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xxviii, 575 p. 24 cm

xxviii, 575 p. 24 cm

In the paper, the author finds an explicit formula for computing Bell numbers in terms of Kummer confluent hypergeometric functions and Stirling numbers of the second kind.

In the paper, the author finds an explicit formula for computing Bell numbers in terms of Kummer confluent hypergeometric functions and Stirling numbers of the second kind.

A possible reason why even functions are called "even" and odd functions are called "odd"

14

Translation of: Logarithmisch-trigonometrisches Handbuch

xxviii, 575 p. 24 cm

xxviii, 575 p. 24 cm

xxviii, 575 p. 24 cm

xxviii, 575 p. 24 cm

In this paper we study the ring $\mathcal{P}$ of combinatorial convex polytopes. We introduce the algebra of operators $\mathcal{D}$ generated by the operators $d_k$ that send an $n$-dimensional polytope $P^n$ to the sum of all its $(n-k)$-dimensional faces. It turns out that $\mathcal{D}$ is isomorphic to the universal Leibnitz-Hopf algebra with the antipode $\chi(d_k)=(-1)^kd_k$. Using the operators $d_k$ we build the generalized $f$-polynomial, which is a ring homomorphism from $\mathcal{P}$ to the ring $\Qsym[t_1,t_2,...][\alpha]$ of quasi-symmetric functions with coefficients in $\mathbb Z[\alpha]$. The images of two polytopes coincide if and only if their flag $f$-vectors are equal. We describe the image of this homomorphism over the integers and prove that over the rationals it is a free polynomial algebra with dimension of the $n$-th graded component equal to the $n$-th Fibonacci number. This gives a representation of the Fibonacci series as an infinite product. The homomorphism is an isomorphism on the graded group $BB$ generated by the polytopes introduced by Bayer and Billera to find the linear span of flag $f$-vectors of convex polytopes. This gives the group $BB$ a structure of the ring isomorphic to $f(\mathcal{P})$. We show that the ring of polytopes has a natural Hopf comodule structure over the Rota-Hopf algebra of posets. As a corollary we build a ring homomorphism $l_{\alpha}\colon\mathcal{P}\to\mathcal{R}[\alpha]$ such that $F(l_{\alpha}(P))=f(P)^*$, where $F$ is the Ehrenborg quasi-symmetric function.

We construct and study a certain zeta function which interpolates multi-poly-Bernoulli numbers at non-positive integers and whose values at positive integers are linear combinations of multiple zeta values. This function can be regarded as the one to be paired up with the $\xi$-function defined by Arakawa and the first-named author. We show that both are closely related to the multiple zeta functions. Further we define multi-indexed poly-Bernoulli numbers, and generalize the duality formulas for poly-Bernoulli numbers by introducing more general zeta functions.

We construct and study a certain zeta function which interpolates multi-poly-Bernoulli numbers at non-positive integers and whose values at positive integers are linear combinations of multiple zeta values. This function can be regarded as the one to be paired up with the $\xi$-function defined by Arakawa and the first-named author. We show that both are closely related to the multiple zeta functions. Further we define multi-indexed poly-Bernoulli numbers, and generalize the duality formulas for poly-Bernoulli numbers by introducing more general zeta functions.

"Symmetric Functions Formed by Systems of Elements of a Finite Algebra and their Connection with Fermat's Quotient and Bernoulli's Numbers" is an article from The Annals of Mathematics, Volume 18 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2007115

We extend the work of A. Ciaffaglione and P. Di Gianantonio on mechanical verification of algorithms for exact computation on real numbers, using infinite streams of digits implemented as co-inductive types. Four aspects are studied: the first aspect concerns the proof that digit streams can be related to the axiomatized real numbers that are already axiomatized in the proof system (axiomatized, but with no fixed representation). The second aspect re-visits the definition of an addition function, looking at techniques to let the proof search mechanism perform the effective construction of an algorithm that is correct by construction. The third aspect concerns the definition of a function to compute affine formulas with positive rational coefficients. This should be understood as a testbed to describe a technique to combine co-recursion and recursion to obtain a model for an algorithm that appears at first sight to be outside the expressive power allowed by the proof system. The fourth aspect concerns the definition of a function to compute series, with an application on the series that is used to compute Euler's number e. All these experiments should be reproducible in any proof system that supports co-inductive types, co-recursion and general forms of terminating recursion, but we performed with the Coq system [12, 3, 14].

