Legal Definition of Sequences

Legal Definition of Sequences

Sequence rules (37 CFR 1.821-1.825) require the use of standard symbols and a standard format for filing sequence data in most patent applications that disclose nucleic acid or amino acid sequences. For the purposes of the sequencing and analysis rules of Section 2400 of the MPEP, the term “disclosure (or disclosure(s) of nucleic acids or amino acid sequences” means the nucleic acid or amino acid sequences described in the patent application by listing their residues and meeting the length thresholds of 37 CFR 1.821(a). Consequences and their limitations (see below) are important concepts for the study of topological spaces. An important generalization of sequences is the concept of networks. A mesh is a function of a (possibly uncountable) set directed to a topological space. Notation conventions for sequences generally apply to meshes as well. That is, infinite sequences of elements indexed by natural numbers. On the other hand, there are Cauchy sequences of rational numbers that are not convergent in rational numbers, for example the sequence defined by x1 = 1 and xn + 1 = xn + 2 / xn / 2 is Cauchy, but has no rational boundary, see here. In general, any sequence of rational numbers that converges to an irrational number is Cauchy sequence, but not convergent if interpreted as a sequence in the set of rational numbers. In cases where all indexing numbers are included, subscript and superscript characters are often omitted. That is, you simply write ( a k ) {displaystyle (a_{k})} for any sequence. It is often assumed that the k-index ranges from 1 to ∞. However, sequences are often indexed from scratch, because defined in is defined as the set of all sequences ( x i ) i ∈ N {displaystyle (x_{i})_{iin mathbb {N} }}, so that for each i x i {displaystyle x_{i}} is an element of X i {displaystyle X_{i}}.

The canonical projections are the maps pi : X → Xi defined by the equation p i ( ( x j ) j ∈ N ) = x i {displaystyle p_{i}((x_{j})_{jin mathbb {N} })=x_{i}}. Then the product topology on X is defined as the coarsest topology (i.e. the topology with the fewest open sets) for which all projections pi are continuous. The product topology is sometimes referred to as the Tychonoff topology. An alternative to writing the domain of a sequence to the subscript sequence is to specify the range of values that the index can assume by listing the highest and lowest allowed values. For example, the notation ( k 2 ) k = 1 10 {displaystyle (k^{2})_{k=1}^{10}} denotes ten times the sequence of squares ( 1 , 4 , 9 ,. , 100 ) {displaystyle (1,4,9,…,100)}. The limits ∞ {displaystyle infty } and − ∞ {displaystyle -infty } are allowed, but do not represent valid values for the index, but only the supremum and infimum of these values, respectively. For example, the sequence ( a n ) n = 1 ∞ {displaystyle (a_{n})_{n=1}^{infty }} is identical to the sequence ( a n ) n ∈ N {displaystyle (a_{n})_{nin mathbb {N} }} , and does not contain an additional term “to infinity”. The sequence ( a n ) n = − ∞ ∞ {displaystyle (a_{n})_{n=-infty }^{infty }} is a biinfinite sequence and can also be written as (. , a − 1 , a 0 , a 1 , a 2 ,.

) {displaystyle (…,a_{-1},a_{0},a_{1},a_{2},…)}. If ( a n ) {displaystyle (a_{n})} and ( b n ) {displaystyle (b_{n})} are convergent sequences, then the following limits exist and can be calculated as follows:[2][7] The most basic sequence types are numeric, that is, sequences of real or complex numbers. This type can be generalized to sequences of elements in a vector space. In analysis, the vector spaces considered are often functional spaces. More generally, one can study sequences with elements in a topological space. In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order plays a role. Like a set, it contains elements (also called elements or terms). The number of clips (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and order is important. Formally, a sequence can be defined as a function whose domain is either the set of natural numbers (for infinite sequences) or the set of the first n natural numbers (for a sequence of finite length n).

The infinite sequences of digits (or characters) of a finite alphabet are of particular interest in theoretical computer science. They are often referred to simply as sequences or streams, as opposed to finite strings. For example, infinite binary sequences are infinite sequences of bits (characters of the alphabet {0, 1}). The set C = {0, 1}∞ of all infinite binary sequences is sometimes called the Cantor space. Metric spaces that satisfy Cauchy`s characterization of convergence for sequences are called complete metric spaces and are particularly well suited for analysis. There are several ways to designate a sequence, some of which are more useful for certain types of sequences. One way to specify a sequence is to list the items. For example, the first four odd numbers form the sequence (1, 3, 5, 7). This notation can also be used for infinite sequences.

