Narrow Results

In this paper, we have introduced a software reliability growth model (SRGM) that integrates a Weibull testing effort function (TEF) within the software fault detection process (FDP) and fault correction process (FCP), effectively capturing the intricacies of software testing. Leveraging the least squared estimation method, we estimated both model parameters and the TEF, leading to a notably improved fit with the dataset and enhancing our comprehension of software reliability dynamics. Our approach also involved employing a reliability-based method to identify the optimal time for software release, further reinforcing the robustness of our model. Subsequently, we devised an optimal control model to minimize testing efforts throughout the software development life cycle. This work provides a comprehensive methodological framework for computing the total cost while ensuring debugging costs follow a learning curve pattern. Through numerical simulations involving hypothetical values and the Weibull TEF, we observed that once the software achieves the desired reliability for release, subsequent costs gradually decline to zero. In essence, both testing and future costs converge to zero post-release, reducing instantaneous expenses.

The aim of this study is to give a deep investigation into the dynamics of the simplified modified Camassa–Holm equation (CHe) for shallow water waves. Taking advantage of the semi-inverse method, we develop the variational principle, based on which the Hamiltonian of the system is extracted. By means of the Galilean transformation, the governing equation is transformed into a planar dynamical system. Then, the bifurcation analysis is presented via employing the theory of the planar dynamical system. Correspondingly, the quasi-periodic and chaotic behaviors of the system are also discussed by introducing two different kinds of perturbed terms. Finally, the variational method is based on the variational principle and Ritz method, and the Kudryashov method is used to construct the diverse solitary wave solutions, which include the bright solitary, dark solitary, kink solitary and the bright–dark solitary wave solutions. The graphic depictions of the obtained diverse solitary wave solutions are presented to elucidate the physical properties. The findings of this research enable us to gain a deeper understanding of the nonlinear dynamic characteristics of the considered equation.

A k-containerC(u, v) of a graph G is a set of k-disjoint paths joining u to v. A k-container C(u, v) is a k*-container if every vertex of G is incident with a path in C(u, v). A graph G is k*-connected if there exists a k*-container between any two distinct vertices u and v. A k-regular graph G is super spanning connected if G is i*-connected for all 1 ≤ i ≤ k. In this paper, we prove that the (n, k)-star graph S_n,k is super spanning connected if n ≥ 3 and (n-k) ≥ 2.

A $k$-container $C(u,v)$ of a graph $G$ is a set of $k$ internally disjoint paths between $u$ and $v$. A $k$-container $C(u,v)$ of $G$ is a $k^{\ast }$-container if it is a spanning subgraph of $G$. A graph $G$ is $k^{\ast }$-connected if there exists a $k^{\ast }$-container between any two different vertices of G. A $k$-regular graph $G$ is super spanning connected if $G$ is $i^{\ast }$-container for all $1\le i\le k$. In this paper, we prove that the arrangement graph $A_{n,k}$ is super spanning connected if $n\ge 4$ and $n-k\ge 2$.

In this note, we use geometric arguments to derive a possible form for the radial part of the "zero-mode Hamiltonian" for the two-dimensional sigma model with target space S³, or more generally a compact simply connected Lie group.

Let H be a self-adjoint operator on a Hilbert space , T be a symmetric operator on and K(t)(t ∈ ℝ) be a bounded self-adjoint operator on . We say that (T, H, K) obeys the generalized weak Weyl relation (GWWR) if e^-itHD(T) ⊂ D(T) for all t ∈ ℝ and Te^-itHψ = e^-itH(T+K(t))ψ, ∀ψ ∈ D(T) (D(T) denotes the domain of T). In the context of quantum mechanics where H is the Hamiltonian of a quantum system, we call T a generalized time operator of H. We first investigate, in an abstract framework, mathematical structures and properties of triples (T, H, K) obeying the GWWR. These include the absolute continuity of the spectrum of H restricted to a closed subspace of , an uncertainty relation between H and T (a "time-energy uncertainty relation"), the decay property of transition probabilities |〈ψ,e^-itHϕ〉|² as |t| → ∞ for all vectors ψ and ϕ in a subspace of , where 〈·,·〉 denotes the inner product of . We describe methods to construct various examples of triples (T, H, K) obeying the GWWR. In particular, we show that there exist generalized time operators of second quantization operators on Fock spaces (full Fock spaces, boson Fock spaces and fermion Fock spaces) which may have applications to quantum field models with interactions.

We develop a mathematical theory of time operators of a Hamiltonian with purely discrete spectrum. The main results include boundedness, unboundedness and spectral properties of them. In addition, possible connections of a time operator of H with regular perturbation theory are discussed.

We give an up-to-date overview of geometric and topological properties of cosymplectic and coKähler manifolds. We also mention some of their applications to time-dependent mechanics.

We use generating function techniques developed by Givental, Théret and ourselves to deduce a proof in $ℂ{\text{P}}^d$ of the homological generalization of Franks theorem due to Shelukhin. This result proves in particular the Hofer–Zehnder conjecture in the nondegenerate case: every Hamiltonian diffeomorphism of $ℂ{\text{P}}^d$ that has at least $d+2$ nondegenerate periodic points has infinitely many periodic points. Our proof does not appeal to Floer homology or the theory of $J$-holomorphic curves. An appendix written by Shelukhin contains a new proof of the Smith-type inequality for barcodes of Hamiltonian diffeomorphisms that arise from Floer theory, which lends itself to adaptation to the setting of generating functions.