One purpose of this paper is to construct twisted q-Euler numbers by using p-adic invariant integral on Zp in the sense of fermionic. Finally, we consider twisted Euler q-zeta function and q-l-series which interpolate twisted q-Euler numbers and polynomials.

Document number CIA-RDP60-00213A000100030090-3 declassified and released through the CIA's CREST database. Previously available only on four computers located outside of Washington D.C., the Agency was successfully pressured into putting the files online as a result of a MuckRock lawsuit and the efforts of Emma Best. The metadata was collected by Data.World, and the files are now being archived and made text searchable by the Internet Archive.

In this research thesis, (part II) we analyze several equations concerning the Black Hole and Wormhole entropies, Mock theta functions and the Partition numbers. New possible mathematical connections with various sectors of Number Theory and String Theory

In this research thesis, (part II) we analyze several equations concerning the Black Hole and Wormhole entropies, Mock theta functions and the Partition numbers. New possible mathematical connections with various sectors of Number Theory and String Theory

Book Source: Digital Library of India Item 2015.164974 dc.contributor.author: W.l.f. Fischer dc.contributor.other: D.van Nostrand Co dc.date.accessioned: 2015-07-06T23:38:23Z dc.date.available: 2015-07-06T23:38:23Z dc.date.copyrightexpirydate: 0000-00-00 dc.date.digitalpublicationdate: 2024-09-03 dc.date.citation: 1916 dc.identifier: CMU dc.identifier.barcode: 1990050037226 dc.identifier.origpath: /data/upload/0037/231 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/164974 dc.description.scanningcentre: AKCE, Universal Digital Library dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 605 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Akce, Universal Digital Library dc.publisher: D.van Nostrand Co dc.rights: Out_of_copyright dc.source.library: Cmu dc.subject.classification: Logarithmic Tables dc.title: Logarithmic Tables Of Numbers And Trigonometrical Functions dc.rights.holder: D.van Nostrand Co

xii p., 1 l., 155 p. 25 cm

xii p., 1 l., 155 p. 25 cm

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E.o`r Release 20011003 on ''Of Danpl icati.an 'of ` ~ncticsas Possible uCtion 'ia ~t bers'af Ba- gaged in OC/4L Supply! Activities ?` At the request of .;the Director' of OMMI i a mad the ?Dds*+c of , IQgistilis,; .rsll~ee~ttl-rrt~,y', *,+~ie s~arsi~ll ~` rbOlti ; ~ ~ .uo Av 7~r47w W Vi sL\i~ W i i.~W.\ii ~.a1i Wi1\iU W ~a Yi~f j.. activities in an effort to detersive v Tether or of d lie ti+oa" 'off ~ tions could, or should, be eli-tnsted and a reduction in the umbers of perst?-nne1 ngaged in supply activities be effected. 2. At the outset, as a result of discussions relating to the OC :esponsi- bilities and general support activities at both Headquarters and overs"as installations, it was detersined thatt (a) OC sqpply activities in the aver-. seas areas would not be a subject of discussion; (b) the OC-I/)WB liait on, library, test and inspection functions were properly an OC responsibility, and the field support functions of the OC-B/)dBB, being directly relates.:... to OC support activities in the overseas areas, dust necessarily be retained in the Office of Communications Vhild OC contiames its present dversewu support activities. 3. In view of the above, the Anly pertinent area for oonsideratican with relation to the subject problea was deemed to be the Stock Maaa,gesent tnd Requirmaents Section of the Materiel Support Brai ob, 00-=. A detailed review and analysis of this activity was oood~netad . and it we dstersi.red that: a. In lisitod areas there appears to be ' sa a dtx lica'tion in records and files between the OL and OC. However, this' d gxlieat or `'does not ceea twat considered necessary for the titvper f nmctioning af` the. tvo within tts Supply Division. ` _ . OC-E/IwB it would be necessary for OL to asks ertain ?reali mmentr idhich sweet certain CC reguireoents. However in ordir'to:, setl" GLASS Olauc CHANGE!) Aiths DDA_RLG. 71763 Tease 2001/s IA-RDP60 ' 00100030090-3 a =,e.. tsxnial i to the Office of Logistics ppr i ig aa4 coord nation Of ice vd` nieatioas requireseents could ' , .~c~i1i~ited vita the to' V14 Im area division lagistIes eev~omaats.` se~c ,`~exc W ss u ciao required, because at responsibility ImWosei by of Personnel with appropriate ' tee 14103_" li sbii3t-as in the cam- The'; Stock Meaasmeetnt and ReQ IIts iIentdoD, OC-1/*m ,z in genero-l., a mission and perforce funetioo samalogoas.'! to the of .. toe, oasOo?ents of the foreign area divisions. ~ &waver. thi^ In 14 *t of the above. it is rec ded that: .. The Material mart Bach. which pegrform for the Offir e of Cama:nieations an essential teehniea. (cassnnications.) logistic. function, continue operations in accordance with ite'tpesent 8tat4rent of It scion and Functions. ene i taial to both offices. It should be ?pointed ` oat that this program would in no. w affect a change in ceiling of the ``ori'fices eomaernm i, nor mould it ckn the career cognizance of the pessoomtl assigned be tween b. There- be no cc oli.dation: of nr /OL nersaaxaex or tuneticsu . ea i1'itb specific reference to the Stock Msnsgeaent and Rsqui re- fs Section .of. CC-E/KBB and Supply Division, OL, a protest be devised to provide; by detail on a tour basis, for the exchange of per- svnnel of the offices of Co winicat io s and Logistdcs, respectivei y, to provides an inter-office type ia84atrinatdon Rich, it is believ ed, would be b ' 0=ended that, _ a wv approatseataly one year after such a pia es omit be established; it be evaluated jointly by the +OtttlA of't?aes+wniCationis vm rw. W. ? aaupt yr uwisian, UL emd the Materiel?8tmport. 8h,nch, ~ r O:. its ontinued; use or the redssitation et and.' t be office of Loiistic s to . de terai.n. the adv ab l cad f d ssi abil

