real analysis – Can we construct a computable sequence of trigonometric polynomials that converges pointwise to a given continuous function defined on the Torus?

Consider any continuous function $f$ on an $m$-dimensional Torus $mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric plynomials), with the band width (degree of the trigonometric polynomial) along any direction, being non decreasing, and additionally each element of the sequence being computable from a finite set of samples of $f$, in such a way that the sequence converges pointwise to the function $f$? The finite set of samples of $f$ for computability of each element of the sequence could be different.

Without this additional computability from finite samples condition, an example given by Yuval Peres here, the multi dimensional Fejer series would be an example. But the computability from finite samples condition would exclude this, due to this non-computability theorem.

This question was refined from here, after the answer from Yuval there.

python – Parsing a sequence of bits out of a bit field

For homework, I had to parse CPUID instruction output in C, which required a lot of specific bit-manipulation like:

(eax & CACHE_LEVEL) >> 5

I calculated the mask CACHE_LEVEL and the amount that was needed to be right-shifted by hand. It was a pain, so I decided to write a couple Python functions to help with this.

For example:

>>> eax = 0x4004121  # Taken from a register after executing a CPUID instruction with leaf-4 indicated
>>> parse_out_bits(eax, 5, 7)  # Return the value at bits 5 to 7 (inclusive)
1  # This is an L1 cache

I’d like notes on anything here, but specifically how this could be done better in terms of bit manipulation.

from typing import Generator

def bit_place_values(start_index: int, end_index: int) -> Generator(int, None, None):
    acc = 1 << start_index
    for _ in range(start_index, end_index + 1):
        yield acc
        acc <<= 1

def parse_out_bits(bit_field: int, start_index: int, end_index: int) -> int:
    mask = sum(bit_place_values(start_index, end_index))
    return (mask & bit_field) >> start_index

postgresql – How to set current sequence while converting primary key serial(integer) to bigserial(bigint)

The table already has few thousands of rows and I want to convert the primary key from Integer to BigInteger. How to change the sequence for biginteger?

--Create new sequence for bigint
CREATE SEQUENCE public.case_audit_case_audit_uid_seq1
    START 5542
    MAXVALUE 9223372036854775807
    CACHE 1;
ALTER SEQUENCE public.case_audit_case_audit_uid_seq1  OWNER TO postgres;
--Alter table
ALTER TABLE public.case_audit ALTER COLUMN case_audit_uid TYPE bigint;  
--Alter pk column Sequence
ALTER TABLE public.case_audit ALTER COLUMN case_audit_uid SET DEFAULT     nextval('indsolv.case_audit_case_audit_uid_seq1'::regclass);
DROP SEQUENCE public.case_audit_case_audit_uid_seq;
--Create new sequence for bigint
CREATE SEQUENCE public.case_audit_case_audit_uid_seq
    START 5542
    MAXVALUE 9223372036854775807
    CACHE 1;
ALTER SEQUENCE public.case_audit_case_audit_uid_seq OWNER TO postgres;
ALTER TABLE public.case_audit ALTER COLUMN case_audit_uid SET DEFAULT nextval('public.case_audit_case_audit_uid_seq'::regclass);
DROP SEQUENCE public.case_audit_case_audit_uid_seq1;

Is there any simple solution compare to this?

real analysis – Is there a known condition for partial sums of a decreasing positive sequence to take all values up to the total sum?

Let $a_0>a_1>cdots>0$ have the property that, for each positive $a<sum_{ninBbb N}a_n$ (admitting $infty$ for the sum), there is $AsubsetBbb N$ such that $a=sum_{nin A}a_n$ . Are there known necessary and sufficient conditions on the $a_n$ (not involving arbitrary partial sums) for this property? To illustrate, $a_n=1/(n+1)$ and $a_n=1/2^n$ possess the required property, but $a_n=1/(2+varepsilon)^n$ does not for any $varepsilon>0$.

(This question is adapted from one asked on Mathematics Stack Exchange two months ago which received no answer.)

Efficient way to import a sequence of data labeled with a parameter value

Suppose we have a number of data in .m format, which are labeled with the value of a parameter $c$, for example, c=0data.m, c=0.2data.m, c=0.4data.m, c=0.6data.m, c=0.8data.m, c=1data.m, …

Now, we need to import these data and assign them to solc0, solc2, solc4,…, solc10 for batch processing. I use the following Do loop with ToString:

Do[solc<>ToString[i] = << "C:\Users\c="<>ToString[i/10]<>data.m", {i, 0, 10, 2}]

which generates lots of errors. Can anyone point out what I overlooked? Thank you.

