On the Proof of the Contradiction of Set Theory

The article is devoted to proving the inconsistency of set theory arising from the existence of strange trees. All steps of the proof rely on common informal set-theoretic reasoning, but they take into account the prohibitions that were introduced into axiomatic set theories in order to overcome the difficulties encountered by the naive Cantor set theory. Therefore, in fact, the article is about proving the inconsistency of existing axiomatic set theories, in particular, the ZFC theory.

Pythagorean Neutrosophic Planar Graphs with an Application in Decision-Making

Graph theory has a significant impact and is crucial in the structure of many real-life situations.To simulate uncertainty and ambiguity,many extensions of graph theoretical notions were created.Planar graphs play a vital role in modelling which has the property of non-crossing edges.Although crossing edges benefit,they have some drawbacks,which paved the way for the introduction of planar graphs.The overall purpose of the study is to contribute to the conceptual development of the Pythagorean Neutrosophic graph.The basic methodology of our research is the incorporation of the analogous concepts of planar graphs in the Pythagorean Neutrosophic graphs.The significant finding of our research is the introduction of Pythagorean Neutrosophic Planar graphs,a conceptual blending of Pythagorean Neutro-sophic and Planar graphs.The idea of Pythagorean Neutrosophic multigraphs and dual graphs are also introduced to deal with the ambiguous situations.This paper investigates the Pythagorean Neutrosophic planar values,which form the edges of the Pythagorean neutrosophic graphs.The concept of Pythagorean Neutrosophic dual graphs,isomorphism,co-weak and weak isomorphism have also been explored for Pythagorean Neutrosophic planar graphs.A decision-making algorithm was proposed with a numerical illustra-tion by using the Pythagorean Neutrosophic fuzzy graph.

Exact Graph Pattern Matching:Applications,Progress and Prospects

Graph pattern matching(GPM)can be used to mine the key information in graphs.Exact GPM is one of the most commonly used methods among all the GPM-related methods,which aims to exactly find all subgraphs for a given query graph in a data graph.The exact GPM has been widely used in biological data analyses,social network analyses and other fields.In this paper,the applications of the exact GPM were first introduced,and the research progress of the exact GPM was summarized.Then,the related algorithms were introduced in detail,and the experiments on the state-of-the-art exact GPM algorithms were conducted to compare their performance.Based on the experimental results,the applicable scenarios of the algorithms were pointed out.New research opportunities in this area were proposed.

About the Strange Tree Paradox and Possible Inconsistency of Set Theory

The existence of “strange trees” is proven and their paradoxical nature is discussed, due to which set theory is suspected of being contradictory. All proofs rely on informal set-theoretic reasoning, but without using elements that were prohibited in axiomatic set theories in order to overcome the difficulties encountered by Cantor’s naive set theory. Therefore, in fact, the article deals with the possible inconsistency of existing axiomatic set theories, in particular, the ZFC theory. Strange trees appear when uncountable cardinals appear.

Isomorphism Continuum Stored Energy Functional for Finite Thermoelastic Deformation

Continuum mechanics for isotropic finite thermoelastic deformations have been reviewed. Thermal effects on mechanical responses of rubbers have been captured by the isomorphism continuum stored energy (CSE) functional with the multiplicative decomposition of deformation gradient while preserving the structure of symmetry for finite structural deformation. The CSE finite thermoelastic model fits and predicts experimental data of SR and NR-C60 rubbers at different external temperatures. For internal temperature effects of both NR and NR-SIC rubbers, the CSE finite thermoelastic model of stored energy and entropy, along with the newly developed CTE and CI models, fits both nominal stress-stretch and temperature change-stretch experimental data in uniaxial extension tests.

An Introduction to the Theory of Matrix Near-Rings

Matrix rings are prominent in abstract algebra. In this paper we give an overview of the theory of matrix near-rings. A near-ring differs from a ring in that it does not need to be abelian and one of the distributive laws does not hold in general. We introduce two ways in which matrix near-rings can be defined and discuss the structure of each. One is as given by Beildeman and the other is as defined by Meldrum. Beildeman defined his matrix near-rings as normal arrays under the operation of matrix multiplication and addition. He showed that we have a matrix near-ring over a near-ring if, and only if, it is a ring. In this case it is not possible to obtain a matrix near-ring from a proper near-ring. Later, in 1986, Meldrum and van der Walt defined matrix near-rings over a near-ring as mappings from the direct sum of n copies of the additive group of the near-ring to itself. In this case it can be shown that a proper near-ring is obtained. We prove several properties, introduce some special matrices and show that a matrix notation can be introduced to make calculations easier, provided that n is small.