E.o`r Release 20011003 on ''Of Danpl icati.an 'of ` ~ncticsas Possible uCtion 'ia ~t bers'af Ba- gaged in OC/4L Supply! Activities ?` At the request of .;the Director' of OMMI i a mad the ?Dds*+c of , IQgistilis,; .rsll~ee~ttl-rrt~,y', *,+~ie s~arsi~ll ~` rbOlti ; ~ ~ .uo Av 7~r47w W Vi sL\i~ W i i.~W.\ii ~.a1i Wi1\iU W ~a Yi~f j.. activities in an effort to detersive v Tether or of d lie ti+oa" 'off ~ tions could, or should, be eli-tnsted and a reduction in the umbers of perst?-nne1 ngaged in supply activities be effected. 2. At the outset, as a result of discussions relating to the OC :esponsi- bilities and general support activities at both Headquarters and overs"as installations, it was detersined thatt (a) OC sqpply activities in the aver-. seas areas would not be a subject of discussion; (b) the OC-I/)WB liait on, library, test and inspection functions were properly an OC responsibility, and the field support functions of the OC-B/)dBB, being directly relates.:... to OC support activities in the overseas areas, dust necessarily be retained in the Office of Communications Vhild OC contiames its present dversewu support activities. 3. In view of the above, the Anly pertinent area for oonsideratican with relation to the subject problea was deemed to be the Stock Maaa,gesent tnd Requirmaents Section of the Materiel Support Brai ob, 00-=. A detailed review and analysis of this activity was oood~netad . and it we dstersi.red that: a. In lisitod areas there appears to be ' sa a dtx lica'tion in records and files between the OL and OC. However, this' d gxlieat or `'does not ceea twat considered necessary for the titvper f nmctioning af` the. tvo within tts Supply Division. ` _ . OC-E/IwB it would be necessary for OL to asks ertain ?reali mmentr idhich sweet certain CC reguireoents. However in ordir'to:, setl" GLASS Olauc CHANGE!) Aiths DDA_RLG. 71763 Tease 2001/s IA-RDP60 ' 00100030090-3 a =,e.. tsxnial i to the Office of Logistics ppr i ig aa4 coord nation Of ice vd` nieatioas requireseents could ' , .~c~i1i~ited vita the to' V14 Im area division lagistIes eev~omaats.` se~c ,`~exc W ss u ciao required, because at responsibility ImWosei by of Personnel with appropriate ' tee 14103_" li sbii3t-as in the cam- The'; Stock Meaasmeetnt and ReQ IIts iIentdoD, OC-1/*m ,z in genero-l., a mission and perforce funetioo samalogoas.'! to the of .. toe, oasOo?ents of the foreign area divisions. ~ &waver. thi^ In 14 *t of the above. it is rec ded that: .. The Material mart Bach. which pegrform for the Offir e of Cama:nieations an essential teehniea. (cassnnications.) logistic. function, continue operations in accordance with ite'tpesent 8tat4rent of It scion and Functions. ene i taial to both offices. It should be ?pointed ` oat that this program would in no. w affect a change in ceiling of the ``ori'fices eomaernm i, nor mould it ckn the career cognizance of the pessoomtl assigned be tween b. There- be no cc oli.dation: of nr /OL nersaaxaex or tuneticsu . ea i1'itb specific reference to the Stock Msnsgeaent and Rsqui re- fs Section .of. CC-E/KBB and Supply Division, OL, a protest be devised to provide; by detail on a tour basis, for the exchange of per- svnnel of the offices of Co winicat io s and Logistdcs, respectivei y, to provides an inter-office type ia84atrinatdon Rich, it is believ ed, would be b ' 0=ended that, _ a wv approatseataly one year after such a pia es omit be established; it be evaluated jointly by the +OtttlA of't?aes+wniCationis vm rw. W. ? aaupt yr uwisian, UL emd the Materiel?8tmport. 8h,nch, ~ r O:. its ontinued; use or the redssitation et and.' t be office of Loiistic s to . de terai.n. the adv ab l cad f d ssi abil

