Mathematical Tricks Research Articles

Einstein's relativity falsified: I. The factor γ can take values larger than √2, which requires a speed of light < c in moving inertial frames

Considering that an Einstein clock can travel at a speed no greater than c, from the principle of relativity, a kinematic time dilation factor can be derived whose value cannot be greater than <mml:math display="inline"> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:math> . In fact, however, the kinematic time dilation factor γ can approach an infinite value. This discrepancy demonstrates that the derivation of the kinematic time dilation factor γ in Einstein's special relativity (SR) cannot be physically justified by the principle of relativity, and that it is not physically possible that the speed of light is constant in any frame of reference. The mathematical method of Einstein's SR, which I refer to as the “mathematical method of relativity,” allows the calculation of constant physical values from different quantities of any physical unit and is thus scientifically worthless. Accordingly, it is not surprising that it is possible to predict so-called general relativistic phenomena, e.g., the phenomena observed at the binary pulsar PSR B1913 + 16, just by applying Kepler's second law and simple quantum physical considerations [R. G. Ziefle, Phys. Essays 33, 99 (2020)]. A careful interpretation of interferometer experiments on Earth clearly shows that there is in fact no need for artificial time acceleration by length contraction. However, today's physicists seem to be lost in mathematics. The aim of this paper is to contribute to a physical theory of relativity that does not require mathematical tricks, such as time acceleration (length contraction), space-time curvature, and other mathematical tricks that follow from Einstein's mathematical methods and uphold the illusion that the belief in a constant speed of light c in any frame of reference is physically justified.

A Maximum Logarithmic Maximum a Posteriori Probability Based Soft-Input Soft-Output Detector for the Coded Spatial Modulation Systems

Soft-input soft-output (SISO) detector for the iterative detection and decoding of coded spatial modulation (SM) signals needs to compute the extrinsic bit log-likelihood ratio (LLR) associated with each detected bit. The maximum logarithmic maximum a posteriori (max-log-MAP) detector is known to perform closely to the optimal logarithmic maximum a posteriori (log-MAP) detector with much lower computational complexity. In this paper, we first apply several mathematical tricks to derive a new and low-complexity algorithm for the max-log-MAP detector. The proposed algorithm features a new tree pruning technique to visit few lattice points and a smart list administration technique to retain a max-log-MAP detector. By following the proposed algorithm, we then design a hardware architecture for SISO detection of coded SM signals over the scenario of 64-QAM symbols, 8 transmit antennas, and 4 receive antennas. Under the TSMC 40 nm CMOS technology, the VLSI implementation results reveal that our architecture requires 46.4K gates and leads to detection throughput 450 Mbps, while operating at clock frequency 400 MHz and consuming power 46 mW. Being the first hardware architecture for SISO detection of coded SM signals, our proposed low-cost max-log-MAP detector is very attractive.

POSSIBLE USE OF MATHEMATICS AND REASONING IN PROBLEM-SOLVING PROCESS OF SEVERAL CHEMICAL PROBLEMS

Nowadays it is important to use various approaches, methods and tricks during teaching process of chemistry helping students gain new knowledge. Using interdisciplinary relations, we show the use of concept of similar triangles and mathematical reasoning during problem-solving process of chemical problems.
 In “Solutions” section it is shown how to apply concept of similar triangles and corresponding formula is derived. Patterns observed here exactly match the properties of well-known concept of similar triangles, which gives possibility to solve problem using mathematical tricks and methods. The formula is used to calculate the mass of precipitate formed caused by the drop of temperature of a saturated solution. In another example, in order to solve the problem, it is needed to solve a triangle equation, which is done using mathematical reasoning and concepts taught at school. It is well known that school knowledge may be insufficient to solve triangle equations. It should be noted, that majority of unknown values in chemical problems can be expressed using discreet values, which allows to solve problem by redefining it to mathematical divisibility problem.
 We provide solutions to chemical problems that don’t involve much of chemical reasoning. Suggested solutions broaden learners’ mind and enhance creative thought.

Mathematical tricks for pseudopotentials in the theories of nonlinear waves in plasmas

In the analysis of nonlinear waves in plasma, especially for the search for periodic waves, shock waves, and solitons, mechanical analogy methods are widely applicable. The most famous of them is the Sagdeev pseudopotential method. However, sometimes mathematical difficulties arise when deriving formulas for pseudopotentials. The author proposes three mathematical tricks to get around these difficulties and obtain exact formulas for pseudopotentials in cases where the direct, Sagdeev method is considered inapplicable: a trick based on the Lambert W-function, a trick based on the inverse function integration, and a trick based on reducing the theory equations to the Bernoulli differential equation (the Bernoulli pseudopotential method). This article, which is methodological by nature, provides detailed examples of the application of each of these tricks when deriving formulas for pseudopotentials.

