Study on “double dawn”

The ancient record, “During the first year of King Yi, the day dawned twice at Zheng”, has provided important clues to early Chinese chronicles. The astronomical conditions and visible area distributions related to such a “double dawn” event are discussed, and the precision and current problems in the calculations of ancient astronomical phenomena are shown. On such a basis, all the solar eclipses from 1000 BC to 840 BC are calculated and their associated “double dawn” features investigated. The conclusion that the “double dawn” was a solar eclipse occurring on April 21st, 899 BC is corfirmed to be the most reasonable. Project supported by the Xia-Shang-Zhou Chronology Project and Director Foundation of the Chinese Academy of Sciences.     

Remarks on “Quasi-contraction on a cone metric space”

Recently, D. Ili? and V. Rako?evi? [D. Ili?, V. Rako?evi?, Quasi-contraction on a cone metric space, Appl. Math. Lett. (2008) doi:10.1016/j.aml.2008.08.011] proved a fixed point theorem for quasi-contractive mappings in cone metric spaces when the underlying cone is normal. The aim of this paper is to prove this and some related results without using the normality condition.     

Some Remarks on “Language-Created Entities”

Some entities, such as fictional characters, propositions, properties, events and numbers are prima facie promising candidates for owing their existence to our linguistic and conceptual practices. However, it is notoriously hard to pin down just what sets such allegedly “language-created” entities apart from ordinary entities. The present paper considers some of the features that are supposed to distinguish between entities of the two kinds and argues that, on an independently plausible account of what it takes to individuate objects, the criteria let in more than friends of the strategy might be happy with.

Special issue on “Innovations on model based clustering and classification”

Advances in Data Analysis and Classification -     

Comments on “Enhancements on the Hyperplanes Arrangements in Mixed-Integer Programming Techniques”

In a recent paper by Prodan et al. (J. Optim. Theory Appl. 154:549–572, 2012), a technique was presented to reduce the number of binary variables needed to represent not convex constraints in a mixed-integer programming (MIP) problem. The proposed technique employs tuples of binary variables, which are associated with feasible regions of the feature space. However, since the number of all possible tuples with a given number of bits is a power of two, there may be several unallocated tuples that must be rendered infeasible by imposing suitable constraints. We show in this paper that it is always possible to partition the tuples so that only one inequality is necessary to render all the unallocated tuples and only them infeasible. Moreover, we develop a systematic procedure to perform this partition and write the referred inequality.     

A remark on “Connections and Higgs fields on a principal bundle”

The authors show that a unipotent vector bundle on a non–Kähler compact complex manifold does not admit a flat holomorphic connection in general. It was also construct examples of topologically trivial stable vector bundle on compact Gauduchon manifold that does not admit any unitary flat connection.     

Comment on “A note on a study on an integrable system of coupled KdV equations”

We analyze the note by Muraved [A note on A study on an integrable system of coupled KdV equations [1], Commun Nonlinear Sci Numer Simulat. doi:10.1016/j.cnsns.2010.06.018]. Author tried to show that our approach is curious. We demonstrate that, author has not shown any error and presented a curious note.     

Note on the Paper “A Note on the SVD of Matrices”

Let F be a field, R be an F-algebra having SVD property['], R+ = {a e Rla = a}. In[2], we proved thatLemma 1 Let R be an F-algebra which having SVD proPerty Then R+ is a sub6e1d inF, R+ is a fOrmaIly rea1 6eId, and R+ is Pythagorean with no zerChdivisor.In the proof of Lernma 1 of [2], thereis this step: "FOr any a E R+ l clearly, there ekistsd E R+ n F such that a2 = ad = a', thus a = la E F, is fOllOws that R+ C F".Recently, paper [11 shows that "The proof of this step is no…     

A note on “Reguralizers for structured sparsity”

In this note, the notion of admissible sets contained in the strictly positive orthant introduced in Micchelli et al. (Adv. Comp. Math. 38(3), 455–489 2013) is analyzed. This notion was used to generalize theoretical results and optimization methods for structured sparsity. Unfortunately, we will prove that there is no generalization using admissible sets.     

Estimates of invariant distances on “convex” domains

Estimates for invariant distances of convexifiable, \(\mathbb{C }\) -convexifiable and planar domains are given.     

Corrigendum to “on the restricted partition function”

The Ramanujan Journal - This article is a corrigendum to our paper (Cimpoea? and Nicolae in Ramanujan J 473:565–588, 2018).     

A note on da Costa-Doria “exotic formalizations”

