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In today’s world, deep learning models are widely used in a variety of fields. Audio applications include speech recognition, audio classification, and music information retrieval. In this paper, we will focus on the classification of music genres using an artificial neural network. The development of audio machine learning techniques has created an independence from traditional, more time-consuming signal processing techniques. Starting with raw audio data, we will gain an understanding of what audio is and its digital representation. Then, the focus will be on obtaining frequency information from audio signals through the use of spectrograms. Transforming the spectrograms into the perceptually relevant mel scale allows us to eventually extract mel frequency cepstral coefficients (MFCC) from audio files. We will then make use of our network architecture to process the MFCC’s. A convolutional neural network, our network of choice here, is trained to classify audio files into one of nine musical genres with an accuracy of 89.1% using the GTZAN dataset, which is only about 4 percentage points below the state-of-the-art performance for this dataset.

In asserting that people in other fields often tend to be afraid to use mathematics, F.K. Mechta, an economist, expressed an uncertainty which contributes to the hesitancy to use mathematics. “Mathematics is tricky, it maintains silence, does its work quietly; and when we do not understand its ways and misinterpret its message, it just smiles. It never loses its temper, never laughs; we can observe a suppressed smile on its lips. Such is mathematics.” It is the purpose of this paper to conduct a brief survey of the applications of difference equations. The use of these equations is often rather elementary, frequently involving little material more technical than college algebra. An individual does not necessarily need a calculus background to be able to apply them. In the research for this work, applications of difference equations were looked for in the social and physical sciences. This paper covers the types of applications found, and the way they were developed. There is particular emphasis on what is left for the mathematician to do; what he can do to help these other fields benefit from mathematics. It will be sometimes noted throughout the paper whither or not the uniqueness of the solution has been checked for by each author. The first section in this paper is an introduction to difference equations. This section includes a brief summary of some of the theory of difference equations with emphasis on methods of solution, as well as a study of difference equations in probability. In the remaining section, the types of applications found in different fields are reviewed.

In this paper we consider second order parabolic partial differential equations on the infinite strip. We confine our attention to the quasi-linear equations, and in particular, the problem (0-1) u͙(t,x) = f(u)uxx(t.x) , (t,x) ε(0,T)x(-∞,∞) ≡ DT, T>0 u(0,x) = u0(x) , x ε (-∞,∞). Existence, uniqueness and properties of the solution are discussed. These results can be immediately generalized to problems where the differential equation is of the form u͙(t,x) = f(t,x,u)u xx(t,x).

The purpose of this paper is to study two particular entire functions which satisfy the conditions set up in a theorem due to Newsom. It is to be hoped that this report may be preliminary to the invention of a method which will lend itself toward the solution of certain general problems.

It is the purpose of this paper to show how vector methods can be applied to the geometry of the triangle. Because of the unlimited amount of material regarding the triangle, the discussion herein is restricted to those considerations involving only points or straight lines, or combinations of the two.

The field of knowledge known as mathematics is composed of the totality of mathematical systems. A mathematical system consists of a body of propositions known as the axioms. The concern of this paper is with the axioms; in fact, with the change which has taken place during the centuries in the ideas of mathematicians about the axioms.

In summary, it should be stressed that the numerous examples of this paper are important only in so much as they illustrate the use of the devices and concepts emphasized in the study. It is our premise that plane geometry, as usually taught, does not possess the value which it might have. Certain changes in content as well as in presentation would materially improve the course. This study, then, presents suggestions for improving the ordinary course in two directions. First, by presenting logical propositions from the students' experiences which are partially isomorphic to the theorems of plane geometry, it is believed that the student will appreciate and better be able to use the principles of deductive logic. Secondly, by placing considerable emphasis upon the concept of variation, the student will gain a greater knowledge of mathematical method and its relation to procedures of empirical science.

The essential purpose of this paper is to obtain further information in regard to the asymptotic representation of a power series.

The purpose of this paper is to exhibit the main properties of Gamma and Beta distributions and show their relation to certain well known distributions. In chapter II the Gamma and Beta distributions are defined in terms of Gamma and Beta functions. The moments of these distributions are calculated, and the moment generating function and cumulant generating function for the Gamma distribution are obtained. The curves are classified with respect to parameter values and the curves are graphically illustrated in Figures 1, 2, and 3. The exponential distribution, as a special case of interest, is shown to be a Gamma distribution with parameter one.

INTRODUCTION This paper will amount to an inquiry into the mathematical structure and properties of a matrix of approximate integration formulas due to Werner Romberg 1 which have found considerable favor in numerical analysis during the past decade. Like other devices for approximate integration the Romberg method has its origin in the exhaustion techniques developed by the Greeks for determining the areas of figures enclosed by lines in the plane. It is particularly closely associated with the scheme employed by Archimedes2 (circa 250 B.C.) for approximating the value of [3.14]. This involves a doubling process which is essentially the basis for the Romberg method. Archimedes began with a circle of radius one within which was inscribed some regular polygon whose area A00 he could readily compute. After doubling the number of sides he was able to compute a new area A01, then a third area A02 after a second doubling and so on to A03, .... ,A0n, ect. It is geometrically clear that such a sequence should converge to [3.14] from below. As an example, if Aon corresponds to a 24-sided polygon it will he found that Aon= 3.(32,(,,Zq) Ao, the latter correct for [3.14] to 3 figures. Much later, in about 1654 in fact, Christian Huygens rather ingeneously deduced an improved version of the Archimedes sequence. This involves combing two successive Archimedes terms with the weiphts,-1/3 and 4/3, to produce a new sequence Ain = 4 Ao,(n+1) -Aon 3 which Huygens showed would converge to [3.14] much more rapidly. Thus for term A1n, corresponding to a 48-sided polygon it will be found that A in = 3. 141590 a value correct for [3.14] to 6 figures! Huygens' numerical innovation lay practically unnoticed for almost three centuries (most likely because of the emphasis on methods of infinitesimal calculus during this period) when in 1936 K. Komqerellˆ3 undertook to extend it by constructing a