In this paper, we give an overview of our previous paper concerning the investigation of the algebraic and $p$-adic properties of Eisenstein-Kronecker numbers using Mumford's theory of algebraic theta functions.

We describe rational knots with any of the possible combinations of the properties (a)chirality, (non-)positivity, (non-)fiberedness, and unknotting number one (or higher), and determine exactly their number for a given number of crossings in terms of their generating functions. We show in particular how Fibonacci numbers occur in the enumeration of fibered achiral and unknotting number one rational knots. Then we show how to enumerate rational knots by crossing number and genus and/or signature. This allows to determine the distribution of these invariants among rational knots. We give also an application to the enumeration of lens spaces.

Book Source: Digital Library of India Item 2015.154738 dc.contributor.author: W.l.f. Fischer dc.contributor.other: D.van Nostrand Co dc.date.accessioned: 2015-07-06T19:40:12Z dc.date.available: 2015-07-06T19:40:12Z dc.date.copyrightexpirydate: 0000-00-00 dc.date.digitalpublicationdate: 2024-09-03 dc.date.citation: 1916 dc.identifier: CMU dc.identifier.barcode: 1990050020429 dc.identifier.origpath: /data/upload/0020/434 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/154738 dc.description.scanningcentre: AKCE, Universal Digital Library dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 605 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Akce Universal Digital Library dc.publisher: D.van Nostrand Co dc.rights: Out_of_copyright dc.source.library: Cmu dc.subject.classification: Logarithmic Tables dc.title: Logarithmic Tables Of Numbers And Trigonometrical Functions dc.rights.holder: D.van Nostrand Co

Book Source: Digital Library of India Item 2015.154738 dc.contributor.author: W.l.f. Fischer dc.contributor.other: D.van Nostrand Co dc.date.accessioned: 2015-07-06T19:40:12Z dc.date.available: 2015-07-06T19:40:12Z dc.date.copyrightexpirydate: 0000-00-00 dc.date.digitalpublicationdate: 2024-09-03 dc.date.citation: 1916 dc.identifier: CMU dc.identifier.barcode: 1990050020429 dc.identifier.origpath: /data/upload/0020/434 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/154738 dc.description.scanningcentre: AKCE, Universal Digital Library dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 605 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Akce Universal Digital Library dc.publisher: D.van Nostrand Co dc.rights: Out_of_copyright dc.source.library: Cmu dc.subject.classification: Logarithmic Tables dc.title: Logarithmic Tables Of Numbers And Trigonometrical Functions dc.rights.holder: D.van Nostrand Co

We give a new construction of q-Genocchi numbers, Euler numbers of higher order, which are different than the q-Genocchi numbers of Cangul-Ozden-Simsek. By using our q-Genoucchi, Euler nimbers of higher order, we can investigate the interesting relationship between w-q-Euler polynomials and w-q-Genocchi polynomials.

