Second-Order Partial Differentiation of Real Binary Functions

Second-Order Partial Differentiation of Real Binary Functions In this article we define second-order partial differentiation of real binary functions and discuss the relation of second-order partial derivatives and partial derivatives defined in [17].

[1]  B. Arneson,et al.  Primitive Roots of Unity and Cyclotomic Polynomials 1 , 2004 .

[2]  Czeslaw Bylinski.  Functions from a Set to a Set , 2004 .

[3]  Czesław Bylí.  Finite Sequences and Tuples of Elements of a Non-empty Sets , 1990 .

[4]  Konrad Raczkowski.  Real Function Differentiability - Part II , 1991 .

[5]  Y. Shidama.  Banach Space of Bounded Linear Operators , 2003 .

[6]  Edmund Woronowicz.  Relations and Their Basic Properties , 2004 .

[7]  Nagano Katsumi Wasaki.  Scalar Multiple of Riemann Definite Integral , 2004 .

[8]  Andrzej Trybulec,et al.  On the Sets Inhabited by Numbers 1 , 2003 .

[9]  Yatsuka Nakamura,et al.  The Definition of Finite Sequences and Matrices of Probability, and Addition of Matrices of Real Elements , 2006 .

[10]  Adam Naumowicz,et al.  Conjugate Sequences , Bounded Complex Sequences and Convergent Complex Sequences , 1996 .

[11]  Józef Bia las.  The σ-additive Measure Theory , 1990 .

[12]  Yasunari Shidama,et al.  Integral of Real-Valued Measurable Function1 , 2006 .

[13]  Konrad Raczkowski,et al.  Topological Properties of Subsets in Real Numbers 1 , 1990 .

[14]  Yasushi Fuwa,et al.  Public-Key Cryptography and Pepin’s Test for the Primality of Fermat Numbers , 2007 .

[15]  Dariusz Surowik,et al.  Cyclic Groups and Some of Their Properties - Part I , 1991 .

[16]  Cleve Moler,et al.  Mathematical Handbook for Scientists and Engineers , 1961 .

[17]  J. Białas.  Several Properties of the σ-additive Measure , 1991 .

[18]  Yasunari Shidama.  Convergence and the Limit of Complex Sequences , 1997 .

[19]  Yasunari Shidama,et al.  The Lebesgue Monotone Convergence Theorem , 2008, Formaliz. Math..

[20]  Wojciech A. Trybulec.  Vectors in Real Linear Space , 1990 .

[21]  Nagano Wakasato,et al.  Trigonometric Functions and Existence of Circle Ratio , 1998 .

[22]  ANDREW GRANVILLE.  THE FUNDAMENTAL THEOREM OF ARITHMETIC , 1998 .

[23]  G. Bancerek,et al.  Ordinal Numbers , 2003 .

[24]  J. Kotowicz.  Properties of Real Functions , 2004 .

[25]  Construction of Finite Sequences over Ring and Left-, Right-, and Bi-Modules over a Ring 1 , 2007 .

[26]  Hongwei Li,et al.  Partial Differentiation of Real Binary Functions , 2008, Formaliz. Math..

[27]  Konrad Raczkowski,et al.  Topological Properties of Subsets in Real Numbers , 1990 .

[28]  Fuguo Ge.  Inner Products, Group, Ring of Quaternion Numbers , 2008, Formaliz. Math..

[29]  The Sum and Product of Finite Sequences of Real Numbers , 1990 .

[30]  Yuzhong Ding,et al.  Formulas and Identities of Trigonometric Functions , 2004 .

[31]  Grzegorz Zwara,et al.  The Divisibility of Integers and Integer Relatively Primes 1 , 1990 .

[32]  Jarosław Kotowicz,et al.  Convergent Sequences and the Limit of Sequences , 2004 .

[33]  Takashi Mitsuishi,et al.  Properties of the Trigonometric Function , 2004 .

[34]  Christoph Schwarzweller.  The Ring of Integers , Euclidean Rings and Modulo Integers , 1999 .

[35]  Bing Xie,et al.  Inverse Trigonometric Functions Arctan and Arccot , 2008, Formaliz. Math..

[36]  J. Harrison.  Formalized Mathematics , 1996 .

[37]  Noboru Endou,et al.  The Definition of the Riemann Definite Integral and some Related Lemmas , 2002 .

[38]  Beata Padlewska,et al.  Families of Sets , 1990 .

[39]  Józef Białas.  Infimum and Supremum of the Set of Real Numbers. Measure Theory , 1990 .

[40]  Yasunari Shidama,et al.  Completeness of the Real Euclidean Space , 2005 .

[41]  Rafa l Kwiatek.  Factorial and Newton coeffitients , 1990 .

[42]  Charles M. Grinstead,et al.  Introduction to probability , 1999, Statistics for the Behavioural Sciences.

[43]  Yasunari Shidama,et al.  The Measurability of Complex-Valued Functional Sequences , 2009, Formaliz. Math..

[44]  Yasunari Shidama,et al.  Integral of Complex-Valued Measurable Function , 2008, Formaliz. Math..

[45]  Piotr Rudnicki,et al.  Multivariate Polynomials with Arbitrary Number of Variables 1 , 2004 .

