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May 30, 2022
05/22

May 30, 2022
by
Petro Kolosov

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Instant messaging systems (IMS) achieve a great success and became the main mean of communication between people via an internet. Thanks to the simplicity and quickness of the message exchanging more and more people over the world start to use instant messengers on daily basis. However, such a great attention forces us to discuss another aspect of these systems, an aspect of the information security and user privacy. The high attention and wide usage of the instant messaging systems in both,...

Topics: IMS, Instant messenger, Bachelor's thesis

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16

May 22, 2022
05/22

May 22, 2022
by
Petro Kolosov

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In this manuscript we provide a new polynomial pattern. This pattern allows to find a polynomial expansion.

Topics: Binomial theorem, Power function, Polynomials, Polynomial identities, Multinomial theorem, Binomial...

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11

May 22, 2022
05/22

May 22, 2022
by
Petro Kolosov

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Discussion on Diffie-Hellman Key Exchange and its implementation via REST

Topics: Diffie-Hellman Key Exchange, DH key exchange, REST

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4.0

May 21, 2022
05/22

May 21, 2022
by
Petro Kolosov

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Simple explanation on the problematics and main idea behind the Rivest- Shamir-Adleman (RSA) encryption.

Topics: Algorithms, Computer science, Cryptography, RSA, Number Theory

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12

May 21, 2022
05/22

May 21, 2022
by
Petro Kolosov

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Goldbach-like conjectures via Bertrand's Postulate

Topics: Bertrand's Postulate, Goldbach Conjecture, Ternary Goldbach Conjecture

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20

May 21, 2022
05/22

May 21, 2022
by
Petro Kolosov

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A study on partial dynamic equation on time scales involving derivatives of polynomials

Topics: Dynamic equations on time scales, Partial differential equations on time scales, Partial dynamic...

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17

May 21, 2022
05/22

May 21, 2022
by
Petro Kolosov

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The relation between Binomial theorem And discrete convolution has been established

Topics: Dynamic equations on time scales, Partial differential equations on time scales, Partial dynamic...

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14

May 21, 2022
05/22

May 21, 2022
by
Petro Kolosov

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Polynomial identities involving Pascal's triangle rows, based on the famous Pascal's triangle property that each row sum is power of 2.

Topics: Binomial coefficients, Binomial theorem, Pascal's triangle, Binomial sums, Binomial distribution,...

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149

Oct 23, 2018
10/18

Oct 23, 2018

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In this paper, we derive and prove, by means of Binomial theorem and Faulhaber's formula, the following identity between m -order polynomials in T ∑ l k=1 ∑ j m=0 A m,j k j (T-k) j = ∑ m k=0 (-1) m-k U m (l,k) ∙ T k = T 2m+1

Topics: Faulhaber's formula, Faulhaber's theorem, Binomial Theorem, Binomial coefficient, Binomial...

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100

May 1, 2017
05/17

May 1, 2017
by
Kolosov Petro

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Calculating the value of C k ∈ { 1 , ∞ } class of smoothness real-valued function's derivative in point of R + in radius of convergence of its Taylor polynomial (or series), applying an analog of Newton's binomial theorem and q -difference operator. ( P , q ) -power difference introduced in section 5. Additionally, by means of Newton's interpolation formula, the discrete analog of Taylor series, interpolation using q -difference and p , q -power difference is shown. ARXIV : 1705.02516 DOI :...

Topics: Mathematics, Quantum calculus, Quantum algebra, Power quantum calculus, Quantum difference,...

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150

Apr 30, 2017
04/17

Apr 30, 2017
by
Kolosov Petro

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The main aim of this paper to establish the relations between forward, backward and central finite and divided differences (that is discrete analog ofthe derivative) and partial and ordinary high-order derivatives of polynomials. DOI : 10.6084/m9.figshare.4955384 , 10.17605/OSF.IO/DQNCJ ORCID: https://orcid.org/0000-0002-6544-8880 Personal Website: https://kolosovpetro.github.io/ Keywords : Finite difference, Derivative, Divided difference, Ordinary differential equation, Partial...

Topics: Mathematics, Finite difference, Derivative, Divided difference, Ordinary differential equation,...

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84

Apr 30, 2017
04/17

Apr 30, 2017
by
Kolosov Petro

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Abstract : In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k, 0

Topics: Series Representation, Power function, monomial, Binomial Theorem, Multinomial theorem, Worpitzky...

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115

Jul 8, 2016
07/16

Jul 8, 2016
by
Kolosov Petro

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Abstract : In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k, 0

Topics: Series Representation, Power function, monomial, Binomial Theorem, Multinomial theorem, Worpitzky...

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99

Apr 7, 2016
04/16

Apr 7, 2016
by
Kolosov Petro

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Abstract : In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k, 0

Topics: Series Representation, Power function, monomial, Binomial Theorem, Multinomial theorem, Worpitzky...

115
115

Mar 16, 2016
03/16

Mar 16, 2016
by
Kolosov Petro

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Abstract : In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k, 0

Topics: Series Representation, Power function, monomial, Binomial Theorem, Multinomial theorem, Worpitzky...