Efficiency analysis of multiple detector effects on MCNP 6.2 simulations

An analysis of some variance reduction methods was introduced and compared with the conventional narrow beam geometry for radiation transport problems. Although the Monte Carlo methods provide accurate results in radiation transport problems, their usability requires a high processing infrastructure. Simulation algorithms need miscellaneous mathematical tricks named as variance reduction to achieve precise results with reduced computing resources. These techniques mostly require experience, intuition, and iteration. For this reason, an alternative approach that may be adopted by all users is needed. Proposed method using source-biased quadruplet and octuplet detection geometries need neither additional code nor advanced experience. The designed geometries were tested on various selected materials, energy values and source definitions to prove its availability. The accuracy and the stability of the method were analyzed with all possible parameters such as material and energy dependence (tasks-error/CPU time, detector fluctuation-NPS, relative error-NPS, deviation-NPS, etc.). Besides, to verify the analysis results, some radiography applications were performed. The efficiency of multiple detector usage was found to be dependent on the complexity of the geometry as expected. It may be concluded that the multi-detector usage is efficient for simplified geometries in terms of relative error and computing time. The firstly introduced geometric designs worked steadily over ~10M histories on all trials. For the advised design, up to ~7 times improvement achieved in terms of computing time based on the figure of merit.

Inverse problem approach to evaluate quadratic Zeeman effect for an atomic fountain frequency standard KRISS-F1

The evaluation of the quadratic Zeeman shift in atomic fountain frequency standards requires information on the magnetic field distribution along the drift region. Plotting Zeeman frequencies against various launching heights provides the temporally averaged magnetic field values seen by the flying atoms. Using those data, we were able to deduce a reasonable field map by taking the approach of solving an inverse problem. We employed a regularization method after establishing a linear set of equations. The analysis enabled us to evaluate the quadratic Zeeman shift and its uncertainty against various launching heights. Our method encompasses and generalizes the deconvolution method. This paper covers the mathematical tricks to solve the inverse problem and its application to the atomic fountain frequency standard KRISS-F1.

Scalable Fully Bayesian Gaussian Process Modeling and Calibration With Adaptive Sequential Monte Carlo for Industrial Applications

Abstract Gaussian process (GP) regression or kriging has been extensively applied in the engineering literature for the purposes of building a cheap-to-evaluate surrogate, within the contexts of multi-fidelity modeling, model calibration, and design optimization. With the ongoing automation of manufacturing and industrial practices as a part of Industry 4.0, there has been a greater need for advancing GP regression techniques to handle challenges such as high input dimensionality, data paucity or big data problems, these consist primarily of proposing efficient design of experiments, optimal data acquisition strategies, sparsifying covariance kernels, and other mathematical tricks. In this work, our attention is focused on the challenges of efficiently training a GP model, which, to the authors opinion, has attracted very little attention and is to-date poorly addressed. The performance of widely used training approaches such as maximum likelihood estimation and Markov Chain Monte Carlo (MCMC) sampling can deteriorate significantly in high-dimensional and big data problems and can lead to cost deficient implementations of critical importance to many industrial applications. Here, we compare an Adaptive Sequential Monte Carlo (ASMC) sampling algorithm to classic MCMC sampling strategies and we demonstrate the effectiveness of our implementation on several mathematical problems and challenging industry applications of varying complexity. The computational time savings of the ASMC approach manifest in large-scale problems helping us to push the boundaries of applicability and scalability of GPs for model calibration in various domains of the industry, including but not limited to design automation, design engineering, smart manufacturing, predictive maintenance, and supply chain manufacturing.

