Differential Geometry Approach to Continuous Model of Micro-Structural Defects in Finite Elasto-Plasticity

This paper concerns finite elasto-plasticity of crystalline materials with micro-structural defects. We revisit the basic concepts: plastic distortion and decomposition of the plastic connection. The body is endowed with a structure of differential manifold. The plastic distortion is an incompatible diffeomorphism. The metric induced by the plastic distortion on the intermediate configuration (considered to be a differential manifold) is a key point in the theory, in defining the defects related to point defects, or extra-matter. The so-called plastic connection is metric, with plastic metric tensor expressed in terms of the plastic distortion and its adjoint. We prove an appropriate decomposition of the plastic connection, without any supposition concerning the non-metricity of plastic connection. All types of the lattice defects, dislocations, disclinations, and point defects are described in terms of the densities related to the elements that characterize the decomposition theorem for plastic connection. As a novelty, the measure of the interplay of the possible lattice defects is introduced via the Cartan torsion tensor. To justify the given definitions, the proposed measures of defects are compared to their counterparts corresponding to a classical framework of continuum mechanics. Thus, their physical meanings can be emphasized at once.

Newton polygons for L-functions of generalized Kloosterman sums

In the present paper, we study the Newton polygons for the L-functions of n-variable generalized Kloosterman sums. Generally, the Newton polygon has a topological lower bound, called the Hodge polygon. In order to determine the Hodge polygon, we explicitly construct a basis of the top-dimensional Dwork cohomology. Using Wan’s decomposition theorem and diagonal local theory, we obtain when the Newton polygon coincides with the Hodge polygon. In particular, we concretely get the slope sequence for the L-function of F ¯ ⁢ ( λ ¯ , x ) := ∑ i = 1 n x i a i + λ ¯ ⁢ ∏ i = 1 n x i - 1 , \bar{F}(\bar{\lambda},x):=\sum_{i=1}^{n}x_{i}^{a_{i}}+\bar{\lambda}\prod_{i=1}% ^{n}x_{i}^{-1}, with a 1 , … , a n {a_{1},\ldots,a_{n}} being pairwise coprime for n ≥ 2 {n\geq 2} .

Orthogonal decomposition of the sum-symmetry model for square contingency tables with ordinal categories: Use of the exponential sum-symmetry model

Summary In the existing decomposition theorem, the sum-symmetry model holds if and only if both the exponential sum-symmetry and global symmetry models hold. However, this decomposition theorem does not satisfy the asymptotic equivalence for the test statistic. To address the aforementioned gap, this study establishes a decomposition theorem in which the sum-symmetry model holds if and only if both the exponential sum-symmetry and weighted global-sum-symmetry models hold. The proposed decomposition theorem satisfies the asymptotic equivalence for the test statistic. We demonstrate the advantages of the proposed decomposition theorem by applying it to datasets comprising real data and artificial data.

Orthogonal decomposition of the sum-symmetry model using the two-parameters sum-symmetry model for ordinal square contingency tables

Summary Studies have been carried out on decomposing a model with symmetric structure using a model with asymmetric structure. In the existing decomposition theorem, the sum-symmetry model holds if and only if all of the two-parameters sum-symmetry, global symmetry and concordancediscordance models hold. However, this existing decomposition theorem does not satisfy the asymptotic equivalence for the test statistic, namely that the value of the likelihood ratio chi-squared statistic of the sum-symmetry model is asymptotically equivalent to the sum of those of the decomposed models. To address this issue, this study introduces a new decomposition theorem in which the sum-symmetry model holds if and only if all of the two-parameters sum-symmetry, global symmetry and weighted global-sum-symmetry models hold. The proposed decomposition theorem satisfies the asymptotic equivalence for the test statistic—the value of the likelihood ratio chi-squared statistic of the sum-symmetry model is asymptotically equivalent to the sum of those of the two-parameters sum-symmetry, global symmetry and weighted global-sum-symmetry models.

