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The ability to perform detailed evolutionary calculations is essential to the development of a well-defined and testable binary model. Unfortunately, traditional evolutionary calculations cannot be used to follow a significant fraction of possible close-binary supersoft sources (CBSS's). It is therefore important to examine the in-put physics carefully, to be sure that all relevant and potentially important physical processes are included. In this paper we continue a line of research begun last year, and explore the role that winds are expected to play in the evolution of CBSS's. We find that at least a subset of the systems that seemed to be candidates for common envelope evolution may survive, if radiation emitted by the white dwarf drives winds from the system. We study the effects of winds on the binary evolution of CBSS's, and compute the number and characteristics of CBSS's expected to be presently active in galaxies such as our own or M31.

We obtain new combinatorial identities for integral values of binary Krawtchouk polynomials $K^{2m}_p(x)$, $0\le p\le 2m$, by computing the characters of the $p$-exterior representations on certain elements of order 2 of $\mathrm{SO}(2m)$. From this identities, we deduce several new relations for binomial coefficients and Catalan numbers.

We present two models of multitype Galton-Watson trees, that we call full binary trees and full binary trees with survivals. We show relevant relations between these trees and the Narayana numbers and the two-dimensional decompositions of integers. We prove further statistical results on our models concerning the related contour processes and offspring distributions.

We present two models of multitype Galton-Watson trees, that we call full binary trees and full binary trees with survivals. We show relevant relations between these trees and the Narayana numbers and the two-dimensional decompositions of integers. We prove further statistical results on our models concerning the related contour processes and offspring distributions.

Sarnak obtained the asymptotic formula of the sum of the class numbers of indefinite binary quadratic forms from the prime geodesic theorem for the modular group. In the present paper, we show several asymptotic formulas of partial sums of the class numbers by using the prime geodesic theorems for the congruence subgroups of the modular group.

We prove some general recursions for the numbers of representations of positive integers as a sum x+y, x in X, y in Y, where X,Y are increasing sequences. In particular, we obtain recursions for the number of the Goldbach, Lemoine-Levy, Chen and other binary partitions.

This research paper is to define new rule to find the divisibility of stream of binary on any prime numbers 3, 5, 7 …… with reminder 0. In general, the divisibility of binary numbers is most important in many digital circuits and mathematical applications. This paper explains this new rule for evaluating the divisibility of any binary number on any prime number greater than 2. This rule will depend on separating the binary number into blocks of bits then processes each block separately in a special procedure to find the possible divisibility on the prime number. After testing this new rule with prime numbers 3, 5, and 7 as a sample of prime numbers, the finding shows that this rule provides fast and true results.

Let $\mathbb{N}_0$ be a class of natural numbers whose binary expansions contain even numbers of ones. Waring problem in numbers of class $\mathbb{N}_0$ is solved.

#### Circular Model for Building Binary (Base-Two) Thinking: 8-bit This is a circular model (8-bit, about 110mm in diameter) for children to play with binary numbers or binary ways of reasoning, in a gaming manner! It does not require a wide base to stand and is funto spin around. You would need a circular board and 8 copies of 1 and 0 each. It works on an integer between 0 and 255, including 0 and 255. There are many ways to use the binary game board. The following is one way to interact with a child or anybody at a party: ####How to Play To begin with, ask a child to think of a base-ten number between 0 and 255 without revealing it to you. Then, you ask, "Is your number bigger or equal to 128?" If the answer is yes, then place a "1" in the 128 slot and subtract 128 from the original number to get the remaining number. Otherwise, place "0" in the 128 slot and move on. Ask similar questions about the left-over using 64, 32, 16, 8, 4, 2, and 1, placing either "1" or "0" in the corresponding slot. The result is the base-two representation of the child's original number. It is essentially binary sorting using the place values. Of course, *by adding up all the place values with 1's*, you would be able to figure out the original base-ten number. The child would be impressed with your math magic! Switch roles and play again and again. I have used the binary game numerous times with children and college students with very positive experiences! Have fun and play with math! #### See my linear design at https://www.thingiverse.com/thing:2662334

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A scaffold for children to learn about binary numbers or numbers in the base two number system (0 to 11111, corresponding to 0 to 31 in the base ten system). The stand has five positions and is about 120mm long. It takes five number pieces to play the binary game (ranging 0 from 31). The number piece has "1" on the front and "0" on the back. I have designed the numbers for easy sliding and 3D printing. Have fun!