[46]  Hiroshi Yamazaki,et al.  Bessel ’ s Inequality , 2004 .

[47]  Michal Trybulec.  Labelled State Transition Systems , 2009, Formaliz. Math..

[48]  Czeslaw Bylinski.  Binary Operations Applied to Finite Sequences , 1990 .

[49]  Józef Bia.  Series of Positive Real Numbers . Measure Theory , 1990 .

[50]  A. Trybulec.  Domains and Their Cartesian Products , 1990 .

[51]  Grzegorz Bancerek,et al.  Segments of Natural Numbers and Finite Sequences , 1990 .

[52]  Victor Shoup,et al.  A computational introduction to number theory and algebra , 2005 .

[53]  Czeslaw Bylinski.  Some Basic Properties of Sets , 2004 .

[54]  Wojciech A. Trybulec.  Non-contiguous Substrings and One-to-one Finite Sequences , 1990 .

[55]  F. Merkl.  Dynkin ’ s Lemma in Measure Theory , 2004 .

[56]  Wojciech A. Trybulec.  Subgroup and Cosets of Subgroups , 1990 .

[57]  Kenneth H. Rosen,et al.  Catalan Numbers , 2002 .

[58]  G. Ivanov.  Definition of Convex Function and Jensen ’ s Inequality , 2004 .

[59]  Yuji Sakai,et al.  Properties of Partial Functions from a Domain to the Set of Real Numbers , 1992 .

[60]  Krzysztof Hryniewiecki,et al.  Basic Properties of Real Numbers , 2004 .

[61]  Nagano Yatsuka Nakamura.  Zero-Based Finite Sequences , 2004 .

[62]  W. Kellaway,et al.  Complex Numbers , 2019, AMS/MAA Textbooks.

[63]  G. Bancerek.  The Fundamental Properties of Natural Numbers , 1990 .

[64]  The Quaternion Numbers , 2006 .

[65]  Beata Perkowska,et al.  Functional Sequence from a Domain to a Domain , 1992 .

[66]  Yasunari Shidama,et al.  Complex Function Differentiability , 2009, Formaliz. Math..

[67]  Yasunari Shidama,et al.  Inferior Limit, Superior Limit and Convergence of Sequences of Extended Real Numbers , 2007 .

[68]  Andrzej Trybulec,et al.  Tuples, Projections and Cartesian Products , 1990 .

[69]  Krzysztof Hryniewiecki.  Recursive Definitions , 1990 .

[70]  Eugeniusz Kusak.  Abelian Groups, Fields and Vector Spaces 1 , 1990 .

[71]  J. K. Hunter,et al.  Measure Theory , 2007 .

[72]  Czeslaw Bylinski.  Functions and Their Basic Properties , 2004 .

[73]  Andrzej Trybulec,et al.  Binary Operations Applied to Functions , 1990 .

[74]  Agata Darmochwa.  Euclidean Space , 2018, How to Pass the FRACP Written Examination.

[75]  R. Milewski.  Trigonometric Form of Complex Numbers , 2004 .

[76]  Konrad Raczkowski.  Real Function Differentiability 1 , 1990 .

[77]  Kenneth Halpern August.  The Cardinal Numbers , 1888, Nature.

[78]  Agata Darmochwa,et al.  Topological Spaces and Continuous Functions , 1990 .

[79]  Noboru Endou.  Definitions and Basic Properties of Measurable Functions , 2004 .

[80]  Yasunari Shidama,et al.  Partial Differentiation on Normed Linear Spaces Rn , 2007 .

[81]  Jaross Law Kotowicz.  Real Sequences and Basic Operations on Them , 1989 .

[82]  Yatsuka Nakamura,et al.  The Inner Product and Conjugate of Finite Sequences of Complex Numbers , 2005 .

[83]  M. Jastrzębska,et al.  On the Properties of the Möbius Function , 2006 .

[84]  Jan Popio,et al.  Real Normed Space , 1991 .

[85]  Yatsuka Nakamura,et al.  Set Sequences and Monotone Class , 2005 .

[86]  Konrad Raczkowski.  Real Function Continuity 1 , 1990 .

[87]  Noboru Endou,et al.  Definition of Integrability for Partial Functions from R to R and Integrability for Continuous Functions , 1994 .

[88]  Piotr Rudnicki,et al.  Abian's Fixed Point Theorem , 2007 .

[89]  Edmund Woronowicz.  Relations Defined on Sets , 1990 .

[90]  Yasunari Shidama,et al.  Uniqueness of Factoring an Integer and Multiplicative Group Z/pZ* , 2008, Formaliz. Math..

[91]  Józef Białas.  Some Properties of the Intervals , 2004 .

[92]  A. Trybulec.  Tarski Grothendieck Set Theory , 1990 .

[93]  Formal Languages - Concatenation and Closure , 2007 .

[94]  Wojciech A. Trybulec.  Lattice of Subgroups of a Group. Frattini Subgroup , 1991 .

[95]  H. Piaggio.  Mathematical Analysis , 1955, Nature.

[96]  Peng Wang,et al.  Several Differentiation Formulas of Special Functions. Part V , 2007 .