56,146 research outputs found
Partitions and their lattices
Ferrers graphs and tables of partitions are treated as vectors. Matrix
operations are used for simple proofs of identities concerning partitions.
Interpreting partitions as vectors gives a possibility to generalize partitions
on negative numbers. Partitions are then tabulated into lattices and some
properties of these lattices are studied. There appears a new identity counting
Ferrers graphs packed consecutively into isoscele form. The lattices form the
base for tabulating combinatorial identities
First-order transition in model with higher-order interactions
The effect of inclusion of higher-order interactions in the {\it XY} model on
critical properties is studied by Monte Carlo simulations. It is found that an
increasing number of the higher-order terms in the Hamiltonian modifies the
shape of the potential, which beyond a certain value leads to the change of the
nature of the transition from continuous to first order. The evidence for the
first-order transition is provided in the form of the finite-size scaling and
the energy histogram analysis. A rough phase diagram is presented as a function
of the number of the higher-order interaction terms.Comment: 6 pages, 5 figures, 6th International Conference on Mathematical
Modeling in Physical Science
On Stability of Non-inflectional Elastica
This study considers the stability of a non-inflectional elastica under a
conservative end force subject to the Dirichlet, mixed, and Neumann boundary
conditions. It is demonstrated that the non-inflectional elastica subject to
the Dirichlet boundary conditions is unconditionally stable, while for the
other two boundary conditions, sufficient criteria for stability depend on the
signs of the second derivatives of the tangent angle at the endpoints
There is Neither Classical Bug with a Superluminal Shadow Nor Quantum Absolute Collapse Nor (Subquantum) Superluminal Hidden Variable
In this work we analyse critically Griffiths's example of the classical
superluminal motion of a bug shadow. Griffiths considers that this example is
conceptually very close to quantum nonlocality or superluminality,i.e. quantum
breaking of the famous Bell inequality. Or, generally, he suggests implicitly
an absolute asymmetric duality (subluminality vs. superluminality) principle in
any fundamental physical theory.It, he hopes, can be used for a natural
interpretation of the quantum mechanics too. But we explain that such
Griffiths's interpretation retires implicitly but significantly from usual,
Copenhagen interpretation of the standard quantum mechanical formalism. Within
Copenhagen interpretation basic complementarity principle represents, in fact,
a dynamical symmetry principle (including its spontaneous breaking, i.e.
effective hiding by measurement). Similarly, in other fundamental physical
theories instead of Griffiths's absolute asymmetric duality principle there is
a dynamical symmetry (including its spontaneous breaking, i.e. effective hiding
in some of these theories) principle. Finally, we show that Griffiths's example
of the bug shadow superluminal motion is definitely incorrect (it sharply
contradicts the remarkable Roemer's determination of the speed of light by
coming late of Jupiter's first moon shadow).Comment: 15 pages, no figure
On the Efficient Gerschgorin Inclusion Usage in the Global Optimization {\alpha}BB Method
In this paper, we revisit the {\alpha}BB method for solving global
optimization problems. We investigate optimality of the scaling vector used in
Gerschgorin's inclusion theorem to calculate bounds on the eigenvalues of the
Hessian matrix. We propose two heuristics to compute good scaling vector d, and
state three necessary optimality conditions for optimal d. Since the scaling
vector calculated by the second presented method satisfies all three optimality
conditions, it serves as a cheap but efficient solution
ThermalSim: A Thermal Simulator for Error Analysis
Researchers have extensively explored predictive control strategies for
controlling heating, ventilation, and air conditioning (HVAC) units in
commercial buildings. Predictive control strategies, however, critically rely
on weather and occupancy forecasts. Existing state-of-the-art building
simulators are incapable of analysing the influence of prediction errors (in
weather and occupancy) on HVAC energy consumption and occupant comfort. In this
paper, we introduce ThermalSim, a building simulator that can quantify the
effect of prediction errors on the HVAC operations. ThermalSim has been
implemented in C/C++ and MATLAB. We describe its design, use, and input format
Eigenvalues of symmetric tridiagonal interval matrices revisited
In this short note, we present a novel method for computing exact lower and
upper bounds of eigenvalues of a symmetric tridiagonal interval matrix.
Compared to the known methods, our approach is fast, simple to present and to
implement, and avoids any assumptions. Our construction explicitly yields those
matrices for which particular lower and upper bounds are attained
Statistical test of Duane-Hunt's law and its comparison with an alternative law
Using Pearson correlation coefficient a statistical analysis of Duane-Hunt
and Kulenkampff's measurement results was performed. This analysis reveals that
empirically based Duane-Hunt's law is not entirely consistent with the
measurement data. The author has theoretically found the action of
electromagnetic oscillators, which corresponds to Planck's constant, and also
has found an alternative law based on the classical theory. Using the same
statistical method, this alternative law is likewise tested, and it is proved
that the alternative law is completely in accordance with the measurements. The
alternative law gives a relativistic expression for the energy of
electromagnetic wave emitted or absorbed by atoms and proves that the
empirically derived Planck-Einstein's expression is only valid for relatively
low frequencies. Wave equation, which is similar to the Schr\"odinger equation,
and wavelength of the standing electromagnetic wave are also established by the
author's analysis. For a relatively low energy this wavelength becomes equal to
the de Broglie wavelength. Without any quantum conditions, the author made a
formula similar to the Rydberg's formula, which can be applied to the all known
atoms, neutrons and some hyperons.Comment: 12 pages, 7 figures, 3 tables, English and French abstrac
On a non-combinatorial definition of Stirling numbers
In Combinatorics Stirling numbers may be defined in several ways. One such
definition is given in [1], where an extensive consideration of Stirling
numbers is presented. In this paper an alternative definition of Stirling
numbers of both kind is given. Namely, Stirling numbers of the first kind
appear in the closed formula for the n-th derivative of ln x. In the same way
Stirling numbers of the second kind appear in the formula for the n-th
derivative of f(e^x), where f(x) is an arbitrary smooth real function. This
facts allow us to define Stirling numbers within the frame of differential
calculus. These definitions may be interesting because arbitrary functions
appear in them. Choosing suitable function we may obtain different properties
of Stirling numbers by the use of derivatives only. Using simple properties of
derivatives we obtain here three important properties of Stirling numbers.
First are so called two terms recurrence relations, from which one can easily
derive the combinatorial meaning of Stirling numbers. Next we obtain expansion
formulas of powers into falling factorials, and vise versa. These expansions
usually serve as the definitions of Stirling numbers, as in [1]. Finally, we
obtain the exponential generating functions for Stirling and Bell numbers. As a
by product the closed formulas for the -th derivative of the functions
f(e^x) and f(ln x) are obtained
Recurrence Relations and Determinants
We examine relationships between two minors of order n of some matrices of n
rows and n+r columns. This is done through a class of determinants, here called
-determinants, the investigation of which is our objective. We prove that
1-determinants are the upper Hessenberg determinants. In particular, we state
several 1-determinants each of which equals a Fibonacci number. We also derive
relationships among terms of sequences defined by the same recurrence equation
independently of the initial conditions. A result generalizing the formula for
the product of two determinants is obtained. Finally, we prove that the Schur
functions may be expressed as -determinants
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