sepehrst

A System-Based Approach for Deriving the Closed-Form Expressions of Arithmetic Series from First Principles

Source: GitHub Gist

Abstract

This note presents a self-contained, constructive algebraic proof for the sum of the first $n$ natural numbers, $f_{\text{all}}(n) = \frac{n(n+1)}{2}$. Rather than treating this series in isolation, or relying on sequence reversal symmetry (Gauss's pairing trick), mathematical induction, or assuming a polynomial formula in advance, this approach constructs a coupled system across three interrelated functions: The total sum $f_{\text{all}}(k)$, the sum of even numbers $f_{\text{even}}(k)$, and the sum of odd numbers $f_{\text{odd}}(k)$. By evaluating the doubled domain $f_{\text{all}}(2k)$ through parity decomposition, scaling, and term-wise translation, we establish a linear system whose solution yields the closed-form expressions for all three sequences

sepehrst

<?xml version="1.0" encoding="UTF-8" standalone="yes" ?>
<CodeBlocksConfig version="1">
	<editor>
		<colour_sets>
			<ACTIVE_COLOUR_SET>
				<str>
					<![CDATA[modnokai night shift v2]]>
				</str>
			</ACTIVE_COLOUR_SET>
			<ACTIVE_LANG>

sepehrst

This archive contains the following programs:

 bfc          The compiler for the 'brainfuck' language (240 bytes!)
 bfc.asm      Source for the compiler
 bfi          The interpreter for the 'brainfuck' language
 bfi.c        Source for the interpreter (portable)
 src/         Some example programs in 'brainfuck'
 src/atoi.b   Reads a number from stdin
 src/div10.b  Divides the number under the pointer by 10
 src/hello.b  The ubiquitous "Hello World!"

sepehrst

FAQ on the xz-utils backdoor (CVE-2024-3094)

This is a living document. Everything in this document is made in good faith of being accurate, but like I just said; we don't yet know everything about what's going on.

Update: I've disabled comments as of 2025-01-26 to avoid everyone having notifications for something a year on if someone wants to suggest a correction. Folks are free to email to suggest corrections still, of course.

Background

sepehrst

People

:bowtie:	😄 :smile:	😆 :laughing:
😊 :blush:	😃 :smiley:	☺️ :relaxed:
😏 :smirk:	😍 :heart_eyes:	😘 :kissing_heart:
😚 :kissing_closed_eyes:	😳 :flushed:	😌 :relieved:
😆 :satisfied:	😁 :grin:	😉 :wink:
😜 :stuck_out_tongue_winking_eye:	😝 :stuck_out_tongue_closed_eyes:	😀 :grinning:
😗 :kissing:	😙 :kissing_smiling_eyes:	😛 :stuck_out_tongue:

sepehrst

# Based on https://gist.github.com/kmatt/71603170556ef8ffd14984af77ff10c5
# prompt ">" indicates Powershell commands
# prompt "$" are Linux shell commands

# https://docs.microsoft.com/en-us/windows/wsl/install-win10

> dism.exe /online /enable-feature /featurename:Microsoft-Windows-Subsystem-Linux /all /norestart
> dism.exe /online /enable-feature /featurename:VirtualMachinePlatform /all /norestart

# install https://wslstorestorage.blob.core.windows.net/wslblob/wsl_update_x64.msi

sepehrst

// example how to set up D3D11 rendering on Windows in C

#define COBJMACROS
#define WIN32_LEAN_AND_MEAN
#include <windows.h>
#include <d3d11.h>
#include <dxgi1_3.h>
#include <d3dcompiler.h>
#include <dxgidebug.h>

sepehrst

This downloads standalone MSVC compiler, linker & other tools, also headers/libraries from Windows SDK into portable folder, without installing Visual Studio. Has bare minimum components - no UWP/Store/WindowsRT stuff, just files & tools for native desktop app development.

Run py.exe portable-msvc.py and it will download output into msvc folder. By default it will download latest available MSVC & Windows SDK from newest Visual Studio.

You can list available versions with py.exe portable-msvc.py --show-versions and then pass versions you want with --msvc-version and --sdk-version arguments.

To use cl.exe/link.exe first run setup_TARGET.bat - after that PATH/INCLUDE/LIB env variables will be updated to use all the tools as usual. You can also use clang-cl.exe with these includes & libraries.

To use clang-cl.exe without running setup.bat, pass extra /winsysroot msvc argument (msvc is folder name where output is stored).

sepehrst

Synchronized Output

Synchronized output is merely implementing the feature as inspired by iTerm2 synchronized output, except that it's not using the rare DCS but rather the well known SM ? and RM ?. iTerm2 has now also adopted to use the new syntax instead of using DCS.

Semantics

When rendering the screen of the terminal, the Emulator usually iterates through each visible grid cell and renders its current state. With applications updating the screen a at higher frequency this can cause tearing.

This mode attempts to mitigate that.

sepehrst

ANSI Escape Sequences

Standard escape codes are prefixed with Escape:

• Ctrl-Key: ^[
• Octal: \033
• Unicode: \u001b
• Hexadecimal: \x1B
• Decimal: 27