Product (mathematics)

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

Singular values of products of random matrices and polynomial ensembles

Abstract: Akemann, Ipsen, and Kieburg showed recently that the squared singular values of a product of M complex Ginibre matrices are distributed according to a determinantal point process. We introduce the notion of a polynomial ensemble and show how their result can be interpreted as a transformation of polynomial ensembles. We also show that the squared singular values of the product of M - 1 complex Ginibre matrices with one truncated unitary matrix is a polynomial ensemble, and we derive a double integral representation for the correlation kernel associated with this ensemble. We use this to calculate the scaling limit at the hard edge, which turns out to be the same scaling limit as the one found by Kuijlaars and Zhang for the squared singular values of a product of M complex Ginibre matrices. Our final result is that these limiting kernels also appear as scaling limits for the biorthogonal ensembles of Borodin with parameter θ > 0, in case θ or 1/θ is an integer. This further supports the conjecture that these kernels have a universal character.

Concurrent design and subsequent partitioning of product and test die

Abstract: One embodiment of the present invention concerns a design methodology for generating a test die for a product die including the step of concurrently designing test circuitry and a product circuitry in a unified design. The test circuitry can be designed to provide a high degree of fault coverage for the corresponding product circuitry generally without regard to the amount of silicon area that will be required by the test circuitry. The design methodology then partitions the unified design into the test die and the product die. The test die includes the test circuitry and the product die includes the product circuitry. The product and test die may then be fabricated on separate semiconductor wafers. By partitioning the product circuitry and test circuitry into separate die, embedded test circuitry can be either eliminated or minimized on the product die. This will tend to decrease the size of the product die and decrease the cost of manufacturing the product die while maintaining a high degree of test coverage of the product circuits within the product die.

Quantum weights of dyons and of instantons with nontrivial holonomy

Abstract: We calculate exactly functional determinants for quantum oscillations about periodic instantons with a nontrivial value of the Polyakov line at spatial infinity. Hence, we find the weight or the probability with which calorons with nontrivial holonomy occur in the Yang-Mills partition function. The weight depends on the value of the holonomy, the temperature, ${\ensuremath{\Lambda}}_{\mathrm{QCD}},$ and the separation between the BPS monopoles (or dyons) that constitute the periodic instanton. At large separation between constituent dyons, the quantum measure factorizes into a product of individual dyon measures, times a definite interaction energy. We present an argument that at temperatures below a critical one related to ${\ensuremath{\Lambda}}_{\mathrm{QCD}},$ trivial holonomy is unstable, and that calorons ``ionize'' into separate dyons.

The refined BPS index from stable pair invariants

Abstract: A refinement of the stable pair invariants of Pandharipande and Thomas for non-compact Calabi-Yau spaces is introduced based on a virtual Bialynicki-Birula decomposition with respect to a C ∗ action on the stable pair moduli space, or alternatively the equivariant index of Nekrasov and Okounkov. This effectively calculates the refined index for M-theory reduced on these Calabi-Yau geometries. Based on physical expectations we propose a product formula for the refined invariants extending the motivic product formula of Morrison, Mozgovoy, Nagao, and Szendroi for local P 1 . We explicitly compute refined invariants in low degree for local P 2 and local P 1 x P 1 and check that they agree with the predictions of the direct integration of the generalized holomorphic anomaly and with the product formula. The modularity of the expressions obtained in the direct integration approach allows us to relate the generating function of refined PT invariants on appropriate geometries to Nekrasov’s partition function and a refinement of Chern-Simons theory on a lens space. We also relate our product formula to wallcrossing.