Subjects: Mathematics >> Statistics and Probability submitted time 2018-11-07
Abstract: Abstract. In studying of a class of random neural network, some of relative researchers have proposed Markov model of neural network. Wherein Markov property of the neural network is based on “assuming”. To reveal mechanism of generating of Markov property in neural network, it is studied how infinite-dimensional random neural network (IDRNN) forms inner Markov representation of environment information in this paper.Because of equivalence between markov property and Gibbsian our conclusion is that knowledge is eventually expressed by extreme Gibbs probability measure—ergodic Gibbs probability measure in IDRNN. This conclusion is also applicable to quantum mechanical level of IDRNN. Hence one can see “ concept “- “ consciousness” is generated at particle(ion) level in the brain and is experienced at the level of the neurons; We have discussed also ergodicity of IDRNN with random neural potential. " "
Peer Review Status:Awaiting Review
Subjects: Physics >> The Physics of Elementary Particles and Fields Subjects: Mathematics >> Mathematical Physics submitted time 2018-10-08
Abstract: In the present paper, we have systematically explored the general rules for all kinds of combination of Hodge star and exterior differentiation operators. We have derived the unified forms of the non-vanishing and independent operators made up of arbitrary numbers of Hodge star and exterior differentiation operators. On basis of this, we have explicitly investigated the interaction of all the combined operators. What is more, all the operators have been classified according to the ranks of the newly generated differential forms. As an application, it has been demonstrated that the Maxwell’s equations for U(1) gauge field can be constructed from the linear combinations of two (n-1)-forms. "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Logic submitted time 2018-09-28
Abstract: In order to fundamentally eliminate all kinds of paradoxes existing in mathematic foundation and make mathematics architecture on a highly reliable basis, it was found that formal logic can only be used in the discussion domain (called the feasible domain) in which all of the three laws, i,e, the law of identity, the law of non-contradictory and the law of excluded middle are hold true. Otherwise, various errors including public opinion will occur. It was concluded that in feasible domain, as long as the premise is reliable and the derivation is strict, no paradox exists. Some historically famous paradoxes such as liar paradox and barber paradox were therefore analyzed. At the same time, the logical mistakes in the application of Piano axiom, in the proofs of Cantor's theorem, the interval method and diagonal argument were pointed out. Suggestion for a uniform definition of natural numbers, rational numbers and irrational numbers to avoid any errors was therefore proposed. " "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics (General) submitted time 2018-09-23
Abstract: " The aim of this paper is to study the heterogeneous optimization problem \begin{align*} \mathcal {J}(u)=\int_{\Omega}(G(|\nabla u|)+qF(u^+)+hu+\lambda_{+}\chi_{\{u>0\}} )\text{d}x\rightarrow\text{min}, \end{align*} in the class of functions $ W^{1,G}(\Omega)$ with $ u-\varphi\in W^{1,G}_{0}(\Omega)$, for a given function $\varphi$, where $W^{1,G}(\Omega)$ is the class of weakly differentiable functions with $\int_{\Omega}G(|\nabla u|)\text{d}x<\infty$. The functions $G$ and $F$ satisfy structural conditions of Lieberman's type that allow for a different behavior at $0$ and at $\infty$. Given functions $q,h$ and constant $\lambda_+\geq 0$, we address several regularities for minimizers of $\mathcal {J}(u)$, including local $C^{1,\alpha}-$, and local Log-Lipschitz continuities for minimizers of $\mathcal {J}(u)$ with $\lambda_+=0$, and $\lambda_+>0$ respectively. We also establish growth rate near the free boundary for each non-negative minimizer of $\mathcal {J}(u)$ with $\lambda_+=0$, and $\lambda_+>0$ respectively. Furthermore, under additional assumption that $F\in C^1([0,+\infty); [0,+\infty))$, local Lipschitz regularity is carried out for non-negative minimizers of $\mathcal {J}(u)$ with $\lambda_{+}>0$. " "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics (General) submitted time 2018-09-22
Abstract: " In this paper, we prove several regularity results for the heterogeneous, two-phase free boundary problems $\mathcal {J}_{\gamma}(u)=\int_{\Omega}\big(f(x,\nabla u)+(\lambda_{+}(u^{+})^{\gamma}+\lambda_{-}(u^{-})^{\gamma})+gu\big)\text{d}x\rightarrow \text{min}$ under non-standard growth conditions. Included in such problems are heterogeneous jets and cavities of Prandtl-Batchelor type with $\gamma=0$, chemical reaction problems with $0<\gamma<1$, and obstacle type problems with $\gamma=1$. Our results hold not only in the degenerate case of $p> 2$ for $p-$Laplace equations, but also in the singular case of $1
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Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics (General) submitted time 2018-09-22