On the Aesthetic Wonder and Unity at the Peak of Literary Theory in the Six Dynasties Key

The study found that the classics born during the turbulent Six Dynasties period,starting from Cao Pi’s“The Classics”to“Wen Fu”,“Wen Xin Diao Long”,“Shipin”and other classics,can all be analyzed to have a common origin.Attributes,that is,the philosophical strategy of dualistic isomorphism of explicit and implicit,is also a pattern;it can be proven that this is the philosophical key to interpreting and studying the classics of the Six Dynasties.It not only governs many aesthetic principles and aesthetic methods,but also emphasizes the need to transcend the appearance of current existence,transcend the current utilitarianism,and pay attention to the hidden and non-existent,just like paying attention to dark matter and antimatter in the universe.Pay attention to the origin of Shipin and expand aesthetic horizons;this philosophical secret not only exists in the classics of the Six Dynasties,it has its unity,graphics and vitality,as well as its historical inheritance and continuous operation.Its trajectory marks the context of literary theory and aesthetics;the modern construction of contemporary literary theory must be based on its operating trends.Only by transcending all utilitarianism can we better demonstrate the natural characteristics of Chinese tradition such as nationality.Otherwise,literary theory,aesthetics,and philosophy will not be self-consistent.

Crossed modules of Lie color algebras

The linear operations of the equivalent classes of crossed modules of Lie color algebras are studied. The set of the equivalent classes of crossed modules is proved to be a vector space, which is isomorphic with the homogeneous components of degree zero of the third cohomology group of Lie color algebras. As an application of this theory, the crossed modules of Witt type Lie color algebras is described, and the result is proved that there is only one equivalent class of the crossed modules of Witt type Lie color algebras when the abelian group Г is equal to Г＋. Finally, for a Witt type Lie color algebra, the classification of its crossed modules is obtained by the isomorphism between the third cohomology group and the crossed modules.

The Lower-bound Estimate of D[X_(2m)] and Fast Construction of Matrix X_(2m)

In this paper, the character matrix x n is studied, fast construction method of matrix X 2m is provided. And it is proved that the lower bound estimate of the number of an Latin squares matrix D[X m] is2m(2m)!+∑mi=2[(2m)!] 2∏ij=1K j!∏rj=1b j!.

ON THE EXISTENCE OF ZERO POINTS OF A CONTINUOUS FUNCTION

A sufficient condition is given to assert that a continuous mapping between Rm and Rn has a zero. The constructive proof of the result is based upon continuation methods and supplies the existence of a path leading to the zero point.

ISOMORPHISMS AND DERIVATIONS IN C*-ALGEBRAS

In this article, we prove the Hyers-Ulam-Rassias stability of the following Cauchy-Jensen functional inequality：‖f （x） ＋ f （y） ＋ 2f （z） ＋ 2f （w）‖ ≤‖ 2f x ＋ y2 ＋ z ＋ w ‖（0.1）This is applied to investigate isomorphisms between C＊-algebras, Lie C＊-algebras and JC＊-algebras, and derivations on C＊-algebras, Lie C＊-algebras and JC＊-algebras, associated with the Cauchy-Jensen functional equation 2f （x ＋ y/2 ＋ z ＋ w） = f（x） ＋ f（y） ＋ 2f（z） ＋ 2f（w）.

Generalized super-Virsoro algebras and their Verma modules

For any additive subgroup M of a field F and α∈F such that 2α∈M ,there are two classes of generalized super-Virasoro algebras denoted by SVir [M,α] and SVir [M,α] by Su and Zhao. The latter is in fact a trivial extension of the former. In this paper,based on the discussion on isomorphisms,the Verma modules of Svir [ M,α] are studied,and the irreducibility of these modules are obtained.

The Homomorphism and Isomorphism on Hypergroup

Basing on the papers from [1] to [4], this paper gives some further research,mainly solves the following problems:(1) It proved that several theorems of subgroup which have been raised to the hypergroup are still true.(2) It proved that the isomorphous relationship of the bottomgroups which guid to the hypergroup can still keep such relationship.(3) It proved that the basic isomorphous Theorem and the homogenous Theorem of the bottomgroup which raise to the hypergroup are still true.(4) It point out that when the isomorphous bottomgroups have been raised to the hypergroup,the conditions which form the isomorphism should be lessened.

Simulated annealing algorithm for detecting graph isomorphism

Evolutionary computation techniques have mostly been used to solve various optimization problems, and it is well known that graph isomorphism problem （GIP） is a nondeterministic polynomial problem. A simulated annealing （SA） algorithm for detecting graph isomorphism is proposed, and the proposed SA algorithm is well suited to deal with random graphs with large size. To verify the validity of the proposed SA algorithm, simulations are performed on three pairs of small graphs and four pairs of large random graphs with edge densities 0.5, 0.1, and 0.01, respectively. The simulation results show that the proposed SA algorithm can detect graph isomorphism with a high probability.