If the list of binary numbers is read by upward-sloping diagonals, the resulting ``sloping binary numbers'' 0, 11, 110, 101, 100, 1111, 1010, ... (or 0, 3, 6, 5, 4, 15, 10, ...) have some surprising properties. We give formulae for the n-th term and the n-th missing term, and discuss a number of related sequences.

We show that the Zagier-Eisenstein series shares its non-holomorphic part with certain weak Maass forms whose holomorphic parts are generating functions for overpartition rank differences. This has a number of consequences, including exact formulas, asymptotics, and congruences for the rank differences as well as $q$-series identities of the mock theta type.

Zundamon and Metan will explore the mystery of sequences composed of 0s and 1s. --- The English voices are improved in the new videos. Thank you for your feedback! [BGM] ほのぼのワルツ【リコーダー】 https://commons.nicovideo.jp/works/nc116360 Caravan https://dova-s.jp/bgm/play2457.html [Materials] VOICEVOX:ずんだもん (Zundamon, illustrated by sakamoto_AHR) VOICEVOX:四国めたん (Shikoku Metan, illustrated by sakamoto_AHR) 効果音ラボ (soundeffect-lab.info) pixabay This video uses synthesized voices. #math

Given an array $X$ of $N+1$ strictly ordered floating point numbers and an array $Z$ of $M$ floating point numbers belonging to the interval $[X_0,X_N)$, a common problem in numerical methods is to find the indices of the largest numbers in the array $X$ which are smaller or equal than the numbers in the array $Z$. The general solution to this problem is to call $M$ times the well known "binary search" algorithm, which has complexity $O(log_2 N)$ per individual search and, in its classic formulation, is not vectorizable, i.e. it is not SIMD friendly. This paper describes improvements to the binary search algorithm, which make it faster and vectorizable. Next it proposes a new vectorizable algorithm applicable to a wide set of $X$ partitions, which is based on an indexing technique and allows to solve the problem with complexity $O(1)$ per individual search at the cost of introducing an initial overhead to compute the index and requiring extra memory for its storage. Test results using streaming SIMD extensions compare the performance of the algorithm versus various benchmarks and demonstrate its effectiveness. Depending on the test case, the algorithm performs up to 43 times faster than the classical binary search in single precision and 39 times faster in double precision. Applicability limitations and cache-friendliness related aspects are also discussed.

Characteristics of comparator circuits for comparison of binary numbers in information processing system

As a continuation of the authors and Wakatsuki's previous paper [5], we study relations among Dirichlet series whose coefficients are class numbers of binary cubic forms. We show that for any integral models of the space of binary cubic forms, the associated Dirichlet series satisfies a simple explicit relation to that of the dual other than the usual functional equation. As an application, we write the functional equations of these Dirichlet series in self dual forms.

We calculate the network fluid regime and phase diagrams of binary mixtures of patchy colloids, using Wertheim's first order perturbation theory and a generalization of Flory-Stockmayer's theory of polymerization. The colloids are modelled as hard spheres with the same diameter and surface patches of the same type, $A$. The only difference between species is the number of their patches -or functionality-, $f_A^{(1)}$ and $f_A^{(2)}$ (with $f_A^{(2)}>f_A^{(1)}$). We have found that the difference in functionality is the key factor controlling the behaviour of the mixture in the network (percolated) fluid regime. In particular, when $f_A^{(2)}\ge2f_A^{(1)}$ the entropy of bonding drives the phase separation of two network fluids which is absent in other mixtures. This changes drastically the critical properties of the system and drives a change in the topology of the phase diagram (from type I to type V) when $f_A^{(1)}>2$. The difference in functionality also determines the miscibility at high (osmotic) pressures. If $f_A^{(2)}-f_A^{(1)}=1$ the mixture is completely miscible at high pressures, while closed miscibility gaps at pressures above the highest critical pressure of the pure fluids are present if $f_A^{(2)}-f_A^{(1)}>1$. We argue that this phase behaviour is driven by a competition between the entropy of mixing and the entropy of bonding, as the latter dominates in the network fluid regime.

1-75 In binary numbers Source: https://www.youtube.com/watch?v=NbVLKP4ZQTI Uploader: Wolfie Numbers

Understanding how numbers are represented. Introduction to binary numbers

Circuit includes a single-word input/output register and a multiword serial memory which are circulated in synchronism. It puts out data at a rate compatible with relatively slow-speed electromechanical devices.

