scholarly journals Affine Extensions of Integer Vector Addition Systems with States

2021 ◽  
Vol Volume 17, Issue 3 ◽  
Author(s):  
Michael Blondin ◽  
Christoph Haase ◽  
Filip Mazowiecki ◽  
Mikhail Raskin
Keyword(s):  
Infinite Matrix ◽  
Integer Vector ◽  
Affine Transformations ◽  
Reachability Problem ◽  
Finite Monoid ◽  
Vector Addition ◽  
The Matrix ◽  
Preliminary Version ◽  
Counter Machines ◽  
Single Matrix

We study the reachability problem for affine $\mathbb{Z}$-VASS, which are integer vector addition systems with states in which transitions perform affine transformations on the counters. This problem is easily seen to be undecidable in general, and we therefore restrict ourselves to affine $\mathbb{Z}$-VASS with the finite-monoid property (afmp-$\mathbb{Z}$-VASS). The latter have the property that the monoid generated by the matrices appearing in their affine transformations is finite. The class of afmp-$\mathbb{Z}$-VASS encompasses classical operations of counter machines such as resets, permutations, transfers and copies. We show that reachability in an afmp-$\mathbb{Z}$-VASS reduces to reachability in a $\mathbb{Z}$-VASS whose control-states grow linearly in the size of the matrix monoid. Our construction shows that reachability relations of afmp-$\mathbb{Z}$-VASS are semilinear, and in particular enables us to show that reachability in $\mathbb{Z}$-VASS with transfers and $\mathbb{Z}$-VASS with copies is PSPACE-complete. We then focus on the reachability problem for affine $\mathbb{Z}$-VASS with monogenic monoids: (possibly infinite) matrix monoids generated by a single matrix. We show that, in a particular case, the reachability problem is decidable for this class, disproving a conjecture about affine $\mathbb{Z}$-VASS with infinite matrix monoids we raised in a preliminary version of this paper. We complement this result by presenting an affine $\mathbb{Z}$-VASS with monogenic matrix monoid and undecidable reachability relation.

Cosserat Modeling of Size Effects in Crystalline Solids

2000 ◽  
Vol 653 ◽  
Author(s):  
Samuel Forest
Keyword(s):  
Stress State ◽  
Size Effects ◽  
Elastic Inclusion ◽  
Characteristic Size ◽  
Infinite Matrix ◽  
Crystalline Solids ◽  
Generalized Continua ◽  
Absolute Size ◽  
Kink Bands ◽  
The Matrix

AbstractThe mechanics of generalized continua provides an efficient way of introducing intrinsic length scales into continuum models of materials. A Cosserat framework is presented here to descrine the mechanical behavior of crystalline solids. The first application deals with the problem of the stress field at a crak tip in Cosserat single crystals. It is shown that the strain localization patterns developping at the crack tip differ from the classical picture : the Cosserat continuum acts as a bifurcation mode selector, whereby kink bands arising in the classical framework disappear in generalized single crystal plasticity. The problem of a Cosserat elastic inclusion embedded in an infinite matrix is then considered to show that the stress state inside the inclusion depends on its absolute size lc. Two saturation regimes are observed : when the size R of the inclusion is much larger than a characteristic size of the medium, the classical Eshelby solution is recovered. When R is much small than the inclusion, a much higher stress is reached (for an inclusion stiffer than the matrix) that does not depend on the size any more. There is a transition regime for which the stress state is not homogeneous inside the inclusion. Similar regimes are obtained in the study of grain size effects in polycrystalline aggregates of Cosserat grains.

scholarly journals Infinite Matrix Products and the Representation of the Matrix Gamma Function

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
J.-C. Cortés ◽  
L. Jódar ◽  
Francisco J. Solís ◽  
Roberto Ku-Carrillo
Keyword(s):  
Gamma Function ◽  
Infinite Product ◽  
Infinite Matrix ◽  
Convergence Results ◽  
Matrix Products ◽  
The Matrix ◽  
Definition Of

We introduce infinite matrix products including some of their main properties and convergence results. We apply them in order to extend to the matrix scenario the definition of the scalar gamma function given by an infinite product due to Weierstrass. A limit representation of the matrix gamma function is also provided.

