The corrected formulation of Lemma 5.2 is as follows. Lemma 5.2 For Lagrange finite elements with a shape-regular family of affine meshes { T n h } h>0 there is a constant C > 0 independent of q and h such that for all w ∈ Wh and q = 2m, m ∈N: CΛq−2 p (∇w,∇Ip h (wq−1))T ∫ T ‖∇w‖l2‖w‖ q−2 0,∞,T dx, ∀T ∈ T n h , (5.1) where Λp = ‖ ∑ndof i=1 |φi|‖0,∞,T is the Lebesgue constant.

This is a mathematical commentary on Teichmuller's paper ``Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flachen" (Determination of extremal quasiconformal maps of closed oriented Riemann surfaces). This paper is among the last (and may be the last one) that Teichmuller wrote on the theory of moduli. It contains the proof of the so-called Teichmuller existence theorem for a closed surface of genus at least 2. For this proof, the author defines a mapping between a space of equivalence classes of marked Riemann surfaces (the Teichmuller space) and a space of equivalence classes of certain Fuchsian groups (the so-called Fricke space). After that, he defines a map between the latter and the Euclidean space of dimension 6g-6 Using Brouwer's theorem of invariance of domain, he shows that this map is a homeomorphism. This involves in particular a careful definition of the topologies of Fricke space, the computation of its dimension, and comparison results between hyperbolic distance and quasiconformal dilatation. The use of the invariance of domain theorem is in the spirit of Poincare and Klein's use of the so-called ``continuity principle" in their attempts to prove the uniformization theorem.

We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions providing analytic continuation through out the whole complex plane. Additionally we demonstrate some series representations for the initial Stieltjes constants appearing in the Laurent expansion of the Hurwitz zeta function. A particular point of elaboration in these developments is the hypergeometric form and its equivalents for certain derivatives of the incomplete Gamma function. Finally, we evaluate certain integrals including $\int_{\tiny{Re} s=c} {{\zeta(s)} \over s} ds$ and $\int_{\tiny{Re} s=c} {{\eta(s)} \over s} ds$, with $\zeta$ the Riemann zeta function and $\eta$ its alternating form., Comment: 17 pages, no figures

We show that paper folding words contain arbitrarily large abelian powers.

In [32], A. Najati and S.-M. Jung obtained the Hyers-Ulam stability of the generalized quadratic functional equation $f(rx+sy)+rsf(x-y)=rf(x)+sf(y)$ with $r+s=1$, $r\neq0$, $s\neq0$, provided that $f$ is an even mapping. The purpose of this paper is to remove this restriction.