If the sequence $ x_n = dfrac {1} { sqrt {n}} left (1+ dfrac {1} { sqrt {2}} + dfrac {1} { sqrt {3}} + ldots + dfrac {1} { sqrt {n}} right) $ monotonous?

Keep that in mind $ x_1 = 1 $ and $ x_2 = dfrac {1} { sqrt {2}} left (1+ dfrac {1} { sqrt {2}} right)> dfrac {1} { sqrt {2}} left ( dfrac {1} { sqrt {2}} + dfrac {1} { sqrt {2}} right) = 1 $.

Consequently, $ x_2> x_1 $. In general we have too $ x_n = dfrac {1} { sqrt {n}} left (1+ dfrac {1} { sqrt {2}} + dfrac {1} { sqrt {3}} + ldots + dfrac {1} { sqrt {n}} right)> dfrac {1} { sqrt {n}} left ( dfrac {1} { sqrt {n}} + dfrac {1} { sqrt {n}} + dfrac {1} { sqrt {n}} + ldots + dfrac {1} { sqrt {n}} right) = 1 $.

Consequently, $ x_n geq 1 $ for all $ n in mathbb {N} $. We also have

$ x_ {n + 1} = dfrac {1} { sqrt {n + 1}} left ( sqrt {n} x_n + dfrac {1} { sqrt {n + 1}} right) = dfrac { sqrt {n}} { sqrt {n + 1}} x_n + dfrac {1} {n + 1} $.

Is it true that $ x_ {n + 1}> x_n $?

reference request – Existence of Analytic Continuation of Dirichlet Series Corresponding to the Indicator Sequence of a Complement of a Special Multiplicative Set

Let $K/ mathbb Q $ be a finite Galois extension and let $X$ be a proper non-empty subset of the Galois group $G=Gal(K/ mathbb Q)$ that is closed under conjugation. Consider a set of integer primes $P$ such that for all sufficiently large primes $p$, the following equivalence holds
$$p in P iff text{ the conjugacy class of the Frobenius element }sigma_p text{ is contained in }X$$

Now let $E$ be a multiplicative set of natural numbers (that is, for all coprime $m, n in mathbb N$, we have the equivalence $mn in E iff m in E$ or $n in E$) such that the set of prime numbers in $E$ is exactly the set $P$ above and let $E’ := mathbb N setminus E$ denote the complement of $E$. Consider the indicator sequence $(a_n)_{n geq 1}$ of $E’$ (so that $a_n := 1 iff n in E’$ and $a_n=0$ otherwise) and let $F(s) := sum_{n geq 1} a_n n^{-s}$ be the Dirichlet Series corresponding to the sequence $(a_n)_{n geq 1}$.

I want to show that the function $F$ analytically continues to a region of the form given in the image The Mentioned Region has been Coloured in Yellow. where $delta>0$ is fixed, the circle around the point $1$ is of radius $epsilon < delta$ and the infinite branches $C$ and $D$ are defined by
$$Re(s) = 1 – frac{a}{(log (2+|Im(s)|))^A}$$
(where $a$ and $A$ are fixed positive numbers, note that the interior of the circle has been excluded from the aforementioned region) such that in this region we have
$$F(s) = O((log |Im(s)|)^A) text{ as } |Im(s)| rightarrow infty$$

The only results of this kind I am somewhat familiar with are those on the analytic continuation of the usual Riemann Zeta Function (which I read in Apostol’s “Introduction to Analytic Number Theory”). Although I have obtained some other immediate observations (for instance: the natural and Dirichlet density of $P$ must both be $|X|/|G| in (0,1)$ by the Chebotarev Density Theorem and that the sequence $(a_n)$ should be multiplicative hence we can get something akin to an `Euler-Product’ representation of the Dirichlet Series $F(s)$), I have no general idea on how to get started on this problem and I would really a proof or a reference containing a complete (and preferably not too inaccessaible) proof of the same. Thank you.

P.S.: It says here (Continuation up to zero of a Dirichlet series with bounded coefficients) that a Dirichlet series with bounded coefficients need not be meromorphically continuable to the right of zero, but I haven’t found any positive results on M.O. in this direction.

abstract algebra – Prove that there exists the exact sequence (over Z) 0 → Z 2 →Z 4 → Z 4 → Z 2 → 0 .


To show that it is exact ,I need f : Z2 -> Z4 monic, and g : Z4 -> Z2 epic.

homomorphism f by f((a))=(2a) , (a) is in Z2 and (2a) is in Z4 and homomorphism g by g((b)2)=(b)4 .

when I take like that by finding all homomorphism between them, image of f is not equal to kernel of h : Z4 -> Z4. Also image of h is not equal to kernel of g.

Could someone help me out?