Product (mathematics)

About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.

Method of designing a product in a virtual environment

Abstract: A method of designing a product for use on a body used to develop a preferred product configuration using a computer-based virtual product development and testing system. A virtual product sub-model is created of a product for use on the body. An environment sub-model is generated so that environmental factors affecting the product are also used in designing or evaluating the product. Instructions defining how the product sub-model and the environment sub-model interact are introduced in an interaction model. The sub-models and the interaction defined by the interaction model are then combined to create a virtual use model simulating the use of the virtual product sub-model by the virtual wearer sub-model. The use model determines the forces, deformations and stresses caused by movement and interaction between the virtual wearer sub-model and the virtual product sub-model using numerical method analysis. The results of the use model are analyzed to evaluate the performance of product features embodied in the virtual product sub-model such as when and exposed to typical movements or forces. The product sub-model is then modified in response to the determined performance of the product feature and the steps of interacting the models and combining the models are reperformed in the use model to design the product.

Calculation of the Hidden Symmetry Operator in PT-Symmetric Quantum Mechanics

Abstract: In a recent paper it was shown that if a Hamiltonian H has an unbroken PT symmetry, then it also possesses a hidden symmetry represented by the linear operator C. The operator C commutes with both H and PT. The inner product with respect to CPT is associated with a positive norm and the quantum theory built on the associated Hilbert space is unitary. In this paper it is shown how to construct the operator C for the non-Hermitian PT-symmetric Hamiltonian $H={1\over2}p^2+{1\over2}x^2 +i\epsilon x^3$ using perturbative techniques. It is also shown how to construct the operator C for $H={1\over2}p^2+{1\over2}x^2-\epsilon x^4$ using nonperturbative methods.

On Weyl function and generalized resolvents of a Hermitian operator in a Krein space

Abstract: A description of generalized resolvents for a densely defined Hermitian operatorA in a Krein space\(\mathcal{K}\) is given under explicit consideration of the number of negative squares of the inner product on the extending space and of the forms [A·,·], [A·,·],A being a selfadjoint extension ofA which corresponds to the generalized resolvent. New classesNkκ of analytic functions are introduced for this purpose. An application to a Sturm-Liouville operator with indefinite weight function is discussed.

Operator Algebras for Multivariable Dynamics

Abstract: Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $\sigma_i:X \to X$ for $1 \le i \le n$. To this the authors associate two conjugacy operator algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra $\mathcal{A}(X,\tau)$ and the semicrossed product $\mathrm{C}_0(X)\times_\tau\mathbb{F}_n^+$. They develop the necessary dilation theory for both models. In particular, they exhibit an explicit family of boundary representations which determine the C*-envelope of the tensor algebra.|Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $\sigma_i:X \to X$ for $1 \le i \le n$. To this the authors associate two conjugacy operator algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra $\mathcal{A}(X,\tau)$ and the semicrossed product $\mathrm{C}_0(X)\times_\tau\mathbb{F}_n^+$. They develop the necessary dilation theory for both models. In particular, they exhibit an explicit family of boundary representations which determine the C*-envelope of the tensor algebra.

On the algorithm of Dirac spurs

Abstract: The spur of the product of any number of Dirac matrices is shown to be square root of a determinant. It can be evaluated directly, however, by means of rules as simple as those of determinant theory. When the number of matrices occurring in the product is ⩾12, further considerations show that many terms of the expansion sum up to give zero total contribution. A final formula is given which takes all such reductions into account. The Appendix contains rules for summing over pairs of λμ which bracket products of Dirac matrices.