Modeling local induction heating of solder balls for flip-chip mounting in COMSOL Multiphysics

For Flip-chip mounting, it is necessary to form contact bumps for 2.5D and 3D electronic modules. Exposure to the energy of high-frequency (HF) electromagnetic oscillations allows for high-performance non-contact heating in various processes for soldering electronic components. This paper presents studies of temperature fields during induction heating of solder balls with several types of copper concentrators made in the COMSOL MULTIPHYSICS software. The dependences of frequency of inductor on heating time and geometric shapes of copper concentrators on quality of the soldered sample were revealed

Operator pencils of the second order and linear fractional relations

The notions of a pencil of the second order and a linear fractional relation (LFR) are defined in spaces of linear bounded operators acting between Banach spaces. It is shown that these notions are closely connected with various theoretical and applied problems and have diverse applications. A number of the open problems, both for pencils and LFR, are posed in this paper. Some of the above problems are solved and applied to the study of dichotomic behavior of dynamical systems

Generalized fractional linear transformations: convexity and compactness of the image and the pre-image; applications.

The convexity and compactness in the weak operator topology of the image and pre-image of a generalized fractional linear transformation is established. As an application the exponential dichotomy of solutions to evolution problems of the parabolic type is proved

Operator fractional-linear transformations: convexity and compactness of image; applications

The present paper consists of two parts. In Section 1 we consider fractional-linear transformations (f.-l.t. for brevity) F in the space $ℒ(X_1,X_2)$ of all linear bounded operators acting from $X_1$ into $X_2$, where $X_1, X_2$ are Banach spaces. We show that in the case of Hilbert spaces $X_1, X_2$ the image F(ℬ) of any (open or closed) ball ℬ ⊂ D(F) is convex, and if ℬ is closed, then F(ℬ) is compact in the weak operator topology (w.o.t.) (Theorem 1.2). These results extend the corresponding results on compactness obtained in [3], [4] under some additional restrictions imposed on F. We also establish that the convexity of the image of f.-l.t. is a characteristic property of Hilbert spaces, that is, if for the f.-l.t. $F:K → (I+K)^{-1}$ the image $F()$ of the open unit ball of the space ℒ(X) is convex, then X is a Hilbert space (Theorem 1.3). In Section 2 we apply the compactness of F(̅) for the closed unit operator ball ̅ to the study of the behavior of solutions to evolution problems in a Hilbert space ℋ. Namely, we establish the exponential dichotomy of solutions for the so-called hyperbolic case (such that the evolution operator is invertible). This is an extension of Theorem 1.1 of [5], where the corresponding assertion was established for the particular case of a Pontryagin space ℋ

Modeling the relationship of the quality of life and the investment attractiveness in grodno region

Comparative analysis of Grodno region districts competitiveness in 2011 was conducted. The system of indicators characterizing the competitiveness of the Republic of Belarus small administrative units was formed. This system consists of two units: the quality of life and investment attraction. District rating for each unit was built. Grouping of districts based on the values of integral indicators was performed. The relationship of quality of life integral indicators and investment attractiveness was analyzed. The integral indicator of competitiveness was built