We study the equidistribution of multiplicatively defined sets, such as the squarefree integers, quadratic non-residues or primitive roots, in sets which are described in an additive way, such as sumsets or Hilbert cubes. In particular, we show that if one fixes any proportion less than $40\%$ of the digits of all numbers of a given binary bit length, then the remaining set still has the asymptotically expected number of squarefree integers. Next, we investigate the distribution of primitive roots modulo a large prime $p$, establishing a new upper bound on the largest dimension of a Hilbert cube in the set of primitive roots, improving on a previous result of the authors. Finally, we study sumsets in finite fields and asymptotically find the expected number of quadratic residues and non-residues in such sumsets, given their cardinalities are big enough. This significantly improves on a recent result by Dartyge, Mauduit and S\'ark\"ozy. Our approach introduces several new ideas, combining a variety of methods, such as bounds of exponential and character sums, geometry of numbers and additive combinatorics.

This game is for the TI-99/4A, which is not emulated directly on the Internet Archive.  Below are directions for using this game with JS99 (which runs in your browser) and Classic99 (which is a Windows application): TO USE JS99:   1. Save the file "ti99.dsk" from this item.   2. Go to https://js99er.net/#/   3. Click on "Load Disk" (at the bottom of the screen).   4. Select the "ti99.dsk" file.   5. Enter TI Extended BASIC (option 2 on the TI start screen).   6. Enter "OLD DSK1.GAME".   7. Enter "RUN". TO USE CLASSIC99:   1. Install Classic99 from http://www.harmlesslion.com/software/Classic99   2. Enable TI Extended BASIC (under Cartridge > Apps).   3. Save the file "game.tifiles" to the "DSK1" directory.   4. Enter TI Extended BASIC (option 2 on the TI start screen).   5. Enter "OLD DSK1.GAME.TIFILES".   6. Enter "RUN". ------------------------------------------------------------------------------ BINARY NUMBERS   A program to help you understand binary numbers An understanding of binary numbers is very important for anyone who is seriously interested in knowing how computers work.  This program will help you get a better understanding of binary numbers and will show you how to convert from binary numbers to decimal numbers. Run the program.  Enter the suggested binary number 1101, which is equal to 13 in decimal.  The computer will show you how it calculated 13 decimal from 1101 binary. Repeat with other binary numbers, such as 111, 101, 1000, etc.  After a little practice, you should become quite familiar with the binary numbering system.

A certain analysis of all possible associative binary operations on N is presented. This is equivalent with an analysis of all possible monoid structures on N. Several results and a conjecture in this regard are given.

This is a set of cards, sorting lid, and keys to teach binary numbers from 0 to 31 OR to use to teach the concept of sorting via dichotomous key with T/F, Y/N questions. The long square keys are for True or 1 bits, the long cylindrical keys are for False or 0 bits. Print 5 each of the T and F keys, one sorting lid, and 32 blank cards, then use diagonal cutters to cut either the square "True" or "1" or "Yes" notches, or the "False" or "0" or "No" notches, out of the upper edge of the cards. An example "10100" or "decimal 20" card is included as an example. (you could design and print 32 different kinds of cards, but really, it's easier to just print them all blank and cut them.) Obviously this is a very simple example of something with a very long history: Jacquard looms, Hollerith cards, the McBee keysort system, etc. This is a simple intro to the concept. With a change in scaling this could also accomodate standard index cards, punched with a one-hole punch. EDIT: finally got to print a trial of this, uploaded resized T and F keys that should fit better in the card slots to allow freer movement. OH NO- problem with the printer, interfering with optimizing this design and getting example prints made. This will be addressed as soon as I'm printing again.

Ultimate Nim: The Use of Nimbers, Binary Numbers and Subpiles in the Optimal Strategy for Nim

We describe various properties of continued fraction expansions of complex numbers in terms of Gaussian integers. Numerous distinct such expansions are possible for a complex number. They can be arrived at through various algorithms, as also in a more general way from what we call "iteration sequences". We consider in this broader context the analogues of the Lagrange theorem characterizing quadratic surds, the growth properties of the denominators of the convergents, and the overall relation between sequences satisfying certain conditions, in terms of nonoccurrence of certain finite blocks, and the sequences involved in continued fraction expansions. The results are also applied to describe a class of binary quadratic forms with complex coefficients whose values over the set of pairs of Gaussian integers form a dense set of complex numbers.

Decoding tree consisting of 40-bit semiconductor read-only memories interconverts binary and decimal numbers 50 to 100 times faster than current methods. Decimal-to-binary conversion algorithm is based on a divided-by-2 iterative equation, binary-to-decimal conversion algorithm utilizes multiplied-by-2 iterative equation.

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Binary numbering 1-12 for a clock I built, had a few people wanting to replicate this themselves so this is being uploaded.