A Spherical Inclusion with Inhomogeneous Interface in Conduction

2003 ◽  
Vol 19 (1) ◽  
pp. 1-8
Author(s):  
T. Chen ◽  
C. H. Hsieh ◽  
P. C. Chuang
Keyword(s):  
Infinite Number ◽  
Effective Conductivity ◽  
Spherical Inclusion ◽  
Cone Angle ◽  
Imperfect Interface ◽  
Infinite Matrix ◽  
Algebraic Equations ◽  
Legendre Series ◽  
Linear Set ◽  
The Matrix

ABSTRACTA series solution is presented for a spherical inclusion embedded in an infinite matrix under a remotely applied uniform intensity. Particularly, the interface between the inclusion and the matrix is considered to be inhomegeneously bonded. We examine the axisymmetric case in which the interface parameter varies with the cone angle θ. Two kinds of imperfect interfaces are considered: an imperfect interface which models a thin interphase of low conductivity and an imperfect interface which models a thin interphase of high conductivity. We show that, by expanding the solutions of terms of Legendre polynomials, the field solution is governed by a linear set of algebraic equations with an infinite number of unknowns. The key step of the formulation relies on algebraic identities between coefficients of products of Legendre series. Some numerical illustrations are presented to show the correctness of the presented procedures. Further, solutions of the boundary-value problem are employed to estimate the effective conductivity tensor of a composite consisting of dispersions of spherical inclusions with equal size. The effective conductivity solely depends on one particular constant among an infinite number of unknowns.

On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences

2021 ◽  
Vol 71 (6) ◽  
pp. 1375-1400
Author(s):  
Feyzi Başar ◽  
Hadi Roopaei
Keyword(s):  
Lower Bound ◽  
Sequence Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Infinite Matrix ◽  
The Matrix ◽  
Alpha Beta ◽  
Necessary And Sufficient ◽  
Bounded Sequences

Abstract Let F denote the factorable matrix and X ∈ {ℓp , c 0, c, ℓ ∞}. In this study, we introduce the domains X(F) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes (ℓ p (F), ℓ ∞), (ℓ p (F), f) and (X, Y(F)) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F. Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.

Matrix transformations between some classes of sequences

1970 ◽  
Vol 68 (1) ◽  
pp. 99-104 ◽  
Author(s):  
C. G. Lascarides ◽  
I. J. Maddox
Keyword(s):  
Matrix Transformation ◽  
Matrix Transformations ◽  
Complex Numbers ◽  
Infinite Matrix ◽  
The Matrix ◽  
Complex Sequences ◽  
Class Of Matrices

Let A = (ank) be an infinite matrix of complex numbers ank (n, k = 1, 2,…) and X, Y two subsets of the space s of complex sequences. We say that the matrix A defines a (matrix) transformation from X into Y, and we denote it by writing A: X → Y, if for every sequence x = (xk)∈X the sequence Ax = (An(x)) is in Y, where An(x) = Σankxk and the sum without limits is always taken from k = 1 to k = ∞. The sequence Ax is called the transformation of x by the matrix A. By (X, Y) we denote the class of matrices A such that A: X → Y.

SOME COMPLEXITY RESULTS FOR RINGS OF PETRI NETS

1994 ◽  
Vol 05 (03n04) ◽  
pp. 281-292
Author(s):  
HSU-CHUN YEN ◽  
BOW-YAW WANG ◽  
MING-SHANG YANG
Keyword(s):  
Petri Nets ◽  
State Machines ◽  
Slight Improvement ◽  
Reachability Problem ◽  
Encoding Scheme ◽  
N Cycle ◽  
Vector Addition ◽  
Finite State ◽  
Boundedness Problem ◽  
Compare And Contrast

We define a subclass of Petri nets called m–state n–cycle Petri nets, each of which can be thought of as a ring of n bounded (by m states) Petri nets using n potentially unbounded places as joins. Let Ring(n, l, m) be the class of m–state n–cycle Petri nets in which the largest integer mentioned can be represented in l bits (when the standard binary encoding scheme is used). As it turns out, both the reachability problem and the boundedness problem can be decided in O(n(l+log m)) nondeterministic space. Our results provide a slight improvement over previous results for the so-called cyclic communicating finite state machines. We also compare and contrast our results with that of VASS(n, l, s), which represents the class of n-dimensional s-state vector addition systems with states where the largest integer mentioned can be described in l bits.

scholarly journals Matrix Transformations between Certain Sequence Spaces over the Non-Newtonian Complex Field

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Uğur Kadak ◽  
Hakan Efe
Keyword(s):  
Linear Operator ◽  
Sufficient Conditions ◽  
Sequence Spaces ◽  
General Linear ◽  
Matrix Transformations ◽  
Infinite Matrix ◽  
Complex Field ◽  
Infinite Matrices ◽  
The Matrix ◽  
Necessary And Sufficient

In some cases, the most general linear operator between two sequence spaces is given by an infinite matrix. So the theory of matrix transformations has always been of great interest in the study of sequence spaces. In the present paper, we introduce the matrix transformations in sequence spaces over the fieldC*and characterize some classes of infinite matrices with respect to the non-Newtonian calculus. Also we give the necessary and sufficient conditions on an infinite matrix transforming one of the classical sets overC*to another one. Furthermore, the concept for sequence-to-sequence and series-to-series methods of summability is given with some illustrated examples.