For example, the infinite sequence of positive odd integers can be written (1, 3, 5, 7, …). The collection is especially useful for infinite sequences with a pattern that can be easily recognized by the first elements. Other ways to designate a sequence are explained after the examples. A sequence can be thought of as a list of items with a certain order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis of series, which are important in differential equations and analysis. The sequences are also interesting in themselves and can be studied in the form of models or puzzles, for example when studying prime numbers. Sequencing rules shall include all unbranched nucleotide sequences with ten or more nucleotide bases and all unbranched sequences of non-D amino acids with four or more amino acids, provided that at least 10 “specifically defined” nucleotides or 4 “specifically defined” amino acids are present. The rules apply to all sequences of a given application, whether claimed or not. All these sequences are relevant for the construction of a complete database and the correct evaluation of the prior art.

It is therefore essential that all sequences, whether only disclosed or claimed, are included in the database. If A is a set, the free monoid on A (denoted A*, also called Kleene`s star of A) is a monoid containing all finite sequences (or chains) of zero or more elements of A, with the binary operation of concatenation. The free semigroup A+ is the lower semigroup of A* that contains all elements except the empty sequence. where c 1 , . , c k {displaystyle c_{1},dots ,c_{k}} are polynomials in n. For most holonomic sequences, there is no explicit formula for explicitly expressing n {displaystyle a_{n}} as a function of n. Nevertheless, holonomic sequences play an important role in various areas of mathematics. For example, many special functions have a Taylor series whose sequence of coefficients is holonomic. The use of the repeat relation allows a quick calculation of the values of these special functions.

A sequence of finite length n is also called n-tuple. Finite sequences contain the empty sequence ( ), which contains no elements. This article formally defines a sequence as a function whose domain is an interval of integers. This definition includes several different uses of the word “sequence”, including one-sided infinite sequences, biinfinite sequences, and finite sequences (see below for definitions of this type of sequence). However, many authors use a narrower definition by requiring that the interval of a sequence be the set of natural numbers. This narrower definition has the disadvantage of excluding finite sequences and biinfinite sequences, both of which are commonly referred to as sequences in standard mathematical practice. Another disadvantage is that if you remove the first terms from a sequence, you must reindex the remaining terms to adjust that definition. In some contexts, to shorten exposure, the codomain of the sequence is determined by the context, for example by requiring it to be the set R of the real numbers,[2] the set C of the complex numbers,[3] or a topological space.

[4] A similar definition can be made for some other algebraic structures. For example, one could have an exact sequence of vector spaces and linear maps or modules and module homomorphisms. For a complete list of integer sequence examples, see Online Encyclopedia of Integer Sequences. 37 CFR 1.821(c) requires applications containing disclosures of nucleotide and/or amino acid sequences that meet the definitions in 37 CFR 1.821(a) to include, as a separate part, disclosure of nucleotide and/or amino acid sequences and related information using the format and symbols set out in 37 CFR 1.822 and 37 CFR 1.823. This separate part of the disclosure is referred to as the “sequence listing” (hereinafter also referred to as the “sequence listing”). The sequence listing required by 37 CFR 1.821(c) is the official copy of the sequence listing and can be submitted as an ASCII text file via EFS-Web, on CD, as a PDF via EFS-Web, or on paper. See MPEP § 2422.03 for more information. Not all sequences can be specified by a repeat relation. An example is the sequence of prime numbers in their natural order (2, 3, 5, 7, 11, 13, 17, …).

For example, the sequence shown on the right a n = n + 1 2 n 2 {displaystyle a_{n}={frac {n+1}{2n^{2}}}}} converges to the value 0. On the other hand, the sequences b n = n 3 {displaystyle b_{n}=n^{3}} (starts 1, 8, 27, …) and c n = ( − 1 ) n {displaystyle c_{n}=(-1)^{n}} (which starts -1, 1, -1, 1, …) are both divergent.

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