A de Rham decomposition type theorem for contact sub-Riemannian manifolds

In this paper we prove a result which can be regarded as a sub-Riemannian version of de Rham decomposition theorem. More precisely, suppose that (M, H, g) is a contact and oriented sub-Riemannian manifold such that the Reeb vector field $$\xi $$ ξ is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on H. Suppose that there exists a point $$q\in M$$ q ∈ M such that the holonomy group $$\Psi (q)$$ Ψ ( q ) acts reducibly on H(q) yielding a decomposition $$H(q) = H_1(q)\oplus \cdots \oplus H_m(q)$$ H ( q ) = H 1 ( q ) ⊕ ⋯ ⊕ H m ( q ) into $$\Psi (q)$$ Ψ ( q ) -irreducible factors. Using parallel transport we obtain the decomposition $$H = H_1\oplus \cdots \oplus H_m$$ H = H 1 ⊕ ⋯ ⊕ H m of H into sub-distributions $$H_i$$ H i . Unlike the Riemannian case, the distributions $$H_i$$ H i are not integrable, however they induce integrable distributions $$\Delta _i$$ Δ i on $$M/\xi $$ M / ξ , which is locally a smooth manifold. As a result, every point in M has a neighborhood U such that $$T(U/\xi )=\Delta _1\oplus \cdots \oplus \Delta _m$$ T ( U / ξ ) = Δ 1 ⊕ ⋯ ⊕ Δ m , and the latter decomposition of $$T(U/\xi )$$ T ( U / ξ ) induces the decomposition of $$U/\xi $$ U / ξ into the product of Riemannian manifolds. One can restate this as follows: every contact sub-Riemannian manifold whose holonomy group acts reducibly has, at least locally, the structure of a fiber bundle over a product of Riemannian manifolds. We also give a version of the theorem for indefinite metrics.

Decomposition of Random Sequences into Mixtures of Simpler Ones and Its Application in Network Analysis

A classic and fundamental result about the decomposition of random sequences into a mixture of simpler ones is de Finetti’s Theorem. In its original form, it applies to infinite 0–1 valued sequences with the special property that the distribution is invariant to permutations (called an exchangeable sequence). Later it was extended and generalized in numerous directions. After reviewing this line of development, we present our new decomposition theorem, covering cases that have not been previously considered. We also introduce a novel way of applying these types of results in the analysis of random networks. For self-containment, we provide the introductory exposition in more detail than usual, with the intent of making it also accessible to readers who may not be closely familiar with the subject.

SOME CHARACTERIZATIONS OF DICHOTOMY FOR IMPULSIVE DYNAMIC SYSTEMS

We study the problem of dichotomy and boundedness for impulsive dynamic equations on arbitrary closed subset of real numbers. The spectral decomposition theorem gives all our main results. The obtained results are fundamentally new, even for the classical case.

The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral Networks

The normalized Laplacian plays an indispensable role in exploring the structural properties of irregular graphs. Let L n 8,4 represent a linear octagonal-quadrilateral network. Then, by identifying the opposite lateral edges of L n 8,4 , we get the corresponding Möbius graph M Q n 8,4 . In this paper, starting from the decomposition theorem of polynomials, we infer that the normalized Laplacian spectrum of M Q n 8,4 can be determined by the eigenvalues of two symmetric quasi-triangular matrices ℒ A and ℒ S of order 4 n . Next, owing to the relationship between the two matrix roots and the coefficients mentioned above, we derive the explicit expressions of the degree-Kirchhoff indices and the complexity of M Q n 8,4 .

Dynamic Analysis of a Deeply Buried Tunnel Influenced by a Newly-built Adjacent Cavity with a Special Emphasis on the Minimum Seismically Safe Tunnel Distance

Contemporary life streams, more often than ever, impose the necessity for construction of new underground structures in the vicinity of existing tunnels, with an aim to accommodate transportation systems and utility networks. A previously uninvestigated case, in which a newly-constructed tunnel opening is closely positioned behind an existing tunnel, referred to as the tunnel–cavity configuration, has been considered in this study. An exact analytical solution is derived considering a pair of parallel circular cylindrical structures of infinite length, with the horizontal alignment, embedded in a boundless homogeneous, isotropic, elastic medium and excited by time-harmonic plane SV-waves under the plane-strain conditions. The Helmholtz decomposition theorem, the wave functions expansion method, the translational addition theorem for bi-cylindrical coordinates, and the pertinent boundary conditions are jointly employed in order to develop a closed-form solution of the corresponding boundary value problem. The primary goal of the present study is to examine the increase in dynamic stresses at an existing tunnel structure due to the presence of a closely driven unlined cavity, as well as in a localized region around the tunnel (at the position of the cavity in close proximity), under incident SV-waves. A new quantity called dynamic stress alteration factor is introduced and the aspect of the minimum seismically safe distance between the two structures is particularly considered.

Soft Frames in Soft Hilbert Spaces

In this paper, we use soft linear operators to introduce the notion of discrete frames on soft Hilbert spaces, which extends the classical notion of frames on Hilbert spaces to the context of algebraic structures on soft sets. Among other results, we show that the frame operator associated to a soft discrete frame is bounded, self-adjoint, invertible and with a bounded inverse. Furthermore, we prove that every element in a soft Hilbert space satisfies the frame decomposition theorem. This theoretical framework is potentially applicable in signal processing because the frame coefficients serve to model the data packets to be transmitted in communication networks.