We analyze N. C. A. da Costa and F. A. Doria’s “exotic formalization” of the conjecture P = NP [3–7]. For any standard axiomatic PA extension T and any number-theoretic sentence ${\varphi }$ , we let ${\varphi ^{\star} := \varphi \vee \lnot \mathsf{Con}\left( \mathsf{T}\right)}$ and prove the following “exotic” inferences 1–3. 1. ${\mathsf{T}+\varphi ^{\star}}$ is consistent, if so is T, 2. ${\mathsf{T}+\varphi}$ is consistent, provided that ${\mathsf{T}+\varphi ^{\star}}$ is ω-consistent, 3. ${\mathsf{T}+\varphi}$ is consistent, provided that T is consistent and has the same provably total recursive functions as ${\mathsf{T}+\left( \varphi \leftrightarrow \varphi ^{\star }\right) }$ . Furthermore we show that 1–3 continue to hold for ${\varphi ^{\star} := \varphi _{S} :=\varphi \vee \lnot S}$ , where ${S=\forall x\exists yR\left( x,y\right)}$ is any ${\Pi _{2}^{0}}$ sentence satisfying: 4. ${\left( \forall n\in \omega \right) \left( \mathsf{T}\vdash S_{x}\left[ \underline{n}\right] \right) }$ , 5. ${\mathsf{Con}\left( \mathsf{T}\right) \Rightarrow \mathsf{T}\nvdash S}$ . We observe that if ${\varphi :=\left[ \mathsf{P}=\mathsf{NP}\right] }$ and ${S:= \left[\digamma total\right] }$ , where ${\digamma=\digamma _{\mathsf{T}}}$ is da Costa-Doria “exotic” function with respect to T, then 4, 5 are satisfied for most familiar (presumably) consistent T in question, while ${\varphi _{S}}$ becomes equivalent to da Costa-Doria “exotic formalization” ${\left[ \mathsf{P}=\mathsf{NP}\right]^{\digamma}}$ . Moreover, the corresponding “exotic” inferences 1–3 generalize analogous da Costa-Doria results. Hence these “exotic” inferences are universal for all number-theoretic sentences and not characteristic to the conjecture P = NP. Nor do they infer relative consistency of P = NP (see Conclusion 15 in the text).     

Comments on “A note on “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets””

In a recently published paper “A note on “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets””, Khalil and Hassan pointed out that assertions (3) and (4) of Theorem 3.2 in our previous paper “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets” are not true [2]. Furthermore, they introduced the notions of a generalized trapezoidal interval type-2 fuzzy soft subset and a generalized trapezoidal interval type-2 fuzzy soft equal and used these two notions to correct the flaw in assertions (3) and (4) of Theorem 3.2 in our previous paper. In this paper, we show by a counterexample that Khalil and Hassan's correction is incorrect and provide the modified versions of assertions (3) and (4) of Theorem 3.2, along with a strict proof. In addition, Khalil and Hassan pointed out by a counterexample that assertions (3)–(6) of Theorem 3.5 in our paper are not true and proposed the corrections of those assertions. In this paper, we show that Khalil and Hassan's counterexample and corrections are incorrect and provide a new example to verify the inaccuracies of assertions (3) and (5) of Theorem 3.5 in our paper. Moreover, we offer the modified versions of assertions (3) and (5) of Theorem 3.5 and prove them. Finally, Khalil and Hassan's statement that assertions (4) and (6) of Theorem 3.5 in our previous paper are not true is proven to be incorrect, i.e. assertions (4) and (6) of Theorem 3.5 in our previous paper are true.     

A note on “A study on an integrable system of coupled KdV equations”

We analyze the paper by Wazwaz [Wazwaz AM. A study on an integrable system of coupled KdV equations. Commun Nonlinear Sci Numer Simul, 2010;15:2846–2850]. Author tried to show that the system of coupled KdV equations is completely integrable but he has used the curious approach for the proof. We demonstrate that, author has taken the relation between dependent variables and has obtained the well known result by Hirota for the KdV equation.     

Some remarks on Smale's “Algorithms for solving equations”

In his report at ICM, Smale posed a problem whether the inequalityα(t, ψ _d )≤1 holds for allt∈(0,1), where . This paper gives a negative answer. In addition, elementary proofs are given for the theorems on the tractability of random algorithm and on the average area of approximate zeros. Meanwhile, discussion about the algorithm of the global Newton's method is devoted in this paper. Supported by the National Natural Science Foundation of China, and of Zhejiang Province Dedicated to the 70th birthday of Prof. Feng Kang     

A Note on Donaldson's “Tamed to Compatible” Question

Recently, Tedi Draghici and Weiyi Zhang studied Donaldson's "tamed to compatible" question(Draghici T, Zhang W. A note on exact forms on almost complex manifolds. arXiv: 1111. 7287v1 [math. SG]. Submitted on 30 Nov. 2011).That is, for a compact almost complex 4-manifold whose almost complex structure is tamed by a symplectic form, is there a symplectic form compatible with this almost complex structure? They got several equivalent forms of this problem by studying the space of exact forms on such a manifold. With these equivalent forms, they proved a result which can be thought as a further partial answer to Donaldson's question in dimension 4. In this note, we give another simpler proof of their result.     

“Laplacians” on finitely ramified,graph directed fractals

A finitely ramified graph directed fractal is approximated by an adapted sequence of increasingly refined graphs. The scaling problem for a corresponding sequence of discrete Laplacians is rephrased via a renormalization map comparing two subsequent graphs. A limit set dichotomy for this map is proved: The forward orbit always accumulates at periodic points, even if the corresponding models are disconnected. Thus these periodic points can be numerically approximated by an iteration scheme based on the Schur complement. This allows to turn numerical information via the short-cut test into theorems on the existence of Laplacians on fractals.Mathematics Subject Classification (2000): 31C25, 60J45, 65N55, 47J10, 28A80Research supported by the DFG research group Spektrale Analysis, asymptotische Verteilungen und stochastische Dynamik.The author is greatful to the anonymous referees for pointing out flaws in two original definitions.     

A Note on the “Smith is Huq” Condition

We show that two known conditions which arose naturally in commutator theory and in the theory of internal crossed modules coincide: every star-multiplicative graph is multiplicative if and only if every two effective equivalence relations commute as soon as their normalisations do. This answers a question asked by George Janelidze.