Multi-Hamiltonian systems are investigated by using the Hamilton–Jacobi method. Integration of a set of total differential equations which includes the equations of motion and the action integral function is discussed. It is shown that this set is integrable if and only if the total variations of the Hamiltonians vanish. Two examples are studied.

First we briefly review our covariant Hamiltonian approach to quasi-local energy, noting that the Hamiltonian-boundary-term quasi-local energy expressions depend on the chosen boundary conditions and reference configuration. Then we present the quasi-local energy values resulting from the formalism applied to homogeneous Bianchi cosmologies. Finally we consider the quasi-local energies of the FRW cosmologies. Our results do not agree with certain widely accepted quasi-local criteria.

We develop the phase-space formulation for the Type IIA superstring on the AdS₄ × ℂℙ³ background in the κ-symmetry light-cone gauge for which the light-like directions are taken from the D = 3 Minkowski boundary of AdS₄. After fixing bosonic light-cone gauge, the superstring Hamiltonian is expressed as a function of the transverse physical variables and in the quadratic approximation corresponds to the light-cone gauge-fixed IIA superstring in flat space.

Cosmological solutions for covariant canonical gauge theories of gravity are presented. The underlying covariant canonical transformation framework invokes a dynamical spacetime Hamiltonian consisting of the Einstein–Hilbert term plus a quadratic Riemann tensor invariant with a fundamental dimensionless coupling constant $g_1$. A typical time scale related to this constant, $\tau =\sqrt{8\pi G{g}_1}$, is characteristic for the type of cosmological solutions: for $t\ll \tau$, the quadratic term is dominant, the energy–momentum tensor of matter is not covariantly conserved, and we observe modified dynamics of matter and spacetime. On the other hand, for $t\gg \tau$, the Einstein term dominates and the solution converges to classical cosmology. This is analyzed for different types of matter and dark energy with a constant equation of state. While for a radiation-dominated universe solution, the cosmology does not change, we find for a dark energy universe the well-known de-Sitter space. However, we also identify a special bouncing solution (for $k=0$) which for large times approaches the de-Sitter space again. For a dust-dominated universe (with no pressure), deviations are seen only in the early epoch. In late epoch, the solution asymptotically behaves as the standard dust solution.

The configuration space of general relativity is superspace, the space of Riemannian three-geometries and the Hamiltonian is just a sum of constraints, with Lagrange multipliers. One can go from this Hamiltonian, via a Legandre transformation, back and forth to the Lagrangian. The Lagrange multiplier (the lapse function) can be eliminated from the Lagrangian and one is left with an action which is a product of square roots. This is the Baierlein-Sharp-Wheeler action for general relativity. This action is unique in that all other square root actions are not self-consistent. This paper shows how to express this result in phase space. The only selfconsistent constrained Hamiltonian on superspace which is ultralocal and quadratic in the momenta in the ADM Hamiltonian for general relativity.

In the one-dimensional stationary case, we construct a mechanical Lagrangian describing the quantum motion of a nonrelativistic spinless system. This Lagrangian is written as a difference between a function T, which represents the quantum generalization of the kinetic energy and which depends on the coordinate x and the temporal derivatives of x up the third order, and the classical potential V(x). The Hamiltonian is then constructed and the corresponding canonical equations are deduced. The function T is first assumed to be arbitrary. The development of T in a power series together with the dimensional analysis allow us to fix univocally the series coefficients by requiring that the well-known quantum stationary Hamilton–Jacobi equation be reproduced. As a consequence of this approach, we formulate the law of the quantum motion representing a new version of the quantum Newton law. We also analytically establish the famous Bohm relation outside the framework of the hydrodynamical approach and show that the well-known quantum potential, although it is a part of the kinetic term, plays really the role of an additional potential as assumed by Bohm.

We investigate classical dynamics of the bosonic string in the background metric, antisymmetric and dilaton fields. We use canonical methods to find Hamiltonian in terms of energy–momentum tensor components. The later are secondary constraints of the theory. Due to the presence of the dilaton field the Virasoro generators have nonlinear realization. We find that, in the curve space–time, opposite chirality currents do not commute. As a consequence of the two-dimensional general covariance, the energy–momentum tensor components satisfy two Virasoro algebras, even in the curve space–time. We emphasize that background antisymmetric and dilaton fields are the origin of space–time torsion and space–time nonmetricity, respectively.

It is shown that the Einstein gravitational equations and the canonical equations, which follow from the Dirac–Schwinger Hamiltonian, in the Faddeev variables coincide. To prove this, first, the Einstein equations are rewritten in canonical variables and then the time derivative of the generalized momenta of the gravitational field in Faddeev form are calculated using canonical Poisson brackets. The results coincide.

Ken Wilson developed powerful renormalization group procedures for constructing effective theories and solving a broad class of difficult physical problems. His insights allowed him to later advance the Hamiltonian approach to quantum dynamics of particles and fields in the Minkowski space–time, motivated by QCD. The latter advances are described in this article, concluding with a remark on Ken's related interest in difficult systemic issues of society.

We study T-duality transformations in canonical formalism for Nambu–Goto action. Then we investigate the relation between worldsheet double Wick rotation and sequence of target space T-dualities and Wick rotation in case of fundamental string and D1-brane.

In a recent work we derived the kinematic Hamiltonian and primary constraints of the new general relativity class of teleparallel gravity theories and showed that these theories can be grouped in 9 classes, based on the presence or absence of primary constraints in their Hamiltonian. Here we demonstrate an alternative approach towards this result, by using differential forms instead of tensor components throughout the calculation. We prove that also this alternative derivation yields the same results and show how they are related to each other.