Abstract: We establish regularity of solutions to the $G$-Laplace equation $-\text{div}\ \bigg(\frac{g(|\nabla u|)}{|\nabla u|}\nabla u\bigg)=\mu$, where $\mu$ is a nonnegative Radon measure satisfying $\mu (B_{r}(x_{0}))\leq Cr^{m}$ for any ball $B_{r}(x_{0})\subset\subset \Omega$ with $r\leq 1$ and $m>n-1-\delta\geq 0$. The function $g(t)$ is supposed to be nonnegative and $C^{1}$-continuous in $[0,+\infty)$, satisfying $g(0)=0$, and for some positive constants $\delta$ and $g_{0}$, $\delta\leq \frac{tg'(t)}{g(t)}\leq g_{0}, \forall t>0$, that generalizes the structural conditions of Ladyzhenskaya-Ural'tseva for an elliptic operator.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics (General) submitted time 2018-09-22
Abstract: "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics (General) submitted time 2018-09-18
Abstract: This paper studies Lyapunov inequalities of a class of higher-order nonlinear differential equations, which is a further discussion and extension of the relevant conclusions of "Lyapunov-type inequalities for ψ-Laplacian equations"
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics (General) submitted time 2018-09-13
Abstract: The aim of this paper is to study the obstacle problem associated with an elliptic operator having degenerate coercivity, and with a low order term and $L^1-$data. We prove the existence of an entropy solution to the obstacle problem and show its continuous dependence on the $L^{1}-$data in $W^{1,q}(\Omega)$ with some $q>1$. " " " "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics (General) submitted time 2018-06-04
Abstract: " In this work, we establish several Lyapunov-type inequalities for a class of nonlinear higher order differential equations having a form \begin{align*} (\psi(u^{(m)}(x)))'+\sum_{i=0}^nr_i(x)f_i(u^{(i)}(x))=0, %\ \ \ \ \text{or}\ \ \ \ (\psi(u^{(m)}))^{(m)}+r_i(x)f(u)=0, \end{align*} with anti-periodic boundary conditions, where $m> n\geq 0$ are integers, $\psi$ and $f_i (i=0,1,2,...,n)$ satisfy certain structural conditions such that the considered equations have general nonlinearities. The obtained inequalities are extensions and complements of the existing results in the literature. "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics (General) submitted time 2018-05-22
Abstract: In this work, we present several Lyapunov-type inequalities for a class of $\psi-$Laplacian equations of the form \begin{align*} (\psi(u'(x)))'+r(x)f(u(x))=0, \end{align*} with Dirichlet boundary conditions, where $\psi$ and $f$ satisfies certain structural conditions with general nonlinearities. We do not require any sub-multiplicative property of $\psi$, and any convexity of $\frac{1}{\psi(t)}$ or $\psi (t)t$ in the establishment of Lyapunov-type inequalities. The obtained inequalities can be seen as extensions and complements of the existing results in the literature. " "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Computational Mathematics. submitted time 2018-04-03
Abstract: In this paper, I found the two reasons of overfitting of logistic regression: boundary samples occupy a larger and larger share as the length of normal vector becomes longer and longer, boundary samples do not fit their probability density function well. With the help of insight in overfitting, I propose a acceleration method for logistic regression and got a training speedup of 38.25 on MNIST dataset, a training speedup of 5.61 on CIFAR10 dataset.
Peer Review Status:Awaiting Review
Subjects: Medicine, Pharmacy >> Other Disciplines of Medicine and Pharmacology Subjects: Mathematics >> Geometry and Topology Subjects: Computer Science >> Integration Theory of Computer Science submitted time 2018-03-31
Abstract: Labeled images are one of the most important means of scientific communication and education. However, traditional markers (arrows, lines) are point markers; do not include information about how large the feature is. We designed an efficient marker system for labeling scientific images (electron or light microscopy, CT, MRI, ultrasonography, camera pictures, etc), called the “T Area Marker, (TAM)”. The basic TAM marker looks like a “T”, composed of a line segment and a small tick on one end; it defines an imagined circle that stands on the tickless end and the diameter of the circle is equal to the length of the line segment. Thus the TAM can define an exact area rather than a single point; and the imagined circle does not break the continuity of the image (unlike traditional visible circles, rectangles, etc). A TAM with N ticks (N>1) means the diameter equals to N times the length of TAM. A TAM may also have a tail and/or several tail branches to define translation of the imagined circle, thus define complicated areas. tAreaMarker.py is free software that combines the drawing and reading of TAMs, although in most cases TAMs are easily interpreted without computer assistance.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Computational Mathematics. submitted time 2018-03-22
Abstract: In this paper, I found the two reasons of overfitting of logistic regression: boundary samples occupy a larger and larger share as the length of normal vector becomes longer and longer, boundary samples do not fit their probability density function well. With the help of insight in overfitting, I propose a acceleration method for logistic regression and got a training speedup of 38.25 on MNIST dataset, a training speedup of 5.61 on CIFAR10 dataset.
Peer Review Status:Awaiting Review