Newtonian quantum gravity

Newtonian Quantum Gravity (NQG) unifies quantum physics with Newton's theory of gravity and calculates the so-called “general relativistic” phenomena more precisely and in a much simpler way than General Relativity, whose complicated theoretical construct is no longer needed. Newton's theory of gravity is less accurate than Albert Einstein's theory of general relativity. Famous examples are the precise predictions of General Relativity at binary pulsars. This is the reason why relativistic physicists claim that there can be no doubt that Einstein's theory of relativity correctly describes our physical reality. With the example of the famous “Hulse-Taylor binary” (also known as PSR 1913 + 16 or PSR B1913 + 16), the author proves that the so-called “general relativistic phenomena” observed at this binary solar system can be calculated without having any knowledge on relativistic physics. According to philosophical and epistemological criteria, this should not be possible, if Einstein's theory of relativity indeed described our physical reality. Einstein obviously merely developed an alternative method to calculate these phenomena without quantum physics. The reason was that in those days quantum physics was not yet generally taken into account. It is not the first time that a lack of knowledge of the underlying physical phenomena has to be compensated by complicated mathematics. Einstein's theory of general relativity indirectly already includes additional quantum physical effects of gravitation. This is the reason why it cannot be possible to unite Einstein's theory of general relativity with quantum physics, unless one uses “mathematical tricks” that make the additional quantum physical effects disappear again in the end.

Modeling of pump wavelength-dependent spectral hole-burning in EDFA

Throughout this paper, considering inhomogeneous broadening in amplification and pump bands, the dynamical behavior of an erbium-doped fiber amplifier is investigated. To this end, equations of propagation and populations are obtained with the erbium atoms being treated as three-level atoms. The governing equations comprise a system of infinitely many PDEs which reduce to finitely many PDEs using some mathematical tricks. The phenomena of spectral hole-burning and pump-mediated inhomogeneity are inspected by solving these equations through using the finite-difference method and calculating the gain difference spectrum.

The variational method in quantum mechanics: an elementary introduction

Variational methods in quantum mechanics are customarily presented as invaluable techniques to find approximate estimates of ground state energies. In the present paper a short catalogue of different celebrated potential distributions (both 1D and 3D), for which an exact and complete (energy and wavefunction) ground state determination can be achieved in an elementary way, is illustrated. No previous knowledge of calculus of variations is required. Rather, in all presented cases the exact energy functional minimization is achieved by using only a couple of simple mathematical tricks: ‘completion of square’ and integration by parts. This makes our approach particularly suitable for undergraduates. Moreover, the key role played by particle localization is emphasized through the entire analysis. This gentle introduction to the variational method could also be potentially attractive for more expert students as a possible elementary route toward a rather advanced topic on quantum mechanics: the factorization method. Such an unexpected connection is outlined in the final part of the paper.

Isometric hashing for image retrieval

Hashing has been attracting much attention in computer vision recently, since it can provide efficient similarity comparison in massive multimedia databases with fast query speed and low storage cost. Since the distance metric is an explicit description of similarity, in this paper, a novel hashing method is proposed for image retrieval, dubbed Isometric Hashing (IH). IH aims to minimize the difference between the distance in input space and the distance of the corresponding binary codes. To tackle the discrete optimization in a computationally tractable manner, IH adopts some mathematical tricks to transform the original problem into a multi-objective optimization problem. The usage of linear-projection-based hash functions enables efficient generating hash codes for unseen data points. Furthermore, utilizing different distance metrics could produce corresponding hashing algorithms, thus IH can be seen as a framework for developing new hashing methods. Extensive experiments performed on four benchmark datasets validate that IH can achieve comparable to or even better results than some state-of-the-art hashing methods.

Wide-Range Closed Formulas for the Wavenumber of Sommerfeld’s Wire

Simple closed formulas are derived for the number of a harmonic wave traveling along a conducting wire, whose diameter may be as large as one wavelength. Mathematical tricks enhance the range of validity of a previous elegant, yet asymptotic formula. The proposed formulas are validated against a numerical method that solves the propagation equation exactly. Their range of validity is estimated.

The Advanced Australian Shuffle

The “classical” Australian under-down shuffle starts with a deck of n cards. Then one proceeds as follows: one card under the deck, one on the table, one under the deck, one on the table, etc. One continues until only one card remains. There is an explicit formula to calculate the number of the card in the original deck that survives, and this is the basis of several mathematical magic tricks.

Is the Free Vacuum Energy Infinite?

Considering the fundamental cutoff applied by the uncertainty relations’ limit on virtual particles’ frequency in the quantum vacuum, it is shown that the vacuum energy density is proportional to the inverse of the fourth power of the dimensional distance of the space under consideration and thus the corresponding vacuum energy automatically regularized to zero value for an infinitely large free space. This can be used in regularizing a number of unwanted infinities that happen in the Casimir effect, the cosmological constant problem, and so on without using already known mathematical (not so reasonable) techniques and tricks.