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We study the class numbers of integral binary cubic forms. For each $SL_2(Z)$ invariant lattice $L$, Shintani introduced Dirichlet series whose coefficients are the class numbers of binary cubic forms in $L$. We classify the invariant lattices, and investigate explicit relationships between Dirichlet series associated with those lattices. We also study the analytic properties of the Dirichlet series, and rewrite the functional equation in a self dual form using the explicit relationship.

From the bitsavers.org collection, a scanned-in computer-related document. mit :: whirlwind :: R-series :: R-090-1 Binary System of Numbers Feb1952

Let $h(d)$ be the class number of indefinite binary quadratic forms of discriminant $d$, and let $\varepsilon_d$ be the corresponding fundamental unit. In this paper, we obtain an asymptotic formula for the $k$-th moment of $h(d)$ over positive discriminants $d$ with $\varepsilon_d\leq x$, uniformly for real numbers $k$ in the range $0

Discusses about the concept and evolution of binary numbers  based on Pingala's Chanda Sutra.

Approved For Release 2008/05/07: CIA-RDP78B04558A000300190011-7 ? S E C R E T 142138Z CORONA 15 Milri 1963 FILE 9963 rAR 15 04 12 Z 1. BINARY CARDS FOR 9039, 9041 AND 9047 WERE AIR MAILED 13 MARCH';1963 IN TWO PACKAGES, REGISTRY NUMBERS 340233 AND 340234. 9039 CARDS ARE LABELED DECK At 9041 CARDS ARE LABELED DECK B AND 9047 CARDS ARE LABELED DECK C. CORRELATION CODE 3 IS TO BE USED FOR DECKS A AND B WITH THE FOLLOWING CHANGES: 22, 39; 35, 37; AND 67, 33. CORRELATION CODE 4 IS TO BE USED FOR DECK C. 2. AN ERROR EXISTS IN OUR COPY OF CORRELATION CODE 3. IT READS --22, 39; --67, 39;--; ONE OF THESE SHOULD READY --, 37. PLEASE ADVISE, ATTENTION 3. WOULD PREFER TO FORWARD BINARY READOUT ON MAGNETIC TAPE. IF ACCEPTABLE PLEASE ADVISE, ATTENTION RECORD FORMATS 25X1 DESIRED. S E C R E T CFN 0589 9039 9041 9047 13 1963 340233 340234 9039 A 9041 B 9047 C 3 A B 22, 39; 35, 37; 67, 33. 4 C 5290 3 --22, 39; --67, 39; --; --, 37. Declassification Review by NGA/DoD 0 u! ,:17V f Approved For Release 2008/05/07: CIA-RDP78B04558A000300190011-7

Given a positive definite binary quadratic form f, let r(n) = |{(x,y): f(x,y)=n}| denote its representation function. In this paper we study linear correlations of these functions. For example, if r_1, ..., r_k are representation functions, we obtain an asymptotic for sum_{n,d} r_1(n) r_2(n+d) ... r_k(n+ (k-1)d).

this is about the good and truthful and kind and real binary numbers 

Using a global optimization approach that directly searches for the composition of greatest stability, we have been able to find the particularly stable structures for binary Lennard-Jones clusters with up to 100 atoms for a range of Lennard-Jones parameters. In particular, we have shown that just having atoms of different size leads to a remarkable stabilization of polytetrahedral structures, including both polyicosahedral clusters and at larger sizes structures with disclination lines.

Using a global optimization approach that directly searches for the composition of greatest stability, we have been able to find the particularly stable structures for binary Lennard-Jones clusters with up to 100 atoms for a range of Lennard-Jones parameters. In particular, we have shown that just having atoms of different size leads to a remarkable stabilization of polytetrahedral structures, including both polyicosahedral clusters and at larger sizes structures with disclination lines.

Using a global optimization approach that directly searches for the composition of greatest stability, we have been able to find the particularly stable structures for binary Lennard-Jones clusters with up to 100 atoms for a range of Lennard-Jones parameters. In particular, we have shown that just having atoms of different size leads to a remarkable stabilization of polytetrahedral structures, including both polyicosahedral clusters and at larger sizes structures with disclination lines.

Using a global optimization approach that directly searches for the composition of greatest stability, we have been able to find the particularly stable structures for binary Lennard-Jones clusters with up to 100 atoms for a range of Lennard-Jones parameters. In particular, we have shown that just having atoms of different size leads to a remarkable stabilization of polytetrahedral structures, including both polyicosahedral clusters and at larger sizes structures with disclination lines.

