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An open research record on the Riemann Hypothesis.

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2 public results for “positive definiteness”.

  1. RESULT 01
    GLOSSARY · ART-RC-0055

    Positive definiteness

    MATCH · TITLE · Title

    A structural positivity condition requiring every permitted finite quadratic combination to be non-negative.

    WHY · TITLE MATCH + BODY MATCH + GLOSSARY DEFINITION
    7 OCCURRENCES
  2. RESULT 02
    EXPLAIN · ART-RC-0003

    What does it mean for a quantity to stay non-negative?

    MATCH · SUMMARY · Summary

    A general-reader explanation of positive, negative and non-negative quantities; pointwise, local and global positivity; numerical testing; and why the MIL-RH-0003 negative region was enough to retire a stronger global route without implying anything stronger about RH.

    WHY · SUMMARY MATCH + BODY MATCH
    38 OCCURRENCES
    Matching occurrences
    1. SUMMARY · Summary

      A general-reader explanation of positive, negative and non-negative quantities; pointwise, local and global positivity; numerical testing; and why the MIL-RH-0003 negative region was enough to retire a stronger global route without implying anything stronger about RH.

    2. METADATA · Public metadata

      EXPLAIN PUBLISHED PASS YES 2026-09-10 2026-09-19 Article ID ART-RC-0003 Content Object ID PUB-RC-0011 Series Explain Summary A general-reader explanation of positive, negative and non-negative quantities; pointwise, local and global positivity; numerical testing; and why the MIL-RH-0003 negative region was enough to retire a stronger global route without implying anything stronger about RH. Math Level GENERAL Index Excerpt A quantity is non-negative if it never falls below zero. This article explains why an everywhere-non-negative claim is strong, why one reliable negative region matters, and how the idea appears in MIL-RH-0003. What does it mean for a quantity to stay non-negative? — Riemann Console A clear general-reader explanation of non-negative quantities, local and global positivity, numerical testing, and why one negative region mattered in MIL-RH-0003 without disproving the Riemann Hypothesis.

    3. PARAGRAPH · Passage

      Mathematicians often care about whether a quantity is positive, negative or zero.

    4. PARAGRAPH · Passage

      (F may be positive or exactly zero, but it is never allowed to be negative.)

    5. PARAGRAPH · Passage

      Positive, negative and non-negative

    6. PARAGRAPH · Passage

      is positive.

    7. PARAGRAPH · Passage

      is neither positive nor negative.

    8. PARAGRAPH · Passage

      A number is called non-negative if it is either positive or zero.

    9. PARAGRAPH · Passage

      The distinction between positive and non-negative matters.

    10. PARAGRAPH · Passage

      Positive means

    11. PARAGRAPH · Passage

      Everywhere else it is positive.

    12. PARAGRAPH · Passage

      For large positive or negative x, it is positive.

    13. PARAGRAPH · Passage

      It does not matter that it is positive for infinitely many other values of x.

    14. FIGURE · Figure · What non-negative allows

      What non-negative allows A small table showing that positive values and zero are allowed by a non-negative condition, while negative values are forbidden. For a global non-negativity claim, every point in the domain must stay in the allowed rows.

    15. PARAGRAPH · Passage

      separates positive from negative values.

    16. PARAGRAPH · Passage

      Positive and non-negative quantities occur throughout mathematics and physics.

    17. PARAGRAPH · Passage

      Suppose we calculate a function near one particular point and find that it is positive there.

    18. PARAGRAPH · Passage

      A function might be positive near the origin and negative much further away.

    19. PARAGRAPH · Passage

      Or positive around most sampled points but dip below zero in a narrow region.

    20. PARAGRAPH · Passage

      Where is the function positive?

    21. PARAGRAPH · Passage

      Would the sign become positive immediately inside the domain?

    22. PARAGRAPH · Passage

      Staying positive at some points is not enough

    23. PARAGRAPH · Passage

      and thousands more positive values.

    24. PARAGRAPH · Passage

      positive examples accumulate evidence;

    25. PARAGRAPH · Passage

      An average can be positive while some values are negative

    26. PARAGRAPH · Passage

      A quantity can have a positive average without being non-negative everywhere.

    27. PARAGRAPH · Passage

      Their average is positive.

    28. PARAGRAPH · Passage

      may be positive even if F(x) dips below zero somewhere.

    29. PARAGRAPH · Passage

      Matrices or kernels may acquire useful definiteness.

    30. PARAGRAPH · Passage

      Non-negative is not the same as positive-definite

    31. PARAGRAPH · Passage

      (This checks one value at a time. Positive definiteness instead tests whether whole collections of values behave consistently together.)

    32. PARAGRAPH · Passage

      Positive definiteness is a deeper structural property.

    33. PARAGRAPH · Passage

      For a function or kernel, it concerns certain quadratic combinations of values at several points, rather than simply asking whether the function itself is positive point by point.

    34. PARAGRAPH · Passage

      So pointwise non-negativity does not automatically imply positive definiteness.

    35. PARAGRAPH · Passage

      And a positive-definite function need not itself stay positive-valued everywhere.

    36. PARAGRAPH · Passage

      We will unpack positive definiteness properly in another Explain article.

    37. PARAGRAPH · Passage

      Positive

    38. PARAGRAPH · Passage

      Whether a real quantity is positive, negative or zero.

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