Selective reduction of carbon dioxide into amorphous carbon over activated natural magnetite

Natural magnetite formed by the isomorphism substitutions of transition metals,including Fe,Ti,Co,etc.,was activated by mechanical grinding followed by H2 reduction.The temperature-programmed reduction of hydrogen(H2-TPR)and temperature-programmed surface reaction of carbon dioxide(CO2-TPSR)were carried out to investigate the processes of oxygen loss and CO2 reduction.The samples were characterized by X-ray diffraction(XRD),field emission scanning electron microscopy(FE-SEM),and energy-dispersive X-ray spectroscopy(EDS).The results showed that the stability of spinel phases and oxygen-deficient degree significantly increased after natural magnetite was mechanically milled and reduced in H2 atmosphere.Meanwhile,the activity and selectivity of CO2 reduction into carbon were enhanced.The deposited carbon on the activated natural magnetite was confirmed as amorphous.The amount of carbon after CO2 reduction at 300°C for 90 min over the activated natural magnetite was 2.87wt%higher than that over the natural magnetite.

Two 3-D Neutral Zn(Ⅱ)-MOFs Generated from the Asymmetric Schiff-base Ligand

Two new Zn（Ⅱ）-MOFs were synthesized based on an asymmetric Schiff-base ligand LH, which is obtained by the condensation of 5-formyl-8-hydroxyquinoline and 3-pyridinecar- boxylic acid hydrazide. In different mixed solvents, we get two new isomorphism compounds, [Zn（L）2]＇CaHsO2-2MeOH （1） and [Zn（L）2]＇CH2CIz＇2MeOH （2）. Compounds 1 and 2 have been characterized by single-crystal X-ray diffraction, elemental analysis and IR. Compound 1 crystallizes in monoclinic, space group P2/c, with a - 13.856（4）, b = 14.156（4）, c = 9.620（3）А, β = 110.167（5）°, V = 1771.3（9） /k3, Dc = 1.500 g/cm3, C3sH3sNsOsZn, Mr = 800.13, F（000） = 832, p（MoKa） = 0.761 mm-1, Z = 2, the final R = 0.0596 and wR = 0.1221 for 2112 observed reflections （I 〉 2σ（0）. Compound 2 belongs to monoclinic, space group P21/c, with a = 13.575（3）, b = 14.130（3）, c = 9.485（2） A, β = 107.908（3）°, V= 1731.2（6） A3, Dc = 1.529 g/cm3, CasH32C12NsO6Zn, Mr = 796.96, F（000） = 820, g（MoKa） = 0.926 mm-1, Z = 2, the final R -- 0.0527 and wR = 0.1395 for 2501 observed reflections （I 〉 2σ-（/））. The Zn（II） centers in both 1 and 2 display a distorted octahedral coordination geometry. The octahedral Zn（lI） nodes and L linkages build up 2-D nets consisting of parallelogram-like grids in the ab plane. The neighboring nets stack through hydrogen bonds to generate 3-D rhombic channels. In 1, the 1,4-dioxane and MeOH guests alternatively arrange in the channels and effectively take up the free space; while in 2, the CH2CI/guests are arranged in the channels. In addition, the encapsulated MeOH molecules are bonded to the framework by hydrogen bonds.

Simplicities and Automorphisms of a Special Infinite Dimensional Lie Algebra

In this paper, a special infinite dimensional Lie algebra is studied. The infinite dimensional Lie algebra appears in the fields of conformal theory, mathematical physics, statistic mechanics and Hamilton operator. The infinite dimensional Lie algebras is pop ularized Virasoro-like Lie algebra. Isomorphisms, homomorphisms, ideals of the infinite dimensional Lie algebra are studied.

GPU acceleration of subgraph isomorphism search in large scale graph

A novel framework for parallel subgraph isomorphism on GPUs is proposed, named GPUSI, which consists of GPU region exploration and GPU subgraph matching. The GPUSI iteratively enumerates subgraph instances and solves the subgraph isomorphism in a divide-and-conquer fashion. The framework completely relies on the graph traversal, and avoids the explicit join operation. Moreover, in order to improve its performance, a task-queue based method and the virtual-CSR graph structure are used to balance the workload among warps, and warp-centric programming model is used to balance the workload among threads in a warp. The prototype of GPUSI is implemented, and comprehensive experiments of various graph isomorphism operations are carried on diverse large graphs. The experiments clearly demonstrate that GPUSI has good scalability and can achieve speed-up of 1.4–2.6 compared to the state-of-the-art solutions.

SMALL INTO ISOMORPHISM FROM(c_0)INTO C(Ω)TYPE

If T is an isomorphism of （c0） into C（Ω） （where Ω is a sequentially compact and paracompact space, or a compact metric space in particular）, which satisfies the condition ｜｜T｜｜·｜｜T^-1｜｜ ≤ 1 ＋ε for some ε ∈ （0,1/5）, then T/｜｜T｜｜ is close to an isometry with an error less than 9ε. The proof of this article is simple without using the dual space or adjoint operator.

P-kernel Normal Systems in Weakly P-inversive Semigroups

In this paper, we first give the concept of weakly P-inversive semigroup S（P）. Then we describe the strong P-congruences on S（P） in terms of their P-kernel normal systems. It is proved that there is a bijection between the strong P-congruences and the P-kernel normal systems. Finally, it is also prove that the lattice of strong P-congruences and the lattice of P-kernel normal systems on S（P） are isomorphic.