This paper describes the straight and common (misère) forms of basic Nim, how binary numbers can be used in three ways to calculate winning positions to execute an optimal strategy for Nim play, and variant end game play for common Nim. In addition, this paper provides a proof of concept for expanding Nimber tables beyond two dimensions for use in play with four or more nim heaps.

A new fast (real time) sorter of binary numbers by one-dimensional cellular automata is proposed. It sorts a list of n numbers represented by k-bits each in exactly nk steps. This is only one step more than a lower bound.

Sisson, Roger Lee, and Alfred Kriss Susskind. 1950. “Devices for Conversion Between Analog Quantities and Binary Pulse-Coded Numbers.” Master’s Thesis, Cambridge: Massachusetts Institute of Technology. A report of the same name was also published under the MIT Servo Lab publication series: Sisson, Roger L., and Alfred L. Susskind. 1950. Devices for Conversion Between Analog Quantities and Binary Pulse-Coded Numbers . Servomechanisms Laboratory, Massachusetts Institute of Technology. I don't know how the contents are different if at all.

A critical analysis of measured partial coordination numbers for binary metallic glasses as a function of composition shows a large scatter of +/-1.5 but clear trends. The current work uses two topological models to predict the influence of relative atomic size and concentration on partial coordination numbers. The equations for partial coordination numbers derived from these two models can reproduce measured data within experimental scatter, suggesting that chemical effects on local structure, although present, may be relatively small. Insights gained from these models show that structural site-filling rules are different for glasses with solute atoms that are smaller than solvent atoms and for glasses where solute atoms are larger than solvent atoms. Specifically, solutes may occupy both beta and gamma intercluster sites when the solute-to-solvent radius ratio R is less than 1.26, but only beta sites can be occupied by solutes when R 1.26. This distinction gives a simple topological explanation for the observed preference for binary metallic glasses with solutes smaller than solvent atoms. In addition to structure-specific equations, simplified phenomenological equations for partial coordination numbers are given as a convenience.

From the bitsavers.org collection, a scanned-in computer-related document. washingtonUniversity :: linc :: LINC Documents :: LD37 Introduction to Binary Numbers and Binary Arithemetic Apr65

The occupation time of an age-dependent branching particle system in $\Rd$ is considered, where the initial population is a Poisson random field and the particles are subject to symmetric $\alpha$-stable migration, critical binary branching and random lifetimes. Two regimes of lifetime distributions are considered: lifetimes with finite mean and lifetimes belonging to the normal domain of attraction of a $\gamma$-stable law, $\gamma\in(0,1)$. It is shown that in dimensions $d>\alpha\gamma$ for the heavy-tailed lifetimes case, and $d>\alpha$ for finite mean lifetimes, the occupation time proccess satisfies a strong law of large numbers.

We investigate the effect of distinct bonding energies on the onset of criticality of low functionality fluid mixtures. We focus on mixtures of particles with two and three patches as this includes the mixture where 'empty' fluids were originally reported. In addition to the number of patches, the species differ in the type of patches or bonding sites. For simplicity, we consider that the patches on each species are identical: one species has 3 patches of type $A$ and the other 2 patches of type $B$. We have found a rich phase behaviour with closed miscibility gaps, liquid-liquid demixing and negative azeotropes. Liquid-liquid demixing was found to preempt the 'empty' fluid regime, of these mixtures, when the $AB$ bonds are weaker than the $AA$ or $BB$ bonds. By contrast, mixtures in this class exhibit 'empty' fluid behaviour when the $AB$ bonds are stronger than at least one of the other two. Mixtures with bonding energies $\epsilon_{BB}=\epsilon_{AB}$ and $\epsilon_{AA}

We investigate the effect of distinct bonding energies on the onset of criticality of low functionality fluid mixtures. We focus on mixtures of particles with two and three patches as this includes the mixture where 'empty' fluids were originally reported. In addition to the number of patches, the species differ in the type of patches or bonding sites. For simplicity, we consider that the patches on each species are identical: one species has 3 patches of type $A$ and the other 2 patches of type $B$. We have found a rich phase behaviour with closed miscibility gaps, liquid-liquid demixing and negative azeotropes. Liquid-liquid demixing was found to preempt the 'empty' fluid regime, of these mixtures, when the $AB$ bonds are weaker than the $AA$ or $BB$ bonds. By contrast, mixtures in this class exhibit 'empty' fluid behaviour when the $AB$ bonds are stronger than at least one of the other two. Mixtures with bonding energies $\epsilon_{BB}=\epsilon_{AB}$ and $\epsilon_{AA}