Germany for the World Cup!?

Abstract“It's what happens on the pitch that matters,” said Adi Preissler and Otto Rehhagel, former German footballers and coaches, in some oft‐quoted words of football wisdom. Statistically, the German national side managed by Jogi Löw is 6.5 times more likely than an “average” team to finally bring home the title again. Here is a scientific approach to a World Cup forecast that uses a few mathematical tricks and computer simulations.

On the Theory of Stochastic Transformation Method

The Stochastic Finite Element Method (SFEM) represents a new approach to solve mechanical systems with stochastic characteristics. The SFEM is based on the deterministic Finite Element Method in which some variables related to the structural state (variables involved in the stiffness matrix) and to the applied actions (involved in the load vector) are uncertain. In other words, the SFEM tries to look for the stochastic properties of the mechanical response. The solution of a stochastic mechanical system is completely defined through the evaluation of the probability density function of the response process. This cannot be analytically achieved through most of the available methods and techniques such as Fokker-Planck equation, Wiener-Hermite expansion, perturbation methods, stochastic linearization, WHEP technique, decomposition method and stochastic finite element methods [1; 2]. Some exact solutions are available for the mean and standard deviation, not for the Probability Density Function (pdf), of the solution process. In this paper, we develop the theory of method named the Stochastic Transformation Method (STM) in order to determine the pdf of the response of a stochastic mechanical system with random excitation and/or stiffness. The Stochastic Transformation Method is based on one-to-one mapping between the random output(s) and input(s) where the transformation Jacobean J can be computed. The pdf of the output(s) is then computed through the known joint pdf of the inputs multiplied by the determinant of transformation Jacobean matrix. The one-to-one mapping condition can be relaxed through some mathematical tricks. This STM allows us to express the “exact” pdf of the mechanical response [3], provided that the transformation Jacobean can be defined. For many cases, the pdf of the response can be obtained in a closed-form in terms of the joint distribution of the input random variables.

Response to the comments by Prof. Swendsen on “Concepts against mathematics: self-inconsistency in thermodynamic evaluations”

The Comments declare that ‘‘Thermodynamics is mathematically rigorous and its predictions are unique’’. Such a statement must be supported by rigorous mathematic operations. Surprisingly, the mathematical transformations in the ‘‘Comments’’ are incorrect and the comments on the transformations more damage thermodynamics than defend it. To explain the damage and mathematical tricks in the ‘‘Comments’’, we have to remind basic equations of thermodynamics. Thermodynamics operates with four energies (U is the internal energy, H is the enthalpy, G is the Gibbs energy, and F is the Helmholtz energy) and four variables (T is the temperature, S is the entropy, P is the pressure, and V is the volume). The functions relate with one another according to equations U 1⁄4 F þ TS ð1aÞ H 1⁄4 U þ PV ð1bÞ G 1⁄4 H TS ð1cÞ F 1⁄4 G PV : ð1dÞ

Stabilization of a spatially non‐causal reaction–diffusion equation by boundary control

SUMMARYStabilization of a reaction–diffusion equation, in which the heat source depends on the temperature of the whole space, is considered by using boundary control. A new backstepping transformation is introduced, in which there are two kernels. Through a series of mathematical tricks, the exact solutions of kernels are obtained, and a control law is obtained specifically. The inverse transformation is derived, and stability of the closed loop system established. Simulation results show that the closed loop system is stable. Copyright © 2012 John Wiley & Sons, Ltd.

The Algorithm of Nonlinear Feature Extraction for Large scale Data set

In order to deal with the storage and computation problem for large-scale data set, an efficient iterative method is proposed. Some mathematical tricks are firstly used to transform the kernel matrix and the sample vector can be gotten in feature space. Then, the sample vector is used for iterative algorithm to extract nonlinear discriminant vectors. The space complexity of proposed method reduces from 2 ( ) m  to ( ) m  . The experimental results validate the effectiveness of the proposed method.

It's a kind of magic: mathematical magic tricks explained, by David Crawford. pp. 258. Members £19-99, Non-members £23-99. 2009. ISBN: 0906588677 (The Mathematical Association).

It's a kind of magic: mathematical magic tricks explained, by David Crawford. pp. 258. Members £19-99, Non-members £23-99. 2009. ISBN: 0906588677 (The Mathematical Association).