In this paper we study (h,q)-zeta functions associated with (h,q)-Bernoulli numbers and polynomials.

In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.

In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.

In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.

xxviii, 575 p. ; 24 cm

The indicial lift and moment functions are determined approximately for sinking and pitching motion at Mach numbers M of 0.3 and 0.6. These functions are determined from a knowledge of the existing oscillatory coefficients at the low reduced frequencies and from approximate expressions of these coefficients at the high reduced frequencies. The beginning portion of the indicial lift function associated with an airfoil penetrating a sharp-edge gust in subsonic flow is evaluated by use of an exact method. By use of an approximate method for determining the remaining portion, the complete indicial gust function is determined for M = 0.3, M = 0.6, and M = 0.7. All the indicial lift and moment functions are approximated by an exponential series; the coefficients which appear in the exponential approximations for each indicial function are tabulated for M = 0.5, M = 0.6, and M = 0.7.

Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulation, and it is shown that this leads to a nonlinear superposition principle for orthogonal, pure-state Wigner functions. It is concluded that the use of complex numbers in quantum mechanics can be regarded as a computational device to simplify calculations, as in all other applications of mathematics to physical phenomena.

Let $T$ be a positive closed current of bidimension (1,1) and unit mass on the complex projective space ${\Bbb P}^n$. We prove that the set $V_\alpha(T)$ of points where $T$ has Lelong number larger than $\alpha$ is contained in a complex line if $\alpha\geq2/3$, and $|V_\alpha(T)\setminus L|\leq1$ for some complex line $L$ if $1/2\leq\alpha

We investigate the regular convergence of the $m$-multiple series $$\sum^\infty_{j_1=0} \sum^\infty_{j_2=0}...\sum^\infty_{j_m=0} \ c_{j_1, j_2,..., j_m}\leqno(*)$$ of complex numbers, where $m\ge 2$ is a fixed integer. We prove Fubini's theorem in the discrete setting as follows. If the multiple series (*) converges regularly, then its sum in Pringsheim's sense can be computed by successive summation. We introduce and investigate the regular convergence of the $m$-multiple integral $$\int^\infty_0 \int^\infty_0...\int^\infty_0 f(t_1, t_2,..., t_m) dt_1 dt_2...dt_m,\leqno(**)$$ where $f: \bar{\R}^m_+ \to \C$ is a locally integrable function in Lebesgue's sense over the closed positive octant $\bar{\R}^m_+:= [0, \infty)^m$. Our main result is a generalized version of Fubini's theorem on successive integration formulated in Theorem 4.1 as follows. If $f\in L^1_{\loc} (\bar{\R}^m_+)$, the multiple integral (**) converges regularly, and $m=p+q$, where $m, p\in \N_+$, then the finite limit $$\lim_{v_{p+1},..., v_m \to \infty} \int^{v_1}_{u_1} \int^{v_2}_{u_2}...\int^{v_p}_{u_p} \int^{v_{p+1}}_0...\int^{v_m}_0 f(t_1, t_2,..., t_m) dt_1 dt_2...dt_m$$ $$=:J(u_1, v_1; u_2, v_2;...; u_p, v_p), \quad 0\le u_k\le v_k

575 p

575 p

575 p

575 p

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

Book Source: Digital Library of India Item 2015.154739 dc.contributor.author: W.l.f. Fischer dc.contributor.other: D.van Nostrand Co dc.date.accessioned: 2015-07-06T19:40:13Z dc.date.available: 2015-07-06T19:40:13Z dc.date.digitalpublicationdate: 2024-09-03 dc.identifier: CMU dc.identifier.barcode: 1990050020430 dc.identifier.origpath: /data/upload/0020/435 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/154739 dc.description.scanningcentre: AKCE, Universal Digital Library dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 600 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Akce Universal Digital Library dc.publisher: D.van Nostrand Co dc.rights: Out_of_copyright dc.source.library: Cmu dc.subject.classification: Logarithmic Tables dc.title: Logarithmic Tables Of Numbers And Trigonometrical Functions Edition 81 dc.rights.holder: D.van Nostrand Co

"On the Relation Between Cauchy's Numbers and Bessel's Functions" is an article from The Annals of Mathematics, Volume 12 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967538

"On the Relation Between Cauchy's Numbers and Bessel's Functions" is an article from The Annals of Mathematics, Volume 12 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967538