Experimental Study of the Fracture and Matrix Effects on Free-Fall Gravity Drainage With Micromodels

SPE Journal ◽  
2014 ◽  
Vol 20 (02) ◽  
pp. 324-336 ◽  
Author(s):  
Mehdi Bahari Moghaddam ◽  
Mohammad Reza Rasaei
Keyword(s):  
Production Rate ◽  
Oil Production ◽  
Free Fall ◽  
Gravity Drainage ◽  
Permeability Ratio ◽  
Fracture Spacing ◽  
Matrix Block ◽  
The Matrix ◽  
Fracture Opening ◽  
Single Matrix

Summary Free-fall gravity drainage (FFGD) is the main production mechanism in the gas-invaded zone of fractured reservoirs. The gravity and capillary forces are two major forces that control the production performance of a fractured system under an FFGD mechanism. Gravity force acts as a driving force to remove oil from the matrix block whereas the resistive capillary force tends to keep oil inside the matrix. In this study, a series of experiments was performed to study the effects of the geometrical characteristics of the fracture and matrix on the oil-production rate under an FFGD mechanism by use of a glass micromodel. The oil-recovery factor (RF) was also obtained for a single matrix block by use of different patterns. Results from the experiments show that different flow regimes occur during the production life of a single matrix block under a FFGD mechanism. The fluid flow is controlled by the capillary-dominated regime at the early stage and late time of production life, whereas it shows a stabilized bulk flow under a gravity-dominated regime is exhibited at other times. Experimental results revealed that for a narrow fracture opening, fracture capillary pressure has a form similar to that of the matrix block. Also, it was observed that the oil-production rate and RF of the matrix block decreased as the permeability ratio between two media (matrix block and fracture) increased. Lower production rate is achieved in larger-fracture-spacing micromodels. In addition, wider vertical fractures lead to an early breakthrough of gas in bottom horizontal fracture that makes up the main portion of oil traps in the matrix block, and this reduces the RF. Results from this study show that in a heterogeneous layered matrix block, both the drainage rate and RF decrease in comparison with a homogeneous matrix block. Finally, a multiple linear-regression analysis was performed to understand the dimensionless groups affecting the RF of the FFGD process. It was found that the Bond number cannot truly describe the process and other parameters such as the fracture-/matrix-permeability ratio; fracture spacing and fracture opening should also be considered.

scholarly journals REACHABILITY PROBLEMS FOR PRODUCTS OF MATRICES IN SEMIRINGS

2006 ◽  
Vol 16 (03) ◽  
pp. 603-627 ◽  
Author(s):  
STÉPHANE GAUBERT ◽  
RICARDO D. KATZ
Keyword(s):  
Matrix Product ◽  
Reachability Problem ◽  
Tropical Semiring ◽  
The Matrix ◽  
Left And Right ◽  
Matrix Vector ◽  
Products Of Matrices

We consider the following matrix reachability problem: given r square matrices with entries in a semiring, is there a product of these matrices which attains a prescribed matrix? Similarly, we define the vector (resp. scalar) reachability problem, by requiring that the matrix product, acting by right multiplication on a prescribed row vector, gives another prescribed row vector (resp. when multiplied on the left and right by prescribed row and column vectors, gives a prescribed scalar). We show that over any semiring, scalar reachability reduces to vector reachability which is equivalent to matrix reachability, and that for any of these problems, the specialization to any r ≥ 2 is equivalent to the specialization to r = 2. As an application of these reductions and of a theorem of Krob, we show that when r = 2, the vector and matrix reachability problems are undecidable over the max-plus semiring (ℤ∪{-∞}, max ,+). These reductions also improve known results concerning the classical zero corner problem. Finally, we show that the matrix, vector, and scalar reachability problems are decidable over semirings whose elements are "positive", like the tropical semiring (ℤ∪{+∞}, min ,+).

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